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Micromechanical Models of Ductile Damage and Fracture

  • A. Amine BenzergaEmail author
Living reference work entry

Abstract

Two classes of micromechanics-based models of void enlargement are presented succinctly with their fundamental hypotheses and synopsis of derivation highlighted. The first class of models deals with conventional void growth, i.e., under conditions of generalized plastic flow within the elementary volume. The second class of models deals with void coalescence, i.e., an accelerated void growth process in which plastic flow is highly localized. The structure of constitutive relations pertaining to either class of models is the same but their implications are different. With this as basis, two kinds of integrated models are presented which can be implemented in a finite-element code and used in ductile fracture simulations, in particular for metal forming processes. This chapter also describes elements of material parameter identification and how to use the integrated models.

Keywords

Representative Volume Element Yield Criterion Void Growth Void Nucleation Microstructural Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

Symbol

Definition

Components

I

Second-order identity tensor

δ ij

\( \mathbb{I} \)

(Symmetric) fourth-order identity tensor

\( \frac{1}{2}\left({\delta}_{ik}{\delta}_{jl}+{\delta}_{il}{\delta}_{jk}\right) \)

T m ≡ 1/3 tr T

Mean part of tensor T

 

T′ ≡ TT m I

Deviator of tensor T

\( {T}_{ij}-\frac{1}{3}{T}_{kk}{\delta}_{ij} \)

\( \mathbb{J}\equiv \mathbb{I}\hbox{--} \frac{1}{3}\mathbf{I}\otimes \mathbf{I} \)

Deviatoric projector, e.g., \( \mathbb{J}:\boldsymbol{\upsigma} ={\boldsymbol{\upsigma}}^{\mathbf{\prime}} \)

\( {I}_{ij kl}-\frac{1}{3}{\delta}_{ij}{\delta}_{kl} \)

||T|| ≡ (3/2 T′ : T′)1/2

von Mises norm of tensor T

 

|| T || h ≡ (3/2 T′ : Open image in new window : T′)1/2

Hill norm of tensor T

Ω

Domain occupied by RVE

ω

Domain occupied by voids

f ≡ |ω|/|Ω|

Void volume fraction (porosity)

W

Void aspect ratio (>1 if prolate)

e 3

Common axis of aligned spheroidal voids

λ

Void spacing ratio (>1 if axial spacing is greatest)

χ

Ligament parameter (=1 if no ligament left)

Λ

Plastic multiplier

σ, d

Microscopic Cauchy stress and rate of deformation

Σ, D

Macroscopic Cauchy stress and rate of deformation

Σeq ≡ ||Σ||

von Mises equivalent stress (similar for σ eq)

D eq = 2/3||D||

Equivalent strain rate (similar for d eq)

\( \overline{\sigma} \)

Microscopic yield stress

Open image in new window

Hill’s anisotropy tensor

Open image in new window

Anisotropy tensor in the space of stress deviators

Open image in new window

Formal inverse of Open image in new window

h i i = 1,6

Components of Open image in new window after Voigt’s condensation

f N

Volume fraction of void-nucleating particles

ϵN

Average nucleation strain

s n

Standard deviation

Introduction

Understanding the fundamental mechanisms of fracture in metal forming is of considerable importance, but the development of physically sound models with a quantitative predictive ability still poses a challenge. First, the plastic strains involved in forming are quite large, and this excludes the use of classical linear elastic fracture mechanics concepts. In addition, nonlinear fracture mechanics has essentially dealt with the fundamental problem of a body containing one or more initial cracks. In metal forming, there usually are no initial cracks. Also, significant microstructural evolutions take place at large plastic deformations. The latter include extreme grain elongations, texture development/evolution as well as the nucleation and growth of microvoids, either on second-phase particles or at stress risers.

The nonproportional loading paths that are inherent to forming operations make the prediction of failure even more challenging. Engineering tools and guidelines based on forming limit diagrams or stress-based criteria that do not embody a set of internal variables are of limited scope. They fail to capture, even qualitatively, the inherent path dependence of failure loci, in whichever way such loci are defined. There is a great potential for metal forming to rely on rational material design, i.e., based on sound physical models that possess the ability to connect processing parameters and microstructural variables to the mechanical properties of interest.

Of paramount importance to some forming operations is the intrinsic ductility of the material. Ductility is often understood as the ability of a material piece to withstand some amount of plastic or viscoplastic strain before the onset of structural instabilities. Necking of a bar under simple tension is the classical example. Viewed as a material property, the necking strain carries more the signature of the hardening capacity of the material than its intrinsic ductility. Yet qualitative correlations are commonly drawn between the two. This may be sufficient in some, but certainly not all forming operations, especially those involving load path changes.

The fundamental mechanisms and mechanics of ductile fracture have recently been reviewed by Benzerga and Leblond (2010); also see Besson (2010) for models. When the microscopic mechanisms involve microvoid growth to coalescence, micromechanical models have the capability to deliver physically sound predictions of structure-property relationships. Even since the last 2010 reviews, significant developments have taken place in this area. The objective of this chapter is to present a synthesis of ductile fracture models that are implementable in finite-element programs for solving metal forming initial- and boundary-value problems. The experimental aspects of the subject are omitted and may be consulted in the above monograph. Also, void nucleation is not addressed for brevity. This lays the focus on the large-deformation phenomena that are void growth and void coalescence. In all models a representative volume element (RVE) is considered according to the classical acception in homogenization theory. The microscale refers to that of the matrix while the macroscale refers to matrix and voids. Situations where either scale separation does not hold or fracture is affected by extreme statistics are not within the scope of this synthesis. The general framework is that of porous metal plasticity with the difference in treatment between void growth and void coalescence lying in the assumed boundary conditions and propensity for microscopic localization.

