constitutive analysis of finite deformation field dislocation mechanics

Journal of the Mechanics and Physics of Solids52 (2004) 301–316

www.elsevier.com/locate/jmps

Constitutive analysis of &nite deformation &elddislocation mechanics

Amit Acharya∗

Department of Civil and Environmental Engineering, Carnegie Mellon University, 103 Porter Hall,Pittsburgh, PA 15213, USA

Received 31 December 2002; received in revised form 27 June 2003; accepted 30 June 2003

Abstract

Driving forces for dislocation motion and nucleation in &nite-deformation &eld dislocationmechanics are derived. The former establishes a rigorous analog of the Peach–Koehler forceof classical elastic dislocation theory in a nonlinear, nonequilibrium &eld-theoretic context; thelatter is a prediction of the theory. The structure of the stress response and permanent distortionare also derived. Su4cient boundary and initial conditions are indicated, and invariance undersuperposed rigid motions is discussed. Hyperelasticity and &nite-deformation elastic theory ofdislocations are shown to be special cases of the framework. Owing to the nonlocal nature ofthe theory, the results as well as the methods used to derive them appear to be novel.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Dislocation mechanics; Driving forces; Finite deformation; Dislocation velocity; Dislocationnucleation

1. Introduction

As is widely recognized, the mechanics of single as well as collective assemblies ofdislocations is important in the understanding of the stressing and deformation of crys-talline materials. Discrete dislocation plasticity approaches with some generality haveonly recently been developed (Kubin et al., 1992; Van der Giessen and Needleman,1995; Zbib et al., 1998; Schwarz, 1999) and such techniques have been shown to ex-plain many complex features of plastic response. However, there are important aspectsin which these methods may not be su4ciently general, most important being the ex-clusion of a fundamental treatment of nonlinear crystal elasticity, &nite deformations,

∗ Tel.: +1-412-268-4566; fax: +1-412-268-7813.E-mail address: [email protected] (A. Acharya).

0022-5096/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0022-5096(03)00093-0

mailto:[email protected]

302 A. Acharya / J. Mech. Phys. Solids 52 (2004) 301–316

and inertial eFects. Moreover, even though the dislocation origin of macroscopic andmesoscopic plasticity is clear, a corresponding &ne-scale theory of dislocation me-chanics incorporating its essential nonlinear attributes and methods for coarse-grainingsuch a dynamical theory are not available at the current time. In this regard, di-rect coarse-graining of molecular dynamics without the intermediate construct of dis-locations sounds appealing in principle but remains elusive. The goal of this pa-per and some recent works (Acharya, 2001, 2003) is to begin to address some ofthese gaps through the formulation of a nonlinear (material as well as geometric),non(thermodynamic)equilibrium theory of &eld dislocation mechanics.A review of the literature on the theory of continuously distributed dislocations sug-

gests that despite its potential, in a generalized form and in combination with moderncomputational methodology, to address current nonlinear problems of inelasticity atsmall scales, the theory has not received as much attention as it deserves—as a canon-ical example, it is su4cient to note that Peierls’ (1940) analysis, ultimately, is ananalysis based on &eld concepts and that discreteness of the dislocation distribution inthat model, albeit not as realistic as it should be due to the simplifying assumptionsmade, is an outcome of the analysis. Consequently, it seems reasonable to attempt atheoretical generalization of the elastic &eld theory of dislocations and an associatedcomputational implementation. Of the things such a theory can be, it can be a rigoroustheory of internal stress and dislocation kinetics without ad hoc assumptions, whensolved at a su4ciently &ne scale. Of the things it cannot be, the theory cannot beexpected to be the correct framework for mesoscopic and macroscopic plasticity with-out coarse-graining. However, till such time as a rigorous coarse-graining methodologybecomes available, it can be heuristically combined with existing proposals for macro-scopic/mesoscopic plasticity that account for hardening due to short-range dislocationinteractions phenomenologically, e.g. Acharya and Beaudoin (2000), while accountingrigorously for long-range stress eFects at the desired scale of resolution. Despite theheuristics involved, such an approach has the potential of contributing positively tothe current debate in the literature on the subject of the correct method for incorporat-ing long-range dislocation stress eFects in phenomenological strain-gradient plasticitytheories (Needleman and Gil Sevillano, 2003).This paper is a completion of theoretical ideas and results presented in Acharya

