Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations

Journal of the Mechanics and Physics of Solids 50 (2002) 1297 – 1329

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Continuum thermodynamic models for crystal plasticity including the e'ects of geometrically-necessary dislocations Bob Svendsen∗ Department of Mechanical Engineering, University of Dortmund, D-44227 Dortmund, Germany Received 12 February 2001; received in revised form 1 July 2001; accepted 5 October 2001

Abstract The purpose of this work is the formulation of constitutive models for the inelastic material behaviour of single crystals and polycrystals in which geometrically necessary dislocations (GNDs) may develop and in4uence this behaviour. To this end, we focus on the dependence of the development of such dislocations on the inhomogeneity of the inelastic deformation in the material. More precisely, in the crystal plasticity context, this is a relation between the density of GNDs and the inhomogeneity of inelastic deformation in glide systems. In this work, two models for GND density and its evolution, i.e., a glide-system-based model, and a continuum model, are formulated and investigated. As it turns out, the former of these is consistent with the original two-dimensional GND model of Ashby (Philos. Mag. 21 (1970) 399), and the latter with the more recent model of Dai and Parks (Proceedings of Plasticity ’97, Neat Press, 1997, p. 17). Since both models involve a dependence of the inelastic state of a material point on the (history of the) inhomogeneity of the glide-system inelastic deformation, their incorporation into crystal plasticity modelling necessarily implies a corresponding non-local generalization of this modelling. As it turns out, a natural quantity on which to base such a non-local continuum thermodynamic generalization, i.e., in the context of crystal plasticity, is the glide-system (scalar) slip deformation. In particular, this is accomplished here by treating each such slip deformation as either (1), a generalized “gradient” internal variable, or (2), as a scalar internal degree-of-freedom. Both of these approaches yield a corresponding generalized Ginzburg– Landau- or Cahn–Allen-type ?eld relation for this scalar deformation determined in part by the dependence of the free energy on the dislocation state in the material. In the last part of the work, attention is focused on speci?c models for the free energy and its dependence on this state. After summarizing and brie4y discussing the initial-boundary-value problem resulting from the current approach as well as its algorithmic form suitable for numerical implementation, the work ends with a discussion of additional aspects of the formulation, and in particular the connection

∗ Tel.: +49-231-755-2686; fax: +49-231-755-2688. E-mail address: [email protected] (B. Svendsen).

0022-5096/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 5 0 9 6 ( 0 1 ) 0 0 1 2 4 - 7

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of the approach to GND modelling taken here with other approaches. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thermodynamics; Crystal plasticity; Dislocations; Hardening; Inhomogeneity; Incompatibility

1. Introduction Standard micromechanical modelling of the inelastic material behaviour of metallic single crystals and polycrystals (e.g., Hill and Rice, 1972; Asaro, 1983; Cuiti˜no and Ortiz, 1992) is commonly based on the premise that resistance to glide is due mainly to the random trapping of mobile dislocations during locally homogeneous deformation. Such trapped dislocation are commonly referred to as statistically stored dislocations (SSDs), and act as obstacles to further dislocation motion, resulting in hardening. As anticipated in the work of Nye (1953) and KrKoner (1960), and discussed by Ashby (1970), an additional contribution to the density of immobile dislocations and so to hardening can arise when the continuum lengthscale (e.g., grain size) approaches that of the dominant microstructural features (e.g., mean spacing between precipitates relative to the precipitate size, or mean spacing between glide planes). Indeed, in this case, the resulting deformation incompatibility between, e.g., “hard” inclusions and a “soft” matrix, is accommodated by the development of so-called geometrically necessary dislocations (GNDs). Experimentally observed e'ects in a large class of materials such as increasing material hardening with decreasing (grain) size (i.e., the Hall–Petch e'ect) are commonly associated with the development of such GNDs. These and other experimental results have motivated a number of workers over the last few years to formulate various extensions (e.g., based on strain gradients: Fleck and Hutchinson, 1993, 1997) to existing local models for phenomenological plasticity, some of which have been applied to crystal plasticity (e.g., strain-gradient-based approach: Shu and Fleck, 1999; Cosserat-based approach: Forest et al., 1997) as well. Various recent e'orts in this direction based on dislocation concepts, and in particular on the idea of Nye (1953) that the incompatibility of local inelastic deformation represents a continuum measure of dislocation density (see also KrKoner, 1960; Mura, 1987), include Steinmann (1996), Dai and Parks (1997), Shizawa and Zbib (1999), Menzel and Steinmann (2000), Acharya and Bassani (2000), and most recently Cermelli and Gurtin (2001). In addition, the recent work of Ortiz and Repetto (1999) and Ortiz et al. (2000) on dislocation substructures in ductile single crystals demonstrates the fundamental connection between the incompatibility of the local inelastic deformation and lengthscale of dislocation microstructures in FCC single crystals. Of these, the approaches of Dai and Parks (1997), Shizawa and Zbib (1999), and Acharya and Bassani (2000) are geared solely to the modelling of additional hardening due to GNDs and involve no additional ?eld relations or boundary conditions. For example, the approach of Dai and Parks (1997) was used by Busso et al. (2000) to model additional hardening in two-phase nickel superalloys, and that of Acharya and Bassani (2000) by Acharya and Beaudoin (2000) to model grain-size e'ects in FCC and BCC polycrystals up to moderate strains. Except for the works of Acharya and Bassani (2000) and

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Cermelli and Gurtin (2001), which are restricted to kinematics, all of these presume directly or indirectly a particular dependence of the (free) energy and=or other dependent constitutive quantities (e.g., yield stress) on the gradients of inelastic state variables, and in particular on that of the local inelastic deformation, i.e., that determine its incompatibility. Yet more general formulations of crystal plasticity involving a (general) dependence of the free energy on the gradient of the local inelastic deformation can be found in, e.g., Naghdi and Srinivasa (1993, 1994), Le and Stumpf (1996), or in Gurtin (2000). From the constitutive point of view, such experimental and modelling work clearly demonstrates the need to account for the dependence of the constitutive relations, and so material behaviour, on the inhomogeneity or “non-locality” of the internal ?elds as expressed by their gradients. In the phenomenological or continuum ?eld context, such non-locality of the material behaviour is, or can be, accounted for in a number of existing approaches (e.g., Maugin, 1980; Capriz, 1989; Maugin, 1990; Fried and Gurtin, 1993, 1994; Gurtin, 1995; Fried, 1996; Valanis, 1996, 1998) for broad classes of materials. It is not the purpose of the current work to compare and contrast any of these with each other in detail (in this regard, see, e.g., Maugin and Muschik, 1994; Svendsen, 1999); rather, we wish to apply two of them to formulate continuum thermodynamic models for crystal plasticity in which gradients of the inelastic ?elds in question in4uence the material behaviour. To this end, we must ?rst identify the relevant internal ?elds. On the basis of the standard crystal plasticity constitutive relation for the local inelastic deformation FP , a natural choice for the principal inelastic ?elds of the formulation is the set of glide-system deformations. In contrast, Le and Stumpf (1996) worked in their variational formulation directly with FP , and Gurtin (2000) in his formulation based on con?gurational forces with the set of glide-system slip rates. In both of these works, a principal result takes the form of an extended or generalized Euler– Lagrange-, Ginzburg–Landau- or Cahn–Allen-type ?eld relation for the respective principal inelastic ?elds. Generalized forms of such ?eld relations for the glide-system deformations are obtained in the current work by modelling them in two ways. In the simplest approach, these are modelled as “generalized” internal variables (GIVs) via a generalization of the approach of Maugin (1990) to the modelling of the entropy 4ux. Alternatively, and more generally, these are modelled here as internal degrees-of-freedom (DOFs) via the approach of Capriz (1989) in the extended form discussed by Svendsen (2001a). In addition, as shown here, these formulations are general enough to incorporate in particular a number of models for GNDs (e.g., Ashby, 1970; Dai and Parks, 1997) and so provide a thermodynamic framework for extended non-local crystal plasticity modelling including the e'ects of GNDs on the material behaviour. After some mathematical preliminaries (Section 2), the paper begins (Section 3) with a brief discussion and formulation of basic kinematic and constitutive issues and relations relevant to the continuum thermodynamic approach to crystal plasticity taken in this work. In particular, as mentioned above, the standard constitutive form for FP in crystal plasticity determines the glide-system slip deformations (“slips”) as principal constitutive unknowns here. Having then established the corresponding constitutive class for crystal plasticity, we turn next to the thermodynamic ?eld formulation and analysis (Sections 4 and 5), depending on whether the glide-system slips are modelled

