Int J Fract (2012) 174:87–94DOI 10.1007/s10704-011-9656-0

ORIGINAL PAPER

Coupled phase transformations and plasticity as a fieldtheory of deformation incompatibility

Amit Acharya · Claude Fressengeas

Received: 31 August 2011 / Accepted: 22 November 2011 / Published online: 13 December 2011© Springer Science+Business Media B.V. 2011

Abstract The duality between terminating discon-tinuities of fields and the incompatibilities of theirgradients is used to define a coupled dynamics of thediscontinuities of the elastic displacement field andits gradient. The theory goes beyond standard transla-tional and rotational Volterra defects (dislocations anddisclinations) by introducing and physically ground-ing the concept of generalized disclinations in solidswithout a fundamental rotational kinematic degree offreedom (e.g. directors). All considered incompatibil-ities have the geometric meaning of a density of linescarrying appropriate topological charge, and a con-servation argument provides for natural physical lawsfor their dynamics. Thermodynamic guidance providesthe driving forces conjugate to the kinematic objectscharacterizing the defect motions, as well as admissibleconstitutive relations for stress and couple stress. Weshow that even though higher-order kinematic objectsare involved in the specific free energy, couple stressesmay not be required in the mechanical descriptionin particular cases. The resulting models are capable

A. Acharya (B)Civil & Environmental Engineering, Carnegie MellonUniversity, Pittsburgh, PA 15213, USAe-mail: acharyaamit@cmu.edu

C. FressengeasLaboratoire d’Etude des Microstructures et de Mécaniquedes Matériaux (LEM3), Université PaulVerlaine-Metz/CNRS, Ile du Saulcy, 57045 Metz Cedex,Francee-mail: claude.fressengeas@univ-metz.fr

of addressing the evolution of defect microstructuresunder stress with the intent of understanding disloca-tion plasticity in the presence of phase transformationand grain boundary dynamics.

Keywords Phase transformation · Plasticity ·Defects

1 Introduction

The dynamic response of solids to mechanical and ther-mal loading involves material-physics phenomena thatare manifested over multiple length and time scales.The material responds to load through atomic level pro-cesses involving crystal defects that mediate plasticity,phase transformations and damage. These processes,however, generally occur in materials whose intrin-sic microstructure affects bulk properties like strength,ductility, and fracture toughness. Although relevantdimensions and time span for the nucleation of theindividual defects can be Å and picoseconds, theircollective interaction, evolution and relaxation occur atlarger spatial and time scales (µm and µs). The objec-tive of the program on which this paper is based isthe development of a kinematically rigorous continuummechanics model to serve as the basis for a simula-tion framework; the goal is to predict deformation-induced microstructure evolution and its effects onmacroscopic properties in materials that undergo cou-pled plasticity and phase transformation. The kinematic

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88 A. Acharya, C. Fressengeas

ingredients of the model naturally include structuraldefects representative of grain boundaries. Such inelas-tic deformation is complex because of the essentiallycollective character of the dynamical behavior of struc-tural defects over engineering time and length scales;nevertheless, understanding the underlying dynamicsof defects is essential for predicting flow and failureat meso- and macroscopic scales. Due to space limita-tions, in this paper we outline only the small deforma-tion kinematics and thermodynamics of our model.

This paper may be considered as an extension of thepioneering work of deWit (1970) on dislocation-discli-nation statics that was generalized by Fressengeas et al.(2011) to account for disclination-dislocation dynam-ics; our work describes the dynamics of a more generalclass of defects than dislocations and disclinations. Italso has broad philosophical similarities with the workof Kleinert (2008). Here, the whole elastic distortion‘gradient’ can be incompatible in order to representterminating/kinking elastic distortion discontinuities asmay arise, e.g., in a martensitic needle or a facet of aphase-transforming inclusion. Through the connectionto the kinematics of phase transformations, our workprovides an unambiguous physical basis to the notion ofgeneralized disclinations (g-disclinations) even in sol-ids without any director degrees of freedom. It is alsomotivated by that of Abeyaratne and Knowles (1990)on dynamic phase transformations that allows for fullfreedom in specifying kinetics of phase front motion,and generalizes it to account for geometric singular-ities in the surfaces of discontinuity. In this last con-text, there appear to be connections between our modeland the fundamental work of Simha and Bhattacharya(1998), and through that, to the configurational forceframework of Gurtin (2000). However, we believe thatthe details of generating approximate solutions to ourtheory based on nonlinear partial differential equations(PDE), while non-trivial, are going to be much sim-pler1 than the complications involved in the configura-tional force setup that, to our knowledge, has not beensolved in full 3-d generality as yet. As we understand,these difficulties stem from applying boundary con-ditions for the field equations on the (solution-depen-dent) moving boundaries of the cylinders representing