(2001, 2003). These eForts build on the pioneering works of KrKoner (1981), and ear-lier references therein, Mura (1963), Willis (1967), Kosevich (1979), and Fox (1966)in constructing a well-set theory of nonequilibrium, &eld dislocation mechanics for bod-ies of &nite extent unavailable until now, despite the long history and importance of thesubject. The theory has the desirable feature that it lends itself to computational imple-mentation through established methods for approximating partial diFerential equations.The small deformation theory has been analytically shown to possess the capability ofpredicting fundamental features of dislocations in solids related to their stress &eldsand motion. Association has been made between the driving force for dislocation mo-tion and the Peach–Koehler force of classical dislocation theory. A nonlocal-in-stressdriving force for nucleation predicted by the theory has produced promising agreementwith computational experiments related to homogeneous nucleation in nanoindentationof nominally 2D atomistic con&gurations (Miller and Acharya, 2002). Additionally,

A. Acharya / J. Mech. Phys. Solids 52 (2004) 301–316 303

&nite-element implementation of the theory (Roy and Acharya, 2002) has producedresults for three-dimensional stress &elds of dislocations in generally anisotropic mediaof small as well as large extent that are in agreement with corresponding analyticalresults. The latter eFort has also produced results on the time evolution of disloca-tion density &elds and permanent deformation (e.g. slip steps) representative of singledislocation activity as well as dislocation walls under loads.Based on this evidence, it appears that a properly formulated &nite-deformation the-

ory should be capable of predicting, in a far-from equilibrium setting, features likeequilibrium dislocation distributions and discreteness eFects in them, Peierls stress ef-fects in dislocation mobility, cross-slip of individual dislocations, and complex dislo-cation interactions, without ad hoc assumptions beyond a speci&cation of nonconvexbulk crystal elasticity (Ortiz and Phillips, 1999) and the constitutive dependence ofthe dislocation velocity and nucleation on their theoretically derived driving forces. Itshould be noted here that a speci&cation of the instantaneous velocity of a dislocationsegment does not by itself provide a condition for cross-slip, namely the velocity byitself cannot describe the process by which the jog segments are produced.The ideas presented in Acharya (2001) referred to the mechanical aspects of the

&nite-deformation theory; no theoretical guidelines for the formulation of constitutiveequations arising from minimal, but essential, thermodynamical grounds were provided.As mentioned in the concluding remarks of that paper, such guidelines are di4cult toderive due to the nonlocal nature of the theory. In this paper, that task is accomplished.Also, as developed in Acharya (2001, 2003), the mechanical structure of the the-

ory relies heavily on the use of an orthogonal decomposition in the space of square-integrable tensor &elds on the body and consequently on the notion of an inner producton this space. While mathematically correct, the reliance on such a device to spell outthe &eld equations of a physical theory seems to be less than satisfactory. In this paper,a minor reformulation is achieved that removes the aforementioned objection relatedto the mathematical structure of the theory while keeping the physical ideas intact.Following the analysis in this paper, the resulting &nite-deformation, nonequilibrium

theory can be set in accord with the minimal (isothermal) thermodynamic requirementthat the global rate of working of the stresses always exceed or equal the globalrate of change of free energy of the body through obvious choices of its constitutiveingredients on their derived driving forces. Although intricate, but physically natural, inits nonlocal structure, the theory is consistent with the conventional notion of classicaldislocation theory that the free energy and stress at a &eld point of a crystalline bodywith defects depend only on the value of the elastic distortion &eld at that point.

