On kinematic, thermodynamic, and kinetic coupling of a damage theory for polycrystalline material

International Journal of Plasticity 26 (2010) 775–793

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On kinematic, thermodynamic, and kinetic coupling of a damage theory for polycrystalline material D.J. Bammann 1, K.N. Solanki * Mississippi State University, MS 39762, United States

a r t i c l e

i n f o

Article history: Received 31 March 2009 Received in final revised form 14 September 2009 Available online 27 October 2009 Keywords: Kinematics Thermodynamics Damage Elastic–plastic Finite-strain

a b s t r a c t The paper proposes a new consistent formulation of polycrystalline finite-strain elastoplasticity coupling kinematics and thermodynamics with damage using an extended multiplicative decomposition of the deformation gradient that accounts for temperature effects. The macroscopic deformation gradient comprises four terms: thermal deformation associated with the thermal expansion, the deviatoric plastic deformation attributed to the history of dislocation glide/movement, the volumetric deformation gradient associated with dissipative volume change of the material, and the elastic or recoverable deformation associated with the lattice rotation/stretch. Such a macroscopic decomposition of the deformation gradient is physically motivated by the mechanisms underlying lattice deformation, plastic flow, and evolution of damage in polycrystalline materials. It is shown that prescribing plasticity and damage evolution equations in their physical intermediate configurations leads to physically justified evolution equations in the current configuration. In the past, these equations have been modified in order to represent experimentally observed behavior with regard to damage evolution, whereas in this paper, these modifications appear naturally through mappings by the multiplicative decomposition of the deformation gradient. The prescribed kinematics captures precisely the damage deformation (of any rank) and does not require introducing a fictitious undamaged configuration or mechanically equivalent of the real damaged configuration as used in the past. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction In many manufacturing processes, ductile polycrystalline materials undergo deformations involving irreversible pathdependent processes such as dislocation generation/motion/annihilation, texture formation, void nucleation, growth, and coalescence. To capture these behaviors in a continuum model, both the kinematics (multiplicative decomposition) and thermodynamics of the state variables are used to describe the multi-dissipation processes. During the past few decades considerable attention has been given to formulating constitutive models for ductile materials using multiplicative decomposition of the deformation gradient (e.g. Rice 1971; Murakami, 1988, 1990; Bammann and Aifantis, 1989; Marin and McDowell, 1996; Steinmann and Carol, 1998; Voyiadjis and Park, 1999; Brünig, 2002; Regueiro et al., 2002; Solanki, 2008). The elastic and plastic parts of the deformation gradient have been used to characterize the kinematics of elastic–plastic materials at the nanoscale using dislocation mechanics (Bilby et al., 1957; Kröner, 1960; Steinmann and Carol, 1998), at the

* Corresponding author. Address: Center for Advanced Vehicular Systems, 200 Research Blvd., Starkville, MS 39759, United States. Tel.: +1 662 325 5454; fax: +1 662 325 5433. E-mail addresses: [email protected] (D.J. Bammann), [email protected] (K.N. Solanki). 1 Mechanical Engineering Department, Mississippi State University, Mail Stop 9552, 210 Carpenter Building Mississippi State University, MS 39762, United States. Tel.: +1 662 325 7308. 0749-6419/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2009.10.006

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mesoscale using crystal plasticity (Mandel, 1973; Rice, 1976; Asaro, 1983; Peirce et al., 1983; Rashid and Nemat-Nasser, 1992; Zbib, 1993; Anand and Kothari, 1996; Regueiro et al., 2002; Potirniche and Daniewicz, 2003; Marin, 2007) and at the macroscale using polycrystalline deformation (Lee, 1969; Bammann and Aifantis, 1989; Bammann et al., 1993, 1996; Bammann, 2001; Brünig, 2002). Continuum models have also been proposed that incorporate dissipative volume changes of a material in the multiplicative decomposition using an implicit representation (Nemat-Nasser, 1979; Marin and McDowell, 1996) and an explicit representation (Murakami, 1988, 1990; Onat and Leckie, 1988; Bammann and Aifantis, 1989; Canova et al., 1996; Park and Voyiadjis, 1998). Although not as popular as multiplicative decompositions, additive decompositions for the deformation gradient have also been employed to describe elastic–plastic behavior (Nemat-Nasser, 1979; Zbib, 1993; Davison, 1995; Armero and Garikipati, 1996), and dissipative volume changes (Zhou and Zhai, 1999; Tvergaard and Niordson, 2004). To model material degradation behavior at continuum level, a scalar damage variable which represents a strength decrease was first introduce by Kachanov (1958). Kachanov postulated that the loss of stiffness and integrity attributed to microcracks can be measured by a deterministic, macroscopic damage parameter and its change may be defined by the evolution of an internal variable that depends on the expected value of the micro-defect density. Through homogenization technique, the micro-defect density can be represented by a tensor of higher order. This idea to describe material degradation with damage variables of different order has led to the appearance of many mathematical formulations for these variables. We briefly mention here only some of the damage models with scalar variables (Cordebois and Sidoroff, 1979; Kachanov, 1980; Lemaitre, 1985; Murakami, 1988, 1990; Onat and Leckie, 1988; Bammann and Aifantis, 1989; Bammann et al. 1996, 1993; Chaboche, 1993; Krajcinovic, 1996; Lubarda and Krajcinovic, 1993; Voyiadjis and Park, 1999; Bonora et al., 2005; Celentano and Chaboche, 2007; Chaboche, 2008), second order tensors (e.g. Betten, 1986; Chow and Chen, 1992; Chaboche, 1993; Lubarda and Krajcinovic, 1993; Halm and Dragon, 1998; Hayakawa et al., 1998; Voyiadjis and Park, 1999; Brünig, 2003; Abu Al-Rub and Voyiadjis, 2003; Hammi et al., 2003, 2004; Hammi and Horstemeyer, 2007; Brünig and Ricci, 2005; Menzel et al., 2005), fourth order tensors (e.g. Ortiz, 1985; Yazdani and Schreyer, 1988; Carol et al., 1994; Carol and Bazant, 1997; Tikhomirov et al., 2001), and higher order tensors (Kanatani, 1984; Lubarda and Krajcinovic, 1993; Krajcinovic and Mastilovic, 1995; Cauvin and Testa, 1999). In the case of tensorially symmetric damage definition, a fictitious deformation gradient method is employed to describe damage using an effective stress concept (equivalence of energy or equivalence of strain concepts) (Cordebois and Sidoroff, 1979; Lemaitre, 1985; Chaboche, 1993; Park and Voyiadjis, 1998; Voyiadjis and Park, 1999; Voyiadjis et al., 2004; Hammi et al., 2003, 2004; Hammi and Horstemeyer, 2007). These fictitious damage mapping approaches, while quite valid for characterizing material integrity and effective stresses, are neither suited nor intended for modeling the kinematic contributions of general damage entities. Damage variables recently got wide applications in numerical modeling of brittle fracture of engineering materials. A good review on some of these models with numerous references can be found in Krajcinovic (1996). In contrast, presented here is a kinematic decomposition where the deformation due to internal defects appears naturally and capable of describing general isotropic or anisotropic behavior. A similar approach for this kinematic decomposition has been taken by Davison (1995), Bammann et al. (1993), Bammann et al., 1996; Bammann (2001), Horstemeyer et al. (2000), Clayton et al. (2005), and Solanki (2008). The idea of utilizing multiplicative decomposition of the deformation gradient to describe finite deformation elastic–inelastic response was introduced by Bilby et al. (1957) and independently by Kröner (1960). These authors were trying to solve the following elasticity problem. Consider a body deformed to a point whereby the body undergoes inelastic deformation (dislocations are introduced) (see Fig. 1). The load is then released to a final configuration described by the total deformation gradient. It is now supposed that this deformation is composed of two parts – an inelastic part associated one-to-one with the dislocations introduced during the motion and elastic part related to the residual elastic strain associated with the presence of the dislocations. During the inelastic part of the deformation gradient, rotations are generally introduced corresponding to the increasing misorientation between grains resulting in an incompatibility or non-Euclidean state. The body is then rotated and twisted by the elastic strain field associated with the dislocations, to restore the compatibility of the final state. The intermediate configuration, being non-Euclidian, could be considered fictitious because it is generally not achievable during loading. This idea has been extended by Davison (1995) and others to include damage, thermal or other deformations, and each intermediate configuration could also be considered fictitious in that they may be unachievable by any real deformation due to the possible existence of incompatibility between the states. This is in contradistinction to the idea of a fictitious configuration for an undamaged configuration as considered by Cordebois and Sidoroff (1979), Lemaitre (1985), Kattan and Voyiadjis (1990), Voyiadjis and Kattan (1990), Chaboche (1993), Park and Voyiadjis (1998), Voyiadjis and Park (1998, 1999), Stumpf and Saczuk (2001), Voyiadjis (2001), Brünig (2003), Saczuk et al. (2003), Voyiadjis et al. (2004, 2008) Hammi et al. (2003, 2004), Hammi and Horstemeyer (2007), Håkansson et al. (2006); Voyiadjis and Dorgan (2007), Brünig et al. (2008), and others. Rather than considering damage as a natural part of the deformation by inclusion in the deformation gradient decomposition, they introduce the concept of a fictitious undamaged configuration of the initial state of the material by mapping through a damage deformation gradient (or reverse mapping). From the undamaged configuration, elastic and plastic deformations are then included, describing the deformation of an undamaged material (see Fig. 2). This is followed by a different damage map back to the final configuration. The relationship between the two damage maps are then approximated through thermodynamic arguments or equivalent strain principles. Additional assumptions must then be made concerning the concentration of stress in the inelastic flow due to damage, as well as the damage dependent degradation of the elastic moduli. In our approach, no additional assumptions are required regarding these apparent physical material responses. By the complete multiplicative decomposition of the