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as generalized internal variables (GIVs) (Section 4), or as internal degrees-of-freedom (DOFs) (Section 5). Next, attention is turned to the formulation of two (constitutive) classes of GND models (Section 6), yielding in particular expressions for the glide-system e'ective (surface) density of GNDs. The ?rst class of such models is based on the incompatibility of glide-system local deformation. To this class belong for example the original model of Ashby (1970) and the recent dislocation density tensor of Shizawa and Zbib (1999). The second is based on the incompatibility of FP and is consistent with the model of Dai and Parks (1997). With such models in hand, the possible dependence of the free energy on quantities characterising the dislocation state of the material (e.g., dislocation densities) and the corresponding consequences for the formulation are investigated (Section 7). Beyond the GND models formulated here, examples are also given of existing SSD models which can be incorporated into models for the free energy, and so into the current approach. After summarizing the results of the general formulation in terms of the initial-boundary-value problem it implies (Section 8), as well as the corresponding algorithmic form, the paper ends (Section 9) with a discussion of additional general aspects of the current approach and a comparison with other related work. 2. Mathematical preliminaries If W and Z represent two ?nite-dimensional linear spaces, let Lin (W; Z) represent the set of all linear mappings from W to Z. If W and Z are inner product spaces, the inner products on W and Z induce the transpose AT ∈ Lin (Z; W ) of any A ∈ Lin (W; Z), as well as the inner product A · B := tr W (AT B) = tr Z (ABT ) on Lin (W; Z) for all A; B ∈ Lin (W; Z). The main linear space of interest in this work is of course three-dimensional Euclidean vector space V . Let Lin (V; V ) represent the set of all linear mappings of V into itself (i.e., second-order Euclidean tensors). Elements of V and Lin+ (V; V ) the subset of all elements of Lin (V; V ) with positive determinant. Elements of V and Lin (V; V ), or mappings taking values in these spaces, are denoted here as usual by bold-face, lower-case a; : : : and upper-case A; : : : ; italic letters, respectively. In particular, I ∈ Lin (V; V ) represents the second-order identity tensor. As usual, the tensor product a ⊗ b of any two a; b ∈ V can be interpreted as an element a ⊗ b ∈ Lin (V; V ) of Lin (V; V ) via (a ⊗ b)c := (b · c)a for all a; b; c ∈ V . Let sym (A) := 12 (A + AT ) and skw(A) := 12 (A − AT ) represent the symmetric and skew-symmetric parts, respectively, of any A ∈ Lin (V; V ). The axial vector axi (W ) ∈ V of any skew tensor W ∈ Lin (V; V ) is de?ned by axi (W ) × a := Wa. Let a; b; c ∈ V be constant vectors in what follows. Turning next to ?eld relations, the de?nition curl u := 2 axi (skw(∇u))

(1)

for the curl of a di'erentiable Euclidean vector ?eld u is employed in this work, ∇ being the standard Euclidean gradient operator. In particular, Eq. (1) and the basic result ∇(fu) = u ⊗ ∇f + f(∇u)

(2)

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for all di'erentiable functions f and vector ?elds u yield the identity curl(fu) = ∇f × u + f(curl u):

(3)

In addition, Eq. (1) yields the identity curl u · a × b = ∇a u · b − ∇b u · a

(4)

for curl u in terms of the directional derivative ∇a u := (∇u)a

(5)

of u in the direction a ∈ V . Turning next to second-order tensor ?elds, we work here with the de?nition 1 (curl T)T a := curl(T T a)

(6)

for the curl of a di'erentiable second-order Euclidean tensor ?eld T as a second-order tensor ?eld. From Eqs. (3) and (6) follows in particular the identity curl(fT) = T(I × ∇f) + f(curl T)

(7)

for all di'erentiable f and T, where (I × a)b := b × a. Note that (I × a)T = a × I with (a × I )b := a × b. Likewise, Eqs. (1) and (6) yield the identity (curl T)(a × b) := (∇a T)b − (∇b T)a

(8)

for curl T in terms of the directional derivative ∇a T := (∇T)a

(9)

of T in the direction a ∈ V . Here, ∇T represents a third-order Euclidean tensor ?eld. Let H be a di'erentiable invertible tensor ?eld. From Eq. (8) and the identity AT (Ab × Ac) = det(A)(b × c)

(10)

for any second-order tensor A ∈ Lin (V; V ), we obtain curl(TH ) = det(H )(curlH T)H −T + T(curl H )

(11) H

for the curl of the product of two second-order tensor ?elds. Here, curl represents the curl operator induced by the Koszul connection ∇H induced in turn by the invertible tensor ?eld H , i.e., ∇H T := (∇T)H −1 :

(12)

The corresponding curl operation then is de?ned in an analogous fashion to the standard form (8) relative to ∇. Third-order tensors such as ∇T are denoted in general in this work by A; B; : : : and interpreted as elements of either Lin (V; Lin (V; V )) or Lin (Lin (V; V ); V ). Note that any third-order tensor A induces one AS de?ned by (AS b)c := (Ac)b:

(13)

1 This is of course a matter of convention. Indeed, in contrast to Eq. (6), Cermelli and Gurtin (2001) de?ne (curl T)a := curl(T T a).

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In particular, this induces the split A = symsS (A) + skwS (A)

(14)

of any third-order tensor A into “symmetric” symS (A) := 12 (A + AS )

(15)

and “skew-symmetric” skwS (A) := 12 (A − AS )

(16)

parts. In addition, the latter of these induces the linear mapping axiS : Lin (V; Lin (V; V )) → Lin (V; V ) | A → A = axiS (A)

(17)

de?ned by axiS (A)(b × c) := 2(skwS (A)b)c = (Ab)c − (Ac)b:

(18)

With the help of Eqs. (13) – (18), one obtains in particular the compact form curl T = axiS (∇T)

(19)

for the curl of a di'erentiable second-order tensor ?eld T as a function of its gradient ∇T from Eq. (8). Lastly, the transpose AT ∈ Lin (Lin (V; V ); V ) of any third-order tensor A ∈ Lin (V; Lin (V; V )) is de?ned here via AT B · c = B · Ac. Finally, for notational simplicity, it proves advantageous to abuse notation in this work and denote certain mappings and their values by the same symbol. Other notations and mathematical concepts will be introduced as they arise in what follows. 3. Basic kinematic, constitutive and balance relations Let B represent a material body, p ∈ B a material point of this body, and E Euclidean point space with translation vector space V . A motion of the body with respect to E in some time interval I ⊂ R takes as usual the form x = (t; p);

(20)

relating each p to its (current) time t ∈ I position x ∈ E in E. On this basis, ˙ represents the material velocity, and F (t; p) := (∇ )(t; p) ∈ Lin+ (V; V )

(21)

the deformation gradient relative to the (global) reference placement  of B into E. Here, we are using the notation ∇ := ∗ (∇(∗ ))

(22)

for the gradient of with respect to  in terms of push-forward and pull-back, where (∗ )(t; r ) := (t; −1 (r )) for push-forward by , with r = (p), and similarly for ∗ . Like ; ˙ and F , all ?elds to follow are represented here as time-dependent ?elds on B. And analogous to that of in Eq. (21), the gradients of these ?elds are all de?ned relative to . More precisely, these are de?ned at each p ∈ B relative to a

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corresponding local reference placement 2 at each p ∈ B, i.e., an equivalence class of global placements  having the same gradient at p. Since  and the corresponding local reference placement at each p ∈ B is arbitrary here, and the dependence of F and the gradients of other ?elds, as well as that of the constitutive relations to follow, on  does not play a direct role in the formulation in this work, we suppress it in the notation for simplicity. In the case of phenomenological crystal plasticity, any material point p ∈ B is endowed with a “microstructure” in the form of a set of n glide systems. The geometry and orientation of each such glide system is described as usual by an orthonormal basis (sa ; na ; ta ) (a = 1; : : : ; n). Here, sa represents the direction of glide in the plane, na the glide-plane normal, and ta := sa × na the direction transverse to sa in the glide plane. Since we neglect in this work the e'ects of any processes involving a change in or evolution of either the glide direction sa or the glide-system orientation na (e.g., texture development), these referential unit vectors, and so ta as well, are assumed constant with respect to the reference placement. With respect to the glide-system geometry, then, the (local) deformation Fa of each glide system takes the form of a simple shear 3 Fa = I + a sa ⊗ na ;

(23)

a being its magnitude in the direction sa of shear. For simplicity, we refer to each a as the (scalar) glide-system slip (deformation). The orthogonality of (sa ; na ; ta ) implies FaT na = na and Fa sa = sa , as well as a = sa · Fa na . In addition, F˙a = sa ⊗ na ˙a (24) follows from Eq. (23). As such, the evolution of the glide-system deformation tensor Fa is determined completely by that of the corresponding scalar slip a . From a phenomenological point of view, the basic local inelastic deformation at each material point in the material body in question is represented by an invertible second-order tensor ?eld FP on I × B. The evolution of FP is given by the standard form F˙ P = LP FP (25) in terms of the plastic velocity “gradient” LP . The connection to crystal plasticity is then obtained via the constitutive assumption m m

LP = ˆ sa ⊗ na ˙a (26) F˙a = a=1

a=1

for LP via Eq. (24), where m 6 n represents the set 4 of active glide-systems, i.e., those for which ˙a = 0. Combining this last constitutive relation with Eq. (25) then yields 2

Referred to by Noll (1967) as local reference con?guration of p ∈ B in E.

