Computational Aspects of Nonlinear Fracture Mechanics

Technical Note GKSS/WMS/02/05 internal report

Computational Aspects of Nonlinear Fracture Mechanics

W. Brocks, A. Cornec and I. Scheider

July 2002

Institute of Materials Research GKSS research centre Geesthacht

CONTENTS 0

Nomenclature.................................................................................................................... 5

1

Introductory Remarks on Inelastic Material Behavior .............................................. 11

2

FE meshes for structures with crack-like defects........................................................ 15

3

2.1

General Aspects and Examples ................................................................................ 15

2.2

Singular Elements for Stationary Cracks ................................................................. 16

2.3

Regular Element Arrangements for Growing Cracks .............................................. 17

Characteristic Parameters of Elasto-Plastic Fracture Mechanics............................. 19 3.1

4

The J Integral............................................................................................................ 19

3.1.1

Foundation........................................................................................................ 19

3.1.2

J as stress-intensity factor: the HRR field ........................................................ 21

3.1.3

The domain integral or vce method.................................................................. 23

3.1.4

Extensions of the J-integral .............................................................................. 25

3.1.5

Path dependence of the J-integral in incremental plasticity............................. 27

3.1.6

The J-integral for growing cracks .................................................................... 28

3.2

The Crack Tip Opening Displacement and the Crack Tip Opening Angle ............. 29

3.3

The Energy Dissipation Rate.................................................................................... 31

Damage Mechanics and “Local Approaches” to Fracture......................................... 35 4.1

Damage and Fracture ............................................................................................... 35

4.2

Damage Indicators.................................................................................................... 36

4.2.1

Ductile tearing .................................................................................................. 36

4.2.2

Cleavage fracture.............................................................................................. 37

4.3 5

Micromechanical Models of Ductile Tearing .......................................................... 42

The Cohesive Model ....................................................................................................... 49 5.1

Introduction .............................................................................................................. 49

5.2

Fundamentals ........................................................................................................... 49

5.2.1

Barenblatt’s Model........................................................................................... 49 3

5.2.2

Advanced cohesive models .............................................................................. 50

5.2.3

Cohesive laws................................................................................................... 53

5.2.4

Cohesive models for mixed mode loading....................................................... 58

5.2.5

Unloading and reverse loading of cohesive elements ...................................... 60

5.2.6

Rate dependent cohesive laws.......................................................................... 62

5.2.7

Work of separation and remote plastic work ................................................... 65

5.2.8

Implementation of cohesive elements in FE codes .......................................... 67

5.3

Relation to micromechanical phenomena of damage and fracture .......................... 69

5.3.1

Ductile crack growth in metals......................................................................... 69

5.3.2

Quasi-brittle fracture of concrete ..................................................................... 71

5.3.3

Crazing in amorphous polymers ...................................................................... 72

5.4

Applications to ductile fracture of metals ................................................................ 73

5.4.1

Simulation of ductile resistance curves by the cohesive model ....................... 73

5.4.2

Embedded ductile layer with center crack under mode-I................................. 76

5.4.3

Interface cracking of dissimilar materials ........................................................ 78

5.4.4

Verification of the cohesive model on homogeneous elastic-plastic structures 83

5.5

Applications to other materials and phenomena ...................................................... 88

5.5.1

Quasi-brittle materials such as concrete........................................................... 88

5.5.2

Heterogeneous compounds .............................................................................. 90

5.5.3

Dynamic fracture.............................................................................................. 92

5.5.4

Rate- and time-dependent fracture ................................................................... 93

5.5.5

Fatigue crack growth........................................................................................ 94

5.6

Summary and Outlook ............................................................................................. 96

6

References ....................................................................................................................... 99

7

Figures ........................................................................................................................... 121

4

0

NOMENCLATURE

A

crack area

B

specimen thickness

BN

specimen net thickness

C*

C* integral in creep fracture

CMOD

crack mouth opening displacement

D

Damage variable

D, Dij

strain rate tensor, components of strain rate tensor

Din , Dijin

inelastic strain rate tensor, components of inelastic strain rate tensor

Dp

plastic strain rate tensor

E

Young's modulus

Et

tangent modulus

E1, E2

elastic moduli for dissimilar material joints

Eve

relaxation modulus for viscoelastic cohesive laws

F

applied force

G

shear modulus

G

Griffith's energy release rate

Gss

Griffith's energy release rate for steady state condition

H

constraint layer height

J

J-integral of Cherepanov and Rice

Ji

J-integral at crack initiation

Jss

J-integral for steady state conditions

K

stiffness matrix

K

stress intensity factor

K0

stress intensity factor at crack initiation

KI, KII, KIII

stress intensity factors for mode I, II, III

Kss

stress intensity factor at steady state

M

mis-match ratio of yield stresses for dissimilar joints 5

Mb

applied bending moment

N

work hardening exponent (N ≤ 1)

Pf

failure probability

Q

Q-stress of O'Dowd and Shih

R

dissipation rate

R(εp)

yield curve of a material

R0

plastic zone size estimation at K0

S

surface, area

S

Cauchy stress tensor

S´

stress deviator

T

three-dimensional traction vector

T

traction

T0

maximum traction, separation strength

TN, TT

normal and tangential direction of the traction, respectively

TN,0, TT,0

normal and tangential direction of the maximum traction, respectively

Udis

(total) dissipated (non-recoverable) mechanical work

Upl

(global) work of plastic deformation

Usep

(local) work of separation

V

volume

Vu

Matrix of shape functions for finite elements

W

specimen width

X

back stress tensor

a

crack length

a0

initial crack length

f

void volume fraction in Gurson and Rousselier model

f*

damage parameter in GTN model

f0

initial void volume fraction

6

fc

critical void volume fraction at beginning void coalescence

ff

void volume fraction at final failure

h

triaxiality, ratio of hydrostatic and effective stress

m

Weibull modulus

n

hardening exponent of Ramberg-Osgood power law (n = 1/N)

ni

i-th component of the normal unit vector

p

norm of inelastic or plastic strain

q

interaction parameter for normal and tangential separation

q1, q2, q3

parameters in GTN model

r

radial coordinate, distance from crack tip

s

arc length

t

plate or sheet thickness

ui

components of displacement vector

[u]

vector of the displacement jump at the cohesive element (equivalent to the separation vector)

vLL

Load-line displacement for C(T) specimens

w

strain energy density

xi

Cartesian coordinates

Δa

crack extension

Δb

diameter reduction in round bars

Δl

global elongation of specimens

Φ

yield function, flow potential

Γ

Integration contour in the definition of the J-integral

Γ0

total cohesive energy, separation energy at failure

Σij

mesoscopic strain components, used for the unit cell

7

β

second Dunders' parameter

δ

separation

δ0

separation at failure (critical separation)

δ1, δ2

shape parameters for several cohesive laws

δ5

crack tip opening displacement defined at 5 mm gauge length across the crack tip, mounted on the specimen surface

δN, δT

normal and tangential direction of the separation, respectively

δN,0, δT,0

normal and tangential direction of the critical separation, respectively

δt, δ5

crack tip opening displacement (CTOD)