2. Notation

The symbol ∈ is shorthand for ‘belongs to’; ∀, that for ‘for all’; ⊂ stands for‘subset’ and ⇒ for ‘implies’. A superposed dot on a symbol represents a materialtime derivative. The statement a := b is meant to indicate that a is being de&nedto be equal to b. The summation convention is implied except for indices appearingbetween parenthesis or when explicitly mentioned to the contrary. We denote by Ab the


action of the second-order (third-order, fourth-order) tensor A on the vector (second-order tensor, second-order tensor) b, producing a vector (vector, second-order tensor).A · represents the inner product of two vectors, a : represents the trace inner productof two second-order tensors (in rectangular Cartesian components, A :B = AijBij) andmatrices. The symbol AB represents tensor multiplication of the second-order tensorsA and B. The curl operation on the current con&guration and the cross product ofa second-order tensor and a vector are de&ned in analogy with the vectorial caseand the divergence (div) of a second-order tensor on the current con&guration: for asecond-order tensor A, a vector v, and a spatially constant vector &eld c,

(A× v)Tc = (ATc) × v ∀c(curlA)Tc = curl(ATc) ∀c: (1)

In rectangular Cartesian components

(A× v)im = emjkAijvk ;

(curlA)im = emjkAik; j ; (2)

where emjk is a component of the third-order alternating tensor X and the spatialderivative, for the component representation, being with respect to rectangular Cartesiancoordinates on the current con&guration.Unless speci&cally mentioned otherwise, we choose the con&guration of the as-

received body, possibly in a state of initial stress, as the reference con&guration. Wedenote positions in this con&guration by x0 and those in the current con&guration byx. The deformation gradient with respect to the reference con&guration will be denotedby F = I + @u=@x0, where u is the displacement of the current con&guration from thereference con&guration and I the second-order identity. While convenient for certainanalyses, the reference con&guration does not play an essential role in de&ning thetheory.

3. Background and mechanical considerations

In this section, the structure of the theory pertaining to pure mechanics is introducedfor background, as developed in Acharya (2001). The minor reformulation indicatedin Section 1 is also incorporated.Our considerations begin with the following equations in local form:

curlW = −�; (3)

divT = 0; T = T(W−1); (4)

◦� = −curl(� × V) + s: (5)

Here, W is the inverse elastic distortion, typically denoted by the symbol F e−1 inconventional plasticity. � is the two-point tensor of dislocation density, between thecurrent con&guration and the ‘intermediate’/lattice con&guration, as de&ned in Willis


(1967), and◦� is a convected derivative of the tensor � following material elements

given by

◦� = (div v)� + � − �LT; (6)

where v is the material velocity and L is the velocity gradient. In addition, T is the(symmetric) Cauchy stress tensor, V is the dislocation velocity vector relative to thematerial, and s is the nucleation rate tensor. BrieRy, Eq. (3) represents the relationshipbetween the incompatibility of the inverse elastic distortion and the dislocation density;Eq. (4) is the standard equation of equilibrium and the general, formal constitutiveequation for T ; and Eq. (5) is a local form of a balance law expressing the rate ofchange of Burgers vector content of dislocation lines threading a deforming materialarea element, as it arises from the entry or exit of dislocation lines into the materialelement with the velocity V , and from the generation of new dislocations at the rate s.In this paper, we do not introduce explicit slip system variables in the belief that

discreteness eFects and slip system-like behavior are to be expected as an outcome ofthe theory when nonlinear crystal elastic descriptions incorporating lattice periodicityand symmetry like, e.g., the EAM potential under the Cauchy–Born hypothesis areused. This not only has a desirable simplifying eFect on the structure of the theory,but is also physically justi&ed. For instance, we note that atomistic simulations do notintroduce slip systems per se but recover behavior indicative of the existence of such.Moreover, there are materials like the b.c.c. structure where the possible operative slipsystems are not known with de&niteness, thus giving credence to the idea of havinga theory where such entities are not fundamental axiomatic ingredients but derivedresults.Eqs. (3) and (4) as developed in Willis (1967) are, arguably, the simplest de&ning

statements of the nonlinear elastic theory of dislocations. In that theory, the displace-ment does not play a role—the current con&guration is assumed known and the goalis to determine the elastic distortions and the corresponding stress—in this sense, itis rightly called a theory of internal stress. However, a dynamical theory of disloca-tion mechanics de&nitely requires a treatment of a displacement &eld to enable, at aminimum, a discussion of physically observed permanent deformation.As discussed in Acharya (2001), given the &eld �, (3) and (4) are not adequate to

determine a unique W and a displacement &eld in any instance. To see this, choosean arbitrary displacement of the initial con&guration satisfying any given displacementboundary conditions. On the so generated current con&guration, let � be a particularsolution of Eq. (3). Clearly, for any vector &eld f ,