1 Indeed, Fressengeas et al. (2011) have already developed anddemonstrated a finite-element implementation of the pure dis-clination-dislocation dynamics case. An attractive feature is thecoherence of the algorithm with that of the pure field dislocationdynamics case first introduced in Roy and Acharya (2005).

the cores of the phase-transformation singularities. Aprimary future goal of our program is to make con-tact with the Topological Model put forward by Hirthand Pond (2011) in the modeling of displacive phasetransformations.

Phase field methodology has been used to studyphase transformations and plasticity (Levitas et al.2009), but the governing dynamical principle thereis of finding local minima of energy. In contrast, thedynamics we propose is based on geometrically rigor-ous conservation laws for line-defect densities carry-ing appropriate topological charge (much like vorticesin fluids and superconductivity) and we have shown,in simple but exact realizations of sub-parts of ourmodel, that the class of equilibria of such conserva-tion laws contains that of phase field models (with thesame energy) but is strictly larger, while in dynamicbehavior it allows defect motion as wave propagationthat simply cannot be predicted by phase field mod-els (Acharya et al. 2010; Acharya and Tartar 2011).Also, generalizing the standard formalism for describ-ing stress (arising from elastic straining) in phase fieldmethodology to the finite deformation setting to incor-porate elastic response developed from atomistic con-siderations appears to be an as yet unsettled matter.

We use more-or-less standard notation. All tensorindices are written with respect to the basis ei , i =1 to 3 of a rectangular Cartesian coordinate system. Thesymbol ei jk represents the components of the alternat-ing tensor X . Vertical arrays of two or three dots repre-sent contraction of the respective number of ‘adjacent’indices on two immediately neighboring symbols (instandard fashion). Also,(curl A)i j or ik j = e jrs∂r Ais or iks

(div A)i or ik = ∂r Air or ikr

(A × V )i j or ik j = e jrs Vs Air or ikr

⎫⎪⎬

⎪⎭

for A a 2nd or 3rd order tensor

2 The physical idea

The fundamental theoretical concern is the modeling ofthe dynamics of the relevant defects that may be viewedas discontinuities in the elastic displacement, rotationand distortion fields (slip, grain and phase boundaries,respectively) across surfaces, terminating in line singu-larities (dislocation, standard disclination and gener-alized disclination, respectively) as sketched in Fig. 1.

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Coupled phase transformations 89

It is well known that when defects are viewed at suf-ficiently small scales there are no discontinuities andsingularities but only appropriately localized smoothfields. Moreover, for the purpose of understanding thenonlinear dynamics of such defects and their collectiveeffects, in particular in the context of nonlinear PDEs,such smooth localized representations of the physicalreality of defects are essential from the point of viewof mathematical analysis and numerical computation.There is a significant duality between the terminat-ing curves of discontinuities of fields on 2-d surfacesand the smooth incompatibilities of the ‘gradients’ ofsuch fields. The essential idea is that the geometricallydefined incompatibilities model defect curves. Each(often closed) curve separates two disjoint pieces ofa 2-d surface. The field in question is discontinuous bydifferent amounts across the two 2-d pieces, and theadvance of the curve represents the spreading of onepiece at the expense of the other. The incompatibilitieshave the rigorous geometric meaning of a density oflines carrying appropriate topological charge, so thata tautological (and therefore, undeniable at the kine-matic level) conservation argument provides most ofthe structural elements of a natural physical law forthe dynamics of the field.2 We exploit this feature todefine coupled PDE-dynamics of the discontinuities ofthe elastic displacement field and its gradient. Whenrestricted to the elastic displacement, a nonlinear the-ory for the dynamics of dislocation and slip results.When applied to the elastic distortion, the theory pro-vides for the dynamics of g-disclinations, going beyondthe Volterra construct by including elastic strain dis-continuities, and allows dealing with phase and grainboundaries. Unlike the Volterra concept, our continu-ously distributed approach seamlessly models singledefects up to a continuously distributed field in theentire body, as might be encountered in the transition ofa solid to a liquid-like state under strong shock loading.These models are capable of addressing phase transfor-mation and dislocation motion-related microstructureevolution under stress. While being based on PDEs inthe set of variables involved, they are non-local modelsin space and time in the standard variables of conven-tional continuum mechanics, thus providing geometri-cally rigorous, physics-based, nonlocal generalization