δvp

viscoplastic portion of the separation

ε

oscillation index for mode mixity

ε0

strain at σ0, used for Ramberg-Osgood hardening law

εij

strain components

ε ein

accumulated inelastic effective strain

εp

plastic strain

εY

yield strain

γ

Griffith's surface energy

η

viscosity coefficient

ϑ

angular coordinate

θ

temperature

κ

acceleration factor for the GTN model

μ

friction coefficient

ν

Poisson's ratio

σ0

yield stress

σc

critical stress, cleavage stress

σe

von Mises effective stress

8

σij

components of Cauchy stress tensor

σ ij′

components of stress deviator

σu

reference stress of Weibull distribution

σw

Weibull stress

τ

normalized T-stress parameter

ψ0

mode mixity parameter, defined at r = R0

ψL

mode mixity parameter, defined at r = L

9

10

1

INTRODUCTORY REMARKS ON INELASTIC MATERIAL BEHAVIOR

In a general sense, nonlinear fracture mechanics can be understood as mechanics of fracture for materials with inelastic stress-strain relations, where inelastic behaviour is any kind of irreversible response to thermo-mechanical loading including phenomena like • temperature and rate dependence, • time dependence: creep and relaxation, • Bauschinger effect, • cyclic hardening and softening, • effects of multi-axial and non-proportional loading. A unified approach to describing inelastic deformations which covers classical elastoplasticity as well as viscoplasticity bases on an incremental formulation < 4>

(

S = C ⋅⋅ D − Din

)

,

(1)

with S and D being an objective derivative of the Cauchy stress tensor and the deformation rate tensor, respectively. The inelastic part of D is described by a "flow rule" Din =

3 2

pN .

(2)

where N denotes the direction of inelastic deformation and p = Din = Dijin Dijin =

3 2

ε ein .

represents the uniaxial effective inelastic strain rate - despite the factor

(3) 3 2

. A number of

scalar and tensorial variables, κ n and X m , capture the load history. Elasto-plastic constitutive theories, i.e. D in = D p , like those of von Mises (1963), Prandtl (1924), and Reuss (1930), Prager (1959), Ziegler (1959), Mròz (1967), Chaboche and Rousselier (1983), introduce a yield condition which limits the set of physically admissible stress states,

Φ (S,X, p) = S′ − X′ − κ = 0 ,

(4)

with evolution equations for the two internal variables,

11

κ = f (S, D p , p, X , κ ,θ ) , ⎫ ⎬ X = ℵ(S, D p , p, X , κ ,θ ) , ⎭

(5)

which have the physical meaning of rules for isotropic and kinematic hardening, respectively, and may depend on temperature, θ, and strain rate, p . S′ = S − 13 σ kk I is the stress deviator, X is called "back stress" tensor, and

S′ − X′ =

(σ ′ − ξ ′ )(σ ′ − ξ ′ ) =

is - despite the factor

ij

2 3

ij

ij

ij

2 3

σe ,

(6)

- the uniaxial "effective" stress which becomes the well-known von

Mises effective stress in the case of pure isotropic hardening, where κ =

2 3

R(ε p ) is the

uniaxial flow stress as obtained in a tensile test. Plastic deformations occur if the loading condition S ⋅⋅ ( S′ − X′) ≥ 0

(7)

is fulfilled in the time increment Δt . The direction of plastic flow is commonly given by the "normality rule" N=

S′ − X′ , (S′ − X′ )

(8)

and the consistency condition Φ=

∂Φ ∂Φ ∂Φ S+ X+ κ =0 , ∂S ∂X ∂κ

(9)

allows for determining p . Visco-plastic constitutive theories basing on the "over-stress model", like those of Bodner and Partom (1975): Robinson (1978) or Chaboche (1993) do not include a yield condition, eq. (4), and hence no consistency condition, eq. (9), either. They introduce a special constitutive law for D in = D vp instead, which commonly also postulates the normality rule, eq. (8) and some power law, ⎛ ( S′ − X ′ ) − κ ⎞ ⎟ p = A⎜ ⎜ ⎟ K ⎝ ⎠

n

,

with A, K, n as material parameters. 12

(10)

The above constitutive equations of continuum mechanics describe deformations, eqs. (2), (8) and (10), under thermo-mechanical loading. The evolution equations for the internal variables, eq. (5), are restricted to material hardening, in general, so that the material behavior is "stable" (Drucker 1964). The corresponding boundary value problem is elliptic, and FE simulations will yield a unique solution which is convergent with refining the mesh. The physical reality, however, is more complex: materials "soften" due to formation of microcracks, initiation, growth and coalescence of voids etc., generally summarized as "damage". This damage may lead to the initiation and growth of macro-cracks in a structure and to final failure in the end. The "crack tip" as addressed in fracture mechanics is a mathematical idealization. In reality, a region of material degradation exists in some process zone ahead of a macro-crack, where finally new surfaces will be created. In this process zone, the microbehavior becomes important for constitutive modeling, and this will raise questions on materials length scales (Sun and Hönig, 1994, Siegmund and Brocks, 1998b) which cannot be answered within the limitations of the theory of "simple materials". The boundary value problem for softening materials may loose ellipticity, and FE simulations will yield mesh dependent results as the element size affects the separation energy (Siegmund and Brocks, 1998a). Three different levels of approaches exist to model damage, material separation and fracture phenomena: 1. no damage evolution is modeled and conventional material models, e.g. elastic-plastic constitutive equations, are applied, the process zone is assumed as infinitesimally small, and special fracture criteria, e.g. based on K, J, C*, for crack extension are required; 2. separation of surfaces is admitted, if some critical value is reached locally, whereas the material outside behaves conventional, the process zone is some surface region, and the fracture criterion is a cohesive law; 3. softening behavior is introduced into the constitutive model, e.g. accumulation of damage, described by additional internal variables, the process zone is a volume, and the evolution equation for damage yields a fracture criterion. Classical elastic-plastic fracture mechanics (EPFM) covers a comparably small part of these constitutive theories and phenomena of inelastic deformation. It has been established under the assumptions of the theory of finite plasticity or "deformation theory" of plasticity by Hencky (1924) and the kinematics of small deformations. It has nevertheless become the most important field of fracture research beside linear-elastic fracture mechanics (LEFM) as it 13

allowed for analytical descriptions and solutions and has been successfully applied to describe crack growth initiation in ductile materials like metals at low and moderate temperatures. The plastic deformation behavior of metals, however, is more realistically described by the theory of incremental plasticity by von Mises, Prandtl and Reuss, eqs. (1) to (9), which accounts for effects of load history, unloading, and local rearrangement of stresses. As no analytical solutions are possible in this case, numerical methods have won great importance. And with the increasing capacities of computers, the constitutive relations used in structural analyses have become more advanced and sophisticated, including more and more of the above mentioned phenomena. But fracture parameters, and criteria for fracture and crack growth, which are used in practice for engineering assessment methods are still the same as in the early times of EPFM, namely • the J-integral of Cherepanov (1967) and Rice (1968) (or its analogon, C*, for creep crack growth, Landes and Begley, 1976) and • crack tip opening displacement (CTOD), δ, (Burdekin and Stone, 1966, Dawes, 1985). Numerical analyses and simulations applying incremental plasticity or more advanced constitutive theories can be used in this context for • determination of classical fracture parameters for complex configurations and boundary conditions including thermo-mechanical loading, if no appropriate analytical solutions are available, • investigations of the applicability and the limitations of these parameters and of engineering assessment methods in general, • application of local criteria of fracture, like those by Beremin (1981, 1983), if the classical parameters fail in giving reliable predictions. The following contribution will essentially restrict to the application of the von Mises theory of incremental plasticity to cracked specimens and components. In particular, the classical parameters of EPFM, J and CTOD, as well as more recently proposed parameters like energy dissipation rate and CTOA and the related computational aspects will be discussed. Some remarks follow on the "local approach to fracture" which bases on continuum field quantities, namely stresses and strains, and the damage models of Gurson (1977) and Rousselier (1987), which have found increasing application, recently, will be shortly addressed in chapter 4. The numerical modeling of decohesion and separation phenomena by "cohesive elements" will be presented in chapter 5. 14

