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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 82 (2015) 287–319

http://d0022-50

n CorrE-m

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Interaction between phase transformations and dislocationsat the nanoscale. Part 1. General phase field approach

Valery I. Levitas a,n, Mahdi Javanbakht b,c

a Iowa State University, Departments of Aerospace Engineering, Mechanical Engineering, and Material Science and Engineering, Ames, IA50011, USAb Isfahan University of Technology, Department of Mechanical Engineering, Isfahan, Iranc Iowa State University, Department of Aerospace Engineering, Ames, IA 50011, USA

a r t i c l e i n f o

Article history:Received 29 December 2014Received in revised form15 April 2015Accepted 15 May 2015Available online 23 May 2015

Keywords:Phase field approachPhase transformationDislocationsLarge strainsInheritance of dislocations

x.doi.org/10.1016/j.jmps.2015.05.00596/& 2015 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ1 515 294 9691.ail address: vlevitas@iastate.edu (V.I. Levitas

a b s t r a c t

Thermodynamically consistent, three-dimensional (3D) phase field approach (PFA) forcoupled multivariant martensitic transformations (PTs), including cyclic PTs, variant–variant transformations (i.e., twinning), and dislocation evolution is developed at largestrains. One of our key points is in the justification of the multiplicative decomposition ofthe deformation gradient into elastic, transformational, and plastic parts. The plastic partincludes four mechanisms: dislocation motion in martensite along slip systems of mar-tensite and slip systems of austenite inherited during PT and dislocation motion in aus-tenite along slip systems of austenite and slip systems of martensite inherited duringreverse PT. The plastic part of the velocity gradient for all these mechanisms is defined inthe crystal lattice of the austenite utilizing just slip systems of austenite and inherited slipsystems of martensite, and just two corresponding types of order parameters. The explicitexpressions for the Helmholtz free energy and the transformation and plastic deformationgradients are presented to satisfy the formulated conditions related to homogeneousthermodynamic equilibrium states of crystal lattice and their instabilities. In particular,they result in a constant (i.e., stress- and temperature-independent) transformation de-formation gradient and Burgers vectors. Thermodynamic treatment resulted in thedetermination of the driving forces for change of the order parameters for PTs and dis-locations. It also determined the boundary conditions for the order parameters thatinclude a variation of the surface energy during PT and exit of dislocations. Ginzburg–Landau equations for dislocations include variation of properties during PTs, which in turnproduces additional contributions from dislocations to the Ginzburg–Landau equations forPTs. A complete system of coupled PFA and mechanics equations is presented. A similartheory can be developed for PFA to dislocations and other PTs, like reconstructive PTs anddiffusive PTs described by the Cahn–Hilliard equation, as well as twinning and grainboundaries evolution.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Interaction between martensitic PTs and dislocational plastic deformation is one of the most fundamental and complexproblem in the theory of PTs in solids, with numerous applications in material science, physics, geophysics, and material

).

Nomenclature

A austeniteM martensiteη PT order parameterξ dislocation order parameterΩ0 reference configurationΩ actual configurationΩt stress-free configuration after elastic

unloadingΩp stress-free configuration after elastic unload-

ing and reverse PTr0 position vector in Ω0

r position vector in Ωei Cartesian unit basis vectort timeF deformation gradientFe elastic part of deformation gradientFp plastic part of deformation gradientUt transformational part of deformation gradientlp plastic velocity gradientε strain tensorεt transformation strainεe elastic strainεp plastic strainω skew-symmetric small rotationωt skew-symmetric transformation rotationωp skew-symmetric plastic rotationnα unit normal to the slip plane αbα Burgers vector for the αth slip systemmα unit vector in the direction of bα

τ resolved shear stress for a single dislocationγ plastic shear for a single dislocationH height of a dislocation bandp traction vectorP first Piola–Kirchhoff stress tensors Cauchy stress tensorf body force per unit massr heat supply per unit massSi˜ local entropy production rateSi total entropy production rates specific entropyρ0 mass density in Ω0

ρ mass density in ΩΘα generalized force conjugate to ξαQ i generalized force conjugate to iη U internal energy per unit massD dissipation rate per unit massX η dissipative force conjugate to η Xξ dissipative force conjugate to ξ Lη kinetics coefficient for PTLξ kinetics coefficient for dislocationsβη gradient energy coefficient for PTβξ gradient energy coefficient for dislocationsψ Helmholtz free energy per unit massψe elastic energy per unit massψθ