2 This is in contrast to dynamics based on the local “principleof virtual power” whose validity has recently been questioned(Fosdick 2011).

of the latter, and leading to well-set models of post-ini-tiation defect mechanics.

3 Motivating kinematical notions

As is well-known, a classical, singular dislocationline in a simply-connected body has a smooth elas-tic (1−)distortion (strain + rotation) field in the nonsimply-connected, punctured domain represented bythe region excluding its core, even though the elas-tic displacement field it is considered to arise fromhas a discontinuity across a (non-unique) 2-d smoothslip surface in the same region. The geometry of anysuch 2-d surface is arbitrary, except that it terminatesalong the curve representing the core of the dislocation.Additionally, the important topological fact about sucha field is that a line integral of the elastic distortionalong any curve encircling the core is non-vanishingand constant over all such curves. This constant repre-sents the strength of the dislocation, i.e., the Burgersvector. Thus, as concerns a classical dislocation, it isonly its elastic distortion field that one can preciselydiscuss, and not the displacement field. The continu-ously distributed dislocations setting eases this classi-cal description by considering smooth elastic distortionfields on simply connected domains in which they arepointwise irrotational outside the core cylinder (possi-bly of non-zero volume), and their curl equals the cor-responding Nye dislocation density tensor fields insidethe cylinder. The distribution of the elastic distortionand the dislocation density tensor fields on the bodyare set up such that the Burgers vector of the disloca-tion being modeled is obtained on integration of thefields along appropriate surface patches and/or closedcurves.

Once understood, this fact enables the definition ofa phase boundary, i.e., a 2-d surface of elastic distortiondiscontinuity (which includes the case of a grain bound-ary), terminating on a g-disclination curve. In this case,we assume the continuous elastic 2-distortion field(i.e., the incompatible elastic distortion ‘gradient’ field)to be irrotational outside the g-disclination core, andthe g-disclination strength to be the second-order ten-sor obtained by integrating the 2-distortion field alonga closed curve encircling the core. In the pure discli-nation case, this second-order tensor is a skew-tensorwhose axial vector is called the Frank vector of the dis-clination. In the non-singular, continuously distributed

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90 A. Acharya, C. Fressengeas

Fig. 1 Discontinuity of adiscontinuity; cross-sectionview of straight defect linesterminatingdisplacement/distortiondiscontinuities

setting, one way of setting up the 3rd order, g-disclina-tion density field for a straight g-disclination would beto assign the tensor product of the 2nd order, strengthtensor and the core line direction divided by the corecross-sectional area as a constant distribution withinthe core cylinder and zero outside it. Considered fromthe dual viewpoint of a discontinuous elastic distortionfield with discontinuity terminating at a core singular-ity, the strength tensor is simply the jump in the elasticdistortion across the surface of discontinuity.

It is important to note explicitly that, in the con-tinuously distributed setting, the elastic distortions areno longer gradients but have incompatible parts thatmay be defined through a Helmholtz decomposition.We would now like to set up a dynamical theoryof such dislocation and g-disclination defect curvesand (meta)slipped regions, taking into account forces,moments and dissipation.

4 Mechanical structure

Let T be the (generally unsymmetric) stress tensor, Λthe couple stress tensor, X the alternating tensor, u thematerial displacement vector, v the material velocityvector, D and � the symmetric and skew-symmetricparts of the velocity gradient, and ω = −(1/2) X :� = curl v the vorticity vector. Let U e be the elastic1-distortion tensor and Ge the elastic 2-distortion ten-sor (in gradient elasticity these would be grad u andgradgrad u, respectively). We define

Ge − grad U e =: Ps (1)

−Ge : X =: αs (2)

curl Ge = curl Ps = Π s, (3)

where αs is the dislocation density (2nd-order) tensorand Π s is the g-disclination density (3rd-order) ten-sor. Both the dislocation density and the g-disclinationdensity tensors are ‘vector(tensor)-valued axial vector’fields representing third (fourth) order tensor fields thatare skew in their last two indices. Thus, they form den-sities that can be integrated over area elements. Phys-ically, such fields represent densities of lines carryingappropriate tensorial attributes.