Structure of Constitutive Relations

All porous metal plasticity models to be presented below share a common structure in their derivation. The macroscopic yield surface is parametrically defined as
$$ {\displaystyle {\varSigma}_{ij}}=\frac{\partial \varPi }{\partial {D}_{ij}}\left(\mathbf{D}\right) $$
(1)
where Π(D) is the macroscopic plastic dissipation associated with D (see Nomenclature):
$$ \Pi \left(\mathbf{D}\right)=\underset{\mathrm{d}\in \mathcal{K}\left(\mathbf{D}\right)}{ \inf}\left\langle \underset{\boldsymbol{\upsigma} *\in \mathcal{C}}{ \sup }{\sigma}_{ij}^{*}{d}_{ij}\right\rangle \Omega $$
(2)
Here, \( \mathcal{C} \) denotes the microscopic (convex) domain of reversibility (the elasticity domain in small transformations) and \( \mathcal{K}\left(\mathbf{D}\right) \) the set of kinematically admissible microscopic deformations. If uniform strain-rate boundary conditions are assumed, then
$$ \mathcal{K}\left(\mathbf{D}\right)=\left\{\left.\mathbf{d}\right|\forall \mathbf{x}\in \Omega \backslash \omega, {d}_{kk}=0\kern1em \mathrm{and}\kern1em \exists \mathbf{v},\forall \mathbf{x}\in \Omega, \kern0.5em {d}_{ij}=\frac{1}{2}\left({v}_{ij}+{v}_{j,i}\right)\kern0.5em \mathrm{and}\kern0.5em \forall \mathbf{x}\in \partial \Omega, \kern0.5em {v}_i={D}_{ij}{x}_{ij}\right\} $$
(3)
and localized modes of deformation within Ω are precluded. To account for these, other types of boundary conditions must be used. The former are typically employed in constructing void growth models, the latter in void-coalescence models.
To obtain expressions for Π in closed form, trial velocity fields are used. Therefore, the basic elements of a micromechanical model are:
  1. (i)

    The geometry of the RVE

     
  2. (ii)

    A microscale plasticity model, i.e., the boundary of \( \mathcal{C} \) with the flow rule being necessarily associative

     
  3. (iii)

    Kinematically admissible microscale velocity fields defining a subset of \( \mathcal{K}\left(\mathbf{D}\right) \)

     

Bounding properties of the macroscopic dissipation are available, as discussed by Benzerga and Leblond (2010). Elimination of D from Eq. 1 leads to a macroscopic yield criterion of the form Φ(Σ; ISVs) = 0 with an associated flow rule. Here “ISVs” refers to a collection of internal state variables with a definite microstructural significance.

Microstructural variables, e.g., the porosity f, enter Π, hence Φ, by way of homogenization. Void growth, rotation, and coalescence result from the evolution of these variables.

For example, the time rate of change of f results from plastic incompressibility at the microscale:
$$ \dot{f}=\left(1-f\right){D}_{kk}=\left(1-f\right)\Lambda \frac{\partial \Phi}{\partial {\varSigma}_{\mathrm{m}}}, $$
(4)
so that \( \dot{f} \) derives directly from the yield criterion by normality. For anisotropic models with one or more void shape parameters, additional evolution equations are required for the void shapes and orientations. Void-coalescence models are inherently anisotropic.

Void Growth

Gurson Model

Gurson (1977) used a different method to obtain his yield function. The same yield function can be arrived at using the approach outlined above:
  1. (i)

    Geometry:

    The RVE is a hollow sphere containing a concentric spherical void. (A variant of the model exists for a cylindrical RVE.) The porosity f is the only microstructural variable entering the model.

     
  2. (ii)

    Plasticity model:

    An associated J 2 flow theory is used for the matrix with the yield criterion and flow rule written as

     
$$ {\sigma}_{\mathrm{eq}}\equiv \left\Vert \boldsymbol{\upsigma} \right\Vert =\overline{\sigma},\kern2.5em \mathbf{d}=\frac{3}{2}\frac{d_{\mathrm{eq}}}{\overline{\sigma}}\boldsymbol{\upsigma} \mathit{\hbox{prime}} $$
(5)
  1. (iii)

    Velocity fields:

     
$$ \forall \mathbf{x}\in \Omega \backslash \omega, \kern1em {v}_i\left(\mathbf{x}\right)=A{v}_i^A\left(\mathbf{x}\right)+{\beta}_{ij}{x}_j,\kern1em {\mathbf{v}}^A\left(\mathbf{x}\right)=\frac{1}{r^2}{\mathbf{e}}_r $$
(6)
where a mix of Cartesian and spherical coordinates is used for convenience. Scalar A and symmetric tensor β are parameters (with β kk = 0).
On that basis, obtain a bounding dissipation function as
$$ \Pi \left(\mathbf{D}\right)=\overline{\sigma}{\left[2{D}_{\mathrm{m}}{ \sinh}^{-1}\left(\frac{2{D}_{\mathrm{m}}x}{D_{\mathrm{eq}}}\right)-\sqrt{4{D}_{\mathrm{m}}^2+\frac{D_{\mathrm{eq}}^2}{x^2}}\right]}_{x=1}^{x=1/f} $$
(7)
Elimination of D from Eq. 1 then leads to the well-known Gurson yield function:
$$ {\boldsymbol{\Phi}}^{\mathrm{Gurson}}\left(\boldsymbol{\Sigma}; f\right)\equiv \frac{{\displaystyle {\varSigma}_{\mathrm{eq}}^2}}{{\overline{\sigma}}^2}+2f\kern1em\cosh \left(\frac{3}{2}\frac{{\displaystyle {\varSigma}_{\mathrm{m}}}}{\overline{\sigma}}\right)-\left(1+{f}^2\right) $$
(8)

In the limit f → 0 (dense matrix) criterion (8) reduces to the von Mises yield criterion (5)1.

Incorporating Plastic Anisotropy

Benzerga and Besson (2001) generalized the Gurson model to a class of plastically anisotropic solids (the cylindrical case was also treated):
  1. (i)

    Geometry:

    The RVE is the hollow sphere model so that the porosity f is the only void-related microstructural variable entering the model.

