Advanced Materials Research Vol. 59 (2009) pp 3-17 online at http://www.scientific.net © (2009) Trans Tech Publications, Switzerland Online available since 2008/Dec/15
Two-Scale Finite Element Modelling of Microstructures Wolfgang Brocks1,a, Alfred Cornec1,b and D. Steglich1,c 1
GKSS Research Centre, Institute of Materials Research, Materials Mechanics, Geesthacht, Germany a
[email protected], b
[email protected],
[email protected]
Keywords: multiscale modelling, micromechanical mode twinning, hcp metals, magnesium, titanium aluminides
lling, crystal plasticity, slip systems,
Abstract. Modelling the constitutive behaviour of metallic materials based on their m icrostructural features and the m icromechanical m echanisms in the fram ework of continuum m echanics is addressed. Deform ation at the lengthscale of gr ains is described by crystal plasticity. The macroscopic behaviour is obtained either by a homogenisation process yielding phenom enological equations or by a subm odel technique. The m odelling processes for two light-weight m aterials, namely magnesium and titanium aluminides are presented. Introduction Phenomenological models for the inelastic deform ation of fcc and bcc m etals are successfully used to describe any kind of irreversible response to thermo-mechanical loading by evolution equations for internal variables like accum ulated plastic strain or back stresses. New structural m aterials, composite m aterials or m aterials under extrem e loading conditions require m ore sophisticated modelling approaches. The m acroscopic phenom ena of deform ation and dam age evolution are caused by m echanisms at the m icroscale, and the m icrostructural f eatures determ ine the macroscopic properties of the m aterial. Understanding and m odelling the m icromechanisms does not only result in im proved constitutive equations at the m acroscale but also helps with the design of optimised microstructures and hence optim ised materials. This is the m otivation for multiscale approaches in m odelling the constitutive behaviour of materials, which have becom e increasingly popular and helpful, not least because of the trem endous increase in com puter power and software capabilities within the last years. A m ost general m ultiscale approach would have to start at the atom ic or m olecular scale and consider the dynam ics of dislocations. This w ould go beyond the scope of the present contribution as it had to address the problem s of relating a pa rticle based and a continuum based approach. The present contribution hence restricts to continuum mechanics and its num erical representation in the framework of the finite elem ent method (FEM). The common procedure consists of sim ulating the behaviour by representative volum e elem ents (RVE ), which characterise the m icrostructure, and establishing a m acroscopic phenom enological m odel based on the overall (averaged) m echanical response of the RVE. Two typical m aterials w ill be addressed as exam ples for a two-scale modelling approach of the deform ation, nam ely m agnesium and γ-titanium alum inides ( γ-TiAl). The presentation has to restrict to the principl e m odelling steps without going into all details, of course. Computational Materials Modelling Multiscale Modelling. Computational modelling of materials behaviour has become a reliable tool in m aterials science to com plement theoretical and experim ental approaches. The structure of matter is of dual nature depending on the length and time scales at which we look at it. It appears continuous when viewed at large length scales and discrete when viewed at an atom istic scale, and respective m odels exist f or both. Multiscale m odelling approaches spanning the bridge f rom All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 141.4.208.26-13/01/09,15:04:05)
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atomistic to continuum analyses are required if an understanding of this dual nature is beneficial for the developm ent and proper design of m aterials, devices and structures under extrem e operating conditions. Respective applications range from m icrometer-sized devices in electronics to large structural components containing the plasma in a fusion reactor. Multiscale m odelling relies on a system atic reduction of the degrees of f reedom on the various natural length scales that can be identif ied in a m aterial from the atom ic dim ensions to the grain size of a polycrystal. A seamless coupling between the various time and length scales is still beyond the capabilities of present m odelling approaches, however. A detailed and sound review of the concepts, theories and their links from computational quantum mechanics and ab initio methods to constitutive m odelling in continuum m echanics, a com prehensive list of related literature and a discussion of the challenges and outstanding i ssues can be found in the overview article by Ghoniem et al. [1]. The present contribution can onl y cover a small section of the full bandwidth of material length scales and restricts to models in the framework of continuum mechanics. Continuum Mechanics and Micromechanical Modelling. Continuum m echanics is a phenomenological field theory, which considers m atter as a dense m anifold of m aterial particles filling the three-dim ensional Euclidean space wit hout any gaps or discontinuities. The field quantities, which are introduced to describe the deform ation under the action of forces and temperature changes, are hence assum ed as continuous and differentiable. The governing equations comprise material independent principles and m aterial specific equations. Deform ation kinematics, namely the compatibility of strain fields or strain-displacem ent relations, and balance equations for mass, linear and angular m omentum or, in the case of static problem s, equilibrium conditions for the stresses are general principles of continuum m echanics. Constitutive equations establish a functional relation between stresses and strains and account for the specific properties of a certain material. Finally, boundary and initial conditions ha ve to be form ulated for the loading of a particular structure. All together, the partial dif ferential and algebraic equations establish an initial boundary value problem , which is nonlinear, in general, and can only be solved num erically. Solutions are obtained by the so-called finite element (FE) method, which is based on the weak form of equilibrium, namely the principle of virtual work, written in the rate form here, − ∫ S ⋅ ⋅δ ( ∇v ) dV + V