2

2.1

FE MESHES FOR STRUCTURES WITH CRACK-LIKE DEFECTS

General Aspects and Examples

Cracks and crack-like defects induce high stress and strain gradients which require a fine discretization resulting in large numbers of elements and degrees of freedom. Nonlinear simulations of components with stress concentrators are therefore expensive with respect to computation time and memory. All possibilities to reduce the number of degrees of freedoms should hence be utilized like • restricting to two-dimensional models of the structure if physically meaningful, • coarsening the mesh away from the defect, • introducing symmetry conditions, • applying singular elements with special shape functions. Modeling does always mean reduction of complexity and simplification. Models should be as simple as possible and only as complex as unavoidable to cover the interesting effects. For plane specimen geometries the possibility of two-dimensional models should always be considered, at least for pre-analyses of a new problem. Thin specimens and sheet metal components are commonly adequately represented by a plane-stress model, and thick or sidegrooved specimens by a plane strain one. Three-dimensional analyses are necessary, of course, if the geometry is not plane or if 3D effects through the thickness and along the crack front are studied. Some examples of typical meshes will be given in the following. Figure 1 shows the two-dimensional model of a compact specimen, C(T), accounting for the symmetry with respect to the ligament. Normal displacements are constrained along the symmetry line. The load pin is modeled by truss elements transferring compression, only. As the pin hole is quite remote from the crack tip, a "correct" modeling of the load pin is not very relevant for the quality of the results as long as no local plasticity is induced by it, which would change the external work and, hence, the J value. Collapsed elements are applied at the crack tip, see figure 2 and section 2.2. Two-fold symmetries exist for center-cracked or double edge-cracked tensile panels, M(T) or DE(T). Figure 3 shows the FE mesh of a center cracked cruciform specimen which is biaxially loaded to study effects of biaxiality on fracture parameters and crack propagation. The ligament is modeled using a regular arrangement of rectangular elements as crack 15

propagation is simulated, see figure 4 and section 3.2. It demonstrates the common strategy used for coarsening the mesh away from the crack. Finally, an example of a three-dimensional model is given in figure 5. It was generated for a tubular joint under 3-point bending having a singular symmetry plane. This is a rather complex geometry as the two pipes are welded and a semi-elliptical crack exists in the weldment. Thus, the model has not only to account for a 3D curved crack but also for the weld which has different material properties than the rest of the structure. It is a convincing example of the necessity to use a coarse mesh for the global structure. The fine mesh in the crack vicinity is displayed in figure 6. Numerous further examples of structures with surface flaws and material gradients can be found in the literature, e.g. Brocks et al. (1989a, 1993), Schmitt et al. (1997). 2.2

Singular Elements for Stationary Cracks

Singular elements have been developed for numerical analyses of fracture problems to increase the accuracy of stress calculations and K-factors at a time when computer capacities were still rather limited. Barsoum (1977) found that triangular or prismatic isoparametric elements which were produced by collapsing one side or plane and shifting the respective mid-side nodes to a quarter position (double distorted elements), see Figure 7, included the 1/√r- singularity of strains in linear elastic fracture mechanics (LEFM) as well as the 1/rsingularity of elastic-plastic fracture mechanics (EPFM) for perfectly plastic material,

ε ij (r, ϑ ) =

α (0) ij (ϑ ) r

+

αij(1) (ϑ ) r

+ α ij (ϑ ) . (2)

(11)

[1] [4] [8] [4] [8] If the tip nodes undergo the same displacement, i.e. u[1] und uy = uy = uy , the x = ux = ux (1) second term in eq. (11) vanishes, α ij = 0 , and a pure 1/√r-singularity of strain and stress

fields as in LEFM is realized. If the mid-side nodes are not shifted to the quarter-point positions and the displacements of the tip nodes are not constrained, the first term vanishes,

αij(0) = 0 , and the 1/r-singularity of strain fields at cracks in perfectly plastic materials is obtained. The strain energy, 2π

w=

r

∫ ∫

ϑ =0 r =0

16

w r dr dϕ =

2π

r

∫ ∫σε

ij ij

ϑ =0 r =0

r d r dϕ ,

(12)

however, remains finite for r → 0 in linear elasticity as well as for HRR-like fields (Hutchinson, 1986a,b, Rice and Rosengren, 1968), see section 3.1.2) because w has a singularity of the order of r-1 in both cases. This is an important attribute for the physical significance of the J-integral, see section 3.1. Numerical studies, see e.g. McMeeking and Rice (1975), Brocks et al. (1985), have shown that triangular or prismatic collapsed 8- or 20-node elements, respectively, with a 1/rsingularity are well suited for elastic-plastic calculations. Together with a large-strain analysis, commonly performed by an updated Lagrangean formulation (Gadala et al., 1980), crack tip blunting can be simulated, see figure 8, and principal stresses show their typical shape exhibiting a maximum ahead of the crack tip (Rice and Johnson, 1970), see figure 9. Singular elements have become less important in recent years, mainly because • J integral calculations by the virtual crack extension method (see section 3.1.3) yield reliable and accurate results even for very coarse meshes, • singular elements can not be applied for crack growth simulations which require a regular arrangement of elements in the ligament as shown in figure 4; these meshes, however, will not provide sufficiently accurate results of CTOD or stresses at the crack tip for stationary cracks. 2.3

Regular Element Arrangements for Growing Cracks

If a critical initiation value of some fracture parameter is exceeded, a crack starts to grow. Different from crack growth in elastic materials, where crack initiation always leads to catastrophic failure of the structure, ductile tearing may occur in a stable manner, i.e. under still growing external forces, or at least deformation controlled even beyond maximum load. Crack growth can be simulated in the following way: • node release techniques, controlled by any fracture mechanics parameter as J, CTOD, CTOA (see section 3), e.g. Siegele and Schmitt (1983), Brocks et al. (1994), Brocks and Yuan (1989), Gullerud et al. (1999). • cohesive elements (see section 5), e.g. Needleman (1990a), Yuan et al. (1996), Lin et al. (1997), Siegmund et al. (1998),

17

• constitutive equations based on damage mechanics concepts (see section 4), e.g. Needleman and Tvergaard (1987), Rousselier et al. (1989), Sun et al. (1988), Brocks et al. (1995a), Xia et al. (1995), Schmitt et al. (1997). Figure 10 illustrates crack growth by node release which is controlled by a criterion assuming constant crack tip opening angle (see section 3.2). The simulation has to be performed under prescribed displacements in order to proceed beyond maximum load. The effect of crack growth and load biaxiality, λ = Fx / Fy, on the plastic zone is displayed in figure 11: plastic deformation is retarded under a tensile load in x-direction (λ = +0.5) and enlarged under a compressive x-load (λ = -0.5). The upper row shows the plastic strain at initiation and the lower at maximum load for a crack growth of Δa = 12mm (λ = +0.5) and Δa = 10mm (λ = 0.5), respectively.

18

3

CHARACTERISTIC PARAMETERS OF ELASTO-PLASTIC FRACTURE MECHANICS

3.1

3.1.1

The J Integral

Foundation

Path-independent integrals are used in physics to calculate the intensity of a singularity of a field quantity without knowing the exact shape of this field in the vicinity of the singularity. They are derived from conservation laws. They have been introduced into fracture mechanics by Cherepanov (1967) and Rice (1968). Budiansky and Rice (1973) also showed that this "Jintegral" is identical with the energy release rate J = G = − ( ∂ U ∂A ) .

