On evolving deformation microstructures in non-convex partially damaged solids

Journal of the Mechanics and Physics of Solids 59 (2011) 1268–1290

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On evolving deformation microstructures in non-convex partially damaged solids 1 ¨ E. Gurses , C. Miehe  Institute of Applied Mechanics, Chair I, University of Stuttgart, 70550 Stuttgart, Pfaffenwaldring 7, Germany

a r t i c l e in f o

abstract

Article history: Received 10 October 2008 Received in revised form 29 December 2010 Accepted 2 January 2011 Available online 1 April 2011

The paper outlines a relaxation method based on a particular isotropic microstructure evolution and applies it to the model problem of rate independent, partially damaged solids. The method uses an incremental variational formulation for standard dissipative materials. In an incremental setting at finite time steps, the formulation defines a quasihyperelastic stress potential. The existence of this potential allows a typical incremental boundary value problem of damage mechanics to be expressed in terms of a principle of minimum incremental work. Mathematical existence theorems of minimizers then induce a definition of the material stability in terms of the sequential weak lower semicontinuity of the incremental functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of weak convexity notions of the stress potential. Furthermore, the variational setting opens up the possibility to analyze the development of deformation microstructures in the postcritical range of unstable inelastic materials based on energy relaxation methods. In partially damaged solids, accumulated damage may yield non-convex stress potentials which indicate instability and formation of fine-scale microstructures. These microstructures can be resolved by use of relaxation techniques associated with the construction of convex hulls. We propose a particular relaxation method for partially damaged solids and investigate it in one- and multi-dimensional settings. To this end, we introduce a new isotropic microstructure which provides a simple approximation of the multi-dimensional rank-one convex hull. The development of those isotropic microstructures is investigated for homogeneous and inhomogeneous numerical simulations. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Material instabilities Damage mechanics Non-convex variational problems Energy relaxation Microstructures

1. Introduction The paper discusses a relaxation technique based on a new isotropic microstructure evolution and its application to the model problem of rate independent partially damaged solids in a geometrically linear setting. The accumulation of damage leads to a softening in the stress response that initiates the formation of narrow patterns of high inelastic deformations in which high strain gradients prevail. This phenomenon is commonly denoted as the strain localization and has been documented for a wide range of materials. The first theoretical contribution is traced back to Hadamard (1903) in the

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E-mail address: [email protected] (C. Miehe). Current address: Computational Solid Mechanics Laboratory (CSML), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia. 1

0022-5096/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2011.01.002

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beginning of the previous century. In late 1950s theoretical investigations on material instability were conducted by Hill (1957, 1958) which are followed later by Thomas (1961) and Rice (1976). It is well known that the application of standard numerical solution schemes based on finite element methods to the simulation of such localization phenomena typically suffers from spurious mesh sensitivity and yields non-objective postcritical results. The reason of this effect is the ill-posedness of the incremental boundary value problem that makes a subsequent standard analysis to be physically meaningless. In the context of finite element formulations the crucial meshdependence was pointed out for example by de Borst (1987) and Belytschko et al. (1988) among others. A possible way to overcome this problem is the introduction of a length scale that enhances the standard continuum description. Length scales may be introduced through (i) a dependence of constitutive functions on gradients of the internal variables (e.g. Peerlings et al., 1996, 2001) or a definition of non-local averaged kinematic variables, (ii) an enhancement of the continuum description by couple stresses and conjugate kinematic quantities, or (iii) a consideration of viscosity (e.g. Needleman, 1988). See for example de Borst (2004) for an overview. In this paper, we propose an alternative approach based on an energy relaxation technique for the treatment of instabilities in strain-softening materials. It is based on the previous works of Miehe and Lambrecht (2003a,b) and Miehe et al. (2004), where strain-softening plasticity and the single-slip crystal plasticity have been investigated. The concept offers the following two perspectives to a localization analysis. Firstly, conditions of the material stability of inelastic solids are based on the weak convexity conditions of incremental energy potentials in analogy to treatments in finite elasticity. Secondly, localization phenomena are interpreted as microstructure developments associated with non-convex incremental energy potentials similar to elastic phase transformation problems. In this context microstructures are considered as complex patterns with length scales much smaller than characteristic macroscopic dimensions of the problem. It is possible to describe these microstructures mathematically by non-convex variational problems. Furthermore, it has been shown that the non-existence of minimizers in these variational problems are closely related to the convergence behavior of fine scale oscillatory infimizing sequences which are interpreted as microstructures. In particular, there is a strong parallelism between the microstructures that develop in martensitic phase transformations and fine scale oscillatory infimizing sequences of energy functionals describing phase transforming crystals, see e.g. Ball and James (1987), Chu and James (1995), James and Hane (2000) and Bhattacharya (2003). Elastic martensitic phase transformation in crystalline solids is one of the most successful application areas of the ¨ relaxation theory, see for example Luskin (1996), Muller (1999), Bhattacharya (2003), Dolzmann (2003) and Carstensen (2001). In recent years, simulation of martensitic microstructures based on different relaxation approaches have been performed by Govindjee and Miehe (2001), Carstensen (2001), Govindjee et al. (2002, 2007), Dolzmann (2003), Aubry et al. (2003), Bartels et al. (2004) and Kruzˇı´k et al. (2005). Furthermore, there are applications of the relaxation theory beyond the analysis of crystalline microstructures in martensitic transformations, see e.g. DeSimone and Dolzmann (2000, 2002) and Conti et al. (2002) for nematic elastomers undergoing a nematic to isotropic phase transformation. Relaxation theories can be applied to inelastic materials provided that a variational formulation of inelasticity is constructed. This is only possible in an incremental sense within discrete time steps. The set up of a general incremental variational formulation of inelasticity has been developed in the recent works (Miehe, 2002; Miehe et al., 2002, which generalized works on plasticity by Simo´ and Honein (1990), Ortiz and Stainier (1999) and Carstensen et al. (2002) towards the broad class of standard dissipative materials driven by threshold functions. Such a formulation of inelasticity is exclusively defined by an energy storage function and the dissipation function, where the latter is constructed form a threshold- or yield-function. A quasi-hyperelastic stress potential at discrete times is obtained from a local minimization problem of the constitutive response. The underlying variational formulation allows to define the stability of the incremental inelastic response in terms of terminologies used in elasticity theory. In other words, classical definitions of localized failure as outlined in Thomas (1961), Hill (1962) and Rice (1976) can be related to the convexity conditions of the incremental stress potential in analogy to treatments in finite elasticity, see e.g. Dacorogna (1989), Krawietz (1986), Ciarlet ˇ ´ (1997). (1988), Marsden and Hughes (1994) and Silhavy In the context of inelasticity, relaxation methods have recently been started to be studied. A relaxation algorithm based on a sequential lamination has been developed by Ortiz and Repetto (1999) and Ortiz et al. (2000) and applied for the modeling of dislocation structures in single crystals. On the basis of experimental observations, they employed strong assumptions with respect to the form and type microstructure, e.g. constant volume fraction of phases. Miehe and Lambrecht (2003a,b) have investigated phenomenological strain-softening plasticity models by using relaxation theory with substantial restrictions on shape of possible microstructures. Furthermore, a one-dimensional strain-softening plasticity model have been studied in Lambrecht et al. (2003b). Carstensen et al. (2002) analyzed the relaxation of single slip multiplicative plasticity which became later a canonical model problem, see for example Hackl and Hoppe (2003), Miehe et al. (2004) and Bartels et al. (2004) for relaxed formulations based on approximations of the quasiconvex envelope by polyconvex or rank-one convex envelopes. Later, Conti and Theil (2005) and Conti (2006) determined analytically the relaxation of a single-slip model based on a quasiconvex envelope in rigid plasticity without and with hardening, respectively. Conti and Ortiz (2005) showed that the relaxed response of single-slip plasticity in the geometrically linear theory is identical to multi-slip ideal plasticity and studied the influence of non-locality through dislocation line energies on types of microstructure patterns. Conti et al. (2007) extended the results of Conti and Ortiz (2005) to the hardening case and established a connection between the relaxation and enhanced-strain finite elements. Note that strain-softening plasticity and slip plasticity models have different origins of non-convexity. The former one has