3

As discussed in Section 6, like FP , and unlike F; Fa is in general not compatible.

4

In standard crystal plasticity models, the number m of active glide system is determined among other things by the glide-system “4ow rule”, loading conditions, and crystal orientation. As such, it is constitutive in nature, and in general variable.

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the basic constitutive expression m m

F˙ P = (sa ⊗ na )FP ˙a F˙a FP =

(27)

a=1

a=1

for the evolution of FP . In turn, this basic constitutive relation implies that m
˙ = ∇F (sa ⊗ na )(∇FP ) ˙a + (sa ⊗ na )FP ⊗ ∇ ˙a P

(28)

a=1

for the evolution of ∇FP , and so that m
curl˙ FP = (sa ⊗ na )(curl FP ) ˙a + sa ⊗ (∇ ˙a × FPT na )

(29)

a=1

for the evolution of curl FP via Eqs. (7) and (8). On this basis, the evolution relation for FP is linear in the set ˙ := ( ˙1 ; : : : ; ˙m ) of active glide-system slip rates. Similarly, the evolution relations for ∇FP and curl FP are linear in ˙ and ∇. ˙ Generalizing the case of curl FP slightly, which represents one such measure, the dislocation state in the material is modelled phenomenologically in this work via a general inelastic state=dislocation measure  whose evolution is assumed to depend quasi-linearly on ˙ and ∇, ˙ i.e., ˙ = K˙ + J ∇˙

(30)

in terms of the dependent constitutive quantities K and J. In particular, on the basis of Eq. (27), FP is considered here to be an element of . In turn, the dependence of this evolution relation on ∇˙ requires that we model the  as time-dependent 0elds on B. As such, in the current thermomechanical context, the absolute temperature , the motion , and the set  of glide-system slips, represent the principal time-dependent ?elds, FP and  being determined constitutively by the history of  and ∇˙ via Eqs. (27) and (30), respectively. On the basis of determinism, local action, and short-term mechanical memory, then, the material behaviour of a given material point p ∈ B is described by the general material frame-indi'erent constitutive form R = R(; C ; ; ∇; ; ˙ ∇; ˙ p)

(31)

for all dependent constitutive quantities (e.g., stress), where C = F T F represents the right Cauchy–Green deformation as usual. In particular, since the motion , as well as ˙ are not Euclidean frame-indi'erent, R is independent of these the material velocity , to satisfy material frame-indi'erence. As such, Eq. (31) represents the basic reduced constitutive form of the constitutive class of interest for the continuum thermodynamic formulation of crystal plasticity to follow. Because it plays no direct role in the formulation, the dependence of the constitutive relations on p ∈ B is suppressed in the notation until needed. The derivation of balance and ?eld relations relative to the given reference con?guration of B is based in this work on the local forms for total energy and entropy balance, i.e., e˙ = div h + s;

(32a)

˙ =  − div  + ;

(32b)

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respectively. Here, e represents the total energy density, h its 4ux density, and s its supply rate density. Likewise, ; , and  represent the production rate, 4ux, and supply rate, densities, respectively, of entropy, with density . In particular, ?eld relations for basic unknowns and  are obtained from Eq. (32a) via its invariance with respect to Euclidean observer. As usual, the thermodynamic analysis is based on Eq. (32b); in addition, it yields a ?eld relation for the temperature, as will be seen in what follows. This completes the synopsis of the basic relations required for the sequel. Next, we turn to the formulation of ?eld relations and the thermodynamic analysis for the constitutive class determined by the form (31). 4. Generalized internal variable model for glide-system slips The modelling of the  as generalized internal variables (GIVs) is based in particular on the standard continuum forms ˙ e =  + 12 % ˙ · ; ˙ h = − q + P T ; ˙ s = r + f · ;

(33)

for total energy density e, total energy 4ux density h, and total energy supply rate density s, respectively, hold. Here, % represents the referential mass density, P the ?rst Piola–Kirchho' stress tensor, and f the momentum supply rate density. Lastly,  represents the internal energy density, and q the heat 4ux density. As in the standard continuum case, P; ; q;  and  represent dependent constitutive quantities in general. Substituting the forms (33) for the energy ?elds into the local form (32a) for total energy balance yields the result ˙ + div q − r = P · ∇ ˙ − z · ˙ + 12 c ˙ · ˙

(34)

for this balance. Appearing here are the ?eld c := %˙

(35)

associated with mass balance, and that z := m˙ − div P − f

(36)

associated with momentum balance, where m := % ˙

(37)

S represents the usual continuum momentum density. As discussed by, e.g., SilhavT y (1997, Chapter 6), in the context of the usual transformation relations for the ?elds appearing in Eq. (34) under change of Euclidean observer, one can show that necessary conditions for the Euclidean frame-indi'erence of Eq. (32a) in the form (34) are the mass c = 0 ⇒ %˙ = 0

(38)

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via Eq. (35), momentum z = 0 ⇒ m˙ = div P + f

(39)

via Eq. (36), and moment of momentum ST = S

(40)

balances, respectively, the latter with respect to the second Piola–Kirchho' stress S = F −1 P. As such, beyond a constant (i.e., in time) mass density, we obtain the standard forms m˙ = 0 + div P + f ;

(41a)

˙ = 12 S · C˙ − div q + r;

(41b)

for local balance of continuum momentum and internal energy, respectively, in the current context via Eqs. (34), (39) and (40). We turn next to thermodynamic considerations. As shown in e'ect by Maugin (1990), one approach to the formulation of the entropy principle for material behaviour depending on internal variables and their gradients can be based upon a weaker form of the dissipation (rate) inequality than the usual Clausius–Duhem relation. This form follows from the local entropy (32) and internal energy (41b) balances via the Clausius–Duhem form  = r=

(42)

for the entropy supply rate  density in terms of the internal energy supply rate density r and temperature . Indeed, this leads to the expression  = 12 S · C˙ − ˙ − ˙ + div( − q) −  · ∇ (43) for the dissipation rate density  := 

(44)

via Eq. (41b), where :=  − 

(45)

represents the referential free energy density. Substituting next the form (31) for into Eq. (43) yields that  = { 1 S − ; C } · C˙ − { + ;  }˙ − ; ∇ · ∇˙ − ; ˙ · K − ; ∇˙ · ∇K 2

+ div( − q − $TV ) ˙ + ($V + div $V ) · ˙ −  · ∇

(46)

for  via Eq. (30). Here, $V := − KT

;;

(47)

$V := ($V1 ; : : : ; $Vm ), and $V := JT

;;

(48)

with $V := (’V1 ; : : : ; ’Vm ). Now, on the basis of Eqs. (30) and (31),  in Eq. (46) ˙ ∇; ˙ K and ∇. is linear in the ?elds C˙ ; ; K Consequently, the Coleman–Noll approach

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to the exploitation of the entropy inequality implies that  ¿ 0 is insured for all thermodynamically admissible processes i' the corresponding coeUcients of these ?elds in Eq. (46) vanish, yielding the restrictions S =2

;C ;

= −

; ;

0=

; ∇ ;

0=

; ˙a ;

0=

; ∇˙a ;

(49a) a = 1; : : : ; m; a = 1; : : : ; m;

on the form of the referential free energy density  = div( − q −

˙ $TV )

(49b) , as well as the reduced expression

+ ($V + div $V ) · ˙ −  · ∇ ln 

(50)

for  as given by Eq. (46), representing its so-called residual form for the current constitutive class. In this case, then, the reduced form = (; C ; )

(51)

of

follows from Eqs. (31) and (49). On the basis of the residual form (50) for , assume next that, as dependent constitutive quantities, $V + div $V and  are de?ned on convex subsets of the non-equilibrium part of the state space, representing the set of all admissible ∇; ˙ and ∇. ˙ If $V + div $V and , again as dependent constitutive quantities, are in addition continuously di'erentiable in ∇; ˙ and ∇˙ on the subset in question, one may generalize the results of Edelen (1973, 1985) to show 5 that the requirement  ¿ 0 on  given by Eq. (50) yields the constitutive results $V + div $V = dV; ˙ − div dV; ∇˙ + V˙ ;

(52a)

− = dV; ∇ ln  + V∇ ln  ;

(52b)

for $V + div $V and , respectively, in terms of the dissipation potential ˙ ∇) ˙ dV = dV (; C ; ; ∇; ;

(53)

and constitutive quantities ˙ ∇); ˙ V˙ = V˙ (; C ; ; ∇; ; V∇ ln  = V∇ ln  (; C ; ; ∇; ; ˙ ∇); ˙

(54)

which satisfy V˙ · ˙ + V∇ ln  · ∇ ln  = 0;

(55)

5 In fact, this can be shown for the weaker case of simply-connected, rather than convex, subsets of the dynamic part of state space via homotopy (see, e.g., Abraham et al., 1988, proof of Lemma 6:4:14).