(13)

for a plane crack extension, ΔA. For linear elastic material, J is hence related to the stress intensity factors by J = G I + G II + G III =

1 1 2 K I2 + K II2 + K III , 2G E'

(

)

(14)

where I, II, III denote the three fracture modes (Irwin, 1957). This relation has become a common technique to calculatate stress intensity factors in LEFM. The J-integral of elasto-statics can be deduced from the equations governing the static boundary value problem for a material body, B, namely equilibrium and boundary conditions and constitutive equations. The components of the material force per thickness acting on the boundary, ∂B, of a plane domain B, Fi =

∫ ⎡⎣ w(ε

∂B

mn

)ni − σ jk nk u j ,i ⎤⎦ d s Fi =

∫ [w(ε

∂B

mn

]

)ni − σ jk nku j ,i ds ,

(15)

are non-zero if and only if B contains a singularity. The closed contour Γ 0 = Γ 1 ∪ Γ + ∪ Γ 2 ∪ Γ in figure 12 does not include a singularity and hence Fi = 0 . Assuming further that the crack faces are straight and stress-free, the first components of the integrals along the respective +

−

contours, Γ , Γ , vanish, and the path independence of the first component of the "J-vector"

19

←

J1 =

∫ [w dx

2

Γ1

]

→

− σ jk nk uj ,1 d s = − ∫

Γ2

←

[ ]= ∫ [ ] .

(16)

Γ2

results. This holds for the other two components, J2, J3, if and only if the contours around the crack tip and the loading are symmetric to the x1-axis. This first component of the J-vector is the "J-integral" used in fracture mechanics, defining that the integration contour runs anti-clockwise, i.e. mathematically positive, around the crack tip,

[

]

J = ∫ w dx2 − σ jknk u j,1 d s . Γ

(17)

Because of its path independence, it can be calculated in the remote field and characterizes also the near tip situation, which establishes its role as a fracture parameter. But note that the path independence does only hold if the conditions (1)

time independent processes, no body forces, σ ij, j = 0 ,

(2)

small strains, ε ij =

(3)

homogeneous hyper-elastic material, σ ij =

(4)

plane stress and displacement fields, i.e. no dependence on x3,

(5)

straight and stress-free crack borders parallel to x1, see figure 13

1 2

(u

i,j

+ uj ,i ), ∂w , ∂ε ij

are met. Extensions of the J-integral will be discussed in section 3.1.4. Analogous considerations as made for the derivation of the J-integral, eqs. (15) to (17) yields the C*-integral C* =

∫ ⎡⎣ w dx

Γ

2

− σ jk nk u j ,1 d s ⎤⎦ .

(18)

for visco-plastic material behavior (Landes and Begley, 1976) if a power (work rate) density, w , exists so that stresses derive from σ ij =

∂w ∂ε ij

. This analogy implies that C* is path

independent under the same conditions which hold for the path independence of J.

20

3.1.2

J as stress-intensity factor: the HRR field

Beside its identification as energy release rate in hyper-elastic materials, eq. (13), J also plays the role of an intensity factor like K in the case of linear elastic materials. Hutchinson (1968a,b) and Rice and Rosengren (1968) derived the singular stress and strain fields at a crack tip in a power law hardening material, the since called HRR-field,

σ ij (r, ϑ ) ⎛ 1 ⎞ =⎜ ⎟ σ0 ⎝ αε 0 In ⎠

1 n +1

⎛ J ⎞ ⎜ ⎟ ⎝ σ 0r ⎠

1

n+1

σ˜ ij (ϑ ) .

(19)

The parameters σ0, ε0, α, n characterize the material's stress-strain curve according to the power law of Ramberg and Osgood (1945) n

⎛σ ⎞ ε σ = + α⎜ ⎟ , ε0 σ0 ⎝ σ0 ⎠

(20)

where σ0 is commonly identified as the initial yield strength of the material and ε0 = σ0 / E, so that the hardening is characterized by two parameters, α, n. Figure 12 gives an example for a Ramberg-Osgood fit of a true stress-strain curve of a ferritic steel. The parameter In and the angular functions, σ˜ ij (ϑ ) , in eq. (19) result from the solution of the respective 4th order ordinary differential equation, see tables given by Shih (1983), Brocks et al. (1990) and depend on the hardening exponent, n. Eq. (19) yields unique stress curves independent of the external loading, if the abscissa is normalized by J. The ratio J / σ0 is proportional to the crack tip opening displacement, δt, in the HRR theory (Shih 1981),

δt = d n

J

σ0

,

(21)

where dn is again a parameter resulting from the solution of the HRR equations which depends on the hardening exponent of the material. Hence, unique stress curves independent of the external loading are also obtained, if the abscissa is normalized by δt. Figure 14 shows the respective normalization for the principal stresses ahead of the crack front of the C(T) specimen, see figures 1 and 9, for two different J-values. A J-dominated stress field can indeed be found by this normalization in a certain region ahead of the crack tip though the stresses can of course not be described by the singular HRR-field correctly in the region of large strains at the blunted crack tip (Rice and Johnson, 1970, McMeeking and Rice, 1975, 21

Brocks and Olschewski, 1986). The maximum principal stress, σyy, shows a sufficiently good agreement between FE and HRR results in this example, whereas the quantitative correspondence is worse for σxx and, hence, also for σ zz =

1 2

(σ

xx

+ σ yy ).

Again, an analogy to the J-dominated singularity in EPFM, eq. (19), holds for the C*-integral, eq. (18), in visco-plastic materials: stress and strain fields have an HRR-like singularity if secondary creep follows a power law (Riedel and Rice, 1980). The HRR-field has been derived under the assumptions of "deformation theory" of plasticity, which actually describes hyper-elastic behavior, and small strains. The actual stress field as calculated in an elastic-plastic FE simulation will therefore more or less deviate from the HRR-field, see figure 14. The amount of deviation indicates the "validity" of J as an intensity parameter of the crack tip field. The comparison with the HRR-field is however significantly affected by the quality of the power-law fit. In particular, materials exhibiting a Lüders plateau in their stress-strain curve cannot be fitted uniquely by a power law, see figure 13, so that it is impossible to distinguish whether a deviation of the FE stresses from the HRR-field indicates a loss of "J-dominance" (McMeeking and Parks, 1979, Shih and German, 1985) or just a poor fit of the material's stress-strain curve. Sharma and Aravas (1991) solved the plane boundary value problem with a two-term power expansion, ⎛ J ⎞ ⎛ J ⎞ HRR (1) σ ij ( r,ϑ ) = σ ijHRR ( r,ϑ ) + Q ⎜ ⎟ ⎟ σ ij (ϑ ) σ ij ( r, ϑ ) = σ ij ( r, ϑ ) + Q ⎜ ⎝ σ 0r ⎠ ⎝ σ 0r ⎠ −q

1

r

σ˜ ij(1) (ϑ ) .

(22)

The idea of a second non-singular term of the stress field - as the T-stress in LEFM - was caught on in the discussion on "constraint effects" on ductile tearing (Garwood, 1979, Brocks et al., 1989b, Link et al., 1991, Sommer and Aurich, 1991, Shih et al., 1993) and resulted in the introduction of a "Q-stress" in EPFM by O'Dowd and Shih (1991, 1994), based on detailed FE analyses,

σ ij (r, ϑ ) = σ ijref (r, ϑ ) + Qσ 0 δ ij .