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777

Fig. 1. Multiplicative decomposition of the deformation gradient into the plastic and elastic parts.

Fig. 2. A schematic of the kinematic of finite elastic-damage deformation using fictitious configuration based on the effective stress concept (Park and Voyiadjis, 1998).

deformation gradient (including damage), the damage dependent stress concentration of the inelastic deformation rate (plastic flow) as well as the degradation of the elastic moduli due to damage are naturally predicted. All material derivatives are calculated by pulling quantities back to the reference configuration, taking the time derivative and pushing forward to the appropriate configuration ensuring compatibility as long as the stretching and rotation of each configuration is specified. The plan of this paper is as follows: to make the derivation simple and easier to understand, we formulate the theory for an elastically damage material state and show how the damage concentrates on the stress and degraded elastic moduli. Later, we will extend the proposed theory by including deformations due to thermal and plastic processes. Finally, this theoretical framework is easily extendable to the addition of other defects (not shown here), and can be generalized to the development of a consistent coupled transport equations for species such as hydrogen, as well as providing a consistent structure for modeling events at diverse length scales. 1.1. Notation Standard notation is used throughout. Boldface symbols denote tensors the orders of which are indicated by the context. All tensor components are written with respect to a fixed Cartesian coordinate system, and the summation convention is used for repeated Latin indices, unless otherwise indicated. Let A and B second order tensors, and C a fourth order tensor; the following definitions are used in the text

ðA  BÞij ¼ Aik Bkj ;

ðA : BÞ ¼ Aij Bij ;

1 devðAÞ ¼ Aij  Akk dij 3

and

ð›A=›BÞijkl ¼ ›Aij =›Bkl ;

jAj ¼ ðAij Aij Þ1=2 :

ðC : AÞij ¼ Cijkl Akl ;

trðAÞ ¼ Aii ;

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2. Kinematics for a elastically damage material In this section, we present the proposed theoretical framework for an elastically damaged material to show how damage concentrates on the stress and degraded elastic modulus. Later in this paper we will revisit the kinematics to include deformations due to plastic, thermal, volumetric, and elastic processes. A material point with location X in the reference configuration (B0) map smoothly to a point x in the current configuration (B) at time t. Then the corresponding deformation gradient is expressed as follows:

F¼

@x @X

ð1Þ

For finite-strain (see Fig. 3) the deformation gradient, F, is decomposed into the volumetric deformation gradient, Fd, associated with dissipative volume change of the material, and the elastic or recoverable deformation, Fe, associated with the lattice rotation and is given by

F ¼ FeFd

ð2Þ

The elastic deformation gradient appears first because we want our material to unload elastically through F 1 e to an inelastically damaged material state. The Jacobian of deformation becomes,

J ¼ detF ¼ detF d detF e

ð3Þ

detF e ¼ J e > 0;

ð4Þ

detF d ¼ J d > 0;

The polar decomposition of the elastic deformation gradient Fe is

F e ¼ V e Re

ð5Þ

where Re and V e are the elastic rotation and the elastic left stretch tensors, respectively. The elastic deformation gradient, F e , describes the material point movement due to elastic motions and can be readily related to elastic unloading. The volumetric deformation gradient due to damage, F d , describes the material point movement caused by dissipative volume changes of the material from nucleation, growth and coalescences. In other words, if a void or defect is present, then enhanced dislocation nucleation, motion, and interaction would occur compared to the case where the void or defect are not present. As new internal free surfaces are created from the applied remote deformation, dislocations nucleate from the voids. One can think of this volume change related to dislocation nucleation to independently act upon void nucleation, growth, and coalescence. As such, each independent mechanism for damage can create internal free surfaces with dislocations independent of each other. The elastic deformation can be unloaded from the current configuration (B) to ~ is a physically obtainable configuration ~ through F 1 . The intermediate configuration ðBÞ the intermediate configuration ðBÞ e by unloading elastically. With the above deformation gradient components, we can define the following stretch tensors:

C ¼ F T F;

C d ¼ F Td F d ;

ee ¼ FT Fe C e

Fig. 3. Multiplicative decomposition of the deformation gradient into the damage and elastic parts.

ð6Þ

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and the corresponding Lagrangian strain tensors

E¼

1 ðC  IÞ; 2

Ed ¼

1 ðC d  IÞ; 2

e e  IÞ e e ¼ 1 ðC E 2

ð7Þ

~ is The total strain in the intermediate configuration B

e¼E ee þ E ed ¼ E e e þ F T Ed F 1 E d d

ð8Þ

Then the total strain in the reference configuration B0 arises by a pull-back operation on Eq. (8)

e d ¼ Ee þ Ed ¼ F T E e E ¼ F Td EF d e F d þ Ed

ð9Þ

_ 1 , separated into an The velocity gradient associated with the deformation gradient in the current configuration B; l ¼ FF elastic and a volumetric parts is given by 1 and ld ¼ F e F_ d F 1 d Fe

le ¼ F_ e F 1 e ;

l ¼ l e þ ld ;

ð10Þ

~ By pulling-back the above equation through F 1 e , the velocity gradient in the intermediate configuration B results,

~l ¼ ~le þ ~l ; d

~le ¼ F 1 F_ e ; e

and ~ld ¼ F_ d F 1 d

ð11Þ

Note that we can decompose any velocity gradient into skew and symmetric parts at any configuration. For example, in the current configuration B

l ¼ d þ w;

d ¼ symðlÞ ¼

1 T ðl þ l Þ; 2

1 T ðl  l Þ 2

ð12Þ

1 ~ ¼ skewð~lÞ ¼ ð~l  ~lT Þ and w 2

ð13Þ

and w ¼ skewðlÞ ¼

~ and in the intermediate configurations B

~ ¼ symð~lÞ ¼ 1 ð~l þ ~lT Þ; d 2

~l ¼ d ~ þ w; ~

With the derived expressions above, we can know map the velocity gradient and stress tensor between different configurations as shown in Table 1. e e Þ with respect to the intermediate configuration B e e E e d; E The time derivative of the strain tensors ð E;

e_ ¼ F T dF e  ð~lT  ~lT ÞE ~  Eð ~ ~l  ~le Þ E e e

ð14Þ

where

F Te dF e ¼ F Te de F e þ F Te dd F e e þ ~lT E ee þ E e e~l FT d Fe ¼ d e

d

F Te de F e

d

d

d

e_ e ¼E

Similarly, we can relate the time derivative of the elastic and the volumetric strain tensors to the elastic and the volumetric rate of deformation tensors as follow:

e  ~lT E e ~ e e_ d ¼ d E d d d  E d ld e p  ð~lT  ~lT Þ E ee  E e e ð~l  ~le Þ e_ e ¼ F T de F e ¼ F T dF e  d E e e e

