a short introduction to basic aspects of continuum ...

ILSB Report 206

hjb/ILSB, 080407 (preliminary version)

ILSB Report / ILSB-Arbeitsbericht 206 (supersedes CDL–FMD report 3–1998)

A SHORT INTRODUCTION TO BASIC ASPECTS OF CONTINUUM MICROMECHANICS

H.J. B¨ ohm Institute of Lightweight Design and Structural Biomechanics (ILSB) Vienna University of Technology

updated: April 7, 2008 c

2007, 2008 Helmut J. B¨ ohm

CONTENTS Notes on this Document List of Variables 1. INTRODUCTION 1.1 Inhomogeneous Materials 1.2 Homogenization and Localization 1.3 Overall Behavior, Material Symmetries 1.4 Basic Modeling Strategies in Continuum Micromechanics of Materials 2. MEAN FIELD APPROACHES 2.1 General Relations 2.2 Eshelby Tensor and Dilute Matrix–Inclusion Composites 2.3 Mean Field Methods for Thermoelastic Composites with Aligned Reinforcements 2.4 Hashin–Shtrikman Estimates 2.5 Other Estimates for Elastic Composites with Aligned Reinforcements 2.6 Mean Field Methods for Inelastic Composites 2.7 Mean Field Methods for Composites with Nonaligned Reinforcements 2.8 Mean Field Methods for Non-Ellipsoidal Reinforcements 2.9 Mean Field Methods for Diffusion-Type Problems 3. VARIATIONAL BOUNDING METHODS 3.1 Classical Bounds 3.2 Improved Bounds 3.3 Bounds for the Nonlinear Behavior 3.4 Some Comparisons of Mean Field and Bounding Predictions 4. GENERAL REMARKS ON APPROACHES BASED ON DISCRETE MICROGEOMETRIES 4.1 Microgeometries 4.2 Numerical Engineering Methods 5. PERIODIC MICROFIELD APPROACHES 5.1 Basic Concepts of Unit Cell Models 5.2 Boundary Conditions, Application of Loads, Evaluation of Microfields 5.3 Unit Cell Models for Continuously Reinforced Composites 5.4 Unit Cell Models for Discontinuously Reinforced Composites 5.5 PMA Models for Some Other Inhomogeneous Materials 5.6 PMA Models for Diffusion-Type Problems

i

6. EMBEDDING APPROACHES AND RELATED METHODS 7. WINDOWING APPROACHES 8. MULTI-SCALE MODELS 9. CLOSING REMARKS APPENDIX: SOME ASPECTS OF MODELING DAMAGE AT THE MICROLEVEL REFERENCES

ii

Notes on this Document The present document is an expanded and modified version of the CDL–FMD report 3–1998, “A Short Introduction to Basic Aspects of Continuum Micromechanics”, which is based on lecture notes prepared for the European Advanced Summer Schools “Frontiers of Computational Micromechanics in Industrial and Engineering Materials” held in Galway, Ireland, in July 1998 and in August 2000. Similar lecture notes were used for graduate courses in a Summer School held at Ameland, the Netherlands, in October 2000 and during the COMMAS Summer School held in September 2002 at Stuttgart, Germany. All of the above documents, in turn, are based on the micromechanics section of the lecture notes for the course “Composite Engineering” (317.003) offered regularly at the Vienna University of Technology. The course notes “A Short Introduction to Continuum Micromechanics” (B¨ ohm,2004) for the CISM Course on Mechanics of Microstructured Materials held in July 2003 in Udine, Italy, may be viewed as an abridged version of the present report that employs a somewhat different notation. The present document is continuously updated to reflect current developments in continuum micromechanics as seen by the author.

For information on the research work in the field of micromechanics of materials currently carried out at the Institut f¨ ur Leichtbau und Struktur-Biomechanik (formerly the Institut f¨ ur Leichtbau und Flugzeugbau) of Vienna University of technology see the web pages http://www.ilsb.tuwien.ac.at/ilfb/ilfb ra3e.html

iii

List of Variables

h

i Volume average a: Aspect ratio of inclusions, reinforcements `: Characteristic length of lower length scale (microscale) L: Characteristic length of higher length scale (macroscale) Ωs : Volume of sample or region Γs : Surface of sample or region

ΩUC : Volume of unit cell ΓUC : Surface of unit cell Ω(p) : Volume occupied by phase V (p) Volume fraction of phase

(p)

(p)

ξ: Inclusion, reinforcement volume fraction η,ζ: Microstructural parameters for higher order bounds ε: Strain tensor (rank 2; vector in Voigt notation) hεi: Tensor of (volume) averaged strains ε0 : Tensor of (local) fluctuating strains εa : Tensor of the applied mechanical (far field) strains hεi(p) : Tensor of averaged strains in phase

(p)

εt : Tensor of unconstrained eigenstrains in inclusion ετ : Tensor of equivalent eigenstrains in equivalent homogeneous inclusion εc : Tensor of constrained strains in inclusion σ: Stress tensor (rank 2; vector in Voigt notation) hσi: Tensor of (volume) averaged stresses σ 0 : Tensor of (local) fluctuating stresses σa : Tensor of applied (far field) stresses hσi(p) : Tensor of averaged stresses in phase

(p)

¯ Phase strain concentration tensor (rank 4; matrix in Voigt notation) A: ¯ Phase stress concentration tensor (rank 4; matrix in Voigt notation) B: ¯ a: Phase thermal strain concentration tensor (rank 2; vector in Voigt notation) ¯ Phase thermal stress concentration tensor (rank 2; vector in Voigt notation) b: C: Compliance tensor (rank 4; matrix in Voigt notation) E: Elasticity tensor (rank 4; matrix in Voigt notation) t: Specific thermal stress tensor (rank 2; vector in Voigt notation)

iv

α: Thermal expansion coefficient tensor (rank 2; vector in Voigt notation) I: Identity tensor (rank 4; matrix in Voigt notation) S: Eshelby tensor (rank 4; matrix in Voigt notation) t: Traction vector ta : Vector of applied tractions nΓ : Surface normal vector E: Young’s modulus G: Shear modulus K: Bulk modulus ν: Poisson’s ratio KT : Transverse bulk modulus T : Temperature σ: Stress component ε: Strain component γ: Shear angle α: Coefficient of thermal expansion

Constituents (or “phases”, i.e. matrix, inclusions, inhomogeneities, reinforcements, . . .) are denoted by superscripts in brackets, e.g. (p) . Axial and transverse properties of transversely isotropic materials are marked by subscripts A and T , respectively, and effective properties are denoted by a superscript asterisk ∗ . Voigt notation (also known as Nye or Voigt–Mandel notation) is used in chapters 1 to 3 of these notes, i.e. the stress and strain tensors are formally denoted as pseudo-vectors with the components    σ(1) σ11  σ(2)   σ22       σ(3)   σ33  σ= =   σ(4)   σ12      σ(5) σ13 σ(6) σ23

   ε(1) ε11  ε(2)   ε22       ε(3)   ε33  ε= =   ε(4)   γ12      ε(5) γ13 ε(6) γ23





,

where γij = 2εij are the shear angles1) . Tensors of rank 4, such as elasticity, compliance, concentration and Eshelby tensors, take the form of 6×6 matrices.

This notation allows strain energy densities to be obtained as U = 21 σε. Furthermore, Θ Θ Θ −1 T the stress and strain “rotation tensors” TΘ ] . σ and Tε follow the relationship Tε = [(Tσ ) 1)

v

vi

1. INTRODUCTION In the present document some basic issues and some important approaches in the field of continuum micromechanics of materials are discussed. The main emphasis lies on application related (or “engineering”) aspects, and neither a comprehensive theoretical treatment nor a review of the pertinent literature are attempted. For more formal treatments of many of the concepts and methods discussed in what follows see e.g. (Mura,1987; Aboudi,1991; Nemat-Nasser/Hori,1993; Suquet,1997; Markov,2000; Bornert et al.,2001; Torquato,2002; Milton,2002; Qu/Cherkaoui,2006). Short overviews were given e.g. by Hashin(1983) and Zaoui (2002). Discussions of the history of the development of the field can be found in (Markov,2000; Zaoui,2002). Due to the author’s research interests, more room is given to the thermomechanical behavior of two-phase materials showing a matrix–inclusion topology and especially to metal matrix composites (MMCs), than to materials with other phase topologies or phase geometries. Extending the methods presented here to other types of inhomogeneous materials, however, in general does not cause principal difficulties. 1.1 Inhomogeneous Materials Many industrial and engineering materials as well as the majority of “natural” materials are inhomogeneous, i.e. they consist of dissimilar constituents (or “phases”) that are distinguishable at some (small) length scale. Each constituent shows different material properties and/or material orientations and may itself be inhomogeneous at some even smaller length scale(s). Typical examples of inhomogeneous materials are composites, concrete, polycrystalline materials, porous and cellular materials, functionally graded materials, wood, and bone. An important aim of theoretical studies of multiphase materials lies in homogenization, i.e. in deducing their overall (“effective” or “apparent”) behavior2) (e.g. stiffness, thermal expansion and strength properties, heat conduction and related transport properties, electrical and magnetic properties, electromechanical properties, . . .) from the corresponding material behavior of the constituents (and of the interfaces between them) and from the geometrical arrangement of the phases. In what follows, the main focus will lie on describing the thermomechanical behavior of inhomogeneous two-phase materials by methods of continuum mechanics. It should be kept in mind, however, that there is a large body of literature applying analogous or related continuum approaches to other physical properties of inhomogeneous materials, see e.g. (Hashin,1983; Markov/Preziosi,2000; Milton,2002) as well as sections 2.9 and 5.6 of the present report. 2)

A number of authors restrict the term “effective material properties” to the response of proper Representative Volume Elements, compare section 1.2, of an infinitely large sample, or of periodic (model) composites, whereas the designation “apparent material properties” is used for results obtained from smaller volume elements such as windows (Huet,1990).

–1–

In micromechanical approaches the stress and strain fields in an inhomogeneous material are split into contributions corresponding to different length scales3) . It is assumed that these length scales are sufficiently different so that for each pair of them • fluctuations of the stress and strain fields at the smaller length scale (microfields, “fast variables”) influence the macroscopic behavior at the larger length scale only via their volume averages and • gradients of the stress and strain fields as well as compositional gradients at the larger length scale (macrofields, “slow variables”) are not significant at the smaller length scale, where these fields appear to be locally constant and can be described in terms of uniform “applied” or “far field” stresses and strains. Formally, this splitting of the strain and stress fields into slow and fast contributions can be written as ε(x) = hεi + ε0 (x) and σ(x) = hσi + σ 0 (x) , (1) where hεi and hσi are the (slow) macroscopic fields, whereas ε0 and σ 0 stand for the microscopic fluctuations. For example, in layered composites (“laminates”) there are at least three length scales that can be described by continuum mechanics, which are often called the macro scale (component or sample), the mesoscale (individual plies), and the microscale (fiber–matrix structure of the plies)4) . Smaller length scales in such material systems may or may not be amenable to continuum mechanical descriptions; for an overview of methods applicable below the continuum range see e.g. (Raabe,1998). Viewed from some given length scale the behavior of a material that is inhomogeneous at some smaller length scale can be described by the behavior of an energetically equivalent homogenized continuum, provided the separation between the two length scales is sufficiently large and the above conditions are met. If the above relations between “fast” and “slow” variables are not fulfilled to a sufficient degree (e.g. due to insufficiently separated length scales, near free surfaces of inhomogeneous materials, at macroscopic interfaces adjoined by at least one inhomogeneous material, or in the presence of marked compositional or load gradients, an example for the latter being inhomogeneous beams under bending loads), special homogenization methods must be used, such as second order schemes explicitly accounting for deformation gradients (Kouznetsova et al.,2002), and the homogenized continuum is nonlocal, see e.g. (Feyel,2003). An important classification scheme for inhomogeneous materials is based on the microscopic phase topology. In matrix–inclusion arrangements (as found in particulate and 3)

Micromechanics of materials deals with the bridging of relative length scales in inhomogeneous materials (as described by the ratio between the characteristic lengths of the upper and lower scales). Continuum micromechanics in most cases cannot per se resolve effects of absolute length scales, such as the absolute size of inhomogeneities, compare chapter 9. 4)

This nomenclature is far from universal, the naming of the length scales of inhomogeneous materials being notoriously inconsistent in the literature.

–2–

fibrous materials, such as most composite materials, porous materials, and closed cell foams5) ) only the matrix shows a connected topology and the constituents play clearly distinct roles. In interwoven phase arrangements (as found e.g. in functionally graded materials or in open cell foams) and in many polycrystals (“granular materials”), in contrast, the phases do not show clearly different topologies. 1.2 Homogenization and Localization For any region of an inhomogeneous material the microscopic strain and stress fields, ε(x) and σ(x), and the corresponding macroscopic responses, hεi and hσi, can be formally linked by localization (or projection) relations of the type ε(x) = A(x)hεi σ(x) = B(x)hσi .

(2)

If, in addition, the region is sufficiently large and contains no significant macroscopic gradients of stress, strain or composition, homogenization relations can be defined as Z Z 1 1 hεi = ε(x) dΩ = (u(x) ⊗ nΓ (x) + nΓ (x) ⊗ u(x)) dΓ Ω s Ωs 2Ωs Γs Z Z 1 1 hσi = σ(x) dΩ = t(x) ⊗ x dΓ , (3) Ω s Ωs Ω s Γs where Ωs and Γs stand for the volume and the surface of the region under consideration, u(x) is the deformation vector, t(x) = σ(x)nΓ (x) are the surface traction vectors, nΓ (x) are surface normal vectors, and ⊗ is the dyadic product of vectors defined as [a⊗b]ij =ai bj . A(x) and B(x) are called mechanical strain and stress concentration tensors (or influence functions; Hill,1963), respectively. The surface integral formulation for hεi given above holds only for perfect interfaces between the constituents, in which case the mean strains and stresses in a control volume, hεi and hσi, are fully determined by the surface displacements and tractions; otherwise correction terms involving the displacement jumps across imperfect interfaces or cracks must be accounted for, see e.g. (Nemat-Nasser/Hori,1993). In the absence of body forces the microstresses σ(x) are self-equilibrated (but not necessarily zero). In the above form, eqn.(2) applies only to linear elastic behavior, but it can be modified to cover thermoelastic behavior, compare eqn.(16), and extended to the nonlinear range (e.g. to elastoplastic materials described by secant or incremental plasticity). For a discussion of homogenization at finite deformations see e.g. (Nemat-Nasser,1999). Equations (1) and (3) immediately imply that volume averages of fluctuations vanish for sufficiently large integration volumes, Z Z 1 1 0 ε (x) dΩ = 0 = σ 0 (x) dΩ . (4) Ω s Ωs Ω s Ωs 5)

With respect to their thermomechanical behavior, porous and cellular materials can usually be treated as inhomogeneous materials in which one constituent has vanishing stiffness, thermal expansion, conductivity etc.

–3–

Similarly, surface integrals over the microscopic fluctuations of appropriate variables typically vanish6) . For inhomogeneous materials that show sufficient separation between the length scales the relation Z

σ ˜ (x) ε˜(x) dΩ = Ωs

Z

σ ˜ (x) dΩ Ωs

Z

ε˜(x) dΩ

(5)

Ωs

can be shown to hold for general statically admissible stress fields σ ˜ and kinematically admissible strain fields ε˜, see (Hill,1952). This equation is known as Hill’s macrohomogeneity condition, the Mandel–Hill condition or the energy equivalence condition, compare (Bornert,2001; Zaoui,2001). For the two special cases of homogeneous stress and homogeneous strain boundary conditions (for which the above relation can be shown to hold by transforming the volume integrals into surface integrals, to which the fluctuations σ 0 and ε0 give zero contributions) it is referred to as Hill’s lemma. Equation (5) implies that the volume averaged strain energy density of an inhomogeneous material can be obtained from the volume averages of the stresses and strains, provided the microand macroscales are sufficiently well separated7) . Accordingly, homogenization procedures can be interpreted as finding a homogeneous comparison material (or “reference material”) that is energetically equivalent to a given microstructured material. The microgeometries of real inhomogeneous materials are at least to some extent random and, in the majority of cases of practically relevance, their detailed microgeometries are highly complex. As a consequence, exact expressions for A(x), B(x), ε(x), σ(x) etc. in general cannot be provided with reasonable effort and approximations have to be introduced. Typically, these approximations are based on the ergodic hypothesis, i.e., the heterogeneous material is assumed to be statistically homogeneous. This implies that sufficiently large volume elements (“homogenization volumes”) selected at random positions within the sample have statistically equivalent phase arrangements and give rise to the same averaged material properties8) . As mentioned above, these material properties are referred to as the overall or effective material properties of the inhomogeneous material. Ideally, the homogenization volume should be chosen to be a proper representative volume element (or reference volume element, RVE), i.e., a subvolume of Ωs that is sufficiently 6)

Whereas integrals over products of slow and fast variables vanish, integrals over products of fluctuating components do not vanish in general, compare section 2.4. 7)

For a discussion of the Hill condition for finite deformations and related issues see e.g. (Khisaeva/Ostoja-Starzewski,2006). 8)

Some inhomogeneous materials are not statistically homogeneous by design (e.g. functionally graded materials in the direction(s) of the gradient(s) and, consequently, may require nonstandard treatment. For such materials it is not possible to define effective material properties in the sense of eqn.(6) and representative volume elements may not exist. Statistical inhomogeneity may also be introduced by manufacturing processes.

–4–

large to be statistically representative of the microgeometry of the material 9) . Such a volume element must be sufficiently large to allow a meaningful sampling of the microfields and sufficiently small for the influence of macroscopic gradients to be negligible and for an analysis of the microfields to be possible10) . For discussions of homogenization volumes and RVEs see e.g. (Hashin,1983; Nemat-Nasser/Hori,1993; Drugan/Willis,1996; Ostoja-Starzewski,1998; Zaoui,2001; Bornert et al.,2001; Kanit et al.,2003; Stroeven et al.,2004; Gitman et al.,2007) as well as section 4.1. Comparisons of RVEs conforming to different definitions can be found in (Trias et al.,2006; Swaminathan et al.,2006). In practice it tends to be rather difficult to study (or, in fact, to identify) proper RVEs of actual inhomogeneous materials (as opposed to simplified phase arrangements) and, accordingly, in models based on discrete microgeometries typically volume elements are used that are only approximations to RVEs. 1.3 Overall Behavior, Material Symmetries The homogenized strain and stress fields of an elastic inhomogeneous material as obtained by eqn.(3), hεi and hσi, can be linked by effective elastic tensors E∗ and C∗ as hσi = E∗ hεi hεi = C∗ hσi ,

(6)

which may be viewed as the elastic tensors of an appropriate equivalent homogeneous material. Using eqn.(3) these effective elastic tensors can be obtained from the local elastic tensors E(x) and C(x) and from the concentration tensors A(x) and B(x) as volume averages Z 1 ∗ E = E(x)A(x) dΩ Ω s Ωs Z 1 ∗ C = C(x)B(x) dΩ . (7) Ω s Ωs 9)

A verbatim interpretation of the above definition leads to the concept of “geometrical RVEs”, which is based directly on the phase geometry of an inhomogeneous material, typˇ ically characterized by statistical descriptors (Pyrz,1994; Zeman/Sejnoha,2001; Jeulin,2001). More generally, a representative volume element may be defined as a volume that shows the same overall response with respect to some given physical property independently of its position and orientation within the sample and irrespective of the applied boundary conditions (Hill,1963; Sab,1992; Huet,1999) with the exception of free surfaces. The sizes of such “physical RVEs” depend on the physical property considered, compare e.g. (Kanit et al.,2003; Swaminathan/Ghosh,2006) and it has been argued that no such RVE exists for softening material behavior (Gitman et al.,2007). The volume averages and higher statistical moments of stresses, strains, and other variables are the same for all physical RVEs and are equal to the corresponding ensemble averages. 10)

This requirement was symbolically denoted as MICROMESOMACRO by Hashin (1983), where MICRO and MACRO have their “usual” meanings and MESO stands for the length scale of the homogenization volume. As noted by Nemat-Nasser (1999) it is the dimension relative to the microstructure relevant for a given problem that is important for the size of an RVE.

–5–

Other effective properties of inhomogeneous materials, e.g. tensors describing their thermophysical behavior, can be obtained by analogous operations. The resulting homogenized behavior of many multiphase materials can be idealized as being statistically isotropic or quasi-isotropic (e.g. for composites reinforced with spherical particles, randomly oriented particles of general shape or randomly oriented fibers, many polycrystals, many porous and cellular materials, random mixtures of two phases) or statistically transversely isotropic (e.g. for composites reinforced with aligned fibers or platelets, composites reinforced with nonaligned reinforcements showing an axisymmetric orientation distribution function, etc.), compare (Hashin,1983). Of course, lower material symmetries of the homogenized response may also be found (e.g. in textured polycrystals). Statistically isotropic multiphase materials show the same overall behavior in all directions, and their effective elasticity tensors (or “material tensors”) and thermal expansion tensors take the form (in Voigt notation) ∗ E11 ∗  E12  ∗ E E∗ =  12  0  0 0



∗ E12 ∗ E11 ∗ E12 0 0 0

∗ E12 ∗ E12 ∗ E11 0 0 0

0 0 0 ∗ E44 0 0

0 0 0 0 ∗ E44 0

∗ E44

 α∗11  α∗11   ∗  α  α∗ =  11   0    0 0

 0 0   0   0   0 ∗ ∗ = 21 (E11 − E12 )



. (8)

Two independent parameters are sufficient for describing such an overall linear elastic ∗ 2 2E12 ∗ behavior (e.g. the effective Young’s modulus E ∗ =E11 − E ∗ +E ∗ , the effective Poisson ∗

11

∗ E12 = E ∗ +E ∗ , the 11 12 ∗ ∗

∗

12

∗ =E44 =E ∗ /2(1

number ν effective shear modulus G + ν ∗ ), the effective bulk modulus K =E /3(1 − 2ν ∗ ), or the effective Lamé constants) and one is required for the effective thermal expansion behavior in the linear range (the effective coefficient of thermal expansion α∗ =α∗11 ). The effective elasticity and thermal expansion tensors for statistically transversely isotropic materials have the structure ∗ E11 ∗  E12  ∗ E ∗ E =  12  0  0 0



∗ E12 ∗ E22 ∗ E23 0 0 0

∗ E12 ∗ E23 ∗ E22 0 0 0

0 0 0 ∗ E44 0 0

0 0 0 0 ∗ E44 0

∗ E66

 0 0   0   0   0 ∗ ∗ = 21 (E22 − E23 )

 α∗11  α∗22   ∗  α  ∗ α =  22   0    0 0 

, (9)

where 1 is the axial direction and 2–3 is the transverse plane of isotropy. Generally, the thermoelastic behavior of transversely isotropic materials is described by five independent elastic constants and two independent coefficients of thermal expansion. Appropriate elastic parameters in this context are e.g. the axial and transverse effective Young’s mod∗2 ∗ ∗2 ∗ ∗2 2E12 E ∗ E ∗2 +E22 E12 −2E23 E12 ∗ ∗ ∗ ∗ =E11 − E ∗ +E uli, EA =E22 − 11 23E ∗ E and ET , the axial and transverse ∗ ∗ −E ∗2 22 23 11 22 12 ∗ ∗ ∗ ∗ effective shear moduli, GA =E44 and GT =E66 , the axial and transverse effective Poisson

–6–

∗ E12 ∗ 22 +E23

∗ numbers, νA = E∗

E ∗ E ∗ −E ∗2

∗ 23 12 and νT = E11 ∗ E ∗ −E ∗2 , as well as the effective transverse bulk mod11

22

12

∗ ∗ ∗ ∗ ∗ ∗2 ulus KT =EA /2[(1 − νT )(EA /ET ) − 2νA ]. The transverse (“in-plane”) properties are ∗ ∗ ∗ related via GT =ET /2(1 + νT ), but for general transversely isotropic materials there is no ∗ ∗ ∗ linkage between the axial properties EA , G∗A and νA beyond the above definition of KT . ∗ ∗ Both an axial and a transverse effective coefficient of thermal expansion, αA =α11 and α∗T =α∗22 , are required. For the special case of materials reinforced by aligned continuous fibers Hill’s (1964) relations (f)

4(νA − ν (m) )2

1 = + (1 − ξ)E + + − ∗ (f) (m) (f) (m) KT (1/KT − 1/KT )2 KT KT (f) νA − ν (m) ξ 1−ξ 1 (f) ∗ (m) + (m) − ∗ νA = ξνA + (1 − ξ)ν + (f) (m) (f) KT 1/K − 1/K K K

∗ EA

(f) ξEA

(m)

T

T

T

1−ξ

ξ

(10)

T

∗ ∗ ∗ allow the effective moduli EA and νA to be expressed by KT , some constituent properties, and the fiber volume fraction ξ. Equations (10) can be used to reduce the number of independent effective elastic parameters required for unidirectional continuously reinforced composites to three.

The the overall material symmetries of inhomogeneous materials and their effect on various physical properties can be treated in full analogy to the symmetries of crystals as discussed e.g. in (Nye,1957). The influence of the overall symmetry of the phase arrangement on the overall mechanical behavior of inhomogeneous materials can be marked 11) , especially on the post-yield and other nonlinear responses to mechanical loads, compare e.g. (Nakamura/Suresh,1993; Weissenbek,1994). Accordingly, it is important to aim at approaching the symmetry of the actual material as closely as possible in any modeling effort. 1.4 Basic Modeling Strategies in Continuum Micromechanics of Materials Homogenization procedures aim at finding a volume element’s responses to prescribed mechanical loads (typically far field stresses or far field strains) or temperature excursions and to deduce from them the corresponding overall properties. The most straightforward application of such studies is materials characterization, i.e. simulating the overall material response under simple loading conditions such as uniaxial tensile tests. Analysis of this kind can be performed by all approaches described here (with the possible exception of bounding methods). Many homogenization procedures can also be employed directly as micromechanically based constitutive material models at higher length scales, i.e. they can link the full homogenized stress tensor to the full homogenized strain tensor or provide the corresponding 11)

Overall properties described by lower order tensors, e.g. the thermal expansion response, are less sensitive to symmetry effects, compare (Nye,1957).

–7–

instantaneous stiffnesses for nonlinear materials12) . This, of course, requires the capability of evaluating the overall response of an inhomogeneous material under any loading condition and for any loading history, see also chapter 8. Compared to semiempirical constitutive laws, as proposed e.g. in (Davis,1996), micromechanically based constitutive models have both a clear physical basis and an inherent capability for “zooming in” on the local phase stresses and strains by using localization procedures. The latter allow the local response of the phases to be evaluated when the macroscopic response or state of a sample or structure is known, e.g. for predicting the onset of plastic yielding or damage of the constituents. Besides materials characterization and constitutive modeling, there are a number of other important applications of continuum micromechanics, among them studying local phenomena in inhomogeneous materials, e.g. the initiation and evolution of damage on the microscale, the nucleation and growth of cracks, the stresses at intersections between macroscopic interfaces and free surfaces, the effects of local instabilities, and the interactions between phase transformations and microstresses. For these behaviors details of the microstructure tend to be of major importance and may determine the macroscopic response, an extreme case being the mechanical strength of brittle inhomogeneous materials. Because for realistic phase distributions the analysis of the spatial variations of the microfields in sufficiently large volume elements tends to be beyond present capabilities, approximations have to be used. For convenience, the majority of the resulting modeling approaches may be treated as falling into two groups. The first of them comprises methods that describe the microgeometries of inhomogeneous materials on the basis of (limited) statistical information13) : • Mean Field Approaches (MFAs) and related methods (see chapter 2): The microfields within each constituent are approximated by their phase averages hεi (p) and hσi(p) , i.e. piecewise (phase-wise) uniform stress and strain fields are employed. Such descriptions typically use information on the microscopic topology, the reinforcements’ shape and orientation, and, to some extent, on the statistics of the 12)

The overall thermomechanical behavior of homogenized materials is typically richer than that of the constituents, i.e. the effects of the interaction of the constituents in many cases cannot be satisfactorily described by simply adapting material parameters without changing the functional relationships in the constitutive laws (for example, a composite consisting of a matrix that follows J2 plasticity and elastic reinforcements has pressure dependent macroscopic plastic behavior, and two constituents showing viscoplastic power law behavior with different exponents in general do not give rise to a power-law type overall response). Accordingly, predicted uniaxial stress–strain curves are not sufficient for calibrating macroscopic constitutive models for describing multiaxial behavior. 13)

General phase arrangements can be described by n-point correlation functions that give the probability that a some geometrical configuration of n points lies within prescribed phases. For statistically isotropic materials 2-point correlations correspond to the volume fractions. For lower overall symmetries they also provide information on the phases’ shapes.

–8–

phase distribution. The localization relations then take the form ¯ (p) hεi hεi(p) = A ¯ (p) hσi hσi(p) = B and the homogenization relations become Z 1 (p) hεi = (p) ε(x) dΩ with Ω Ω(p) Z 1 (p) σ(x) dΩ with hσi = (p) Ω Ω(p)

(11)

hεi =

X

V (p) hεi(p)

p

hσi =

X

V (p) hσi(p)

(12)

p

where (p) denotes a given phase of the material, Ω(p) is the corresponding phase P volume, and V (p) =Ω(p) / k Ω(k) is the volume fraction of the phase. In contrast to ¯ and B ¯ are not functions of eqn.(2), for MFAs the phase concentration tensors A the spatial coordinates. Mean field approaches tend to be formulated in terms of the phase concentration tensors, they pose relatively low computational requirements, they have been highly successful in describing the thermoelastic response of inhomogeneous materials, and their use for modeling nonlinear inhomogeneous materials is a subject of active research. • Variational Bounding Methods (see chapter 3): Variational principles are used to obtain upper and (in many cases) lower bounds on the overall elastic tensors, elastic moduli, secant moduli, and other physical properties of inhomogeneous materials. Many analytical bounds are obtained on the basis of phase-wise constant stress (polarization) fields. Bounds — in addition to their intrinsic value — are important tools for assessing other models of inhomogeneous materials. Furthermore, in many cases one of the bounds provides good estimates for the physical property under consideration, even if the bounds are rather slack (Torquato,1991). Many bounding methods are closely related to MFAs. The second group of approximations is based on studying discrete microstructures and includes • Periodic Microfield Approaches (PMAs), also referred to as periodic homogenization schemes or unit cell methods, see chapter 5: The inhomogeneous material is approximated by an infinitely extended model material with a periodic phase arrangement14) . The resulting periodic microfields are usually evaluated by analyzing unit cells (which may describe microgeometries ranging from rather simplistic to highly complex ones) via analytical or numerical methods. Unit cell methods 14)

Whereas mean field methods make the implicit assumption that there is no long range order in the inhomogeneous material, periodic phase arrangements imply the existence of just such an order.

–9–

are often used for performing materials characterization of inhomogeneous materials in the nonlinear range, but they can also be employed as micromechanically based constitutive models. The high resolution of the microfields provided by PMAs can be very useful for studying the initiation of damage at the microscale. However, because they inherently give rise to periodic configurations of damage, PMAs are not suited to investigating phenomena such as the interaction of the microstructure with macroscopic cracks. Periodic microfield approaches can give detailed information on the local stress and strain fields within a given unit cell, but they tend to be computationally expensive. • Embedded Cell or Embedding Approaches (ECAs; see chapter 6): The inhomogeneous material is approximated by a model material consisting of a “core” containing a discrete phase arrangement that is embedded within some outer region to which far field loads or displacements are applied. The material properties of this outer region may be described by some macroscopic constitutive law, they can be determined self-consistently or quasi-self-consistently from the behavior of the core, or the embedding region may take the form of a coarse description and/or discretization of the phase arrangement. ECAs can be used for materials characterization, and they are usually the best choice for studying regions of special interest, such as crack tips and their surroundings, in inhomogeneous materials. Like PMAs, embedded cell approaches can resolve local stress and strain fields in the core region at high detail, but tend to be computationally expensive. • Windowing Approaches (see chapter 7): Subregions (“windows”) — typically of rectangular or hexahedral shape — are chosen (“cut out”) from a given phase arrangement and subjected to boundary conditions that guarantee energy equivalence between the micro- and macroscales. For the special cases of macrohomogeneous stress and strain boundary conditions, respectively, lower and upper estimates for and bounds on the overall behavior of the inhomogeneous material can be obtained. • Other homogenization approaches employing discrete microgeometries, such as the statistics-based non-periodic homogenization scheme of Cui et al. (Li/Cui,2005). Because this group of methods explicitly study mesodomains (in the sense of section 1.2) they are sometimes referred to as “mesoscale approaches”. Figure 1 shows a sketch of a microstructure as well as PMA, ECA and windowing approaches applied to it. For small inhomogeneous samples (i.e. samples the size of which exceeds the microscale by not more than, say, an order of magnitude) the full microstructure can be modeled and studied, see e.g. (Guidoum,1994; Papka/Kyriakides,1994; Silberschmidt/Werner,2001; Luxner et al.,2005). In such cases boundary effects (and thus the boundary conditions applied to the sample as well as the sample’s size in terms of the characteristic length of the inhomogeneities) typically play a prominent role in the mechanical behavior. Note that, even though such problems certainly pertain to continuum micromechanics, they

– 10 –

ORIGINAL CONFIGURATION

PERIODIC APPROXIMATION

EMBEDDED CONFIGURATION

WINDOW

FIG.1: Schematic sketch of a random matrix–inclusion microstructure and of the modified microgeometries used by periodic microfield, embedded cell and windowing approaches for modeling it. do not involve a scale transition and describe somewhat different physical problems compared to mesoscale approaches. Further descriptions of inhomogeneous materials, such as rules of mixture (isostrain and isostress models) and semiempirical formulae like the Halpin–Tsai equations (Halpin/Kardos,1976), are not discussed here due to their typically rather weak physical background and their rather limited predictive capabilities. For brevity, a number of models with solid physical backgrounds, such as expressions for self-similar composite sphere assemblages (CSA, Hashin,1962) and composite cylinder assemblages (CCA, Hashin/Rosen,1964), information theoretical approaches (Kreher,1990), and the semiempirical methods proposed for describing the elastoplastic behavior of composites reinforced by aligned continuous fibers (Dvorak/Bahei-el-Din,1987), are not covered within the present discussions, either. For studying materials that are inhomogeneous at a number of (sufficiently widely spread) length scales (e.g. materials in which well defined clusters of inhomogeneities are present), hierarchical procedures that use homogenization at more than one level are a natural extension of the above concepts. Such multi-scale models are the subject of a short discussion in chapter 8.