It is helpful for physical interpretation to invokea Stokes–Helmholtz-like orthogonal decomposition ofthe field Ps into compatible and incompatible parts:

Ps = Ps⊥ + grad Zs

div Ps⊥ = 0with Ps⊥n = 0 on boundary of the body B

⎫⎬

⎭(4)

It is clear from (3) and (4) that when Π s = 0 thenPs⊥ = 0. Thus, referring back to our physical pictureof a single terminating elastic distortion discontinuity,the expression

Ge = Ps⊥ + grad U e + grad Zs (5)

implies that a smooth representation of elastic dis-tortion discontinuities across surfaces that do notterminate or have discontinuities in their tangent planeorientation field is embodied in the compatible partof Ge, characterized by grad U e + grad Zs . Thus,

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Coupled phase transformations 91

the compatible part of Ge characterizes smoothedphase/grain boundaries without kinks or corners. Ps⊥,on the other hand, characterizes phase/grain boundarieswith kinks or corners. Also, recognizing curl U e =−grad U e : X as representative of slip dislocationsarising from slip discontinuities, the expression

αs = −Ge : X = −Ps⊥ : X − grad U e : X

−grad Zs : X

implies that −grad Zs : X is representative of puretransformation dislocations, while −Ps⊥ : X is repre-sentative of g-disclination-induced transformation dis-locations. Thus, it becomes clear why an infinite grainboundary/incoherent phase boundary can often be rep-resented by slip dislocations. In addition, (5) also makesclear what parts of the structure of a general grain/phaseboundary cannot be represented by slip dislocations.

We now assume that the dislocation and the g-dis-clination densities representing line-like objects havevelocity fields, V α and V π , associated with them. In sit-uations when these fields represent individual defectsthrough non-singular localization in core cylinders, thecorresponding local value of the velocity field repre-sents the velocity of the movement of the core cylinders.Furthermore, because these defect fields are line densi-ties, conservation laws for their evolution are an imme-diate consequence (Acharya 2011). The local form ofthe conservation law for Π s is given by

Π s = −curl(Π s × V π

). (6)

As for the evolution of the line density field curl U e,we note from (5) that

αs + Ps : X = curl U e (7)

which we set equal to −curl (αs × V α) from the kine-matics of flux of lines moving into an area patch throughits bounding curve3 to obtain

αs = −curl(αs × V α

) − Ps : X, (8)

which implies

div αs = Π s : I.

Physically, this means that in contrast to the pure dislo-cation case, dislocation lines can terminate within thebody on phase/grain boundaries, but the locations at

3 An equally plausible expression would be to set −curl(curl U e ×V α), but the one we choose appears to have a definiteadvantage for the modeling of disclination and dislocation freephase transformations.

which they do so necessarily involve a variation in thestrength of the elastic distortion discontinuity along thephase/grain boundary.

The statements (3) and (6) imply

Ps = −Π s × V π + grad K s, (9)

where K s is a strain rate (2nd-order tensor) andgrad K s is a compatible contribution to the elastic2-distortion rate associated with the transverse motionof (physically identifiable) phase/grain boundaries. K s

requires specification based on geometric and consti-tutive considerations.

We now introduce the definition

U p := grad u − U e. (10)

Then, (7), (8), (10) imply

U p = αs × V α, (11)

where, unlike the 2-distortion case, we do not includethe free gradient (representing an inelastic strain rate)that arises in deducing (11) due to the assumptionthat slip-boundaries are often not physically identifi-able. Even when they are, as in the case of stackingfaults, their evolution typically consists of expansion orshrinkage in the slip plane governed by the motion oftheir bounding dislocation curve(s), so that no genera-tion of independent inelastic strain rate may be assignedto this evolution.