ð15Þ ð16Þ

3. Thermodynamics for a elastically damage material In this work, a thermodynamic approach with internal state variables to describe the underlying irreversible processes ~ and then mapped to the current configuration B. (damage) has been formulated at the intermediate configuration B

Table 1 Mapping of stress tensor and velocity gradient between respective configurations. Configuration

Stress

Velocity gradient

B

r

e B B0

T e S ¼ J e F 1 e rF e 1 e T S ¼ J F SF

_ 1 l ¼ FF ~l ¼ F 1 lF e e L ¼ F 1~lF

d d

d

d

d

where r is the Cauchy stress in the current configuration, e S is the second Piola–Kirchhoff stress in the intermediate configuration, and S is the second Piola– Kirchhoff stress in the reference configuration.

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The Clausius–Duhem (or dissipation) inequality which expresses the assumption that the local entropy production is none for an isothermal deformation as negative, can be defined in the intermediate configuration B

q~ ~e_  S~ : F Te symð~lÞF e  0

ð17Þ

~ is the mass density. where ~ e_ is the internal energy, and q ~ as the following for the isothermal case: After Coleman and Gurtin (1967), we assume the Helmholtz free energy, w,

~_ ¼ ~e_ w

~ ¼ ~e; w

ð18Þ

~ may be defined as a function of a local state, which may be characterized by observable variThe Helmholtz free energy, w ables such as strain, and hidden variables (internal state variables) such as damage. In the case of isothermal deformation processes, the Helmholtz free energy is composed of purely reversible stored energy and irreversible energy associated with the change in microstructure produced by damage. Substituting Eq. (18) into Eq. (17), results in

~_ þ e ~w q S : F Te symð~lÞF e  0

ð19Þ

We know that

e þ ð~lT  ~lT Þ E e_ e þ d ee þ E e e ð~l  ~le Þ F Te symð~lÞF e ¼ F Te dF e ¼ E d e

ð20Þ

Substituting Eq. (20) into Eq. (19), we get

e þ ð~lT  ~lT Þ E e_ e þ d ee þ E e e ð~l  ~le Þ  0 ~_ þ S~ : ½ E ~w q d e

ð21Þ

The internal state variables are introduced into the free energy to describe the effect of ‘‘defects” such as voids, dislocations, diffusing species (i.e., hydrogen atoms) on the energetic state of the material. In general, this will lead to an observable change in mechanical response, but in some cases, for example, the energy storage of misorientation that leads to observable recrystallization, resulting in small changes in stress–strain response. The form of the internal state variable may be as important as the choice of the defect. In the case of the damage, the circumstances are complicated by the fact that the effect of damage is configurational in nature. If we consider a hole in a lattice, it is easy to visualize the dissipation associated during growth of the void by the creation of a free surface. The energy storage appears more complicated. If there is no strain in the lattice, the presence of the void has little effect. However, if the lattice is in a state of elastic strain (from either internal or external sources), the effect of the void is to concentrate this elastic strain. Therefore, the damage and the elastic strains cannot enter the free energy independently, but the form of the state variable is also subject to some obvious physical constraints. In this framework, we assume that the free energy is the function of the elastic strain and damage, i.e.,

e d Þ; ee; C ~ ¼ wð ~^ E w

e d ¼ F T F 1 where C e d

ð22Þ

~ upon these variables is constrained by the physical assumptions that, (1) if However, the dependence of the free energy, w, the elastic strain vanishes, the free energy must go to zero (or some base state), and (2) if the damage is zero, the free energy must reduces to the value resulting from the elastic strain.

^~ e ^~ e e ^~ e e d Þ ¼ 0 and wð wð E e ¼ 0; C E e ; C d ¼ IÞ ¼ wð E eÞ

ð23Þ

i.e., the first approximation to satisfying these conditions is simply the product of the elastic strain and the damage.

^~ e e ^~ e e ) wð E e ; C d Þ ¼ wð Ee C dÞ

ð24Þ

e d (elastic strain – damage state variable). ee C where E e d is the pulled back of ee and the pushed forward of C d to the natural configuration. It is a meaee C Note that the product E sure of the undamaged elastic strain with respect to the damage configuration. For the case of isotropic damage, where 1 F d ¼ ð1/Þ 1=3 I, one obtains:

e dF d ¼ F T E ee C e T 1 e F Td E d eF d F d F d ¼ Ee

ð25Þ

The material time derivative of Eq. (24) is

~_ ¼ w

~ ~ @w e d þ @w : E e_ d e_ e C ee C :E ~ e C~ d Þ ~ e C~ d Þ @ðE @ðE

ð26Þ

By substituting Eq. (26) into Eq. (21) and rearranging we get

" ~ S~  q

# h  i ~ ~ @w @w T e þ ð~lT  ~lT Þ E e_ d þ S~ : d e ee þ E e e ~l  ~le  0 ee C e_ e  q ~ Cd : E :E d e ~ ~ ~ ~ @ðEe C d Þ @ðEe C d Þ

ð27Þ

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Following classical arguments (Coleman and Noll, 1963; Kratochvil and Dillon, 1969), one can obtain from Eq. (27) the constitutive law for e S as

e ~ S ¼q

~ @w eT C e dÞ d e @ð E e C

ð28Þ

The dissipation inequality then reads as

h i e þ ^lT E e e ee ~ S: d d d e þ Ee ld  q

~ @w e_ d  0 ee C :E e dÞ ~e C @ðE

ð29Þ

The first term in Eq. (29) represents the dissipative work rate due to damage, and the second term gives the internal work associated with the field of the residual stresses which accompanies the increase in defects. Let us assume that the applied e e Þ is small, an assumption typically valid for metals. Then, a quadratic form for the Helmholtz free energy is elastic strain ð E adequate to describe the thermodynamic state of the material

 2 e d : Ce : E e d ¼ l trð E ed : E e d Þ þ k trð E e dÞ ee C ee C ee C ee C ee C ~ ¼ 1E w 2 2

ð30Þ

where K is the bulk modulus and l is the shear modulus. The isotropic fourth rank elastic tensor is given by Ce ¼ 2lI þ kI  I and the Lame constant k ¼ K  23 l. At this point, we assumed a linearized Hooke’s law both to simplify the math and because the macroscopic elastic deformation of many metals is adequately described as linear elastic. Taking the derivative of the free energy with respect to elastic strain – damage state variable we get

~ @w e dÞ ee C @ð E

e d þ k trð E e d ÞI ¼ 2l Devð E e d Þ þ K trð E e d ÞI ee C eeC ee C eeC ¼ 2l E

ð31Þ

Based on Eqs. (28) and (31), the constitutive law for S~ can be derived as

h i h i e d þ k trð E e d ÞI C e d Þ þ K trð E e d ÞI C e d ¼ 2l Devð E ed ee C ee C ee C ee C S~ ¼ 2l E

ð32Þ

By pushing Eq. (32) into the current configuration, we get T T T T T T J e F 1 e rF e ¼ ½2lF e ee F e F e c d F e þ k trðF e ee F e F e c d F e ÞIF e c d F e

ð33Þ

ee ¼ e where E e is The material time derivative of the elastic constitutive law in the intermediate configurations B F Te e F e .