(23)

The second crack tip field parameter, Q, is obtained as the difference between the "full" stress field, i.e. the FE results, and any "reference" (HRR or small scale yielding) solution,

22

Q=

ref σ ϑϑ − σ ϑϑ σ0

at

r J σ0

= 2, ϑ = 0 .

(24)

Figure 14 shows that the location of r = 2 J/σ0 is beyond the stress maximum and hence outside the region of large plastic strain. Yuan and Brocks (1998) found from FE analyses of fracture mechanics specimens that a linear relation exists between Q-stress and stress triaxiality, h=

σh 2 σ kk , = σ e 3 3 σ ′ijσ ij′

(25)

which is hardly affected by the load amplitude, though the stress field is in general not fully described by eq. (23) as the differences between the stresses from the FE analysis and the respective stresses of the HRR field are not the same for all components, see figure 14 again. Beside T- and Q-stresses, stress triaxiality, h, has also been used directly as a second parameter to quantify crack tip constraint (Brocks et al., 1989b, Sommer and Aurich, 1991, Brocks and Schmitt, 1993, 1995) as micromechanical considerations have shown its physical significance for ductile void growth (Rice and Tracey, 1969). If Q has to be determined by a finite element analysis for a specific structure, there seems to be no necessity of determining the stress triaaxiality via Q anyway, as the hydrostatic stress is a direct outcome of the FE calculation. 3.1.3 The domain integral or vce method Calculating a contour integral like eq. (17) is quite unfavorable in finite element codes as coordinates and displacements refer to nodal points and stresses and strains to Gaussian integration points. Stress fields are generally discontinuous over element boundaries and extrapolation of stresses to nodes requires additional assumptions. Hence, a domain integral method is commonly used to evaluate contour integrals, see e.g. ABAQUS (2000). Applying the divergence theorem, the contour integral can be re-formulated as an area integral in two dimensions or a volume integral in three dimensions over a finite domain surrounding the crack front. The method is quite robust in the sense that accurate values are obtained even with quite coarse meshes, because the integral is taken over a domain of elements, so that errors in local solution parameters have less effect. This method was first suggested by Parks (1974, 1977) and further worked out by deLorenzi (1982a, b). 23

The J-integral is defined in terms of the energy release rate, eq. (13), associated with a fictitious small crack advance, Δa, see Fig 15, J=

1

⎡⎣σ ij u j ,k − wδ ik ⎤⎦ Δ xk ,i dS , Δ A ∫∫ B

(26)

0

where Δxk is the shift of the crack front coordinates, ΔA the correspondent increase in crack area and the integration domain is the grey area in Fig. 15. Because of this physical interpretation, the domain integral method is also known as "virtual crack extension" (vce) method. Eq. (26) allows for an arbitrary shift of the crack front coordinates, Δxk, yielding the energy release rate, Gϕ, in the respective direction, which can be applied for investigations of mixed mode fracture problems. The common J-integral, i.e. the first component of the J-vector, J 1 = Gϕ

ϕ =0

, is obtained if and only if Δxk has the direction of x1 (or ξ1 in three-dimensional

cases, see Fig. 16), which means that it has to be both, perpendicular to the crack plane normal, x2, and (in three-dimensional cases) to the crack front tangent, ξ3, see section 3.1.4. In a case where the crack front intersects the external surface of a three-dimensional solid, the virtual crack extension must lie in the plane of the surface. If the vce is chosen perpendicular to the crack plane, i.e. in x2-direction, one obtains the second component of the J-vector, J 2 = Gϕ

ϕ =π 2

.

For 2-D plane strain or plane stress conditions, the extended crack area is simply Δ A = t ⋅ Δa , where t is the specimen thickness. In a 3-D analysis, the vce has to be applied to a single node on the crack front if the local value of the energy release rate is sought. For a constant strain element, like the 8-noded 3D isoparametric element, the interpolation functions are linear and a shift of a node on the crack front will result in a triangle, Δ A = 12 ( 1 + 2 ) ⋅ Δa , where

1

,

2

are

the lengths of the adjacent elements. For the 20-noded isoparametric element, the interpolation functions are of second order, and a node shift will produce a crack area increase of parabolic shape which differs for corner nodes and mid-side nodes. In any case, ΔA is linear in Δa and, hence, in |Δxk|. Note also, that the crack extension is "virtual" in a sense that it does not change the stress and strain fields at the crack tip.

24

3.1.4 Extensions of the J-integral

The three-dimensional J

Assuming plane crack surfaces, the J-integral may be applied to three-dimensional problems. It is defined locally, J(sc), sc being the curved crack front coordinate, following the concepts of Kikuchi et al. (1979), Amestoy et al. (1981) and Bakker (1984). Suppose, the crack is in the (x1, x3)-plane, then a local coordinate system (ξ1, ξ2=x2, ξ3) is introduced in any point P tangential to the crack front, see Fig. 6, so that the (ξ1, ξ3)-plane is perpendicular to the crack. The domain, B0 = B - BS , is again a material sheet of constant thickness, t, with t → 0, but as this is a three-dimensional problem, its border now also contains the upper and lower faces, S + and S − , in the (ξ1, ξ2)-plane, ∂B0 = Γ 1 ∪ Γ + ∪ Γ 2 ∪ Γ - ∪ S + ∪ S − . Eq. (15) becomes ←

Fi =

⎫

⎧

→

1⎪ ⎪ ∫Γ [ ] d s + ∫ [ ] d s + Γ∫ [ ] d s + ∫ [ ] d s + h ⎨⎪ ∫∫ [ ] dS + ∫∫ [ ] dS ⎬⎪ = 0 , Γ+

1

⎩ S+

Γ−

2

S−

⎭

(27)

and for an infinitesimal thickness, t, a Taylor expansion can be applied so that under the same assumptions and arguments as above, the first component of the three-dimensional J-integral is obtained. ←

J ( sc ) =

∫ ⎡⎣ w d x Γ

2

− σ jk nk u j ,i d s ⎤⎦ − ∫∫ S−

1

∂ ⎡ w d x2 − σ jk nk u j ,i d s ⎤⎦ dS ∂ξ3 ⎣

,

(28)

which is a local value, i.e. it varies along the crack front. The second term vanishes if J is constant with respect to the crack front coordinate; it may contribute significantly if strong gradients occur, e.g. at the specimen surface. Applying the domain integral method, the respective volume integral of eq. (26) already includes three-dimensional "effects". If the whole crack front is shifted by the same amount, Δa, an average value, J =

1

c

∫ J (s )d s , c

c

for the total structure is obtained as in the

c 0

experimental procedures where J is evaluated from the area under the load vs displacement curve.

25

Body forces, surface tractions and thermal loading

The fundamental equation for deriving the path independence postulates that the stress tensor is divergence free. These equilibrium conditions are restricted to static and stationary processes without body forces or heat sources acting in B. Constant body forces like gravitational forces, which have a potential not explicitly depending on the coordinates, xi, can easily be included in the w-term and do not affect path independence. In all other cases, J becomes path dependent unless an extra term is added (deLorenzi, 1982b, Siegele 1989): J=

1 ⎡ (σ ij u j ,k − wδ ik ) Δ xk ,i − f i ui , j Δ x j ⎤ dv ⎣ ⎦ Δ A ∫∫∫ V

.