W = � +@f@x

= � +@f@x0

F−1 (7)

is also a solution of Eq. (3). Now, with � and u &xed, consider the solution of

div T

({� +

@f@x0

F−1}−1

)= 0 (8)


for the vector &eld f , with appropriate traction boundary conditions (any displacementboundary condition has already been taken care of). Even when Eq. (8) can be solveduniquely for the &eld f , because of the arbitrariness in the choice of the &eld u wesee that W and u cannot both be uniquely determined.It is to resolve this nonphysical nonuniqueness that the notion of an orthogonal

projection associated with an inner-product was introduced in Acharya (2001). Anadditional &eld equation involving this projection was then introduced that eliminatedthe nonuniqueness. In the following, we achieve the same goal without the use of theinner-product or the projection. However, the price for the greater physical elegance isthat the number of &eld variables of the theory is increased due to the addition of anadded point (position vector) &eld.We pose the mechanical theory as follows:

curl � = −�; (9)

div � = 0; (10)

W = � +@f@x

= � +@f@x0

F−1; (11)

divT = 0; T = T(W−1); (12)

◦� = −curl(� × V) + s (13)

on V , with the &elds f , V , and s speci&ed constitutively. The following boundaryconditions are also required:

� n = 0 on @V; (14)

f speci&ed at a single (arbitrarily chosen) point of the body; (15)

{�(V · n) = prescribed Rux or � = prescribed} on @Vi; (16)

where n is the outward unit normal to the boundary of the current con&guration @V ,and @Vi is the inRow set of points on the boundary where V · n¡ 0 (Acharya, 2003).Eqs. (14)–(16) can be shown to be su4cient for uniqueness in su4ciently simplerealizations of the theory. Since it is the gradient of f that aFects the response of thebody as can be seen from Eqs. (9)–(12), the essential boundary condition (15) is insome sense simply a matter of convenience, without aFecting the physical generalityof the theory. In addition, standard displacement and/or traction boundary conditionson @V or appropriate parts of it are also required. An initial condition on the &eld � isrequired. An additional initial condition on f is determined by solving the problem ofinitial stress, corresponding to the initial dislocation density &eld, on the as-received—and possibly loaded—con&guration of the body (Acharya, 2001).Eqs. (9), (10), and (14) allow for a unique determination of � when � and the

current con&guration are treated as data, as can be seen from an elementary proof.


To obtain a general idea of the expected nature of the constitutive speci&cation of f ,consider the situation where some plastic Row (dislocation motion) has taken place andthen the body is completely free of dislocations (e.g., an idealization of the process ofannealing) with no applied loads. Since this is just an illustrative thought experiment,we also consider the case where the stress-response derives from a convex strain energyfunction. Under these conditions

� = 0 (17)

and

W =@f@x

=@f@x0

F−1; (18)

so that the equilibrium equation

div T

(F(

@f@x0

)−1)

= 0 (19)

with zero traction boundary condition implies

F =@f@x0

(20)

up to a spatially uniform orthogonal tensor &eld. Hence, we conclude that f should berelated to the plastic (permanent) deformation that arises due to dislocation motion.As described in Acharya (2001, 2003), � × V represents the local plastic distortion

rate produced by the ‘motion’ of �. Consequently, we would like to relate

@(f )@x

(21)

to the rate of plastic distortion while making sure that the choice does not violate anythermodynamical requirements. In order to derive these thermodynamical requirements,we will need the following mathematical de&nitions and properties (adapted from Weyl,1940):The notation A== represents the orthogonal projection of the second-order tensor &eld

A on the null space of the operator curl de&ned in the weak sense. Let the problemcurlW = � be de&ned weakly as follows:∫

VW : curlQ dv=

∫V� :Q dv ∀Q ∈T; (22)

where T is the set of continuous test functions on the current con&guration V thatvanish on the boundary @V of V , and have at least piecewise continuous &rst derivativesin V . The null space, N (curl), of the operator curl is de&ned as the set of all squareintegrable tensor &elds Y on V that satisfy∫

VY : curlQ dv= 0; ∀Q ∈T: (23)


In other words, N (curl) could be referred to as the set of all weakly irrotational tensor&elds on the current con&guration.Let D be the set of all square integrable tensor &elds on the domain V , endowed

with the standard L2 inner product—the inner product of two tensor &elds A, B isgiven by