∫ t ⋅ δ v dA + ∫ f ⋅ δ v dV = 0 , (1)
∂V
V
where S is Cauchy’s stress tensor, t the vector field of stre sses acting on the body surface, f the vector field of volum e forces like grav itation, m agnetic or electric forces, and δv any virtual velocity field. The classical phenom enological continuum mechanics deals with idealised m aterials in which the material properties as well as stresses and stra ins within an infinitesim al neighbourhood of any material point can be regarded as unif orm. This justifies the constitutive theory of simple materials, where a local relation between the two f ield quan tities is assum ed, i.e. the state of stress in a material point is supposed to be uniquely determ ined by the deform ation history of this point only. The boundary-value problem and its num erical representation by an FE m odel do not include any material length scale. W hether the structural dim ensions are m icrometers or m eters does not affect the solution. As already m entioned, matter is discretely structur ed and any m aterial element consists of various constituents with differing properties and shapes, wh ich means that it has its own and, in general, evolving m icrostructure. For describing the m echanical behaviour of m aterials with strong local gradients the classical approach of sim ple materials is obviously not sufficient. A second, phenomenological m otivation to account for m aterial length scales results from num erous size effects that have been experim entally observed. The particle size of com posites or m etals containing second phase inclusions affects thei r fracture toughness, the strength of thin wires depends on their diameter and the measured microhardness increases with decreasing indenter size. One has to be careful with the term inology wh en talking about m aterial length-scales. The heterogeneity of the m icrostructure and the size e ffects mentioned above are related but have to be
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discriminated, nevertheless. Non-local constitutiv e theories, m ostly based on strain-gradient approaches [2], have been developed in order to capture size effects. The m aterial length scales, however, which enter via the strain gradients are mostly still phenom enological quantities without any microstructural equivalent. The latter can only be achieved by an approach, which considers the actual m icrostructure of a m aterial and esta blishes m acroscopic constitutive equations by a homogenisation procedure. On the other hand, constitutive equations that are m icromechanically based are m ostly form ulated as local m odels on the m acroscale, i.e. assum ing a sim ple m aterial. This has been done for the examples given below. Damage models, which exhibit a strain-softening behaviour of the m aterial, require an intrin sic length scale, which m ay or m ay not be micromechanically m otivated. Otherwise, they s how a pathological m esh dependence as soon as localisation of deformation and damage occurs [3, 4]. Micromechanics has been established as an expandi ng field of research in m aterials science and quantities associated with a constitutive m odeling [5]. It is a technique to express continuum material elem ent in term s of the param eters which characterise the m icrostructure and the microconstituents of this elem ent. Micromechanical modeling takes place on a meso level between a full representation of the microstructure and a phenomenological approach. It aims at • Describing the evolution of the microstructure during processing and service conditions; • Developing constitutive equations representing the mechanical behaviour of the m aterial on a homogenised mesoscale level; • Realising the general schem e of interdependence between the m icrostructure of a m aterial, the deformation and degradation phenom ena on th e m icroscale and the overall (m acroscopic) strength and toughness properties. As in every m odelling process, the generally com plex m icrostructure of a m aterial has to be idealised. To this end, the concept of a represen tative volume element (RVE), which is considered as statistically representative of the m aterial neighbourhood of any m aterial point, has been introduced and successf ully utilised. An RVE m ay contain several grains, dif ferent phases, inclusions and m icro defects like voids or cracks a nd thus defines a length scale, which is specific for the m icrostructure of a m aterial. In order to be representative, it should include a large num ber of such micro-heterogeneities. However, the number of involved parameters will seriously interfere with a system atic treatm ent of the problem . An operational definition regards an RVE as the smallest m aterial unit cell that contains reasonably sufficient information about the characteristic phenomena of the constitutive behaviour [ 6]. A com mon sim plification is to assum e a periodic microstructure, which fulfils the compatibility condition, see Fig. 1.