     
  2. (ii)

    Plasticity model:

    The matrix is taken to obey Hill’s quadratic associated yield criterion:

     
where \( \overline{\sigma} \) is the yield stress of the material in some reference direction and
Fourth-rank tensors Open image in new window and Open image in new window are symmetric, positive definite.
  1. (iii)

    Velocity fields:

     
$$ \forall \mathbf{x}\in \varOmega \backslash \omega, \kern1em {v}_i\left(\mathbf{x}\right)=A{v}_i^A\left(\mathbf{x}\right)+{\beta}_{ij}{x}_j,\kern1em {\mathbf{v}}^A\left(\mathbf{x}\right)=\frac{1}{r^2}{\mathrm{e}}_r $$
(11)
where A and β (β kk = 0) are parameters. These are the same velocity fields used by Gurson; cf. Benzerga and Leblond (2010) for a discussion.
An upper bound of the dissipation potential is then
$$ \prod \left(\mathbf{D}\right)=\overline{\sigma}{\left[h{D}_{\mathrm{m}}{ \sinh}^{-1}\left(\frac{h{D}_{\mathrm{m}}x}{D_{\mathrm{eq}}}\right)-\sqrt{h^2{D}_{\mathrm{m}}^2+\frac{D_{\mathrm{eq}}^2}{x^2}}\right]}_{x=1}^{x=1/f} $$
(12)
so that Benzerga and Besson’s yield function reads
h being an invariant of tensor Open image in new window . In axes pointing toward the principal directions of matrix orthotropy, h admits the following expression (Benzerga and Besson 2001):
$$ h=2{\left[\frac{2}{5}\frac{h_1+{h}_2+{h}_3}{h_1{h}_2+{h}_2{h}_3+{h}_3{h}_1}+\frac{1}{5}\left(\frac{1}{h_4}+\frac{1}{h_5}+\frac{1}{h_6}\right)\right]}^{\frac{1}{2}} $$
(14)

For an isotropic matrix, Open image in new window = \( \mathbb{I} \) and h = 2 so that the yield function reduces to that of Gurson. For a dense matrix (f = 0) criterion (13) reduces to Hill’s quadratic criterion.

Quite recently, Stewart and Cazacu (2011) generalized the above model to a class of anisotropic materials exhibiting tension-compression asymmetry, e.g., hexagonal closed-packed polycrystals. The matrix was taken to obey an associated quadratic yield criterion of a general family of non-quadratic criteria (Cazacu et al. 2006):
$$ {\sigma}_{\mathrm{eq}}\equiv \sqrt{\left(\left|{\widehat{\sigma}}_i\right|-k{\widehat{\sigma}}_i\right)\left(\left|{\widehat{\sigma}}_i\right|-k{\widehat{\sigma}}_i\right)}=\overline{\sigma},\kern1em \widehat{\boldsymbol{\sigma}}=\mathbb{L}:{\boldsymbol{\sigma}}^{\boldsymbol{\prime}} $$
(15)
where \( \overline{\sigma} \) as above and \( \mathbb{L} \) an invertible tensor invariant with respect to the orthotropy group satisfying major and minor symmetries such that L iikl = const for k = l. Under axisymmetric loadings, their approximate macroscopic yield function takes the same form as Eq. 13 with the quadratic term replaced with Σeq. defined as in Eq. 15 and a variant of coefficient h in Eq. 14 appears in the exponential term.

Void Shape Effects: Case of Spheroids

Gologanu and coworkers incorporated the anisotropy due to void shape in a series of models: for prolate voids (Gologanu et al. 1993) and for oblate ones (Gologanu et al. 1994). Later, Gologanu et al. (1997) developed an improved model whose general lines are recalled next:
  1. (i)

    Geometry:

    The RVE is a hollow spheroid containing a confocal spheroidal cavity. In addition to the porosity f, the model involves one void aspect ratio, W, and the common void axis, e3, as microstructural variables.

     
  2. (ii)

    Plasticity model:

    The isotropic, associated J 2 flow theory is used for the matrix; cf. Eq. 5.

     
  3. (iii)
    Velocity fields:
    $$ \forall \mathbf{x}\in \Omega /\omega, \kern1.5em {v}_i\left(\mathrm{x}\right)=A{v}_i^A\left(\mathrm{x}\right)+{\beta}_{ij}{x}_j, $$
    (16)
    where v A gives rise to a nonuniform deformation field. It is given by four terms of the axisymmetric expansion field derived by Lee and Mear (1992) involving associated Legendre functions of the first and second kinds. As above, scalar A and symmetric tensor β are parameters (with β kk = 0).
     
On that basis, an estimate of the dissipation function is obtained in an implicit form. After a series of approximations, the GLD yield function Φ GLD(Σ; f, W, e3) is given by
$$ {\Phi}^{\mathrm{GLD}}=C\frac{{\left\Vert \varSigma \mathit{\hbox{prime}}+\eta {\varSigma}_h\mathbf{Q}\right\Vert}^2}{{\overline{\sigma}}^2}+2\left(g+1\right)\left(g+f\right) \cosh \left(\kappa \frac{\varSigma :\mathbf{X}}{\overline{\sigma}}\right)-{\left(g+1\right)}^2-{\left(g+f\right)}^2 $$
(17)
Here, Q and X are transversely isotropic tensors given by
$$ \mathbf{X}\equiv {\alpha}_2\left({\mathrm{e}}_1\otimes {\mathrm{e}}_1+{\mathrm{e}}_2\otimes {\mathrm{e}}_2\right)+\left(1-2{\alpha}_2\right){\mathrm{e}}_3\otimes {\mathrm{e}}_3 $$
(18)
$$ \mathbf{Q}\equiv -\frac{1}{3}\left({\mathrm{e}}_1\otimes {\mathrm{e}}_1+{\mathrm{e}}_2\otimes {\mathrm{e}}_2\right)+\frac{2}{3}{\mathrm{e}}_3\otimes {\mathrm{e}}_3 $$
(19)

Σ h ≡ Σ : X is a weighted average of the normal stresses along the principal axes of the void and e1, e2 are arbitrarily chosen transverse unit base vectors. Also, κ, α 2, g, C and n are scalar-valued functions of microstructural parameters f and W. In the limit of a spherical void W → 1, Eq. 17 reduces to Gurson’s yield function (8), whereas for W → ∞ it reduces to Gurson’s criterion for cylindrical cavities. The von Mises yield criterion is obtained when setting f = 0 for W > 1 (prolate voids). In the case of oblate voids, the limit f → 0 corresponds to a material with a distribution of penny-shaped cracks.

The evolution of porosity is obtained by specializing Eq. 4 to Φ = ΦGLD. Void shape evolution is governed by
$$ \dot{S}=\frac{3}{2}\left[1+\left(\frac{9}{2}-\frac{T^2+{T}^4}{2}\right)\left(1-\sqrt{f}\right)\frac{\alpha_1-{\alpha}_1^{\mathrm{G}}}{1-3{\alpha}_1}\right]\kern0.1em {\mathrm{e}}_3\cdot {{\mathbf{D}}^{\mathbf{\prime}}}^{\mathrm{p}}\cdot {\mathrm{e}}_3+\left(\frac{1-3{\alpha}_1}{f}+3{\alpha}_2-1\right)\mathbf{I}:{\mathbf{D}}^{\mathrm{p}} $$
(20)
where S = ln W, T is the stress triaxiality ratio and α 1(f, W) and α 1 G (f, W) are given in Appendix A. The evolution of the void axis e3 is given by
$$ {\dot{\mathrm{e}}}_3=\mathbf{W}\cdot {\mathrm{e}}_3 $$
(21)
which assumes that the voids rotate with the material, W being the total material spin. This is clearly an approximation. An improved representation may be found in (Keralavarma and Benzerga 2010) on the basis of earlier work by Kailasam and Ponte Castaneda (1998).