(29)

The forces fi can be body forces like electro-magnetic forces or "acceleration forces", f i = − ρ xi

, in the case of dynamic loading. +

−

In addition, the boundary conditions postulate that the crack faces, Γ , Γ , are traction free. If this condition is not met, path independence has to be re-established again by a surface correction term, J=

⎫⎪ 1 ⎪⎧ ⎨ ∫∫∫ [ ] dv − ∫∫ ti ui , j Δ x j dS ⎬ , Δ A ⎩⎪ V ∂Vc ⎭⎪

(30)

where the vector ti represents the surface tractions (or pressure) acting on the crack faces, ∂Vc , (deLorenzi 1982b, Siegele 1989). The correction term for thermal fields is J=

1 ⎪⎧ ⎨

Δ A ⎩⎪ ∫∫∫ V

⎫ ∂α ∂θ (θ − θ0 ) + α ⎤⎥ δ ij Δ xk dv ⎪⎬ θ ∂ ∂ x ⎣ ⎦ k ⎭⎪

[ ] dv + ∫∫∫ σ ij ⎡⎢ V

.

(31)

where θ(xi) is the temperature field, θ0 the reference temperature and α the coefficient of thermal expansion (Muscati and Lee 1984, Siegele 1989). Multi-phase materials Path independence of J holds only if the material is homogeneous. However, the assessment of defects in composite or gradient materials or in welded structures requires an extension of J to multi-phase materials. Again, correction terms have to be added to re-establish path 26

independence. Moreover, the boundary conditions become asymmetric in these cases (mixed mode problem), so that a single component of J is insufficient to characterize the crack field and the complete "J-vector" has to be considered. In a 2-D problem, this reduces to J1 and J2. If the contour Γ passes a phase boundary between two materials near the crack tip, it includes an additional singularity of stresses and strains. This contribution has to be eliminated by a closed contour integral along this interface, see Kikuchi and Miyamoto (1982) and Fig. 17,

[

]

Ji = ∫ wni − σ jk nk uj ,i d s − Γ

∫ [wn − σ i

Γ pb

jk

]

nk u j,i d s .

(32)

3.1.5 Path dependence of the J-integral in incremental plasticity The severest restriction for J results from the assumed existence of a strain energy density, w, as a potential from which stresses can be uniquely derived. This assumption also conceals behind frequently used expressions like "deformation theory of plasticity" (Hutchinson, 1968a,b, Rice and Rosengren, 1968) or theory of "finite plasticity" (Hencky, 1924). But it actually does not describe irreversible plastic deformations as in the "incremental theory of plasticity" of von Mises, Prandtl and Reuss, but "hyperelastic" or non-linear elastic behavior. It does not only exclude any local unloading processes but also any local re-arranging of stresses, i.e. changing of loading direction in the stress space, resulting from the yield condition. All loading paths in the stress space are supposed to remain "radial" so that the ratios of principal stresses do not change with time. The condition of monotonous global loading of a structure is of course not sufficient to guarantee radial stress paths in nonhomogeneous stress fields. Hence, the J-integral will become path dependent as soon as plasticity occurs and the contour Γ passes the plastic zone (McMeeking, 1977). For small scale and contained yielding, a path independent integral can be computed outside the plastic zone. This means that Γ - or the respective evaluation domain - has to be large enough to surround the plastic zone and pass through the elastic region only. In gross plasticity, this is not possible, and some more or less pronounced path-dependence will always occur, so that the evaluation of a "path-independent" integral is a question of numerical accuracy. Because of its relation to the global energy release rate, eq. (13), which is used to evaluate J from fracture mechanics test results, J has to be understood as a "saturated" value reached in the "far-field" remote from the crack tip.

27

Significant stress re-arrangements occur at a blunting crack tip, see figures 9 and 14 as well as results by McMeeking and Rice (1975), Brocks et al. (1986), and the path dependence increases strongly, see figure 17. Thus, a small strain analysis is advantageous if only the Jintegral and no stresses at the crack tip shall be calculated as only little path dependence occurs. But note that stresses at the crack tip lack physical significance in this case. As the work dissipated by plastic deformation always has to be positive, the calculated J values have to increase monotonically with the size of the domain (Yuan and Brocks, 1991), which is confirmed by Fig. 18 - except contour #21 touching the boundary. The highest calculated J-value with increasing domain size is always the closest to the "real" far-field J,

J tip ≤ J(r ) ≤ J far field .

(33)

Any different result would indicate an error in the definition of contours or in the evaluation of J. Fig 19 illustrates the approach to a saturation value of J for two different load-line displacements. Moreover, J will keep a finite value in the limit of a vanishingly small contour if and only if the strain energy density, w, has a singularity of the order of r-1,

[

]

J tip = lim ∫ wdx 2 − σ jk nk uj ,1 d s = lim Γ →0

Γ

r →0

+π 2

−

∫π

wr cosϑ d ϑ .

(34)

2

This holds in linear elasticity where stresses and strains have a 1

r -singularity and for

HRR-like fields. As the stress singularity at the blunting crack tip vanishes under the assumption of finite strains and incremental plasticity, J will not even have a finite value any more,

[

]

J tip = lim ∫ wdx 2 − σ jk nk uj ,1 d s = 0 . Γ →0

Γ

(35)

3.1.6 The J-integral for growing cracks J is also used as a fracture parameter in EPFM for growing cracks (Rice et al., 1973, Garwood et al., 1975). Resistance curves against ductile crack propagation are determined by plotting J versus crack growth, Δa, see ASTM E 1737 (1996). These JR-curves however are subject to a lot of size, geometry and other "constraint" effects which have filled conferences, journals

28

and books over years (e.g. Garwood, 1979, Sommer and Aurich, 1991, Shih et al. 1993, Brocks and Schmitt, 1993, 1995, Yuan and Brocks, 1998). A serious objection against J as fracture parameter for growing cracks comes from computational mechanics. J has once become a fracture parameter because of its path independence stating that global and local energy release rates are equal. If path independence is lost, an important argument in favour of J is gone. J is actually accumulated plastic work in the specimen or structure during crack growth and this work results mostly from global plastic deformation and hence depends on size, geometry and loading configuration. But J is not the driving force for a growing crack in the sense that it does not equal the local energy release rate at the tip any more. As was shown in Figure 18 already, that blunting of a stationary crack results in a significant path dependence, and eq. (35) states that not even a finite value of Jtip exists. The same effect occurs - even stronger - at growing cracks as was shown by Brocks and Yuan (1989) and Yuan and Brocks (1991). Though stresses and strains are still singular, their singularity is not strong enough to provide a non-zero local energy release rate. This was addressed long ago by Rice (1965, 1979) as the "paradox of elastic-plastic fracture mechanics", stating that no "energy surplus" exists for crack propagation. Thus, any kind of "near-field" integrals as proposed by Atluri et al. (1984) and Brust et al. (1985) are physically meaningless. What really happens in the process zone, namely damage induced strain softening, is not covered by classical plasticity with hardening material behavior. Continuum damage or cohesive zone models have to be applied in this region to capture the respective effects properly, see sections 4 and 5. 3.2

The Crack Tip Opening Displacement and the Crack Tip Opening Angle

The crack tip opening displacement, CTOD, for a stationary crack can be determined from the numerically simulated blunted crack tip, see figure 8, which requires a large strain analysis. The mesh around the crack tip should consist of collapsed elements (section 2.2, figures 2 and 7). However, no unique definition of CTOD exists, neither for the numerical nor for the experimental determination. A comparison between experimental and numerical results does only make sense, of course, if a unique definition is applied to both. Various evaluation procedures are used:

29

• Linear extrapolation of the deformed crack faces in the remote field to the crack tip. This definition allows to use a rather coarse mesh at the crack tip, as CTOD is mainly based on far field displacements which are rather unaffected by the discretization in the crack tip vicinity. Comparison with test results is easy because the latter are commonly determined by similar procedures (BS 7448 Part 1 1991, ASTM E 1290 1999). • Displacement of the crack tip node of the respective element at the free crack surface. This definition is easy to handle, but the results depend strongly on the FE mesh at the crack tip and no experimental equivalent exists. • Displacement at the intersection of two secants originating from the crack tip under angles of ±45o with the opening profile of the blunted crack. This definition is in accordance with the HRR theory (Shih 1981) and yields a simple analytical relation of CTOD with the Jintegral, eq. (21). However, it cannot be realized in fracture mechanics testing. • Displacement measured over a gauge length of 5mm at the location of the initial crack tip. This definition was proposed as a simple experimental procedure which can also be applied to growing cracks for the determination of δ5 based resistance curves (Hellmann and Schwalbe, 1984, Schwalbe, 1995, Schwalbe et al. 2002). It can be easily applied to FE data. In the course of crack growth after initiation no significant blunting of the crack tip occurs any more, see figure 10. Instead, the crack tip opening angle, CTOA, characterizes the near field deformation. CTOA is applied as a crack growth criterion especially for thin metal sheets (Newman at al. 1992, O'Donoghue et al. (1997), Gullerud et al. 1999). Numerical simulations of CTOA-controlled crack growth require a regular mesh in the ligament, see figures 4 and 10. As for CTOD, no unique definition exists for the opening angle neither in testing nor in numerical simulations. Averaging of deformation over some elements appears necessary but the number of elements does of course affect the results. An approximate relation exists to the derivative of the δ5 R-curve (Yuan, 1990),

ψ≈ see figure 20.

30

dδ 5 , da

(36)

3.3

The Energy Dissipation Rate

The cumulative quantity J, which rises with increasing crack length, is not the true driving force for ductile tearing as Turner (1990) has pointed out in a basic discussion on the necessity of defining an alternative measure of tearing toughness. He proposed to define tearing resistance in terms of energy dissipation rate 1

R=

dUdis dWext dUel = − , dA dA dA

(37)

where Wext is external work and Uel the (recoverable) elastic strain energy. This definition is a straight transfer of Griffith's elastic energy release rate (Griffith, 1920) to plastic processes which is consistent with the incremental theory of plasticity. The dissipation rate has the same dimension as J and characterizes the increment of irreversible work per crack growth increment. It falls with increasing crack length in gross plasticity and consists of two contributions, namely work of remote plastic deformation and local work of separation, R=

dUpl dU sep + dA dA

(38)

For specimens with constant thickness B and straight crack front ("plane" problems) the increment of crack area is d A = B d a . The dissipation rate, R, is more appropriate for characterizing crack growth in plastically deformed structures than the conventionally used J integral (Memhard et al., 1993, Kolednik et al., 1997, Atkins et al., 1998, Sumpter, 1999). It is, in fact, the true "driving force" which has to equalize the structural resistance in order to propagate the crack by some amount, Δa, whereas J accumulates the plastic work done along a given loading path. It is, however, not a material but a structural property as it contains the work of remote plastic deformation. When introducing R, Turner generally doubted that splitting it into local and global contributions will ever be possible. Thus, every measured ductile crack growth resistance will necessarily contain remote plastic work which in general is much larger than the local work of separation. In fact, only external work and elastic energy can be measured. Models of damage mechanics, however, provide ideas how to perform this separation.

1

The term "dissipated energy" means "non-recoverable mechanical work".

31

For quasi-static processes, the dissipation rate can be simply evaluated from the area under the measured or calculated load vs. displacement curve, FΔv pl Δ Udis = lim , Δa → 0 B Δa Δa → 0 B Δ a N N

R = lim

(39)

if R is supposed to include the whole irreversible part of the work done and, hence, may also include dissipated energy in zones far remote from the crack tip, e.g. around load points or supports. As no splitting into local and global contributions according to eq (38) is possible, "dissipated" work equals total "plastic" work as determined from the area under the load vs. displacement record. If the respective information is not available any more, R can be simply re-evaluated from existing JR-curves by inverting the procedure of calculating J from test data. The respective formula are (Memhard et al., 1993) R=

W − a dJpl γ + J pl η da η

(40)

for bend type specimens, C(T) and SE(B), with the well-known geometry factors ⎧2.0 + 0.522 (1 − a / W ) ⎩2.0

for C(T) ⎫ ⎬, for SE(B)⎭

(41)

⎧1.0 + 0.76 (1 − a / W ) ⎩1.0

for C(T) ⎫ ⎬, for SE(B) ⎭

(42)

η=⎨

γ =⎨

and

R = (W − a)

dJ pl da

(43)

for tension type specimens, M(T) and DE(T). R versus ∆a curves are obtained as crack growth resistance curves for the respective specimens and materials, see figure 21. Similar to CTOA R-curves, they decrease from higher values at initiation in a transition region to stationary values. In an elastic-plastic FE simulation of crack growth the dissipation rate can be calculated directly from stresses and strains (Memhard et al., 1993, Siegmund and Brocks, 1999) U pl = ∫ σ ij ε ijp dV = ∫ σ e ε ep dV Vpl

32

Vpl

(44)

σe and ε ep being von Mises effective stress and effective plastic strain, eqs. (6) and (3), respectively. The volume integral may either be performed over the whole body, thus yielding the result of eq. (37), or just over the plastic zone at the crack tip, how large it may be. If the FE simulation reflects plastic processes, only, i.e. it is simply based on the Mises-PrandtlReuss constitutive equations and follows experimental records of either J(Δa) or VL(Δa), see e.g. Siegele and Schmitt (1983), dissipated work equals total plastic work as in the experimental procedure. Recent approaches to modeling of ductile rupture refer to Barenblatt's idea (Barenblatt 1962) of introducing a "process zone" ahead of the crack tip where material degradation and separation occur. This approach requires a constitutive description of the material behavior in the process zone which can mirror the local loss of stress carrying capacity. In general, two alternatives are applied: • phenomenological "cohesive zone models" describing the decohesion process by a traction strength and the work of separation per unit area, e.g. Needleman (1990a), Yuan et al. (1995), Lin et al. (1997), Siegmund et al. (1998), see section 5, and • models based on the micromechanisms of ductile failure, namely the nucleation, growth and coalescence of voids, as e.g. the most commonly used models of Gurson (1997), Tvergaard and Needleman (1983), the GTN model, or Rousselier (1987), see section 4. For the cohesive zone model UÝsep is calculated from δ0

U sep = Γ 0 A = ∫ T (δ ) dδ ⋅ A

,

(45)

0

where T is the "surface traction", i.e. the stress acting on the surface of a continuum element, and δ is the "separation", i.e. the displacement jump between adjacent continuum elements, see section 5.2. For the GTN or the Rousselier model the energy needed for material separation, U sep , in an incremental crack advance, A , is given by: U sep =

∫ (1 − f )σ

e

ε ep dV .

(46)

Vsep

Here, the volume integral is performed over the "separation zone" only, i.e. a single row of elements ahead of the crack tip being described by the GTN model. As the damaged zone in fracture specimens is generally restricted to one row of elements in the ligament, the rest of 33

the structure may be described by classical von Mises plasticity, which reduces computation time but does not affect the macroscopic behavior. Models like this have been addressed as "computational cells" by Xia and Shih. (1995), Xia et al. (1995), Ruggieri et al. (1996), Faleskog et al. (1998) and Gao et al. (1998). Both approaches allow for splitting the total dissipated work into the (local) work of separation in the process zone and the (global) plastic work in the embedding material and, thus, solves a classical problem of elastic-plastic fracture mechanics (Siegmund and Brocks, 2000a,b). Examples will be given in section 5.2.7.