Figure 1: Unit cell of a material containing particles or voids assuming a periodic microstructure The homogenisation process starts with the definition of mesoscopic stresses and strains or strain rates by averaging the respective microscopic field quantities over the cell volume,
S=
1 1 S(x) dV = ∫ VV V
∫ xt dA
∂V
1 1 D = ∫ 12 ( ∇v + v∇ ) dV = VV V
∫ ( nv + vn ) dA
∂V
, (2)
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where t are the stress vectors acting on the cell surface, D is the m esoscopic stretching rate tensor and n are the surface norm al vectors. Based on these quantities, constitutive equations representing the average structural behaviour of the cell can be established on the m esoscale. Hom ogenised constitutive equations have to f ulfil the principle of m acro hom ogeneity f or the strain energy density, 1 1 S ⋅ ⋅∇v dV = ∫ t ⋅ v dA , (3) ∫ VV V ∂V resulting f rom the Hill-Mandel lem ma [ 7]. Any vi olation of this principle will result in homogenisation errors. Homogenisation methods require not only constitutiv e equations for the individual constituents but also appropriate rules to m ake the transition be tween scales. Som e of the m ost widely used approaches to link local fields with m acroscale phenom ena are Taylor type, assum ing uniform strains and fulfilling com patibility at the boundaries but not equilibrium , or Sachs-type, assum ing homogeneous stresses but violating com patibility. In self-consistent averaging approaches, the interaction between a cell and its neighbours is treat ed as that between an inclusion em bedded in a homogeneous equivalent m edium, which has th e sam e, however unknown, properties as the macroscopic aggregate [8]. Porous Metal Plasticity. The m ost prom inent and successf ul exam ple of m icromechanical modelling is the Gurson m odel [9]. The plasti c potential has been derived by a hom ogenisation process for a unit cell containing a spherical void of volume fraction f, see Fig. 1, S ⋅ ⋅D =
⎛ tr S ⎞ 3S ⋅ ⋅S 2 + 2 f cosh ⎜ ⎟ − 1 − f = 0 , (4) 2σ 0 ⎝ 2σ 0 ⎠ where σ0 denote the yield strength of the m atrix material surrounding the void. Together with an evolution law for void growth derived from the principle of mass conservation,
Φ ( S, f ) =
f = (1 − f ) tr Dp , (5) and som e em pirical m odifications introduced by Tv ergaard and Needlem an [10, 11] it is widely used to sim ulate damage evolution and failure of ductile m etals. As discussed above, it is m ostly formulated in the framework of a local theory, which results in a pathological mesh dependence [4]. The required internal length, which lim its localisation and perm its the local dissipation rate to remain finite, depending on the average spacing of inclusions, is com monly introduced via the height of the finite elem ents in the crack li gament, see discussion in [12], however nonlocal formulations of the Gurson model have been proposed [13]. The Gurson m odel is based on the classical von Mises theory of plasticity assum ing an isotropic material, i.e. the deform ation behaviour of a pol ycrystal f or the m atrix m aterial of the RVE. Investigating the deformation of single grains requires a model appropriate for single crystals. Crystal Plasticity. The kinematical theory for the mechanics of crystals has been established in the pioneering work of Taylor [ 14] and the theory by Hill [ 15], Rice [ 16] and Hill and Rice [ 17]. The model of crystal plasticity used here em ploys the framework of Peirce et al. [18] and Asaro [19, 20]. The crystalline m aterial undergoes elastic st retching, rotation and plastic deform ation. The latter is assum ed to arise solely f rom crystalline slip. The total def ormation gradient F is decomposed as F = F* ⋅ F p , (6) where F p denotes plastic shear of the m aterial to an intermediate reference configuration in which lattice orientation and spacing are the same as in the initial configuration, and F* denotes stretching and rotation of the lattice, see Fig. 2.