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a non-monotonic stress function governed by non-convex incremental potentials while in the latter one geometrical constraints of the slip system cause the non-convexity. A softening mechanism different from plasticity is due to damage, which provides an alternative area for the application for relaxation methods. Francfort and Marigo (1993) were the first who investigated in the two-dimensional setting a brutal damage model with the help of an energetic stability criterion and the relaxation theory. They considered a non-smooth brutal damage evolution such that the elastic modulus upon satisfying a criterion suddenly drops to a lower but still positive value. They showed that minimizers in the form of a local fine mixtures of the damaged and undamaged material appear as a result of the relaxation. Furthermore, it has been demonstrated that the relaxed problem behaves as a progressive (continuous) damage model contrary to the original brutal damage model. We also refer to Schmidt-Baldassari and Hackl (2003), Lambrecht et al. (2003a), Francfort and Garroni (2006) and Mielke and Roubı´cˇek (2006) for alternative variational formulations of damage mechanics in connection with the relaxation theory. The key contribution of this paper is the outline of a novel relaxation technique for non-convex (unstable), partially damaged solids, which builds upon the introduction of a new isotropic microstructure evolution. The proposed method substantially enhances the work by Lambrecht et al. (2003a), where a first-order relaxation algorithm has been developed for a strictly two-dimensional isotropic damage mechanics model problem. In Lambrecht et al. (2003a), laminate-type microstructures have been employed in the relaxation algorithm. It has been observed that the laminate orientation and the principal shear directions are co-axial, if the damages affects only the isochoric part of the deformation. Based on these results, we propose in this work the evolution of particularly simple isotropic microstructures, which are composed of fine mixtures of two co-axial phases. It is shown that in a two-dimensional setting the relaxed response based on first-order laminates is identical to the proposed isotropic microstructures. Contrary to the model of Francfort and Marigo (1993), we consider a continuous saturation-type damage evolution. The maximum damage of the isochoric part of the deformation is controlled by a material parameter (saturation value), which is set such that the stiffness of the solid does not diminish completely. After the saturation value is reached, the material behaves elastically with a reduced stiffness. This response can micromechanically be motivated by the existence of two types of micro-bonds in the material. The first type of bonds, which is susceptible to damage and breakable, is responsible for the softening and the formation of isotropic microstructures, whereas the second one is unbreakable and gives the final elastic response upon reaching a saturation of damage. Such type of material response is conceptually observed in filled rubber-like materials, where chain–particle bonds may break but the ground-state network remains undamaged. In this spirit, we discuss scalar and multidimensional damage models and construct algorithms of the relaxation, which are shown to be very robust within finite element solutions of boundary value problems. 2. Variational formulation of local constitutive response The setting up of a general incremental variational formulation of inelasticity in this section follows closely the works Miehe (2002) and Miehe et al. (2002). 2.1. Internal variable formulation of inelasticity 2.1.1. General formulation Let e 2 SymðdÞ with d 2 ½1,2,3 be the symmetric strain tensor, governing a homogeneous local macro-deformation of a material at time t 2 R þ . Focusing on purely mechanical problems, the local response at x 2 B of the body B  Rd is assumed to be physically constrained by the Clausius–Planck inequality: _ Z 0, D :¼ r : e_ c

ð1Þ

where r denotes the stress tensor. The local energy storage is governed by an energy storage function c, that depends on the strains e and an internal variables array q 2 Rn , consisting of n scalar fields defined on B. Insertion of the free energy cðe,qÞ into (1) yields by a standard argument the constitutive equation for the stresses

r ¼ @e cðe,qÞ

ð2Þ

and the reduced dissipation inequality D ¼ p  q_ Z0 n

with p :¼ @q cðe,qÞ,

ð3Þ

where p 2 R is the internal force conjugate to q. A local model of inelasticity is required to be supplemented by an additional constitutive function which determines the evolution of the internal variables. A broad spectrum of inelastic _ solids is covered by the so-called standard dissipative media, where q_ is governed by a dissipation function fðq,qÞ depending on the flux q_ and q. Rate-independent models, treated in this work, are governed by dissipation functions that _ i.e. fðeq,qÞ _ _ ¼ efðq,qÞ are positively homogeneous of degree one with respect to the flux q, for all e 2 R þ . Such a function has a cone-like graph and is not differentiable at the point q_ ¼ 0. The subsequent formulation then needs a generalization of the differential operator of smooth functions to the notion of a sub-differential operator @ of non-smooth convex functions, see Moreau (1974, 1976), Halphen and Nguyen (1975) and Nguyen (2000). We define a convex set of admissible _ _ denoted as the elastic domain. The element p 2 E of the sub-differential forces E :¼ @q_ fð0,qÞ :¼ fpjp  q_ r fðq,qÞ for all qg,

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is a sub-gradient of f at q_ ¼ 0. The dissipation function governs the evolution of q by the differential equation _ 0 2 @q cðe,qÞ þ@q_ fðq,qÞ

with qðx,0Þ ¼ q0 ðxÞ

ð4Þ

often referred to as the Biot equation of standard dissipative systems, see Biot (1965), Ziegler (1963), Ziegler and Wehrli (1987), Germain (1973), Halphen and Nguyen (1975) and Nguyen (2000). The two constitutive equations (3) and (4) determine the stress response of a normal-dissipative material in a strain-driven process. Convexity of the dissipation function with respect to the first slot ensures the thermodynamic consistency (3)1. 2.1.2. Introduction of threshold functions An elastic domain of the internal forces is a key feature of rate-independent inelastic material models. In classical treatments of these models, the elastic domain E in the space of the internal driving forces p is directly described in the constitutive modeling. Let the level surface f ðpÞ ¼ cðqÞ with the threshold function cðqÞ describe the boundary @E of a convex, elastic domain E :¼ fpjf ðpÞ rcðqÞg. The level set function f is assumed to be convex, positively homogeneous of degree one and normalized to zero with respect to internal forces p. For a known elastic domain E, i.e. given functions f and c, the dissipation function f may be defined by a generalization of the classical principle of maximum dissipation _ ¼ supfp  qg: _ fðq,qÞ

ð5Þ

p2E

The problem (5) with inequality constraints can be solved by a Lagrange method and the associated Karush–Kuhn–Tucker equations determine the evolution of the internal variables along with the loading–unloading conditions q_ ¼ l@p f ðpÞ 4 l Z0 4 f ðpÞ r cðqÞ 4 lðf ðpÞcðqÞÞ ¼ 0:

ð6Þ

Insertion of (6) into (3)1 and exploiting the homogeneity of the level set function f results in the constitutive expression for the dissipation D ¼ cðqÞl Z0:

ð7Þ

Thermodynamic consistency is ensured by choosing a threshold function cðqÞ Z0 with positive image. 2.1.3. Specification to damage mechanics Isotropic damage mechanics described by a scalar internal damage variable q often uses simple threshold function f ðpÞ ¼ p,

ð8Þ

where the driving force p Z 0 dual to q is a positive energetic expression. Then (6) attains the simple form _ q_ Z 0 4 p rcðqÞ 4 qðpcðqÞÞ ¼0

ð9Þ

and the dissipation (7) reads D ¼ cðqÞq_ Z 0:

ð10Þ

2.2. Incremental variational formulation of inelasticity 2.2.1. General formulation A variational formulation of the above local constitutive problem of inelasticity, giving a consistent approximation of the differential equation (4) in a finite time increment ½tn ,tn þ 1  2 R þ , is constructed as follows. The key point is the definition of an incremental stress potential W, depending on the strains en þ 1 :¼ eðx,tn þ 1 Þ at time tn + 1, that determines the stresses rn þ 1 :¼ rðx,tn þ 1 Þ at tn + 1 by a quasi-hyperelastic function evaluation

rn þ 1 ¼ @e Wðen þ 1 Þ:

ð11Þ

The stress potential W covers within the time increment characteristics of both the storage function c as well as the dissipation function f. The potential is defined by the local constitutive variational problem Z tn þ 1  _ þ f dt ½c with qðx,tn Þ ¼ qn ðxÞ, ð12Þ Wðen þ 1 Þ ¼ inf q

tn

which optimizes an incremental work expression with respect to the path of the internal variables. In practical applications, the path within a typical time increment is governed by the choice of a certain integration algorithm. 2.2.2. Specification to damage mechanics In the strain-driven incremental setting of isotropic damage mechanics, the incremental update of the scalar internal variable can be written qn þ 1 ¼ pðen þ 1 Þ

ð13Þ

as a closed-form function of the current strain en þ 1 . Here, we denote p as the path function associated with the time increment [tn, tn + 1], see for example (69) and (86) below. With the path function p at hand, the incremental stress potential