34

DAMAGE MECHANICS AND “LOCAL APPROACHES” TO FRACTURE

4

4.1

Damage and Fracture

"Local approaches" and "micromechanical modeling" of damage and fracture (Pineau 1981) have found increasing interest within the last 20 years. The general advantage, compared with classical fracture mechanics, is that, in principle, the parameters of the respective models are only material and not geometry dependent. Thus, these concepts guarantee transferability from specimens to structures over a wide range of sizes and geometries and can still be applied when only small pieces of material are available which do not allow for standard fracture specimens. It is not even necessary to consider specimens with an initial crack as, of course, also initially uncracked structures will break if the local degradation of material has exceeded some critical state. The identification and determination of the "micromechanical" parameters require a hybrid methodology of combined testing and numerical simulation. Different from classical fracture mechanics, this procedure is not subject to any size requirements for the specimens as long as the same fracture phenomena occur. Micromechanical modeling encounters a new problem, however, namely that – different from the assumption of continuum mechanics - the material is not uniform on the microscale but consists of various constituents with differing properties and shapes. A material element has its own complex and evolving microstructure. Micromechanics is a general methodology of expressing continuum quantities in terms of the parameters which characterize the microstructure and properties of the microconstitutents of the material neighborhood (NematNasser and Hori 1993). To this end, the concept of a representative volume element (RVE) has been introduced by Hill (1963), Hashin (1964) and others. An RVE for a material point is a material volume which is statistically representative of a material neighborhood of that material point. By this, a lengthscale is introduced in the continuum. Many constitutive models for damage evolution exist for various phenomena of material behavior, see also section 5.4. Fracture phenomena in ductile metals occur by either • formation of microcracks and their extension with little global plastic deformation ("brittle" or cleavage fracture), or • the nucleation, growth, and coalescence of microvoids with significant plastic deformation (ductile rupture). 35

Cleavage processes are stress controlled and consume little deformation energy. Hence, the crack grows fast and unstable and a high amount of kinetic energy can be released especially in large structures. Local fracture criteria are based on a critical cleavage stress. Compared with this, the strain controlled process of ductile rupture consumes much more energy by plastic deformation. The crack grows slowly and deformation-controlled or even stable, i.e., neglecting creep effects, growth stops if the load does not increase. Local failure criteria base on a critical failure strain, a critical void growth ratio, or a yield condition for porous materials. Numerical models exist for both failure phenomena, in particular 1. the Beremin (1983) model which is based on a critical fracture stress concept together with the "weakest link" assumption and Weibull (1939a, 1939b, 1951) statistics, 2. the void growth law of Rice and Tracey (1969), and 3. the models of Gurson (1977) or Rousselier (1987) for porous metal plasticity. Beremin's model for cleavage fracture and the model of Rice and Tracey for crack initiation due to ductile tearing yield "damage indicators", only, by an a-posteriori evaluation of stress and strain fields obtained in a conventional elastic-plastic finite element analysis. They are thus easy to handle, but they do not account for the effect of damage on plastic deformation, and no crack growth can be simulated. The constitutive models of Gurson and Rousselier on the other hand allow for both but require specific material subroutines for performing the FE analysis as the evolution of damage affects the yield behavior. In most of the commercial FE codes, these subroutines have to be provided by the user. 4.2

Damage Indicators

4.2.1 Ductile tearing Damage indicators are obtained by a pure postprocessing of stress and strain data from a conventional elastic-plastic analysis. Damage does not change the constitutive equations. According to the analytical solution of Rice and Tracey (1969), void growth follows the equation D=

36

⎛ 3σ ⎞ r = 0.283 ε p exp ⎜ h ⎟ r ⎝ 2σ e ⎠

(47)

for high triaxialities. The rate of damage, D , is defined as the growth rate of an average spherical void of radius, r, with increasing plastic strain, where σh and σe are the hydrostatic stress and the von Mises effective stress, respectively. This evolution law can be integrated as ε ⎡ ⎛ 3σ ⎞ p ⎤ r = exp(D − D0 ) = exp ⎢0.283 ∫ exp⎜ h ⎟ dε ⎥ . ⎝ 2σ e ⎠ r0 0 ⎣ ⎦ p

(48)

The initial void radius, r0. is not explicitly needed. Void coalescence and ductile crack extension is supposed to start, as soon as r r0 reaches its critical value, (r r 0 )c . This value is calibrated by testing of notched tensile bars and a simple numerical simulation of an axisymmetric model with elastic-plastic material behavior, see figure 22. The respective procedure is described in an official ESIS procedure (ESIS P6 1998). Eq. (48) is evaluated for all elements which have plastically deformed, and the highest value obtained in any element at crack initiation in the test yields (r r 0 )c , which is supposed to be a material parameter which can be used to assess crack initiation in any structure. The assumption of a single scalar damage parameter is equivalent to the assumption of spherical voids. Hence, the relation 3

⎛r⎞ f = ⎜ ⎟ f0 , ⎝ r0 ⎠

(49)

holds between the void volume fraction used in the Gurson or Rousselier model and the damage parameter of the Rice and Tracey model. 4.2.2 Cleavage fracture The critical fracture stress concept states that fracture occurs for a critical value, σc, of the maximum principal stress, σI. This criterion is applicable to any notched structure as long as the notch root radius remains finite. It has to be modified for structures with macroscopic cracks which induce singular stress fields by introducing a characteristic distance, xc, from the crack tip. The critical stress is related to Griffith's surface energy by

σc =

4Eγ π (1 − ν 2 )

(50)

37

where E is Young's modulus, ν is Poisson's ratio, γ is the surface energy, and l is the crack length. The cleavage fracture strength, σc, can be determined by elastic-plastic finite element (FE) analyses of notched bend or tensile bar tests (ESIS P6 1998). In body centered cubic (bcc) materials a transition region from cleavage to ductile tearing exists where fracture toughness increases and becomes subject to scatter. In order to describe the latter phenomenon Beremin (1983) and Mudry (1987) applied the concept of the failure probability for brittle materials by Weibull (1939a, 1939b, 1951) to ferritic steels. The basic idea is the so-called "weakest link" assumption that the probability of having cleavage fracture of a structure at any given load is equal to the probability that its weakest element ("link") fails at this load. The failure criterion, eq. (49), is established on a "mesoscopic" level for every material element, (i), which is subject to a stress σ I . The "critical length", (i )

(c) i

, of an assumed microcrack in this element derives from

Griffith's criterion as (c) i

=

4E γ 1 2 (i ) 2 π(1− ν ) (σ c )

(51)

Now assume that the probability of having a crack of length between

i

and

i

+d

i

in this

element is P(

i

)d

i

=

α β

d i ,

(52)

i

where α and β are parameters depending on the material's microstructure and mechanism of microcrack formation. Then its failure probability becomes Pi (σ f

m

∞

( i) I

)= ∫ P ( ) d i

(c) i

i

⎛ σ (i ) ⎞ =⎜ I ⎟ , ⎝ σu ⎠

(53)

with the two "Weibull parameters"

m = 2β − 2

(54)

and 1

⎛ β − 1⎞ m σu = ⎝ α ⎠

38

4E γ . π (1− ν 2 )

(55)

According to the "weakest link" assumption, the failure probability of the whole structure ("chain") with n elements ("links") is the product n

[

]

Pf =1 − ∏ 1− Pi f (σ (iI ) ) i= 1

(56)

and its "survival probability" is (1-Pf). Since the failure probability is supposed to be small, f

Pi