Summarizing, one minimal representation of thefield equations of the model consist of Eqs. (1), (2),(3), (9), (11) along with

div T = 0 (12)

div Λ− X : T = 0 (13)

representing balance of linear momentum and angu-lar momentum, ignoring body forces and couples andinertia terms (without loss of essential generality forthe present purpose). The last two equations serve todetermine the displacement field, when appended withconstitutive equations for the stress and couple stresstensors. In addition, constitutive equations are alsorequired for the dislocation and g-disclination veloc-ities as well as the potential K s for the compatible partof the elastic 2-distortion rate, Ge.

5 Constitutive guidance from thermodynamics

Following Mindlin and Tiersten (1962), we considerthe following statement to characterize the work ofexternal agents on the body:

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92 A. Acharya, C. Fressengeas

∫

∂B(T n) · v dv +

∫

∂B(Λn) · ω dv,

where B is the body, ∂B its boundary and n the outwardunit normal to the boundary. Denoting by ψ the freeenergy per unit volume field on the body, we define the(mechanical) dissipation to be

D =∫

∂B(T n) · v dv +

∫

∂B(Λn) · ω dv −

∫

Bψ dv

=∫

BTsym : D dv − 1

2

∫

BΛ : grad (X : �) dv

−∫

Bψ dv. (14)

We now consider three possible constitutive depen-dencies for ψ and show that even though ‘higher-order’ kinematic objects are involved in the specificfree energy, in two of these cases no couple stressis required, at least based on thermodynamic consid-erations. We also derive the driving forces for thedissipative mechanisms of dislocation motion (char-acterized by V α), g-disclination motion (characterizedby V π ) and phase/grain boundary motion (character-ized by K s , up to further future geometric refinementto go from a strain-rate to a phase-front velocity).

By substituting these constitutive hypotheses into(14) we identify the dissipative power-conjugates tothe defect motion variables as driving forces. We alsoidentify constitutive equations for the stress and couplestress tensors by requiring that D = 0 for all possiblemotions of the body in the absence of the dissipativemechanisms.

5.1 ψ = ψ (U e, Ps,Π s) implies no couple stress

The considered constitutive class is one that containselastic 2-distortion contributions in the specific freeenergy arising only from disclination and phase/grainboundary-induced fields, but none from the gradient ofthe elastic 1-distortion. We require that

ψ(U e, Ps,Π s) = ψ

(U e + W, Ps,Π s) ∀ W skew,

which implies that ∂U eψ has to be symmetric (andmore). Also,

ψ =(∂U e ψ

)

sym: grad v −

(∂U e ψ

)

sym

: U p + ∂Ps ψ : Ps + ∂Π s ψ : Π s

which, along with (6, 9, 11) and (14) implies,

D =∫

B

[

Tsym −(∂U e ψ

)

sym

]

: D dv (15)

−1

2

∫

BΛ : grad (X : �) dv

+∫

B

[

X(∂U e ψ

)

symαs

]

· V α dv

+∫

B

[

X

{(∂Ps ψ + curl ∂Π s ψ

)

i jl

Π si jkel ⊗ ek

}]· V π dv

+∫

Bdiv

(∂Ps ψ

): K s dv

−∫

∂B

(K s : ∂Ps ψ

)n da +

∫

∂B

(Π s × V π

)

...(∂Π s ψ × n

)da.

Thus we conclude that

T =(∂U e ψ

)

sym(16)

Λ = 0 (17)

X(

T Tαs)↪→ V α in bulk (18)

X

{(∂Ps ψ + curl ∂Π s ψ

)

i jlΠ s

i jkel ⊗ ek

}

↪→ V π

(19)

K s ↪→ div(∂Ps ψ

)in bulk, (20)

form a consistent set of constitutive guidelines that sat-isfies balance of angular momentum identically. Here↪→ is used as shorthand for “is the driving force for.”In addition, the boundary terms in (15) may be inter-preted as providing driving forces for K s and V π onthe boundary of the body.

The driving ‘force’ in (18) is the analog of the Peach–Koehler force of classical dislocation theory; it hasphysical units of f orce/volume, to be interpreted asthe force acting on unit volume of the dislocation corecylinders. The driving force in (19) is the correspondingthermodynamic force on unit volume along the lengthof a g-disclination cylinder; this is a new constructwhose analog for the pure disclination case arises inFressengeas et al. (2011). Finally, (20) shows the driv-ing force for the inelastic strain-gradient rate produceddue to the motion of phase/grain boundaries.