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W in (12) appears in the closed form Wðen þ 1 Þ ¼ cðen þ 1 ,pðen þ 1 ÞÞcðen ,qn Þ þ Dðen þ 1 Þ in terms of the increment of dissipated work Z tn þ 1 Z pðen þ 1 Þ Dðen þ 1 Þ :¼ f dt ¼ cðqÞ dq tn

ð14Þ

ð15Þ

qn

obtained by integration of (10). 3. Stability of the incremental constitutive response The incremental variational formulation for inelastic solids outlined above provides a perspective towards a distinctive definition of material instabilities based on weak convexity notions. Existence results for boundary value problems of finite ˇ ´ (1997). The introduction of the ¨ elasticity are reviewed in Dacorogna (1989), Krawietz (1986), Muller (1999) and Silhavy incremental stress potential W in (14) allows an application of these results to boundary value problems of incremental inelasticity. This is achieved by applying statements on the weak convexity of the storage function c of elasticity to the incremental stress potential W of inelasticity. 3.1. The global incremental variational problem The variational formulation of the local constitutive response outlined above induces a variational formulation of the incremental boundary value problem of inelasticity. Let uðx,tÞ denote the displacement field on B at time t. A global incremental potential of the inelastic continuum associated with the increment [tn, tn + 1] may be defined by the functional Z Pðun þ 1 Þ ¼ Wðrs un þ 1 Þ dVPext ðun þ 1 Þ ð16Þ B

governed by is the incremental stress potential function W defined in (12) or (14), and the incremental loading contribution Z Z t n þ 1  ðun þ 1 un Þ dA: ð17Þ Pext ðun þ 1 Þ ¼ c n þ 1  ðun þ 1 un Þ dV þ B

@Bt

c denotes a given body force field on B and t a given traction field on @Bt . As usual, we consider a decomposition of the surface @B into a part @Bu , where the displacements are prescribed and a part @Bt where the tractions are given. The current displacement field of the inelastic solid is then determined by the incremental variational principle   inf Pðun þ 1 Þ , ð18Þ un þ 1 ¼ ARG un þ 1 2W

which minimizes the incremental potential for all admissible displacement fields u 2 W :¼ fu 2 W 1,p ðBÞ j u ¼ uðxÞ on @Bu g,

ð19Þ

which satisfy the Dirichlet-type boundary conditions. The Euler equations of the variational problem (18) div½@e Wðrs un þ 1 Þ ¼ c n þ 1 in B

and

½@e Wðrs un þ 1 Þ  n ¼ t n þ 1 on @Bt

ð20Þ

characterize the stress equilibrium and the Neumann-type boundary conditions at the current time tn + 1 in terms of the stresses rn þ 1 :¼ @e W obtained via (11) from the incremental constitutive potential W. 3.2. Weak convexity notions and check of stability We consider the sequentially weakly lower semicontinuity (swlsc) of the functional (16) as the key property for the existence of sufficiently regular minimizers of the variational problem (18). The internal part of the functional (16) is assumed to be sequentially weakly lower semicontinuous, if the incremental stress potential W defined in (14) is quasiconvex and satisfies some growth conditions, see for example Dacorogna (1989) and Acerbi and Fusco (1984). As ˇ ´ (1997, p. 387), the quasiconvexity of W is a necessary, and under technical hypotheses also a sufficient noted in Silhavy condition for the swlsc of the functional Pðun þ 1 Þ, i.e. Wðen þ 1 Þ quasiconvex þgrowth conditions ) Pðun þ 1 Þ swlsc:

ð21Þ

The quasiconvexity notion introduced by Morrey (1952), is the key property for the existence of sufficiently regular minimizers of the variational problem (18). For details of existence theorems in elasticity we refer to Ball (1977), Ciarlet ˇ ´ (1997) and the references cited therein. In the small-strain (1988), Dacorogna (1989), Marsden and Hughes (1994), Silhavy

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Fig. 1. Interpretation of incremental energetic stability conditions of an inelastic material in terms of quasiconvexity. A given homogeneous deformation state e of the material is stable if superimposed fine scale perturbation patterns rs w with support on the boundary @D increase the averaged incremental stress potential on D.

case under consideration, we define W to be quasiconvex at en þ 1 if Z 1 Wðen þ 1 þ rs wÞ dV Wðen þ 1 Þ r jDj D

ð22Þ

holds for all perturbation fields w defined on D  B such that w 2 W 0 :¼ fw 2 W 1,1 ðDÞ j w ¼ 0 on @Dg. Here, we use the notation rs ðÞ :¼ ½rðÞ þ rT ðÞ=2 for the symmetrization of the gradient operator. Fig. 1 gives an illustration of this condition, which may be interpreted as a fundamental criterion for material stability. The well-motivated concept of quasiconvexity is based on an integral condition that is hard to verify in practice. This motivates the introduction of the slightly weaker, but more handleable condition of the rank-one convexity. In addition to its practical applicability, as noted ˇ ´ (1997, p. 278): ‘the experience shows that the difference between global quasiconvexity and rank-one by Silhavy convexity is relatively small’. The classical rank-one convexity condition may be written in the small-strain context Wðxe þ þð1xÞe Þ r xWðe þ Þ þð1xÞWðe Þ

ð23Þ

with the rank-one connection

e þ e ¼ aðm  nÞs

ð24Þ T

in terms of the laminate strains e , e and the fraction 0 r x r 1. Here, m and n are unit vectors and ðÞs :¼ ðÞ þðÞ is a tensor symmetrization operator. The above condition represents a superordinate criterion that requires the rank-one convexity of the incremental potential W for any laminate strains e þ and e . However, in the context of non-convex inelasticity, the potential W may consist of convex and non-convex ranges. In order to detect whether W is rank-one convex for a given strain en þ 1 , we consider it to be decomposed þ



en þ 1 ¼ xe þ þ ð1xÞe

ð25Þ þ



into fractions of laminate strains e and e . This equation can be regarded as a compatibility condition that is required to be satisfied by the strains e þ and e . We introduce the appropriate ansatz

e :¼ en þ 1 xaðm  nÞs and e þ :¼ en þ 1 þ ð1xÞaðm  nÞs 

ð26Þ

þ

that parametrizes e and e in terms of the laminate unit directors n and m, respectively, such that the requirements (24) and (25) are fulfilled. For two-dimensional problems, the unit vectors m ¼ sinae1 þ cosae2 and n ¼ sinbe1 þcosbe2 can be parametrized by two angles a and b, respectively. Then, the state vector of micro-variables for two-dimensional problems reads c :¼ fx,a, a, bg 2 C

ð27Þ

with admissible range C. Clearly, for three-dimension problems, it has to be extended by two further directional angles. Insertion of (25) and (26) into the right-hand-side of (23) induces the function W ðen þ 1 ,cÞ ¼ xWðe þ Þ þ ð1xÞWðe Þ

ð28Þ

representing a volume average of two strain phases. Then, the rank-one convexity condition (23) can be represented in the form Wðen þ 1 Þ r inf fW ðen þ 1 ,cÞg: c2C

ð29Þ

en þ 1 is not a rank-one convex point of the incremental stress potential W, if the minimum of the function W ðen þ 1 ,cÞ with respect to c is smaller than the potential Wðen þ 1 Þ, i.e. Wðen þ 1 Þ 4 inf c2C fW ðen þ 1 ,cÞg. Depending on whether condition (29) is

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Fig. 2. Convexification of a non-convex stress potential in strain-softening damage. (a) At en þ 1 the incremental stress potential W is not convex (dashed line). As a consequence, the macroscopic deformation state en þ 1 is not stable and decomposes into two micro-phases (+ ) and (  ) which describe the convex envelope (solid line). (b) Due to the constant slope of the convex envelope WC the relaxed stress r n þ 1 is constant in the non-convex range. Note the loading–unloading curves, governed by the path functions (69) or (86).

satisfied, the strains en þ 1 lie in a convex range of the incremental potential function W or not. The weak convexity condition (29) says that the homogeneous deformation state en þ 1 is stable as long as no combination of two phases ( +) and (  ) exists that possesses a lower energetic level than en þ 1 . Fig. 2a shows the shape of a non-convex incremental stress potential W for the one-dimensional case. Obviously, the incremental potential Wðen þ 1 Þ is greater than the interpolation of the incremental potentials Wðe þ Þ and Wðe Þ corresponding to the two phases (+ ) and ( ). As a consequence, the homogeneous deformation state is not stable and decomposes into the micro-strains e þ and e . Then, according to (25), the strain en þ 1 is the volume average of the two strains e and e þ . 3.3. Convexity based on an assumed isotropic microstructure The microstructure given by (26) generate an anisotropy due to the laminate vectors m and n. With regard to the treatment of isochoric and isotropic damage mechanics, we assume in what follows a simplified microstructure that preserves the isotropy. We focus on a class of problems, suitable for the treatment of damage in polymers, where the damage affects isotropically only the isochoric part of the deformation. This assumption has the following consequence for the mathematical representation of the incremental stress potential W. Firstly, the potential decomposes into volumetric and isochoric contributions iso Wðen þ 1 Þ ¼ Wvol ðevol n þ 1 Þ þ Wiso ðen þ 1 Þ