Void Shape Effects: Case of Ellipsoids

More recently, Madou and Leblond (2012a, b) have implemented the homogenization approach of section Structure of Constitutive Relations to general ellipsoids:
  1. (i)

    Geometry:

    The RVE is an ellipsoidal volume containing a confocal ellipsoidal void. In addition to the porosity f, the model involves two void aspect ratios, W 1 and W 2, and the common void axes as microstructural variables.

     
  2. (ii)

    Plasticity model:

    The isotropic, associated J 2 flow theory is used for the matrix; cf. Eq. 5.

     
  3. (iii)

    Velocity fields:

    The authors used the fields discovered by Leblond and Gologanu (2008) provided in ellipsoidal coordinates and involving elliptic integrals.

     

The outcome of their analyses is a general yield function whose expression is omitted here. Even more recently, Madou and Leblond (2012a, b) have developed evolution laws for the microstructural variables of the model. They proposed heuristic corrections to the evolution of the void strain-rate and void axes. Their corrections are based on a large series of computationally efficient limit analyses.

Combined Plastic Anisotropy and Void Shape Effects

The homogenization problem combining the two kinds of anisotropies has been addressed by a number of authors in recent years. Thus, Monchiet et al. (2006, 2008) developed a solution based on consideration of the velocity fields used by Gologanu et al. (1993, 1994) in their earlier versions of the GLD model, and Keralavarma and Benzerga (2008) developed an improved solution using the richer Lee-Mear fields used by Gologanu et al. (1997). The latter model is, however, restricted to axisymmetric loadings and microstructures for which the void axis is aligned with one direction of material orthotropy.

Keralavarma and Benzerga (2010) developed a porous plasticity model for materials containing spheroidal voids embedded in a Hill matrix thus generalizing the GLD model to plastically anisotropic matrices. Their model is also a generalization of Benzerga and Besson’s (2001) model accounting for void shape effects:
  1. (i)

    Geometry:

    The RVE is a hollow spheroid containing a confocal spheroidal cavity. Porosity f, void aspect ratio, W, and void axis, e3, are microstructural variables.

     
  2. (ii)

    Plasticity model:

    The orthotropic, associated Hill flow theory is used for the matrix; cf. Eq. 9. The orthotropy axes are not necessarily aligned with the voids, L, T, and S referring to the principal directions.

     
  3. (iii)
    Velocity fields:
    $$ \forall \mathrm{x}\in \Omega /\omega, \kern2.5em {v}_i\left(\mathbf{x}\right)=A{v}_i^A\left(\mathbf{x}\right)+{\beta}_{ij}{x}_j, $$
    (22)
    where the inhomogeneous part v A of Eq. 6 2 is replaced with four terms of the axisymmetric expansion field due to Lee and Mear (1992) involving associated Legendre functions of the first and second kinds. As above, scalar A and symmetric tensor β are parameters (with β kk = 0). Here, β is not necessarily axisymmetric if one admits the ensuing approximations.
     
Their approximate yield function ΦKB(Σ; f, W, e3, Open image in new window ), applicable to non-axisymmetric loadings, reads
$$ {\boldsymbol{\Phi}}^{\mathrm{KB}}=C\frac{3}{2}\frac{\varSigma :\mathrm{\mathbb{H}}:\varSigma }{{\overline{\sigma}}^2}+2\left(g+1\right)\left(g+f\right) \cosh \left(\kappa \frac{\varSigma :\mathbf{X}}{\overline{\sigma}}\right)-{\left(g+1\right)}^2-{\left(g+f\right)}^2 $$
(23)
where the macroscopic anisotropy tensor ℍ is given by

Here, X and Q are defined as in Eq. 19 and criterion parameters κ, C, and η are scalar-valued functions of microstructural parameters (f and W) and of Open image in new window , whereas α 2 and g are only functions of f and W; cf. Appendix B.

For example, a simplified expression of κ is
$$ \kappa =\left\{\begin{array}{ll}\frac{3}{h}{\left\{1+\frac{h_{\mathrm{t}}}{h^2\mathrm{In}\ f}\mathrm{In}\frac{1-{e}_2^2}{1-{e}_1^2}\right\}}^{-1/2}\hfill & \left(\mathrm{p}\right)\hfill \\ {}\frac{3}{h}{\left\{1+\frac{\left({g}_f-{g}_1\right)+\frac{4}{5}\left({g}_f^{5/2}-{g}_1^{5/2}\right)-\frac{3}{5}\left({g}_f^5-{g}_1^5\right)}{\mathrm{In}\left({g}_f/{g}_1\right)}\right\}}^{-1}\hfill & \left(\mathrm{o}\right)\hfill \end{array}\right. $$
(25)
where (p) and (o) stand for prolate and oblate, respectively, and g x = g/(g + x). The dependence of the criterion parameters upon anisotropy tensor Open image in new window enters through one invariant, h, and two transversely isotropic invariants, h t and h q, of that tensor. When expressed in the basis associated with the principal directions of orthotropy (in the context of this section, this means replacing indices 1 to 6 in Eq. 14 with L, T, S, TS, SL, and LT, respectively.), invariant h is given by Eq. 14, while h t and h q are given by
$$ {h}_{\mathrm{t}}=\frac{1}{5}\left[-\frac{13}{12}\left({\widehat{h}}_{\mathrm{L}}+{\widehat{h}}_{\mathrm{T}}\right)+\frac{8}{3}{\widehat{h}}_{\mathrm{S}}+4\left({\widehat{h}}_{\mathrm{T}\mathrm{S}}+{\widehat{h}}_{\mathrm{S}\mathrm{L}}\right)-\frac{7}{2}{\widehat{h}}_{\mathrm{L}\mathrm{T}}\right] $$
(26)
and

Here, the \( {\widehat{h}}_i \) are the components of Open image in new window expressed using Voigt’s condensation. h q only appears in the expressions of C and η (it was denoted \( {\widehat{h}}_q \) in Keralavarma and Benzerga (2010).