_ e d þ ½2l E e d þ 2l E e d ÞI þ k trð E e d þ k trð E e d ÞI C e e_ d þ k trð E e_ d ÞI C e_ d e_ e C ee C ee C e_ e C ee C ee C S ¼ ½2l E

ð34Þ

and the current configuration B is

r_ þ

je T T T T T T T _ _ _T _T _ _ r  le r  rle ¼ J 1 e s½2lfF e ee F e þ F e ee F e þ F e ee F e gF e c d F e þ 2lF e ee F e fF e c d F e þ F e c d F e þ F e c d F e g Je þ k trðfF_ T ee F e þ F T e_ e F e þ F T ee F_ e gF T c d F e ÞI þ k trðF T ee F e fF_ T c d F e þ F T c_ d F e þ F T c d F_ e gÞI e

 ðF Te c d F e ÞT

e

þ ½2l

e

F Te e F e F Te c d F e

e

e

þk

e

e

trðF Te e F e F Te c d F e ÞIfF_ Te c d F e

e

e

þ

F Te c_ d F e

e

þ

F Te c d F_ e gT t

ð35Þ

This equation simplifies considerably when one considers small elastic strains. 3.1. In the case of small applied elastic strain Typically in metals, the elastic strains are orders of magnitude less than plastic strains in well-developed plastic flow. The small elastic strain assumption is introduced through Fe as

F e ¼ I þ He

kH e k < ee  1

T e e ¼ H e þ H e ¼ ee E 2

ð36Þ ð37Þ

e_ e and F e ¼ I þ H e along with rate form of Eq. (37) yields Using relationship F Te de F e ¼ E

e_ e ¼ e_e ðI þ H e ÞT de ðI þ H e Þ ¼ H Te de þ H Te de H e þ de þ de H e ¼ de ¼ E

ð38Þ

Any second rank tensor must satisfy the following condition based on Cayley–Hamilton’s theorem, which states, for a second rank tensor A

A3  trðAÞA2 þ trðAÞ1 detAA  detAI ¼ 0 In the above Eq. (39), substituting A ¼ F e ¼ I þ H e , for linearized elasticity we get

ð39Þ

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J e ¼ detF e ¼ 1

ð40Þ

By substituting Eq. (40) into Eq. (35) and rearranging we get the constitutive law in the current configuration as

r_  we r  rwTe ¼ ½2lde c d þ 2lee c_ d þ k trðde c d ÞI þ k trðee c_ d ÞIc d þ ½2lee c d þ k trðee c d ÞIc_ d

ð41Þ

The Eq. (41) is valid for anisotropic damage as long as it does not induce anisotropic elasticity. Notice that at this point Eq. (41) is valid for any damage describable up to a second rank tensor. 3.2. In the case of isotropic damage The development of the continuum damage mechanics begins with the introduction of a scalar damage variable (Kachanov, 1958) which represents the degradation of strength in a one-dimensional tensile bar due to creep. The idea to describe material degradation with damage variables of different order has led to the appearance of many mathematical formulations for these variables. We briefly mention here only some of the damage models with scalar variables (e.g. Kachanov, 1958; Lemaitre, 1985; Bhattacharya and Ellingwood, 1998). Consider a uniform bar subjected to a uniaxial tensile stress, r, as shown in Fig. 4. The cross-sectional area of the bar in the stressed configuration is A. The uniaxial tensile stress acting on the bar is easily expressed using the formula

r¼

P A

ð42Þ

In order to use the principles of continuum damage mechanics, the following expression for the effective uniaxial stress r, (Kachanov, 1958; Rabotnov, 1968) is derived such that:

rD ¼

P Að1  uÞ

ð43Þ

where u damage is defined as a loss of cross-sectional area or a decrease of load carrying capacity. Similarly, a relationship between the effective stress tensor, rD, and the nominal stress tensor, r, for the case of isotropic damage (i.e., scalar damage variable) can be written as follows from Eqs. (42) and (43):

rD ¼

r ð1  uÞ

ð44Þ

For Hookean elasticity, from Eq. (44), we get

rD ¼ Eð1  uÞee

ð45Þ

Therefore damage tends to degrade the elastic moduli. A more detailed three dimensional approximation has been derived by Budiansky and O’Connell (1976) using self-consistent techniques. They found the elastic moduli were degraded according to

  2 rD ¼ Kð/Þ  lð/Þ I þ 2lð/Þee 3

ð46Þ

n

Fig. 4. A square bar subjected to uniaxial tension.

D.J. Bammann, K.N. Solanki / International Journal of Plasticity 26 (2010) 775–793

783

where the coefficient K(/) is the bulk modulus and l(/) is the shear modulus. Budiansky and O’Connell (1976) found expressions for the dependence of K and l on / as

  3ð1  m0 Þ Kð/Þ ¼ K 0 1  / ; 2ð1  2m0 Þ where K0, l0, and



lð/Þ ¼ l0 1 



15ð1  m0 Þ / 7  5m0

ð47Þ

l0 m0 ¼ 12 ½3K3K002  are the values of the moduli for undamaged material. þl0

In present theory, we will show how naturally damage concentrated on the stress and degrade elastic modulus though proposed multiplicative decomposition theory. We can defined damage, /, as the ratio of the change in volume of an element e from its volume in the initial reference state to its volume in the elastically unloaded in the elastically unloaded state ð BÞ state,

/¼

Vm Ve B

ð48Þ

where V e is the total volume in the intermediate configurations and Vm is the added volume due to voids. The change in volB e is ume from the reference configuration B0 to the intermediate configuration B

V e ¼ V B0 þ V m

ð49Þ

B

where V B0 is the initial volume in the reference configuration. Using Eqs. (48) and (49) we can show that

V B0 ¼ V eð1  /Þ B

ð50Þ

where now the Jacobian is determined by the damage parameter, /,

J d ¼ detF d ¼

Ve 1 B ¼ V B0 1  /

ð51Þ

From this definition, we get

Fd ¼

1 ð1  /Þ1=3

~l ¼ F_ F 1 ¼ d d d

I

ð52Þ

/_ I 3ð1  /Þ

ð53Þ

We know

e d ¼ ðF T F 1 Þ ¼ ð1  /Þ2=3 I C d d

ð54Þ

In the rate form

/_ e_ d ¼  2 I C 3 ð1  /Þ2=3

ð55Þ

In case of linearized elasticity with isotropic damage, substitute Eqs. (54) and (55) into Eq. (41) and we get

r_  we r  rwTe ¼ 

i /_ 4h 2lee ð1  /Þ4=3 þ k trðee ÞIð1  /Þ4=3 þ ½2lde þ k trðde ÞIð1  /Þ4=3 3 ð1  /Þ

ð56Þ

Taking a trace of the above equation, we get

4 /_ p_ ¼ Ktrðde Þð1  /Þ4=3  p 3 1/

ð57Þ

Eqs. (56) and (57) shows the damage is concentrated on the stress and degrade elastic modulus which appear naturally though proposed multiplicative decomposition theory. In the next section, we will review the theory of fictitious configuration based on the effective stress concept to show the underlying physical difference with the proposed theory. 4. Kinematic description of damage using the fictitious configuration based on the effective stress concept The kinematics of finite elastic-damage deformation based on the fictitious effective concept is shown in Fig. 2. In this concept, the initial undeformed configuration that consists of the initial material damage is denoted by B0. The configuration B represents the final elastically deformed and damaged configuration of the body after being subjected to a set of external stimuli. The configuration B0 undergoes a sequence of deformations starting with an elastic deformation without damage Be, followed by a damage deformation. The configuration B0 represents the initial configuration of the body that is obtained by fictitiously removing the initial damage from the B0 configuration. If the initial configuration is

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undamaged consequently there is no difference between configurations B0 and B0 . Similarly, the configuration B is obtained by fictitiously removing the damage from configuration B. The deformation gradient referred to the undeformed configuration, B0, denoted by F is multiplicatively decomposed into the elastic deformation gradient denoted by Fe and the damage deformation gradient denoted by Fd. In the case of the fictitious effective concept, the total deformation gradient is decomposed into