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i.e., they do not contribute to . To simplify the rest of the formulation, it is useful to work with the stronger constitutive assumption that dV exists, in which case V˙ and V∇ ln  vanish identically. On the basis then of the constitutive form = ˆ  −1 q +  −1 ($V + dV; ∇˙ )T ˙

(56)

for the entropy 4ux density,  is determined by the form of dV alone, i.e.,  = dV; ˙ · ˙ + dV; ∇˙ · ∇˙ + dV; ∇ ln  · ∇ ln :

(57)

Among other things, Eq. (57) implies that a convex dependence of dV on the nonequilibrium ?elds is suUcient, but not necessary, to satisfy  ¿ 0. Indeed, with dV (; C ; ; 0; 0; 0) = 0, dV is convex in ∇, ˙ and ∇˙ if  ¿ dV (i.e., with  given by Eq. (57)) for given values of the other variables. So, if dV is convex in ∇ and , ˙ and dV ¿ 0, then  ¿ 0 is satis?ed. On the other hand, even if dV ¿ 0,  ¿ 0 does not necessarily require  ¿ dV , i.e., dV convex. Lastly, in the context of the entropy balance (32b), the constitutive assumption (42), together with Eq. (44) and the results (49a), (49b), (50) and (52), lead to the expression c˙ = 12 S;  · C˙ + !V + div dV; ∇ ln  + r

(58)

for the evolution of  via Eqs. (51) and (53). Here, c := − 

(59)

; 

represents the heat capacity at constant , C , and so on, 12 S;  · C˙ =  (density) of heating due to thermoelastic processes, and !V := (dV; ˙ + KT

;  ) · ˙ +

(dV; ∇˙ + JT

;  ) · ∇˙

; C

· C˙ the rate (60)

that due to inelastic processes via Eq. (30). In addition, Eq. (52a) implies the result dV; ˙ = div(JT

;

+ dV; ∇˙ ) − KT

;

(61)

for the evolution of  via Eqs. (47) and (48). Finally, −q = dV; ∇ ln  + (JT

;

+ dV; ∇˙ )T ˙

(62)

follows for the heat 4ux density q follows from Eqs. (56) and (52b). As such, the ˙ lead in general to additional dependence of on , as well as that of dV on ∇, contributions to q in the context of the modelling of the  as GIVs. This completes the formulation of balance relations and the thermodynamic analysis for the modelling of the  as GIVs. Next, we carry out such a formulation for the case that the  are modelled as internal DOFs. 5. Internal degrees-of-freedom model for glide-system slips Alternative to the model for the glide-system slips as GIVs in the sense of the last section is that in which they are interpreted as so-called internal degrees-of-freedom

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

1309

(DOFs). In this case, the degrees-of-freedom 6 of the material consist of (i), the usual “external” continuum DOFs represented by the motion , and (ii), the “internal” DOFs . Or to use the terminology of Capriz (1989), the  are modelled here as scalar-valued continuum microstructural ?elds. Once established as DOFs, the modelling of the  proceeds by formal analogy with that of , the only di'erence being that, in contrast to external DOFs represented by , each internal DOF a is (i.e., by assumption) Euclidean frame-indi'erent. Otherwise, the analogy is complete. In particular, each a is assumed to contribute to the total energy, the total energy 4ux and total energy supply, of the material in a fashion formally analogous to , i.e., ˙ e =  + 12 ˙ · % ˙ + 12 ˙ · %I; h = − q + P T ˙ + $TF ; ˙ s = r + f · ˙ + & · ; ˙

(63)

for total energy density e, total energy 4ux density h, and total energy supply rate density s. Here,   !11 · · · !1m  . . ..   . . (64) I :=  . . .   !n1 · · · !mm represents the matrix of microinertia coeUcients, $F := (’F1 ; : : : ; ’Fm ) the array of 4ux densities, and & := (&1 ; : : : ; &Fm ) the array of external supply rate densities, associated with . For simplicity, we assume that I is constant in this work. Next, substitution of Eq. (63) into the general local form (32a) of total energy balance yields ˙ ˙ + div q − r = P · ∇ ˙ + $F · ∇˙ − z · ˙ − $F · ˙ + 12 c( ˙ · ˙ + ˙ · I)

(65)

via Eqs. (35) and (36). Here, $F := "˙ − div $F − &

(66)

is associated with the evolution of , " := %I˙

(67)

being the corresponding momentum density. Consider now the usual transformation relations for the ?eld appearing in Eqs. (63) and (65) under change of Euclidean observer, and in particular the assumed Euclidean frame-indi'erence of the elements of , I, and $F . As discussed in the last section, using these, one can show that necessary conditions for the Euclidean frame-indi'erence of Eq. (32a) in the form (65) are the mass (38), momentum (39), and moment of momentum (40) balances, respectively. As such, beyond a constant (i.e., in time) mass density, we obtain the set m˙ = 0 + div P + f ; 6

(68a)

This entails a generalization of the classical concept of “degree-of-freedom” to materials with structure.

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B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

"˙ = $F + div $F + &;

(68b)

˙ = 12 S · C˙ + $F · ∇˙ − $F · ˙ − div q + r;

(68c)

of ?eld relations via Eqs. (39), (40), (65) and (66). Since we are modelling the  as (internal) DOFs in the current section, the relevant thermodynamic analysis is based on the usual Clausius–Duhem constitutive forms  = q=;

(69a)

 = r=;

(69b)

for the entropy 4ux  and supply rate  densities, respectively. Substituting these into the entropy balance (32b), we obtain the result  = 12 S · C˙ + $F · ∇˙ − $F · ˙ − ˙ − ˙ −  −1 q · ∇

(70)

for the dissipation rate density  :=  via Eq. (68c) via Eq. (45). In turn, substitution of the constitutive form (31) for the free energy into Eq. (70), and use of that Eq. (30) for , yields  = { 12 S −

˙ − { +

;C } · C

+ $FN · ∇˙ − $FN · ˙ −

˙−

;  }

; ∇

· ∇˙ −  −1 q · ∇

; ˙ · K − ; ∇˙ · ∇K

(71)

with $FN := $F − JT $FN := $F + KT

;; ;;

(72a) (72b)

the non-equilibrium parts of $F and $F , respectively. On the basis of Eq. (31),  is ˙ ∇, ˙ K and ∇. linear in the independent ?elds C˙ , , K As such, in the context of the Coleman–Noll approach to the exploitation of the entropy inequality,  ¿ 0 is insured for all thermodynamically admissible processes i' the corresponding coeUcients of these ?elds in Eq. (71) vanish, yielding S =2

;C ;

= −

; ;

0=

; ∇ ;

0=

; ˙a ;

0=

; ∇˙a ;

a = 1; : : : ; m; a = 1; : : : ; m:

(73)

As in the last section, these restrictions also result in the reduced form (51) for . Consequently, the constitutive ?elds S,  and  are determined in terms of as given by Eq. (51). On the other hand, the $FN , $FN as well as q still take the general form (31). These are restricted further in the context of the residual form  = $FN · ∇˙ − $FN · ˙ −  −1 q · ∇

(74)

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

1311

for  in the current constitutive class from Eq. (73). Treating $FN , $FN and q constitutively in a fashion analogous to $V + div $V and  from the last section in the context of Eq. (52), the requirement  ¿ 0 results in the constitutive forms $FN = dF; ∇˙ + F∇#˙;

(75a)

−$FN = dF; ˙ + F#˙;

(75b)

−q = dF; ∇ ln  + F∇ ln  ;

(75c)

for these in terms of a dissipation potential dF and corresponding constitutive quantities F∇˙, F˙ and F∇ ln  , all of the general reduced material-frame-indi'erent form (31). As in the last section, the latter three are dissipationless, i.e., F˙ · ˙ + F∇˙ · ∇˙ + F∇ ln  · ∇ ln  = 0

(76)

analogous to Eq. (55) in the GIV case. Consequently,  reduces to  = dF; ˙ · ˙ + dF; ∇˙ · ∇˙ + dF; ∇ ln  · ∇ ln 

(77)

via Eqs. (75) and (76), analogous to Eq. (57) in the GIV case. In what follows, we again, as in the last section, work for simplicity with the stronger constitutive assumption that dF exists, in which case F#˙, F∇#˙ and F∇ ln  vanish identically. On the basis of the above assumptions and results, then, the ?eld relation c˙ = 12 S;  · C˙ + !F + div dF; ∇ ln  + r

(78)

for temperature evolution analogous to Eq. (58) is obtained in the current context via Eq. (59), with !F := (dF; ˙ + KT

;  ) · ˙ +

(dF; ∇˙ + JT

;  ) · ∇˙

(79)

the rate of heating due to inelastic processes analogous to !V from Eq. (60). Finally, Eqs. (75a), (75b) lead to the form %IK + dF; ˙ = div(JT