∫V A :B dv. Then, any A∈D can be uniquely written as the sum

A= A⊥ + A== ; (24)

where:

i. A== belongs to N (curl) ⊂ D.ii. A⊥ belongs to N⊥(curl)=

{B ∈D such that

∫V B :V dv= 0;∀V ∈N (curl)

}. The

space N⊥(curl) is the closure of the set of all tensor &elds of the type curlQ∗ onV , Q∗ ∈T , and Q∗ su4ciently smooth for curlQ∗ to make sense.

iii:∫VA :B dv=

∫VA== :B== dv+

∫VA⊥ :B⊥ dv: (25)

iv. For every A∈D there exists a WA ∈D that satis&es∫VWA : curlQ dv=

∫VA⊥ :Q dv ∀Q ∈T with WA = 0 on @V: (26)

WA may be determined from

curlWA = A⊥

divWA = 0on V (27)

WA = 0 on @V

or the ‘weak’ equivalent of Eq. (27) when A is not su4ciently smooth.v. For A∈D and a su4ciently smooth B ∈D∫

VcurlWA :B dv=

∫VWA : curlB dv: (28)

4. Thermomechanics and constitutive considerations

We analyze the dissipation implied by the theory in the isothermal case in order toidentify the general form of the stress response, the structure of the permanent distor-tion, and the driving forces for dislocation motion and nucleation. Assuming balanceof mass holds in its usual form in classical continuum mechanics—a discussion of thevalidity of which, in a continuum theory representing dislocation phenomena, we donot venture into in this paper—the dissipation D is given by

D =∫VT :D dv −

∫V� dv: (29)


In the above, is the free energy per unit mass of the body and D is the rate ofdeformation tensor given by the symmetric part of the velocity gradient. We assume,consistent with classical notions in dislocation theory and continuum plasticity, that at any material point of the body is a function of the elastic distortion W−1= : F e atthat point:

= (F e) ⇒ =@ @F e : (−F eWF e): (30)

De&ning

� := �F eT @ @F e F

eT; (31)

D =∫VT :D dv+

∫V� : W dv: (32)

We now proceed to an analysis in the ‘relative’ description, i.e., we choose thecurrent con&guration at time t (arbitrarily &xed instant of time) as the &xed referencecon&guration and consider the motion of the body (parametrized by a time-like variable,say, �) ensuing from this con&guration, i.e., x(�=0)=x(t). We refer to the deformationgradient &eld based on this reference as Ft and derivatives of &elds with respect tothis con&guration as @()=@xt . Then,

W = � +@f@xtF−1t ⇒ W = ˙�FtF−1

t +@(f )@xt

F−1t −WL: (33)

Now, Eq. (9) implies∫c� dx = −

∫a�n da (34)

for any surface patch a in the body with n as the unit normal &eld (with some chosenorientation) on it and c as its boundary curve. Consequently,∫

ct�Ft dxt = −

∫atJt�F−T

t nt dat ⇒ curlt ˙�Ft = − ˙Jt�F−T

t : (35)

In the above, curlt is the curl operator on the con&guration at time t, Jt = det Ft , ct ,at and nt (unit normal &eld) are the images, in the con&guration at time t, of theclosed curve c, the surface patch a and the &eld n, respectively, and dxt and dat arethe corresponding line and area element images. Similarly, Eq. (13) implies (and wasderived from)

˙∫a�n da= −

∫c� × V dx+

∫asn da (36)

so that∫at

˙Jt�F−T

t nt dat = −∫ct� × V F t dxt +

∫aJtsF

−Tt nt dat (37)


and hence

˙Jt�F−T

t = −curlt(� × V F t) + JtsF−Tt : (38)

We now evaluate Eq. (33) at � = 0 (or, in the parlance of continuum plasticity,choose the current con&guration as reference) and substitute in Eq. (32) to obtain

D =∫VT :D dv+

∫V� :

(˙�Ft +

@(f )@x

−WL)

dv; (39)

where it is understood that all terms are evaluated at time t or �= 0, followed by

D=∫VT :D dv −

∫VWT� :L dv+

∫V�⊥ :

(˙�Ft +

@(f )@x

)dv

+∫V�== :