ð30Þ

driven by the spherical and deviatoric parts of the strain tensor iso vol evol n þ 1 :¼ tr½en þ 1 1=d and en þ 1 :¼ en þ 1 en þ 1

ð31Þ

for the dimension d 2 f2,3g. Secondly, the volumetric part Wvol ¼ Wvol ðen þ 1 Þ

with en þ 1 :¼ tr½evol n þ 1

ð32Þ

is purely elastic and directly related via Wvol ¼ cvol ðen þ 1 Þcvol ðen Þ to the increment of a convex volumetric strain energy function cvol . Thirdly, the isochoric part is isotropic and assumed to be a function of only one invariant of strain deviator Wiso ¼ Wiso ðen þ 1 Þ

with en þ 1 :¼ jeiso n þ 1j

ð33Þ

namely the norm of the deviator. Note that making the isochoric part dependent only on the norm en þ 1 of the deviator is exact for two-dimensional and a reasonable assumption for three-dimensional isotropic solids. If the strain-softening damage mechanism affects only the isochoric part, the isochoric contribution Wiso may have non-convex regions, while the volumetric part Wvol is a priori convex. Now, introduction of an isotropic microstructure related to the amount of the deviatoric strains

e :¼ en þ 1 xa and e þ :¼ en þ 1 þð1xÞa

ð34Þ

in analogy to (26), reduces the micro-variables (27) to c~ :¼ fx,ag 2 C~

ð35Þ

with an admissible range C~ :¼ fc~ j 0 r x r1,a Z 0g. Then, when introducing the function ~ ðen þ 1 , cÞ ~ ¼ Wvol ðen þ 1 Þ þ xWiso ðe þ Þ þð1xÞWiso ðe Þ W

ð36Þ

as a specification of (28) for the isochoric and isotropic damage problem under consideration, the convexity condition (29) is forced to be restricted in an isotropic manner to ~ ðen þ 1 , cÞg: ~ Wðen þ 1 Þ r inf fW c~ 2C~

ð37Þ

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Note that the assumed microstructure (34) affects only the amount en þ 1 of the deviator. The principal strain directions are forced to remain constant iso eiso =jeiso j ¼ e þ iso =je þ iso j ¼ eiso n þ 1 =jen þ 1 j: 

ð38Þ

þ

As a consequence, e and e are coaxial and have the same principal orientations as the total strain en þ 1 . Combining this with (34), we may recast (26) for the above specific assumption into the form iso iso iso e ¼ evol n þ 1 þ fjen þ 1 jxagen þ 1 =jen þ 1 j, iso iso iso e þ ¼ evol n þ 1 þfjen þ 1 jþ ð1xÞagen þ 1 =jen þ 1 j:

ð39Þ

This parameterization defines the formation of an a priori isotropic microstructure, which we use for testing of the material ~ in (36) is isotropic due to the scalar nature of stability in isochoric damage mechanics. Clearly, the averaged potential W the micro-variables c~ defined in (35). With this reduced step of scalar microvariables and the conditions (39), we have enforced the isotropy property ~ ðen þ 1 , cÞ ~ ðQ en þ 1 Q T , cÞ ~ ¼W ~ W

for all

Q 2 SOðdÞ

ð40Þ

on the volume averaged incremental potential (36). In this sense, condition (37) essentially checks the convexity of the isochoric contribution Wiso to the stress potential, which is considered to be a stronger restriction than the rank-one convexity condition (29). 3.4. Relationship to a laminate microstructure for two-dimensional problems Note that, following the above argumentation, the assumed isotropic microstructure (39) may formally be obtained form (26) by replacing the laminate basis iso ðm  nÞs ) eiso n þ 1 =jen þ 1 j

ð41Þ

by the a priori known direction of the strain deviator. Clearly, this violates in the general three-dimensional case the rankone connectivity condition (24). However, as already proposed in Lambrecht et al. (2003a), this is not the case for twodimensional problems, where the above assumption turns out to be an exact solution of the minimization problem (29) with respect to the laminate orientation. This is proved as follows. Isochoric microstructures are defined by (26), however, with the additional a priori constraint mn¼0

ð42Þ

of orthogonal laminate vectors. As a consequence of this constraint and due to the unit length characteristics of m and n, we may set for two-dimensional problems without loss of generality m, a ¼ n

and

n, b ¼ m,

ð43Þ

where a and b are the microstructural laminate orientation parameters defined in (27). Furthermore, consider the spectral representations of the two-dimensional deviators

e eþ eiso ¼ pffiffiffi ðs1  s2 Þs and e þ iso ¼ pffiffiffi ðs1þ  s2þ Þs 2

ð44Þ

2

with e 7 :¼ je 7 iso j and the principal shear directions s17 and s27 of both isochoric deformation phases. Then, when taking into account the assumptions (30)–(33) of an isotropic stress potential W with volumetric-isochoric decoupling, the necessary conditions of minimization problem on the right hand side of (29) with respect to the laminate orientations require W , a ðen þ 1 ,cÞ ¼ xð1xÞafWiso, e ðe þ iso ÞWiso, e ðeiso Þg : ðm, a  nÞs ¼ 0, W , b ðen þ 1 ,cÞ ¼ xð1xÞafWiso, e ðe þ iso ÞWiso, e ðeiso Þg : ðm  n, b Þs ¼ 0:

ð45Þ

Note that, due to the isotropy assumption for Wiso in (33), the deviatoric micro-stresses in the two phases appear in the spectral form Wiso, e ðeiso Þ ¼

0 Wiso ðe Þ  pffiffiffi ðs 1  s2 Þ s 2

and

Wiso, e ðe þ iso Þ ¼

0 Wiso ðe þ Þ pffiffiffi ðs1þ  s2þ Þs 2

ð46Þ

in terms of the principal shear axes introduced in (44). Insertion of this into (45) and taking into account (43), we obtain for the general case xaf0,1g and aa0 the constraints 0 0  ðe þ Þðs1þ  s2þ Þs Wiso ðe Þðs fWiso 1  s2 Þs g : ðn  nÞ ¼ 0, 0 0  ðe þ Þðs1þ  s2þ Þs Wiso ðe Þðs fWiso 1  s2 Þs g : ðm  mÞ ¼ 0:

ð47Þ

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They are satisfied for any non-zero amounts e and e þ of the isochoric strains only for co-axial strains in the two phase, i.e. for a choice s1þ ¼ s 1 ¼n

and

s2þ ¼ s 2 ¼ m:

ð48Þ

Next, considering the isochoric parts of (26), we have

eiso ¼ eiso and e þ iso ¼ eiso n þ 1 xaðm  nÞs n þ 1 þ ð1xÞaðm  nÞs ,

ð49Þ

where the total strain deviator reads in two-dimensional spectral form

en þ 1 eiso n þ 1 ¼ pffiffiffi ðs1  s2 Þs :

ð50Þ

2

Then, insertion of (44) and (50) into (49) and taking into account the result (48), we finally identify the isochoric laminate directors for two-dimensional problems n ¼ s1

and

m ¼ s2

ð51Þ

with the principal shear orientations s1 and s2 of the total strain deviator eiso n þ 1 . This a priori identification of the laminate orientations proves the assumption (41) for the two-dimensional problem. However, we emphasize that in the general three-dimensional setting, the subsequently assumed convexity (37) provides a stronger constraint than the rank-one convexity condition (29).

4. Relaxation of a non-convex constitutive response A key advantage of the variational formulation for the constitutive response is the opportunity to apply the concept of relaxation of non-convex variational problems to strain-softening inelastic solids. In the case of a non-convex incremental potential W, a microstructure develops characterized by a fluctuation field w as depicted in Fig. 1. The relaxation is concerned with the determination of microstructures arising as a consequence of material instabilities. We refer ˇ ´ (1997) and Muller ¨ to Dacorogna (1989), Silhavy (1999) for a sound mathematical basis.