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m*( α )
F* ⋅ F p
F*
n*( α )
m( α )
Fp n( α )
7
m( α ) n( α )
Figure 2: Kinematics of crystal plasticity The rate of change of F p is related to the slip rate γ (α ) of the α slip system by F p ⋅ F p −1 = ∑ γ (α ) m (α ) n (α ) , (7) α
where the sum ranges over all activated slips system s, and the unit vectors m (α ) , n (α ) are the slip direction and the slip-plane norm al, respectivel y. The elastic properties are assum ed to be unaffected by slip, i.e. the stress is determ ined solely by F*. The crystalline slip is assumed to obey Schmid's law, i.e. the slipping rate γ (α ) depends on the stress tensor S solely through Schm id's resolved shear stresses,
τ (α ) = ( ρ 0 ρ ) n*(α ) ⋅ S ⋅ m*(α ) , (8) where ρ0 and ρ are the m ass densities in the ref erence and current states. According to Peirce et al. [16], the constitutive equation of slip is assumed as a viscoplastic power law, n
⎛ τ (α ) ⎞ γ (α ) τ (α ) sign = ⎜ (α ) ⎟ , (9) γ0(α ) τ Y(α ) ⎝τY ⎠ (α ) where γ0 is a reference strain rate, τ Y(α ) characterises the current strength of the α slip system, and
n is the rate sensitivity exponent. Strain hardening is characterised by the evolution of the strengths
τY(α ) = ∑ hαβ ( γ ) γ (α ) , (10) β
with hαβ being the self ( α = β) and latent ( α ≠ β) hardening m oduli depending on Taylor's cumulative shear strain on all slip systems, t
γ = ∑ ∫ γ (α ) dτ . (11) α
0
The implementation in the commercial finite element code ABAQUS i s based on t he user-material routine of Huang [21]. Extensions to hexagona l lattices have been im plemented by W erwer & Cornec [22] and Graff et al. [23]. Deformation of hcp Metals Motivation and Significance. Magnesium alloys have attracted attention in recent years as lightweight m aterials for the transportation i ndustry. A broad application requires reliable simulation tools for predicting the form ing capabilities, the structural behaviour under m echanical loads and the lifetim e of the com ponent. The respective constitutive m odels have to account for
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several anomalies of the mechanical behaviour. Rolled or extruded Mg products show a pronounced strength differential effect at low hom ologous tem peratures: the yield stress in tension is m uch higher than in compression. Furthermore, Mg exhibits a low ductility as well as strong deform ation anisotropy. Both phenom ena dem and for non stat e-of-the-art sim ulation techniques. The mechanical anom alies originate from its m icrostructural def ormation m echanisms, w hich are determined by its hexagonal close-packed (hcp) crystallographic structure. Metals with hcp crystalline structures hold a reduced num ber of av ailable slip system s com pared to body-centred cubic (bcc) and face-centred cubic (fcc) lattices, making plastic deform ation m ore difficult. W ith the asym metric distribution of slip system s ove r the crystallographic reference sphere, various primary and secondary slip and twinning mechanisms can and have to be activated at the same time. In contrast, usually only system s of one fam ily become active in fcc m etals. Therefore, m odelling of the m acroscopic deform ation of hcp m aterials requires a careful investigation of the micromechanisms. Understanding the m echanisms of dislocation gliding and deform ation twinning for single crystals and polycrystalline aggregat es constitutes the foundation for m odelling of the macroscopic mechanical behaviour. Deformation Mechanisms in Magnesium. Planes and orientations of the hexagonal lattice are usually described w ith the Miller-Bravais indices related to a coordinate system of three basal vectors, a1 , a 2 , a3 , and the longitudinal axis, c, which is the axis of hexagonal sym metry, see Fig. 3. Within this system , four axes are used rather than three orthogonal ones. Dislocations in a hexagonal lattice m ay be grouped in three fam ilies, a , c and a + c , depending on the orientation of the slip plane and th e slip directions, with the respective Burgers vectors of lengths a, c and a 2 + c 2 . The basal plane is characterised by its norm al vector n {0001} , the prism atic planes by n {1100} and the pyram idal planes by n {01 11} (π1), n {1122} (π2) and n {10 12} (π3). The corresponding slip directions are m ⎡⎣1120 ⎤⎦ f or basal, prism atic and π1 slip. Straining in direction of the c-axis can only be accom modated by c + a -systems, with a slip direction of m ⎡⎣1123⎤⎦ f or the three pyram idal slip system s, π1, π2 and π3. Beside the deformation caused by slip along crystallographic planes, deform ation twinning is an im portant deformation m echanism f or hcp m etals. It m ight generally be of tensile or com pressive nature, depending whether it results in an elongation or reduction of the c-axis length.