;

+ dF; ∇˙ ) − K T ;  + &

(80)

for the evolution of  via Eqs. (67), (68b) and (72). With the general thermodynamic framework established in the last two sections now in hand, the next step is the formulation of speci?c models for GND development and their incoporation into this framework, our next task. 6. E(ective models for GNDs The ?rst model for GNDs to be considered in this section is formulated at the glide-system level. As it turns out, this model represents a three-dimensional generalization of the model of Ashby (1970), who showed that the development of GNDs in a given glide system is directly related to the inhomogeneity of inelastic deformation in this system. In particular, in the current ?nite-deformation context, this generalization

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B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

is based on the incompatibility of Fa with respect to the reference placement. To this end, consider the vector measure 7   lGa (C) := Fa tC = a (na · tC )sa (81) C

C

of the length of glide-system GNDs around an arbitrary closed curve or circuit C in the reference con?guration, the second form following from Eq. (23). Here, tC represents the unit tangent to C oriented clockwise. Alternatively, lGa (C) is given by 8  Fa tC = (curl Fa )nS (82) lGa (C) := S

C

with respect to the material surface S bounded by C via Stokes theorem. Here, curl Fa = (sa ⊗ na )(I × ∇a ) = sa ⊗ (∇a × na )

(83)

from Eqs. (7), (23), and the constancy of (sa ; na ; ta ). On the basis of Eq. (82), lGa (C) can also be interpreted as a vector measure of the total length of GNDs piercing the material surface S enclosed by C. The quantity curl Fa determines in particular the and Zbib (1999) as dislocation density tensor (I ) worked with recently by Shizawa

n (I ) based on the incompatibility of their slip tensor  :=  s a=1 a a ⊗ na . Indeed, we

n have (I ) := curl (I ) = a=1 curl Fa in the current notation. Now, from Eq. (81) and the constancy of sa , note that lGa (C) is parallel to the slip direction sa , i.e., lGa (C) = lGa sa with

 lGa (C) :=

(84)

C

 a na · t C =

S

(curl Fa )T sa · nS

(85)

the scalar length of GNDs piercing S via Eq. (82). With the help of a characteristic Burgers vector magnitude b, this length can be written in the alternative form (86) lGa (C) = b gGa · nS S

in terms of the vector ?eld gGa determined by gGa := b−1 (curl Fa )T sa = b−1 ∇a × na :

(87)

From the dimensional point of view, gGa represents a (vector-valued) GND surface (number) density. As such, the projection gGa · nS of gGa onto S gives the (scalar) 7

Volume dv, surface da and line d‘ elements are suppressed in the corresponding integrals appearing in what follows for notational simplicity. Unless otherwise stated, all such integrals to follow are with respect to line, surfaces and=or parts of the arbitrary global reference placement of the material body under consideration. 8 Note that curl F appearing in Eq. (82) is consistent with the form (8) for the curl of a second-order a Euclidean tensor ?eld.

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

1313

surface (number) density of such GNDs piercing S. The projection of Eq. (87) onto the glide-system basis (sa ; na ; ta ) yields sa · gGa = − b−1 ta · ∇a ;

(88a)

ta · gGa = b−1 sa · ∇a ;

(88b)

na · gGa = 0;

(88c)

for the case of constant b. In particular, the ?rst two of these expressions are consistent with two-dimensional results of Ashby (1970) for the GND density with respect to the slip direction and that perpendicular to it in the glide plane generalized to three dimensions. Such three-dimensional relations are also obtained in the recent crystallographic approach to GND modelling of Arsenlis and Parks (1999). Likewise in agreement with the model of Ashby (1970) is the fact that Eq. (88c) implies that there is no GND development perpendicular to the glide plane (i.e., parallel to na ) in this model. From another point of view, if ∇a were parallel to na , there would be no GND development at all in this model; indeed, as shown by (83), in this case, Fa would be compatible. The second class of GND models considered in this work is based on the vector measure  lG (C) := FP tC = (curl FP )nS (89) C

S

of the length of GNDs from all glide systems around C in the material as measured by the incompatibility of the local inelastic deformation FP . In particular, the phenomenological GND model of Dai and Parks (1997), utilized by them to model grain-size e'ects in polycrystalline metals, applied as well recently by Busso et al. (2000) to model size e'ects in nickel-based superalloys, is of this type. In a di'erent context, the incompatibility of FP has also been used recently by Ortiz and Repetto (1999), as well as by Ortiz et al. (2000), to model in an e'ective fashion the contribution of the dislocation self- or core energy to the total free energy of ductile single crystals. In what follows, we refer to the GND model based on the measure (89) as the continuum (GND) model. To enable comparison of this continuum GND model with the glide-system model discussed above, it is useful to express the former in terms of glide-system quantities formally analogous to those appearing in the latter. To this end, note that the evolution relation (27) for FP induces the glide-system decomposition lG (C) =

n


lGa (C)sa

(90)

a=1

of lG (C) in terms of the set lG1 (C); : : : ; lGn (C) of glide-system GND lengths with respect to C formally analogous to those (85) in the context of the glide-system GND model. In contrast to this latter case, however, each lGa here is determined by an evolution relation, i.e.,  ˙lGa (C) = ˙a FPT na · tC = curl(F˙a FP )T sa · nS (91) C

S

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B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

with curl(F˙a FP ) = (sa ⊗ FPT na )(I × ∇˙a ) + sa ⊗ (curl FP )T na ˙a

(92)

via Eqs. (7) and (23). Alternatively, we can express lGa (C) as determined by Eq. (91) in the form (86) involving the vector-valued GND surface density gGa , with now g˙Ga = b−1 curl(F˙a FP )T sa = b−1 ∇˙a × FPT na + b−1 (curl FP )T na ˙a

(93)

in the context of Eq. (89). As implied by the notation, g˙Ga from Eq. (96) in the current context is formally analogous to the time-derivative of Eq. (87) in the glide-system GND model. Now, from the results (29) and (93), we have curl˙ FP = b

m


sa ⊗ g˙a ;

(94)

a=1

and so the expression 9 curl FP = b

n


sa ⊗ g a

(95)

a=1

for the incompatibility of FP in terms of the set (g1 ; : : : ; gn ) of vector-valued GND densities. Substituting this result into Eq. (93) then yields
g˙Ga = (na · sb )gGb ˙a + b−1 ∇˙a × FPT na (96) b=a

with na · sa = 0 and −1

sa · g˙Ga = b

FPT na

:=

m

Relative to (sa ; na ; ta ), note that
× sa · ∇˙a + (na · sb )sa · gGb ˙a ;

b=a

b=1; b=a .

b=a

ta · g˙Ga = b−1 FPT na × ta · ∇˙a +




(na · sb )ta · gGb ˙a ;

b=a

na · g˙Ga = b−1 FPT na × na · ∇˙a +




(na · sb )na · gGb ˙a ;

(97)

b=a

via Eqs. (8) and (24), analogous to Eq. (88). In contrast to the glide-system GND model, then, this approach does lead to a development of (edge) GNDs perpendicular to the glide plane (i.e., parallel to na ). To summarize then, we have the expressions  −1 ˙  glide-system model (98a)  b ∇a × na
g˙Ga = −1 ˙ T (na · sb )gGb ˙a continuum model (98b)   b ∇a × FP na + b=a

9 Assuming the integration constant to be zero for simplicity, i.e., that there is no initial inelastic incompatibility.

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

1315

for the evolution of the vector-valued measure gGa of GND density from the glidesystem and continuum models discussed above. With these in hand, we are now ready to extend existing models for crystal plasticity to account for the e'ects of GNDs on their material behaviour, and in particular their e'ect on the hardening behaviour of the material. In the current thermodynamic context, such extensions are realized via the constitutive dependence of the free energy on GND density, and more generally on the dislocation state in the material, our next task. 7. Free energy and GNDs With GND models such as those from the last section in hand, the question arises as to how these can be incoporated into the thermodynamic formulation for crystal plasticity developed in the previous sections. Since this formulation is determined predominantly by the free energy density and dissipation potential d, this question becomes one of (i), which quantities characterize e'ectively (i.e., phenomenologically) the GND, and more generally dislocation, state of each material point p ∈ B, and (ii), how do and d depend on these? The purpose of this section is to explore these issues for the case of the referential free energy density . In particular, this involves the choice for . Among the possible measures of the inelastic=dislocation state of each material point, we have the arrays %S = (+S1 ; : : : ; +Sn ) and gG = (gG1 ; : : : ; gGn ) of glide-system e'ective SSD and GND densities, respectively. Choosing then  = (FP ; %S ; gG ), takes the form (; C ; ) = for

D (; C ; FP ; %S ; gG )

(99)

from Eq. (51), with m


+˙Sa =

KSab ˙b ;

b=1

g˙Ga =

m


kGab ˙b + JGab ∇˙b ;