– 11 –

2. MEAN FIELD APPROACHES In this section relations are given for two-phase materials only, extensions to multiphase materials being rather straightforward in most cases. Special emphasis is put on effective field methods of the Mori–Tanaka type, which may be viewed as the simplest mean field approaches for modeling inhomogeneous materials that encompass the full physical range of phase volume fractions15) . Unless specifically stated otherwise, the material behavior of both reinforcements and matrix is taken to be linear (thermo)elastic. Perfect bonding between the constituents is assumed in all cases. There is an extensive body of literature covering mean field approaches, most of which can be classified as effective field or effective medium theories, and related methods, so that the following treatment is far from complete. 2.1 General Relations For linear thermoelastic materials the overall stress–strain relations can be denoted in the form hσi = E∗ hεi + t∗ ∆T hεi = C∗ hσi + α∗ ∆T

,

(13)

where t∗ = −E∗ α∗ is the overall specific thermal stress tensor (i.e. the overall stress response of the fully constrained material to a purely thermal unit load) and ∆T is the (spatially uniform) temperature difference with respect to some stress-free reference temperature. The constituents, here a matrix (m) and inhomogeneities (i) , are also taken to behave thermoelastically, i.e. hσi(m) = E(m) hεi(m) + t(m) ∆T

hσi(i) = E(i) hεi(i) + t(i) ∆T

hεi(m) = C(m) hσi(m) + α(m) ∆T

hεi(i) = C(i) hσi(i) + α(i) ∆T

,

(14)

where the relations t(m) = −E(m) α(m) and t(i) = −E(i) α(i) hold16) . From the definition of volume averaging, eqn.(3), and from eqn.(12) one obtains the following relations for the phase averaged fields hεi = ξhεi(i) + (1 − ξ)hεi(m) = εa hσi = ξhσi(i) + (1 − ξ)hσi(m) = σa

,

(15)

15)

Because Eshelby and Mori–Tanaka methods are specifically suited for matrix–inclusiontype microtopologies, the expression “composite” is often used in the present chapter instead of the more general designation “inhomogeneous material”. 16)

In mean field theories the constitutive relations such as eqns.(13) and (14) are typically formulated in terms of total averaged strains and thus of the averaged fields defined in eqns.(3) and (12). Contributions of inelastic strains in Hooke’s law are accounted for by appropriate eigenstresses s, which in the case of thermoelasticity take the form s = t∆T = −Eα∆T .

– 12 –

provided no displacement jumps are present. Here ξ=V (i) =Ω(i) /(Ω(i) + Ω(m) ) stands for the volume fraction of the inhomogeneities, εa is the applied (far field) strain and σa is the applied (far field) stress. In the isothermal case, the equalities hεi=εa and hσi=σa correspond to homogeneous strain and homogeneous stress boundary conditions, respectively17) , perfectly bonded interfaces being also assumed in the former case. The phase averaged strains and stresses can be related to the overall strains, stresses, and temperature changes by the elastic and thermal phase strain and stress concentration ¯ (Hill,1963), respectively, which for thermoelastic inhomogeneous ¯ ¯ ¯ and b tensors A, a, B, materials are defined by the expressions ¯ (m) hεi + ¯ hεi(m) = A a(m) ∆T ¯ (m) ∆T ¯ (m) hσi + b hσi(m) = B

¯ (i) hεi + ¯ hεi(i) = A a(i) ∆T ¯ (i) ∆T ¯ (i) hσi + b hσi(i) = B

(16)

(compare eqn.(11) for the isothermal elastic case). By using eqns.(15) and (16), the phase strain and stress concentration tensors (or strain and stress concentration tensors for short) can be shown to fulfill the relations ¯ (i) + (1 − ξ)A ¯ (m) = I ξA ¯ (i) + (1 − ξ)B ¯ (m) = I ξB

ξ¯ a(i) + (1 − ξ)¯ a(m) = 0 ¯ (i) + (1 − ξ)b ¯ (m) = 0 ξb

.

(17)

From eqn.(7) the effective elasticity and compliance tensors of the composite can be obtained from the properties of the phases and from the concentration tensors as18) ¯ (m) ¯ (i) + (1 − ξ)E(m) A E∗ = ξE(i) A ¯ (i) + (1 − ξ)C(m) B ¯ (m) C∗ = ξC(i) B

.

(18)

The effective specific thermal stress tensor and the effective thermal expansion coefficient tensor can be written as t∗ = ξ[E(i) ¯ a(i) + t(i) ] + (1 − ξ)[E(m) ¯ a(m) + t(m) ] ¯ (i) + α(i) ] + (1 − ξ)[C(m) b ¯ (m) + α(m) ] , α∗ = ξ[C(i) b

(19)

respectively. Alternatively, the overall coefficients of thermal expansion can be obtained from the Mandel–Levin formula (Mandel,1965; Levin,1967) as ¯ (m) ]T α(m) + ξ[B ¯ (i) ]T α(i) α∗ = (1 − ξ)[B

,

(20)

17)

By using the Gauss theorem it can be shown theorem that the average stresses in a volume element of an inhomogeneous material with perfect phase boundaries are fully determined by the tractions on its boundaries and the average strains are determined by the displacements at the boundaries also for more general boundary conditions, compare eqn.(3). 18)

Because E∗ and its inverse C∗ depend not only on the properties of the constituents, but also on the concentration tensors and thus on the microscopic phase arrangement, it can be argued that overall properties of inhomogeneous materials, e.g. their effective Young’s modulus, are not material parameters in the strict sense.

– 13 –

¯ T denotes the transpose of B. ¯ where B The stress and strain concentration tensors for a given phase are linked to each other by expressions such as ¯ (m) = C(m) B ¯ (m) E(i) ]−1 = C(m) B ¯ (m) E(i) [I + (1 − ξ)(C(m) − C(i) )B ¯ (m) E∗ A ¯ (m) = E(m) A ¯ (m) C(i) [I + (1 − ξ)(E(m) − E(i) )A ¯ (m) C(i) ]−1 = E(m) A ¯ (m) C∗ B

.

(21)

In addition, relations can be derived that connect the thermal strain and stress concentration tensors of a given phase with the elastic strain and stress concentration tensors, respectively (see e.g. Benveniste/Dvorak,1990; Benveniste et al.,1991), typical expressions being19) ¯ (m) ][E(i) − E(m) ]−1 [t(m) − t(i) ] ¯ a(m) = [I − A ¯ (m) = [I − B ¯ (m) ][C(i) − C(m) ]−1 [α(m) − α(i) ] . b

(22)

From eqns.(18) to (22) it is evident that the knowledge of one elastic phase concentration tensor is sufficient for describing the full thermoelastic behavior of a two-phase inhomogeneous material within the mean field framework20) . There are a number of additional relations that allow e.g. the phase concentration tensors to be obtained from the overall and phase elastic tensors, see e.g. (B¨ ohm,1998). The macrohomogeneity condition and Hill’s lemma, eqn.(5), can be denoted for elastic inhomogeneous materials as X 1 Z T T T hσ εi = εT (x) E(p) ε(x) dΩ = hεi hσi = hεi E∗ hεi . (23) ( p ) Ω Ω(p) p For detailed discussions of Hill’s lemma for the case of periodicity boundary conditions see e.g. (Suquet,1987; Michel et al.,1999). 2.2 Eshelby Tensor and Dilute Matrix–Inclusion Composites A large proportion of the mean field descriptions used in continuum micromechanics of materials are based on the work of Eshelby (1957), who studied the stress and strain distributions in homogeneous media containing a subregion that spontaneously changes its shape and/or size (undergoes a “transformation”) so that it no longer fits into its previous space in the parent medium. Eshelby’s results show that if an elastic homogeneous ellipsoidal inclusion in an infinite linear elastic matrix is subjected to a uniform strain εt (called the “stress-free strain”, “unconstrained strain”, “eigenstrain”, or “transformation 19)

Relations for the inclusion concentration tensors corresponding to eqns.(21) and (22) take analogous forms. 20)

Similarly, n−1 elastic phase concentration tensors must be known for describing the overall thermoelastic behavior of an n-phase material.

– 14 –

strain”), uniform stress and strain states are induced in the constrained inclusion21) . The uniform strain in the constrained inclusion (the “constrained strain”), εc , is related to the stress-free strain εt by the expression εc = Sεt

,

(24)

where S is called the (interior point) Eshelby tensor. For eqn.(24) to hold, εt may be any kind of eigenstrain which is uniform over the inclusion (e.g. a thermal strain, or a strain due to some phase transformation which involves no changes in the elastic constants of the inclusion). For mean field descriptions of dilute matrix–inclusion composites, of course, it is the stress and strain fields in inhomogeneous inclusions (“inhomogeneities”) embedded in a matrix that are of interest. Such cases can be handled on the basis of Eshelby’s theory for homogeneous inclusions, eqn.(24), by introducing the concept of equivalent homogeneous inclusions. This strategy involves replacing an actual perfectly bonded inhomogeneous inclusion, which has different material properties than the matrix and which is subjected to a given unconstrained eigenstrain εt , with a (fictitious) “equivalent” homogeneous inclusion on which an appropriate (fictitious) “equivalent” eigenstrain ε τ is made to act. This equivalent eigenstrain must be chosen in such a way that the same stress and strain fields, hσi(i) and hεi(i) =εc , are obtained in the constrained state for the actual inhomogeneous inclusion and the equivalent homogeneous inclusion. Following (Withers et al.,1989) this process can be visualized as consisting of “cutting and welding” operations as shown in fig. 2. The concept of the equivalent homogeneous inclusion can be extended to cases where a uniform mechanical strain εa or external stress σa is applied to a perfectly bonded inhomogeneous elastic inclusion in an infinite matrix. Here, the strain in the inclusion, hεi(i) , will be a superposition of the applied strain and of an additional term εc due to the constraint effects of the surrounding matrix. A fair number of different expressions for concentration tensors obtained by such procedures have been reported in the literature, among the simplest being those proposed by Hill (1965) and elaborated by Benveniste (1987). For deriving them, the condition of equal stresses and strains in the actual inclusion (elasticity tensor E(i) ) and the equivalent inclusion (elasticity tensor E(m) ) under an applied far field strain εa is denoted as hσi(i) = E(i) hεi(i) = E(m) (hεi(i) − ετ ) , hεi(i) = εa + εc = εa + Sετ

(25)

where eqn.(24) is used to describe the constrained strain of the equivalent homogeneous inclusion. From these expressions and from eqn.(16) the inclusion strain concentration tensor can be obtained as ¯ (i) = [I + SC(m) (E(i) − E(m) )]−1 A dil 21)

(26)

For dilute inclusions this “Eshelby property” is limited to ellipsoidal shapes (Lubarda/Markenscoff,1998). For nondilute periodic arrangements of inclusions, however, nonellipsoidal shapes give rise to homogeneous fields (Liu et al.,2007).

– 15 –

FIG.2: Sketch depicting an equivalent inclusion procedure for a matrix–inclusion system loaded by a transformation strain εt (Withers et al.,1989). and the inclusion stress concentration tensor is found as ¯ (i) = E(i) [I + SC(m) (E(i) − E(m) )]−1 C(m) B dil = [I + E(m) (I − S)(C(i) − C(m) )]−1

.

(27)

compare (Hill,1965; Benveniste,1987). Alternative expressions for dilute mechanical and thermal inclusion concentration tensors were given e.g. by Mura (1987), Wakashima et al. (1988), and Clyne/Withers (1993). All of the above relations were derived under the assumption that the inhomogeneities are dilutely dispersed in the matrix and thus do not “feel” any effects due to their neighbors (i.e. they are loaded by the unperturbed applied stress σa or applied strain εa , the so called dilute case). Accordingly, they are independent of the reinforcement volume fraction ξ. The stress and strain fields outside a transformed homogeneous or inhomogeneous inclusion in an infinite matrix are not uniform on the microscale22) (Eshelby,1959). Within the framework of mean field theories, which aim to link the average fields in matrix and inhomogeneities with the overall response of inhomogeneous materials, however, it is only the average matrix stresses and strains that are of interest. For dilute composites, such expressions follow directly by combining eqns.(26) and (27) with eqn.(17). Estimates for 22)

The fields outside a single inclusion can be described via the (exterior point) Eshelby tensor, see e.g. (Ju/Sun,1999). From the (constant) interior point fields and the (position) dependent exterior point fields the stress and strain jumps at the interface between inclusion and matrix can be evaluated.

– 16 –

the overall elastic and thermal expansion tensors can, of course, be obtained in a straightforward way from the concentration tensors by using eqns.(19) and (22). It must be kept in mind, however, that the dilute expressions are strictly valid only for vanishingly small inhomogeneity volume fractions and give dependable results only for ξ 0.1. The Eshelby tensor S depends only on the material properties of the matrix and on the aspect ratio a of the inclusions, i.e. expressions for the Eshelby tensor of ellipsoidal inhomogeneities are independent of the material symmetry and properties of the inhomogeneities. Expressions for the Eshelby tensor of spheroidal inclusions in an isotropic matrix are given e.g. in (Pedersen,1983; Tandon/Weng,1984; Mura,1987; Clyne/Withers,1993) 23) , the formulae being very simple for continuous fibers of circular cross-section (a → ∞), spherical inclusions (a = 1), and thin circular disks (a → 0). Closed form expressions for the Eshelby tensor have also been reported for spheroidal inclusions in a matrix of transversely isotropic (Withers,1989) or cubic material symmetry (Mura,1987), provided the material axes of the matrix constituents are aligned with the orientations of nonspherical inclusions. In cases where no analytical solutions are available, the Eshelby tensor can be evaluated numerically, compare (Gavazzi/Lagoudas,1990). 2.3 Mean Field Methods for Thermoelastic Composites with Aligned Reinforcements Theoretical descriptions of the overall thermoelastic behavior of composites with reinforcement volume fractions of more than a few percent must explicitly account for collective interactions between inhomogeneities, i.e. for the effects of all surrounding reinforcements on the stress and strain fields experienced by a given fiber or particle. Within the mean field framework such reinforcement interaction effects as well as the concomitant perturbations of the stress and strain fields in the matrix are accounted for via suitable phase-wise constant approximations. Beyond these “background effects”, there are interactions between individual reinforcements which, on the one hand, give rise to inhomogeneous stress and strain fields within each inhomogeneity (“intra-particle fluctuations”). On the other hand, they cause the levels of the average stresses and strains in individual inhomogeneities to differ (“interparticle fluctuations”). These interactions are not accounted for by most mean field methods24) . Mori–Tanaka-Type Estimates One way for achieving this consists of approximating the stresses acting on an inhomogeneity, which may be viewed as the perturbation stresses caused by the presence of other 23)

Instead of evaluating the Eshelby tensor for a given configuration, the so called mean polarization factor tensor Q=E(m) (I − S) may be evaluated instead, see e.g. (Ponte Casta˜ neda,1996). 24)

In the literature the designator “interacting models” is typically used for approaches that explicitly account for interactions between pairs or higher numbers of inhomogeneities, i.e., not for Mori-Tanaka and self-consistent methods.

– 17 –

inhomogeneities (“image stresses”, “background stresses”, “mean field stresses”, “effective stresses”) superimposed on the applied far field stress, by an appropriate average matrix stress hσi(m) . The idea of combining this concept of an average matrix stress with Eshelby-type equivalent inclusion approaches goes back to Brown and Stobbs (1971) as well as Mori and Tanaka (1973). Effective field theories of this type are generically called Mori–Tanaka methods. It was recognized by Benveniste (1987) that in the isothermal case the central assumption involved in Mori–Tanaka approaches can be expressed as (i)

(i)

(m)

¯ hεi(m) = A ¯ A ¯ hεi(i) = A dil dil M hεi ¯ (m) ¯ (i) hσi(m) = B ¯ (i) B hσi(i) = B dil dil M hσi .

(28)

These relations can be viewed as modifications of eqn.(11) in which the applied strain and stress, εa and σa , are replaced by the (unknown) matrix strain or stress, hεi(m) and hσi(m) , respectively, according to the effective field strategy. Based on eqn.(28) simple and straightforward Mori–Tanaka-type expressions for the matrix strain and stress concentrations tensors were proposed by (Benveniste,1987) as ¯ (m) = [(1 − ξ)I + ξ A ¯ (i) ]−1 A M dil ( m ) ( ¯ = [(1 − ξ)I + ξ B ¯ i) ]−1 , B M dil

(29)

the corresponding relations for the inclusion strain and stress concentration tensors following from eqns.(28) as ¯ (i) = A ¯ (i) [(1 − ξ)I + ξ A ¯ (i) ]−1 A M dil dil (i) (i) (i) −1 ¯ ¯ ¯ BM = Bdil [(1 − ξ)I + ξ Bdil ] ,

(30)

Equations (29) and (30) may be evaluated with any strain and stress concentration ¯ (i) and B ¯ (i) pertaining to dilute inhomogeneities in a matrix. If, for example, tensors A dil

dil

eqns.(26) and (27) are employed, the Mori–Tanaka matrix strain and stress concentration tensors take the form ¯ (m) = {(1 − ξ)I + ξ[I + SC(m) (E(i) − E(m) )]−1 }−1 A M (m) ¯ BM = {(1 − ξ)I + ξE(i) [I + SC(m) (E(i) − E(m) )]−1 C(m) }−1

,

(31)

compare (Benveniste,1987; Benveniste/Dvorak,1990; Benveniste et al.,1991). A number of authors gave different but essentially equivalent Mori–Tanaka-type expressions for the phase concentration tensors and effective thermoelastic tensors of inhomogeneous materials, among them (Pedersen,1983; Wakashima et al.,1988; Taya et al.,1991; Pedersen/Withers,1992; Clyne/Withers,1993). Alternatively, the Mori–Tanaka method can be formulated (Tandon/Weng,1984) to directly give the overall elasticity tensor as −1 E∗M = E(m) I − ξ[(E(i) − E(m) ) S − ξ(S − I) + E(m) ]−1 [E(i) − E(m) ] – 18 –

,

(32)

FIG.3: Sketch of an aligned ellipsoidally distributed spatial arrangement of aligned ellipsoidal inhomogeneities that is closely related to Mori–Tanaka-type approaches (a=2.0). which can be easily modified for porous materials by letting E(i) → 0 to give E∗M,por = E(m) I +

−1 ξ (I − S)−1 1−ξ

.

(33)

Equation (33) should not, however, be used for void volume fractions that are much in excess of, say, ξ = 0.2525) . In accordance with their derivation, Mori–Tanaka-type theories at all volume fractions describe composites consisting of aligned ellipsoidal inhomogeneities embedded in a matrix, i.e. inhomogeneous materials with a distinct matrix–inclusion microtopology. More precisely, it can be shown (Ponte Casta˜ neda/Willis,1995) that Mori–Tanaka methods are a special case of Hashin–Shtrikman variational estimates (compare section 2.4) in which the reference material is the matrix and the spatial arrangement of the (aligned) inhomogeneities follows an “ellipsoidal distribution” with the same aspect ratio and orientation as the inhomogeneities themselves, compare fig. 326) . For two-phase composites Mori–Tanaka estimates coincide with Hashin–Shtrikman-type bounds (compare section 3.1) and, accordingly, their predictions for the overall Young’s and shear moduli are always on the low side for composites reinforced by stiff aligned or spherical reinforcements (see the comparisons in section 3.4 as well as tables 2 to 6) and on the high side for materials containing compliant reinforcements in a stiffer matrix. In high contrast situations they tend to under- or overestimate the effective 25)

Mori–Tanaka expressions for two-phase materials are based on the assumption that the shape of the inhomogeneities can be described by ellipsoids of a fixed aspect ratio throughout the deformation history. In the case of porous materials with high void volume fractions, e.g. closedcell cellular materials and foams, deformation at the microscale takes place mainly by bending and buckling of the matrix “walls” between voids, see e.g. (Gibson/Ashby,1988). Such large strain effects cannot be properly described by Mori–Tanaka approaches, which, consequently, tend to markedly overestimate the effective stiffness of cellular materials. 26)

Geometries with ellipsoidal symmetry as shown in fig. 3 can be thought of as affinely deformed versions of the composite sphere assemblage (CSA) geometries that were proposed by Hashin (1962).

– 19 –

elastic properties by a considerable margin (compare table 6). Similar tendencies hold for multi-phase composites. For discussions of further issues with respect to the range of validity of Mori–Tanaka theories for elastic inhomogeneous two-phase materials see (Christensen et al.,1992). Mori–Tanaka-type theories can be implemented into computer programs in a straightforward way. Because they are explicit algorithms, all that is required are matrix additions, multiplications, and inversions plus expressions for the Eshelby tensor. Despite their limitations, Mori–Tanaka approaches provide useful accuracy for the elastic contrasts pertaining to most practically relevant composites. This combination of features makes them important tools for evaluating the stiffness and thermal expansion properties of inhomogeneous materials that show a matrix–inclusion topology with aligned inhomogeneities or voids. Mori–Tanaka-type approaches for thermoelastoplastic materials are discussed in section 2.4 and “extended” Mori–Tanaka approaches for nonaligned reinforcements in section 2.7. Classical Self-Consistent Estimates Another group of estimates for the overall thermomechanical behavior of inhomogeneous materials are self-consistent schemes (or effective medium theories), in which an inhomogeneity or some phase arrangement is embedded in the effective material (the properties of which are not known a priori)27) . Figure 4 shows a schematic comparison of the material and loading configurations underlying Eshelby, Mori–Tanaka, classical self-consistent, and generalized self-consistent (compare section 3.4) approaches.

i m

i m

i

i eff m

CSCS

GSCS

eff

σa

σ(m) tot

dilute

MTM

σa

σa

FIG.4: Schematic comparison of the Eshelby method for dilute composites, Mori– Tanaka approaches, and classical as well as generalized self-consistent schemes. If the kernel consists of an inhomogeneity only, classical (or two-phase) self-consistent schemes are obtained (CSCS, see e.g. Hill,1965), which formally can be described by relations such as ¯ i,∗ . E∗SCS = E(m) + ξ[E(i) − E(m) ]A (34) dil ¯ i,∗ , the strain concentration tensor of an inhomogeneity embedded in the Because A dil effective medium in analogy to eqn.(26), is a function of E∗SCS , this expression is implicit 27)

Within the classification of micromechanical methods given in section 1.4, self-consistent mean field methods can also be viewed as as analytical ECAs.

– 20 –

and E∗SCS has to be solved for by self-consistent iteration using a scheme of the type En+1 = E(m) + ξ[E(i) − E(m) ][I + Sn Cn (E(i) − En )]−1 −1 Cn+1 = En+1 .

(35)

The Eshelby tensor used in this expression, Sn , pertains to an inclusion embedded in the n-th iteration for the effective medium and has to be recomputed for each iteration. Generally, classical self-consistent schemes are best suited to describing the overall elastic properties of two-phase materials that do not show a matrix–inclusion microtopology in at least part of their range of volume fractions28) , but they have also been employed for studying particle reinforced MMCs in which the grain size of the matrix and the size of the particulates are similar (Corvasce et al.,1991). When microstructures described by the CSCS show a geometrical anisotropy of the transversely isotropic type (e.g. in terms of the free lengths in the phases in metallographical sections), this anisotropy determines the “inhomogeneity” aspect ratio to be used. For porous materials classical self-consistent schemes predict a breakdown of stiffness due to the percolation of the pores at ξ = 12 for spherical voids and at ξ = 13 for aligned cylindrical voids (Torquato,2002). Because self-consistent schemes are by definition implicit methods, their computational requirements are in general higher than those of Mori–Tanaka-type approaches. Like Mori–Tanaka approaches, they can form the basis for describing the behavior of nonlinear inhomogeneous materials. Differential Schemes A further group of mean field approaches besides effective field and self-consistent methods are differential schemes (McLaughlin,1977; Norris,1985). They can be envisaged as involving repeated cycles of adding small concentrations of inhomogeneities to a material followed by homogenizing. According to (Hashin,1988) the resulting overall elastic tensors can be described by the differential equations 1 dE∗D ¯ i,∗ = (E(i) − E∗D )A dil dξ 1−ξ 1 dC∗D ¯ i,∗ = (C(i) − C∗D )B dil dξ 1−ξ

(36)

with the initial conditions E∗D =E(m) and C∗D =C(m) , respectively, at ξ=0. In analogy to ¯ i,∗ and B ¯ i,∗ depend on E∗ . Differential schemes describe aligned matrix– eqn.(34) A D dil dil 28)

Classical self consistent schemes can be shown to correspond to perfectly disordered materials (Kr¨ oner,1978) or self-similar hierarchical materials (Torquato,2002). In the context of two-phase materials typical targets are interwoven composites and functionally graded materials. Multi-phase expressions analogous to eqn.(34) are often used to describe the mechanical behavior of polycrystals.

– 21 –

inclusion microgeometries with markedly polydisperse distributions of inhomogeneity sizes (corresponding to the repeated homogenization steps) 29) . Because of their association with specialized microgeometries (that are not typical of “classical composites”, in which reinforcements tend to be monodispersely sized) and due to their higher mathematical complexity (as compared e.g. to Mori–Tanaka-approaches) Differential Schemes have not been used extensively for studying the mechanical behavior of composite materials. 2.4 Hashin–Shtrikman Estimates The stress field in an inhomogeneous material can be expressed in terms of the one in a homogeneous comparison (or reference) medium as σ(x) = E0 ε(x) + τ (x) = E0 (ε0 + ε˜ 0 (x)) + τ (x)

(37)

where τ (x) is known as the polarization tensor, E0 is the elastic tensor of the comparison material, ε0 is the homogeneous strain in the reference material, and ε˜ 0 (x) is the local strain fluctuation tensor associated with τ (x). Equation (37) can be interpreted as using the polarizations for describing the “fast” contributions to the actual stress field, σ(x), while the “slow” contributions are covered by the behavior of the reference material. Phase-wise constant polarization tensors τ (p) = (E(p) − E0 )hεi(p)

(38)

can be obtained by phase averaging τ over phase (p). Equation (38) can be used as the starting point for deriving general mean field strain concentration tensors of the type ¯ (p) = (L0 + E(p) )−1 ξ(L0 + E(i) )−1 + (1 − ξ)(L0 + E(m) )−1 −1 A HS

.

(39)

This expression leads to estimates of the overall elastic stiffness of microstructured twophase materials in the form −1 E∗HS = E(m) + ξ(L0 − E(m) ) I + (1 − ξ)(L0 + E(m) )−1 (E(i) − E(m) )

,

(40)

which are referred to as Hashin–Shtrikman elastic tensors. The tensor L0 appearing in eqns.(39) and (40) is known as the overall constraint tensor (Hill,1965) and is defined as L0 = E0 [(S0 )−1 − I] 29)

.

(41)

With the exception of the CSCS for two-dimensional conduction (Torquato/Hyun,2001), the mean field methods discussed in section 2.3 do not exactly correspond to monodisperse inclusion–matrix microstructures, i.e., they are not realizable as single-scale dispersions.

– 22 –

Here the Eshelby tensor S0 must be evaluated with respect to the comparison material. L0 can be shown to depend on the two-point statistics of the microgeometrical arrangement, see (Bornert,2001). Standard mean field models can be obtained as special cases of Hashin–Shtrikman methods by appropriate choices of the comparison material. For example, using the matrix as the comparison material results in Mori–Tanaka methods, whereas the selection of the effective material as comparison material leads to classical self-consistent schemes. Generalized Self-Consistent Estimates The above Hashin–Shtrikman formalism can be extended in a number of ways, one of which consists in not only studying inhomogeneities but more complex geometrical entities, called patterns or motifs, embedded in a matrix, see (Bornert,1996; Bornert,2001). Typically the stress and strain fields in such patterns are inhomogeneous even in the dilute case and numerical methods have to be resorted to in evaluating the corresponding overall constraint tensors. A simple example of such a pattern is a spherical particle or cylindrical fiber surrounded by a layer of matrix at an appropriate volume fraction, which, when embedded in the effective material, allows to recover the three-phase or generalized self-consistent scheme (GSCS) of (Christensen/Lo,1979; Christensen/Lo,1986), compare fig. 4. GSCS-expressions for the overall elastic moduli are obtained by considering the differential equations describing the elastic response of such three-phase regions under appropriate boundary and loading conditions. The GSCS in its original version is limited to inhomogeneities of spherical or cylindrical shape, where it describes microgeometries of the CSA and CCA types, respectively, and gives rise to third order equations for the shear modulus G and the transverse shear modulus GT (for the other moduli, CSA and CCA results are available). Extensions of the GSCS to multi-layered spherical inhomogeneities (Hervé/Zaoui,1993) and to aligned ellipsoidal inhomogeneities (Huang/Hu,1995), were reported in the literature. Generalized self-consistent schemes give excellent results for inhomogeneous materials with matrix–inclusion topologies and are accordingly highly suited for obtaining estimates for the thermoelastic moduli of composite materials reinforced by (approximately) spherical particles or aligned continuous fibers. They are, however, not mean-field schemes 30) in the sense of section 2.3. 2.5 Other Estimates for Elastic Composites with Aligned Reinforcements A number of interpolative schemes have been proposed, which generate estimates between mean-field and related estimates describing, on the one hand, some composite and, on the other hand, a material in which the roles of matrix and reinforcements are interchanged (referred to, e.g., as Mori–Tanaka and Inverse Mori–Tanka schemes), see e.g. (Lielens 30)

Even though GSCS schemes are not a mean field methods in the strict sense, pertinent phase concentration tensors can easily be obtained from the overall elastic tensors, so that mean field formalisms may be used to generate e.g. effective coefficients of thermal expansion.

– 23 –

et al.,1997). Such approaches, while being capable of providing improved estimates for given composites, suffer from the drawback of not being associated with realizable microgeometries. Recently three-point estimates for the thermoelastic properties of two-phase materials were developed by Torquato (1998), which correspond to the same phase arrangements and use the same three-point microstructural parameters η and ζ as discussed for threepoint bounds in section 3.2. These estimates lie between the corresponding bounds and give excellent analytical predictions for the overall thermoelastic response of inhomogeneous materials that show the appropriate microstructures and are free of flaws. 2.6 Mean Field Methods for Inelastic Composites Models for microstructured materials with viscoelastic constituents are closely related to those for elastic composites. Relaxation moduli and creep compliances can be obtained by applying micromechanical approaches in the Laplace transformed domain, where the problems are analogous to elastic ones for the same microgeometries. Standard micromechanical methods can be used in conjunction with complex moduli for steady state vibration analysis of viscoelastic composites. For correspondence principles between descriptions pertaining to elastic and viscoelastic inhomogeneous materials, respectively, see e.g. (Hashin,1983). The extension of mean field estimates and bounding methods to elastoplastic, viscoelastoplastic, and damaging inhomogeneous materials, however, has proven to be challenging. (Thermo)Elastoplastic Composites The aim of mean field models of microstructured materials with at least one elastoplastic constituent consists in providing accurate estimates for the material response for any load state and load history at reasonable computational cost. The main difficulties in attaining this goal lie in the typically strong intra-phase fluctuations of the microstress and microstrain fields in elastoplastic inhomogeneous materials and in the path dependence of plastic behavior. The responses of the constituents can thus vary markedly at the microscale, with each material point essentially following a different trajectory in stress space. Accordingly, even a 2-phase elastoplastic composite effectively behaves as a multiphase material and phase averages over matrix and reinforcements are less useful descriptors than in the linear elastic case. Mean field models of elastoplastic inhomogeneous materials31) typically are based on solving sequences or sets of linearized problems in terms of linear inhomogeneous reference media. Accordingly, choices have to be made with respect to the linearization procedure, the linear homogenization model, and the phase-wise equivalent stresses and 31)

The present chapter is restricted to continuum mechanical descriptions of thermoelastoplastic composites. In a separate line of development, phase averaged stress fields obtained by Eshelby-type methods have been used in dislocation-based descriptions of nonlinear matrix behavior, e.g. for studying the response of MMCs to temperature excursions, see e.g. (Taya/Mori,1987).