In the special case of an isotropic von Mises matrix ( Open image in new window = Open image in new window = \( \mathbb{I} \)), the yield condition (23) reduces to the GLD criterion. In the case of spherical voids in a Hill matrix, one obtains \( \underset{W\to 1}{ \lim }{\alpha}_2=1/3,\kern0.5em C=1,\kern0.5em \eta =0 \) and η = 0. Also, Eq. 25 reduces to κ BB = 3/h and the upper-bound yield criterion of Benzerga and Besson (2001) is recovered. In particular, the Gurson yield function is obtained in the limit of spherical voids in an isotropic matrix since Open image in new window = \( \mathbb{I} \) implies k bb = 3/2. In the limit of cylindrical voids in a Hill matrix with eS = e3, we have \( \underset{W\to \infty }{ \lim }{\alpha}_2=1/2,\kern0.5em C=1,\kern0.5em \eta =0 \) and Eq. 25 reduces to
$$ {\kappa}^{\mathrm{cyl}}=\sqrt{3}{\left[\frac{1}{4}\frac{h_{\mathrm{L}}+{h}_{\mathrm{T}}+4{h}_{\mathrm{S}}}{h_{\mathrm{L}}{h}_{\mathrm{T}}+{h}_{\mathrm{T}}{h}_{\mathrm{S}}+{h}_{\mathrm{S}}{h}_{\mathrm{L}}}+\frac{1}{2{h}_{\mathrm{L}\mathrm{T}}}\right]}^{-\frac{1}{2}} $$
(28)
which is the result obtained by Benzerga and Besson (2001). In particular, the Gurson yield function for cylindrical cavities in a von Mises matrix is recovered with \( {\kappa}^{\mathrm{cyl}}=\sqrt{3} \) in that case.

Keralavarma and Benzerga (2010) supplemented yield criterion (23) with evolution laws for the microstructural variables f, W, and the void axis e3. The first two are in essence similar to those used in the GLD model, but the latter one employs an Eshelby concentration tensor for the spin following a proposal by Kailasam and Ponte Castaneda (1998).

Void Coalescence

If void growth could proceed until failure (complete loss of stress carrying capacity) as modeled in the previous section, then a good estimate of void coalescence would be when the lateral void size (along x 3) has reached the lateral void spacing. The void size relative to its initial value is typically what a void growth model delivers. The current void spacing can directly be inferred from the initial void spacing and deformation history. This approach would lead to a considerable overestimation of ductility and other fracture properties. This holds even if anisotropic void growth models are employed. To illustrate this, recall that typical values of the critical void growth ratio needed in failure models (Beremin 1981; Johnson and Cook 1985) fall between 1.2 and 2.0. On the other hand, typical values of the ratio of initial void spacing to void size are within the range 10–100, possibly larger. Even after considering the deformation-induced decrease in lateral spacing, an important gap remains. The reason for this is that the void growth models of Section 3 assume that plastic flow takes place in the whole RVE. It is now established that certain modes of localized plastic deformation would deliver lower values of the plastic dissipation Π, hence are more likely to prevail after sufficient microstructural evolution.

In this context, microstructure evolution refers to changes in the geometrical configuration of voids, as described by their relative size and spacing. Void coalescence is an inherently directional void growth process. In all the models presented below, void coalescence is assumed to take place in the x 1x 2 plane with the major applied normal stress being along the x3 direction.

Coalescence Under Predominately Tensile Loads

Thomason’s Model

Thomason (1985) posed the following limit-analysis problem:
  1. (i)

    Geometry:

    The RVE is a square-prismatic cell containing a cylindrical void with a square basis, the height of the void being smaller than the cell’s height. This geometry is determined by the void aspect ratio W (i.e., the height to breadth ratio), the cell aspect ratio λ, and a relative ligament size χ, which is the ratio of void breadth to cell breadth, the latter representing the void spacing transverse to the major stress.

     
  2. (ii)

    Plasticity model:

    The isotropic, associated J 2 flow model (5) is used for the matrix, but only in the central region Ωlig containing the intervoid ligament. The regions above and below the void are modeled as rigid.

     
  3. (iii)
    Velocity fields: (in the intervoid ligaments only)
    $$ \forall \mathbf{x}\in {\Omega}_{\mathrm{lig}}/\omega, \kern1em \mathbf{v}=\frac{A}{2}\left[\left(\frac{L^2}{x_1}-{x}_1\right){\mathbf{e}}_1+{x}_2\left(\frac{L^2}{x_1^2}-1\right){\mathbf{e}}_2+2{x}_3{\mathbf{e}}_3\right] $$
    (29)
    where A is a constant set by the boundary conditions. The above velocity field gives rise to a state of uniaxial extension of the cell (D 11 = D 22 = 0 and D 33 ≠ 0). As a consequence, the dissipation is only a function of D 33 so that the yield criterion only depends on Σ33. Note that under such circumstances, the criterion will be insensitive to variations in λ.
     
Thomason did not solve the above problem in closed form. He obtained numerical solutions to which he proposed an empirical fit. His yield function may be expressed as follows:
$$ {\boldsymbol{\Phi}}^{\mathrm{T} \hom}\left(\varSigma; W,\upchi \right)=\frac{\varSigma_{33}}{\overline{\sigma}}-\left(1-{\upchi}^2\right)\left(0.1{\left(\frac{\upchi^{-1}-1}{W}\right)}^2+1.2\sqrt{\upchi^{-1}}\right)\equiv \frac{\varSigma_{33}}{\overline{\sigma}}-\frac{\varSigma_{{}^{33}}^{\mathrm{T}}}{\overline{\sigma}} $$
(30)

In the limit χ → 1 the square void fills the ligament and criterion (30) reduces to Σ33 = 0 so that all stress-bearing capacity vanishes. In the limit W → 0 (flat void) Σ 33 T → ∞ and the criterion is never met. This deficiency is believed to have limited consequences in materials failing after some significant void growth.

Thomason did not supplement his yield criterion with evolution equations for the microstructural variables W and χ. As a consequence, his yield criterion has essentially been used as a criterion for the onset of void coalescence, which is often sufficient to estimate strains to failure as a function of loading parameters, such as the stress-state triaxiality (Lassance et al. 2007). For ductile fracture simulations, however, criterion (30) must be supplemented with evolution laws for W and χ. This task was undertaken by Pardoen and Hutchinson (2000) and Benzerga (2002) who have proposed additional heuristic extensions of the above criterion.