F ¼ F 0d F e F d

ð58Þ

where F 0d and F d are the fictitious effective initial and final damage deformation gradients, respectively, and F e is the fictitious effective elastic deformation gradient. In the fictitious effective concept (see Park and Voyiadjis, 1998), the Green deformation tensor cannot be expressed in the classical way, because two effective configurations B0 and B are obtained fictitiously removing damage from the two real configurations, therefore the green deformation tensor in the fictitious effective concept is define as follows: T

C ¼ F Td F Te F 0d F 0d F e F d  F Td F d þ F 0d F 0d

ð59Þ

The Lagrangian damage strain tensor measured with respect to the fictitious configuration B is given by

Ed ¼

1 T F Fd  I 2 d

ð60Þ

and the corresponding Lagrangian effective elastic strain tensor measured with respect to the fictitious configuration B is given by

Ee ¼

1 T ðF F e  IÞ 2 e

ð61Þ

The total effective Lagrangian stress tensor in the reference configuration B0 is therefore expressed as follows

E¼

1 T  1  0T 0 1 F F  I þ F 0d F Te F e  I F 0d þ F Te F 0d T F Td F d  I F 0d F e 2 d d 2 2

ð62Þ

T F 0d T F 0d  I is the Lagrangian initial damage strain tensor, 12 F 0d F Te F e  I F 0d is the measure of the Lagrangian elastic  strain with respect to the current configuration, and 12 F Te F 0d T F Td F d  I F 0d F e is the measure of the Lagrangian final damage strain with respect to the current configuration. The Jacobian of the damage deformation based on the fictitious effective concept can be written as follows:

where

1 2



J d ¼ detF d ¼

1 ð1  /Þ1=3

ð63Þ

Thus one can assume the following relationship without loss of generality:

Fd ¼

1 ð1  /Þ1=3

I

ð64Þ

Then a quadratic form for the Helmoltz free energy in the current configuration is to describe the thermodynamic state of the material as

w¼

1 ee : Ce : ee 2

ð65Þ

where Ce is the fourth rank effective stiffness of the material, which is defined as follows: 1 1 T Ce ¼ F T d F d Ce F d F d

ð66Þ

Substituting Eq. (64) into Eq. (66), we get

Ce ¼ ð1  /Þ4=3 Ce

ð67Þ

Also we know that

@w r¼q ¼ Ce : ee ¼ ð1  /Þ4=3 Ce : ee @e e

ð68Þ

Thee above equation is consistent with Eq. (56), which was derived using the proposed theory of multiplicative decomposition. This suggests that the use of the proposed theory is consistent with not only the physics of damage but representing the material damage due to the internal defects. The advantage of the proposed theory is to avoid the empirical removal of damage fictitiously and adding it back. The effective stress concept shows consistent behavior in case of isotropic damage when compared to the proposed theory but may differ in the case of higher order damage. Also in the proposed theory, in addition to the degradation of elastic moduli, the stress in the flow rule is naturally concentrated by the damage in a

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manner consistent with that proposed by Kachanov (1958), as will be discussed later. This generally requires an additional independent empirical constituent assumption in the previous theory. In the next section, we will extend the proposed theory by incorporating deformation due to the thermal and inelastic processes to show how the internal state variables related to inelastic processes are physically affected by the presences of elastic strain due to damage. 5. Kinematics (thermal, plastic, volumetric, and elastic deformations) The finite-strain deformation gradient (see Fig. 5), F, is decomposed into a thermal deformation, Fh, associated with the thermal expansion; the deviatoric plastic deformation, Fp, attributed to the history of dislocation glide/movement; the volumetric deformation gradient, Fd, associated with dissipative volume change of the material; and the elastic or recoverable deformation, Fe, associated with the lattice rotation/stretch and is given by

F ¼ FeF dF pF h

ð69Þ F 1 e

The elastic deformation gradient appears first because we want our material to unload elastically through to an inelastically damaged material state. The position of the thermal deformation gradient is chosen somewhat arbitrarily and could effectively have been placed before the elastic deformation gradient. The position of the plastic and damage deformation gradients have been chosen in order to naturally concentrate the stress in the plastic flow rule by the damage state and also degrade the elastic moduli. This will be demonstrated in great detail in the following. For most polycrystalline materials of interest, the thermal deformation gradient associated with thermal expansion is small; as such the thermal portion of the deformation gradient can be given in terms of the linear coefficient of thermal expansion f and the temperature change Dh as

F h ¼ F h I ¼ ð1 þ fDhÞI

ð70Þ

And as a result the placement of thermal deformation gradient becomes irrelevant. The Jacobian of the deformation becomes

J ¼ detF ¼ detF h detF p detF d detF e detF e ¼ J e > 0;

detF d ¼ J d > 0;

ð71Þ detF p ¼ 1;

detF h ¼ F 3h ¼ ð1 þ fDhÞ3

ð72Þ

The elastic deformation gradient, Fe, describes the material point movement due to elastic motions and can be readily related to elastic unloading. The plastic deformation gradient, Fp, describes the material point movement due to the distortion caused by dislocation movement. The thermal deformation gradient, Fh, is due to thermal variation. The volumetric deformation gradient due to damage, Fd, describes the material point movement caused by dissipative volume changes of the material from nucleation, growth and coalescences. In other words, if a void or defect is present, then enhanced dislocation nucleation, motion, and interaction would occur compared to the case if the void or defect were not present. As new internal free surfaces are created from the applied remote deformation, dislocations nucleate from the voids. One can think of this volume change related to dislocation nucleation to independently act upon void nucleation, growth, and coalescence. As such, each independent mechanism for damage can create internal free surfaces with dislocations independent of each other.

Fig. 5. Multiplicative decomposition of the deformation gradient into the thermal, plastic, damage and elastic parts.

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~ The interThe elastic deformation can be unloaded from the current configuration (B) to the intermediate configuration ðBÞ. ~ is a physically obtainable configuration by unloading elastically, while with a given damage, the mediate configuration ðBÞ ^ and B0 are not obtainable (i.e., damage deformation is irrecoverable). configurations B; B With the above deformation gradient components, we can define the following stretch tensors:

C ¼ F T F;

C h ¼ F Th F h ;

C^ p ¼ F Tp F p ;

ee ¼ FT Fe C e

C d ¼ F Td F d ;

ð73Þ

and the corresponding Lagrangian strain tensors,

E¼

1 ðC  IÞ; 2

Eh ¼

1 ðC h  IÞ; 2

^ p ¼ 1 ðC^ p  IÞ; E 2

Ed ¼

1 ðC d  IÞ; 2

e e  IÞ e e ¼ 1 ðC E 2

ð74Þ

Similar expressions for the damage strain have been discussed by Steinmann and Carol, 1998; Hayakawa et al., 1998; Voyiadjis and Park, 1999; Brünig, 2002, Brünig, 2003; and others. Then the total strain is obtained by pulling all back to the configuration B0,

E ¼ Ee þ E d þ Ep þ E h

ð75Þ

The fact that the total strain decomposes additively into the sum of elastic, damage, plastic and thermal parts is true with respect to every configuration as long as each strain tensor has been properly ‘‘pushed forward” or ‘‘pulled back” between configurations. _ 1 , is separated into The velocity gradient associated with the deformation gradient in the current configuration B; l ¼ FF an elastic, plastic, thermal and a volumetric part, and is given by

l ¼ l e þ ld þ lp þ lh ;

le ¼ F_ e F 1 e ;

1 ld ¼ F e F_ d F 1 d Fe ;

1 1 1 1 1 lp ¼ F e F d F_ p F 1 lh ¼ F e F d F p F_ h F 1 p F d F e and h Fp Fd Fe

ð76Þ

As with the strain tensor, similar additive equation holds for the velocity gradient with respect to every configuration. ~ By pulling back the above equation through F 1 e , the velocity gradient in the intermediate configuration B results,