– 24 –

equivalent strains to be used in evaluating the elastoplastic constituent material behavior, see (Zaoui,2001). In the literature several lines of development of mean field and Hashin–Shtrikman-type approaches for elastoplastic inhomogeneous materials can be found. Historically, the most important of them have been secant plasticity concepts based on deformation theory, see e.g. (Tandon/Weng,1988; Dunn/Ledbetter,1997), and incremental plasticity models, compare e.g. (Hill,1965; Lagoudas et al.,1991; Karayaka/Sehitoglu,1993; Pettermann,1997). Other relevant methods are the “tangent” concept (Molinari et al.,1987) and mean field applications of the Transformation Field Analysis (TFA; Dvorak/Benveniste,1992; Dvorak,1992). The latter describes inelastic strains via transformation strains and uses elastic accommodation for stresses and strains in the elastoplastic regime, resulting in computationally efficient algorithms. A further recent development are affine formulations (Masson et al.,2000), in which the behavior of viscoplastic composites is studied on the basis of thermoelastic problems. A fairly recent overview of mean field methods for elastoplastic composites can be found in (Ponte Casta˜ neda/Suquet,1998). Secant models aim at directly arriving at solutions in terms of the overall response for a given load state. They are based on the deformation theory of plasticity and can be formulated in terms of potentials, which allows for a concise mathematical presentation (Bornert/Suquet,2001). Secant models treat elastoplastic composites as nonlinearly elastic materials and are, accordingly, limited to monotonic loading and radial (or approximately radial) trajectories of the constituents in stress space during loading 32) , which precludes their use as micromechanically based constitutive models in multi-scale analysis. Advanced secant formulations, however, are highly suitable for materials characterization, where they give excellent results. Incremental mean field models, affine approaches and the TFA are not subject to limitations with respect to loading paths. However, in their standard forms incremental MFAs and the TFA tend to overestimate the overall strain hardening in the postyield regime to such an extent that they are of limited practical applicability, especially for matrix dominated deformation modes, compare e.g. (Gilormini,1995; Suquet,1997; Chaboche/Kanouté,2003). Recent developments involve the use of tangent operators that reflect the macroscopic symmetry of the composite, e.g. “isotropized” operators 33) for statistically isotropic materials, see e.g. (Bornert,2001), and algorithmic modifications, compare (Doghri/Ouaar,2003; Doghri/Friebel,2005). These improvements have succeeded in markedly reducing the above weaknesses of incremental mean field mod32)

The condition of radial loading paths in stress space at the constituent level is generally violated by a considerable margin in the phases of elastoplastic inhomogeneous materials, even for macroscopic loading paths that are are perfectly radial, see e.g. (Pettermann,1997). This effect is due to changes in the accommodation of the phase stresses and strains in inhomogeneous materials upon the changes in tangential stiffness in any constituent upon yielding and, to a lesser extent, in the strain hardening regime. 33)

For a discussion of a number of issues pertaining to the use of “isotropic” versus “anisotropic” tangent operators see e.g. (Chaboche/Kanouté,2003).

– 25 –

els such as incremental Mori–Tanaka methods for particle reinforced composites, making them suitable for practical applications. Modifications to the Transformation Field Analysis have also been proposed, see e.g. Chaboche et al.(2001). Major improvements to the TFA can be achieved at limited computational cost by describing the inelastic strains at the microscale via nonuniform transformation fields, the plastic strain field being approximated by suitable generalized strains referred to as plastic modes that are extracted from unit cell models (Michel/Suquet,2004). A problem common to all mean field approaches for elastoplastic composites lies in handling the intraphase variations of the stress and strain fields within the assumption of phase-wise uniformity34) . Improvements in this respect have been obtained by evaluating the phase averages of the von Mises equivalent stresses from energy considerations (Qiu/Weng,1992; Hu,1996) or by using statistics-based theories (Buryachenko,1996; Ju/Sun,2001). Within mean field methods that use secant plasticity, accounting for fluctuations in the effective equivalent stress gives rise to “modified secant models”, see e.g. (Ponte Casta˜ neda/Suquet,1998), which have shown excellent agreement with predictions from multi-particle unit cell models in a materials characterization context (Segurado et al.,2002). This approach, however, is not suitable for incremental methods, for which an empirical correction was proposed by Delannay et al.(2007). In what follows a short discussion is given of the Incremental Mori–Tanaka (IMT) method developed by Pettermann (1997). Incremental mean field methods can be formulated in terms of phase averaged strain and stress rate tensors, dhεi(p) and dhσi(p) , which can be expressed in analogy to eqn.(16) as (p) ¯ (tp) dhεi + ¯ dhεi(p) = A at dT ¯ (tp) dhσi + b ¯ (tp) dT dhσi(p) = B

,

(42)

where dhεi stands for the far field mechanical strain rate tensor, dhσi for the far field (p) ¯ (p) ¯ (tp) , ¯ stress rate tensor, and dT for a spatially uniform temperature rate. A at , B t , and ( p ) ¯ are instantaneous strain and stress concentration tensors, respectively. Using the b t assumption that the inhomogeneities show elastic and the matrix elastoplastic material behavior35) , the overall instantaneous (tangent) stiffness tensor can be written in terms of the phase properties and the instantaneous concentration tensors as (m) ¯ (tm) E∗t = E(i) + (1 − ξ)[Et − E(i) ]A (m) ¯ (tm) ]−1 = [C(i) + (1 − ξ)[Ct − C(i) ]B

,

(43)

34)

In general, the phase average of the equivalent stress in a given phase is greater than the equivalent stress computed from the phase averages of the stress components, which leads to a tendency of simple mean field methods to overestimate overall yield and flow stresses. As an extreme case, evaluating the von Mises equivalent stresses from the phase averaged stress components in such approaches leads to predictions that materials with spherical reinforcements will not yield under overall hydrostatic loads and homogeneous temperature loads. 35)

Analogous expressions can be derived for elastoplastic inhomogeneities in an elastic matrix or, in general, for composites containing any required number of elastoplastic phases.

– 26 –

in analogy to eqn.(18). Note that eqn.(43) is formulated such that dhσi = E ∗t dhεi, i.e. E∗t is a “continuum” tangent operator, compare (Doghri/Ouaar,2003). Using the Mori–Tanaka formalism of Benveniste (1987), compare eqn.(31), the instantaneous matrix concentration tensors take the form ¯ (tm) = {(1 − ξ)I + ξ[I + St C(tm) (E(i) − E(tm) )]−1 }−1 A ¯ (tm) = {(1 − ξ)I + ξE(i) [I + St C(tm) (E(i) − E(tm) )]−1 Ct(m) }−1 B

.

(44)

The instantaneous Eshelby tensor St depends on the current state of the matrix material and in general has to be evaluated numerically due to the anisotropic structure of the (m) instantaneous tangent operator Et . Expressions for the instantaneous coefficients of thermal expansion and for the instantaneous thermal concentration tensors can be derived in analogy to the corresponding thermoelastic relations, e.g. eqns.(19) and (22). Formulations of eqns.(42) to (44) that are directly suitable for implementation as micromechanically based constitutive models at the integration point level within Finite Element codes can be obtained by replacing rates such as dhσi(p) with finite increments such as ∆hσi(p) . It is worth noting that in the resulting incremental Mori–Tanaka methods no assumptions on the overall yield surface and the overall flow potential are made, the effective material behavior being entirely determined by the incremental mean field equations and the constitutive behavior of the phases. However, mapping of the stresses onto the yield surface cannot be handled at the level of the homogenized material and stress return mapping must be applied to the matrix at the microscale instead. Accordingly, the constitutive equations describing the overall behavior cannot be integrated directly (as is the case for homogeneous elastoplastic materials), and iterative algorithms are required. For example, Pettermann (1997) used an implicit Euler scheme in an implementation of an incremental Mori–Tanaka method as a user supplied material routine (UMAT) for the commercial Finite Element code ABAQUS (Simulia, Pawtucket, RI). Algorithms of this type can also handle thermal expansion effects36) and temperature dependent material parameters. Incremental Mori–Tanaka methods, especially when they incorporate improvements as proposed by Doghri and Friebel (2005), offer a combination of useful accuracy, flexibility in terms of inhomogeneity geometries, and relatively low computational requirements. Accordingly, they are attractive candidates for use at the lower length scale in multi-scale models, compare chapter 8, of ductile matrix composites. Similar methods have also been employed to study the nonlinear response of inhomogeneous materials due to microscopic damage or to combinations of damage and plasticity, see e.g. (Tohgo/Chou,1996; Guo et al.,1997). 36)

Elastoplastic inhomogeneous materials such as metal matrix composites typically show a hysteretic thermal expansion response, i.e. the “coefficients of thermal expansion” are not material properties in the strict sense. This dependence of the thermal expansion behavior on the instantaneous mechanical response requires special treatment within the IMT framework, compare (Pettermann,1997).

– 27 –

Self-consistent schemes can also be combined with secant, incremental and affine formulations to describe elastoplastic inhomogeneous materials, see e.g. (Hill,1965; Hutchinson,1970; Berveiller/Zaoui,1979; Bornert,1996). They have also been extended to account for microscopic damage in addition to plasticity (Estevez et al.,1999; Gonz´ alez/LLorca,2000). Such procedures (as well as the weaknesses and strengths of the approaches) are closely related to those discussed above for the Mori–Tanaka method 37) . Analogous statements hold for Hashin–Shtrikman estimates (and bounds, compare section 3.3) in general, which have e.g. been developed to account for geometrical nonlinearities due to the evolution of of pore shapes and orientations in porous materials, compare (Kailasam et al.,2000). Finally, it is worth mentioning that in the elastoplastic range the “Eshelby property” (i.e. uniformity) of the stress and strain fields within the inhomogeneities is lost even at low reinforcement volume fractions, both intra-inhomogeneity and inter-inhomogeneity fluctuations being caused by interactions with neighboring fibers or particles, see e.g. (B¨ ohm/Han,2001). 2.7 Mean Field Methods for Composites with Nonaligned Reinforcements The overall symmetry of composites reinforced by nonaligned short fibers in many cases is isotropic (random fiber orientations) or transversely isotropic (fiber arrangements with axisymmetric orientation distributions, e.g., fibers with planar random orientation). However, processing conditions can give rise to a wide range of fiber orientation distributions and, consequently, lower overall symmetries, compare e.g. (Allen/Lee,1990). The statistics of the microgeometries of nonaligned matrix–inclusion composites can be described on the basis of orientation distribution functions (ODFs) and aspect ratio or length distribution functions (LDFs) of the reinforcements, which can be determined experimentally. In addition, information is required on the type of phase geometry to be described. At elevated fiber volume fractions local domains of (nearly) aligned fibers are typically observed, which give rise to a “grain-type” mesostructure, referred to as an aggregate system (Eduljee/McCullough,1993). Composites of this type are appropriately modelled via two-step homogenization schemes that involve meso-level averages at the grain level, see e.g. (Camacho et al.,1990; Pierard et al.,2004). At low to moderate fiber volume fractions, in contrast, the orientations of neighboring fibers are essentially independent within the geometrical constraints of non-penetration. Such dispersed systems (Eduljee/McCullough,1993) can be studied via single-step mean field schemes involving configurational averaging procedures, which may encompass aspect ratio averaging in addition to orientational averaging. Orientational averaging of tensor valued variables can be done by direct (numerical) integration, see e.g. (Pettermann et al.,1987), or on the basis of expansions of the 37)

In analogy to the elastic case, Mori–Tanaka based methods are more suitable for describing elastoplastic matrix–inclusion composites, whereas two-phase self-consistent schemes are more suited for modeling inhomogeneous materials with topologically indistinguishable phases, e.g. polycrystals or interwoven materials.

– 28 –

ODF in terms of generalized spherical harmonics (Viglin expansions), compare e.g. (Advani/Tucker,1987). The latter approach can be formulated in a number of ways, e.g. via texture coefficients or texture matrices, compare (Siegmund et al.,2004). For a discussion of a number of issues relevant to configurational averaging see Eduljee/McCullough (1993). Mori–Tanaka methods have been adapted to describing the elastic behavior of rather general microstructures consisting of nonaligned inhomogeneities embedded in a matrix. A typical starting point for such “extended” Mori–Tanaka algorithms are dilute stress concentration tensors for reinforcements that have some given orientation with respect ¯ (i),Θ , which can be expressed as to the global coordinate system, B dil (m) (i) (m) −1 Θ −1 ¯ (i),Θ = TΘ I + E (I − S)(C − C ) Tσ B σ dil

.

(45)

on the basis of eqn.(27). Here TΘ σ stands for the stress transformation tensor from the local (reinforcement based) to the global coordinate system. A phase averaged dilute fiber stress concentration tensor can then be obtained by orientational averaging as Z D E (i) (i),Θ (i),Θ ¯ ¯ ρ (Θ)Bdil dΘ = ρ Bdil (46) Bdil = Θ

where Θ stands for the orientation angle and the orientation distribution function ρ(Θ) is assumed to be normalized to give hρi = 1. The core statement of the Mori–Tanaka approach according to Benveniste (1987), eqn.(28), can then be written in the form (i)

(i)

(m)

(i)

hσi(i) = BM hσi = Bdil BM hσi = Bdil hσi(m)

(47)

This allows the phase averaged Mori–Tanaka concentration tensors to be expressed as (m)

(i)

BM = [(1 − ξ)I + ξBdil ]−1 (i)

(i)

(i)

BM = Bdil [(1 − ξ)I + ξBdil ]−1

(48)

in analogy to standard Mori–Tanaka methods, compare eqns.(29) and (30). “Extended” Mori–Tanaka methods for modeling the elastic behavior of dispersed microstructures that contain nonaligned inhomogeneities were developed by a number of authors, compare e.g. (Benveniste,1990; Dunn/Ledbetter,1997; Pettermann et al.,1997; Mlekusch,1999). These models differ mainly in the algorithms employed for orientational or configurational averaging. In addition to providing estimates for the overall thermoelastic behavior of composites containing nonaligned reinforcements, “extended” Mori–Tanaka theories can also be used for describing the dependence of the microscopic stress and strain tensors in individual reinforcements in such materials on the latter’s orientation. For this purpose eqn.(28) is rewritten as m) ¯ (i),Θ = B ¯ (i),Θ B(M B M dil

– 29 –

,

(49)

4

σ1 (MPa)

3 2 1 σ1 - MTM σax - MTM σ1 - UC

0 -1 0

10

20

30

40

50

60

70

80

90

Θ(°) FIG.5: “Extended” Mori–Tanaka and unit cell predictions for the angular dependence of the maximum principal stresses σ1 (solid line) in the randomly oriented fibers (aspect ratio a = 5) of a SiC/Al MMC subjected to uniaxial tension. In addition, the variation of the axial stresses σxx is given (dashed line; note that the maximum principal stresses show little variation for fibers oriented at angles in excess of Θ≈65◦ to the loading direction.) Numerical results are presented in terms of the mean values (solid circles) and the corresponding standard deviations within individual fibers; from (Duschlbauer et al.,2003). (i),Θ

(m)

¯ where B and BM are defined by eqns. (45) and (48). Results obtained with the above M relation are in good agreement with numerical predictions for moderate reinforcement volume fractions and elastic contrasts, compare fig. 5. Even though they tend to be highly useful for practical applications, “extended” Mori– Tanaka methods38) are of an ad-hoc nature and can lead to nonsymmetric effective “elastic tensors” in a number of scenarios, especially in multi-phase materials or when anisotropic phases are present (Benveniste et al.,1991; Ferrari,1991) 39). “Extended” Mori–Tanaka methods based on orientational averaging have also been employed in studies of the nonlinear behavior of nonaligned composites. Secant plasticity schemes of the above type were used for describing ductile matrix materials, see e.g. (Bhattacharyya/Weng,1994; Dunn/Ledbetter,1997), and incremental approaches for modeling composites consisting of an elastoplastic matrix reinforced by nonaligned or random short fibers were proposed by Doghri/Tinel (2005) and Lee/Simunovic (2000), respectively, with the latter work also accounting for debonding between reinforcements and matrix. 38)

Alternative modified Mori–Tanaka models for nonaligned composites can be obtained by using spatially averaged Eshelby tensors (Johannesson/Pedersen,1998). Elastic composites containing woven, braided, and knitted reinforcements may be treated in analogy to “extended” Mori–Tanaka models, the the orientation of the fiber tows (“yarns”) being described by appropriate “equivalent orientations” (Gommers et al.,1998). 39)

The reason for this behavior lies in the fact, mentioned in section 2.3, that Mori–Tanaka methods inherently describe aligned ellipsoidal symmetries of the type shown in fig. 3.

– 30 –

A rigorous approach for studying the effects of orientation and aspect ratio distributions on the overall elastic behavior of inhomogeneous materials was proposed by Ponte Casta˜ neda and Willis (1995) on the basis of Hashin–Shtrikman estimates (compare section 2.4). It is based on “ellipsoid-in-ellipsoid” phase arrangements in which different Eshelby tensors are used for the ellipsoids describing the two-point correlations of the phase arrangement and for the ellipsoids describing the shapes of the inhomogeneities. The Hashin–Shtrikman estimates of Ponte Casta˜ neda and Willis are limited, however, to reinforcement volume fractions that follow the requirement that the inner ellipsoids (shapes of the inhomogeneities) must stay subvolumes of the outer ellipsoids (“positions” of the inhomogeneities). The resulting range of unrestricted applicability is rather small for markedly prolate or oblate reinforcements that show a considerable degree of misalignment. For a discussion of other micromechanical methods related to these estimates see (Hu/Weng,2000). For the special case of randomly oriented fibers or platelets of a given aspect ratio, “symmetrized” dilute strain concentration tensors may be constructed, which are known as Wu tensors (Wu,1966). They can be combined with Mori–Tanaka methods40) , see e.g. (Benveniste,1987), or classical self-consistent schemes (Berryman,1980) to describe composites with randomly oriented phases of the matrix–inclusion and interwoven types. Another mean field approach for such materials, the Kuster–Toks¨ oz (1974) model, essentially is a dilute description applicable to matrix–inclusion topologies and tends to give unphysical results at high reinforcement volume fractions. In addition, the Hashin– Shtrikman approach of Ponte Casta˜ neda and Willis (1995) can be adapted to randomly oriented reinforcements. For recent discussions on the relationships between some of the above approaches see e.g. (Berryman/Berge,1996; Hu/Weng,2000). Due to the overall isotropic behavior of composites reinforced by randomly oriented fibers or platelets, their elastic response must comply with the “standard” Hashin–Shtrikman bounds (Hashin/Shtrikman,1963), see also table 4 in section 5.4. Besides mean field and unit cell models (compare section 5.4) a number of other methods have been proposed for studying composites with nonaligned reinforcements. Most of them are based on the assumption that the contribution of a given fiber to the overall stiffness and strength depends solely on its orientation with respect to the applied load and on its length, interactions between neighboring fibers being neglected. The paper physics approach (Cox,1952) and the Fukuda–Kawata theory (Fukuda/Kawata,1974) are based on summing up stiffness contributions of fibers crossing an arbitrary normal section on the basis of fiber orientation and length distribution functions, compare also (Jayaraman/Kortschot,1996). Such theories use shear lag models (Cox,1952; Fukuda/Chou,1982) or improvements thereof (Fukuda/Kawata,1974) to describe the behavior of a single fiber embedded in matrix, the results being typically given as modified rules of mixtures with fiber direction and fiber length corrections. Laminate analogy approaches, see e.g. (Fu/Lauke,1998), approximate nonaligned reinforcement arrangements 40)

When used for describing randomly oriented reinforcements via a Wu tensor, Mori– Tanaka methods are not liable to giving nonsymmetric overall elastic tensors.

– 31 –

by a stack of layers each of which handles one fiber orientation and, where appropriate, one fiber length. Like related models that are based on orientational averaging of the elastic tensors corresponding to different fiber orientations, see e.g. (Huang,2001; Siegmund et al.,2004), and the so-called aggregate models (Toll,1992), laminate analogy approaches use Voigt (strain coupling) approximations. 2.8 Mean Field Methods for Non-Ellipsoidal Reinforcements When non-ellipsoidal, inhomogeneous or coated inclusions are subjected to a homogeneous eigenstrain the resulting stress and strain fields are in general inhomogeneous. As a consequence, the interior field Eshelby tensors and dilute inclusion concentration tensors depend on the position wthin such inhomogeneities. Analytical solutions are available only for certain inclusion shapes, see e.g. (Mura,1987). A straightforward extension of the mean field concept for such cases consists in using Eshelby or dilute inclusion concentration tensors that are averaged over the inhomogeneity (Duschlbauer,2004). Averaged (or “replacement”) dilute inclusion concentration tensors, ¯ (i,r) and B ¯ (i,r) , can be obtained by setting up models with a single dilute inhomogeneity, A dil dil generating solutions for six linearly independent load cases e.g. by the Finite Element method, and using eqn.(75) to evaluate the average strains or stresses, respectively, in the phases. In order to achieve a workable formulation it is necessary to replace, in addition, the elastic tensors of the inhomogeneities, E(i) and C(i) , with “replacement elastic tensors” 1 ¯ (i,r) ]−1 E(i,r) = E(m) + (E∗ − E(m) )[A dil ξdil dil 1 ¯ (i,r) ]−1 , (50) C(i,r) = C(m) + (C∗dil − C(m) )[B dil ξdil where E∗dil and C∗dil are the effective elastic tensors of the dilute single-inhomogeneity configurations, and ξdil is the corresponding reinforcement volume fraction. The replacement elastic tensors ensure consistency within the mean field framework by fulfilling eqns.(18). An alternative ( but related) method of handling non-ellipsoidal inhomogeneities in a mean field framework is provided by the compliance contribution tensor formalism proposed by Kachanov et al. (1994). It uses the difference between the compliance tensors of the matrix and of the dilutely reinforced material, ¯ (i,m) = C∗dil − C(m) , (51) H dil

which is referred to as the compliance contribution tensor, to formulate mean field expressions for the overall responses of inhomogeneous materials containing reinforcements or holes of general shape. The C∗dil , like above, must be evaluated from appropriate dilute single-inhomogeneity configurations. Both the replacement inclusion approach and thee compliance contribution formalism are approximations applicable to non-ellipsoidal inhomogeneities of general shapes, and both inhomogeneities and matrix may have general elastic symmetries.

– 32 –

2.9 Mean Field Methods for Diffusion-Type Problems There are strong conceptual and algorithmic analogies between inclusion problems describing the elastostatic behavior of and diffusive transport processes in inhomogeneous materials, see e.g. (Hashin,1983) and Table 1. The ranks of the involved tensors, however, are lower in the diffusion problems and the underlying differential equations are different.

elasticity

thermal conduction

electrical conduction

species diffusion

σ stress [N m−2 ]

q th heat flux [W m−2 ]

j current density [A m−2 ]

qc species flux [kg m−2 s−1 ]

ε strain []

∆T temperature gradient [K m−1 ]

E electrical field [V m−1 ]

∆C concentration gradient [m−1 ]

u displacements [m]

T temperature [K]

V electrical potential [V]

C concentration []

E “elasticity” [N m−2 ]

Kth thermal conductivity [W m−1 K−1 ]

Kel electrical conductivity [Ω−1 m−1 ]

ρD D: diffusivity [m2 s−1 ]

C=E−1 compliance [N−1 m2 ]

−1 Rth =Kth thermal resistivity [W−1 m K]

−1 Rel =Kel electrical resistivity [Ω m]

—

S elastic Eshelby tensor

S diffusion Eshelby tensor



TABLE 1: Analogies between the mathematical descriptions of elastic and diffusion problems (ρ stands for the mass density). Specifically, in diffusion-type problems (such as heat conduction, electrical conduction, or diffusion of moisture)41) the effects of dilute ellipsoidal inhomogeneous inclusions embed41)

The differential equations describing antiplane shear in elastic solids, d’Arcy creep flow in porous media, and equilibrium properties such as overall dielectric constants and magnetic permeabilities are also of the Poisson type and thus mathematically equivalent to diffusive transport problems.

– 33 –

ded in a matrix can be described in analogy to eqns.(24) to (27) via “diffusion Eshelby tensors”42) . The Eshelby property takes the form of constant fluxes and constant gradients within dilute inclusions, and dilute gradient and flux concentration tensors can be obtained in analogy to eqns.(26) and (27) as (i) A¯dil = [I + S (m) R(m) (K(i) − K(m) )]−1 (i) B¯dil = K(i) [I + S (m) R(m) (K(i) − K(m) )]−1 R(m)

.

(52)

Similar descriptions can also be devised for a number of coupled problems, such as the electromechanical behavior of inhomogeneous materials with at least one piezoelectric constituent, see e.g. (Huang/Ju,1994). For diffusive transport in nondilute materials direct analoga of the Mori–Tanaka and self-consistent approaches introduced in sections 2.3 and 2.7 can be derived, see e.g. (Hatta/Taya,1986; Miloh/Benveniste,1988; Dunn/Taya,1993; Chen,1997). For example, the equivalents of eqns.(29) and (30) take the form (m) (i) A¯MT = [(1 − ξ)I + ξ A¯dil ]−1 (i) (m) B¯MT = [(1 − ξ)I + ξ B¯dil ]−1

(i) (i) (i) A¯MT = A¯dil [(1 − ξ)I + ξ A¯dil]−1 (i) (i) (i) B¯MT = B¯dil [(1 − ξ)I + ξ B¯dil ]−1 .

(53)

The classical self-consistent scheme takes the form Kn+1 = K(m) + ξ[K(i) − K(m) ][In + Sn Rn (K(i) − Kn )]−1 −1 Rn+1 = Kn+1 ,

(54)

compare eqns.(35).

Replacement inhomogeneity models in analogy to section 2.8 can be used to handle non-ellipsoidal and inhomogeneous reinforcements as well as reinforcements with finite interfacial conductances. In the latter case provision must be make for the “replacement inhomogeneity” to account for the interfacial temperature jumps, see e.g. the discussion in (Duschlbauer,2004). Replacement conductivities or resistitivities must be evaluated from consistency conditions equivalent to eqns.(50). Alternatively, resistivity contribution tensors may be used to handle non-ellipsoidal or inhomogeneous inclusions in analogy to the compliance contribution tensor formalism, eqn.(51). Due to the lower order of the tensors involved in diffusion-type problems the latter tend to be more benign and easier to handle. Extensive discussions of diffusion-type problems in inhomogeneous media can be found e.g. in (Markov/Preziosi,2000; Torquato,2002; Milton,2002). 42)

In contrast to the “mechanical Eshelby tensor” S, which (like the elasticity tensor E) is of rank 4, the “diffusion Eshelby tensor” S is of rank 2 (like the conductivity tensor K), and, for spheroidal inclusions, it depends only on their aspect ratios. Explicit expressions for the elements of “diffusion Eshelby tensors” are given e.g. in (Hatta/Taya,1986; Duan et al.,2006).

– 34 –

3. VARIATIONAL BOUNDING METHODS Whereas mean field methods, unit cell approaches and embedding strategies can typically be used for both homogenization and localization tasks, bounding methods are limited to homogenization. Discussions in this chapter again are restricted to materials consisting of two perfectly bonded constituents. Rigorous bounds for the overall elastic properties of inhomogeneous materials are typically obtained from appropriate variational (minimum energy) principles. In what follows only outlines of bounding methods are given; formal treatments can be found e.g. in (Nemat-Nasser/Hori,1993; Bornert,1996; Ponte Casta˜ neda/Suquet,1998; Markov,2000; Bornert et al.,2001; Milton,2002). 3.1 Classical Bounds Hill Bounds Classical expressions for the minimum potential energy and the minimum complementary energy in combination with uniform stress and strain trial functions lead to the simplest variational bounding expressions, the upper bounds of Voigt (1889) and the lower bounds of Reuss (1929). In their tensorial form (Hill,1952) they are known as the Hill bounds and can be written as E∗H−

=

hX

V

(p)

(p)

C

(p)

i−1

≤ E∗ ≤

X

V (p) E(p) = E∗H+

.

(55)

(p)

These bounds, while universal and very simple, do not contain any information on the microgeometry of an inhomogeneous material beyond the phase volume fractions, and are typically too slack to be of much practical use43) , but in contrast to the Hashin– Shtrikman and higher order bounds they also hold for control volumes that are too small to be proper RVEs. Hashin–Shtrikman-Type Bounds Considerably tighter bounds on the overall behavior of inhomogeneous materials can be obtained from a variational formulation due to Hashin and Shtrikman (1962). It is formulated in terms of a homogeneous reference material and of polarization fields τ (x) that describe the difference between the microscopic stress fields in the inhomogeneous material and a homogeneous reference material, compare eqn.(37). In combination with the macrohomogeneity condition (compare section 2.1) polarization fields allow the 43)

The bounds on the Young’s and shear moduli obtained from eqn.(55) are equivalent to Voigt and Reuss expressions in terms of the corresponding phase moduli only if the constituents have appropriate Poisson’s ratios. Due to the homogeneous stress and strain assumptions used for obtaining the Hill bounds, the corresponding phase strain and stress concentration tensors ¯ (p) =I and B ¯ (p) =I, respectively. are A V R

– 35 –

complementary energy of the composite to be formulated in such a way that the “fast” and “slow” contributions are separated, giving rise to the Hashin–Shtrikman variational principle. The latter can be written in the form Z 1 FHS (τ ) = {τ (x)[E0 − E(x)]−1 τ (x) + τ (x) − hτ i∗ ε˜ 0 (τ (x)) + 2τ ε0 } dΩ , (56) V V compare e.g. (Gross/Seelig,2001). The stationary values of FHS take the form FHS = ε0 (E∗ − E0 ) ε0 and are maxima when (E(x) − E0 ) is positive definite and minima if (E(x) − E0 ) is negative definite. In order to allow an analytical solution for the above variational problem, the highly complex position dependent polarizations τ (x) are approximated by phase-wise constant polarizations τ (p) , compare eqn.(38). This has the consequence that the strain fluctuations ε˜ 0 (τ (p) ) can be evaluated by an Eshelby-type eigenstrain procedure. It can be shown that for the case of isotropic constituents and isotropic overall behavior the proper Eshelby tensor to be used is the one for spheres, i.e. for inclusion aspect ratio a = 1. By optimizing the Hashin–Shtrikman variational principle with respect to the τ (p) the tightest possible bounds within the Hashin–Shtrikman scheme are found. In the case of two-phase materials in which the matrix is softer than the reinforcements, i.e. for (K (i) − K (m) ) > 0 and (G(i) − G(m) ) > 0, the above procedure leads to lower bounds for the elasticity tensors of the form h i−1 E∗HS− = E(m) + ξ (E(i) − E(m) )−1 + (1 − ξ)S(m) C(m) ¯ (i) = E(m) + ξ(E(i) − E(m) ) A M

,

(57)

(i)

¯ corresponds to the Mori–Tanaka strain concentration tensor given in eqn.(30). where A M The upper bound Hashin–Shtrikman elasticity tensors are obtained as h i−1 E∗HS+ = E(i) + (1 − ξ) (E(m) − E(i) )−1 + ξS(i) C(i)

(58)

by exchanging the roles of matrix and inhomogeneities. The Hashin–Shtrikman bounds proper, eqn.(57) and (58), were stated by Hashin and Shtrikman (1963) in terms of bounds on the effective bulk modulus K ∗ and the effective shear modulus G44) . They apply to essentially all practically relevant inhomogeneous 44)

Whereas engineers tend to describe elastic material behavior in terms of Young’s moduli and Poisson’s ratios (which can be measured experimentally in a relatively straightforward way), many homogenization expressions are best formulated in terms of bulk and shear moduli, which directly describe the physically relevant hydrostatic and deviatoric responses of materials. For obtaining bounds on the effective Young’s modulus E ∗ from the results on K ∗ and G∗ see (Hashin,1983), and for bounding expressions on the Poisson numbers ν ∗ see (Zimmerman,1992).