(a)
(b)
Figure 3: Crystallographic unit cell of an hcp metal: (a) basal and prismatic plane, (b) pyramidal planes π1, π2, π3, The knowledge about the relevant deform ation m echanism in m agnesium has im proved over the years. Modeling activities started recently and concentrate on the alloy AZ31 as the m ost common wrought m agnesium alloy. Agnew et al. [24] stud ied the relation between m echanical behaviour and texture evolution. In order to reproduce sim ilar textures as those observed experim entally, they had to account for a + c slip. Furtherm ore, the authors concluded that prism atic a slip should be kept marginal for avoiding undesired effects in the simulated texture. Thus, they have considered
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only basal, pyramidal a + c and tensile twinning on the π3-plane. This limitation turned out to be satisfying for sim ulating uniaxial com pression te sts of a plate for both in-plane and throughthickness orientations. In later studies, Agnew et al. [25] and Agnew and Duygulu [26] added prismatic a slip to the previous set of def ormation m odes. Non-basal slip in a direction has been shown to be necessary for m odeling the in-p lane anisotropy of AZ31B rolled plates at low temperatures. Staroselsky and Anand [27] descri bed the m acroscopic m echanical behaviour in tension and com pression as well as the respective texture evolution of AZ31B extruded rods and rolled plates without considering any slip syst em having a deform ation component oriented in c direction. The considered deformation mechanisms, basal, prismatic, pyramidal a slip and tensile twinning along the π3-plane, were sufficient to get good agreem ent between experim ent and simulation. The systematic texture simulations of AZ31 conducted by Styczynski et al. [28] showed that the best agreem ent between sim ulated and ex perimental textures is obtained by considering basal, prismatic, pyramidal a + c slip and tensile twinning, which is the sam e as in [25] and [26]. Yi et al. [29] selected all three a slip m odes: basal, prism atic, pyramidal, as well as tensile twin and pyramidal a + c -slip. In all cases Schm id’s law for crystalline slip was applied, which seem s to be the appropriate choice. Based on these investigations, Gr aff et al. [23] chose basal a , prismatic a , pyramidal a + c slip plus tensile twinning on {10 12} , which is treated like slip, to m odel the def ormation of pure magnesium single crystals in the f ramework of crystal plasticity. Table 1 sum marises the deformation modes used. Table 1: Deformation modes for magnesium [23] Name
Number of Slip Systems
Slip Plane n
Slip Direction m
Basal a
3
{0001}
Prismatic a
3
⎡⎣1120 ⎤⎦ ⎡⎣1120 ⎤⎦
Pyramidal a + c (π2)
6
Tensile Twin (π3)
6
{1100} {1122} {10 12}
⎡⎣1123⎤⎦ ⎡⎣10 11⎤⎦
The signif icant def ormation m echanisms acting in Mg and its alloys are still subjected to discussion. Recent investigations [30 - 33] suggest that contraction twinning on {10 11} planes in ⎡⎣10 12 ⎤⎦ direction plays a signif icant role and is m ore im portant than pyram idal a + c -slip. In addition, the lattice rotation due to twinning, which has not taken into account by Graff et al. [23], affects the activities of the rem aining slip system s in a single crystal, see Fig. 4. Neither has detwinning been modelled yet, which may occur in unloading.
(a)
(b)
(c)
Figure 4: Deformation by twinning of (a) a lattice domain and respective rotations of the hexagonal cell for (b) tensile and (c) contraction twinning [32]
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Parameter Identification. Apart f rom the m aterial constants f or isotropic elasticity, nam ely Young’s modulus and Poisson's ratio, the constitutive model presented above comprises three kinds of parameters per slip system, α, • two parameters of the viscoplastic law, γ0(α ) and n, eq. (9), • between two and four parameters of direct hardening, depending on the respective law, eq. (10), • parameters of latent hardening, describing the interaction between the various slip systems. For sim plification, it has been assum ed, that direct and latent hardening are identical f or all slip systems of any of the four basic m echanisms, namely basal slip, prism atic slip, pyram idal slip and twinning. In total, 12 param eters f or direct hard ening and 16 interaction param eters have to be determined, which requires respective tests on si ngle crystals, which allow f or identif ying the activation of specific slip systems. Kelley and Hosford [34, 35] performed channel-die compression tests on pure m agnesium single crystals and on textured m agnesium sam ples cut out of a rolled plate. The authors used a channel die experi ment, see Fig. 5. Sm all sam ples of approx. 6×10×13 mm3 were compressed in a rigid steel channel in one direction, while the second direction is constrained in displacement and the third one is free. By changing the initial orientation, different slip and twinning m odes are activated. This set of experiments has been used for identification of the above mentioned parameters [23].
(a)
(b)
Figure 5: Channel die tests for single crystals and polycrystals: (a) experiment and (b) FE model Fig. 6 shows the com parison between the experim ental results of Kelley and Hosford [34] and the simulations of the single crystal tests. The qualitative dif ferences in the def ormation behaviour of different orientations indicated by the letters A to G, which are due to the activation of different slip or twinning m echanisms, are obvious and m et by the sim ulations. A sim ilarly good agreem ent between tests [ 35] and sim ulations [ 23] has b een obtained for the polycrystal sam ples. The discretisation of the FE m odel has an im portant effect on the results. If the m esh is too coarse, the structure will behave too stif f; if it is too f ine, the com putation tim e m ay becom e unreasonably high, so that a com promise has to be found. In th e present case, just a single elem ent has been used to m odel the single crystal, which doubtlessly im poses def ormation constraints. The sam e discretisation was applied to the polycr ystal, however, which consisted of 8 ×8×8 elem ents, each representing one grain of individual crystallogr aphic orientation. Thus, the discretisation is consistent between the single crystal and the polyc rystal. A finer m esh of the single crystal would affect the values of the m aterial parameters that have been obtained. But then the FE m esh of one grain in the polycrystal has to be chosen id entically. A larger num ber of elem ents for the polycrystal would have m eant a reduction of the possible number of param eter studies and virtual testing, however, see next section.