~l ¼ ~le þ ~l þ ~lp þ ~lh ; d

~le ¼ F 1 F_ e ; e

~l ¼ F_ F 1 ; d d d

~lp ¼ F F_ p F 1 F 1 d p d

1 1 1 and ~lh ¼ F d F p F_ 1 h Fh Fp Fd

ð77Þ

Note that we can decompose any velocity gradient into skew and symmetric parts in any configuration. For example, in the current configuration B

l ¼ d þ w;

d ¼ symðlÞ ¼

1 T ðl þ l Þ; 2

1 T ðl  l Þ 2

ð78Þ

1 ~ ¼ skewð~lÞ ¼ ð~l  ~lT Þ and w 2

ð79Þ

and w ¼ skewðlÞ ¼

~ and in the intermediate configuration B

~l ¼ d ~ þ w; ~

~ ¼ symð~lÞ ¼ 1 ð~l þ ~lT Þ; d 2

With the derived expressions above, we can now map the velocity gradient and stress tensor between different configurations as shown in Table 2. e e; E e h Þ living in the intermediate configuration B ~ e E e p; E The material time derivative of the strain tensors ð E;

e_ ¼ F T dF e  ð~lT  ~lT Þ E e  Eð e ~l  ~le Þ E e e

ð80Þ

where

F Te dF e ¼ F Te de F e þ F Te dd F e þ F Te dp F e þ F Te dh F e e p þ ~lT E ee þ E e p~lp F T dp F e ¼ d e

p

e þ ~lT E e e~ ¼d h h e þ E e lh e þ ~lT E e~ e F Te dd F e ¼ d d d e þ E e ld _ ee F Te de F e ¼ E F Te dh F e

Table 2 Mapping of stress tensor and velocity gradient between respective configurations. Configuration

Stress

Velocity gradient

B

r

e B

T e S ¼ Je F 1 e rF e SF T S ¼ J F 1 e

_ 1 l ¼ FF ~l ¼ F 1 lF e e l ¼ F 1~lF

T S^ ¼ Jp F 1 p SF p 1 ^ T S ¼ J F SF

^l ¼ F 1lF p p L ¼ F 1^lF h

B b B B0

d d

h h

d

h

d

h

d

D.J. Bammann, K.N. Solanki / International Journal of Plasticity 26 (2010) 775–793

e p  ð~lT þ ~lT Þ E e ep  E e p ð~lp þ ~ld Þ  ~lT E e ~ e_ p ¼ d E p d p d  E d lp e h  ð~lT  ~lT Þ E e_ h ¼ d eh  E e h ð~l  ~le Þ  ~lT ð E ep þ E e dÞ  ðE ep þ E e d Þ~lh E e h e  ~lT E e ~ e e_ d ¼ d E d d d  E d ld ep  d eh  d e  ð~lT  ~lT Þ E ee  E e e ð~l  ~le Þ e_ e ¼ F T de F e ¼ F T dF e  d E d e e e

787

ð81Þ ð82Þ ð83Þ ð84Þ

ep; d e h and d e in Eq. (84) as well as ~l  ~le that To complete this system of equations it is necessary to write expressions for d d e h , and w e d . These equations have been discussed in Regueiro et al. (2002) e p; w necessitates the additional representation of w and more physically based expressions will be given in future works. The rotational parts are necessary to specify the convective parts of the derivatives in Eq. (84), that in conjunction with the expression of companion stretches fixes or specifies the position of each configuration at any given instant as can been seen by properly taking the material derivative of these kinematic quantities by pulling back to the current configuration, taking the time derivative and pushing forward. The required rotation and rotational rate naturally appear to render the derivatives properly objective. 6. Thermodynamics (thermal, plastic, volumetric and elastic deformations) In this work, a thermodynamic approach, with internal state variables to describe the underlying irreversible processes ~ and then mapped to the current configuration B. (plastic and damage) was formulated at the intermediate configuration B, ~ The rate of change of internal energy or the First Law of Thermodynamics in the intermediate configuration B,

~ q ~  ~r ¼ 0 q~ ~e_  S~ : F Te symð~lÞF e þ r

ð85Þ

~ is the heat flux per unit area, q ~ is the mass density, and ~r is the heat source per unit volume. where ~ e_ is the internal energy, q Similarly the entropy inequality or the Second Law of Thermodynamics is

~r h

1 h

~ q ~þ q~ ~S_ g  q~  r

1 h2

~h ~r q

ð86Þ

where ~ Sg is the arbitrary history dissipation function related to entropy, and h is the absolute temperature. ~ as the following: After Coleman and Gurtin (1967), we assume the Helmholtz free energy, w,

~ ¼ ~e  h~Sg ; w

~_ ¼ ~e_  h_ ~Sg  h~S_g w

ð87Þ

~ may be defined as a function of a local state, which may be characterized by observable variThe Helmholtz free energy, w ables such as temperature and strain, and hidden variables (internal state variables) such as isotropic hardening, and kinematic hardening. In case of isothermal deformation processes, the Helmholtz free energy is composed of purely reversible stored energy and irreversible energy associated with the change in microstructure produced by plastic deformation and damage. Substituting Eq. (87) into Eq. (85) and using relation Eq. (86) results in

1 ~h  0 ~_  q ~r ~ h_ ~Sg þ S~ : F Te symð~lÞF e  q ~w q h

ð88Þ

We know that

ep þ d eh þ d e þ ð~lT  ~lT Þ E e_ e þ d ee þ E e e ð~l  ~le Þ F Te symð~lÞF e ¼ F Te dF e ¼ E d e

ð89Þ

Substituting Eq. (89) into Eq. (88), we get

h i ep þ d eh þ d e þ ð~lT  ~lT Þ E ~h  0 e_ e þ d ee þ E e e ð~l  ~le Þ  1 q ~_  q ~r ~ h_ ~Sg þ S~ : E ~w q d e h

ð90Þ

e e , the absolute temperature h, the In this frame work, we assume that the free energy is the function of the elastic strain, E e i , (where i = 1 to number of internal ~ h, and a set of i number of strain-like internal state variables, A temperature gradient, r state variables) i.e.,

e iÞ e d ; h; r ~ h; A eeC ~ ¼ wð ~^ E w

ð91Þ

e i , are incorTo model evolving internal structure during plastic deformation, the set of strain-like internal state variables, A porated to represent the irreversible mechanism of the inelastic material. In present study, the description of structural change was incorporated with two internal state variables which represent the internal elastic strain field induced by the presence of dislocations created by deformation. These state variables are denoted here as ~eS , a scalar variable represent lattice deformation due to the presence of statistically stored dislocations, and e b, a symmetric deviatoric tensorial variable represent incompatible lattice curvature due to presence of geometrically necessary dislocations (GNDs) at grain boundaries eS , represents induced non-directional (isotropic) hardening, while eb, and around second phase particles. In this context, ~