– 36 –

materials with isotropic phases that show (statistically) isotropic overall symmetry 45) and fulfill the condition (K (i) − K (m) )(G(i) − G(m) ) > 0. The Hashin–Shtrikman formalism also allows to recover the Voigt and Reuss bounds when the reference material is chosen to be much stiffer or much softer than either of the constituents. Another set of Hashin–Shtrikman bounds is applicable to (statistically) transverse isotropic composites reinforced by aligned continuous fibers46) . For a discussion of evaluating these bounds for composites reinforced by transversely isotropic fibers see (Hashin,1983). Further developments led to the Walpole (1966) and Willis (1977) bounds, which pertain to thermoelastically anisotropic overall behavior due to anisotropic constituents, statistically anisotropic phase arrangements, and/or non-spherical reinforcements, a typical example being the aligned ellipsoidal microgeometry sketched in fig.3. These bounds include the Hashin–Shtrikman bounds discussed above as special cases and they can also handle situations where a “softest” and “stiffest” constituent cannot be identified unequivocally so that a fictive reference material must be used. Hashin–Shtrikman-type bounds are not restricted to materials with matrix–inclusion topologies, but hold for any phase arrangement of the appropriate symmetry and phase volume fractions. In addition, it is worth noting that Hashin–Shtrikman-type bounds are sharp, i.e. they are the tightest bounds that can be given for the type of geometrical information used (volume fraction and overall symmetry, corresponding to two-point correlations). From the practical point of view it is of interest that for two-phase materials Mori–Tanaka estimates are equal to one of the Willis bounds, compare e.g. (Weng,1990), while the other bound can be obtained after a “color inversion” (i.e. after exchanging the roles of inhomogeneities and matrix)47) . Accordingly, the bounds can be evaluated for fairly general aligned phase geometries by simple matrix algebra such as eqn.(32). For elevated phase contrasts the lower and upper Hashin–Shtrikman-type bounds tend to be rather far apart, however. 45)

Avellaneda and Milton (1989) constructed hierarchical laminates that are statistically isotropic and the overall conductivities of which do not fall within the pertinent Hashin– Shtrikman bounds. Accordingly, statistical isotropy by itself is not a sufficient condition for guaranteeing the validity of expressions such as eqns.(57) and (58). For a discussion of Hashin–Shtrikman-type bounds of macroscopically isotropic composites with anisotropic constituents see e.g. Nadeau/Ferrari (2001). 46)

These bounds apply to general (transversely isotropic) two-phase materials, the (“inplane”) phase geometries of which are translation invariant in the “out-of-plane” or “axial” direction. 47)

Put more precisely, the lower bounds for two-phase materials are obtained from Mori– Tanaka methods when the more compliant material is used as matrix phase, and the upper bounds when it is used as inclusion phase. For matrix–inclusion materials with more than two constituents, Mori–Tanaka expressions are bounds if the matrix is the stiffest or the softest phase, and fall between the appropriate bounds otherwise (Weng,1990). Note, however, that extended Mori–Tanaka methods for nonaligned composites as discussed in section 2.7 in general do not coincide with bounds.

– 37 –

Composites reinforced by randomly oriented fibers or platelets follow the Hashin– Shtrikman bounds for statistically isotropic materials, eqns.(57) and (58). A discussion of Hashin–Shtrikman-type bounds on the elastic responses of composites with more general fiber orientation distributions can be found in (Eduljee/McCullough,1993). In the case of porous materials (where the pores may be viewed as an ideally compliant phase) only upper Hashin–Shtrikman bounds can be obtained, the lower bounds for the elastic moduli being zero. The opposite situation holds in case one of the constituents is rigid. Hashin–Shtrikman-type variational formulations can also be employed to generate bounds for more general phase arrangements. Evaluating the polarizations for “composite regions” consisting of inhomogeneities embedded in matrix gives rise to Hervé–Stolz– Zaoui bounds (Hervé et al.,1991). When complex phase patterns are to be considered (Bornert,1996; Bornert et al.,1996) numerical methods must be employed for evaluating the polarization fields. Bounds for Discrete Microstructures Hashin–Shtrikman-type bounds can also be derived for (simple) periodic microgeometries, see e.g. (Nemat-Nasser/Hori,1993). Among other bounding approaches for such microstructures are those of Bisegna and Luciano (1996), which uses approximate variational principles evaluated from periodic unit cells via Finite Element models, and of Teply and Dvorak (1988), which evaluates bounds for the elastoplastic behavior of fiber reinforced composites with periodic hexagonal phase arrangements. In addition, windowing methods as discussed in chapter 7 may be used to generate sequences of bounds on the overall behavior of given phase arrangements. 3.2 Improved Bounds When more complex trial functions are used in variational bounding, their optimization requires statistical information on the phase arrangement in the form of n-point correlation functions48) . This way improved bounds can be generated that are significantly tighter than Hashin–Shtrikman-type expressions. Three-point bounds for statistically isotropic two-phase materials can be formulated in such a way that the information on the phase arrangement statistics is contained in two three-point microstructural parameters, η(ξ) and ζ(ξ). The evaluation of these parameters for given arrangement statistics is a considerable task. However, analytical expressions or tabulated data in terms of the reinforcement volume fraction ξ are available for a number of generic microgeometries of practical importance, among them statistically homogeneous isotropic materials containing identical, bidisperse and polydisperse impenetrable (“hard”) spheres (that describe matrix–inclusion composites) as well as monodisperse interpenetrating spheres (“boolean models” that can describe many interwoven phase arrangements), and statistically homogeneous transversely isotropic materials reinforced by impenetrable or interpenetrating aligned cylinders. 48)

For discussions on statistical descriptions of phase arrangements see e.g. (Torquato,1991; Ghosh et al.,1997; Pyrz/Bochenek,1998; Torquato,1998; Torquato,2002).

– 38 –

References to some three-point bounds and to a number of expressions for η and ζ applicable to some two-phase composites can be found in section 3.4, where results from mean field and bounding approaches are compared. For reviews of higher order bounds for elastic (as well as other) properties of inhomogeneous materials see (Torquato,1991; Torquato,2002). Improved bounds typically provide highly useful information for low and moderate phase contrasts (as typically found in technical composites), but even they tend to be rather slack for elevated phase contrasts. 3.3 Bounds for the Nonlinear Behavior Analoga to the Hill bounds for nonlinear inhomogeneous materials were introduced by Bishop and Hill (1951). For polycrystals the nonlinear equivalents to Voigt and Reuss expressions are usually referred to as Taylor and Sachs bounds, respectively. In analogy to mean field estimates for elastoplastic material behavior, nonlinear bounds are typically obtained by evaluating a sequence of linear bounds. Talbot and Willis (1985) extended the Hashin–Shtrikman variational principles to obtain one-sided bounds (i.e. upper or lower bounds, depending on the material combination) on the nonlinear mechanical behavior of inhomogeneous materials. An important development was the derivation of a variational principle by Ponte Casta˜ neda (1992) that allows upper bounds on the effective nonlinear behavior of inhomogeneous materials to be generated on the basis of upper bounds for the elastic response49) . It uses a series of inhomogeneous reference materials, the properties of which have to be obtained by optimization procedures for each strain level. Essentially, the variational principle guarantees the best choice for the comparison material at a given load. The Ponte Casta˜ neda bounds are closely related to mean field approaches using improved secant plasticity methods (compare the remarks in section 2.4). For higher order bounds on the nonlinear response of inhomogeneous materials see also (Talbot/Willis,1998). The study of bounds — like the development of improved estimates — for the overall nonlinear mechanical behavior of inhomogeneous materials has been an active field of research during the last decade, see the recent reviews by Suquet(1997), Ponte Casta˜ neda and Suquet (1998), and Willis (2000). 3.4 Some Comparisons of Mean Field and Bounding Predictions In order to give some idea of the type of predictions that can be obtained by different mean field (and related) approaches and by bounding methods for the thermomechanical responses of inhomogeneous thermoelastic materials, results for the overall elastic moduli 49)

The Ponte Casta˜ neda bounds are rigorous for nonlinear elastic inhomogeneous materials and, on the basis of deformation theory, are very good approximations for materials with at least one elastoplastic constituent. Applying the Ponte Casta˜ neda variational procedure to elastic lower bounds does not necessarily lead to a lower bound for the inelastic behavior.

– 39 –

and coefficients of thermal expansion are presented as functions of the reinforcement volume fraction ξ in this section. All comparisons are based on E-glass particles or fibers embedded in an epoxy matrix. The elastic contrast of these constituents is c ≈ 21 and the thermal expansion contrast takes a value of approximately 0.14, see table 2.

matrix reinforcements

E [GPa]

ν []

3.5 74.0

0.35 0.2

α [1/K ×10

−6]

36.0 4.9

TABLE 2: Constituent material parameters of epoxy matrix and E-glass reinforcements used in generating figs. 6 to 12. Figures 6 and 7 show predictions for the overall Young’s and shear moduli of a particle reinforced glass/epoxy composite based on the above constituent material parameters. The Hill bounds can be seen to be very slack. The Mori–Tanaka results (MTM) — as mentioned before — coincide with the lower Hashin–Shtrikman bounds (H/S LB), whereas the classical self-consistent scheme (CSCS) shows a typical behavior in that it is close to one Hashin–Shtrikman bound at low volume fractions, approaches the other at high volume fractions, and displays a transition behavior in the form of a sigmoid curve inbetween. The three-point bounds (3PLB and 3PUB) given in figs. 6 and 7 apply to monodisperse impenetrable spherical inhomogeneities, are based on formalisms developed by Beran/Molyneux (1966), Milton (1981) and Phan-Thien/Milton (1983), and use expressions for the statistical parameters η and ζ reported in (Miller/Torquato,1990) and (Torquato et al.,1987), respectively. Like Torquato’s second order estimates (3PE) these expressions are available for reinforcement volume fractions up to ξ=0.6 50) . Interestingly, for the case considered here, the results from the generalized self-consistent scheme (GSCS), which as usual predicts a slightly stiffer behavior than the Mori–Tanaka method (taking the “standard” case of stiff inhomogeneities in a compliant matrix), are very close to the lower three-point bounds even though microgeometries corresponding to the GSCS do not show monodisperse inhomogeneities. Predictions for the effective coefficients of thermal expansion of statistically isotropic glass-epoxy composites are given in fig. 8, the bounds being obtained from a relation proposed by Levin (1967) and Schapery (1968) that can be evaluated with any bounds or estimates for the effective bulk modulus. Here the Hashin–Shtrikman bounds and the three-point bounds are used as basis for bounding the overall thermal expansion behavior51) . The CTEs corresponding to the self-consistent schemes were obtained by 50)

This value approaches the maximum volume fraction achievable in random packings of spheres, which is ξ = 0.64 (Weisstein,2000). 51)

Upper bounds for the bulk modulus give rise to lower bounds for the CTEs and vice

versa.

– 40 –

40.0 0.0

EFFECTIVE YOUNGs MODULUS [GPa]

CSCS 3PE (mono/h) GSCS 3PUB (mono/h) 3PLB (mono/h) H/S UB H/S LB; MTM Hill/Voigt UB Hill/Reuss LB

0.0

0.2

0.4

0.6

0.8

1.0

PARTICLE VOLUME FRACTION []

20.0


0.0

EFFECTIVE SHEAR MODULUS [GPa]

FIG.6: Bounds and estimates for the effective Young’s moduli of glass/epoxy particle reinforced composites as functions of the particle volume fraction.

0.0

0.2

0.4

0.6

0.8

1.0

PARTICLE VOLUME FRACTION [] FIG.7: Bounds and estimates for the effective shear moduli of glass/epoxy particle reinforced composites as functions of the particle volume fraction.

– 41 –

40.0 20.0 0.0

EFFECTIVE CTE [1/K x 10 −6 ]

CSCS 3PE (mono/h) GSCS 3PLB (mono/h) 3PUB (mono/h) H/S LB H/S UB; MTM

0.0

0.2

0.4

0.6

0.8

1.0

PARTICLE VOLUME FRACTION [] FIG.8: Bounds and estimates for the effective CTEs of glass/epoxy particle reinforced composites as functions of the particle volume fraction.

using the mean field formalism discussed in section 2.1. Because they are based on the same estimates for the effective bulk modulus, the results given for the GSCS and the Mori–Tanaka-scheme agree with the upper Levin/Hashin–Shtrikman bounds. In figs. 9 to 12 results are presented for the overall transverse Young’s moduli, overall axial and transverse shear moduli, as well as overall axial and transverse coefficients of thermal expansion of glass/epoxy composites reinforced by aligned continuous fibers 52) . The three-point bounds shown correspond to the case of aligned identical impenetrable circular cylinders, follow (Silnutzer,1972; Milton,1981), and use statistical parameters evaluated by Torquato and Lado (1992). A qualitatively similar behavior to the particle reinforced case is evident. It is noteworthy, however, that the overall transverse CTEs in fig. 12 (marked as “tr”) at low fiber volume fractions are predicted by all models to exceed the CTEs of both constituents. Such behavior is typical for continuously reinforced composites and is caused by the axial constraint enforced by the stiff fibers. Nearly the same axial CTEs (“ax”) are obtained for all models considered. No bounds for the CTEs are given in fig. 12 because there is no equivalent to the Levin formula for overall transversely isotropic materials. 52)

In the case of the axial Young’s moduli the different estimates and bounds are essentially indistinguishable from each other (and from the rules of mixture result) for the material parameters and scaling used in fig. 9 and are, accordingly, not shown. It is, however, worth noting that the effective axial Young’s modulus of a composite reinforced by aligned continuous fibers is always equal to or larger than the rules of mixture result.

– 42 –

40.0 0.0

EFFECTIVE YOUNGs MODULUS [GPa]


0.0

0.2

0.4

0.6

0.8

1.0

FIBER VOLUME FRACTION []

30.0 10.0

20.0

CSCS 3PE (mono/h) GSCS 3PUB (mono/h) 3PLB (mono/h) H/S UB; MTM H/S LB; MTM Hill/Voigt UB Hill/Reuss LB

0.0


FIG.9: Bounds and estimates for the effective transverse Young’s moduli of glass/epoxy composites with continuous aligned fiber reinforcement as functions of the fiber volume fraction.

0.0

0.2

0.4

0.6

0.8

1.0

FIBER VOLUME FRACTION [] FIG.10: Bounds and estimates for the effective axial shear moduli of glass/epoxy composites with continuous aligned fiber reinforcement as functions of the fiber volume fraction.

– 43 –

30.0 20.0 10.0 0.0


CSCS 3PUB (mono/h) GSCS 3PUB (mono/h) 3PLB (mono/h) H/S UB H/S LB; MTM Hill/Voigt UB Hill/Reuss LB

0.0

0.2

0.4

0.6

0.8

1.0

FIBER VOLUME FRACTION []

40.0 20.0

CSCS tr. CSCS ax. 3PE (mono/h) tr. 3PE (mono/h) ax. GSCS tr. GSCS ax. MTM tr. MTM ax.

0.0

EFFECTIVE CTE [1/K x 10

−6

]

FIG.11: Bounds and estimates for the effective transverse shear moduli of glass/epoxy composites with continuous aligned fiber reinforcement as functions of the fiber volume fraction.

0.0

0.2

0.4

0.6

0.8

1.0

FIBER VOLUME FRACTION [ ] FIG.12: Estimates for the effective axial and transverse CTEs of glass/epoxy composites with continuous aligned fiber reinforcement as functions of the fiber volume fraction.

– 44 –

In figs. 6 to 12, the classical self-consistent scheme is not in agreement with the threepoint bounds shown, which explicitly correspond to matrix–inclusion topologies. Considerably better agreement with the CSCS is found by using three-point parameters of the interpenetrating sphere or cylinder type (which can also describe cases where both phases percolate, but are not as symmetrical with respect to the constituents as the CSCS). From a practical point of view it is worth noting that despite their sophistication improved bounds (and Torquato’s second order estimates) may give overly optimistic predictions for the overall moduli because they describe ideal two-phase materials, whereas in actual materials it is practically impossible to avoid flaws such as porosity. Before closing this chapter it should be mentioned that more complex responses may be obtained when one or both of the constituents are transversely isotropic with a high stiffness contrast between the axial and transverse directions (note that e.g. carbon fibers typically exhibit transverse Young’s moduli that are much smaller than the Young’s moduli of, say, metallic matrices, whereas the opposite holds true for the axial Young’s moduli). Such effects may be especially marked in the nonlinear range. Also, the angular dependences of stiffnesses and CTEs in fiber reinforced materials can be quite rich, compare e.g. (Pettermann et al.,1997).

– 45 –

4. GENERAL REMARKS ON APPROACHES BASED ON DISCRETE MICROGEOMETRIES In the present context, micromechanical approaches based on discrete microgeometries encompass unit cell, embedding and windowing methods. Broadly speaking, these approaches trade off restrictions to the generality of the microstructures that can be studied for the capabilities of using fine grained geometrical models and of resolving details of the stress and strain fields at the length scale of the inhomogeneities. The main fields of application of these methods are studying the nonlinear behavior of inhomogeneous materials and evaluating the microscopic stress and strain fields of “model materials” at high resolution. This information, on the one hand, may be required when the local stress and strain fields fluctuate strongly and are, accordingly, difficult to describe by averaged fields. On the other hand, it is important for understanding the damage and failure behavior of inhomogeneous materials, which can depend on details of their microgeometry. 4.1 Microgeometries The generation of “simply periodic” microgeometries, such as periodic hexagonal arrangements of fibers or cubic arrangements of spherical particles, compare chapter 5, is rather straightforward. In order to obtain more realistic, and thus more complex, microgeometries for modeling inhomogeneous materials, one of two — often complementary — modeling philosophies is typically followed, viz., using computer generated or “real structure” phase arrangements. Computer-Generated Microgeometries Computer generated microgeometries may be classified into two groups, one of which is based on generic random arrangements of a small to a fair number of reinforcements, whereas the other aims at approximating some given phase distribution statistics. The generation of generic random microgeometrical models typically involves random addition methods53) and random perturbation or relaxation techniques. The reinforcement volume fractions that can be reached by RSA methods tend to be moderate due to jamming (geometrical frustration) and their representativeness for phase arrangements generated by mixing processes is open to question (Stroeven et al.,2004). Major improvements on both counts can be obtained by using RSA geometries as starting configurations for random perturbation or hard core shaking models (HCSM, also referred to as dynamic simulations), that apply small random displacements to each inhomogeneity to enforce minimum spacings between individual reinforcements as the size of the volume element 53)

In random addition (RSA) algorithms (also known as random sequential insertion or, in two-dimensional cases, as random sequential adhesion models, and sometimes referred to as static simulations), positions for new reinforcements are created by random processes, a candidate inhomogeneity being accepted if it does not “collide” with (approach too closely or overlap with) any of the existing ones and rejected otherwise.

– 46 –

is reduced54) . Methods of the above types can be used to generate statistically homogeneous distributions of reinforcements, see e.g. (Gusev,1997; B¨ ohm/Han,2001), as well as idealized clustered microgeometries (Segurado et al.,2003). Another modeling strategy, based on simulated annealing and related procedures, aims at generating “statistically reconstructed” microstructures55) . This way accurate representations of the microgeometry of a given composite can be provided on the base of experimentally determined statistical descriptors. Computer generated microgeometries have tended to employ idealized reinforcement shapes, equiaxed particles embedded in a matrix, for example, being typically represented by spheres, and fibers by cylinders or prolate spheroids56) of appropriate aspect ratio57) . Microgeometries of composites reinforced by continuous fibers can be well described this way, but for reinforcements with aspect ratio close to unity such simplified phase arrangements may constitute a less satisfactory approximation, giving rise, for example, to considerable weaker macroscopic hardening in models of ductile matrix composites compared to polyhedral reinforcements of equivalent orientation and aspect ratio, compare (Chawla/Chawla,2006). By appropriately adjusting the algorithms, computer generated microgeometries may be periodic or non-periodic, as required by the analysis method to be employed. Enforcing periodicity (for use with periodic microfield methods) introduces long-range order into phase arrangements and may make them less realistic. Real Structure Microgeometries Instead of generating them by computer algorithms, microgeometries may be chosen to follow as closely as possible the phase arrangement in part of a given sample of the material to be modeled, obtained from metallographic sections and serial sections, tomographic data, etc., see e.g. (Fischmeister/Karlsson,1977; Terada/Kikuchi,1996; Li et 54)

Alternatively, explicit models of production processes may be used for achieving realistic microstructures with elevated reinforcement volume fractions. 55)

Statistically reconstructed microstructures are equivalent to real ones in a statistical sense, i.e. they show equal or very similar phase distribution statistics. Algorithms for reconstructing matrix–inclusion and more general microgeometries that approximate predefined statistical descriptors are discussed e.g. in (Rintoul/Torquato,1997), (Torquato,1998), ˇ (Roberts/Garboczi,1999), (Zeman/Sejnoha,2001) and (Bochenek/Pyrz,2002). For in-depth discussions of many of the underlying issues, such as statistical descriptions of inhomogeneous materials, see e.g. (Torquato,2002; Zeman,2003). 56)

For uniform boundary conditions it can be shown that the overall elastic behavior of matrix–inclusion type composites can be bounded by approximating the actual shape of particles by inner and outer envelopes of “smooth” shape (e.g., inscribed and circumscribed ellipsoids). This is known as the Hill modification theorem (or Hill’s comparison theorem, Hill’s auxiliary theorem) see (Hill,1963; Huet et al.,1991). Approximations by ellipsoids typically work considerably better for convex than for non-convex particle shapes (Kachanov/Sevostionov,2005). 57)

As an alternative to approximating the aspect ratios of real reinforcements, equal surface vs. volume ratios S 3 /V 2 may be used in models, see e.g. (Kenesei et al.,2005).

– 47 –

al.,1999; Babout et al.,2004; Chawla/Chawla,2006). The resulting descriptions are often referred to as “real structure” models. Recently the generation of real structure models from pixel (digital images) or voxel (tomographic) data on the geometries of inhomogeneous materials has been the focus of considerable research efforts. It typically involves the steps of selecting an appropriate volume element for analysis (“registration”) and of identifying the areas or volumes, respectively, occupied by the material’s constituents by thresholding of the grey values of the pixels or voxels (“segmentation”). At this stage pixel or voxel models (compare section 4.2) can be generated directly or contouring procedures may be used for obtaining “smooth” phase domains, the latter operation typically being manpower intensive and difficult to automatize. Sufficiently large real structure models can provide excellent discrete microstructures for micromechanical modeling, but care58) is required to avoid bias in the choice of the regions (“windows”) extracted for modeling. Typically some simplification of the experimental data is necessary within the framework of discretizing engineering methods, e.g., with respect to the shapes of the reinforcements and by not explicitly modeling reinforcements the sizes of which fall below some threshold. Real structure microgeometries typically are non-periodic, so they cannot be used directly in periodic microfield models. Sizes of Volume Elements For both computer generated phase arrangements and for real structure models the question immediately arises what geometrical complexity (and thus what size of volume element) is required for adequately representing the physical behavior of the inhomogeneous material to be studied59) . There are two main approaches to assessing adequate sizes of volume elements. On the one hand, the arrangement statistics of the microgeometry can be used, in correspondence with the idea of “geometrical RVEs”, compare section 1.2. In this context adequate sizes of model geometries may be estimated on the basis of experimentally obtained correlation lengths (Bulsara et al.,1999) or covariances (Jeulin,2001) of the phase arrangement or by posing the requirement of having at least two statistically independent inhomogeneities ˇ in the cell (Zeman/Sejnoha,2001). On the other hand, statistical descriptors (e.g., the integral range of a second order correlation function, compare (Jeulin,2001)), hierarchies of bounds (resulting, e.g., from windowing analysis, compare section 7), the variations in the predictions for some physical property (e.g., an effective modulus or the extremal value of some microfield) or differences between predictions for some macroscopic behavior (e.g., a homogenized stress vs. strain curve) can be employed for checking the adequacy of volume elements following the concept of “physical RVEs”. In addition, 58)

Statistical assessments of the phase geometries of given volume elements are suitable for establishing their suitability within the concept of geometrical RVEs, whereas at present there appears to be no low-cost procedure for checking the compliance with the concept of physical RVEs. 59)

This requirement is sometimes phrased as finding a “minimum RVE” (Ren/Zheng,2004).

– 48 –

deviations from the required material symmetry of the overall response can be used for checking the adequacy of the size of a given volume element. A comparison of different geometry and microfield based criteria for choosing the size of model geometries was given by Trias et al. (2007). Volume elements fulfilling one or more of the above criteria with a given degree of accuracy are in general not RVEs in the strict sense as discussed in section 1.2 and are sometimes referred to as Statistical Representative Volume Elements (SRVEs). With growing size of SRVEs the predicted micro and macro fields tend to become less sensitive to the boundary conditions actually applied, compare (Shen/Brinson,2006). Assessments based on geometrical parameters or on the overall elastic behavior typically give rise to relatively small volume elements. For example, Zeman (2003) reported that the transverse elastic behavior of continuously reinforced composites can be satisfactorily described by unit cells containing reconstructed arrangements of 10 to 20 fibers. The concept of “physical volume elements” implies that the size of model geometries depends on the specific physical property to be studied. For the case of elastic statistically isotropic composites with matrix–inclusion topology and sphere-like particles, Drugan and Willis (1996) estimated that for approximating the overall moduli with errors of less than 5% or less than 1%, respectively, (nonperiodic) volume elements with sizes of approximately two or five particle diameters are sufficient for any volume fraction; compare also (Drugan,2000). When inelastic constituent behavior is present, however, a number of numerical studies (Zohdi,1999; Jiang et al.,2001; B¨ ohm/Han,2001; Gitman,2007) have indicated that larger volume elements may be required for satisfactorily approximating the overall symmetries and for obtaining good agreement between the responses of (nominally) statistically equivalent phase arrangements, especially at elevated strains. The reason for this behavior lies in the marked inhomogeneity of the strain fields typically present in such regimes, where contiguous zones of high strains tend to appear. These often evolve to become considerably larger than individual reinforcements and thus effectively introduce a new length scale into the problem. Analogous behavior was reported for models of composites that explicitly account for microscopic damage, see e.g. (Swaminathan/Ghosh 2006). Accordingly, the size of satisfactory multi-particle unit cells tends to depend on the phase material behavior60) . In general, each model microgeometry may be viewed as being a member of a family of phase arrangements, all of which are (approximate) realizations of some statistical process. In the typical case, where individual model geometries obtained by a computer model or from real structures are smaller than proper RVEs, ensemble averaging over a number of results obtained from such volume elements can be used to estimate the 60)

For elastoplastic matrices, weaker strain hardening tends to translate into geometrically more complex volume elements, and very large model geometries are necessary when one of the phases shows softening, e.g. due to damage (Zohdi/Wriggers,2001). For the latter case, in fact, eliminating the dependence of the homogenized response on the size of the SRVE may be impossible (Gitman et al.,2007). In the case of elastic polycrystals, the anisotropy and shape of the grains were also reported to influence the required size of volume elements (Ren/Zheng,2004).

– 49 –

effective material properties, compare e.g. (Kanit et al.,2003; Stroeven et al., 2004). The number of volume different elements required for a given accuracy of the ensemble averages decreases as the sizes of the models increase (Khisaeva/Ostoja-Starzewski,2006). 4.2 Numerical Engineering Methods The majority of published micromechanical studies of discrete microstructures have employed standard numerical engineering methods for resolving the microfields. Studies using Finite Difference (Adams/Doner,1967) and Finite Volume (Bansal/Pindera,2006) algorithms, spring lattice models (see e.g. Ostoja-Starzewski,2002), the Boundary Element Method (BEM; see e.g. Achenbach/Zhu,1989; Liu et al.,2005), the Finite Element Method (FEM) and its developments such as Extended Finite Element Methods (XFEM; see e.g. Sukumar et al.,2001), meshfree and particle methods (see e.g. Missoum-Benziane et al., 2007), techniques using Fast Fourier Transforms (FFT; see e.g. Moulinec/Suquet,1994) and Discrete Fourier Transforms (DFT; M¨ uller,1996)61) , as well as FE-based discrete dislocation models (Cleveringa et al.,1997) have been reported. A number of more specialized approaches are discussed in connection with periodic microfield analysis, see section 5.1. Generally speaking, spring lattice models tend to have advantages in handling pure traction boundary conditions and in modeling the progress of microcracks due to local (brittle) failure. Boundary elements tend to be at their best in studying geometrically complex linear elastic problems. For all the above methods the characteristic length of the discretization (“mesh size”) must, of course, be considerably smaller than the microscale of a given problem in order to obtain spatially well resolved results. At present, the FEM is the most commonly used numerical scheme for evaluating discrete microgeometries, especially in the nonlinear range, where its flexibility, efficiency and capability of supporting a wide range of constitutive models for the constituents and for the interfaces between them are especially appreciated62) . An additional asset of the FEM in the context of continuum micromechanics is its ability to handle discontinuities in the stress and strain components (which typically occur at interfaces between different constituents) in a natural way via appropriately placed element boundaries. Applications of the FEM to micromechanical problems tend to fall into four main groups, compare fig. 13. In most published works the phase arrangements are discretized by an often high number of “standard” continuum elements, the mesh being designed in such a way that element boundaries (and, where required, special interface elements) are positioned at all interfaces between constituents. Such an approach has the advantage that in principle any microgeometry can be handled and that readily available commercial 61)

FFT and DFT based methods are limited to periodic configurations, for which they tend to be highly efficient, compare e.g. (Michel et al.,1999). Due to their use of a fixed regular grid they are well suited for real structure models and for studying the evolution of microstructures, see e.g. (Dreyer et al.,1999). 62)

Constitutive models for constituents used in FEM based micromechanics have included a wide range of elastoplastic, viscoelastoplastic and continuum damage mechanics type descriptions as well as crystal plasticity models (McHugh et al.,1993; Bruzzi et al.,2001).

– 50 –

a)

b)

c)

d)

FIG.13: Sketch of FEM approaches used in micromechanics: a) discretization by standard elements, b) special hybrid elements, c) pixel/voxel discretization, d) “multiphase elements” FE packages may be used. On the one hand, the actual modeling of complex phase configurations in many cases requires sophisticated and/or specialized preprocessors for generating the mesh, a task that has been difficult to automatize for microgeometries. The resulting stiffness matrices may show unfavorable conditioning due to suboptimal element shapes and the requirement of satisfactory resolution of the microfields at local “hot spots” (e.g. between closely neighboring reinforcements) can lead to very large models indeed63) . On the other hand, the possibility of providing for a refined mesh where it is required is one of the strength of this discretization strategy. Alternatively, a smaller number of special hybrid elements may be used, which are specifically formulated to model the deformation, stress, and strain fields in an inhomogeneous region consisting of a single inhomogeneity together with the surrounding matrix on the basis of some appropriate analytical theory, see e.g. (Accorsi,1988). The most highly developed approach of this type at present is the Voronoi Finite Element Method (Ghosh et al.,1996), in which the mesh for the hybrid elements is obtained by Voronoi tesselations based on the positions of the reinforcements. Large planar multi-inhomogeneity arrangements can be analyzed this way using a limited number of (albeit rather complex) elements, and good accuracy as well as significant gains in efficiency have been claimed. Numerous damage models have been implemented into the method and an extension to three dimensions was reported recently (Ghosh/Moorthy,2004). Computational strate63)

Note that even when quadratic shape functions are used, discretizing “thin” regions of a phase (such as the matrix between two reinforcements) with two “layers” of elements restricts the variation of stresses and strains there to piecewise linear functions consisting of linear sections. This is, at best, the barest minimum for capturing the local fields, which tend to show marked gradients exactly in such regions, especially in the elastoplastic regime. The macroscopic and phase averaged fields tend to be much more forgiving with respect to the fineness of the discretization.

– 51 –

gies of this type are, of course, specifically tailored for inhomogeneous materials with matrix–inclusion topologies. A computational strategy that occupies an intermediate position between between the above two approaches uses static condensation to remove the degrees of the interior nodes of the inhomogeneities (or, in the case of coated reinforcements, of inhomogeneities and interface) from the stiffness matrix of a heterogeneous volume element (Liu et al.,2005). This method is, however, limited to linear analysis. Especially when the phase arrangements to be studied are based on experimentally obtained digital representations of actual microgeometries, a third approach for discretizing microgeometries of interest. It consists of using a regular mesh that consists of rectangular or hexahedral elements of fixed size mesh and has the same resolution as the digital data, each element being assigned to one of the constituents by operations such as thresholding of the grey values of the corresponding pixel or voxel, respectively. Such models have the advantage of being suitable for (semi-)automatic generation from appropriate experimental data (metallographic sections, tomographic scans) and of avoiding ambiguities in smoothing the digital data (which may be an issue if “standard” FE meshes are employed to discretize data of this type). Obviously “voxel element” strategies lead to ragged phase boundaries, which may give rise to some oscillatory behavior of the solutions (Niebur et al.,1999), can cause very high local stress maxima (Terada et al.,1997), may degrade accuracy, and restrict modeling of interfacial behavior. Such “digital image based” (DIB) models are, however, claimed not to cause unacceptably large errors even for relatively coarse discretizations, at least in the linear elastic range (Guldberg et al.,1998). Good spatial resolution, i.e. a high number of pixels or voxels, of course, is very beneficial in such models. A fourth approach also uses regular FE meshes, but assigns phase properties at the integration point level of standard elements (“multiphase elements”, see e.g. Schmauder et al.,1996; Quilici/Cailletaud,1999). Essentially, this amounts to trading off ragged boundaries at element edges for smeared-out (and typically degraded) microfields within those elements that contain a phase boundary, because stress or strain discontinuities within elements cannot be adequately handled by standard FE shape functions. With respect to the element stiffnesses the latter concern can be much reduced by overintegrating elements containing phase boundaries, which leads to good approximations of integrals involving non-smooth displacements by numerical quadrature, see (Zohdi/Wriggers,2001). The resulting stress and strain distributions, however, remain smeared-out approximations in elements that contain phase boundaries, the rate of convergence of the microfields being accordingly slow. The above drawbacks are avoided by algorithmic developments such as Extended Finite Element methods, which use a background mesh that does not have to conform to phase boundaries, which are accounted for, e.g., by appropriately enriching the shape functions (Moës et al.,2003). A fairly recent development for studying microgeometries involving a large number of inhomogeneities by Finite Element methods involves programs specially geared towards solving micromechanical problems. Such codes may be based on matrix-free iterative

– 52 –

solvers such as Conjugate Gradient (CG) methods, analytical solutions for the microfields (e.g. constant strain approximations corresponding to the Hill upper bounds, eqn.(55)) being used as starting solutions to speed convergence64) . For studies involving such codes see e.g. (Gusev et al.,2000; Zohdi/Wriggers,2001). Fast solvers of this type open the possibility of solving inverse problems, e.g. for finding optimal particle shapes for given load cases and damage modes, compare (Zohdi,2003). Special Finite Element formulations are typically required for asymptotic homogenization (compare section 5.2) and for other schemes that concurrently solve the macroscopic and microscopic problems, see e.g. (Urba´ nski,1999).