A Complete Void-Coalescence Model

Benzerga (2002) posed the following limit-analysis problem:
  1. (i)

    Geometry:

    The RVE is a cylindrical cell containing a spheroidal void. This geometry is determined by the void aspect ratio W, the cell aspect ratio λ, and the relative ligament size, χ, which is the ratio of void diameter to cell diameter. The latter represents the void spacing transverse to the major stress.

     
  2. (ii)

    Plasticity model:

    The isotropic, associated J 2 flow model (5) is used for the matrix, but only in the central region Ωlig containing the intervoid ligament. The regions above and below the void are modeled as rigid.

     
  3. (iii)
    Velocity fields: (in the intervoid ligaments only)
    $$ \forall \mathbf{x}\in {\Omega}_{\mathrm{lig}}\backslash \omega, \kern2.5em {v}_i\left(\mathbf{x}\right)=A{v}_i^A\left(\mathbf{x}\right)+{\beta}_{ij}{x}_j, $$
    (31)
    where v A contains the same four terms of the axisymmetric Lee-Mear field, Eq. 16. Here, the constants A and β are in principle determined by the boundary conditions, which are not of the homogeneous kind.
     
The above problem is mathematically more involved than Thomason’s. Benzerga used some numerical solutions to which he proposed an empirical fit. The numerical results were taken from Gologanu (1997) who assumed the GLD model to hold in the central porous layer. Benzerga’s approximate yield function reads
$$ \begin{array}{lll}{\boldsymbol{\Phi}}^{\mathrm{B}\mathrm{enz}}\left(\varSigma; W,,,\upchi \right)=\frac{\varSigma_{33}}{\overline{\sigma}}-\left(1-{\upchi}^2\right)\left(\alpha {\left(\frac{\upchi^{-1}-1}{W^2+0.1{\upchi}^{-1}+0.02{\upchi}^{-2}}\right)}^2+\beta \sqrt{\upchi -1}\right) \\ \equiv \frac{\varSigma_{33}}{\overline{\sigma}}-\frac{\varSigma_{33}^{\mathrm{B}}}{\overline{\sigma}} \end{array} $$
(32)
with α = 0.1 and β = 1.3. This approximation is better than Thomason’s for W < 0.5 and removes the deficiency in the limit W → 0 (penny-shape crack) since Σ 33 B admits a finite limit in that case. This correction is believed to have important consequences in materials failing after some limited void growth. Pardoen and Hutchinson (2000) proposed another heuristic extension of criterion (30) in which the factors α and β were taken to vary with the strain-hardening exponent. Such a fit was based on a series of finite-element cell model calculations.
Benzerga (2002) also derived (the velocity fields used in the limit analysis were not used to derive the evolution equations) evolution equations for the state variables W and x on the basis of matrix incompressibility, boundary conditions, and cell model phenomenology. A shape factor γ was introduced in addition to W. The void shape was taken to evolve from spheroidal (γ = 1/2) at the onset of internal necking (χ = χ c ) to conical (γ = 1) at complete coalescence (χ = 1). The evolution equations are as follows:
$$ \dot{\upchi}=\frac{3}{4}\frac{\uplambda}{W}\left[\frac{3\upgamma}{\upchi^2}-1\right]{D}_{\mathrm{eq}}+\frac{\upchi}{2\upgamma}\dot{\upgamma}, $$
(33)
$$ \dot{W}=\frac{9}{4}\frac{\uplambda}{\upchi}\left[1-\frac{\upgamma}{\upchi^2}\right]{D}_{\mathrm{eq}}-\frac{W}{2\upgamma}\dot{\upgamma}, $$
(34)
$$ \dot{\upgamma}=\frac{1}{2\left(1-{\upchi}_c\right)}\dot{\upchi} $$
(35)
where λ represents the current value of the void spacing ratio, which is updated through
$$ \dot{\uplambda}=\frac{3}{2}\uplambda \kern0.5em {D}_{\mathrm{eq}}. $$
(36)

The void and cell axes were tacitly taken to rotate with the material as per Eq. 21.

Thomason’s Model Revisited

Recently, Benzerga and Leblond (2014) have revisited Thomason’s analysis by considering a circular cylindrical geometry and a velocity field appropriate for the constrained plastic flow configuration. They obtained a fully analytical expression for the effective yield criterion. Their closed-form expression can be used instead of Thomason’s empirical relation (30). It also constitutes a first step toward some useful generalizations to other geometries and general loadings, which are lacking to date.

Coalescence Under Combined Tension and Shear

Very recently, Tekoglu et al. (2012) have proposed a void-coalescence model applicable under combined shear and tension:
  1. (i)

    Geometry:

    The RVE is a “sandwich” made of three superposed planar layers. Only the central layer contains some porosity. The void shape need not be specified in this model. This geometry is determined by the volume fraction of the porous layer, c, and the porosity within it, f b . If the geometry is further specified as that considered by Thomason or Benzerga and Leblond (2014), then f b = χ2 and c = Wχ/λ using the same notations as above.

     
  2. (ii)

    Plasticity model:

    The isotropic, associated J 2 flow model (5) is used for the matrix but only in the central region Ωlig containing the intervoid ligament. The top and bottom layers are modeled as rigid.

     
  3. (iii)
    Velocity fields: (in the intervoid ligaments only)
    $$ \forall \mathbf{x}\in {\Omega}_{\mathrm{lig}}\backslash \omega, \kern1em {v}_i\left(\mathbf{x}\right)=A{v}_{{}^i}^A\left(\mathbf{x}\right)+{\beta}_{ij}{x}_j, $$
    (37)
    where the second field accommodates shear deformation with β being a constant, traceless symmetric tensor. The only nonzero components of β are
    $$ {\beta}_{13}=2c{D}_{13};\kern2em {\beta}_{23}=2c{D}_{23} $$

    Above, v A is the field that would prevail under pure triaxial tension (no shear). It is not explicitly specified but could be taken as Thomason’s field, Eq. 29, if the geometry is made explicit. The constant A is fully determined by the boundary conditions, which are not of the homogeneous kind.