(100)

b=1

from Eq. (30). This choice induces the decompositions KT

JT

;

= (F˙ P; ˙ )T

;

D; FP

= 0 + 0 + JGT

+ KTS D; gG ;

D; %S

+ KTG

D; gG ;

(101a) (101b)

from Eq. (27) of the constitutive quantities KT ;  and JT ;  determining the form (61) or (80) of the ?eld relation for . In particular, the models (98) for g˙Ga yield  glide-system model  0
kGab =  (102) (nb · sc )gGc continuum model   ab c=b

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B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

and

 −1

JGab = b

ab

I × nb I×

FPT nb

glide-system model continuum model

for kGab and JGab , respectively. And from Eq. (103), we have  glide-system model na × D; ga T T −1 (J ;  )a = (JG D; gG )a = b T FP na × D; ga continuum model

(103)

(104)

for the 4ux contribution appearing in the evolution relation (61) or (80) for . From Eqs. (100), (101a) and (102), we have (KT

;  )a

= − .a + xa + sgn(˙a )ra ;

(105)

where .a = − ((F˙ P; ˙)T

; F P )a

= − (sa ⊗ na )FP ·

;FP

represents the glide-system Schmid stress via Eq. (27),  glide-system model  0
xa := (na · sb ) D; gGa · gGb continuum model  

(106)

(107)

b=a

(a contribution to) the glide-system back stress, and m
ra := ISba D; +Sb

(108)

b=1

the glide-system yield stress, with KSba = ISba sgn(˙b ). Note that sgn(˙a ) is a constitutive quantity in existing crystal plasticity models. For example, in the case of the (non-thermodynamic) glide-system 4ow rule    .a n  sgn(.a ) ˙a = ˙a0  (109) .Ca  of Teodosiu and Sidero' (1976) (similar to the form used by Asaro and Needleman, 1985; see also Teodosiu, 1997), we have sgn(˙a ) =sgn(. ˆ a ). Here, .Ca represents the critical Schmid stress for slip. In particular, such a constitutive assumption insures that the contribution .a ˙a = |.a ||˙a | to the dissipation rate density remains greater than or equal to zero for all a ∈ {1; : : : ; m}. Such a constitutive assumption is made for other types of glide-system 4ow rules, e.g., the activation form   XGa (|.a |; .Ca ) ˙a = ˙a0 exp − sgn(.a ) (110) kB  used by Anand et al. (1997) to model the inelastic behaviour of tantalum over a much wider range of strain rates and temperatures than possible with Eq. (109). Here, XGa (|.a |; .Ca ) represents the activation Gibbs free energy for thermally induced dislocation motion. Consider next the dependence of +˙Sa on ˙b , i.e., KSab . As it turns out, a number of existing approaches model this dependence. For example, in the approach of Estrin

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

1317

(1996, 1998) to dislocation-density-based constitutive modelling (see also 10 Estrin et al., 1998; Sluys and Estrin, 2000), this dependence follows from the constitutive relation   n
√ +˙Sa = jSab +Sb − kSa +Sa sgn(˙a )˙a (111) b=1

for the evolution of +Sa in terms of magnitude |˙a | = sgn(˙a )˙a of ˙a for a = 1; : : : ; m. In Eq. (111), jS11 ; jS12 ; : : : represent the elements of the matrix of athermal dislocation storage coeUcients, which in general are functions of %S , and ka the glide-system coeUcient of thermally activated recovery. In this case, then,   n
√ (112) ISab = ab jSbc +Sc − kSb +Sb c=1

holds, and so (KTS

D; %S )a

= KSaa

D; +Sa

= sgn(˙a )ISaa

(113)

D; +Sa

from Eq. (108). Other such models for KSab can be obtained analogously from existing ones for +˙Sa in the literature, e.g., from the dislocation-density-based approach of Teodosiu (1997). Models such as those (98b) for gGa , or that (111) in the case of +Sa , account in particular for dislocation–dislocation interactions. At least in these cases, then, such interactions are taken into account in the evolution relations for the dislocation measures, and so need not (necessarily) be accounted for in the form of . From this point of view, D could take for example the simple “power-law” form n n

2g −1 2s −1 1 = E · C E + s c 3  + g c 3 Ga D E E S G Sa 2 E a=1

=:

DE

+

DS

+

DG

a=1

(114)

in the case of ductile single crystals, perhaps the simplest possible. Here, CE represents the referential elasticity tensor, and EE := 12 (CE − I )

(115)

the elastic Green strain determined by the corresponding right Cauchy–Green tensor CE := FP−T CFP−1 :

(116)

Further, cS and cG are (scaling) constants, s and g exponents, 3 the average shear modulus, and √ Sa := ‘S +Sa ;  Ga := ‘G |gGa |; (117) 10 Because their model for SSD 4ux includes a Fickian-di'usion-like contribution due to dislocation cross-slip proportional to na · ∇+Sa , the approach of Sluys and Estrin (2000) does not ?t into the current framework as it stands. The necessary extension involves treating the SSD densities %S as, e.g., (independent) GIVs, analogous to the .

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B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

non-dimensional deformation-like internal variables associated with SSDs and GNDs, respectively, involving the characteristic lengths ‘S and ‘G , respectively. In particular, the GND contribution DG to D appearing in Eq. (114) is motivated by and represents a power-law generalization of the model of Kuhlmann-Wilsdorf (1989) for dislocation self-energy (see also Ortiz and Repetto, 1999) as based on the notion of dislocation line-length. From Eq. (114), we have in particular the simple expression .a = sa ⊗ na · 2CE

(118)

DE; CE

for the Schmid stress .a from Eq. (106) in terms of the Mandel stress − 2CE DE; CE . In addition, D; +a

= cS 3‘S2s +s−1 Sa ;

D; ga

= cG 3‘G2g |gGa |g−2 gGa ;

T D; FP FP

=

(119)

then hold. From these, we obtain in turn   0
xa = c 3‘2g |g |g−2 (na · sb )gGa · gGb G Ga G  

glide-system model continuum model

(120)

b=a

from Eq. (107) for xa , (JGT

D; gG )a

= cG 3b−1 ‘G2g |gGa |g−2



na × gGa

glide-system model

FPT na

continuum model

× gGa

from Eq. (104), as well as the result m
2s ra = cS 3‘S ISba +s−1 Sb

(121)

(122)

b=1

from Eq. (108) for ra . On the basis of models like that (111) of Estrin (1998) for +˙Sa , this last form for ra is consistent with and represents a generalization of distributed dislocation strength models (e.g., Kocks, 1976, 1987) to account for the e'ects of GNDs on glide-system (isotropic) hardening. Indeed, for s = 1, ra becomes proportional √ to +Sb in the context of Eq. (111). The simplest case of the formulation as based on Eq. (114) arises in the context of the glide-system model for GNDs when we set s = 1 and g = 2. Then  −1 (123a) b na × (∇a × na ) glide-system model T 4 −1 (JG D; gG )a = cG 3‘G b T continuum model (123b) FP na × gGa follows from Eq. (121) for the 4ux contribution via Eqs. (87) and (98). Note that Eq. (123a) follows from the fact that Eq. (98a) is integrable. Then, the corresponding reduction of ra from Eqs. (122), (61) and (123a) implies in particular the evolution-?eld relation m
dV; ˙a = cG 3‘G4 b−2 diva (∇a ) + .a − cS 3‘S2 KSba (124) b=1

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

1319

for a modelled as a GIV via Eq. (61), again in the context of the glide-system GND model, assuming cG , 3 and ‘G constant. The perhaps simplest possible non-trivial form of Eq. (124) for the evolution of a in the current context follows in particular from the corresponding simplest (i.e., quasi-linear) form 11 dV; ˙a = 4a ˙a for dV; ˙a in terms of the glide-system damping modulus 4a ¿ 0 with units of J s m−3 or Pa s (i.e., viscosity-like). In addition, diva (∇a ) := (I − na ⊗ na ) · ∇(∇a ) = (sa ⊗ sa + ta ⊗ ta ) · ∇(∇a )

(125)

represents the projection of the divergence operator onto the glide plane spanned by (sa ; ta ). Given suitable forms for the constitutive quantities, then, the ?eld relation (124) can in principal be solved (i.e., together with the momentum balance in the isothermal case) for a . On the other hand, since FP does not depend explicitly on , and, in contrast to the glide-system model, gGa does not depend explicitly on  and ∇ in the continuum GND model, no “simple” expression like Eq. (124) for the evolution of a is obtainable in this case. Indeed, in all other cases, one must proceed more generally to solve initial-boundary-value problems for , the , and . We return to this issue in the next section. A second class of free energy models can be based on the choice  = (FP ; ); curl FP ), i.e., (; C ; ) =

C (; C ; FP ; ); curl FP );

(126)

with #˙a = |˙a |

(127)

the glide-system accumulated slip rate. In this case, we have (KT

;  )a

(JT

= − .a + xa + sgn(˙a )ra ;

(128a)

= FPT na × (

(128b)

;  )a

C; curl FP )

T

sa ;

via Eqs. (29) and (106), where now xa := sa ⊗ na ·

C; curl FP (curl FP )

T

(129)

and ra :=

C; #a :

(130)

Consider for example the particular form C

= =:

for

11

C

1 2 EE CE

· C E EE + +

CS

+

CS ())

+ g−1 cG 3 ‘Gg |curl FP |g

CG

analogous to Eq. (114) for D . In this context, the choice    (.S − .0 )2 h0 = ln cosh # CS h0 .S − . 0

The coupling with ∇ in dV , and the dependence of dV on ∇, ˙ is neglected here.