64)

Essentially, in such a scheme the initial guess gives a good estimate of “long wavelength” contributions to the solution, and the CG iterations take care of short “wavelength” variations.

– 53 –

5. PERIODIC MICROFIELD APPROACHES The following discussion concentrates on Periodic Microfield Approaches (PMAs) as used in continuum micromechanics of materials, where they are used to describe the macroscopic and microscopic behavior of inhomogeneous materials by studying model materials that have periodic microstructures. Closely related methods are used in other fields of engineering for handling problems that involve two or more appropriate length scales, e.g. for describing beam, plate and shell structures made up of repeating periodic units65) . The first two sections of this chapter cover some basic concepts of unit cell based PMAs. Following this, applications to continuously reinforced composites, discontinuously reinforced composites, and some other inhomogeneous materials are discussed. In the following discussion the main emphasis is on the small strain elastic and elastoplastic behavior; for homogenization methods geared specifically towards finite strain problems see e.g. (Miehe et al.,1999) or (Terada et al.,2003). 5.1 Basic Concepts of Unit Cell Models Periodic microfield approaches analyze the behavior of infinite (one, two- or threedimensional) periodic phase arrangements66) under the action of far field mechanical loads or uniform temperature fields67) . The most common approach for studying the stress and strain fields in such periodic configurations is based on describing the microgeometry by a periodically repeating unit cell to which the investigations may be limited without loss of information or generality, at least for static analysis68) . The literature on 65)

For unified treatments of continuum, shell-like and beam-like periodic structures see e.g. (Urba´ nski,1999; Pahr/Rammerstorfer,2006). 66)

In the present work the designation “unit cell” is used for any phase arrangement that generates a periodic microgeometry. Accordingly, a unit cell may comprise a part of a simple periodic base unit, a simple periodic base unit, a collective of simple periodic base units, or a phase arrangement of arbitrary geometrical complexity that shows translational periodicity. 67)

Standard PMAs cannot handle macroscopic gradients in mechanical loads, temperature or composition in any direction in which periodicity of the fields is prescribed. Gradients and free boundaries can be handled, however, in directions where periodicity is not prescribed, a typical case being layer-type models that are nonperiodic in one direction and periodic in the other(s), see e.g. (Wittig/Allen,1994; Reiter et al.,1997; Weissenbek et al.,1997). 68)

Periodic phase arrangements typically are not well suited to dynamic analysis, because they act as filters that exclude all waves with frequencies outside certain bands, see e.g. (Suzuki/Yu,1998). In addition, due to the boundary conditions required for obtaining periodicity, unit cell methods can only handle wavelengths that are smaller than or equal to the relevant cell dimension, and only standing waves can be dealt with. In analogy, in stability analysis unit cells can directly resolve only buckling modes of specific wavelengths, so that considerable care is required in using them for such purposes, compare (Vonach,2001; Pahr/Rammerstorfer,2006). However, bifurcation modes with wavelengths exceeding the size of the unit cell can be resolved in periodic linear models by using the Bloch theorem from solid state physics (Gong et al.,2005).

– 54 –

such PMAs is fairly extensive, and well developed mathematical theories are available on scale transitions in periodic structures, compare (Michel et al.,2001). A wide variety of unit cells have been employed in published PMA studies, ranging from geometries that describe simple periodic arrays of inhomogeneities to highly complex phase arrangements, such as multi-inhomogeneity cells (“supercells”). For some simple periodic phase arrangements it is also possible to find analytical solutions via series expansions that make explicit use of the periodicity (see e.g. Sangani/Lu,1987; Cohen,2004) or via appropriate potential methods (Wang et al.,2000). Even though most PMA studies in the literature have used standard numerical engineering methods as described in chapter 4, some more specialized approaches for evaluating microscopic stress and strain fields are worth mentioning. One of them, known as the Method of Cells (Aboudi,1989; Aboudi,1991), discretizes unit cells that correspond to square arrangements of square fibers into four subcells, within each of which displacements are approximated by low-order polynomials. Traction and displacement continuity conditions at the faces of the subcells are imposed in an average sense and analytical and/or semi-analytical approximations to the deformation fields are obtained in the elastic and inelastic ranges. While using highly idealized microstructures and providing only limited information on the microscopic stress and strain fields, the resulting models pose relatively low computational requirements but have limited capabilities for handling axial shear. The Method of Cells has been used as a constitutive model for analyzing structures made of continuously reinforced composites, see e.g. (Arenburg/Reddy,1991). Developments of the algorithm led to the Generalized Method of Cells (Aboudi,1996), which allows finer discretizations of unit cells for fiber and particle reinforced composites, reinforcement and matrix being essentially split into a number of “subregions” of rectangular or hexahedral shape. For some comparisons with microfields obtained by Finite Element based unit cells see e.g. (Iyer et al.,2000; Pahr/Arnold,2002). Recently, further improvements of the Method of Cells have been reported, see (Aboudi et al.,2003; Pindera et al.,2003). Another micromechanical approach, the Transformation Field Analysis (Dvorak,1992), allows the prediction of the nonlinear response of inhomogeneous materials based on either mean field descriptions (compare the remarks in section 2.4) or on periodic microfield models for the elastic behavior. High computational efficiency is claimed for TFA models involving piecewise uniform transformation fields (Dvorak et al.,1994), but very fine discretizations of the phases are required to achieve high accuracy. A further group of solution strategies for PMAs reported in the literature (Axelsen and Pyrz,1995; Fond et al.,2001; Schjødt-Thomsen and Pyrz,2004) use numerically evaluated equivalent inclusion approaches that account for interacting inhomogeneities as provided e.g. by the work of Moschovidis and Mura (1975). In typical periodic microfield approaches strains and stresses are split into constant macroscopic strain and stress contributions (“slow variables”), hεi and hσi, and periodically varying microscopic fluctuations (“fast variables”), ε0 (z) and σ 0 (z), in analogy to eqn.(1). The position vectors, however, are denoted as z to indicate that the unit cell

– 55 –

ε, u

us

εs s

∆us

lU A

B

A

B

A

B

A

s FIG.14: Schematic depiction of the variation of the strains εs (s) and the displacements us (s) along a section line (coordinate direction s) in some hypothetical inhomogeneous material consisting of constituents A and B.

“lives” on the microscale. The volume integrals used to obtain averages, eqns.(3) and (4), must, of course, be solved over the volume of the unit cell, ΩUC . Formal derivations of the above relationships for periodically varying microstrains and microstresses show that the work done by the fluctuating stress and strain contributions vanishes, compare (Michel et al.,2001). Evidently in periodic microfield approaches each unit of periodicity (unit cell) contributes the same increment of the displacement vector ∆u and the macroscopic displacements vary (multi)linearly. An idealized depiction of such a situation is presented in fig. 14, which shows the variations of the strains εs (s) = hεs i + ε0s (s) and of the corresponding displacements us (s) = hεs i s + u0s (s) along some section line s in a hypothetical periodic two-phase material consisting of constituents A and B. Obviously, the relation hε s i = ∆us /lU holds for linear displacement–strain relations, where lU stands for the length of a unit cell in direction s and ∆us for the corresponding displacement increment. The periodicity of the strains and of the displacements is immediately apparent. 5.2 Boundary Conditions, Application of Loads, Evaluation of Microfields Boundary Conditions Unit cells together with the boundary conditions (B.C.s) prescribed on them must generate valid tilings both for the undeformed geometry and for all deformed states pertinent to the investigation. Accordingly, gaps and overlaps between neighboring unit cells as

– 56 –

well as unphysical constraints on the deformations must not be allowed. In order to achieve this, the boundary conditions for the unit cells must be specified in such a way that all deformation modes appropriate for the load cases to be studied can be attained. The three major types of boundary conditions used in periodic microfield analysis are periodicity, symmetry, and antisymmetry (or point symmetry) B.C.s69) .

D

B A

E C G F

H

I

J

2

1

periodic boundary symmetry boundary point symmetry boundary

symmetry center (pivot point)

FIG.15: Periodic hexagonal array of circular inhomogeneities in a matrix and 11 unit cells that can be used to describe at least some of the mechanical responses of this arrangement under loads acting parallel to the coordinate axes. Generally, for a given periodic phase arrangement unit cells are non-unique, the range of possible shapes being especially wide when point or mirror symmetries are present in the microgeometry. As an example, fig. 15 depicts a (planar) periodic hexagonal array of circular inhomogeneities (fibers with out-of-plane orientation) and some of the unit cells that can be used to study aspects of the mechanical behavior of this arrangement. There are considerable differences in the sizes and capabilities of the unit cells shown. In fig. 16 schematic sketches of periodicity, symmetry and antisymmetry B.C.s, are given for two-dimensional unit cells. The sides and corner points of the cells are annotated; an extension of this nomenclature to three dimensions is trivial. 69)

For more formal treatments of boundary conditions for unit cells than given here see e.g. (Anthoine,1995; Michel et al.,1999). In the case of layer-type models, which can be generated by specifying free surfaces at appropriate boundaries, macroscopic bending and twisting deformations can be studied by appropriately modifying the above standard B.C.s. For models where such macroscopic rotational degrees of freedom are present, “out-of-plane” shear loads must always be accompanied by appropriate bending moments to achieve stress equilibrium, see e.g. (Urba´ nski,1999).

– 57 –

u’ NW N NE u’ W E u’ SW SE NW

NE

z2

SE

S

u’ u’ N NE NW W E u’ SE SW S NE

NW

SE

N u’ NW U NE E W u’ P u’ L SW S u’NW

U

P

L

z1

FIG.16: Sketch of periodicity (left), symmetry (center), and antisymmetry (right) boundary conditions as used with two-dimensional unit cells.

The most general boundary conditions for unit cells are periodicity (“toroidal”, “cyclic”) B.C.s, which can handle any physically valid deformation state of the cell and, consequently, of the inhomogeneous material to be modeled. Periodicity boundary conditions make use of translatory symmetries of a geometry; in fig. 15 cells A to E belong to this group. Because such unit cells tile the computational space by translation, neighboring cells (and, consequently, the “opposite” faces of a given cell) must fit into each other like the pieces of a jigsaw puzzle in both undeformed and deformed states. For the case of quadrilateral two-dimensional unit cells this can be achieved by pairing opposite faces and linking the corresponding degrees of freedom within each pair of faces. Using the conventions for naming the vertices and faces shown in fig. 16 (left), the resulting equations for the periodically varying microscopic displacement vectors at the boundaries 70) u0 , can be symbolically written as u0N (˜ z1 ) = u0S (˜ z1 ) + u0NW

u0E (˜ z2 ) = u0W (˜ z2 ) + u0SE

and

(59)

which implies u0NE = u0NW + u0SE

.

Here z˜1 and z˜2 denote “corresponding positions” on the N and S faces and on the E and W faces of the unit cell, respectively. Equation (59) slaves the microscopic displacements of nodes on faces N and E to those of corresponding nodes on the “master faces” S and W, with the nodes SE and NW acting as “master nodes”. The undeformed geometry must fulfill the compatibility conditions zE (˜ z2 ) − zW (˜ z2 ) = zSE − zSW = lEW

and

zN (˜ z1 ) − zS (˜ z1 ) = zNW − zSW = lNS (60)

70)

In principle, all variables (i.e., for mechanical analysis, the displacements, strains and stresses) must be linked by appropriate periodicity conditions. When a displacement based FE code is used, however, such conditions can be specified explicitly only for the displacement components (including, where appropriate, rotation D.O.F.s), and the periodicity of the stresses and strains is fulfilled in an integral sense only. When stresses are periodic, unit cell boundary tractions are antiperiodic. For periodicity and antisymmetry boundary conditions additional errors may be introduced if the nodal stresses and strains are not averaged across cell boundaries, even though they ought to be.

– 58 –

˜W as well as ˜ ˜S , i.e. it must be posfor master–slave pairs of nodes ˜ zE and z zN and z sible to translate a master face into its corresponding slave face. In addition, for each master–slave pair of faces the phase arrangements and the FE discretizations must be compatible, compare (Swan,1994). With the exception of these conditions there is wide latitude in choosing the shapes and the microgeometries of unit cells that use periodicity boundary conditions71) . Periodicity B.C.s generally are the least restrictive option for multi-inhomogeneity unit cell models using phase arrangements obtained by statisticsbased algorithms (“supercells”) or on the basis of experimental techniques (real structure based models). In practice FE-based unit cell studies using periodicity boundary conditions can be rather expensive in terms of computing time and memory requirements, because the multi-point constraints required for implementing eqn.(59) or its equivalents tend to degrade the band structure of the system matrix, especially in three-dimensional problems. In addition it is worth noting that — especially for simple phase arrangements — considerable care may be required to prevent over- and underconstraining due to inappropriate selection of regions with periodicity boundary conditions, compare (Pettermann/Suresh,2000). For rectangular and hexahedral unit cells in which the faces of the cell coincide with symmetry planes of the phase arrangement and for which this property is retained for all deformed states that are to be studied, periodicity B.C.s simplify to symmetry boundary conditions. For the arrangement shown in fig. 16 (center) these B.C.s take the form u0E (z2 ) = u0SE

0 0 vN (z1 ) = vNW

u0W (z2 ) = 0

vS0 (z1 ) = 0 ,

(61)

where u0 and v 0 are the components of the microscopic displacement vector u0 . The roles of slaves and masters are the same as in eqn.(59). Evidently, eqn.(61) enforces the condition that pairs of master and slave faces must stay parallel throughout the deformation history. Note that for symmetry B.C.s there are no requirements with respect to compatibility of phase arrangements or meshes at different faces. Symmetry boundary conditions are fairly easy to use and tend to give rise to small unit cells for simply periodic phase arrangements, but the load cases that can be handled are limited to uniform thermal loads, mechanical loads that act in directions normal to one or more pairs of faces, and combinations of the above72) . In fig. 15, unit cell F 71)

Conditions analogous to eqn.(59) can be specified for regular area-filling two-dimensional cells having an even number of sides, i.e. squares, rectangles and hexagons (compare cell E in fig. 15), and for regular space-filling three-dimensional cells, i.e. right hexahedra, cubes (sc symmetry), rhombic dodekahedra (fcc symmetry) and regular tetrakaidekahedra (truncated octahedra, bcc symmetry). More complicated conditions have to be used for unit cells of less regular shape, compare e.g. (Cruz/Patera,1995; Estrin et al.,1999; Xia et al.,2003). If appropriate symmetries are present within periodic unit cells, as may be the case with textile composites, these can be used to reduce the problem to arrays of smaller subcells, see e.g. (Tang/Whitcomb,2003). 72)

Because these load cases include different normal loads acting on each pair of faces, loading by macroscopic extensional shear (obtained e.g. in the two-dimensional case by applying normal stresses σa to the vertical and −σa to the horizontal faces), hydrostatic loading, and

– 59 –

uses boundary conditions of this type73) . Symmetry boundary conditions are typically very useful for describing relatively simple microgeometries, but tend to be somewhat restrictive for modeling phase arrangements generated by statistics-based algorithms or obtained from micrographs. Antisymmetry boundary conditions are even more limited in terms of the microgeometries that they can handle, because they require the presence of centers of point symmetry (“pivot points”). In contrast to symmetry boundary conditions, however, unit cells employing them on all faces are subject to few restrictions to the load cases that can be studied. Among the unit cells shown in fig. 15, cells G and H use point symmetry B.C.s on all faces and can handle any in-plane deformation74) . Alternatively, antisymmetry B.C.s can be combined with symmetry B.C.s to obtain very small unit cells that are restricted to loads acting normal to the symmetry faces, compare unit cells I and J in fig. 15. The right hand sketch in fig. 16 shows such a “reduced” unit cell, the antisymmetry boundary conditions being applied on the E-side where a pivot point P is present. For this configuration the boundary conditions u0U (˜ s) + u0L (−˜ s) = 2u0P

0 0 0 vN (z1 ) = vNW = 2vP

vS0 (z1 ) = 0

u0W (z2 ) = 0

(62)

must be fulfilled, where u0U (˜ s) and u0L (−˜ s) are the deformation vectors of pairs of points U and L that are positioned symmetrically on face E (where a local coordinate system s˜ centered on the pivot P is defined in analogy to fig. 24) and deform symmetrically with respect to the pivot P. The undeformed geometry of such a face also must be antisymmetric with respect to the pivot point P, i.e. zL (˜ s) + zU (−˜ s) = 2zP

,

(63)

and the phase arrangements as well as the discretizations on both halves of the face must be compatible. Three-dimensional unit cells employing combinations of symmetry and point symmetry B.C.s can e.g. be used to advantage for studying cubic arrays of particles, see (Weissenbek et al.,1994). (hygro)thermal loading, symmetry B.C.s are typically sufficient for materials characterization. However, stress triaxialities obtained this way may correspond to different orientations with respect to the macroscopic material coordinate system (for example, extensional shear corresponds to simple shear in a coordinate system rotated by 45◦ ). 73)

Whereas for a given periodic phase arrangement there are infinitely many equivalent unit cells with periodicity B.C.s, the number of “smallest” unit cells with symmetry B.C.s is always limited. It is also worth noting that for an n-dimensional problem 2 n unit cells with symmetry B.C.s can be assembled (by mirroring operations) into a unit cell that is suitable for the application of periodicity B.C.s. 74)

Triangular unit cells similar to cell H in fig. 15 were used e.g. by Teply and Dvorak (1988) to study the transverse mechanical behavior of hexagonal arrays of fibers. Rectangular cells with point symmetries on each boundary were introduced in (Marketz/Fischer,1994) for perturbed square arrangements of inhomogeneities. The periodic cells D and E also show point symmetry on their faces.

– 60 –

Within the nomenclature shown in fig. 16 the displacements at the master nodes SE and NW, u0SE and u0NW , control the overall deformation of the unit cell in the 1- and 2-directions and can be used to evaluate the macroscopic strain of the corresponding periodic model material, compare eqn.(72). With respect to unit cell models that use the FEM for evaluating the microfields, it is worth noting that all of the above types of boundary conditions can be handled by any code that provides linear multipoint constraints between degrees of freedom and thus allows the linking of three or more D.O.F.s by linear equations. Obviously, the description of real materials, which in general are not periodic, by periodic model materials entails some geometrical approximations. These take the form of periodicity constraints on computer generated generic phase arrangements or of appropriate modifications in the case of real structure microgeometries75) , compare fig. 1. The effects of such approximations in the vicinity of the cell surfaces, of course, diminish in importance with growing size of the model. Equations (59) to (63) refer to the usual case where local material descriptions are used on the macroscale and the scale transition is handled via homogenized stress and strain fields. Periodicity boundary conditions that are conceptually similar to eqn.(59) can be devised for cases where gradient (nonlocal) theories are employed on the macroscale and higher order stresses as well as strain gradients figure in coupling the length scales (Geers et al.,2001). Linking Macroscale and Microscale The primary practical challenge in using periodic microfield approaches for modeling inhomogeneous materials lies in choosing and generating suitable unit cells that — in combination with appropriate boundary conditions — allow a realistic representation of the actual microgeometries within available computational resources. The unit cells must then be subjected to appropriate macroscopic stresses, strains and temperature excursions. Whereas loads of the latter type generally do not pose major difficulties, applying far field stresses or strains is not necessarily straightforward (note that the variations of the stresses along the faces of a unit cell in general are not known a priori, so that it is not possible to prescribe the boundary tractions via distributed loads). The most versatile and elegant strategy for linking the macroscale and the microscale in periodic microfield models is based on a mathematical framework known as asymptotic homogenization, asymptotic expansion homogenization, or homogenization theory, see e.g. (Suquet,1987). Instead of formulating the unit cell problem solely on the microscale, 75)

The simplest way of “periodifying” a non-periodic rectangular or cuboidal volume element consists in mirroring it with respect to 2 or 3 of its surfaces, respectively, and doing periodic homogenization on the resulting larger unit cell. Such operations, however, may lead to undesired shapes of reinforcement and will change the arrangement statistics at least to some extent. The simpler expedient of specifying symmetry B.C.s for rectangular or cuboidal real structure cells amounts to an even larger modification of the microgeometry and introduces marked local constraints. For sufficiently large multi-inhomogeneity cells Terada et al. (2000) obtained useful results by applying periodicity boundary conditions to nonperiodic geometries.

– 61 –

macroscopic and microscopic coordinates, Z and z, respectively, are explicitly introduced. They are related by the expression zi = Zi /

,

(64)

where = `/L 1 is a scaling parameter, L and ` being characteristic lengths of the macro- and microscales76) . The displacement field in the unit cell can then be represented by an asymptotic expansion of the type (0)

(1)

(2)

ui (Z, z, ) = ui (Z) + ui (Z, z) + 2 ui (Z, z)

+ H.O.T. ,

(0)

(65)

(1)

where the ui are the effective or macroscopic displacements and ui stands for the 1st order displacement perturbations due to the microstructure, which are assumed to vary periodically77) . Using the chain rule, i.e. Z ∂ f (Z, z = ) ∂x

→

∂ 1 ∂ f+ f ∂Z ∂z

,

the strains can be related to the displacements (in the case of small strains) as 1 n ∂ (0) ∂ (0) ∂ (1) o ∂ (1) εij (Z, z, ) = u + u u + u + 2 ∂Zj i ∂Zi j ∂zj i ∂zi j n ∂ (1) ∂ (2) ∂ (1) ∂ (2) o + + + H.O.T. u + u u + u 2 ∂Zj i ∂Zi j ∂zj i ∂zi j (1)

(2)

= εij (Z, z) + εij (Z, z) (0)

+ H.O.T. ,

(66)

(67)

(0)

where terms of the type εij = 1 ∂z∂ j ui are deleted due to the underlying assumption that the variations of slow variables are negligible at the microscale. The stresses are expanded in analogy to the strains so that the equilibrium condition can be written as ∂ i 1 ∂ h (1) (2) σij (Z, z) + σij (Z, z) + fi (Z) = 0 , (68) + ∂Zj ∂zj the fi being macroscopic body forces. Working out eqn.(68) and sorting the contributions by order of gives rise to a hierarchical system of partial differential equations. ∂ (1) σ =0 ∂zj ij ∂ (2) ∂ (1) σij + σ + fi = 0 ∂Zj ∂zj ij

(order −1 ) (order 0 ) ,

(69)

76)

Equation (64) may be viewed as “stretching” the microscale so it becomes comparable to the macroscale, f (x) → f (Z, Z/) = f (Z, z). 77)

The nomenclature used in eqns.(64) to (70) follows typical usage in asymptotic homogenization. It is more general than, but can be directly compared with, the one used in eqns.(1) to (63), where only microscopic coordinates are employed. Zero order terms can generally be identified with macroscopic quantities and first order terms generally correspond to periodically varying microscopic quantities.

– 62 –

the first of which gives rise to a boundary value problem at the unit cell level that is referred to as the “micro equation”. By making a specific ansatz for the strains at unit cell level and by volume averaging over the second equation in the system (69), the “macro equation”, the microscopic and macroscopic fields can be linked such that the homogenized elasticity tensor is obtained as H Eijkl

1 = ΩUC

Z

ΩUC

Eijmn

∂ δmk δnl + δml δnk + χkmn dΩ 2 ∂zl

1

.

(70)

Here ΩUC is the volume of the unit cell, Eijkl (z) is the phase-level (microscopic) elasticity tensor, and the “characteristic function” χkmn (z) describes the deformation modes of the unit cell78) and, accordingly, relates the micro- and macrofields. Analogous expressions can be derived for tangent modulus tensors in nonlinear analysis, compare e.g. (Ghosh et al.,1996).

The above relations can be used as the basis of Finite Element algorithms that solve for the characteristic function χkmn . For detailed discussions of asymptotic homogenization methods within the framework of FEM-based micromechanics see e.g. (Jansson,1992; Ghosh et al.,1996; Hassani/Hinton,1999; Chung et al.,2001). Asymptotic homogenization allows to directly couple FE models on the macro- and microscales, compare e.g. (Terada et al.,2003), an approach that has been used in a number of multi-scale studies (compare chapter 8) and is sometimes referred to as FE2 method (Feyel/Chaboche,2000). A treatment of homogenization in the vicinity of macroscopic boundaries can be found in (Schrefler et al.,1997), an asymptotic homogenization procedure for elastic composites that uses standard elements within a commercial FE package was proposed by Banks-Sills et al. (1997). Asymptotic homogenization schemes have also been involved for problems involving higher order stresses and strain gradients (Kouznetsova et al.,2002)79). When asymptotic homogenization is not used, it is good practice to apply far field stresses and strains to a given unit cell via concentrated nodal forces or prescribed displacements at the master nodes and/or pivots, an approach termed the method of macroscopic degrees of freedom by Michel et al. (1999). Employing the divergence theorem and using the nomenclature and configuration of fig. 16 (left), the forces to be applied to the master nodes SE and NW, PSE and PNW , of a two-dimensional unit cell with periodicity boundary conditions can be shown to be given by the surface integrals PSE =

Z

ta (z) dΓ

PNW =

ΓE

Z

ta (z) dΓ ,

(71)

ΓN

78)

Even though eqns.(69) and (70) are derived from an explicit two-scale formulation neither of them contains the scale parameter , see the discussion in (Chung et al.,2001). 79)

Such methods are especially useful for problems in with the length scales are not well spread; in them the “unit cells” do not necessarily remain periodic during the deformation process.

– 63 –

compare (Smit et al.,1998). Here ta (z) = σa nΓ (z) stands for the homogeneous surface traction vector corresponding to the applied (far field) stress field80) at some given point z on the cell’s surface ΓUC , with nΓ (z) being the local surface normal. Equation (71) can be generalized to require that each master node is loaded by a force corresponding to the surface integral of the applied traction vectors over the face slaved to it via an equivalent of eqn.(59). An analogous procedure holds for three-dimensional cases, and symmetry as well as antisymmetry boundary conditions as described by eqns.(61) and (62) can be handled similarly. For applying far field strain boundary conditions within the method of macroscopic degrees of freedom, displacements are prescribed to the master nodes. These nodal displacements must be obtained from the macroscopic strains via the appropriate displacement– strain relations. For example, following the notation of eqns.(59) to (62), the displacements to be prescribed to the master nodes 2 and 4 of the unit cells shown in fig. 16 can be evaluated as u0SE = hεi11 lEW

0 vSE = γ12 lEW

u0NW = γ21 lNS

0 vNW = hεi22 lNS

(72)

for the case of linear strain–displacement relations81) , the reference lengths of the rectangular unit cells being defined in eqn.(60). When FE methods are used, strain controlled analysis is typically considerably easier to carry out by the method of macroscopic degrees of freedom than stress controlled analysis82) . In order to obtain the elastic tensors of periodic phase arrangements, a sufficient number (depending on the overall symmetry of the unit cell, compare section 1.3) of linearly independent load cases must be evaluated. In general, the overall stress and strain tensors within a unit cell can be evaluated by volume averaging83) (e.g. using some numerical integration scheme) or by using the equiv80)

Note that the ta (z) are not identical with the actual local values of the tractions, t(z) at the cell boundaries, but are equal to them over the cell face in an integral sense. For geometrically nonlinear analysis eqn.(71) must be applied to the current configuration. Because σ a is constant it can be factored out of the R surface integrals, so that the force applied to a master node P M takes the form PM = σa Γ nΓ (z) dΓ, which is the projection of the applied stress on the S

current average surface normal of the slave face ΓS times the current area of the face. 81)

0 and u0NW , is an integration constant One of the two displacement components, vSE describing solid body rotations of the unit cell and, accordingly, can be chosen at will. The displacements at node NE are fully determined by those of the master nodes, compare eqn.(59). Analogous relations hold for three-dimensional cases. Equations (72) can be directly extended to nonlinear strain–displacement relations and to handling deformation gradients. 82)

Even though the loads acting on the master nodes obtained from eqn.(71) are in principle equilibrated, in stress controlled analysis solid body rotations may be induced through small numerical errors. When these solid body rotations are suppressed by deactivating degrees of freedom beyond those required for enforcing periodicity, the quality of the solutions may be degraded or transformations of the averaged stresses and strains may be required. 83)

For large strain analysis the consistent procedure is to first average the deformation gradients and then evaluate the strains from these averages.

– 64 –

alent surface integrals given in eqn.(3), i.e. hσi =

1

1

Z

Z

σ(z) dΩ = t(z) ⊗ z dΓ ΩUC ΩUC ΩUC ΓUC Z Z 1 1 hεi = ε(z) dΩ = (u(z) ⊗ nΓ (z) + nΓ (z) ⊗ u(z)) dΓ ΩUC ΩUC 2ΩUC ΓUC

. (73)

In the case of rectangular or hexahedral unit cells that are aligned with the coordinate axes, averaged engineering stress and strain components can, of course, be evaluated by dividing the applied or reaction forces at the master nodes by the appropriate surface areas and by dividing the displacements at the master nodes by the appropriate cell lengths, respectively. Typical relations of this kind are hσ11 i =

P1,SE ΓE

and

hεi11 =

u0SE lEW

,

(74)

where P1,SE is the 1-component of P1,SE . For evaluating phase averaged quantities from unit cell models, it is usually possible to use direct volume integration of eqn.(12) or similar expressions84) . In many FE codes this can be done by approximate numerical quadrature according to 1 hf i = Ω

Z

N

1 X f (z)dΩ ≈ fl Ωl Ω Ω

,

(75)

l=1

where fl and Ωl are the function value and the integration weight (in terms of an integration point volume), respectively, associated with the l-th integration point within a given integration volume Ω that contains N integration points85) , When effective values or phase averages are to be generated of variables that are nonlinear functions of the stress or strain components (e.g. equivalent stresses, equivalent strains, stress triaxialities, stress principals), the variable must be evaluated first (e.g. at the integration points) and volume averaging of the results performed subsequently, compare also the remarks in section 2.4. Besides generating overall and phase averages, microscopic variables can also be evaluated in terms of higher statistical moments and of distribution functions (“stress spectra”, see e.g. Bornert et al.,1994; B¨ ohm/Rammerstorfer,1995; B¨ ohm/Han,2001).

84)

It is of some practical interest that volume averaged and phase averaged microfields obtained from unit cells must fulfill all relations given in section 2.1 and can thus be conveniently used to check the consistency of a given model by inserting them into eqns.(15). For finite deformations, however, appropriate stress and strain measures must be used for this purpose (Nemat-Nasser,1999). 85)

Such procedures may also be used for evaluating other volume integrals, e.g. in computing Weibull-type fracture probabilities for reinforcements (Antretter,1998; Eckschlager,2002).

– 65 –

5.3 Unit Cell Models for Continuously Reinforced Composites Composites reinforced by continuous aligned fibers, which typically show a statistically transversely isotropic overall behavior, can be studied relatively easily and efficiently with unit cell methods. Materials characterization — with the exception of the overall axial shear behavior — can be carried out with two-dimensional unit cell models employing generalized plane strain elements that use one global degree of freedom to describe the axial deformation of the whole model86) . Such elements are implemented in a number of commercial FE codes. For handling the overall axial shear response, however, special elements (Adams/Crane,1984; Sørensen,1991) or three-dimensional models with appropriate boundary conditions (Pettermann/Suresh,2000) are required87) . In all of the above cases generalized plane strain states of some type are described, i.e. the axial strains are constant over the unit cell, while axial out-of-plane deformations are present. Three-dimensional modeling is also relevant for the materials characterization of composites reinforced by continuous aligned fibers when the effects of fiber misalignment, of fiber waviness, or of broken fibers88) are to be studied, see e.g. (Mahishi,1986; Garnich/Karami,2004). Unit cell models used for nonlinear constitutive modeling, of course, must be fully three-dimensional and employ periodicity boundary conditions. Figure 17 shows a number of periodic fiber arrangements for two-dimensional unit cell modeling of aligned continuously reinforced composites, the periodic hexagonal (PH0) and periodic square (PS0) arrays being “classical” simple periodic models. The models with hexagonal symmetry (PH0,CH1,RH2,CH3) show the appropriate transversely isotropic thermoelastic overall behavior, whereas the other fiber arrangements shown in fig. 17 are macroscopically tetragonal (PS0,CS7,CS8) or monoclinic (MS5). None of the eight configurations gives a transversely isotropic response in the elastoplastic regime under mechanical loads that are not axisymmetric with respect to the fiber direction, because the hexagonal symmetry of the phase arrangements is broken due to the resulting distributions of the local material properties (the strain hardening and thus the instantaneous moduli at a given position depend on the load history the point has undergone), compare also fig. 19. Fiber arrangements of the type shown in fig. 17 can reproduce many features and effects of the fiber arrangements found in continuously reinforced composites. They are, however, not statistically representative of the microstructures of most real composites, 86)

Because the axial stiffness of composites reinforced by continuous aligned fibers can usually be satisfactorily described by Voigt type models, unit cell models of such materials have tended to concentrate on the transverse behavior. Plane strain models, however, cannot provide any information on the induced axial response of continuously reinforced composites and do not adequately describe the axial microstresses. 87)

For linear elastic material behavior there is the additional option of making use of the formal analogy between out-of-plane (antiplane) shear and diffusion problems. 88)

For investigating the axial failure behavior of continuously reinforced MMCs, statistical methods concentrating on fiber fragmentation or more elaborate spring lattice models, see e.g. (Zhou/Curtin,1995), have been used successfully.