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gives the direction dependent (kinematic) hardening effects. We strongly believe that the states of elastic strain (from either internal or external sources) are affected by the elastic strain concentrated by the void. Thus, assuming

e i ¼ f~eS~t d ; b ~C~ d g A where ~t d ¼

ð92Þ

~ Þ trðC d 3

e d ; ~eS~td ; b ~ hÞ ee C ~ ¼ wð ~^ E ~C~ d ; h; r w

ð93Þ

Notice that the free energy is function of elastic strain-like internal variables and thermal internal energy variables. We view plastic deformation as the motion of dislocations and the state of the material as a freeze-frame of the deformed state which is represented by the elastic lattice deformation due to the presence of dislocations and due to the external loading. In comparison with Voyiadjis and Park, 1999; Brünig, 2003; and others who have used the approach described in these papers, we have not included any dependence of the Helmholtz free energy upon inelastic strains. We believe that the plastic deformation does not explicitly contain any information about the current energetic state of the material and as such cannot be a state variable. A material point may have experienced a very large deformation yet contain few geometric defects such as dislocations and therefore the free energy is similar to that of an undeformed material. For an example, at high temperatures dislocations may recover as fast as they are created resulting in little or no change of state and apparent hardening. As illustrated in Fig. 6, we see a defect free lattice with a reference free energy and stress. As the dislocation passes through the representative volume element (RVE) the free energy and associated stress (internal elastic strain) attains a finite value and plastic strain is one half of burgers vector. After the dislocation has passed through the RVE the free energy and stress revert to their reference value but the plastic strain is now equal to one burgers vector. Therefore the plastic strain is not an appropriate choice as a state variable except in cases of a one to one correspondence between plastic strain and dislocation density. For example, this occurs when the RVE is so small as to contain one dislocation whereby all dislocations are b represent the elastic strains geometrically necessary dislocations (GNDs). In this vein, the internal state variables ~eS , and e associated with internal defects and the damage enters the free energy as an elastic strain concentration factor as described in detail in the following. In the case of statically stored dislocations (SSDs), it is important to understand that the elastic strain associated with the network of dislocations is the proper choice to describe energy changes associated with the observed energy changes. Energy stored in dipole networks is proportional to the density of dislocations. If we assume the free energy depends linearly upon the SSDs then the thermodynamic conjugate force is a constant. To recover the Taylor theory of hardening requires an assumption that the free energy depends upon the SSD density to the 3/2 power, which does not correspond to any physical dislocation structure. This can be resolved by introducing the appropriate state variable associated with the presence of SSDs. Consider a dislocation in a lattice (see Fig. 7). The extra plane of atoms introduces an elastic strain field in the crystal. A scalar representation of this strain for a Frank network is shown to be of the form quadratic

^~ ~ ~ ~ ¼ wð eS td Þ w

ð94Þ

~ s . In an increment of strain, We relate this strain-like internal variable, ~eS , to the density of statistically-stored dislocations q dislocations are stored inversely proportional to the mean free path, which in a Taylor lattice is inversely proportional to the

Fig. 6. The state variable present in the model must be physically motivated as illustrated by: (a) a RVE made of defect free lattice with a reference free energy and stress, (b) as the dislocation passes through the RVE the free energy and associated stress attains a finite value and plastic strain is one half of burgers vector, and (c) after the dislocation has passed through the RVE the free energy and stress revert to their reference value but the plastic strain is now equal to one burgers vector.

D.J. Bammann, K.N. Solanki / International Journal of Plasticity 26 (2010) 775–793

789

Fig. 7. Statistically stored dislocations (SSD) provide observed hardening.

square root of dislocation density. Dislocations are annihilated or ‘‘recover” due to cross-slip or climb in a manner proportional to the dislocation density. Following the Taylor assumption (Bammann, 2001), the lattice deformation due to the presence of statistically stored dislocations, ~eS can be defined as

pffiffiffiffiffi ~eS ¼ b q ~s

ð95Þ

where b is the magnitude of Burger’s vector (see Fig. 7) Assuming energy is quadratic in elastic strains for SSDs

1 ^~ ~ ~ wð eS td Þ ¼ lð~eS~td Þ2 2

ð96Þ

where l is the temperature dependent shear modulus. Therefore the thermodynamic force conjugate associated with the internal state variable related to SSDs can be defined by as:

j~ S ¼

^~ ~ ~ pffiffiffiffiffi 2 @ wð eS td Þ ~ s ½~t d  ¼ l~eS ½~td 2 ¼ lb q ~ ~ @ðeS td Þ

ð97Þ

As we can see from the above equation, the Taylor theory is recovered by an appropriate choice of internal state variable. The material time derivative of Eq. (93) yields

~_ ¼ w

~ ~ ~ ~ ~ ~ _ ~ ~ @w @w @w e d þ @w : b e d þ @w E e_ d e_ d þ @ w  h_ þ @ w  r ~ h þ @w : E e_e C ee C ~_ C ~C  ~e_ s~t d þ  ~es~t_ d þ :b ~ e e e e e e ~ ~ @h @ð~es~t d Þ @ð~es~t d Þ @ r h @ðb C d Þ @ðb C d Þ @ð E e C d Þ @ð E e C d Þ ð98Þ

By substituting Eq. (98) into Eq. (90) and rearranging we get

" e ~ S q ~ q

# " # ~ ~ ~ @w @q @w _ _ T ~p þ d ~h þ d ~ þ ð~lT  ~lT Þ E e e e ee þ E e e ð~l  ~le Þ e e ~ ~ S : ½d þ S g  h_ þ e C : Ee  q : Ee C d  q d e e dÞ d e dÞ ee C ee C @h @ð E @ð E

~ ~ ~ ~ _ @w @w @w @w @w ed  q e_ d  q ~h P 0 ~h  1q ~_ C ~C ~ :r ~ ~ ~ ~  ~es~t_ d  q  ~e_ s~td  q r :b :b ~h e dÞ e dÞ ~C ~C h @ð~es~t d Þ @ð~es~t d Þ @r @ðb @ðb

ð99Þ

Because the increasing strong interaction between these individual dislocation strain fields and those of neighboring dislob, with cations impedes further dislocation motion (material hardening), one could phenomenologically associate ~es , and e b are hardening variables. The stress-like internal state variables work-conjugate to ~es and e

j~ s ¼ q~

~ @w ~t d ; ~ @ðes~td Þ

~ ~a ¼ q

~ @w C~ Td ~ @ðbC~ d Þ

ð100Þ

790

D.J. Bammann, K.N. Solanki / International Journal of Plasticity 26 (2010) 775–793

Following classical arguments of (Kratochvil and Dillon, 1969)

e ~ S ¼q

~ @w eT ; C e dÞ d ee C @ð E

ð101Þ

~ @w e Sg ¼  @h ~ @w ¼0 ~h @r

ð102Þ ð103Þ

The dissipation inequality then reads as

~ p þ ~lT E e e e~ S : ½d p e þ E e lp  þ

j~ s  ~e_ s þ q~



~ @w e_ d ee C :E e dÞ e @ð E e C ! ~_ þ e ~C~_ d þ ~a : b S :b

!

~ þ ~lT E e e e~ ~ S : ½d d d e þ E e ld   q

~ ~ @w @w ~  ð~es~t_ d Þ þ q _ ~ @ðbC~ d Þ @ð~es~t d Þ

1 ~ þ ~lT E e e~ ~ ~ : ½d h h e þ E e lh   q  rh  0 h

ð104Þ

The first term in Eq. (104) represents the plastic work rate due to irreversible dislocation motion, the second term represents the dissipative work rate due to damage, the third term gives the internal work associated with the field of the residual stresses which accompanies the increase in defect density, and the last terms appears due to thermal deformation. The Helmoltz free energy for small elastic and internal strains in a quadratic form to describe the thermodynamic state of the material is,

ed : b ed : E e d Þ þ kðhÞ ðtrð E e d ÞÞ2 þ C j lðhÞð~es~t d Þ2 þ C b lðhÞb eeC eeC ee C ~ ~ ¼ lðhÞtrð E ~C ~C~ d þ wðhÞ w 2

ð105Þ

where kðhÞ ¼ KðhÞ  23 lðhÞ

~ @w e d þ kðhÞtrð E e d ÞI ee C ee C ¼ 2lðhÞ E e dÞ ee C @ð E

ð106Þ

e ed e d þ kðhÞtrð E e d ÞI C ee C ee C S ¼ ½2lðhÞ E

ð107Þ

By pushing Eq. (107) into the current configuration, we get T T T T T T J e F 1 e rF e ¼ ½2lðhÞF e ee F e F e c d F e þ kðhÞtrðF e ee F e F e c d F e ÞIF e c d F e