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400
Exp. A Exp. B Exp. C Exp. D Exp. E Exp. F Exp. G
single crystal
True stress [MPa]
300
200
100
0
0
2
4
6
strain [%]
8
10
Figure 6: Channel die tests for single crystals and polycrystals: (a) experiment and (b) FE model Yielding of Textured Polycrystals. Applying the material parameters for the single crystal, virtual uniaxial or biaxial tests on RVEs with arbitrarily textured polycry stals can now be perform ed [23]. The RVE consists of 8 ×8×8 elem ents, each representing one grai n of individual crystallographic orientation, subject to periodic boundary cond itions. As de-twinning is still an open issue, no unloading or non-proportional loading has been sim ulated, yet. The mesoscopic yield surfaces are actually iso-strain contours in the stress space. They exhibit the typical features observed in tests, namely anisotropy and different yield strengths in tension and com pression. A phenom enological yield surface has to account for both effects. Cazacu and Barlat [36] proposed a respective yield function with 16 m odel parameters, which includes the third stress invariant, J3, to account for the sign of stresses,
(J ) 0 2
3
2
− cJ 30 = τ Y3 . (11)
The superscript “0” denotes the anisotropic form of the second and third invariant. For plane stress states, the num ber of param eters reduces to 8, which can be determ ined from the num erical simulations of polycrystals under biaxial tension [37]. Appropriate hardening laws have still to be found, however. A phenomenological mesoscopic yield function will finally allow for sim ulations of magnesium forming [38] or the structural performance components. Thus, the m odelling chain from micro to macro is complete, see Fig. 7. Material Testing
Microstructural Models
Processing
Crystal Plasticity
10-9
10-3
101
l [m]
Figure 7: Micro to macro transition in the modelling of deformation of hcp metals
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Lamellar γ-TiAl Motivation and Significance. Two-phase γ-based titanium alum inides (TiAl) are light-weight alloys for structural applications at high tem peratures such as turbine blades, turbocharger rotors and automotive valves. Its m icrostructure can be va ried by processing and heat treatm ents as fully lamellar, near lam ellar, duplex and near- γ. Fine grained lam ellar m icrostructures show the m ost balanced properties with respect to creep resistance and fracture toughness for the desired applications. They suffer from poor ductility and low fracture toughness at room tem perature, however. The m echanical behaviour of lam ellar TiAl has been extensively studied by m eans of polysynthetically twinned (PST) crystals, the lam ellar analogues to single crystals [39, 40]. The deformation and fracture properties are highly anis otropic. Yield stress and fracture toughness are very low for shearing and crack growth (delam ination) parallel to the lam ellar planes but m uch higher for other deformation or fracture modes, respectively [41, 42], see Fig. 8.
Figure 8: Dependence of the yield stress (0.2% proof stress) on the lamellae orientation, φ, of a PST with respect to loading [22]. Microstructure. Lamellar TiAl consists of an interm etallic γ-phase (TiAl) and, depending on the chemical composition, a f ew percent α2-phase (Ti 3Al). Their respective crystallographic structures are tetragonal and hexagonal, see Fig. 9.
(a)
(b)
Figure 9: Crystallographic structure of TiAl: (a) α2-phase (Ti3Al), (b) γ-phase (TiAl)
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The two phases f orm colonies of thin, parallel plat es (lamellae). In the lam ellar plane, the closely packed planes {0001} and {111} of α2 and γ are parallel. Due to the sixf old sym metry of the {111}γ-plane this relation is m et by six orientations. W ith respect to their stacking orders, three matrix (m ) and three twin (t) va riants are distinguished. The γ-lamellae show a dom ain structure [43]. Within a γ-lamella either γ-matrix or γ-twin occurs, see Fig. 10.