4.1. Variational problem of the relaxed response If the above outlined material stability analysis detects a non-convex incremental stress potential W, then the existence of solutions of (18) is not ensured and energy-minimizing deformation microstructures develop. A relaxation is associated with a quasiconvexification of the non-convex function W by constructing its quasiconvex envelope WQ. On the physical side, quasiconvexity is the passage from a microscopic energy to a macroscopic energy that is obtained by an averaging over fine scale oscillations. The relaxed formulation has two important properties: (i) the infimum of the relaxed problem is equal to the infimum of the original problem (which may have no solution) and (ii) any minimum of the relaxed problem corresponds to the limit of a minimizing sequence of the original problem. Note that the framework developed by relaxation yields a mathematically well-posed problem without introduction of any length scale or viscosity into the formulation. Following Dacorogna (1989) and Acerbi and Fusco (1984), we consider the relaxed energy functional Z PQ ðun þ 1 Þ ¼ WQ ðrs un þ 1 Þ dVPext ðun þ 1 Þ, ð52Þ B

where W in (16) is replaced by its quasiconvex envelope WQ. The external loading part Pext ðun þ 1 Þ is given in (17). The current displacement field is then determined by the relaxed incremental variational principle   un þ 1 ¼ ARG inf PQ ðun þ 1 Þ ð53Þ un þ 1 2W

that minimizes the relaxed incremental potential for the admissible deformation field defined in (19). The Euler equations of the relaxed problem (53) div½@e WQ ðrs un þ 1 Þ ¼ c n þ 1 in B

and

½@e WQ ðrs un þ 1 Þ  n ¼ t n þ 1 on @Bt

ð54Þ

characterize the stress equilibrium and the Neumann-type boundary conditions at the current time tn + 1 in terms of the relaxed stresses r n þ 1 :¼ @e WQ , obtained from the relaxed incremental constitutive potential WQ. This quasiconvexified incremental stress potential is defined by the minimization problem  Z  1 WQ ðen þ 1 Þ ¼ inf Wðen þ 1 þ rs wÞ dV ð55Þ w2W 0 jDj D with respect to the microscopic fluctuation field w defined on D  B such that w 2 W 0 :¼ fw 2 W 1,1 ðDÞjw ¼ 0 on @Dg.

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4.2. Convexification based on the assumed isotropic microstructure Since the quasiconvexification is very difficult to obtain for general material models, we approximate in what follows the quasiconvexified potential WQ by a convexified potential WC based on the assumed microstructure (39) outlined in Section 3.3, suitable for an isotropic and isochoric response. Hence, we replace PQ in (52) by the approximated convexified potential Z PC ðun þ 1 Þ ¼ WC ðrs un þ 1 Þ dVPext ðun þ 1 Þ ð56Þ B

in terms of the specific envelope ~ ðen þ 1 , cÞg ~ WC ðen þ 1 Þ ¼ inf fW

ð57Þ

c~ 2C~

based on the function ~ ðen þ 1 , cÞ ~ ¼ Wvol ðen þ 1 Þ þ xWiso ðe þ Þ þ ð1xÞWiso ðe Þ W

ð58Þ

already introduced in (36) in terms of the reduced set c~ of microvariables defined in (35). Recall that the volumetric part ~ ðen þ 1 , cÞ ~ contains the volume average of the isochoric potentials in Wvol is assumed to be a priori convex. The function W two micro-phases. The first derivative of the convexified potential WC defines the relaxed stresses

r n þ 1 :¼ @e WC ðen þ 1 Þ:

ð59Þ

A further derivative determines the relaxed moduli in the closed form representation ~ ,ce , ~ , ee W ~ , ec ½W ~ ,cc 1 W C n þ 1 :¼ @2ee WC ðen þ 1 Þ ¼ W

ð60Þ

where the second term characterizes a softening part due to the developed microstructure. The basic steps of the relaxation procedure are summarized in Box 1. Fig. 2a gives a one-dimensional interpretation of the convexification by depicting the shape of a non-convex potential function and its convex envelope, that is described by the micro-strains e and e þ . In the non-convex range ½e , e þ , the non-convex incremental potential function W is replaced by its convex envelope WC. The convexified incremental potential WC is obtained by a fictitious projection of the non-convex incremental potential W onto the convex envelope. As shown in Fig. 2b, the relaxed stresses r n þ 1 at en þ 1 is associated with a Maxwell-type line similar to classical treatments in phase-decompositions of real gases, see e.g. Krawietz (1986) or Dacorogna (1989). Note that the loss of convexity of the incremental stress potential marks the transition from a homogeneous one-phase analysis to a two-phase relaxation analysis. Within the following time-increments the minimizing phases are determined, which characterize the convexified stress response. 4.3. Thermodynamic consistency of the relaxed model As pointed out above, the loss of convexity initiates a decay of the unstable homogeneous state and a two-phase relaxation analysis. The two micro-phases e 7 defined in (39) are uniquely determined by the state vector of microvariables c~ ¼ fx,ag introduced in (35), where x corresponds in the subsequent treatment of damage mechanics to the volume fraction of the phase ( + ) which is more damaged. Due to the irreversibility of the damage process, the volume fraction should not decrease. In other words, the damaged fraction of the material cannot be healed. This induces a natural restriction for

Box 1–Basic steps of incremental two-phase relaxation procedure. 1. Homogeneous analysis: Define the incremental stress potential Wðen þ 1 Þ ¼ cðen þ 1 ,pðen þ 1 ÞÞcðen ,qn ÞþDðen þ 1 Þ and compute stresses and moduli rn þ 1 ¼ @e Wðen þ 1 Þ and Cn þ 1 ¼ @ee2 Wðen þ 1 Þ: 2. Material stability: Check convexity of W for microstructure c˜ :¼ fx,ag ˜

Wðen þ 1 Þ inf fW ðen þ 1 ,c˜ Þg: ˜

˜

c2C

If Wðen þ 1 Þ is convex, continue next time increment with step 1. 3. Two-phase relaxation analysis: Determine the convexified potential ˜

WC ðen þ 1 Þ ¼ inf fW ðen þ 1 ,c˜ Þg, ˜

c˜ 2C

and compute relaxed stresses and moduli rn þ 1 ¼ @e WC ðen þ 1 Þ and C n þ 1 ¼ @ee2 WC ðen þ 1 Þ: 4. Recovery of material stability: Check if volume fraction x ¼ 1.

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the microstructural volume fraction

x_ Z0 3 xn þ 1 Z xn :

ð61Þ

As a consequence, the admissible domain C~ of the micro-variables introduced in (35) is re-defined by C~ :¼ fc~ j xn r xn þ 1 r1,a Z 0g:

ð62Þ

Here, the volume fraction x can be regarded as an internal variable that measures the irreversibility of the phase decomposition. Note in this context that an anisotropic, laminate-type microstructure of the form (26) may rotate. Since the two phases correspond to highly damaged and slightly damaged microstructural fractions of the material, this laminate rotation would lead to a sudden healing of some highly damaged parts. This is not consistent with the above definition of a thermodynamical consistency. In contrast, the proposed isotropic microstructure (39) does satisfy the thermodynamical consistency as long as the volume fraction x is non-decreasing. We consider this as a physically based argument for the choice of the a priori isotropic microstructure (39) in isotropic damage mechanics.

5. A constitutive model for a partially damaged solid In the sequel, we consider first a one-dimensional model in order to point out the key features of the convexification algorithm. It is later extended to the multi-dimensional setting. Here, the relaxation method with isotropic microstructure development is applied.

5.1. A one-dimensional partial damage model 5.1.1. Model definition We consider a model with a scalar damage variable q and its dual internal force p. The model problem is completed by the definition of the fundamental constitutive functions c, f and c for the free energy, the level set of the elastic domain and the threshold, respectively. The elastic response is governed by the energy storage function

cðe,qÞ ¼ ð1dðqÞÞc0 ðeÞ with c0 ðeÞ ¼ 12Ee2 ,

ð63Þ

where c0 represents the effective elastic storage mechanism and d is the damage degradation function. Here, E4 0 denotes the elasticity modulus. The damage law visualized in Fig. 3 is assumed to have the particular form dðqÞ ¼ d1 ½1expðq=ZÞ

ð64Þ

with Z 4 0, d1 2 ½0,1Þ, specifying the rate of saturation and the maximum damage, respectively. The material parameters used in computations are E ¼ 1  104 MPa, d1 ¼ 0:99 [–] and Z ¼ 2:5 [–]. Exploitation of (2) and (3)2 yields the stress and the internal force

s ¼ ð1dðqÞÞEe and p ¼ c0 ðeÞd0 ðqÞ,

ð65Þ

where d0 denotes the derivative of the function d. With regard to the level set and the threshold we consider the functions f ðpÞ ¼ p

and

cðqÞ ¼ qd0 ðqÞ:

ð66Þ

Fig. 3. A model of one dimensional partial damage with remaining stiffness. A saturation type damage law dðqÞ ¼ d1 ½1expðq=ZÞ with a saturation value d1 ¼ 0:99.