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Coupled phase transformations 93

5.2 ψ = ψ(U e, Ps,Π s, curl U e) implies no couplestress

Define α = curl U e. Then, an additional term to thedissipation (15) is

−∫

B∂αψ : ˙α dv =

∫

B∂αψ : curl

(αs × V α

)dv.

The results (16), (17), (19) remain unchanged and (18)changes to

X(

T + curl ∂αψ)Tαs ↪→ V α in bulk,

with an additional boundary term in the dissipation (15)given by

−∫

∂B∂αψ : [(

αs × V α) × n

]da.

It is interesting to note that the presence of a specialcombination of grad U e, i.e., the part only related toslip dislocation content, allows a theory without couplestress and a symmetric stress tensor on thermodynamicgrounds. A special case of the constitutive class consid-ered here was treated, with similar results, in Acharya(2010).

5.3 ψ = ψ(U e, Ps,Π s, grad U e) implies modelwith couple stress

We use the notation grad U e =: ∇U e and observethat a functional dependence of the specific freeenergy on the whole elastic 2-distortion, i.e., ψ =ψ(U e,Ge,Π s) is a special case of the constitutiveclass being considered for which

∂∇U e ψ (U e, Ps,Π s,∇U e) = ∂Ge ψ (U e,Ge,Π s).

In this case, the additional terms in the dissipation (15)are

−∫

B∂∇U e ψ

... gradgrad v dv

+∫

B∂∇U e ψ

... grad(αs × V α

)dv.

Noting that

Λ : grad (X�) = −emniΛi jvm,nj ,

(17) is replaced by the relation

emniΛi j+emjiΛin=−[(∂∇U e ψ

)

mnj+

(∂∇U e ψ

)

mjn

]

,

(18) is replaced by

X(

T − div ∂∇U e ψ)Tαs ↪→ V α in bulk,

with an additional boundary term in the dissipation (15)given by∫

∂B

[(αs × V α

) : ∂∇U e ψ]

n da.

Thermodynamical considerations leave the followingparts of the stress and couple stress tensors,

Ti j − Tji

emniΛi j − emjiΛin,

constitutively unspecified. Balance of linear momen-tum (12) and angular momentum (13) along withboundary conditions serve as two constraints for theirdetermination.

6 Concluding remarks

A ‘small-deformation’ theory of coupled plasticity andphase transformation accounting for the dynamics ofgeneralized defects, i.e., dislocations and g-disclina-tions, has been proposed. When the elastic strain-gra-dient is compatible, the present theory reduces to thatof Fressengeas et al. (2011), which involves only dis-locations and pure disclinations.

A fundamental premise of our work is that it isnot necessary, neither efficient, to resolve atomic levelvibrations to model defect dynamics leading to inelas-ticity at meso- and macroscopic scales. Our PDE-basedmodeling paradigm is to account for the defects inatomic configurations in (material-specific) nonlinearelastic media, focusing on the dynamics of distribu-tions of defects in such configurations, rather thanthe dynamics of the atoms themselves. This requiresminimal, but essential, input on the mobility of sin-gle defects and nonlinear (and non-monotone) elasticbehavior from MD/Quantum mechanics. Such trans-fer of information has the potential for computationalefficiency because one does not have to resolve atomicvibrations anymore. Our general philosophy is moti-vated by the great success of the Peierls model inelucidating basic dislocation physics, and is a general-ization of that approach to deal with unrestricted geo-metric nonlinearity, full-fledged non-monotone bulkelasticity and inertia. Moreover, the most noteworthy

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94 A. Acharya, C. Fressengeas

merit of such a PDE-based approach is that it suggestsrational and natural ways for developing coarse-grainedmeso/macroscale models of averaged defect densitiesinteracting with stress and leading to inelastic flow andfailure, a subject of great current interest.

Acknowledgments CF gratefully acknowledges funding fromCMU and UPV-M for his visits to CMU. AA acknowledges thehospitality and support of the LEM3, UPV-M, and the T4-Divi-sion of the Los Alamos National Laboratory.

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