ð108Þ

The resultant internal stress associated with the elastic strain energy due to the presences of the SSDs can be derived by taking derivative of the free energy (Eq. (105)) with respect to ð~es~td Þ and we get

j~ s ¼ q~

~ @w ~t d ¼ 2C j lðhÞ~es ð~td Þ2 ~ @ðes~td Þ

ð109Þ

Similarly, one can derived the following equation for the resultant internal stress due to the presences of the GNDs by e d Þ, we get ~C taking derivative of the free energy (Eq. (105)) with respect to ðb

~ ~a ¼ q

~ @w e dC eT e T ¼ ½cb lðhÞb ~C C d d ~ @ðbe c dÞ

ð110Þ

e is The material time derivative of the elastic constitutive law in the intermediate configurations B

  @ lðhÞ _ e e _ e d þ 2lðhÞ E e d ÞI þ kðhÞtrð E e d ÞI þ kðhÞtrð E e e_ d þ @kðhÞ htrð e_ d ÞI C ed e_ e C ee C e_ e C ee C ee C _ E h E e C d þ 2lðhÞ E S¼ 2 @h @h e d þ kðhÞtrð E e d ÞI C e_ d ee C ee C þ ½2lðhÞ E

ð111Þ

and in the current configuration B is

r_ þ

 je @ lðhÞ _ T T r  le r  rle ¼ J 1 hF e ee F e F Te c d F e þ 2lðhÞfF_ Te ee F e þ F Te e_ e F e þ F Te ee F_ e gF Te c d F e þ 2lðhÞF Te ee F e fF_ Te c d F e e s 2 @h Je @kðhÞ _ þ F Te c_ d F e þ F Te c d F_ e g þ htrðF Te ee F e F Te c d F e ÞI þ kðhÞtrðfF_ Te ee F e þ F Te e_ e F e þ F Te ee F_ e gF Te c d F e ÞI @h i þ kðhÞtrðF Te ee F e fF_ Te cd F e þ F Te c_ d F e þ F Te c d F_ e gÞI ðF Te c d F e ÞT þ ½2lðhÞF Te ee F e F Te c d F e þ kðhÞtrðF Te ee F e F Te c d F e ÞIfF_ Te c d F e þ F Te c_ d F e þ F Te c d F_ e gT t

ð112Þ

D.J. Bammann, K.N. Solanki / International Journal of Plasticity 26 (2010) 775–793

791

e is The rate form of the internal stress associated with SSDs in the intermediate configurations B

j~_ s ¼ 2

@ lðhÞ _ hC j ~es ð~t d Þ2 þ 2C j lðhÞ~e_ s ð~td Þ2 þ 4C j lðhÞ~es~td~t_ d @h

ð113Þ

and in the current configuration B is

j_ s ¼ 2

@ lðhÞ _ hC j es ðt d Þ2 þ 2C k lðhÞe_ s ðt d Þ2 þ 4C j lðhÞes t d t_ d @h

ð114Þ

e as Similarly, we can derive the constitutive law related to the GNDs in the current configuration B

  @ lðhÞ _ ~ e e_ d þ C b lðhÞb e dC e_ T e T þ ½C b lðhÞb ed C ~C ~C ~_ C a~_ ¼ C b hb C d þ C b lðhÞb d d @h

ð115Þ

and in the current configuration B is

" je @ lðhÞ T 1 a_ þ a  le a  ale ¼ ðJe Þ C b lðhÞ Je @h

# h_ T T_ T _ T _ F bF þ F bF þ F bF þ F bF F T c F ðF T c F ÞT lðhÞ e e e e e e e e e d e e d e   T þ ðJ e Þ1 C b lðhÞF Te bF e F_ Te cd F e þ F Te c_ d F e þ F Te cd F_ e F Te cd F e þ F Te cd F e ðF_ Te cd F e þ F Te c_ d F e þ F Te cd F_ e ÞT ð116Þ

In the case of small applied elastic strain, the constitute Eqs. (112), (114), and (116) in the current configuration (B) reduced to as follows:

  @ lðhÞ _ @kðhÞ _ r_  we r  rwTe ¼ 2 hee c d þ 2lðhÞde c d þ 2lðhÞee c_ d þ htrðee c d ÞI þ kðhÞtrðde c d ÞI þ kðhÞtrðee c_ d ÞI c d @h @h _  ½2lðhÞee c d þ kðhÞtrðee c d ÞIc d @ lðhÞ _ j_ s ¼ 2 hC j es ðt d Þ2 þ 2C j lðhÞe_ s ðtd Þ2 þ 4C j lðhÞes td t_ d @h   @ lðhÞ _ T _ d cT þ ½C b lðhÞbc d c_ T a_  le a  ale ¼ C b hbc d þ cb lðhÞbc_ d þ C b lðhÞbc d d @h

ð117Þ ð118Þ ð119Þ

In the case of isotropic damage, substituting Eqs. (54) and (55) into Eq. (117) yields constitutive law in the current configuration (B) as

  4 @ lðhÞ @kðhÞ /_ _  /Þ4=3 ee þ r_  we r  rwTe ¼  ½2lðhÞee ð1  /Þ4=3 þ kðhÞtrðee Þð1  /Þ4=3 I trðee ÞI hð1 þ 2 3 @h @h ð1  /Þ þ ½2lðhÞde þ kðhÞtrðde ÞIð1  /Þ4=3

ð120Þ

Taking a trace of Eq. (120), we get

p_ ¼

_ @KðhÞ _  /Þ4=3 þ KðhÞtrðde Þð1  /Þ4=3  4 p / trðee Þhð1 @h 3 1/

ð121Þ

Then, the deviatoric stress (r0 ) can be obtained as

r_ 0  we r0  r0 wTe ¼

@ lðhÞ r0 _ 4 /_ 0 hð1  /Þ4=3 þ 2lðhÞde ð1  /Þ4=3  r0 @h lðhÞ 3 1/

ð122Þ

Equations (120)–(122) show not only the degradation of elastic moduli, the stress is naturally concentrated by the damage in a manner consistent with that proposed by Kachanov (1958). This generally requires an additional independent empirical constitutive assumption in previous theory. Similarly, we can show how the damage is concentrated on the kinematic and isotropic hardening internal state variables using the proposed theory. In the case of isotropic damage, substituting Eqs. (54) and (55) into Eqs. (118) and (119) yields constitutive law in the current configuration (B) as

# 4 h_ /_ _ j_ s ¼ 2C j lðhÞð1  /Þ e þe  e lðhÞ s s 3 s 1  / " # i /_ @ lðhÞ h_ 4h T _ a_  le a  ale ¼ C b lðhÞ b þ b ð1  /Þ4=3  C b lðhÞbð1  /Þ4=3 @h lðhÞ 3 1/ 4=3

" @ lðhÞ @h

ð123Þ ð124Þ

The above Eqs. (123) and (124) show how naturally we can derive the damage concentration on the kinematic and isotropic hardening evolution equations using the proposed theory without considering a fictitious effective configuration.

792

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7. Conclusion In this paper, we were able to present a theory that is consistent in not only representing the material damage but also modeling the material damage coupled with the internal elastic strain fields of internal defects. We believe this new consistent formulation of polycrystalline finite-strain elastio-plasticity coupling kinematics and thermodynamics with damage using an extended multiplicative decomposition of the deformation gradient that accounts for temperature effects will help in modeling deformation due to complex combination of mechanisms, such as thermally-activated dislocation motion and generation, dislocation annihilation, dislocation drag, texture, void nucleation, growth, and coalescence. We showed that such a macroscopic decomposition of the deformation gradient is physically motivated by the mechanisms underlying lattice deformation, plastic flow, and evolution of damage at voids in polycrystalline material. Finally, we showed that prescribing plasticity and damage evolution equations in their physical intermediate configurations leads to physically justified evolution equations in the current configuration. In the past, these equations have been modified in order to represent experimentally observed behavior with regard to damage evolution, whereas in this paper, these modifications appear naturally through mappings by the multiplicative decomposition of the deformation gradient. The prescribed kinematics captures precisely the damage deformation (of any rank) and does not require introducing a fictitious undamaged configuration or mechanically equivalent of the real damaged configuration as used in past. 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