     
Under such circumstances and without specifying the field v A , Tekoglu et al. (2012) obtain an approximate yield function of the quadratic type
$$ {\boldsymbol{\Phi}}^{\mathrm{TLP}}\left(\varSigma; W,,,\upchi \right)=\frac{\varSigma_{33}}{\varSigma_{33}^A}+\frac{3\left({\varSigma}_{13}^2+{\varSigma}_{23}^2\right)}{{\left(1-{f}_{\kern-0.25em b}\right)}^2{\overline{\sigma}}^2}-1 $$
(38)
where Σ 33 A refers to either Σ 33 T in Eq. 30 or Σ 33 B in Eq. 32. Hence, in the absence of any shear loading, the criterion (38) reduces to either Eqs. 30 or 32.

Description of Two Integrated Models

GTN Model

The most widely used model of ductile damage is the Gurson-Tvergaard-Needleman (GTN) model . It is based on the Gurson model with some heuristic, often micromechanically motivated extensions to incorporate hardening and viscous flow, void interactions, void nucleation, and void coalescence. Within a convective representation of finite deformation viscoplasticity, additive decomposition of the total rate of deformation D is assumed with the plastic part Dp obtained from the flow potential (Gurson 1977; Pan et al. 1983)
$$ {\mathrm{\mathcal{F}}}^{\mathrm{GTN}}=\frac{\varSigma_{\mathrm{eq}}^2}{{\overline{\sigma}}^2}+2{q}_1\kern0.2em {f}^{*} \cosh \left(\frac{3{q}_2{\varSigma}_{\mathrm{m}}}{2\overline{\sigma}}\right)-1-{\left({q}_1\kern0.2em {f}^{*}\right)}^2=0 $$
(39)
by assuming equality of macroscopic plastic work rate and matrix dissipation
$$ {\mathbf{D}}^{\mathrm{p}}=\left[\frac{\left(1-f\right)\overline{\sigma}\dot{\overline{\mathit{\in}}}}{\varSigma :\frac{\partial \mathrm{\mathcal{F}}}{\partial \varSigma }}\right]\frac{\partial \mathrm{\mathcal{F}}}{\partial \varSigma } $$
(40)
Here, \( \overline{\sigma} \) is the matrix flow strength and q 1 and q 2 are parameters introduced by Tvergaard (1981). The function f *(f) was introduced by Tvergaard and Needleman (1984) to account for the effects of rapid void coalescence at failure
$$ {f}^{*}=\left\{\begin{array}{ll}f\hfill & f<{f}_c\hfill \\ {}{f}_c+\left({f}_u^{*}-{f}_c\right)\left(f-{f}_c\right)/\left({f}_f-{f}_c\right)\hfill & f\ge {f}_c\hfill \end{array}\right. $$
(41)

The constant f u * = 1/q 1 is the value of f * at zero stress. As ff f and f *f u *, the material loses all stress carrying capacity. Equation 41 is a phenomenological description of coalescence involving two parameters, f c and f f , which are both material dependent and stress-state dependent (Koplik and Needleman 1988). Appropriate values for f c and f f can be derived using predictive micromechanical models (Benzerga et al. 1999; Benzerga 2002), which assume that coalescence occurs through an internal necking mechanism. Strain-rate effects can be accounted for through a relation \( \dot{\overline{\epsilon}}\left(\overline{\sigma},\overline{\epsilon}\right) \) , e.g., Benzerga et al. (2002b), where \( \dot{\overline{\epsilon}} \) is the effective strain rate and \( \overline{\epsilon}={\displaystyle \int \dot{\overline{\epsilon}}\ \mathrm{d}t} \) is the effective plastic strain.

To account for void nucleation, the rate of increase of the void volume fraction is given by

$$ \dot{f}={\dot{f}}_{\mathrm{growth}}+{\dot{f}}_{\mathrm{nucleation}} $$
(42)
where the first term accounts for the growth of existing voids through Eq. 4 and the second term represents the contribution from void nucleation. For example, the nucleation of voids by a strain controlled mechanism is modeled using (Chu and Needleman 1980)
$$ {\dot{f}}_{\mathrm{nucleation}}=\mathbf{\mathcal{D}}\dot{\overline{\epsilon}} $$
(43)
with
$$ \mathbf{\mathcal{D}}=\frac{f_N}{s_N\sqrt{2\pi }} \exp \left[-\frac{1}{2}{\left(\frac{\overline{\epsilon}-{\epsilon}_N}{s_N}\right)}^2\right] $$
(44)

Proposed Model

Within a finite deformation framework, a corotational formulation of the constitutive equations can be used (Benzerga et al. 2004). The total rate of deformation D is written as the sum of an elastic and a plastic part:
$$ \mathbf{D}={\mathbf{D}}^{\mathrm{e}}+{\mathbf{D}}^{\mathrm{p}} $$
(45)
Elasticity is included through a hypoelastic law:
$$ {\mathbf{D}}^{\mathrm{e}}={\mathrm{\mathbb{C}}}^{\hbox{--} 1}:\dot{\mathbf{P}} $$
(46)
where ℂ is the rotated tensor of elastic moduli and P is the rotated stress:
$$ \mathbf{P}=J{\boldsymbol{\Omega}}^T\cdot \varSigma \cdot \kern0.5em \boldsymbol{\Omega} $$
(47)

Here, Ω is an appropriate rotation tensor; it is identified with the rotation R resulting from the polar decomposition of the deformation gradient F if the Green-Naghdi rate of Σ is used and \( \dot{\boldsymbol{\Omega}}\cdot {\boldsymbol{\Omega}}^T=\mathbf{W} \) if the Jaumann rate is used, W being the spin tensor. Also, J = det F.