(˙�Ft +

@(f )@x

)dv: (40)

Noting that curlW�=�⊥ Eqs. (26)–(27), and that the curl of a gradient vanishes, thethird integral in Eq. (40) can be written as∫

V�⊥ : ˙�Ft dv=

∫VW� : curl ˙�Ft dv=

∫V�⊥ : � × V dv −

∫VW� : s dv; (41)

where Eqs. (35) and (38), evaluated at � = 0, have been used as well as Eq. (28).Finally, noting that ˙�Ft = � + �L when evaluated at �= 0, dissipation takes the form

D=∫VT :D dv −

∫VWT� :L dv

+∫V�⊥ : � × V dv −

∫VW� : s dv

+∫V�== : (� + �L) dv+

∫V�== :

@(f )@x

dv (42)

and this holds for all t as this instant was chosen arbitrarily. Henceforth, we refer to@(f )=@x simply as @f [email protected] mentioned earlier, one of our primary goals is to relate the &eld @f =@x to the

rate of plastic distortion produced due to dislocation motion given by �×V . To do so,we &rst note that the // component of any &eld is a ‘weak’ gradient merely from itsde&nition Eqs. (23) and (24). We now postulate the constitutive equation for the &eld fto be ‘as close as possible’ to the &eld (�×V) without violating the requirement that thedissipation be non-negative, and consistent with the desire that constitutive input shouldcome into the theory only through the speci&cation of the elasticity of the material andthe dependence of the dislocation velocity and nucleation rates on their, as yet to be


identi&ed, driving forces. We require that the &eld f satisfy

@f@x

= (� × V)== − (� + �L)== (43)

so that∫V�== : (� + �L) dv+

∫V�== :

@f@x

dv=∫V�== : (� × V) dv; (44)

where Eq. (25) has been used. With Eq. (43) enforced, the dissipation takes the form

D=∫VT :D dv −

∫VWT� :L dv

+∫V� : � × V dv −

∫VW� : s dv: (45)

Based on physical grounds, the &eld W = F e−1 transforms to WRT under a super-posed rigid body motion of the body, where R is the time dependent, spatially uniformorthogonal tensor &eld de&ning the superposed rigid motion. Consequently, F e trans-forms to RF e and this implies, akin to conventional plasticity, that the free energyfunction has to be a function of the elastic right Cauchy–Green tensor C e = F eTF e,i.e., = (C e). Hence, @ =@F e = 2F e(@ =@C e) so that

WT�= �@ @F eF

eT = 2�F e @ @C eF

eT (46)

using de&nition Eq. (31). Substituting Eq. (46) in Eq. (45) and requiring no dissipationin the absence of any dislocations and dislocation nucleation in the body, we arrive atthe constitutive equation for the (symmetric) stress in the theory as

T = 2�F e @ @C eF

eT: (47)

With the constitutive choices Eqs. (43) and (47) and the de&nition

�∗ := F e� (48)

for the ‘pull-back,’ to the current con&guration, of the two-point tensor &eld �, thedissipation reduces to the form

D =∫V� : � × V dv −

∫VW� : s dv=

∫V{X(T�∗)} · V dv −

∫VW� : s dv; (49)

thus exposing the driving forces for the dislocation velocity and nucleation rate to be

V → X(T�∗) (50)

and

s → −W�: (51)

In the above, X is the third-order alternating tensor.


The above results are analogous, with appropriate generalizations for the varioustensors involved, to those derived in Acharya (2003) for the geometrically linear theory.In particular, there the driving force for dislocation velocity is analyzed in detail toestablish its relationship with the classical Peach–Koehler force of elastic dislocationtheory, as well as to notions of Schmid and non-Schmid behavior and the capability ofthe theory in describing the onset of bowing of pinned segments and that of cross-slip.All of those conclusions apply in the present case of the nonlinear theory with almostno conceptual change.An important requirement of the theory is that

div � = 0; (52)

so that solutions exist to Eq. (9). It also corresponds to the physical observation thatdislocation lines do not end within the body (assuming no disclinations to be present).This requirement implies a restriction on the constitutive form of the dislocation nu-cleation rate s, as we now show.Consider a ball B0 of material particles of arbitrarily small, but &nite, radius about

some material point as the center. Let us imagine the boundary sphere, @B0, of this ballas cut along any closed material curve, thus generating two material surfaces that &ttogether to form the whole bounding sphere. Applying Eq. (13) to the deforming imageof the two surfaces so formed, and adding the result of these operations we obtain (dueto a cancellation of the ‘curl’ terms when transformed to the cut via Stokes’ theorem)∫