– 66 –

PH0

PS0

CH1

MS5

RH2

CS7

CH3

CS8

FIG.17: Eight periodic fiber arrangements of fiber volume fraction ξ=0.475 for modeling continuously fiber reinforced composites (B¨ ohm/Rammerstorfer,1995). Note that the minimum fiber distances are the same for all configurations except the simple periodic arrangements PH0 and PS0. the exception being materials reinforced by undirectional monofilaments and produced by fiber–foil techniques. Major improvements with respect to the realism of the microgeometries can be be obtained by multi-fiber unit cells (see e.g. Nakamura/Suresh,1993; ˇ Moulinec/Suquet,1997; Gusev et al.,2000; Sejnoha/Zeman,2002) that use random inplane positions of the fibers, compare fig. 18. In table 3 some thermoelastic properties of an aligned continuously reinforced ALTEX/Al MMC as predicted by bounding methods, MFAs and unit cells using periodic hexagonal (PH0), clustered hexagonal (CH1), and periodic square (PS0) arrangements as well as the multi-fiber cell shown in fig. 18 (MFUC) are listed. For this material combination all unit cell results fall within the Hashin–Shtrikman bounds for continuously reinforced composites89) , but the square arrangement gives predictions for the overall transverse properties that, besides being clearly anisotropic, significantly violate the three-point bounds for materials reinforced by identical aligned continuous fibers. Table 3 also shows 89)

The constituents’ material properties underlying table 3 have a low elastic contrast of c ≈ 3. In general the elastic stiffnesses obtained from square-type arrangements may to violate the Hashin–Shtrikman bounds and usually lie outside the three-point bounds.

– 67 –

FIG.18: Multi-fiber unit cell for a continuously reinforced MMC (ξ=0.453) using 60 quasi-randomly positioned fibers and symmetry B.C.s (Nakamura/Suresh,1993)

minor deviations of the results of the multi-fiber model from transversely isotropic elastic behavior. The configurations depicted in figs. 17 and 18 give nearly identical results for the overall elastoplastic behavior of continuously reinforced composites under axial mechanical loading, and the predicted overall axial and transverse responses under thermal loading are very similar. The overall elastoplastic behavior under transverse mechanical loading, however, tends to be very sensitive to the fiber arrangement. As to simply periodic models, the behavior of hexagonal arrangements is typically sandwiched between the stiff (in 0◦ –direction) and the compliant (in 45◦ –direction) responses of periodic square arrangements in both the elastic and elastoplastic ranges, see fig. 19. Multi-fiber unit cells that approach statistical transverse isotropy such as fig. 18 tend to show noticeably stronger strain hardening than periodic hexagonal arrangements of the same fiber volume fraction, compare also (Moulinec/Suquet,1997). Generally the macroscopic yield surfaces of uniaxially reinforced MMCs are not of the Hill type, but rather follow bimodal descriptions (Dvorak/Bahei-el-Din,1987). The distributions of microstresses and microstrains in fibers and matrix tend to depend markedly on the fiber arrangement, especially under thermal and transverse mechanical loading90) . In the plastic regime, the microscopic distributions of equivalent stresses 90)

Irrespective of the types of load applied, for technologically relevant reinforcement volume fractions Eshelby’s (1957) result of uniform stresses and strains in dilute inhomogeneities holds neither in the elastic nor the inelastic regimes for nondilute composites.

– 68 –

∗ ∗ EA ET [GPa] [GPa]

fibers matrix HS/lo HS/hi 3PBm/lo 3PBm/hi MTM GSCS 2OEm PH0 CH1 PS0/00 PS0/45 MFUC/00 MFUC/90

180.0 67.2 118.8 119.3 118.8 118.9 118.8 118.8 118.8 118.8 118.7 118.8 118.8 118.8 118.8

180.0 67.2 103.1 107.1 103.8 104.5 103.1 103.9 103.9 103.7 103.9 107.6 99.9 104.8 104.6

∗ νA []

∗ νT []

0.20 0.35 0.276 0.279 0.278 0.279 0.279 0.279 0.279 0.279 0.279 0.279 0.279 0.278 0.278

0.20 0.35 0.277 0.394 0.326 0.347 0.342 0.337 0.338 0.340 0.338 0.314 0.363 0.334 0.333

α∗A α∗T −6] −1 [K ×10 [K ×10−6] −1

6.0 23.0 — — — — 11.84 11.84 11.89 11.84 11.90 11.85 11.85 11.90 11.90

6.0 23.0 — — — — 16.46 16.46 16.40 16.46 16.42 16.45 16.45 16.31 16.46

TABLE 3: Overall thermoelastic properties of a unidirectional continuously reinforced ALTEX/Al MMC (ξ=0.453) as predicted by the Hashin–Shtrikman (HS) and three-point (3PB) bounds, by the Mori–Tanaka method (MTM), by the generalized self-consistent scheme (GSCS), by Torquato’s second order estimates (2OE), as well as by unit cell analysis using periodic arrangements shown in fig. 17 (PH0, CH1, PS0) and the multifiber cell displayed in fig. 18 (MFUC). For arrangement PS0 responses in the 0◦ and 45◦ directions, and for the multi-fiber cell responses in the 0◦ and 90◦ directions are given. and equivalent plastic strains91) , of hydrostatic stresses, and stress triaxialities tend to be markedly inhomogeneous92) , see e.g. fig. 20. This type of behavior typically is most marked for applied transverse shear loads and for elastic–ideally plastic matrix behavior. As a consequence of the inhomogeneity of the microfields, there tend to be strong constrained plasticity effects in continuously reinforced MMCs and the onset of damage in the matrix, of fracture of the fibers, and of interfacial decohesion at the fiber– matrix interfaces show a strong dependence on the fiber arrangement. Marked local irregularities in the microgeometry, such as clusters of closely approaching fibers and/or “matrix islands”, as present e.g. in arrangement RH2 or in the multi-fiber unit cell shown in fig. 18, are especially critical in this respect. A further group of materials that are reinforced by continuous fibers and can be studied by unit cell methods are composites using woven, knitted, and braided reinforce91)

Note that for regions of concentrated strains in inhomogeneous materials to be shear bands in the strict sense the constituent material behavior must display strain softening, e.g. due to damage. 92)

The corresponding distribution functions, phase averages, and higher statistical moments are also significantly influenced by the fiber arrangement, compare (B¨ ohm/Rammerstorfer,1995).

– 69 –

6.00E+01 5.00E+01

DN

4.00E+01

PH

3.00E+01

PS/45

2.00E+01

Matrix

1.00E+01

APPLIED STRESS [MPa]

PS/00

Fiber

0.00E+00

DN/00 PS0/45 PS0/00 PH0/90 PH0/00 Al99.9 MATRIX ALTEX FIBER 0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

3.00E-03

3.50E-03

4.00E-03

STRAIN []

FIG.19: Transverse uniaxial response of a unidirectional continuously reinforced ALTEX/Al MMC (ξ=0.453, elastoplastic matrix with linear isotropic hardening) as predicted by unit cell models based on periodic hexagonal fiber arrangements (PH0) and periodic square fiber arrangements (PS0), compare fig. 17, as well as by the multi-fiber unit cell shown in fig. 18 (MFUC).

EPS.EFF.PLASTIC 2.25E-02 1.75E-02 1.25E-02 7.50E-03 2.50E-03

FIG.20: Microscopic distributions of the accumulated equivalent plastic strains in the matrix of a transversely loaded unidirectional continuously reinforced ALTEX/Al MMC (ξ=0.453, elastoplastic matrix) under uniaxial loading in the horizontal direction as predicted by the multi-fiber unit cell shown in fig. 18.

– 70 –

ments. Such materials are typically modeled by considering bundles of fibers (tows), the properties of which are obtained by treating them as unidirectional composites of high fiber volume fraction, that are embedded in pure matrix. Models of this type, see e.g. (Chang/Kikuchi,1993; Dasgupta et al.,1996, Wang et al.,2007), can be considered as multiscale approaches, compare chapter 8, in which the weave geometry proper is considered at the mesoscale. The unit cells used for studying laminated and woven composites are typically three-dimensional, usually have two free surfaces, and may have macroscopic rotational degrees of freedom to describe warping and twisting. They tend to be fairly complex geometrically and to be computationally expensive, especially for nonlinear constituent behavior. One way of mitigating the computational costs consists of making use of the symmetries of the unit cells of the weaves by introducing special boundary conditions (Tang/Whitcomb,2003). Finally, it is worth mentioning that unit cell approaches can be used for investigating the local stress and strain fields in cross-ply laminates reinforced by monofilaments, see e.g. (Lerch et al.,1991; Ismar/Schr¨ oter,2000). 5.4 Unit Cell Models for Discontinuously Reinforced Composites The present section concentrates on unit cell models for many practically important types of discontinuously reinforced composites, viz. short fiber and particle reinforced materials that do not exhibit major compositional gradients or variations93) . Due to the complexity and irregularity of their three-dimensional phase arrangements discontinuously reinforced composites tend to be considerably more difficult to model by techniques based on discrete microgeometries than are materials reinforced by aligned continuous fibers, two-dimensional descriptions rarely being satisfactory. Composites Reinforced by Aligned Short Fibers Because of their relatively simple microgeometries short fiber composites with aligned reinforcements have been the focus of considerable modeling efforts over the past decades. The overall symmetry of such materials is transversely isotropic. Most three-dimensional unit cell models of aligned short fiber reinforced composites reported in the literature describe non-staggered or staggered cylindrical fibers in periodic square or hexagonal arrangements94) , compare fig. 21. Such approaches are relatively simple conceptually and, when employing periodicity B.C.s, can in principle be used to model the full thermomechanical response of aligned short fiber reinforced composites. When limitations are accepted with respect to the load cases that can be studied, considerably smaller unit cells that employ symmetry boundary conditions may be used 93)

Platelet reinforced composites can generally be described in analogy to short fiber reinforced ones. 94)

Like in the case of models for continuously reinforced composites, hexagonal arrangements are overall transversally isotropic in the elastic range, whereas square arrangements give rise to tetragonal overall symmetry, i.e. the transverse overall properties are direction dependent.

– 71 –

FIG.21: Sketch of fiber positions in non-staggered hexagonal (left) and staggered square (right) periodic arrangements of aligned short fibers.

for staggered and non-staggered fiber arrangements, see e.g. (Levy/Papazian,1991) and compare fig. 22. Analogous microgeometries based on periodic hexagonal arrangements of non-staggered (J¨ arvstr˚ at,1992) or staggered (Tucker/Liang,1999) fibers as well as ellipsoidal unit cells aimed at describing composite ellipsoid assembly microstructures (J¨ arvstr˚ at,1993) were also proposed95) .

FIG.22: Three-dimensional unit cells for short fibers employing symmetry boundary conditions used for modeling non-staggered (top) and staggered (bottom) square arrangements of fibers (Weissenbek,1994). Staggered and non-staggered unit cells of the above types, however, are rather restrictive in terms of fiber arrangements and shapes, a difficulty that can be resolved by using much larger multi-fiber unit cells that contain randomly positioned aligned fibers of different lengths see e.g. (Ingber/Papathanasiou,1997; Gusev,2001; Liu et al.,2005). For many materials characterization studies, a more economical alternative to the above three-dimensional unit cells are axisymmetric models describing the axial behavior of non-staggered or staggered arrays of aligned cylindrical short fibers in an approximate way. The basic idea behind these models is to replace unit cells for square or hexagonal arrangements by circular composite cylinders of equivalent cross sectional area (and volume fraction) as sketched in fig. 23. 95)

For a comparison between unit cell and analytical predictions for the overall elastic properties for short fiber reinforced composites see e.g. (Tucker/Liang,1999).

– 72 –

FIG.23: Periodic arrays of aligned non-staggered (top) and staggered (bottom) short fibers and corresponding axisymmetric cells (left: cross sections in transverse plane; right: sections parallel to fibers). The resulting axisymmetric cells are not unit cells in the strict sense, because they overlap and are not space filling96) . In addition, they do not have the same transverse fiber spacing as the corresponding three-dimensional arrangements and they are restricted to handling load cases that lead to axisymmetric deformations (i.e. they cannot be used to study the response to transverse normal or shear loading). They have the advantage, however, of significantly reduced computational requirements. NE

N

~s

U

NW

NE

N NW

U

E E

P

W

W

z

P

z

L SW

S

~ −s

L

SW

S

SE

r UNDEFORMED

r

SE

DEFORMED

FIG.24: Axisymmetric cell for staggered arrangement of short fibers: undeformed and deformed shape. Axisymmetric cell models employ symmetry boundary conditions for the top and bottom faces of the cells, and the B.C.s at the cylinder surface are chosen to maintain the same 96)

Analogous axisymmetric cells may also be used for studying aspects of the behavior of continuously reinforced composites.

– 73 –

cross sectional area along the axial direction for an aggregate of cells. In the case of non-staggered fibers this can be easily done by specifying symmetry-type boundary conditions, whereas for staggered arrangements a pair of cells with “opposite” fiber positions is considered, for which the total cross sectional area is required to be independent of the axial coordinate97) . Using the nomenclature of fig. 24 and the notation of eqns.(59) to (62), nonlinear relations are obtained for the radial displacements, u0 , together with linear constraints for the axial displacements, v 0 , (rU + u0U )2 + (rL + u0L )2 = 2(rP + u0P )2

and

0 0 vU + vL0 = 2vP

.

(76)

Here U and L are nodes on the E-side of the unit cell that are positioned symmetrically with respect to the pivot point P. Nonlinear constraints of this type may not be available or can be cumbersome to use in standard FE codes98) . However, for most applications the B.C.s for the radial displacements in eqn.(76) can be linearized without major loss in accuracy, so that antisymmetry-type boundary conditions analogous to eqn.(62) are obtained for the radial surface, i.e. u0U + u0L = 2u0P

and

0 0 vU + vL0 = 2vP

.

(77)

For the past 15 years axisymmetric cell models have been the workhorses of PMA studies of short fiber reinforced composites, see e.g. (Povirk et al.,1992; Tvergaard,1994), and they have been extended to cover composites containing aligned inhomogeneities of different sizes, shapes and aspect ratios (essentially by coupling two different cells instead of two equal ones as shown in fig. 24), compare (B¨ ohm et al.,1993; Weissenbek,1994; Tvergaard,2004). Typically, descriptions using staggered arrangements allow a wider range of microgeometries to be covered and give somewhat more realistic descriptions of actual composites. Composites Reinforced by Nonaligned Short Fibers Whereas a considerable number of unit cell studies of materials reinforced by aligned discontinuous fibers have been reported, the literature contains few such models dealing with nonaligned fibers, among them three-dimensional studies of composites reinforced by alternatingly tilted misaligned fibers (Sørensen et al.,1995) and plane-stress models describing planar random short fibers (Courage/Schreurs,1992). The situation with respect to mean field models for such materials is not fully satisfactory for the elastic range, either, compare section 2.7, while little work has been done on the inelastic thermomechanical behavior. 97)

As originally proposed by Tvergaard (1990), the nonlinear displacement boundary conditions in eqn.(76) were combined with antisymmetry traction B.C.s for use with a hybrid FE formulation. 98)

For an example of the use of the nonlinear BCs described in eqn.(76) with a commercial FE code see e.g. (Ishikawa et al.,2000), where a cell of truncated cone shape is employed to describe bcc arrangements of particles.

– 74 –

FIG.25: Unit cell for a short fiber reinforced MMC (ξ=0.15 nominal) containing 15 cylindrical short fibers of aspect ratio a = 5 in a quasi-random arrangement suitable for using periodicity B.C.s (B¨ ohm et al.,2002).

At present the most promising modeling strategy for nonaligned short fiber reinforced composites consists of using three-dimensional multi-fiber unit cells in which the fibers are randomly positioned and oriented such that the required ODF is approximately fulfilled, see e.g. (B¨ ohm et al.,2002; Lusti et al.,2002). Such multi-fiber unit cells can also account for the distributions of fiber sizes and/or aspect ratios. This modeling approach, however, tends to pose considerable challenges in generating appropriate fiber arrangements at nondilute volume fractions due to geometrical frustration. The meshing of the resulting phase arrangements for use with the FEM also is typically difficult, compare e.g. (Shephard et al.,1995), and analyzing the mechanical response of the resulting cells requires considerable computing power, especially for nonlinear constituent behavior. The first studies of this type were accordingly restricted to the linear elastic range, where the BEM has been found to answer well, see e.g. (Banerjee/Henry,1992). As an example of a multi-fiber unit cell for a composite reinforced by nonaligned short fibers, fig. 25 shows a cube-shaped cell that contains 15 randomly oriented cylindrical fibers of aspect ratio a = 5, supports periodicity boundary conditions, was generated by random sequential insertion, and is discretized by tetrahedral elements. The phase arrangement is set up in such a way that spheroidal fibers of the same aspect ratio, volume fractions, center points, and orientations can also be used in order to allow a comparison of fiber shapes (B¨ ohm et al.,2002). Unit cells corresponding to higher fiber volume fractions can be generated (as well as meshed and solved in the linear range) by the commercial micromechanics code PALMYRA (MatSim Ges.m.b.H., Zurich, Switzerland).

– 75 –

SiC fibers Al2618-T4 matrix HS/lo HS/hi MTM CSCS KTM MFUC/sph MFUC/cyl

E∗ [GPa]

ν∗ []

450.0 70.0 87.6 106.1 89.8 91.2 90.3 89.4 90.0

0.17 0.30 0.246 0.305 0.285 0.284 0.285 0.285 0.284

TABLE 4: Overall elastic properties of a SiC/Al MMC reinforced by randomly oriented fibers (a = 5, ξ=0.15) as predicted by the Hashin–Shtrikman (HS) bounds, by Benveniste’s (1987) Mori–Tanaka method (MTM), by Berryman’s (1980) classical selfconsistent scheme (CSCS), by the Kuster–Toks¨ oz (1974) model (KTM) and by multi-fiber unit cells of the type shown in fig. 25 (MFUC), which contain 15 spheroidal or cylindrical fibers (B¨ ohm et al.,2002.

In table 4 predictions are given for the overall elastic response of a composite reinforced by randomly positioned and oriented fibers obtained by analytical estimates and bounds as well as results of the above type of unit cell, data of the latter type being provided for both cylindrical and spheroidal inhomogeneities. Interestingly, considerable differences were found between the predictions for reinforcement by spheroidal and spheroidal fibers that have the same fiber volume fraction and aspect ratio, occupy the same positions and show the same orientations, the spheroidal reinforcements leading to a softer response, especially in the inelastic regime. Generally very satisfactory results were obtained despite the low number of fibers in the unit cell (chosen in view of the available computational resources) and excellent agreement with analytical descriptions has been achieved in terms of the orientation dependence of the stresses in individual fibers, compare fig. 5. It should be noted, however, that from the point of view of the representativeness of the phase arrangement and for use in the elastoplastic range unit cells containing a considerably higher number of fibers are certainly desirable. Composites Reinforced by Particles Materials reinforced by statistically uniformly distributed particles show isotropic overall behavior, but there exist no simple periodic arrangements of particles that are both inherently elastically isotropic and space filling99) . In actual materials, which often show rather irregular particle shapes, anisotropy in the microgeometry and in the overall response may be introduced by processing effects, e.g. extrusion textures. Accordingly, unit cell studies of particle reinforced composites that use generic phase arrangements 99)

Even though pentagonal dodekahedra and icosahedra have the appropriate symmetry properties (Christensen,1987), they are not space filling.

– 76 –

are subject to nontrivial tradeoffs between keeping computational requirements at moderate levels (which favors simple particle shapes combined with two-dimensional or simple three-dimensional microgeometries) and obtaining sufficient level of realism for a given purpose (which often leads to requirements for unit cells containing a considerable number of particles of complex shape at random positions in three dimensions). In many respects periodic microfield analysis of particle reinforced composites is subject to similar constraints and use analogous approaches as work on short fiber reinforced composites.

1-part.cell 1-particle cell

2-particle cell

2-particle cell

2-particle cell

Reference Volume

1-particle cell

FIG.26: Simple cubic, face centered cubic, and body centered cubic arrangements of spheres with some simple unit cells (Weissenbek et al.,1994) Most three-dimensional unit cell studies of generic microgeometries for particle reinforced composites have been based on simple cubic (sc), face centered cubic (fcc) and body centered cubic (bcc) arrangements of spherical, cylindrical and cube-shaped reinforcements, see e.g. (Hom/McMeeking,1991). By invoking the symmetries of these arrangements and using symmetry as well as antisymmetry boundary conditions, relatively simple unit cells can be obtained for materials characterization100) , see e.g. (Weissenbek et al.,1994; Sanders/Gibson,2003) and compare figs. 26 and 27. In addition, work employing hexagonal or tetrakaidekahedral particle configurations, see e.g. (Rodin,1993), and Voronoi cells for cubic arrangements of particles (Li/Wongsto,2004), was reported.

FIG.27: Some unit cells for particle reinforced composites using cubic arrangements: s.c. arrangement of cubes, b.c.c. arrangement of cylinders and f.c.c. arrangement of spheres (Weissenbek et al.,1994) 100)

Again, larger unit cells with periodicity rather than symmetry boundary conditions are required for unrestricted modeling of the full thermomechanical response of cubic phase arrangements.

– 77 –

The overall elastic moduli obtained from cubic arrangements of particles101) do not necessarily fulfill the Hashin–Shtrikman bounds and they typically lie outside the three-point bounds, compare table 5. Of the unit cells shown in fig. 26, simple cubic models are the easiest to handle, but they show a more marked anisotropy than bcc and fcc ones. Recently, three-dimensional studies based on more complex phase arrangements have appeared in the literature. Gusev (1997) used Finite Element methods in combination with unit cells containing up to 64 randomly distributed particles to describe the overall behavior of elastic particle reinforced composites. Hexahedral unit cells containing up to 10 particles in a perturbed cubic configuration (Watt et al.,1966) as well as cube shaped cells incorporating 15 to 20 spherical particles at quasi-random positions (B¨ ohm et al.,1999, B¨ ohm/Han,2001), compare fig. 28, were proposed for studying elastoplastic particle reinforced MMCs and related materials. Three-dimensional simulations involving high numbers of particles for investigating composites with statistically uniform phase arrangements have been reported for models of elastic composites (Michel et al.,1999) and for studying brittle matrix composites that develop damage (Zohdi/Wriggers,2001). In addition composites with statistically nonuniform arrangements of spherical reinforcements were investigated via unit cells containing 7 clusters of 7 particles each (Segurado et al.,2003).

FIG.28: Unit cell for a particle reinforced MMC (ξ=0.2 nominal) containing 20 spherical particles in a quasi-random arrangement suitable for using periodicity B.C.s (B¨ ohm/Han,2001) 101)

Cubic symmetry is a special case of orthotropy with equal responses in all principal directions and requires three independent moduli for describing the elastic behavior. For a discussion of the elastic anisotropy of cubic materials see e.g. (Cazzani/Rovati,2003). The diffusion behavior of materials with cubic symmetry is isotropic.

– 78 –

In analogy to short fiber reinforced materials, axisymmetric cell models with staggered or non-staggered particles (compare fig. 21) can be used for materials characterization of particle reinforced composites. Such approaches have been a mainstay of PMA modeling of these materials, see e.g. (Bao et al.,1991; LLorca,1996). It is interesting to note that by appropriate choices of the cells’ aspect ratios axisymmetric cell models can be generated that in many ways correspond to simple, face centered, and body centered cubic arrangements (Weissenbek,1994). E∗ [GPa]

E ∗ [100] [GPa]

E ∗ [110] [GPa]

ν∗ []

α∗ [K−1 ×10−6]

SiC particles Al99.9 matrix HS/lo HS/hi 3PBm/lo 3PBm/hi MTM GSCS 2OEm sc fcc bcc axi/sc axi/fcc axi/bcc

429.0 67.2 90.8 114.6 91.4 93.9 90.8 91.5 91.8 — — — 95.4 88.1 87.9

— — — — — — — — — 96.4 89.0 90.0 — — —

— — — — — — — — — 90.5 91.7 90.6 — — —

0.17 0.35 0.286 0.340 0.323 0.328 0.329 0.327 0.327 — — — 0.318 0.342 0.352

4.30 23.0 16.8 18.6 18.5 18.6 18.6 18.6 18.6 18.7 18.6 18.6 18.6/18.6 18.1/19.0 18.4/18.8

3/D MPUC

92.4

—

—

0.326

2/D PST MPUC 2/D PSE MPUC

85.5 98.7

— —

— —

0.334 0.500

TABLE 5: Overall thermoelastic properties of a particle reinforced SiC/Al MMC (spherical particles, ξ=0.197) as predicted by the Hashin–Shtrikman (HS) and three-point (3PB) bounds, by the Mori–Tanaka method (MTM), by the generalized self-consistent scheme (GSCS), by Torquato’s second order estimates (2OE), as well as by unit cell analysis using three-dimensional cubic arrangements, axisymmetric cells, three-dimensional multi-particle models, and two-dimensional multi-particle models using plane stress (2/D PST) and plane strain (2/D PSE) states. Due to their relatively low computational requirements, planar unit cell models of particle reinforced materials can also be found in the literature. Generally, plane stress models (which actually describe thin “reinforced sheets”) show a more compliant and plane strain models (which correspond to reinforcement by aligned continuous out-of-plane fibers rather than particles) show a stiffer overall response than three-dimensional descriptions, compare table 5. With respect to modeling the elastoplastic overall behavior of particle reinforced materials, plane stress kinematics tend to give slightly better results than

– 79 –

plane strain descriptions, see e.g. (Weissenbek,1994). It must be stressed, however, that no two-dimensional model can reliably provide satisfactory results in terms of the predicted microstress and microstrain distributions102) , see table 5 and (B¨ ohm/Han,2001). Axisymmetric cell models typically provide considerably better results than planar ones. Accordingly, extreme care on the part of the analyst is required in using planar unit cell models for particle reinforced composites and quantitative agreement with experiments cannot be expected. Evidently, three-dimensional (or axisymmetric) models should be used for discontinuously reinforced materials wherever possible. In table 5 bounds, MFA results, and PMA predictions for the overall thermoelastic moduli of a particle reinforced SiC/Al MMC (elastic contrast c ≈ 6) are compared, the loading directions for the cubic arrangements being identified by Miller indices. It is interesting that — for the loading directions considered here, which do not span the full range of possible responses of the configurations — none of the results from the cubic arrangements falls within the three-point bounds for identical spherical inhomogeneities103) . The overall anisotropy of the simple cubic arrangement is marked, whereas the body centered and face centered arrays deviate much less from overall isotropy. The predictions of the axisymmetric models can be seen to be of comparable quality to those of the corresponding cubic arrays. The results given for the three-dimensional multi-particle models are ensemble averages over a number of unit cells and loading directions (compare the discussion in section 4.1), and they show very good agreement with the three-point bounds and second order estimates, whereas there are marked differences to the plane stress and plane strain models104) . For the composite studied here, predicted overall moduli evaluated from the individual multi-particle cells differ by about 2%, and due to the use of a periodic pseudo-random rather than a statistically homogeneous particle distribution, the Drugan–Willis estimates for the errors in the overall moduli are exceeded slightly. Table 6 lists predictions for the overall behavior of a particle reinforced glass-epoxy composite (elastic contrast c ≈ 23) at much higher volume fraction, ξ=0.389, where the three-point bounds are no longer particularly sharp. Both the three-point bounds and Torquato’s second order estimates show some sensitivity to the size distribution of the spherical reinforcements, monodisperse (m) and polydisperse (p) cases being compared. 102)

Plane stress configurations tend to predict much higher levels of equivalent plastic strains in the matrix and weaker overall hardening than do plane strain and generalized plane strain models using the same phase geometry. For a given particle volume fraction, results from threedimensional arrangements typically lie between those of the above planar models. Evidently, the configurations of regions of concentrated strains that underlie this behavior depend strongly on the geometrical constraints, compare also (Iung/Grange,1995; G¨ anser et al.,1998; B¨ ohm et al.,1999, Shen/Lissenden,2002). For continuously reinforced composites under transverse loading the equivalent plastic strains tend to concentrate in bands oriented at 45 ◦ to the loading directions, whereas the patterns are qualitatively different in particle reinforced materials. 103)

For cubic symmetries the extremal values of the Young’s modulus are found for loading in the [100] and [111] directions (Nye,1957). 104)

The comparisons given in tables 3 to 6 are strictly applicable only for the material parameters used there, but do show typical trends.

– 80 –

E∗ [GPa]

ν∗ []

G∗ [GPa]

K∗ [GPa]

α∗ [K ×10−6]

glass particles expoxy matrix

73.1 3.16

0.18 0.35

30.1 1.17

38.1 3.51

4.90 40.0

HS/lo HS/hi 3PB(m)/lo 3PB(m)/hi 3PB(p)/lo 3PB(p)/hi MTM GSCS 2OE(m) 2OE(p)

6.80 20.9 7.05 10.0 7.17 12.1 6.80 7.23 7.29 7.52

0.024 0.404 0.236 0.334 0.195 0.356 0.315 0.303 0.304 0.304

2.59 8.53 2.69 3.95 2.74 4.79 2.59 2.77 2.80 2.88

6.11 12.7 6.16 7.22 6.26 8.56 6.11 6.11 6.22 6.39

12.1 23.5 20.1 23.6 17.2 23.0 23.5 23.5 23.2 22.6

3/D MPUC(m) 3/D MPUC(p)

7.95 8.06

0.291 0.288

3.11 3.11

6.34 6.36

22.8 22.7

−1

TABLE 6: Overall elastic properties of a particle reinforced glass/epoxy composite (spherical particles, ξ=0.389) as predicted by the Hashin–Shtrikman (HS) and three-point (3PB) bounds, by the Mori–Tanaka method (MTM), by the generalized self-consistent scheme (GSCS), by Torquato’s second order estimates (2OE), as well as by multi-particle unit cells (MPUC) using monodisperse and polydisperse particle sizes. Again there is very good agreement between the analytical results and the predictions of multi-particle unit cells, which used 80 monodispersely or polydispersely sized quasirandomly positioned spherical particles. Both cells were generated and evaluated with the FE-based program PALMYRA. On closer inspection the moduli listed for the unit cells can be seen to deviate somewhat from the relationships pertaining to isotropic materials, because ensemble averaged moduli from slightly anisotropic cells are given. Like fiber reinforced MMCs, particle reinforced composites typically display highly inhomogeneous distributions of the microstresses and microstrains in the nonlinear range, compare fig. 29 (which shows the predicted equivalent accumulated plastic strains inside a multi-particle model of an MMC). This behavior tends to give rise to microscopic “structures” that can be considerably larger than individual particles, leading to longer ranged interactions between inhomogeneities105) . The latter, in turn, underlie the need for larger control volumes for studying composites with nonlinear matrix behavior mentioned in chapter 4. If the particles are highly irregular in shape or if their volume fraction markedly exceeds 0.5 (as is typically the case for cermets such as WC/Co), generic unit cell models 105)

Due to the filtering effect of periodic phase arrangements mentioned in section 5.1, the length of such features can exceed the size of the unit cell only in certain directions (parallel to the faces and diagonals of the cells), which may induce a spurious direction dependence into the predicted large-strain behavior of sub-RVE unit cells.