(a)
(b)
(c)
Figure 10: Microstructure of lamellar TiAl: (a) micrograph, (b) schematic, (c) deformed periodic unit cell (PUC) Deformation Mechanisms. The deform ation anisotropy has been described by Lebensohn et al. [40] by a relaxed constraint m odel. W erwer an d Cornec [22] introduced a periodic unit cell as shown in Fig. 10c, which will be applied in the following. The plastic def ormation in the α2-phase occurs by basal {0001} ⎡⎣1120 ⎤⎦ , prismatic {1 100} ⎡⎣1120 ⎤⎦ and pyramidal {1121} ⎡⎣ 1 126 ⎤⎦ slip systems. As the volume fraction of α2 is small, the sensitivity of the deformation behaviour to the values of the critical resolved shear stresses is low, and hence their values have been taken f rom the literature [ 44]. The plastic def ormation in the γ-phase occurs by {111} [110] ordinary dislocations, {111} [101] super dislocations and {111} ⎡⎣112 ⎤⎦ twinning [40]. Altogether, 16 individual slip and twinning systems have to be taken into account. Since the f ree slip length of any slip system depends on the orientation to the lam ellar plane, three morphological slip m odes, nam ely longitudinal, m ixed and transverse, can be distinguished, see Fig. 11 and Table 2 [ 22]. It has been assum ed that there is no dif ference between the critica l resolved shear stresses of ordinary slip, super slip and dislocations [40]. Therefore only three critical resolved shear stresses for the three morphological modes remain to be identified [45]. Table 2: Deformation modes for γ-TiAl Mixed
Transverse
Slip type
Longitudinal
Ordinary
{111} ⎡⎣1 10⎤⎦
{11 1} ⎡⎣1 10⎤⎦
{1 11}[110]
Super
{111} ⎡⎣01 1 ⎤⎦ {111} ⎡⎣10 1 ⎤⎦
{1 1 1} ⎡⎣01 1 ⎤⎦ {1 11} ⎡⎣10 1 ⎤⎦
Twinning
{111} ⎡⎣112 ⎤⎦
{11 1} ⎡⎣0 1 1 ⎤⎦ {11 1} ⎡⎣ 10 1 ⎤⎦ {1 11} ⎡⎣0 1 1 ⎤⎦ {1 1 1} ⎡⎣ 10 1 ⎤⎦ {11 1} ⎡⎣ 1 12 ⎤⎦ {1 11} ⎡⎣ 112 ⎤⎦ {1 1 1} ⎡⎣ 112 ⎤⎦
-
{111} [110]
14
1st International Conference On New Materials for Extreme Environments
(a)
(b)
(c)
Figure 11: Morphological slip modes of lamellar TiAl: (a) longitudinal, (b) mixed and (c) transverse Two-Scale Model. The periodic unit cell (PUC) shown in Fig. 10c consists of three lam ellae, α2, γ-matrix and γ-twin, where both γ-lamellae consist of three matrix- or twin- dom ains, respectively, but other, m ore complex configurations can be de signed [45, 46]. The Taylor assum ption has been adopted, that all domains within one lamella undergo the same strain. After identif ying the param eters f or the three slip m odes [46], the PUC can be directly used for simulations of the constitutive behaviour of PST crys tals with arbitrary lam ellar orientation. Fig. 8 displays the dependence of the 0.2% proof stress on the lamellae orientation. The sim ulations meet the test results quite well. The experim ental data for the orientations 0°. 45° and 90° were used for the parameter fitting, the result s for the other orientations are model predictions. Some stress-strain curves of PST crystals of varying orientations ar e shown in Fig 12. The strain-hardening behav iour of th e orientations 15°. 75° and 85° is not perfec tly but satisfactorily matched, considering also the difficulties of specimen preparations and ex ecution of te sts. The graphs include illustrations of the inhomogeneous strain fields of the specimens under compressive loading.
Figure 12: Stress vs. plastic strain curves of compression tests on TiAl PST crystals for various orientations compared to model predictions including illustrations of the strain fields in the deformed specimens [46] Instead of a phenom enological constitutive law obta ined by a hom ogenisation process, the PUC is now directly im plemented as a sub-m odel for the in tegration points of a superordinate FE m odel, which is known as the FE 2 technique [47]. If only one PUC is implemented in each FE, zero-energy modes may occur. To avoid this, eight PUCs form one FE. To account for trans- or inter-lam ellar
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fracture, additional non-linear springs have been app lied to the ref erence nodes, but f or the sake of simplicity, this will not be f urther addressed, here. Finally, FEs are taken to f orm lamellar colonies in a polycrystal [46]. Fig 13 summarises the transition from the micro- to the macro-scale.