Bdiv s dv=

∫@Bsn da=

˙∫@B�n da=

˙∫Bdiv � dv= 0; (53)

where B and @B are deformed images. Owing to the arbitrariness of the radius of theball, Eq. (53) implies the requirement

div s = 0: (54)

Conversely, if Eq. (54) is satis&ed, then Eq. (52) is too, as long as the initial dislocationdensity is divergence free.In a crystal it is often possible to identify the (undeformed) Burgers vectors of the

possible dislocations that can be formed in it based on crystallographic considerations.If m is such an (undeformed, unit) direction in the unstretched lattice, then the drivingforce for the nucleation of a density of such dislocations may be taken to be the tensor

m ⊗ (−WT�m): (55)

It may be of use in the formulation of constitutive equations for the nucleation rate forparticular densities to note that the tensor indicated in Eq. (55) is indeed solenoidal,since m may be considered as spatially uniform, being a direction in the unstretched lat-tice con&guration, and W� is divergence free. Alternatively, a nucleation rate potential� may be introduced (Acharya, 2003) that satis&es

curl�= s: (56)

Then

−∫VW� : s dv=

∫V

−�⊥ :� dv; (57)


so that the driving force for � is

� → −�⊥: (58)

In the constitutive context, � may be speci&ed to be a function of −�⊥ such thatthe integrand of the right-hand side of Eq. (57) is nonnegative, and then s derivedfrom taking the curl of this &eld, thus satisfying Eq. (54) automatically. However,the connection between the nucleation rate potential and the driving force for speci&cfamilies of dislocation densities characterized by their Burgers vectors is not easilyestablished.In the context of a nonlinear &eld theory of nonequilibrium dislocation mechanics,

the results presented here appear to be new. In this connection, Gurtin (2002) alsorefers to the microforces of his theory as analogs of the Peach–Koehler force; ourdriving force for dislocation velocity (physical units of force/volume) does not appearto share any common ground with the microforces of Gurtin. Amongst the diFerences,the microforces do not depend upon the Cauchy stress whereas our driving force does;an essential aspect of Gurtin’s theory is the explicit dependence of the free energyon the dislocation density tensor and the consequent need to characterize the natureof this dependence whereas in the present theory a dependence on only the elasticstrain is required. Additionally, no other &eld theory of continuum plasticity, includingGurtin (2002), has been shown to predict elastic stress &elds of dislocations whereasthe present theory incorporates the elastic theory of dislocations exactly.

5. Some observations

To summarize, we collect the fundamental statements of the theory in one place.The governing equations are

curl � = −�div � = 0(F e−1 =W = � +

@f@x

= � +@f@x0

F−1)

divT = 0◦� = −curl(� × V) + s:

on V (59)

The constitutive ingredients are (an arrow representing a driving force):

= (C e);

T = 2�F e @ @C eF

eT;

@f@x

= (� × V)== − (� + �L)== ;

V → X(T�∗);

s → −W� or � → −�⊥:

(60)


The boundary conditions are given by

�n = 0 on @V;

f speci&ed at a single (arbitrarily chosen) point of the body;

{�(V · n) = prescribed Rux or � = prescribed}on inRow boundary @Vi;

Standard displacement and=or traction boundary conditions:

(61)

Finally, initial conditions are required on the &elds � and f . The initial condition on fis deduced from solving the problem of initial stress on the reference con&guration, i.e.,(59)1;2;3;4 are solved with �(t = 0; x0) = �0(x0) speci&ed, u(t = 0; x0) ≡ 0, (61)1;2 andtraction boundary conditions on the as-received con&guration of the body. It should benoted that the reference con&guration does not play an essential role in the governingequations and constitutive equations of the theory, as should be the case in a theoryof inelastic response.As a conceptual guide, it is convenient to think of a pointwise collection of time-

dependent, unstretched lattice con&gurations as an ‘intermediate con&guration,’ similarto conventional &nite-deformation continuum plasticity. This time-dependent ‘con&g-uration’ has the property that it is identical for any two actual motions of the bodythat diFer by a rigid deformation at any instant. The tensors �;W ; @f =@x;W�, and �map vectors of the current con&guration to this lattice con&guration. Consequently,their transformation rules, under a superposed rigid transformation characterized by theproper-orthogonal tensor R, are given by