(131)

(132)

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B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

yields the simple model for isotropic or Taylor hardening (i.e., due to SSDs) proposed by Hutchinson (1976), with n
# := #a (133) a=1

the total accumulated inelastic slip in all glide systems. Here, .S represents a characteristic saturation strength, .0 a characteristic initial critical resolved shear stress, and h0 a characteristic initial hardening modulus, for all glide systems. Another possibility for CS is the form n
ra0 #a + s−1 h0 # s = (134) CS a=1

consistent with the model of Ortiz and Repetto (1999) for latent hardening in single crystals, with now  
 n # :=  !Sab #a #b (135) a;b=1

the e'ective total accumulated slip in all glide systems in terms of the interaction coeUcients !Sab , a; b = 1; : : : ; n. In particular, this model is based on the assumptions that (i), hardening is parabolic in single slip (i.e., for s = 2=3), and (ii), the hardening matrix CS; #a #b is dominated by its o'-diagonal components. Beyond such models for glide-system (isotropic) hardening, Eq. (131) yields the expression xa = cG 3‘Gg |curl FP |g−2 sa ⊗ na · (curl FP )(curl FP )T

(136)

for glide-system back-stress from Eq. (129), as well as that (JT

;  )a

= cG 3‘Gg |curl FP |g−2 FPT na × (curl FP )T sa

(137)

for (JGT C; curl FP )a from Eq. (128b). Analogous to FP and gGa in the case of the continuum GND model, because FP and curl FP do not depend explicitly on  and ∇, no ?eld relation for a of the type (124) follows from Eq. (131), and we are again forced to proceed numerically. 8. Algorithmic formulation for quasi-static conditions: basic aspects Returning now to the general formulation, the purpose of this section is to discuss basic aspects of its numerical implementation via, e.g., the ?nite-element method. This is carried out here for the internal DOF model for the ; the corresponding results for the GIV model can be obtained analogously. For simplicity, all dynamic and inertial e'ects, as well as all supplies, are neglected in the following. On this basis, the internal DOF model for the  yields the set ∗ 2F ; C · ∇ = p⊥ · ∗ ; B0

9B0

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

B0



B0

{(KT ;  )a + dF; ˙a }∗a + {c˙ − 

B0

∗ ˙ ; C  · C − !F } +

{(JT ;  )a + dF; ∇˙a } · ∇∗a = B0

dF; ∇ ln  · ∇ ∗ =

9B0

9B0

1321

’a⊥ ∗a ;

q⊥  ∗ ;

(138)

of weak ?eld relations from Eq. (68a) via Eq. (73a), from Eq. (80), and from Eq. (78), respectively, for , each a , and , respectively. These relations are expressed with respect to the corresponding test functions ∗ , ∗a and  ∗ relative to a reference-initial con?guration B0 of the material body under consideration with boundary 9B0 . As usual, test functions vanish on those parts of 9B0 where the corresponding ?elds are speci?ed. In addition, p⊥ = Pn9B0

on 9B0

’a⊥ = ’a · n9B0 q⊥ = q · n9B0

on 9B0

on 9B0

(139)

represent as usual the normal momentum 4ux, glide-system slip-4ux, and “heat” 4ux, boundary conditions on 9B0 with unit normal n9B0 . Given such relations for K and J the system Eq. (138) of weak ?eld relations is completed by speci?c forms for the constitutive potentials and dF . In the simplest case of no coupling, for example, this latter potential could take the “power-law” form  na +1 m
|˙a | na na 1 &a #˙a0 dF = k∇ ln  · ∇ ln  + 2 na + 1 #˙a0 a=1

m

+

1
9a ∇˙a · ∇˙a : 2

(140)

a=1

Here, k represents the thermal conductivity with units of J K −1 m−1 s−1 , &a a yieldstress-like quantity with units of J m−3 or Pa, #˙a0 a characteristic value of |˙a |, na the glide-system slip-rate exponent, and 9a a material property with units of viscosity times length-squared. In particular, these latter three material properties in4uence possible activation- or relaxation-type processes during glide-system slip. The relation (140) for dF is referred to as a power-law form in the sense that, in the classical case, for which we have dF; ˙a = .a − xa − sgn(˙a )ra , Eq. (140) yields  n |.a − xa | − ra a ˙a = #˙a0 sgn(.a − xa ) (141) &a for the evolution of a via the constitutive assumption sgn(˙a ) = ˆ sgn(.a − xa ) common in crystal plasticity. A second possibility is the “activation” form dF = 12 k∇ ln  · ∇ ln  +

m
a=1

−1 −1 &a #˙a0 [:(1 + n−1 a ; cosh (1 + #˙a0 |˙a |)

(142)

1322

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329 −1 −1 + (−1)1=na :(1 + n−1 a ; −cosh (1 + #˙a0 |˙a |)] m

1
9a ∇˙a · ∇˙a ; (143) 2 a=1 ∞ where :(a; x) = x z a−1 e−z d z represents the incomplete gamma function. Again in the classical context with dF; ˙a = .a − xa − sgn(˙a )ra , Eq. (143) implies   n   |.a − xa | − ra a ˙a = #˙a0 cosh − 1 sgn(.a − xa ) (144) &a +

for the evolution of a , again via Eq. (142). In the context of the dislocation-density based model (114) for discussed in the last section, for example, and the GND models in Eq. (98), the form of is determined by the set %S of e'ective SSD densities, that gG of e'ective GND densities, and the local inelastic deformation FP . In turn, these ?elds are determined by simultaneous solution of Eq. (138) and the set m
+˙Sa = KSab ˙b ; a=1

g˙Ga =

m


KSab ˙b + JGab ∇˙b ;

b=1

F˙ P =

m


(sa ⊗ na )FP ˙a ;

(145)

a=1

from Eqs. (100) and (27). On the other hand, the model (131) for involving curl FP as a continuum measure of GND density involves simultaneous solution of Eq. (138) and the set m
F˙ P = (sa ⊗ na )FP ˙a ; a=1

curl˙ FP =

m


(sa ⊗ na )(curl FP )˙a + sa ⊗ (∇˙a × FPT na );

(146)

a=1

of evolution relations for FP and its curl from Eqs. (27) and (29). The algorithmic formulation and numerical implementation of any of these models in the context of the ?nite element method can be based for example on the incremental weak form !( ; ; ; ∗ ; ∗ ;  ∗ ; Xt; 0 ; ∗0 ; 0 ) = 2F ; C · ∇ ∗ B0

+

m
a=1

B0

{(KT

;  )a

+ dF; ˙a }∗a + {(JT

;  )a

+ dF; ∇˙a } · ∇∗a

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

 +

B0

+

B0

c

X − Xt

1323



;C

·

dF; ∇ ln  · ∇ ∗ −

XC − !F  ∗ Xt 9B0

p⊥ · ∗ −

9B0

q⊥  ∗ −

m
a=1

9B0

’a⊥ ∗a

=0

(147)

of the system (138) with respect to the time increment Xt = t−t0 . Here, Xa := a −a0 , X :=  − 0 , and similarly for the other increments. Further, we have   X X(∇) : (148) dF = dF ; C ; ; ∇; ; Xt Xt In addition, !F =

m


[dF; ˙a + (KT

a=1

;  )a ]

Xa + [dF; ∇˙a + (JT Xt

;  )a ] ·

X(∇a ) Xt

(149)

follows from Eqs. (79) and (142) for the algorithmic form of the heating rate due to inelastic processes. For example, the set (146) of evolution relations associated with the form (131) of the free energy yields the algorithmic system   m
−1 FP FP0 = exp (sa ⊗ na )Xa ; a=1