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With the free energy function (63), the level set and threshold functions (66) at hand, the evolution equations (6) appear in simple form _ c0 qÞ ¼ 0 q_ Z 0 4 c0 r q 4 qð

ð67Þ

with the simple closed-form solution for the internal variable qðtÞ ¼ maxfc0 ðeðsÞÞg s2½0,t

ð68Þ

as used for example in Miehe (1995). As shown in Fig. 4b, after reaching the damage saturation value, the material behaves elastically with a reduced stiffness. Such a response can micromechanically be motivated by the existence of two types of micro-bonds in the material, where one group breaks and the other does not. This is a typical feature of filled rubber-like materials, where chain–particle bonds may break but the ground-state network remains undamaged. 5.1.2. Incremental path of internal variable The path of the internal variable q within a finite step-sized time increment is defined by an update algorithm, which attains for the model problem under consideration a simple closed-form representation. Based on (68), in the algorithmic, deformation-driven setting, the current internal variable of the discontinuous damage evolution is defined by qn þ 1 ¼ pðen þ 1 Þ ¼ ð1sÞqn þ sc0 ðen þ 1 Þ, where we introduced the damage loading flag  1 for c0 ðen þ 1 Þ 4 qn , s¼ 0 otherwise

ð69Þ

ð70Þ

that characterizes the discontinuity of the damage evolution. Note that, as a result of above defined functions, the internal variable q can be identified as the maximum effective strain energy c0 obtained in the history of the deformation process. Furthermore, observe that (69) characterizes the current damage variable as a discontinuous function of the strain en þ 1 . We made this dependence precise by introducing the path function p associated with the incremental setting. 5.1.3. Incremental stress potential With this result at hand, we are able to construct an exact expression for the incremental dissipation. Exploitation of (15) gives Z pðen þ 1 Þ cðqÞdq ð71Þ Dðen þ 1 Þ ¼ qn

yielding with the given threshold function (66)2 the closed-form result Dðen þ 1 Þ ¼ dðpÞðp þ ZÞdn ðqn þ ZÞd1 ðpqn Þ,

ð72Þ

where pðen þ 1 Þ is the path function defined in (69). With this value at hand, we obtain for the incremental stress potential W ¼ ccn þD introduced in (15) the exact closed-form expression Wðen þ 1 Þ ¼ ð1dðpÞÞc0 cn þdðpÞðp þ ZÞdn ðqn þ ZÞd1 ðpqn Þ:

ð73Þ

Note again that W depends critically on the incremental path function pðen þ 1 Þ defined in (69). Exploitation of (11) yields the closed-form expression for the stress

sn þ 1 :¼ @e W ¼ ð1dðpÞÞEen þ 1 ,

ð74Þ

Fig. 4. A one-dimensional partial damage model with remaining stiffness. (a) Non-convex potential W and convexified potential WC. (b) Non-convex stress s and convexified stress s .

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which governs the balance equations (20) of the non-relaxed problem. A further derivative gives the tangent modulus

Cn þ 1 :¼ @2ee W ¼ ð1dðpÞÞEsd0 ðpÞðEen þ 1 Þ2 ,

ð75Þ

which governs the linearization of the balance equations (20). 5.1.4. Convexification of the incremental stress potential For a non-convex incremental stress potential W, we perform the convexification procedure outlined in Section 4. This requires at each time step the solution of the optimization problem (57), that determines the convexified potential WC in terms of the current micro-variables x and a. The relaxed stress response obtained from the optimization (57) with respect to the two variables x and a is plotted in Fig. 4b, together with the original (non-relaxed) response. As a result of the numerical convexification it is found that the micro-variable a remains constant during the whole process. The evolution of x and a are plotted in Fig. 5a and b, respectively. The development of the phases and the corresponding internal variables are shown in Fig. 5c and d. From Fig. 5b it is clear that micro-variable a remains constant. Hence, it is not necessary to solve the minimization problem for the two variables c~ :¼ fx,ag at each time step. In fact, one needs to determine the micro-variable a once numerically and then it is possible to compute x directly. Note that for a general relaxation algorithm the internal variables q and q þ in each phase have to be monitored separately. However, for the special case considered here, i.e. for constant a, e and e þ , the internal variables q 7 remain constant as well. Thus, they do not need to be monitored separately during a microstructural evolution. This numerical observation is consistent with a straightforward convexification of the one-dimensional non-convex function W given in (73). This can be performed by a determination of the start and end points e and e þ of non-convex domain, as depicted in Fig. 2, based on the two conditions

(1) Tangents to the energy W must be equal at e and e þ , stating that the convexified stress at both point is equal, i.e. W 0 ðe Þ ¼ W 0 ðe þ Þ:

ð76Þ

(2) The convexified stress divides the stress–strain strain curve in the non-convex range into two equal areas, i.e. Z eþ

½W 0 ðeÞW 0 ðe Þ de ¼ Wðe þ ÞWðe ÞW 0 ðe Þ½e þ e  ¼ 0:

ð77Þ

e

Fig. 5. Development of the volume fraction, the intensity of micro-bifurcation, total strains and the internal damage variables in the micro-phases. (a) The volume fraction increases linearly from x ¼ 0 to the final value x ¼ 1. (b) The constant intensity of micro-bifurcation a= 0.22135 determines the distance between the two phases (+ ) and ( ). (c) The total strains e þ , e do not change during the convexification analysis. (d) Also the internal damage variables q þ , q remain constant in the non-convex span.

E. G¨ urses, C. Miehe / J. Mech. Phys. Solids 59 (2011) 1268–1290

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Box 2–Generic formulation of two-phase relaxation algorithm.

1. Determine numerically a priori phases e þ , e from conditions (76) and (77) for 1D or (94) and (95) for 3D and set micro-bifurcation a ¼ e þ e . 2. Given en þ 1 and xn determine xn þ 1 by a case distinction scheme A. Initial stable homogeneous state xn ¼ 0: (i) en þ 1  e : xn þ 1 ¼ 0: (ii) en þ 1 4 e : xn þ 1 ¼ ðen þ 1 e Þ=ðe þ e Þ. B. Phase decay of homogeneous state 0o xn o 1: (i) en þ 1  xn e þ þð1xn Þe : xn þ 1 ¼ xn . (ii) xn e þ þð1xn Þe o en þ 1 o e þ : xn þ 1 ¼ ðen þ 1 e Þ=ðe þ e Þ. (iii)en þ 1 Z e þ : xn þ 1 ¼ 1. C. Recovery of stable homogeneous state xn ¼ 1 : xn þ 1 ¼ 1. 3. Compute the relaxed stress and tangent modulus from (79) for 1D and (97) for 3D. r n þ 1 ¼ @e WC ðen þ 1 Þ and C n þ 1 ¼ @ee2 WC ðen þ 1 Þ:

Fig. 6. Localization of a bar in tension. The test specimen under consideration consists of two parts: B2 with length k and B1 with length 1k. This may be considered as a FE discretization with two elements. In B1 the maximum damage d1 is decreased by 0.0001%.