– 81 –

SECTION FRINGE PLOT EPS.EFF.PLASTIC INC.12 5.0000E-02 4.0000E-02 3.0000E-02 2.0000E-02 1.0000E-02

SCALAR MIN: 6.6202E-04 SCALAR MAX: 8.1576E-02

FIG.29: Predictions for the equivalent plastic strains in a particle reinforced MMC (ξ=0.2 nominally) subjected to uniaxial tensile loading obtained by a unit cell with 20 spherical particles in a quasi-random arrangement (B¨ ohm/Han,2001)

employing relatively regular particle shapes may not result in very satisfactory microgeometries. For such materials a useful approach consists of basing PMA models on modified real structures obtained from metallographic sections, compare e.g. (Fischmeister/Karlsson,1977). Due to the nature of the underlying experimental data, models of this type have often taken the form of planar analysis, the limitations of which are discussed above. This difficulty can be overcome e.g. by voxel-based discretization schemes using digitized serial sections, see (Terada/Kikuchi,1996) or microtomography data, see e.g. Kenesei et al. (2006). Alternatively, more or less irregular particles identified by such procedures may be approximated by equivalent ellipses or ellipsoids, compare (Li et al.,1999). 5.5 PMA Models for Some Other Inhomogeneous Materials In this section some additional applications of periodic microfield approaches are presented, which were selected mainly in view of discussing modeling concepts. The reader should be aware that it touches only upon a very small fraction of the research activities in which unit cell methods have been applied to the study of the thermomechanical properties of materials. Generic PMA Models for Clustered, Graded, and Interwoven Materials In some steels, the overall strength is limited by the presence of weak inhomogeneities which must, accordingly, be avoided. In others, among them high speed tool steels (HSS), carbidic particles are used to increase strength, hardness and wear resistance. Both of

– 82 –

the above types of steel may be modeled as matrix–inclusion-type composites, the inhomogeneities being dilutely dispersed in the former group. The carbide volume fraction in HSS typically is non-dilute and the phase arrangements range from statistically homogeneous to highly clustered or layered mesogeometries. Limitations of available computing power, however, have mostly restricted studies of clustered or layered configurations, in which a considerable number of particles have to be modeled, to two-dimensional descriptions. A rather straightforward, but nevertheless flexible strategy for setting up generic planar model geometries for such studies consists of tiling the computational plane with regular hexagonal cells which are assigned to one of the constituents by statistical or deterministic rules. If required, the shapes of the cells can be modified or randomly distorted and their sizes can be changed to some extent to adjust phase volume fractions. Such a Hexagonal Cell Tiling (HCT) concept can be used e.g. to generate unit cells for analyzing layer structured HSSs, compare (Plankensteiner et al.,1997) and see fig. 30 (left), as well as clustered or uniform arrangements of particles in a matrix106) .

FIG.30: HCT unit cell models for a layer structured high-speed steel containing equiaxed and elongated carbides (left; Plankensteiner et al.,1997) and for a functionally graded material (right). Similar HCT concepts can be used to obtain generic planar geometries for modeling functionally graded materials (FGMs), in which the phase volume fractions run through the full physically possible range (Weissenbek et al.,1997; Reiter et al.,1997), so that 106)

HCT and related models of composites with matrix–inclusion topology are, however, typically restricted to moderate reinforcement volume fractions and tend to be much more regular than actual materials. The direct 3D equivalent to HCT consists of filling space with cube-shaped, dodekahedral or tetrakaidekahedral cells, which are assigned phase properties in analogy to HCT. A related, but geometrically more flexible approach based on tetrahedral regions was reported by Galli et al.(2008).

– 83 –

there are regions that show matrix–inclusion microtopologies and others that do not, compare fig. 30 (right)107) . A further application of such models has been the study of the microstructure–property relationships in duplex steels, for which matrix–inclusion and interwoven (bi-connected) phase arrangements were compared108) , see (Siegmund et al.,1995; Silberschmidt/Werner,2001). An FFT-based equivalent to the HCT models was developed by Michel et al. (1999) explicitly for studying interwoven microgeometries. PMA Models for Polycrystals Most materials that are of technological interest are polycrystals and, accordingly, are inhomogeneous at the length scale of the crystallites. In the specific case of metals and metallic alloys phenomena associated with plastic flow and ductile failure can be investigated at a number of length scales, compare e.g. (McDowell,2000), some of which are amenable to study by continuum micromechanical methods, among them unit cell analysis. Micromechanical methods, especially analytical self-consistent approaches, have been applied to investigating the mechanical behavior of polycrystals since the 1950s. The more recent concept of studying polycrystals by periodic microfield approaches is quite straightforward in principle — a planar or three-dimensional unit cell is partitioned into subregions corresponding to grains the behavior of which is described by appropriate anisotropic material models and to which suitable material as well as orientation parameters are assigned109) . In practice the generation of realistic grain geometries may, however, require sophisticated algorithms. In addition, the use of crystal plasticity formulations for describing the material behavior of the individual grains tends to pose considerable demands on computational resources, especially for three-dimensional models, see e.g. (Quilici/Cailletaud,1999). Resolved microstructure models of polycrystals have, for example, successfully described the evolution of texture in metals during forming processes. For a general discussion of the issues involved in micromechanical models in crystal plasticity and related fields see e.g the overview by Dawson (2000). PMA Models for Two-Phase Single Crystal Superalloys Nickel-base single crystal superalloys, which consist of a γ matrix phase containing aligned cuboidal γ 0 precipitates, are of major technical importance due to their creep 107)

In contrast to the other inhomogeneous materials discussed in this report, FGMs and layer-structured HSSs are not statistically homogeneous. 108)

In HSSs and FGMs particulate inhomogeneities are present at least in part of the volume fraction range, so that planar models are less than ideal for them. Duplex steels, in contrast, typically show elongated aligned grains, for which generalized plane strain kinematics are satisfactory. Analogous considerations hold for polycrystalline aggregates consisting of aligned columnar grains, see e.g. (Chimani/M¨ orwald,1999). 109)

Symmetry boundaries passing through grains must be avoided in such models because they give rise to “twin-like” pairs of grains that are unphysical in most situations.

– 84 –

resistance at high temperatures. The most important field of application of superalloys is in the hot sections of gas turbines. Because these materials show relatively regular microstructures that change under load, a process known as rafting, there has been considerable interest in micromechanical modeling for elucidating their thermomechanical behavior and for better understanding their microstructural evolution. The elastic behavior of two-phase superalloys can be handled relatively easily by hexahedral unit cells with appropriately oriented anisotropic phases. In the inelastic range, however, the highly constrained plastic flow in the γ channels is difficult to describe even by crystal plasticity models, see e.g. (Nouailhas/Cailletaud,1996). PMA Models for Porous and Cellular Materials, Cancellous Bone, Wood Elastoplastic porous materials have been the subject of a considerable number of PMA studies due to their relevance to the ductile damage and failure behavior of metals110) . Generally, modeling concepts are closely related to those employed for particle reinforced composites, compare section 2.4, the main difference being that the shapes of the voids may evolve significantly through the loading history 111) . Accordingly, axisymmetric cells, see e.g. (Koplik/Needleman,1988; Pardoen/Hutchinson,2000; G˘ ar˘ ajeu et al.,2000) and three-dimensional unit cells based on cubic arrangements of voids, compare e.g. (McMeeking/Hom,1990; Segurado et al.,2002), have been used in the majority of published unit cell studies of porous materials. Recently, studies of void growth in ductile materials based on multi-void cells and using geometry data from serial sectioning were reported (Shan/Gokhale,2001). In cellular materials, such as foams, wood and cancellous bone, the volume fraction of the solid phase is low (often amounting to a few percent or less). The voids may be topologically connected (open cell foams), unconnected (closed cell foams, syntactic foams), or both connected and unconnected voids may be present (as in hollow sphere foams). The macroscopic tensile and compressive stress–strain responses of cellular materials usually are very different, with the latter typically showing three regimes. At small strains the materials display a finite stiffness, the linear elastic range often being very small. At higher stains gross shape changes of the cells take place, with bending, elastic 110)

Many models describing ductile damage in metals, see e.g. (Rice/Tracey,1969; Gurson,1977; Tvergaard/Needleman,1984; Gologanu et al.,1997; Kailasam et al.,2000; Benzerga/Besson,2001), are based on micromechanical studies of porous ductile materials and on the growth of (pre-existing) voids in elastoplastic matrices. 111)

Additional complexity is introduced by void size effects (Tvergaard,1996) and void coalescence (Faleskog/Shih,1997). The evolution of the shapes of initially spheroidal voids under non-hydrostatic loads has also been the subject of intensive studies by mean field type methods, see e.g. (Kailasam/Ponte Casta˜ neda,1998; Kailasam et al.,2000). Models of the latter type are typically based on the assumption that initially spherical pores will stay ellipsoids throughout the deformation history, which studies using axisymmetric cells (G˘ ar˘ ajeu et al.,2000) have shown to be an excellent approximation for axisymmetric tensile load cases. For compressive loading, however, initially spherical pores may evolve into markedly different shapes (Segurado et al.,2002).

– 85 –

buckling, plastic buckling, and brittle failure of cell walls or struts playing major roles on the microscale. On the macroscale this regime is characterized by a stress plateau, which gives rise to the typically excellent energy absorption properties of cellular materials. At even higher strains the effective stiffness rises sharply as the cellular structure is crushed so that cell walls or struts are in contact. No plateau region is present under tensile or shear loading. Periodic microfield methods are well suited to studying the thermomechanical behavior of cellular materials. The widely used results of Gibson and Ashby (1988), in which thermomechanical moduli and other overall physical properties of cellular materials are expressed as power laws in terms of the relative density, e.g., ρ ∗ n E∗ ∝ E (m) ρ(m)

,

(78)

were derived by analytically studying specific arrangements of beams (for open cell foams) and plates (for closed cell foams)112) . When the finite strain behavior of cellular materials is to be studied by periodic microfield methods, large deformations of cells and contact between cell walls/struts must be typically provided for113) . In addition, models must be sufficiently large so that appropriate nontrivial deformation patterns can develop 114) . The geometrically most simple cellular materials are regular honeycombs, which can be modeled by planar hexagonal cell models. Despite their geometrical simplicity regular honeycombs can show complex deformation patterns, the resolution of which requires the provision of sufficiently large unit cells, compare e.g. (Daxner et al.,2002). Furthermore, they tend to be subject to marked localization effects under compressive “in-plane” loading, see e.g. (Papka/Kyriakides,1994). Somewhat less ordered two-dimensional arrangements have been used for studying the crushing behavior of soft woods, see e.g. (Holmberg et al.,1999), and highly irregular planar arrangements, compare fig. 31, can be used to study aspects of the geometry dependence of the mechanical response of cellular materials such as metallic foams. For three-dimensional studies of closed cell foams various generic microstructures based on cubic cells, see e.g. (Hollister et al.,1991), truncated cubes plus small cubes (Santosa/Wierzbicki,1998), rhombic dodekahedra, regular tetrakaidekahedra (Grenestedt,1998; Simone/Gibson,1998; Ableidinger,2000), compare fig. 32, as well as Kelvin, 112)

The majority of these geometries can be extended to give proper unit cells that describe periodic phase arrangements. The behavior of the Gibson–Ashby models, however, is dominated by local bending. As a consequence, n=2 in eqn.(78) is obtained for open cell foams and the predicted overall moduli tend to be lower than those of many actual microgeometries. 113)

Note that this may imply providing for periodic contact.

114)

For perfectly regular structures such as hexagonal honeycombs the minimum size of a unit cell for capturing bifurcation effects can be obtained from extended homogenization theory (Saiki et al.,2002), and Bloch wave theory (Gong et al.,2005) may be used to study long wave buckling modes. Symmetry B.C.s must not be specified on cell walls that can be expected to bend or buckle.

– 86 –

Y

Z

X

FIG.31: Planar periodic unit cell for studying irregular cellular materials (Daxner et al.,2002).

Williams and Wearie–Phelan115) (Kraynik/Reinelt,1996; Bitsche,2005) geometries have been used. In models of this type the cell walls are typically described via thin shells. Analogous regular microgeometries have formed the basis for analytical and numerical models for open cell foams, see e.g. (Zhu et al.,1997; Shulmeister et al.,1998; Vajjhala et al.,2000), the struts connecting the nodes of the cellular structures being typically modeled as beams116) . In addition, tetrahedral arrangements (Sihn/Roy,2004) and cubic arrangements with additional “reinforcements” as well as perturbed strut configurations (Luxner et al.,2005) have been used for studying open cell microstructures. A recent development in the modeling of metallic and polymer foams have been unit cell models that employ voxel-type discretizations for open-cell and closed-cell foams 117) , which are based on three-dimensional microstructures of the actual materials obtained by tomographic methods, see e.g. (Maire et al.,2000; Roberts/Garboczi,2001). Alternatively, image processing methods my be used to generate a surface model of the solid phase in the volume element to allow meshing with “standard FE techniques, compare (Youssef et al.,2005). Because the microstructures extracted from the experiments are, in general, 115)

In Kelvin, Williams and Wearie–Phelan foams some of the cell walls are spatially distorted in order to minimize the total surface, whereas in polyhedral foams all cell walls are planar. These small distortions lead to noticeable differences in the overall elastic response at low solid volume fractions. 116)

For unit cell models employing structural finite elements such as shells and beams periodicity, symmetry and antisymmetry boundary conditions, eqns.(59) to (63), must be extended to also cover the rotational degrees of freedom of the boundary nodes, see e.g. (Bitsche,2005). 117)

Roberts and Garbozci (2001) estimated the “systematic discretization error” to be of the order of 10% in terms of the overall moduli for voxel-based models of elastic open-cell and closed-cell foams.

– 87 –

FIG.32: Microstructure of a closed cell foam modeled by regular tetrakaidekahedra (truncated octahedra). not periodic, windowing approaches using homogeneous strain boundary conditions are often preferred to unit cells in such models. A further group of cellular materials amenable to PMA modeling are syntactic foams (i.e. hollow spheres embedded in a solid matrix) and hollow sphere foams (in which the spaces between the spheres are “empty”), for which axisymmetric or three-dimensional cell models describing cubic arrangements of spheres can be used, see e.g. (Rammerstorfer/B¨ ohm,2000; Sanders/Gibson,2003). The effects of details of the microgeometries of cellular materials (e.g. thickness distributions and geometrical imperfections or flaws of cell walls or struts, the Plateau borders formed at the intersections of cell walls), which can considerably influence the overall behavior (a typical example being the very small elastic ranges of metallic foams), have been an active field of research, see e.g. (Grenestedt,1998; Daxner,2002). Even though unit cell analysis of cellular materials with realistic microgeometries tends to be rather complex and numerically demanding due to these materials’ tendency to deform by local mechanisms and instabilities118) , analytical solutions have been reported for the linear elastic behavior of some simply periodic phase arrangements, see e.g. (Warren/Kraynik,1991). A further type of cellular material, cancellous (or spongy) bone, has attracted considerable research interest for the past twenty years119) . Cancellous bone shows a wide range of microstructures, which can be idealized as beam or beam–plate configurations (Gibson,1985), and the solid phase again is an inhomogeneous material at a lower length 118)

Interestingly, instability phenomena in the cell walls of closed cell foams may even occur under uniaxial tensile macroscopic loading, see (Ableidinger,2000). 119)

Like many materials of biological origin, bone is an inhomogeneous material at a number of length scales. The solid phase of spongy bone is inhomogeneous and can also be studied with micromechanical methods.

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scale. In studying the mechanical behavior of cancellous bone, large three-dimensional unit cell models based on tomographic scans of actual samples and using voxel-based discretization schemes have become fairly widely used, see e.g. (Hollister et al.,1994; van Rietbergen et al.,1999). 5.6 PMA Models for Diffusion-Type Problems Periodic microfield methods analogous to those discussed in sections 5.1 to 5.5 can be used to study linear diffusion-type problems of the types mentioned in section 2.9, Laplace solvers (typically for use with heat conduction problems) being available with many FE packages. Specialized solvers are, however, required in some cases, e.g. when studying the frequency dependent dielectric properties of composites via complex potentials, see e.g. (Krakovsky/Myroshynchenko,2002). There are, however, intrinsic difficulties in employing periodic microfields for transport problems where nonlinear conductivity/resistivity behavior of the constituents is present. Whereas in solid mechanics, material nonlinearities are typically formulated terms of the “generalized intensity” or “generalized flux” fields, i.e. the microscopic strains and stresses, nonlinear resistivities or diffusivities in transport problems typically depend on the direct variable, e.g. the temperature in heat conduction, compare table 1. As is evident from fig. 14, in PMAs the generalized intensities and fluxes are periodic, whereas the direct variables consist of fluctuating plus linear contributions, so that they accumulate from cell to cell. As a consequence, in solid mechanics problems the position dependence of material properties is periodic also in the presence of nonlinearities, but this is not the case in transport problems. Accordingly, even though seemingly workable solutions, e.g. in terms of temperature fields, can be obtained from unit cells in which temperature dependent material properties are specified, the results do not correspond to proper periodic solutions. Periodic homogenization approaches should not be used in such cases, embedded cell models being more suitable for studying such problems.

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6. EMBEDDING APPROACHES AND RELATED METHODS Like periodic microfield methods, Embedded Cell Approaches (ECAs) aim at predicting the microfields in inhomogeneous materials at high spatial resolution. For this purpose they use models consisting of a geometrically fully resolved core (“local heterogeneous region”), which can range from relatively simple configurations to complex phase arrangements, that is embedded in an outer region for which a smeared out behavior is used and in which the phases are not (or at least not fully) resolved, compare fig. 33. This embedding region (“effective region”, “effective layer”) serves mainly for transmitting the applied loads into the core. The material description used for the outer region must be chosen such that it is consistent with the homogenized behavior and symmetry of the material described by the core, so that errors in the macrosocpic accommodation of stresses and strains are kept to a minimum.

CORE (discrete phase arrangement)

INTERFACE between core and embedding region

EMBEDDING REGION

FIG.33: Schematic depiction of the arrangement of core, embedding region, and interface in an embedded cell approach. The resulting modeling strategy avoids some of the drawbacks of PMAs, especially the requirement that the geometry and all microfields must be strictly periodic120) . An intrinsic feature of embedding models are “boundary layers” that occur at the “interfaces” between the core and the embedding region121) . and perturb the local stress and 120)

Embedding models can be used without intrinsic restrictions at or in the vicinity of free surfaces and interfaces, they can handle gradients in composition and loads, they can be used to study the interaction of macrocracks with microstructures, and they support temperature dependent conductivities in thermal analysis. The cores of embedding models do not necessarily have to be RVEs. 121)

The interfaces between core and embedding region obviously are a consequence of the modeling approach and do not have any physical significance. The resulting boundary layers typically have a thickness of, say, a particle diameter for elastic materials, but they may be longer ranged for nonlinear material behavior.

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strain fields. Provision must be made to keep regions of specific interest, such as crack tips and process zones, away from the core’s boundaries122) . Although some analytical methods such as classical and generalized self-consistent schemes, see section 2.3, may be viewed as embedding schemes, most ECAs with more complex core microgeometries have used numerical engineering methods in order to resolve discrete microstructures in the core region. Three basic types of embedding approaches can be found in the literature. One of them uses discrete phase arrangements in both the core region and in the surrounding material, the latter, however, being discretized much more coarsely, see e.g. (Sautter et al.,1993). Mesh superposition techniques, which use a coarse mesh over the macroscopic model of some structure or sample together with a geometrically independent, much finer mesh in regions of interest (Takano et al.,2001) are conceptually similar to the above modeling strategy. If appropriately graded meshes are used models of the above types are not subject to marked boundary layers between core and embedding regions. In the second group of embedding methods the behavior of the outer regions is described via appropriate smeared-out material models that suitably approximate the macroscopic behavior of the inhomogeneous material. These typically take the form of semiempirical or micromechanically based constitutive laws that are prescribed a priori for the embedding zone. Such constitutive laws and the material parameters used with them must be be chosen to correspond in a sufficiently close way to the homogenized behavior of the core123) . This way, conceptually simple models are obtained that are very well suited for studying local phenomena such as the stress and strain distributions in the vicinity of crack tips or at macroscopic interfaces in composites (Chimani et al.,1997), the growth of cracks in inhomogeneous materials (Wulf et al.,1996; Ableidinger,2000), or damage due to the processing of composites (Monaghan/Brazil,1998). The third type of embedding scheme uses the homogenized thermomechanical response of the core to determine the effective behavior of the surrounding medium, giving rise to models of the self-consistent type, which are mainly employed for materials characterization. The use of such approaches is predicated on the availability of suitable parameterizable constitutive laws for the embedding material that show the appropriate 122)

When crack tips are to be modeled they obviously must not be allowed to progress into the boundary layers. Similarly, the boundary layers and embedding regions may interfere with extended regions of concentrated strains, with shear bands, and with the progress of damage. In transient analysis the boundary layers will in general lead to reflection and/or refraction of waves. Special transition layers can, however, be introduced to mitigate boundary layer effects in the latter case. 123)

Care is required in modeling the behavior of elastoplastic inhomogeneous materials with continuous reinforcements via two-dimensional models, because the homogenized response of the material (and thus, of the core) is anisotropic and, consequently, cannot be well described by von Mises plasticity. It is worth noting that by restricting damage effects to the core of a model and prescribing the behavior of the damage-free composite to the embedding material, the volume fractions of damaged regions can in principle be controlled in ECA models, compare e.g. (Bao,1992).

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material symmetry and that can follow the core’s instantaneous homogenized behavior with sufficient accuracy for all load cases and for any loading history. This requirement typically can be fulfilled easily in the linear range (see e.g. Chen et al.,1994), but leads to considerable difficulties when at least one of the constituents shows elastoplastic or viscoplastic material behavior124) . Accordingly, approximations have to be used (the consequences of which may be difficult to assess in view of the nonlinearity and path dependence of elastoplastic material behavior)125) , and models of this type are best termed “quasi-self-consistent schemes”. Such approaches are discussed e.g. in (Bornert et al.,1994; Dong/Schmauder,1996). When a multi-particle unit cell is used as the core in a quasi-self-consistent embedding scheme, the relaxation of the periodicity constraints tends to make the overall responses of the embedded configuration softer than that of the same periodic arrangement in a PMA (Bruzzi et al.,2001)126).

2 3

1

FIG.34: Embedded cell model of a CT specimen for studying crack progress in an open cell metallic foam (Ableidinger,2000). 124)

Typically, the effective yielding behavior of the (inhomogeneous) core shows some dependence on the first stress invariant, and, for low plastic strains, the homogenized response of the core tends to be strongly influenced by the fraction(s) of the elastoplastic constituent(s) that have actually yielded. In addition, in many cases anisotropy of the yielding and hardening responses is introduced by the phase arrangement of the core (e.g. aligned fibers). At present, there appears to be no constitutive law that, on the one hand, can fully account for these phenomena (compare e.g. Dvorak,1991) and, on the other hand, has the capability of being adapted to the instantaneous response of the core by adjusting appropriate free parameters. 125)

In case a uniaxial stress–strain diagram is adapted to the core’s behavior and multiaxial stresses are handled via standard metal plasticity relations (e.g. using von Mises or Hill-type plasticity), such approximations are implicitly invoked. 126)

In the study mentioned above, the difference between PMA and ECA results is relatively small. Essentially identical responses can be expected from the two types of model if the phase arrangement is a proper RVE and the embedding material can fully describe the homogenized behavior of the core.

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For materials characterization embedded cells can be loaded by homogeneous stresses or strains applied to the outer boundaries. If problems involving macrocracks (or related questions) are to be studied, however, it is often preferable to impose displacement boundary conditions corresponding to the far field behavior of suitable analytical solutions (e.g. for the displacement field around a crack tip). Embedded cell models also allow complete samples to be considered in “simulated experiments”, see e.g. fig. 34, which shows an embedding model consisting of one half of a CT specimen made of an open cell foam (Ableidinger,2000). In it, the embedding region is described as a homogenized elastic continuum, in the core region a highly idealized beam model is used for the open cell foam, and the crack is assumed to run along the symmetry plane. The core and embedding regions may be planar, axisymmetric or fully three-dimensional, and symmetries present in the geometries can, as usual, be employed to reduce the size of the model, compare (Peters,1997). Effective and phase averaged stresses and strains from embedded cell models are best evaluated from eqn.(73) or its equivalents, and in the case of geometrically complex phase arrangements it is good practice to use only the central regions of the core for this purpose in order to avoid boundary layers. A modeling approach that is closely related to embedding models uses coupling conditions between a macromodel that typically employs smeared-out material data with a geometrically fully resolved micromodel. Such submodeling techniques may, on the one hand, be implemented “off-line” such that the macromodel is solved for first and the micromodel is studied in a separate analysis, the macromodel being only used to provide suitable boundary conditions, see e.g. (Heness et al.,1999; V´ aradi et al.,1999). On the other hand, geometrically fully resolved regions may be activated automatically in regions where the conditions for periodic homogenization fail to be met (e.g. near crack tips), see e.g. Ghosh et al. (2001), where loads are introduced into the core regions via transition elements. Like unit cell models, embedding and related approaches approaches are also well suited for handling diffusion-type problems.

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7. WINDOWING APPROACHES A further group of methods for studying microstructures that were obtained experimentally or were computer generated to follow some given phase distribution statistics is subsumed here as “windowing approaches”. These methods are based on extracting simply shaped mesoscopic regions (“test windows”) at random positions (and with random orientations) from an inhomogeneous material, compare fig. 35, and subjecting them to nonperiodic boundary conditions that are guaranteed to fulfill the macrohomogeneity condition, eqn.(23).

FIG.35: Schematic representation of a windowing approach — four square windows of different size obtained from a material consisting of a matrix reinforced by aligned continuous fibers are shown. By using the Gauss theorem and assuming small strain conditions as well as perfect interfaces between the constituents the macrohomogeneity condition can be formulated in terms of a surface integral as Z

Ωs

t(x) − hσinΓ (x)

u(x) − hεix dΓ = 0

(79)

(Hazanov,1998). This condition trivially holds when homogeneous stress (“uniform static”, “uniform traction”, “uniform natural”) or homogeneous strain (“uniform kinematic”, “uniform displacement”, “uniform essential”) boundary conditions are used, both of which set one of the vectors in the integrand of eqn.(79) identically to zero. In addition, eqn.(79) can be fulfilled by mixed uniform boundary conditions that make the two vectors orthogonal127) and by periodic B.C.s, in which the contributions of coupled 127)

Mixed uniform boundary conditions can be strictly realized only in materials that have orthotropic or higher symmetry (Hazanov,1998) and must be applied in the appropriate material coordinate system.

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pairs of faces cancel out. In general, homogeneous strain and mixed uniform boundary conditions are more straightforward to apply than periodic B.C.s because there are no dependences between phase geometries or discretizations on “opposite” faces, and the simpler constraint equations can make for reduced computational requirements. Prescribing uniform homogeneous tangential tractions, however, is not supported by most preprocessors. Windowing approaches using homogeneous strain and traction boundary conditions allow to obtain estimates for and rigorous bounds on the overall responses of inhomogeneous materials on the basis of randomly extracted volume elements that may be too small to be proper RVEs128) . In fact, windowing methods typically aim at describing the apparent responses of inhomogeneous materials when it is known that the available volume elements are too small to be proper RVEs. When “non-pathological” SRVEs are subjected to homogeneous stress and homogeneous strain boundary conditions, respectively, nontrivial lower and upper estimates are obtained, compare e.g. (Nemat-Nasser/Hori,1993), and ensemble averages of such estimates provide lower and upper bounds on the overall elastic (and diffusion-type129) ) behavior of the material. It was shown by Hazanov and Huet (1994) that for elastic material behavior estimates obtained with mixed uniform boundary conditions are bounded by the lower and upper estimates generated with macrouniform boundary conditions for the same volume element, and Ostoja-Starzewski (2006) gave order relations for apparent elastic moduli obtained with the above types of B.C.s. Suquet (1987) proved that elastic tensors obtained with periodicity boundary conditions always lie between results generated with uniform traction and uniform strain boundary conditions. Pahr and Zysset (2008) proposed a set of mixed uniform boundary conditions that leads to equal predictions as do periodicity B.C.s for periodic volume elements with phase arrangements of orthotropic or higher symmetry; these B.C.s are referred to as “periodicity compatible mixed uniform boundary conditions”. By definition, for proper RVEs the lower and upper estimates and bounds as well as estimates obtained with mixed uniform and periodic boundary conditions must coincide130) , (Hill,1963), compare also section 1.2. The closeness of the upper and lower bounds ob128)

Such “sub-RVE” volume elements are also referred to as Test Volume Elements (TVEs) or Statistical Reference Volume Elements (SRVEs). Bounds obtained from such volume elements are known as mesoscale bounds. 129)

For diffusion-type problems homogeneous normal flux and homogeneous gradient boundary conditions must be applied, see e.g. (Jiang et al.,2002; Kanit et al.,2006). These give rise to lower and upper estimates and bounds, respectively, of the effective conduction or diffusion tensors. 130)

Note that, when a given volume element is to be checked as to how closely it approaches being an RVE, either the complete apparent/effective tensors must be evaluated or different loading conditions (e.g. load cases with different macroscopic stress triaxialities or different load orientations) must be sampled. Equal lower and upper mesoscale bounds are, however, difficult to obtain in practice, the convergence of the two bounds being especially slow for softer inclusions in a stiffer matrix.

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tained for the macroscopic properties can be used to check the suitability of volume elements of a given size for describing the effective (as opposed to apparent) mechanical response of an inhomogeneous material, see e.g. (Ostoja-Starzewski,1998). Hierarchies of bounds for a given microstructure can be generated by using series of windows of increasing size, which allows to assess the dependence of the predicted overall moduli on the size of the windows. This concept of hierarchical bounds obtained from samples that are smaller than RVEs was verified by planar (Amieur,1994) and three-dimensional (Guidoum,1994) modeling schemes and by experimental studies (Huet,1999). Volume elements of increasing size subjected to periodicity or symmetry boundary conditions have been reported to give rise to hierarchies of estimates, see e.g. (Sautter et al.,1993; Terada et al.,2000). Recently windowing approaches giving rise to hierarchical bounds have been extended into the nonlinear range (Jiang et al.,2001, Jiang et al.,2002), where the bounding properties of the two types of homogeneous boundary conditions can be proven within the context of nonlinear elasticity (and thus also hold for the deformation theory of plasticity). No rigorous proofs, however, appear to be available for incremental plasticity. A discussion of windowing bounds in a finite strain elasticity setting is given in Khisaeva/Ostoja-Starzewski (2006). Windowing approaches employing mixed uniform boundary conditions can be used for generating estimates of the macroscopic behavior of elastoplastic inhomogeneous materials. The restrictred load paths that can be followed by such an approach, however, allow little more than materials characterization to be carried out. For relatively small windows (having a ratio between inhomogeneity diameter and window size of, say, 5) different boundary conditions, especially uniform kinematic and uniform static B.C.s, can give rise to marked differences in the distributions of the microfields, especially the plastic strains. When larger windows are used, however, increasingly similar statistical distributions of the microfields are obtained (Shen/Brinson,2006). Again, improved predictions of the overall properties can be obtained by ensemble averaging over a number predictions obtained from small volume elements with the same B.C.s. A conceptually different class of windowing models were reported by Soppa et al.(2003), who subjected rectangular test windows to experimentally determined boundary displacements. The main interest in such approaches lies in correlating microstrain fields obtained by experiments and by modelling. Windowing concepts have also been used to determine statistical material property fields, compare e.g. (Ostoja-Starzewski,1993), and to studying the statistics of the apparent material properties of inhomogeneous materials obtained from moving-window mesoscale TVEs, see e.g. (Graham/Baxter,2001).

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8. MULTI-SCALE MODELS The micromechanical methods discussed in chapters 2 to 7 are each designed to handle a single scale transition between a lower and a higher length scale (“microscale” and “macroscale”), overall responses being obtained by homogenization and local fields by localization. Many inhomogeneous materials, however, show more than two clearly distinct characteristic length scales, typical examples being laminated composites, woven composites reinforced by tows containing many thin fibers, materials in which there are well defined clusters of particles, as well as many biomaterials. In such cases an obvious modeling strategy is a hierarchical (or multi-scale) approach that uses a sequence of scale transitions, i.e. the material response at any given length scale is described on the basis of the homogenized behavior of the next lower one131) . Figure 36 schematically shows such a hierarchical model for a particle reinforced composite with a clustered mesostructure.

scale transition #2

scale transition #1

MACROSCALE (sample)

MESOSCALE (particle clusters)

MICROSCALE (particles in matrix)

FIG.36: Schematic representation of a multi-scale approach for studying a material consisting of clustered particles in a matrix. Two scale transitions, macro←→meso and meso←→micro, are used. From a technical point of view, homogenization and/or localization of the thermomechanical responses of an inhomogeneous material to link a given pair of length scales is equivalent to finding an energetically equivalent homogeneous material by micromechanical approaches as discussed in chapters 2 and 5 to 7. A multi-scale model can be viewed as a sequence of such scale transitions and, accordingly, all of the above methods, i.e. mean field, unit cell, embedding, and some windowing approaches, may be used as 131)

Describing the material behavior at lower length scales by a homogenized model implies that characteristic lengths differ by, say, an order of magnitude or more, so that valid volumes can be defined for homogenization. It is not technically correct to employ hierarchical approaches within “bands” of more or less continuous distributions of length scales, as can be found e.g. in some metallic foams with highly disperse cell sizes.