(·) → (·)RT; (62)

which also implies that the point &eld f should remain unchanged under the rigidtransformation. Since the dissipation at any point of the body should remain unchangeddue to the superposed rigid transformation, the invariance rules for the dislocationvelocity and the nucleation rate should be the same as for their driving forces givenin Eqs. (50) and (51).It can be shown that, granted the constitutive equations for V and s satisfy their

transformation rules, the theory is indeed invariant under superposed rigid motions asde&ned in the sense above. The basic idea in this demonstration is that if all the &eldequations are satis&ed in the one motion of the body, then the transformed &elds are thesolution of the &eld equations on the ‘rotated’ motion. For equations involving the //and ⊥ projections, the demonstration requires a consideration of the variational partialdiFerential equations de&ning them with test functions satisfying the transformationproperties mentioned above.An important special case of the theory is when no dislocations are present in the

body. In such a case the theory should reduce to nonlinear elasticity and it indeed does.For simplicity, let us assume that the initial con&guration is the ‘natural’ con&gurationof the unstressed elastic body and the elastic strain energy is convex. In the absenceof dislocations, Eqs. (9) and (10) with Eq. (14) imply that � vanishes on the body.The initial stress problem and Eq. (43) imply that @f =@x0=I (up to an inconsequentialspatially uniform orthogonal tensor &eld) and remains so for the entire motion. Conse-quently, W ≡ F−1 and we note that Eq. (47) is the standard constitutive equation for


hyperelastic response. If the restrictive assumption on the strain energy is not made,then the conclusion remains unchanged if the initial condition on @f =@x0 is simplyassumed to be the identity since no dislocations are present. A more interesting spe-cial case is that of the elastic theory of dislocations on a generally deforming currentcon&guration, to be recovered when V ≡ 0 and s ≡ 0, i.e., dislocations are present inthe deforming body under loads, but do not move with respect to the material and arenot nucleated. The appropriate condition indicative of this idealized physical situationis

WF = 0 (63)

or, in other words, the inverse elastic distortion as measured from any reference con-&guration — in this discussion chosen to be the initial con&guration to make contactwith Fp = F e−1F ≡ WF of conventional continuum plasticity — remains unchangedin time.Owing to the hypothesis on the dislocation velocity and nucleation, reasoning iden-

tical to that made in deriving Eqs. (35) and (38) implies that ˙�F can be representedas a gradient of a vector &eld on the initial con&guration. But

� + �L= ˙�FF−1 (64)

so that �+�L is a gradient on the current con&guration in this situation, and this alongwith Eq. (43) imply

@f@x

= −(� + �L)== = −(� + �L) = − ˙�FF−1 ⇒ @f@x0

= − ˙�F : (65)

But then,

W = � +@f@x

= � +@f@x0

F−1 ⇒ WF = ˙�F +@f@x0

= 0 (66)

and we have shown that the theory reduces to the elastic theory of dislocations inappropriate circumstances. The theory implies no dissipation in this case, of course.It is a trivial observation with signi&cant physical and practical importance that the

theory generalizes seamlessly to incorporate inertial eFects. All that is required is toadd the inertia term to the right-hand side of Eq. (59)4. In the context of compu-tational algorithms, such a dynamical theory may be much easier to solve than thecorresponding static theory, especially keeping in mind the fact that even in the statictheory time-step limitations arising from the dislocation velocity are quite stringentfor a well-resolved calculation of transients. This is in contrast to discrete dislocationmethodology where even a conceptual extension does not seem to be possible (to thisauthor) to the nonlinear elastic and inertial cases.

Acknowledgements

Financial support for the work from the Program in Computational Mechanics of theUS O4ce of Naval Research, Grant No. N00014-02-1-0194 is gratefully acknowledged.


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constitutive analysis of finite deformation field dislocation mechanics

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