To account for rate dependence, the plastic part of the rate of deformation, D p, is obtained by normality from the gauge function:
$$ \phi ={\sigma}_{\star }-\overline{\sigma}\left(\overline{\varepsilon}\right) $$
(48)
where \( \overline{\sigma} \) is the matrix flow stress, \( \overline{\varepsilon} \) is the effective plastic strain, and σ is an effective matrix stress which is implicitly defined through an equation of the type ℱ(Σ; ISVs, σ ) = 0 where “ISVs” refers to a collection of internal state variables with a definite microstructural significance. For a rate-independent material (standard plasticity), plastic flow occurs for ϕ = 0 and ϕ = 0. For a rate-dependent material (viscoplasticity), one has ϕ > 0 during plastic flow.
Prior to the onset of void coalescence, the potential ℱ admits an expression of the type Open image in new window where the relevant ISVs are the porosity f, the void aspect ratio W, the void axis e3, and Hill’s tensor Open image in new window , which describes the effect of the current texture. With the KB model as reference (cf. Eq. 23), the following expression may be used:
$$ {\mathbf{\mathcal{F}}}^{\left(c-\right)}=C\frac{3}{2}\frac{\varSigma :\mathrm{\mathbb{H}}:\varSigma }{\sigma_{\star}^2}+2{q}_w\left(g+1\right)\left(g+f\right) \cosh \left(\kappa \frac{\varSigma :\mathbf{X}}{\sigma_{\star }}\right)-{\left(g+1\right)}^2-{q}_w^2{\left(g+f\right)}^2 $$
(49)
where ℍ is given by Eq. 24, X and Q by Eq. 19, κ by Eq. 25, h by Eq. 14, and h t by Eq. 26 and the remaining criterion parameters C(f, W, Open image in new window ), η(f, W, Open image in new window ), α 2(f, W), and g(f, W) are given in Appendix B. Also, q W is a heuristic void-shape-dependent factor that was determined by Gologanu et al. (1997) to fit unit-cell results:
$$ {q}_w=1+\left(q-1\right)/ \cosh S $$
(50)
where q = 1.6 is the value taken by q w for a spherical void. The evolution laws of f, W and e3 are given by Eqs. 4, 20, and 21, respectively, where D should be replaced with D p and W with \( \dot{\boldsymbol{\Omega}}\cdot {\boldsymbol{\Omega}}^T,\kern0.5em \boldsymbol{\Omega} \) being the rotation tensor used in Eq. 47.
After the onset of void coalescence, the flow potential is given bywhere Σ 33 B is given by Eq. 32. For an arbitrary void shape between a spheroid and a cone, χ is exactly related to the void spacing ratio, λ, through a shape factor γ as
$$ \upchi =\left\{\begin{array}{ll}{\left[3\upgamma \frac{f}{W}\uplambda \right]}^{1/3}\hfill & \left(\mathrm{P}\right)\hfill \\ {}W{\left[3\upgamma \frac{f}{W}\uplambda \right]}^{1/3}\hfill & \left(\mathrm{T}\right)\hfill \end{array}\right. $$
(52)
where (P) and (T) are a shorthand notation for parallel and transverse loading, respectively. As χ → 1 the material loses all stress carrying capacity. At the onset of coalescence, we have ℱ(c−) = ℱ(c+) = 0. The evolution laws of the microstructural variables are given by Eqs. 33, 34, 35, and 36 along with Eq. 21.

A variant of the integrated model was implemented in a finite-element code and used to model ductile fracture in notched bars (Benzerga et al. 2004) and slant fracture in plane strain (Benzerga et al. 2002a). A fully implicit time integration procedure was used for the local behavior in conjunction with an iterative Newton-Raphson method. The consistent tangent matrix was computed as detailed in the case of prolate voids by Benzerga et al. (2002a).

Identification of Material Parameters

GTN Model

The following parameters enter the GTN model: q 1, q 2, f c , f f , f N, s N and ϵN. There is no straightforward procedure for identifying these parameters using standard experiments. In addition, some of the above parameters play interdependent roles. In practice, the first four of these parameters could be fixed based on micromechanical models or cell model calculations. Other model parameters pertain to the elastoplastic behavior of the matrix. The hardening response of the matrix is determined using uniaxial testing with appropriate large-strain corrections.

Proposed Model

One advantage of using the proposed model is that it involves fewer parameters, void nucleation set aside. Examination of the constitutive equations of section “Proposed Model” reveals actually no adjustable fracture parameter, unless the value of q = 1.6 is treated as a free parameter. When this model is used to predict crack initiation, only deformation-related parameters need to be calibrated on experiments. Account for plastic anisotropy may be necessary depending on the material. Anisotropic response is common in metal forming applications although it is often restricted to two-dimensional measurements. Within the confines of the integrated model above, this first step will deliver the basic hardening curve \( \overline{\sigma}\left(\overline{\varepsilon}\right) \) as well as the anisotropy tensor Open image in new window .

With the basic flow properties of the matrix calibrated, the next step is to determine the volume fraction, aspect ratio and relative spacing of damage initiation sites (inclusions, precipitates, etc.) in the average sense. Practically, this can be achieved by examining three perpendicular cross sections in optical microscopy, carrying out the needed two-dimensional measurements using digital image analysis, and finally operating standard stereology transformations to infer their 3D counterparts. The outcome of this step is the set of parameters f 0, W 0, and λ0 needed to initialize the state of the microstructure in constitutive Eqs. 48, 49, 50, 51, 52, 33, 34, 35, and 36. Other details on how to account for 3D aspects and void nucleation may be found in the review by Benzerga and Leblond (2010).

How to Use the Model

The proposed model as well as the GTN model may be used to model the initiation of a crack in an initially crack-free specimen or to model crack growth. In a first step, it is recommended to assume that void nucleation is instantaneous and occurs at a fixed value of the effective strain. Therefore, in simulations of ductile fracture, voids are considered to be present from the outset of plastic deformation. With all deformation-related quantities calibrated on experiments, model predictions can be compared with a sufficiently discriminating set of experiments (notched bars, CT specimens, plane strain bars, etc.) If the model predicts larger than measured fracture properties, one should attempt a threshold strain for nucleation. The hypothesis of a delayed or continuous nucleation could then be checked by metallographic examinations. If the latter do not corroborate the hypothesis, a parameter sensitivity analysis should be carried out by varying the initial values of microstructural parameters.

When failure by void coalescence is predicted at a material point, the material loses its stress-bearing capacity at this point. In a finite-element simulation, it is typical to represent this in terms of an element-vanish technique whereby the stresses and stiffness in the element are subsequently disregarded. This procedure allows computational simulations of crack growth. Obtaining mesh-independent predictions requires a length scale to be incorporated in the problem formulation. Several approaches with varying levels of refinement have been developed to that end but remain underutilized in fracture simulations for metal forming applications.

Notes

Acknowledgments

This research was supported by NPRP grant No 4-1411-2-555 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the author. Partial support from the National Science Foundation (Grant Number DMR-0844082) is gratefully acknowledged.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringTexas A&M UniversityCollege StationUSA
  2. 2.Department of Materials Science and EngineeringTexas A&M UniversityCollege StationUSA

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