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“building blocks” at any level within hierarchical schemes132) . Such multi-scale modeling strategies have the additional advantage of allowing the behavior of the constituents at all lower length scales to be assessed via the corresponding localization relations 133) . Because local stresses and strains at the lower length scales can differ markedly from the applied macroscopic conditions in both magnitude and orientation, it is advisable that the homogenized descriptions of the thermomechanical responses at all length scales (with the possible exception of the top one) take the form of proper constitutive models that are capable of handling any loading conditions and any loading history. In most cases, this requirement can be readily fulfilled by micromechanically based constitutive models that employ mean field methods or periodic microfield approaches. Among the continuum mechanical multi-scale descriptions of the thermomechanical behavior of inhomogeneous materials reported in the literature, some combine mean field methods at the higher length scale with mean field (Hu et al.,1998; Tszeng,1998) or periodic microfield approaches (Gonz´ alez/LLorca,2000) at the lower length scale. The most common strategy for multi-scale modeling, however, uses Finite Element based unit cell or embedding methods at the topmost length scale, which implies that the homogenized material models describing the lower level(s) of the hierarchy must take the form of micromechanically based constitutive laws that can be evaluated at each integration point. This requirement typically does not give rise to problematic computational workloads in the elastic range, where the superposition principle can be used to obtain the full homogenized elasticity and thermal expansion tensors from a limited number of load cases applied to an appropriate unit cell134) . For simulating the thermomechanical response of nonlinear inhomogeneous materials, however, essentially a full micromechanical submodel has to be maintained and solved at each integration point in order to account for the history dependence of the local responses135) . Even though within such a framework the use of sophisticated unit cell based models at the lower length scales usually is a very expen132)

Micromechanical methods that by construction describe inhomogeneous materials with a wide range of length scales, e.g. differential schemes, are, however, of limited suitability for multi-scale analysis. 133)

It should be noted, however, that generating predictions for all relevant fields at two or more length scales can give rise to very large amounts of data, especially if models based on numerical discretizing methods are used at one or more of the length scales. 134)

In the elastic range domain decomposition techniques can be used to formulate the problems at the lower length scales in such a way that they are well suited for parallel processing, allowing the development of computationally highly efficient multi-scale procedures, see (Oden/Zohdi,1997). 135)

Some approaches have been reported that aim at partially “decoupling” models at the higher and lower length scales by parameterizing results from unit cell analyses and using the stored data together with some interpolation scheme as an “approximate constitutive model”, see e.g. (Terada/Kikuchi,1996; G¨ anser,1998; Schrefler et al.,1999; Ghosh et al.,2001). The main difficulty in such modeling strategies lies in handling load history and load path dependences to account for elastoplastic constituents or microscopic damage to constituents. Appropriate representation of general multiaxial stress and/or strain states also tends to be a challenge.

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Eps_eff,p_(m) 3.5000E-03 3.0000E-03 2.5000E-03 2.0000E-03 1.5000E-03

SCALAR MIN: -6.8514E-04 SCALAR MAX: 1.3667E-02

FIG.37: Phase averaged microscopic equivalent plastic strains in the matrix within the particle-poor regions of a cluster-structured high speed steel under mechanical loading as predicted by a mesoscopic unit cell model combined with an incremental Mori–Tanaka model at the microscale (Plankensteiner,2000). sive proposition in terms of computational requirements, work of this type was reported e.g. by (Wu et al.,1989), and asymptotic homogenization has been used successfully in this context for two-dimensional descriptions (Ghosh et al.,1996). Lower (but by no means negligible) computational costs can be achieved by using mean field models in lieu of constitutive models at the integration point level. Applications for elastoplastic composites136) have included incremental Mori–Tanaka methods, see e.g. (Pettermann,1997), mean field based versions of the Transformation Field Analysis, see e.g. (Fish/Shek,1998; Plankensteiner,2000), and the nonuniform TFA (Michel/Suquet,2004). It is also possible to handle the “lowest” scale transition by a combination of a mean field or TFA-type homogenization procedure and unit-cell based localization algorithms, the latter being used only in regions of special interest (Fish et al.,1997). Figure 37 shows a result obtained by applying a relatively straightforward multi-scale model to the materials characterization of a cluster-structured high speed steel. At the mesoscale, a unit cell approach is used in which particle-rich clusters and particle-poor regions are described as separate phases. Each of the latter is treated as a particle reinforced MMC with appropriate reinforcement volume fraction at the microscale, an incremental Mori–Tanaka model being employed for homogenization. Such an approach 136)

Multi-scale approaches using micromechanical models at the integration point level may lead to spurious stress and strain fields close to interior boundaries at the higher length scale, e.g. at interfaces between reinforced and monolithic regions in selectively reinforced components, because in such regions the conditions for standard homogenization methods typically are not fulfilled locally, compare e.g. (Chimani et al.,1997).

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not only predicts macroscopic stress–strain relationships, but also allows the distributions of phase averaged microscopic variables to be evaluated, compare (Plankensteiner et al.,1998)137). Alternatively, multiscale approaches may be rather “loosely coupled”, essentially linking together quite different models to obtain an overall result, see e.g. (Onck/van der Giessen,1997). For a number of years there has been a surge of research interest in developing and applying algorithms for the multi-scale and hierarchical continuum modeling of inhomogeneous materials, see e.g. (Lee/Ghosh,1996; Zohdi et al.,1996; Fish/Shek,2000; Feyel/Chaboche,2000; Ibrahimbegović/Markoviˇc,2003; Moës et al.,2003). A recent development are multi-scale methods in which the computational domain is adaptively split into regions resolved at appropriate length scale. In an FE-based framework developed Ghosh et al. (2001) “non-critical” regions are described by continuum elements with homogenized material properties, and at zones of high macroscopic stress and strain gradients PMA models are automatically activated at the integration points to monitor the phase behavior at the microscale. If violations of the homogenization conditions (e.g. onset of damage, mesoscopic interfaces, free edges) are detected, embedded models of the fully resolved microstructure are activated at the appropriate positions. This strategy allows detailed studies of critical regions in inhomogeneous materials, e.g. near free edges. In an alternative approach, homogenization models employing generalized continua have been proposed for handling regions close to free surfaces and zones with high local field gradients (Feyel,2003). Finally, it should be noted that multi-scale approaches are not limited to using the “standard” methods of continuum micromechanics as implicitly assumed above, see e.g. the discussions on metal plasticity in (Tadmor et al.,2000). Especially the capability of the Finite Element method of handling highly complex constitutive descriptions has been used to build multi-scale descriptions that employ, for example, material models based on atomistics (Shenoy et al.,1998). Evidently, the distinction between micromechanical studies using sophisticated constitutive models for the phases on the one hand and multiscale modeling on the other hand as used here is to some extent a matter of definition (and personal preferences).

137)

When assessing results from multi-scale analysis using mean field methods at the smallest length scale, such as fig. 37, it must be kept in mind that the maximum stresses and strains shown are the maxima of the local phase averaged values and not maximum microscopic stresses as obtained by “standard” micromechanical unit cell studies.

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9. CLOSING REMARKS The research field of continuum micromechanics of materials has enjoyed considerable success in the past four decades in furthering the understanding of the thermomechanical behavior of inhomogeneous materials and in providing predictive tools for engineers and materials scientists. However, its methods are subject to some practical limitations that should be kept in mind when employing them. All the methods discussed in sections 2 to 7 implicitly use the assumption that the constituents of the inhomogeneous material to be studied can be treated as homogeneous at the lower length scale, which, of course, does not necessarily hold. When the length scale of the inhomogeneities in a constituent is much smaller than the length scale of the phase arrangement to be studied, the above assumption is valid and multi-scale models (“successive homogenization”) as discussed in chapter 8 can be used, see e.g. Carrere et al. (2004). However, no rigorous theory appears to be available at present for handling scale transitions in materials that do not fulfill the above requirement (e.g. particle reinforced composites in which the grain size of the matrix is comparable to or even larger than the size of the particles138) ). Typically the best that can be done in such cases is either to use sufficiently large models with resolved microgeometry (a computationally very expensive proposition) or to employ homogenized phase properties and be aware of the approximation that is introduced139) . A major practical difficulty in the use of continuum micromechanics approaches (which in many cases is closely related to the questions mentioned above) has been identifying appropriate constitutive models and obtaining material parameters for the constituents. Typically, available data pertain to the behavior of the bulk materials (as measured from homogeneous macroscopic samples), whereas the actual requirement is for parameters and, in some cases, constitutive theories that describe the in-situ response of the constituents at the microscale. For example, with respect to MMCs, on the one hand there is considerable evidence that classical continuum plasticity theories (in which there is no absolute length scale) cannot adequately describe the behavior of metallic materials in the presence of strong strain gradients, where geometrically necessary dislocations can markedly influence the material behavior, see e.g. (Hutchinson,2000). On the other hand, it is well known that the presence of reinforcements can lead to accelerated aging and refinement of the grain size in the matrix (“secondary strengthening”). Furthermore, in many cases reasonably accurate material parameters are essentially not available (e.g. strength parameters for most interfaces in inhomogeneous materials). In fact, the dearth of dependable material parameters is one of the reasons why predictions of the strength of inhomogeneous materials by micromechanical methods tend to be a considerable challenge. 138)

For this special case unit cell or embedding models employing anisotropic or crystal plasticity models for the phases may be used which, however, can become very large when the statistics of grain orientation are accounted for. 139)

It is worth noting that bounds evaluated using such an ad-hoc procedure are no longer rigorous — such results should be treated as lower and upper estimates.

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Another point that should be kept in mind is that continuum micromechanical descriptions in most cases do not have absolute length scales unless a length scale is introduced, typically via the constitutive model(s) of one (or more) of the constituents140) . In this vein, absolute length scales, on the one hand, may be provided explicitly via discrete dislocation models (Cleveringa et al.,1997), via gradient or nonlocal constitutive laws, see e.g. (Tomita et al.,2000; Niordson/Tvergaard,2002), or via (nonlocal) damage models. On the other hand, they may be introduced in an ad-hoc fashion, e.g. by adjusting the phase material parameters to account for grain sizes via the Hall–Petch effect. Analysts should also be aware that absolute length scales may be introduced inadvertently into a model by mesh dependence effects of discretizing numerical methods, a “classical” example being strain localization due to softening141) material behavior of a constituent, compare also the discussions in the Appendix. When multi-scale models are used special care may be necessary to avoid introducing inconsistent length scales at different modeling levels. In addition it is worth noting that usually the macroscopic response of inhomogeneous materials is much less sensitive to the phase arrangement (and to modeling approximations) than are the distributions of the microfields. Consequently, whereas good agreement in the overall behavior of a given model with “benchmark” theoretical results or experimental data typically indicates that the phase averages of stresses and strains are described satisfactorily, this does not necessarily imply that the higher statistical moments of the stress and strain distributions (and thus the heterogeneity of these fields) are captured correctly. It is important to be aware that work in the field of micromechanics of materials invariably involves finding viable compromises in terms of the complexity of the models, which, on the one hand, have to be able to account (at least approximatively) for the physical phenomena relevant to the given problem, and, on the other hand, must be sufficiently simple to allow solutions to be obtained within the relevant constraints of time, cost, and computational resources. Obviously, actual problems cannot be solved without recourse to various approximations and tradeoffs — the important point is to be aware of them and to account for them in interpreting and assessing results. It is worth keeping in mind that there is no such thing as a “best micromechanical approach” for all applications and that models, while indispensable in understanding the behavior of inhomogeneous materials, are “just” models and do not reflect the full complexities of real materials.

140)

One important exception are models for studying macroscopic cracks (e.g. via embedded cells), where the crack length does introduce an absolute length scale. Note the behavior of nanocomposites the behavior typically is “interface dominated”. Appropriately accounting for this behavior may require the introduction of an absolute length scale. Within the framework of mean field methods this may be achieved by introducing surface terms into the Eshelby, see e.g. Duan et al. (2005). 141)

Note that here the expression “localization” is used to describe a physical phenomenon in materials and not a procedure in micromechanical modeling strategy as in section 1.2.

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APPENDIX: SOME ASPECTS OF MODELING DAMAGE AT THE MICROLEVEL Modeling efforts aimed at studying the damage and failure behavior of inhomogeneous materials may take the form of microscopic or macroscopic approaches. Methods belonging to the latter group often are based to some extent on micromechanical concepts and encompass, among others, the “classical” failure surfaces for continuously reinforced laminae such as the Tsai–Wu criterion (Tsai/Wu,1971) and macroscopic continuum damage models of various degrees of complexity, see e.g. (Chaboche et al.,1998; Voyiadjis/Kattan,1999). Also on the macroscopic level the Considére criterion may be used in combination with micromechanical models to check for (macroscopic) plastic instability when studying the ductility of inhomogeneous elastoplastic materials, see e.g. (Gonz´ alez/LLorca,1996). In what follows, however, only the modeling of damage on the microscale in the context of (mainly Finite Element based) micromechanical methods of the types discussed in chapters 4 to 8 are considered. General Remarks When damage and failure of inhomogeneous materials are to be studied at the microscale, it is physically intuitive to represent them as acting at the level of the constituents and of the interfaces between them, where appropriate damage and fracture models have to be provided. All of the resulting microscopic failure mechanisms contribute to the energy absorption during crack growth. Taking MMCs as an example, this concept translates into viewing damage and failure as involving a mutual interaction and competition between ductile damage of the matrix, brittle cleavage of the reinforcements, and decohesion of the interfaces, whereas in polycrystals intergranular or intragranular failure are typical damage mechanisms. The dominant microscopic damage process is determined by the individual constituents’ strength and stiffness parameters, by the microgeometry, and by the loading conditions, compare Needleman et al. (1993). In fact, it may be markedly influenced by details of the microgeometry, such as particle shape and size, see e.g. Miserez et al. (2004), a behavior which strongly supports the concept of using micromechanics for damage modeling. Generally, macrocracks as well as microcracks that have lengths comparable to the microscale of a given phase arrangement must be resolved explicitly in micromechanical studies142) . Smaller cracks and voids, however, can be treated in a homogenized manner as damage. For describing cracks in micromechanical models “standard” FE models of cracks may be used; for an overview covering some of the relevant methods see e.g. (Liebowitz et al.,1995). “Standard” criteria for the initiation and/or growth of cracks, such as J -integral, CTOD or CTOA criteria, etc., can also be used. If only information on the initiation of damage is required it may be sufficient to evaluate appropriate “damage relevant fields”, such as the maximum principal stress in brittle 142)

Crack-like features that develop due to the failure of contiguous sets of elements in a damage-type model fulfill this requirement.

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materials, a ductile damage indicator, or the traction vectors at interfaces, without accounting for the loss of stiffness that accompanies the evolution of damage. For the use of a ductile damage indicator in this context see e.g. (Gunawardena et al.,1993; B¨ ohm/Rammerstorfer,1996). Alternatively, it may be of interest to study microstructures containing flaws (e.g. fiber breaks, matrix flaws) in order to assess the resulting microfields, see e.g. (Tsotsis/Weitsman,1990), where J -integrals were evaluated for specific matrix crack configurations. If, however, the progress of damage and the interaction of damage modes at the constituent level as well as final failure are to be studied, two major modeling strategies are available at present. They are, on the one hand, the “volume oriented” damage-based approaches used in continuum damage mechanics and related models and, on the other hand, “surface oriented” methods that aim at following the progress of individual cracks. The remainder of the present section is devoted to a short discussion of some aspects of these two groups of methods. When constituent damage at the microscale is modeled by continuum damage mechanics (CDM) and related approaches (lumped together as CDM-type models in what follows), the implicit assumption is made that the characteristic length scale of inhomogeneities underlying the damage (pores, microcracks, etc.) is much smaller than the characteristic length scale of phase arrangement to be studied. Essentially such models amount to multi-scale methods as discussed in chapter 8. This group of approaches includes ductile rupture models for describing ductile damage, see e.g. (Gurson,1977; Rousselier,1987), and damage indicator triggered (DDIT) techniques, in which the stiffness of an integration point is drastically reduced or the corresponding element is removed when some critical value of an appropriate damage indicator is reached. CDM-type methods can account for the stiffness degradation due to the evolution of damage, which tends to give rise to localization and may lead to the formation of proper shear bands within the microscopic phase arrangement. A numerical difficulty typically encountered with CDM-type models lies in their tendency to provide solutions that show a marked mesh dependence143) . The characteristic size of the localized or failed region is typically found to be comparable to the dimensions of one element, regardless of the actual degree of mesh refinement, and the predicted energies for crack formation accordingly tend to decrease with decreasing mesh size, compare fig. 38. This behavior can be avoided or at least mitigated by using theories that are suitably regularized by enrichment with higher order gradients, see e.g. (Geers et al.,2001), by nonlocal averaging of the rate of an appropriate damage variable, compare (Tvergaard/Needleman,1995; Jir´ asek/Rolshoven,2002), or by the use of time dependent formulations involving rate effects (e.g. viscoelasticity, viscoplasticity), compare (Needleman,1987). All of the above procedures explicitly introduce an absolute length scale 143)

Within classical continuum theory strain softening material behavior leads to ill-posed boundary value problems, i.e. the governing equations lose their ellipticity or hyperbolicity, giving rise to the mesh sensitivity of the solutions, compare e.g. (Baaser/Gross,2002). In order to provide an objective description of localization, the problem has to be regularized.

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into the problem144) A rather ad-hoc alternative to the above regularization methods consists in selecting a mesh size that leads to physically reasonable sizes of localization features145) .

force-ratio RF2 /RF2,max

1.2

≈ 2.500 el. ≈ 6.000 el. ≈ 21.000 el.

1 0.8 0.6 increasing number of elements

0.4 0.2 0

0

0.2 0.4 0.6 0.8 displacement-ratio U2 /U2,max

1

FIG.38: Normalized overall force–displacement responses obtained for a column with a hole under transverse loading with a local Gurson–Tvergaard-Needleman ductile rupture model (Drabek,2006). A marked mesh dependence of the results is evident. Surface-oriented numerical strategies for simulating the progress of cracks emphasize the concept that the nucleation and growth of cracks correspond to the creation of new free surface. In most such models crack growth involves breaking the connectivity between neighboring elements, e.g. by node release techniques (Bakuckas et al.,1993; Narasimhan,1994) or by cohesive surface elements, compare e.g. (Elices et al.,2002). Due to the restriction of the growth of cracks to element boundaries, in models of this type a mesh dependence of the predicted crack paths tends to arise unless the latter are known a priori. This difficulty can be counteracted by using very fine and irregular discretizations or by introducing remeshing schemes that locally adapt the element geometry for each crack growth increment, compare e.g. (Fish et al.,1999; Bouchard et al.,2003) and thus provide realistic crack paths. Both of these remedies tend to lead to complex and/or large models, so that in micromechanics of materials crack opening approaches have been reported mainly for interfacial cracks and for phase arrangements where the crack paths stay can be assumed to follow some plane of symmetry. The mesh dependence of the crack paths predicted by surface oriented approaches can be removed by using specially formulated solid elements, known as embedded discontinuity 144)

A precondition for the proper regularization by higher order gradients or by nonlocal averaging is that the discretization must be finer than than this absolute length scale. 145)

Local damage procedures may be interpreted as introducing a field of characteristic lengths via the local element sizes. Material parameters identified on the basis of such procedures pertain to a given element size only.

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elements, which employ appropriately enriched shape functions. Such elements, on the one hand, must allow discontinuities to develop at any position and in any direction within each element and, on the other hand, must provide for continuous crack paths across element boundaries, see e.g. (Jir´ asek,2000; Alfaiate et al.,2002). The actual opening of cracks is typically handled by formulations of the cohesive surface type. Embedded discontinuity elements at present are in a developmental stage, are typically not available in standard FE packages, and up till have been little used in the context of continuum micromechanics. Modeling of Reinforcement Damage and Failure Particulate and fibrous reinforcements in composites typically fail by brittle cleavage. Within the framework of micromechanical models using the FEM, the failure of such phases can be modeled by following the extension of a crack through the the reinforcement, which implies that complex crack paths may have to be resolved, or by employing descriptions in which the crack opens instantaneously along some predefined “crack plane”. There are two groups of approaches for controlling the activation of cracks that are modeled as opening instantaneously. When deterministic (Rankine) criteria are used, a particle or fiber is induced to fail once a critical level of some stress or strain measure is reached or some fracture probability is exceeded. Studies using normal and maximum principal stress criteria, respectively, within such a framework were published e.g. by Berns et al. (1998) and Ghosh/Moorthy (1998). In “fully probabilistic” models a random process (Monte-Carlo-type procedure) is provided which “decides” if a given reinforcement will fail at a given load level on the basis of appropriate fracture probabilities, compare e.g. (Pandorf,2001; Eckschlager,2002). Within micromechanical descriptions such fracture probabilities can be evaluated from the predicted distributions of the microstresses by Weibull-type weakest-link models146) that were originally developed for describing the failure statistics of bulk brittle materials. A simple Weibull expression for evaluating the failure probability Pn of a given particle n may take the form (Eckschlager,2002) (

1 Pn = 1 − exp − V0

Z

σ (x) m 1 dΩ σ0 Ωn :σ1 >0

)

,

(80)

where σ1 (x) is the microscopic field of maximum principal stresses in the inhomogeneity and Ω:σ1 >0 is that part of the particle volume in which the maximum principal stress 146)

Whereas there is agreement that the weakest-link behavior of long fibers is largely due to surface defects, the physical interpretation is not as clear for Weibull models of particles embedded in a ductile matrix. Weibull models based on eqn.(80) or similar expressions, however, have a number of advantages for the latter class of applications, compare (Wallin et al.,1987), not the least of which is their capability for describing particle size effects. It should be noted, however, that simple Weibull-type models were reported to give predictions for the dependence of the number of fracture events per volume on particle size and volume fraction that do not agree with acoustic emission data from ductile matrix composites (Mummery et al.,1993).

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is tensile. Equation (80) involves three material parameters, viz. σ0 , which stands for the particle’s average strength, m, which is known as the Weibull modulus, and V0 , a control volume. Expressions such as eqn.(80) assign a failure probability to a particle or short fiber as a whole147) , which fits well with the concept of “instantaneous cleavage” of particles. Typically procedures of this type predict the onset of damage at rather low applied loads, with some of the reinforcements failing at low fracture probabilities. Because each simulation run provides a different realization of the underlying statistics, the results from fully probabilistic models for a given microgeometry are best evaluated in terms of ensemble averages over a number of anlyses. In models of discontinuously reinforced composites that use axisymmetric cells, a natural choice for the position of a predefined crack is a symmetry plane that is oriented normal to the axis of rotation (which coincides with the loading direction) and runs through the center of the inhomogeneity, see e.g. (Bao,1992; Davis et al.,1998; Gonz´ alez/LLorca,2000). For more general arrangements of particles or short fibers the most realistic approach would be to consider the stress state of each particle in selecting the crack plane 148) , so that perturbations due to (possibly failed) neighboring particles can be accounted for. Due to their high computational requirements up till now such algorithms have only been reported for planar arrangements of ellipsoidal reinforcements, the crack being taken to pass through the position of maximum principal stress in the particle and to be oriented normally to the local maximum principal stress at this point (Ghosh et al.,2001). For three-dimensional particle arrangements a simplified concept for approximating the orientations and positions of cracks within spherical particles can be based on the assumption that they pass through the particle center and are oriented normal to the average maximum principal stress in the inhomogeneity or to the macroscopic maximum principal stress (Eckschlager et al.,2001). Tomography data obtained on a model system consisting of aluminum reinforced by spheres of zirconia/silica at ξ=0.04 (Babout et al.,2004), however, show cracks that are oriented normally to the macroscopic maximum principcal stress but do not tend to run through the centers of particles. The alternatives to the above approach of instantaneous particle cracking consist in using either cohesive surface models for opening predefined cracks in reinforcements, see e.g. (Finot et al.,1994; Tvergaard,1994), or in applying continuum damage models to describing failure of the reinforcements, see e.g. (Zohdi/Wriggers,2001; Mishnaevsky,2004). Such models are capable of resolving the growth sequence of a crack rather than “popping it open” instantaneously. Whereas predefined crack paths in the reinforcements have typ147)

In modeling the fracture of continuous fibers data on the positions of transverse cracks are obviously required. Such information can be obtained by evaluating Weibull fracture probabilities for cylindrical subvolumes of the fibers. 148)

For small monocrystalline particulates the crystallographic orientations can also be expected to play a role in determining the crack plane, thus introducing a statistical element into the problem. In particles of irregular shape local stress concentrations must be expected to play a role in determining the positions of cracks and for elongated particles or fibers the appearance of multiple cracks must be provided for. For experimental results on the orientation of cracks in particles see e.g. (Mawsouf,2000).

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ically been used with cohesive surfaces, no such provision is required in the CDM-type models, which can describe geometrically complex cracks149) . Both approaches, however, do not introduce effects of the absolute size of the reinforcements into micromechanical models. Modeling of Matrix Failure The type of matrix failure found in composites obviously depends on the matrix material and thus on the composite system considered. In the case of metallic matrices ductile damage and failure must be considered, which either takes place by void nucleation, growth, and coalescence (void impingement) or by the formation of void sheets, compare e.g. (Horstemeyer et al.,2000). The former mechanism dominates at high stress triaxialities σm /σeq , whereas the latter one leads to shear failure in the low triaxiality regime. Over the past two decades the dominant approach for modeling the failure of ductile matrices in metal matrix composites have been descriptions based on the ductile rupture model of Gurson (1977) and its improved versions (Tvergaard/Needleman,1984; G˘ ar˘ ajeu,1995). Models of this type are implemented in a number of commercial Finite Element codes150) . For micromechanical studies that employ Gurson-type models for describing ductile failure of the matrix of MMCs see e.g. Needleman et al. (1993), Geni and Kikuchi (1999), and Hu et al. (2007). A version of the ductile damage model of Rousselier (1987) was used in micromechanical models by Drabek and B¨ ohm (2004). In the context of micromechanics element elimination models triggered by a ductile damage indicator (“DDIT models”) have been used as an alternative to “ductile rupture” models of the above type151) . The damage indicator most commonly used for such purposes was proposed by Gunawardena et al. (1991) on the basis of earlier studies on void growth (Rice/Tracey,1969; Hancock/Mackenzie,1976) and takes the form Did =

εZeq,p 0

exp

3 2 σm /σeq

1.65 εf

dεeq,p

,

(81)

149)

From the physical point of view, continuum damage models are most appropriate in situations where a process zone, e.g. due to microcracks, is present. Because particle cleavage tends to produce “clean” cracks, it should preferably be handed by surface based models. 150)

Gurson models are are based on the behavior of spherical voids and are best suited for handling ductile damage in situations of high stress triaxaiality. A Gurson-type model that allows for evolution of the void shapes was developed by Gologanu et al.(1997), and Nahshon and Hutchinson (2008) extended a Gurson-type model to the low-triaxiality situation of shear failure. 151)

In contrast to the physically interpretable damage variables used in ductile rupture and standard CDM models, which allow the effect of damage on all material properties to be handled consistently, only two values of the damage parameter Did defined in eqn.(81) have a direct physical interpretation: 0 indicates no damage and 1 ductile failure. Accordingly, ductile damage indicators per se cannot follow the degradation of stiffness as damage develops.

– 108 –

where εeq,p stands for the accumulated equivalent plastic strain, σm for the mean (hydrostatic) stress, σeq for the equivalent (von Mises) stress, and εf for the uniaxial failure strain of the material152) . Like most ductile rupture models the damage indicator, eqn.(18), was derived for the high stress triaxiality regime153) , which is the more relevant one for micromechanical analysis of composites on account of the pronounced tendency towards highly constrained plasticity. Recently ductile damage indicators have been developed that can also account for local shear failure, see e.g. (Wilkins,1999). DDIT-based element elimination schemes in the context of continuum micromechanics of inhomogeneous materials were reported e.g. in (Schmauder et al.,1996; Berns et al.,1998). Other criteria used for controlling element elimination in ductile matrices have included strain energy criteria (Mahishi/Adams,1982), the so called s-criterion (Spiegler et al.,1990), plastic strain criteria (Adams,1974), and stress triaxiality criteria (Steinkopff et al.,1995). Wulf et al. (1996) reported comparisons between predictions obtained by planar embedded cell models using element elimination models controlled by stress triaxiality, plastic strain, and Rice–Tracey criteria, the latter giving the best agreement with experimental results for that special case. The Finite Element code ABAQUS provides a set of hybrid schemes for modeling ductile and shear damage in metals, which combine a damage indicator-type concept for detecting “damage initiation”154) with phenomenological CDM-like algorithms for describing the evolution of damage, leading finally to element elimination. The use of nonlocal versions of ductile rupture models and DDIT schemes, which allow the mesh dependence of the solutions to be reduced or eliminated, was first proposed for micromechanical applications by Tvergaard and Needleman (1995). These methods have seen further development recently155) see e.g. (Hu et al.,2004; Drabek/B¨ ohm,2004; Drabek,2006). 152)

The main advantage of this ductile damage indicator lies in requiring only one material parameter, εf , which can be evaluated experimentally in a relatively straightforward way, compare the discussion in Fischer et al. (1995). Alternative ductile damage indicators were developed e.g. by McClintock (1978) and Oyane (1972). 153)

For a discussion of the validity of eqn.(81) in the low to negative triaxiality regime, where the damage mechanism is shear failure rather than void growth and coalescence, see (G¨ anser et al., 2001). 154)

In this context the expression “damage initation” is not used in the sense of metals physics, but rather signifies a state where damage effects on the behavior of some material volume become noticeable. 155)

As mentioned before, all elements subject to damage must be smaller than the characteristic length in nonlocal averaging. The average mesh size, however, should not be too much smaller than the characteristic length, either, because otherwise the high number of neighbors contributing to the nonlocal averages will make the algorithms too slow. These restrictions can considerably constrain the applicability of nonlocal averaging methods. Note that at present there is no agreement whether the characteristic length in ductile damage is a physical material property or a numerical modeling parameter. If, however, the characteristic length is assumed to be a physical parameter determined by void nucleation from small defects it must be inversely proportional to the third root of the defects’ volume density (Rousselier,1987).

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Alternatively, cohesive surface models may be used for modeling ductile failure, especially in cases where the crack path is known a priori, a typical example being ductile cracks propagating along a plane of symmetry, see e.g. Finot et al. (1994). A possible drawback of cohesive surface models for studying ductile failure in a continuum micromechanics framework, however, has been revealed by recent studies, which indicate that the decohesion parameters tend to depend on the stress triaxialities, see e.g. (Chen eta al.,2001). Approaches for modeling matrix failure that do not fall into the above groups were reported, among others, by Biner (1996), who studied creep damage to the matrix of an MMC using Tvergaard’s (1984) grain boundary cavitation model, and by Mahishi (1986), who combined a node release technique with an energy release criterion to study cracks in polymer matrices156) . Composites with brittle elastic matrix behavior can be investigated via fracture mechanics based methods see e.g. (Ismar/Reinert,1997), or Weibull-type criteria may be employed (Brockm¨ uller et al.,1995). Finally, it may be mentioned that in the case of cyclic loading shakedown theorems may be used on the microscale to assess if elastic shakedown may take place in an inhomogeneous material or if low cycle fatigue will act on the microscale, see (B¨ ohm,1993; Schwabe,2000). Modeling of Interface Failure Probably the most commonly used description for the debonding of a reinforcement from a matrix in numerical micromechanics has been a cohesive surface model developed by Needleman (1987), It is based on an interface potential that specifies the dependence of the normal and tangential interfacial tractions on the normal and tangential displacement differences across the interface and was extended by Tvergaard (1990) to handle the modeling of tangential separation. Interfacial behavior of this type is typically described in FE codes via special interface elements, which in many cases are also capable of handling contact and friction between reinforcement and matrix once interfacial decohesion has occurred. Some examples of micromechanical studies employing algorithms of the Needleman–Tvergaard type are (Nutt/Needleman,1987; Needleman et al.,1993; Haddadi/Teodosiu,1999). A modified version of the above interfacial decohesion model was developed by Michel and Suquet (1994), and another related technique is reported in (Zhong/Knauss,2000). An alternative approach suitable for describing very weakly bonded reinforcements consists in using standard contact elements at the interface, the composite being held together in a “shrink fit” by the thermal stresses caused by cooling down from the processing temperature, see e.g. (Gunawardena et al.,1993; Sherwood/Quimby,1995). As an extension of this type of description, Coulomb-type models, i.e. a linkage between normal and shear stresses analogous to that employed for dry friction, may be used for the interface (Benabou et al.,2002). 156)

Because crack paths in inhomogeneous materials may be rather complicated, appropriate criteria for determining the crack direction have to be employed and remeshing typically is necessary when node release techniques are used, compare (Vejen/Pyrz,2002).

– 110 –

Other interfacial decohesion models reported in the literature include strain softening interfacial springs (Vosbeek et al.,1993; Li/Wisnom,1994), a node release technique based on fracture mechanics solutions for bimaterial criterion (Moshev/Kozhevnikova,2002). Interphases, i.e. interfacial layers of finite thickness, were studied by various fictitious crack techniques, see e.g. (Ismar/Schmitt,1991; McHugh/Connolly,1994; Walter et al.,1997) and thermomechanically based continuum damage models (Benabou et al.,2002). A rather different approach has been developed specifically for accounting for the stiffness loss due to debonding within mean field models. It is based on describing debonded reinforcements by perfectly bonded (transversally isotropic) “fictitious inhomogeneities” of appropriately reduced stiffness. Within such a framework, the fraction of debonded reinforcements may be controlled via Weibull-type models, see e.g. (Zhao/Weng,1995; Sun et al.,2003). Models Involving Multiple Failure Modes Whereas an appreciable number of the works mentioned above involve two interacting micromechanical failure modes, only few attempts to concurrently model matrix, reinforcement and interfacial failure have been published. For two-dimensional studies of this type describing ductile matrix composites see e.g. (Berns et al.,1998; Pandorf,2000), where generic particle arrangements in tool steels are compared with respect to their fracture properties, and (Hu et al.,2007), where particle reinforced metal matrix composites are investigated.

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