X(curl FP ) =

m


(sa ⊗ na ) (curl FP )Xa + sa ⊗ (X(∇a ) ×

(150) FPT na );

a=1

via standard backward-Euler integration. In work in progress, the isothermal special case of this system is being investigated numerically with the help of the ?nite-element method. This includes among other things the investigation of the in4uence of boundary conditions for ’a⊥ or a , e.g., the “natural” 4ux-boundary condition ’a⊥ = 0. The model for a on the boundary is obtained from the dependence of dF on ˙a in a fashion analogous to the derivation of, e.g., Eq. (141) from Eq. (140), or Eq. (144) from Eq. (143), above. The corresponding FE-implementation of these, as well as initial simulation results, will be reported on in future work. 9. Discussion From a phenomenological point of view, the concrete form (114) for D , and in particular that of DE , or that of CE in Eq. (131), is contingent upon the modelling of FP as an elastic material isomorphism (e.g., Wang and Bloom, 1974; Bertram, 1993; Svendsen, 1998), i.e., inelastic processes represented by FP do not change the form of the elastic constitutive relation. Such an assumption, quite appropriate and basically universal for single-crystal plasticity, may be violated in the case of strong texture development, induced anisotropy and=or anisotropic damage in polycrystals. As

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B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

discussed by Svendsen (1998), one consequence of the modelling of FP as an elastic isomorphism is the identi?cation of FE := FFP−1

(151)

as the local (elastic) deformation in the material, and in particular that of the crystal lattice in single-crystal plasticity. More generally, FP can be modelled as a material uniformity (Maugin and Epstein, 1998; Svendsen, 2001b) in the case of simple materials. In the current context, Eq. (151) implies the connection curlFP FE = − det(FP−1 )FE (curl FP )FPT

(152)

via Eqs. (10) and (8) between incompatibility of the local lattice deformation FE with respect to the intermediate (local) “con?guration” and that of FP with respect to the reference (local) “con?guration” (i.e., placement) at each p ∈ B via Eqs. (11), (12), and the compatibility of F. Alternatively, we have curlFP FE = − FE GI ;

(153)

where 12 GI := det(FP )−1 (curl FP )FPT = det(FE )(curlF FE−1 )FE−T

(154a) (154b)

represents the geometric dislocation tensor recently introduced by Cermelli and Gurtin (2001). As shown by them, GI represents the incompatibility of FP relative to the surface element nI daI := det(FP ) FP−T nS daS

(155)

in the intermediate con?guration. Indeed, relative to this element, the equivalence (curl FP )nS daS = GI nI daI

(156)

holds. As such, curl FP gives the same measure of GNDs with respect to surface elements in the reference con?guration as does GI with respect to such elements in the intermediate con?guration. Note that GI , like curl FP , has units of inverse length. The de?nition (154a) implies the form G˙ I = curlFP LP + LP GI + GI LTP − GI (I · LP )

(157)

for the evolution of GI via Eq. (25). Alternatively, this can be expressed “objectively” as ˙ (158) det(FP )−1 FP [det(FP )FP−1 GI FP−T ]; FPT = det(FP )−1 FP G˙ R FPT = curlFP LP relative to the “upper” Oldroyd–Truesdell derivative of GI with respect to FP , where GR := det(FP )FP−1 GI FP−T = FP−1 (curl FP )

(159)

12 Recall that we have de?ned the curl of a second-order tensor ?eld in Eq. (6) via (curl T)T a := curl (T T a), rather than in the form (curl T)a := curl (T T b) used by Cermelli and Gurtin (2001).

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

1325

represents the referential form of GI via Eq. (154). In the current crystal plasticity context, the right-hand side of Eq. (158) reduces to FP

curl LP = b

n


sa ⊗

FP−T g˙Ga

=

a=1

n


sa ⊗ FP−T [∇˙a × na ]

(160)

a=1

via Eqs. (6) and (26) in terms of the evolution of the vector-valued GND surface density gGa for the glide-system GND model from Eq. (87). As such, Eq. (157) implies G˙ I =

m


[(sa ⊗ na )GI + GI (na ⊗ sa )]˙a +

m


sa ⊗ FP−T [∇˙a × na ]

(161)

a=1

a=1

for the evolution of GI in the case of crystal plasticity via Eq. (26) and the fact that I · LP = 0 in this context. So, another class of speci?c forms for from Eq. (51) can be based on the choice  = (FP ; ); GI ), implying (; C ; ; p) = g(; CE ; ); GI ; p)

(162)

via Eq. (116). In turn, g itself is a member of the class de?ned by the choice  = (FP ; ); ∇FP ), as can be concluded directly from Eq. (28) and the fact that curl FP is a function of ∇FP via Eq. (19). As shown by Cermelli and Gurtin (2001), constitutive functions for any p ∈ B depending on ∇FP must reduce to a dependence on GI for their form to be independent of change of compatible local reference placement at p ∈ B, i.e., one induced by a change 13 of global reference placement. This requirement is in turn based on the result of Davini (1986), and Davini and Parry (1989) that such changes leave dislocation measures such as GI unchanged, representing as such “elastic” changes of local reference placement. As it turns out, one can show more generally (Svendsen, 2001c) that reduces to g for all p ∈ B, i.e., for B as a whole, under the assumption that FP represents a particular kind of material uniformity. Consider next the results of the two approaches to the modelling of the glide-system slips  from Sections 4 and 5. Formally speaking, these di'er in (i), the respective forms (61) and (80) for the evolution of the , (ii), those (60) and (79) for the rate of heating ! due to inelastic processes, and (iii), those (62) and (75c) for the heat 4ux density q. In particular, in view of the corresponding forms (58) and (78) for temperature evolution, this latter di'erence is of no consequence for the ?eld relations. Indeed, except for the contribution F∇˙ to $N in the internal DOF model for the , the total energy 4ux density h has the same form in both cases, i.e., h = −q + P T ˙ = dV; ∇ ln  + V∇ ln  + (JT

;

˙ + dV; ∇˙ )T ˙ + P T ;

= −q + P T ˙ + $TF ˙ = dF; ∇ ln  + F∇ ln  + (JT

13

;

˙ + dF; ∇˙ + F∇˙ )T ˙ + P T ;

(163)

Since local changes of reference placement for any p ∈ B represent equivalence classes of corresponding changes of global reference placement there, any local change of reference placement for p ∈ B is induced by a corresponding change of global placement.

1326

B. Svendsen / J. Mech. Phys. Solids 50 (2002) 1297 – 1329

from Eqs. (33b), (62), (72a), and (75a). This fact is related to the observation of Gurtin (1971, footnote 1) in the context of classical mixture theory concerning the interpretation of “entropy 4ux” and “heat 4ux” in phenomenology and the relation between these two. There, the issue was one of whether di'usion 4ux is to be interpreted as a 4ux of energy (e.g., Eckhart, 1940; Gurtin, 1971) or a 4ux of entropy (e.g., Meixner and Reik, 1959; DeGroot and Mazur, 1962; MKuller, 1968). In the current context, the 4ux (density) of interest is that (JT ;  +d;∇˙ )T . ˙ In the GIV approach, the constitutive form (56) shows that this 4ux (i.e., divided by ) is being interpreted as an entropy 4ux. On the other hand, Eqs. (63b) and (69b) imply that it is being interpreted as an energy 4ux in the internal DOF approach. In this point, then, both approaches are consistent with each other. As it turns out, the ?eld relation (61) for the  derived on the basis of the modelling these as generalized internal variables, represents a generalized form of the Cahn– Allen ?eld relation (e.g., Cahn, 1960; Cahn and Allen, 1977) for non-conservative phase ?elds, itself in turn a generalization of the Ginzburg–Landau model for phase transitions. In particular, Eq. (61) would reduce to the Cahn–Allen form (i), if dV were proportional to a quadratic form in ˙ and independent of ∇, ˙ and (ii), if JT ;  T and K ;  were reduceable to ;∇ and ;  , respectively. In particular, this latter case arises only for monotonic loading and small deformation. The Cahn–Allen relation has been studied quite extensively from the mathematical point of view (see, e.g., Brokate and Sprekels, 1996). As such, one may pro?t from the corresponding literature on the solution of speci?c initial-boundary value problems in applications of the approach leading to Eq. (61), or more generally that leading to Eq. (80), which are currently in progress. Acknowledgements I sincerely thank a reviewer of the ?rst version of this work for pointing out a basic problem with my original formulation which has been corrected here. In addition, I would like to thank Paolo Cermelli for helpful discussions and for drawing my attention to his work and that of Morton Gurtin on gradient plasticity and GNDs. References Abraham, R., Marsden, J.E., Ratiu, T., 1988. Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences, Vol. 75. Springer, Berlin. Acharya, A., Bassani, J.L., 2000. Lattice incompatibility and a gradient theory of crystal plasticity. J. Mech. Phys. Solids 48, 1565–1595. Acharya, A., Beaudoin, A.J., 2000. Grain-size e'ect in viscoplastic polycrystals at moderate strains. J. Mech. Phys. Solids 48, 2213–2230. Anand, L., Balasubramanian, S., Kothari, M., 1997. Constitutive modeling of polycrystalline metals at large strains, application to deformation processes. In: Teodosiu, C. (Ed.), Large Plastic Deformation of Crystalline Aggregates, CISM, Vol. 376. Springer, Berlin, pp. 109–172. Arsenlis, A., Parks, D.M., 1999. Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Mater. 47, 1597–1611.

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