Based on these two conditions, the constants e and e þ can be determined numerically. The micro-bifurcation then has the constant value a ¼ e þ e : With these results at hand, we obtain the closed-form 8 Wðen þ 1 Þ, > > > e e e e > > : Wðen þ 1 Þ,

ð78Þ solution for the convexified global stress potential

en þ 1 o e , e r en þ 1 r e þ ,

ð79Þ

en þ 1 4 e : þ

The two-phase incremental relaxation analysis can be summarized in two steps: (i) We first determine the micro-phases

e 7 and a numerically based on (76)–(78). (ii) Next, we set up a case-distinction scheme depending on the volume fraction xn of the previous time-interval and the current strain en þ 1 . The algorithm is summarized in Box 2. Such an algorithm obviously does not require any numerical solution of non-convex minimization problem at each time step and consequently is very robust and computationally efficient. 5.1.5. Mesh objectivity of the relaxed problem We close the one-dimensional considerations by an example that treats a computational localization analysis in a strain-softening material. The main goal is the demonstration of the mesh-invariance of the proposed relaxation technique in a finite element discretization. For this purpose we consider the bar depicted in Fig. 6 of length and cross section area 1, subject to a tensile stress s~ . The bar is fixed at its left boundary. This example was previously investigated in the context of strain-softening elastoplasticity by Lambrecht et al. (2003b). In order to point out the mesh-dependence of the nonrelaxed formulation we discretize the bar with two elements B1 and B2 for different lengths k ¼ 0:2=0:4=0:6=0:8=1:0. A localization of the homogeneous boundary value problem is triggered by decreasing the maximum damage d1 in the element B1 by 0.0001%. Fig. 7a depicts the stress–displacement curves for the different discretizations mentioned above. We start at the origin of the diagram and proceed on the loading branch. At the peak of the displacement curves in Fig. 7a we observe a loss of local structural stability documented by a change of sign of the tangent. After the peak the element B1 switches into a post-critical path while the element B2 switches back to the elastic unloading path. The non-convex

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Fig. 7. Global stress–strain curves of imperfect test specimen. (a) Visualization of the non-objective length- or FE-mesh-dependent stress response for different choices k within the standard formulation. (b) Invariant solution due to the convexification analysis.

analysis yields the spectrum of equilibrium paths in Fig. 7a. They document the well-known strong mesh-dependence of the non-objective post-critical analysis. These post-critical results are physically meaningless. The ill-posed boundary value problem can be transformed into a well-posed one by means of the relaxation method suggested in Section 4. The relaxed analysis yields an identical result for all mesh densities. The mesh-invariant post-critical equilibrium path is documented in Fig. 7b. Note that the stability condition based on a global convexity of the incremental potential detects instability before the tangent changes its sign, i.e. before the start of global softening. 5.2. A multi-dimensional partial isochoric damage model 5.2.1. Model definition We now extend the above formulation to a multi-dimensional isochoric damage model. As above, we consider a scalar damage variable q and its dual internal force p. The construction of the fundamental constitutive functions c, f and c for the free energy, the level set of the elastic domain and the threshold are slight modifications of the one-dimensional functions (63)–(68). The elastic response is governed by the isotropic energy storage function iso 2 cðe,qÞ ¼ 12ktr2 ½e þ ð1dðqÞÞciso 0 ðeÞ with c0 ðeÞ ¼ me , vol

ð80Þ

iso

where e :¼ tr½e  and e :¼ je j are the invariants of the strain tensor e defined in (32) and (33), respectively. k 4 0 denotes the bulk modulus and m 40 the shear modulus. Note that the damage function dðqÞ affects only the isochoric contribution of c and assumed to have the particular form dðqÞ ¼ d1 ½1ð1þ WqÞn 

ð81Þ

in terms of the material parameters d1 2 ½0,1, W 40 and n 40. Exploitation of (2) and (3)2 yields the stress and the internal force 0 r ¼ ke1 þð1dðqÞÞ2meiso and p ¼ ciso 0 ðeÞd ðqÞ:

ð82Þ

With regard to the level set and the threshold we consider again the functions f ðpÞ ¼ p

and

cðqÞ ¼ qd0 ðqÞ:

ð83Þ

With the free energy function (80), the level set and threshold functions (83) at hand, the evolution equations (6) appear in simple form _ ciso q_ Z 0 4 c0 rq 4 qð 0 qÞ ¼ 0

ð84Þ

with the simple closed-form solution for the internal variable iso

qðtÞ ¼ maxfc0 ðeðsÞÞg:

ð85Þ

s2½0,t

5.2.2. Incremental path of internal variable The path of the internal variable q within a finite step-sized time increment is defined by an update algorithm. In analogy to (69), we obtain for the multidimensional model the simple closed-form representation iso

qn þ 1 ¼ pðen þ 1 Þ ¼ ð1sÞqn þ sc0 ðen þ 1 Þ

ð86Þ

E. G¨ urses, C. Miehe / J. Mech. Phys. Solids 59 (2011) 1268–1290

with the damage loading flag ( iso 1 for c0 ðen þ 1 Þ 4qn , s¼ 0 otherwise

1283

ð87Þ iso

that is now driven by the isochoric energy c0 . Note that the algorithmic dependence of the current internal variable qn þ 1 on the current strain tensor en þ 1 is made precise by introducing the path function p. 5.2.3. Incremental stress potential With this result at hand, as in the one-dimensional case, we are able to construct an exact expression for the incremental dissipation. Exploitation of (15) gives Z pðen þ 1 Þ cðqÞ dq ð88Þ Dðen þ 1 Þ ¼ qn

yielding with the given threshold function (83)2 and the damage function (81) the closed-form result en þ 1 Þ Dðen þ 1 Þ ¼ ½qdðqÞd1 fqð1 þ WqÞ1n =ðWð1nÞÞgpð , qn

ð89Þ

where pðen þ 1 Þ is the path function defined in (86). With this value at hand, we obtain for the incremental stress potential W ¼ ccn þD introduced in (14) the exact closed-form expression en þ 1 Þ : Wðen þ 1 Þ ¼ cðen þ 1 ,pðen þ 1 ÞÞcn þ ½qdðqÞd1 fqð1 þ WqÞ1n =ðWð1nÞÞgpð qn

ð90Þ

Note again that W depends critically on the incremental path function pðen þ 1 Þ defined in (86). Exploitation of (11) yields the closed-form expression for the stress

rn þ 1 :¼ @e W ¼ ken þ 1 1 þ ð1dðpðen þ 1 ÞÞÞ2meiso n þ 1,

ð91Þ

which governs the balance equations (20) of the non-relaxed problem. A further derivative gives the tangent modulus iso Cn þ 1 :¼ @2ee W ¼ k1  1 þ ð1dðpðen þ 1 ÞÞÞ2mPsd0 ðpðen þ 1 ÞÞ4m2 eiso n þ 1  en þ 1 ,

ð92Þ

which governs the linearization of the balance equations (20). Here, P denotes the fourth order deviatoric projection tensor. 5.2.4. Convexification of the incremental stress potential Note that the incremental stress potential W in (90) decomposes according to (30)–(31) into volumetric and isochoric contributions, where the isochoric part appears in closed form en þ 1 Þ Wiso ðen þ 1 Þ ¼ ð1dðpðen þ 1 ÞÞÞme2n þ 1 þ ½qdðqÞd1 fqð1 þ WqÞ1n =ðWð1nÞÞgpð qn

ð93Þ

with an isotropic dependence on en þ 1 :¼ jeiso n þ 1 j as assumed in (31). For a non-convex potential Wiso, we perform the convexification procedure outlined in Section 4. As discussed above, the solution of the optimization problem (57) can be simplified for one-dimensional problems to the steps (76)–(79). As a consequence, we may a priori compute the convexification of Wiso by first solving numerically the conditions, in analogy to (76) and (77) 0 0 ðe Þ ¼ Wiso ðe þ Þ Wiso

ð94Þ

for equal stresses in the relaxed range and Maxwell-type condition 0 ðe Þ½e þ e  ¼ 0 Wiso ðe þ ÞWiso ðe ÞWiso

ð95Þ

for the constants e and e . The isotropic, isochoric micro-bifurcation then has the constant value 

þ

a ¼ e þ e : and we obtain the closed-form solution for the convexified global stress 8 Wðen þ 1 Þ, > > > < en þ 1 e e þ en þ 1 WC ðen þ 1 Þ ¼ Wvol ðen þ 1 Þ þ þ  Wiso ðe þ Þ þ þ  Wiso ðe Þ, > e  e e e > > : Wðen þ 1 Þ,

ð96Þ potential

en þ 1 o e , e r en þ 1 r e þ ,

ð97Þ

en þ 1 4 e : þ

The two-phase incremental relaxation analysis can be summarized in two steps: (i) We first determine the micro-phases e 7 and a numerically based on (94)–(96). (ii) Next, we set up a case-distinction scheme depending on the volume fraction xn of the previous time-interval and the current strain en þ 1 . The algorithm is identical to the one-dimensional case summarized in Box 1.

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Fig. 8. Visualization of a deformation microstructure during a loading process.