Figure 13: Micro to macro transition in the modelling of deformation of lamellar TiAl As l amellar TiAl is a two-phase m aterial with a rather complex microstructure, modeling is e ven more dem anding than for a single-p hase m aterial. Crystal plas ticity applies to the α2- and the γphases. The concept of periodic unit cells has been applied to m odel the lam ellar structure. The PUCs, which represent the m echanical behaviour of PST crystals, are the equivalent to the single crystals in the m icromechanical m odel of m agnesium described above, and the lam ellar colonies correspond to the grains in a single-phase polycrystal. Summary and Conclusions Two exam ples of a two-scale m odelling approach for the m echanical behaviour ha ve been presented. For both m aterials, an understanding of their m acroscopic properties requires an insight into the deform ation m echanisms on the m icroscale, nam ely crystallographic slip and twinning. Though just two length scales have been consid ered and m odelling has been restricted to the framework of continuum s m echanics, the com plexity of the m odelling process is obvious. It requires the identif ication of crystal structures and slip system s of the phases, first of all. Subsequently, crystallographic m odels for each phase to describe the dom inant deform ation processes have to be form ulated. Finally ho mogenisation schem es to obtain the m acroscopic mechanical response of the material must be developed. Modelling is thus not solely the task of experts in continuum mechanics. Expertise in experim ental mechanics is essential likewise. Particularly, micromechanics requires non-standard tests and advanced experim ental techniques ranging from in-situ testing, local def ormation m easurements, micro- and nano indentation to m odern analysis methods of neutron scattering, high-energy X-ray tomography and focused ion beam (FIB). Some essential inf ormation on the m icroscale m ay not be directly m easurable at all like, e.g., properties of interfaces or grain boundaries. Reasona ble assumptions have to be m ade instead and verified by simulations. In addition, sim plifications and idealisations have to be introduced to keep the model manageable. One should never forget, that a m odel is never a direct im age of physical reality, even if one goes down the lengthscales to explain m acroscopic properties by their causes on the micro- or nanoscale. Hence, the m ultiscale modelling approach is both a respectable vision and a challenge, but should not be m istaken with the illusion, that one day m aterial science will be able to derive everything ab initio. The sm aller length and tim e scales becom e, the higher is the experimental complexity and the more arguable the assumptions may become.
Accepted forNew publication in: Advanced Engineering Materials 1st International Conference On Materials for Extreme Environments
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Acknowledgement Contributions by Stéphane Graff (PhD thesis), Ma lek Homayonifar and Malte Werwer (PhD thesis) are gratefully acknowledged. The TiAl m odelling has been financially supported by the Germ an Research Foundation (DFG) within the Collaborative Research Centre (SFB) 371 “Micromechanics of multi-phase materials”. References [1]
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[29] S.B. Yi, C.H.J. Davies, H.G. Brokm eier, R.E. Bolmaro, K.U. Kainer and J. Hom eyer: Acta Mater. Vol. 54 (2006), 549. [30] L. Jiang, J.J. Jonas, R.K. Mishras, A.A. Luo , A.K. Sachdev and S. Godet: Acta Mater. Vol. 55 (2007), 3899. [31] M.R. Barnett: Mat. Sci. Engng. A, Vol. 404 (2007), 8. [32] L. Wu, A. Jain, D.W . Brown, G.M. Stoica, S.R. Agnew, B. Clausen, D.E. Fielden and P.K. Lian: Acta Mater. Vol. 56 (2008), 688. [33] A. Jain, O. Duygulu, D.W. Brown, C.N. Tomé and S.R. Agnew: Mat. Sci. Engng. A, in press. [34] E.W. Kelley and W.F. Hosford: Trans. Metall. Soc. AIME Vol. 242 (1968), 5. [35] E.W. Kelley and W.F. Hosford: Trans. Metall. Soc. AIME Vol. 242 (1968), 654. [36] O. Cazacu and F. Barlat: Int. J. Plasticity Vol. 20 (2004), 2027 [37] D. Steglich, S. Graff and W. Brocks: Mater. Sci. Forum Vol. 539 (2007), 1741. [38] S. Graff, D. Steglich and W. Brocks: Advanced Eng. Mater. 9 (2007), 803. [39] T. Fujiwara, A. Nakam ura, M. Hosomi, S.R. Nishitani, Y. Shirai and M. Yam aguchi: Philos. Mag. A Vol. 61 (1990), 591. [40] R. Lebensohn, H. Uhlenhut, C. Hartig and H. Mecking: Acta Mater. Vol 46 (1998), 4701. [41] T. Nakano, T. Kawanaka, H.Y. Yasuda and A. Umakoshi: Mater. Sci. Eng. A Vol. 194 (1995), 43.
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1st International Conference On New Materials for Extreme Environments
1st International Conference On New Materials for Extreme Environments doi:10.4028/3-908454-01-8 Two-Scale Finite Element Modelling of Microstructures doi:10.4028/3-908454-01-8.3