5.3. Visualization and interpretation of isotropic microstructures We simply assume that the microstructures develop such that a bifurcation of the intensity of the isochoric deformation occurs. A visualization of evolving deformation microstructure is illustrated in Fig. 8. The outer circle in Fig. 8 represents the intensity of the deviatoric strain, while the inner circle denotes the highly damaged (+ ) phase. During loading both circles expand but the inner one faster, which corresponds to increase in the overall damage of the material. On the other hand, during unloading both circles start to shrink, however, relative to the outer circle, the inner one always grows or remains constant. This property satisfies the requirement (61) and asserts the thermodynamic consistency. In other words, damage is never recovered. Note that for any process damage levels dðq þ Þ and dðq Þ of the phases are constant but the volume fraction x of the highly damaged phase ( + ) is an increasing function of deformation satisfying the physical requirement, namely the increase in the macroscopic damage variable in a volume averaged sense. 6. Numerical examples We demonstrate the performance of the above outlined relaxation technique by means of representative numerical examples. The main goals of the numerical investigations are the analysis of the developing microstructures and the demonstration of the objectivity of the relaxation technique proposed. In the first example we investigate in a twodimensional setting a strain-driven cyclic local loading test and document the development of the microstructures. The second numerical example is concerned with the tension test of a perforated plate in two-dimensional and threedimensional plane strain frameworks. We report on the arising microstructures which are resolved by a convexification of the non-convex incremental stress potential. Comparison of the relaxed and the non-relaxed load–displacement curves underlines the objectivity of the relaxed stress response. The material parameters governing the energy storage function and the level set function for two- and three-dimensional damage models are summarized in Tables 1 and 2, respectively. 6.1. Volume preserving cyclic loading test The first example is concerned with a volume preserving cyclic loading test described by the macroscopic strain tensor " # pffiffiffi pffiffiffi 1 2cos½L 1 þ 2sin½L pffiffiffi pffiffiffi ð98Þ e ¼ ai 1 þ 2sin½L 1 þ 2cos½L with a1 = 0.062 for 0 r L r2p and a2 = 0.102 for 2p o L r 4p. The loading parameter L is increased in increments DL ¼ 0:01 up to the final value Lmax ¼ 4p. Fig. 9a visualizes the isochoric loading path of the test in the e11 e12 -plane. In Fig. 10 the Table 1 Material parameters for two-dimensional examples. Bulk modulus Shear modulus Maximum damage Saturation intensity Exponent

k = 150.000 N/mm2 m = 60.000 N/mm2 d1 = 0.999/0.975 W = 0.100/0.390 n = 2.000/0.950

Table 2 Set of material parameters for three-dimensional examples. Bulk modulus Shear modulus Maximum damage Saturation intensity

k = 150.000 N/mm2 m = 60.000 N/mm2 d1 = 0.78 Z = 500

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Fig. 9. (a) Cyclic loading test. Isochoric loading path in the e11 e12 -strain plane. (b) Perforated plate in tension. Geometry, loading and boundary conditions.

development of the microstructures is documented at the points plotted in Fig. 9a. The first eight microstructures correspond to the first cycle (a1 = 0.062), the following seven to the second cycle (a2 =0.102). After the loss of material stability two micro-phases e 7 emerge. The fraction x increases if the intensity e ¼ Jdev½eJ increases as well. Note that for the case of a decrease in intensity e the fraction x remains constant as pointed out before. During the second cycle a recovery of a stable single phase deformation occurs. The stress components of the non-relaxed and relaxed computations are plotted in Fig. 11. Note that after the recovery of stable state the stress components of both formulations coincide. 6.2. Tension test of a perforated plate in two-dimensional framework The second example analyzes the deformation of a perforated plate in tension. The geometry, loading and boundary conditions are plotted in Fig. 9b. Owing to the symmetry only one fourth of the specimen is discretized with 24, 83, 109 and 204 unstructured finite elements. We use the four-node enhanced incompatible-mode Q1E4 finite element formulation developed by Simo´ and Armero (1992) and treat the problem in a deformation-driven analysis with increments Du ¼ 0:001 mm up to a maximum displacement of u =0.1 mm. The mesh-dependent response of the non-relaxed formulation is evident by considering the load-displacement curves plotted in Fig. 12a. In contrast to the non-relaxed formulation, application of the proposed relaxation technique yields a mesh-invariant response. The loaddeflection curves do not depend on the mesh-size, but are identical for all different mesh-densities, see Fig. 12b. Note that the objectivity of the material behavior is obtained without the introduction of an internal length scale parameter or viscosity into the model. 6.3. Tension test of a perforated plate in 3-D plane strain In the last example previous boundary value problem with the geometry specified in Fig. 9b is investigated in threedimensional plane strain framework. One fourth of the specimen is discretized with 109, 204 and 352 unstructured finite

Fig. 10. Cyclic loading test. Development of isotropic microstructures after the loss of material stability. Isotropic microstructures are interpreted as a decomposition of the intensity of strain into two phases. The first eight microstructures correspond to the inner cycle whereas the following seven to the outer one in Fig. 9a.

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Fig. 11. Comparison of stress components obtained with (a) non-relaxed formulation, and (b) proposed relaxation technique.

Fig. 12. Tensile test of a perforated plate in two-dimensional setting. Load–displacement curves for four different finite-element meshes based on (a) the non-relaxed (non-objective) formulation (the finer the mesh the softer the response) and on (b) the proposed relaxation technique.

elements due to symmetry of the problem. The single phase homogeneous computations are performed with an arclength method whereas the relaxation computations are done in a deformation driven context and both up to a maximum displacement of u =0.7 mm. The mesh-dependent response of the non-relaxed formulation is evident by considering the overall load–displacement curves plotted in Fig. 13a for three different mesh densities (109, 204 and 352 elements).

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Fig. 13. Tensile test of a perforated plate in three-dimensional plane strain setting. Zoomed in load–displacement curves for three different finiteelement meshes based on (a) the non-relaxed (non-objective) formulation (the finer the mesh the softer the response) and on (b) the proposed relaxation technique.

In contrast to the non-relaxed formulation, application of the proposed relaxation technique yields a mesh-invariant response, see Fig. 13b for overall load–deflection diagram. Note that relaxed load deflection curves show some deviation for different discretizations when compared with the two-dimensional results. This may be due to the assumption of isotropic microstructures which, contrary to the two-dimensional problem, does not coincide with first order rank-one laminates in the three-dimensional setting. However, the relaxed response clearly removes the oscillatory behavior observed in Fig. 13a and provides a numerical evidence about the quality of the numerical approximation of the quasiconvex hull. Fig. 14 demonstrates the localization bands for considered discretizations. The objectivity of the results can be seen also from the thicknesses of these bands which are independent of the mesh density. In Fig. 15 the evolution of unstable regions are visualized together with the microstructures at the selected elements for the 204 finite element mesh at u =0.4,0.5 and 0.6 mm.

7. Conclusion We proposed a new relaxation technique for partially damaged solids based on particularly chosen isotropic microstructures. At first we outlined an incremental variational formulation of the local constitutive response that determines a quasi-hyperelastic stress potential. In analogy to treatments in elasticity, the existence of this variational formulation allows for the definition of the material stability of dissipative solids based on the weak convexity of the stress potential. In this context, material instabilities are assumed to give rise to the formation of microstructures. The arising microstructures are then resolved by the relaxation of non-convex incremental stress potential. Here, convexification of the potentials requires the solution of a local minimization problem. A new relaxation algorithm was developed for partially damaged solids which is based on a particularly chosen isotropic microstructure evolution. Contrary to numerical relaxation methods based on sequential lamination algorithms, which require the solution of non-convex optimization problems, the proposed algorithm is robust and computationally very efficient. Based on the objective numerical results obtained for inhomogeneous boundary value problems with different finite element mesh sizes, we conclude that the proposed relaxation algorithm provides a quite close approximation to the quasiconvex hull of the incremental potential.

Fig. 14. Tensile test of a perforated plate in three-dimensional plane strain setting. Comparison of relaxation analysis for different mesh densities. (a) 109, (b) 204 and (c) 352 elements for distribution of instable regions at u =0.7 mm.

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Fig. 15. Tensile test of a perforated plate in three-dimensional plane strain setting. Visualization of the instable regions and microstructures for the selected elements at (a) u= 0.4 mm; (b) u =0.5 mm and (c) u= 0.6 mm.

Acknowledgment Support for this research was provided by the Deutsche Forschungsgemeinschaft (DFG) under Grant FOR 509 Mi 295/10. References Acerbi, E., Fusco, N., 1984. Semicontinuity problems in the calculus of variations. Archive for Rational Mechanics and Analysis 86, 125–145. Aubry, S., Fago, M., Ortiz, M., 2003. A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials. Computer Methods in Applied Mechanics and Engineering 192, 2823–2843.

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