η thermal energy per unit masscψξ crystalline energy per unit massintψξ the energy of interaction of dislocation cores

per unit mass

ψη∇ gradient energy per unit mass related to PT

ψξ∇ gradient energy per unit mass related to

dislocationsCk elastic moduli tensor of the kth rankR orthogonal lattice rotation tensorEe Lagrangian elastic strain tensorAα coefficient characterizing the magnitude of

the multi-well crystalline energyA kα coefficient of the energy of interaction of dis-

location coresωα width of the thin layer between dislocation

bandsy coordinate along the normal to the slip planek ratio of Aα for the thin layer between disloca-

tion bands and that for dislocation bandZ ratio of the coefficients for the gradient energy

normal to and along the slip planeA0 coefficient characterizing the magnitude of

the double well barrier between M and AGΔ θ difference between the thermal part of free

energy of M and AsΔ difference between the entropy of M and A

A coefficient characterizing the magnitude ofthe double well barrier between martensiticvariants

θ temperatureθe equilibrium temperatureθc critical temperatureB parameter controlling energy away from M

and A minimaD parameter controlling energy away from M

and A minimab mixed term coefficient in the gradient energy

for PTγs surface energyΛ function characterizing surface energy chan-

ges associated with the exit of a single dis-location at the surface

p equivalent order parameter for PTJ Jacobian determinantJt ratio of elementary volumes of M and AV0 volume in Ω0

V volume in ΩS0 surface in Ω0

S surface in Ωn0 normal to S0n normal to SVt volume in the unloaded configurationI unit tensor∀ for all≔ equal by definition∇ gradient operator in Ω0

2∇ Laplacians in Ω0

⊗ dyadic productδij Kronecker delta

Subscripts and Superscripts

e elasticp plastic

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319288

t transformationθ thermalA austeniteM martensiteT transposes symmetric part of a second rank tensora skew-symmetric part of a second rank tensor

η PTξ dislocations

MA slip system of A in A

AM slip system of M in A

MM slip system of M in M

AA slip system of A in A

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 289

technologies. The interaction with dislocations drastically changes PT thermodynamics, nucleation and growth kinetics,microstructure, PT hysteresis and irreversibility, i.e., the region of metastability of phases. PT in turn causes transformation-induced plasticity (Cherkaoui et al., 1998; Fischer et al., 2000), which is utilized, e.g., to obtain an optimal combination ofstrength and ductility of steels and ceramics (Olson, 1997). Heat and thermomechanical treatment of materials to obtaindesired a structure and properties is based, in particular, on a combination of martensitic PT and plastic deformation. PTsunder large plastic deformations of materials under high pressure are utilized for the synthesis of various materials em-ploying ball milling (Delogu, 2011) and compression and shear in rotational Bridgman anvils (Bridgman, 1935) and rota-tional diamond anvil cells (Blank et al., 1994; Ji et al., 2012; Levitas, 2004a). One of the mechanisms of deep earthquakes issupposed to be related to the material instability caused by a shear strain-induced PT (Green and Burnley, 1989; Kirby, 1987).An analysis of the experimental results (Blank et al., 1982, 1983; Estrin, 1993) on high pressure-induced PTs in variousmaterials, led to the conclusion in Levitas (1997) and Levitas et al. (1998) that an athermal threshold for PT is proportional tothe yield strength, and consequently it increases with plastic deformation. PTs during friction, indentation, surface treat-ment, and projectile penetration are also related to the interaction between PT and plasticity.

For the description of PT in an elastic solid, the principle of a minimum of Gibbs free energy is usually used to determinestationary microstructure. Due to plastic dissipation and threshold-type dissipation due to athermal interface friction, thisprinciple cannot be applied to elastoplastic materials. That is why the formulation of the governing principles and equationsis a nontrivial problem within continuum thermodynamics. The appearance of the spherical nucleus in an elastoplasticsphere under pressure was analyzed in Roitburd and Temkin (1986). An ellipsoidal nucleus with elastoplastic deformation inan infinite space was treated in Kaganova and Roitburd (1989). In both these papers, as well as in various other in-vestigations of PT in elastoplastic materials (Fischer et al., 1994; Marketz and Fischer, 1994, 1995) the principle of theminimum of Gibbs free energy was used. Olson and Cohen (1972, 1976, 1986) developed a dislocation model of martensite(M) nucleation. Various physical aspects of nucleation theory are reviewed in Olson and Roytburd (1995).

The condition for PT in the finite region (“macroscopic nucleation”) in the form that dissipation increment due to PT only(excluding plastic and other types of dissipation) reaches an athermal threshold was suggested in Levitas (1995a–c, 1997). Alocal description of PTs was developed in Levitas (1996a,b, 1998), which gave independent justification of this criterion.Using the postulate of realizability (Levitas, 1995d, 1997, 1998), an extremum principle for determining all unknownparameters of a nucleus (position, orientation, shape, etc.) was derived in Levitas (1995a–c, 1996a, 1997, 1998) that sub-stituted the principle of the minimum Gibbs energy. For Arrhenius-type kinetics, the postulate of realizability results in theprinciple of minimum of transformation time (Levitas, 2000).

Various approaches to M growth within elastoplastic materials were considered in Olson and Cohen (1986), Marketz andFischer (1994), Roitburd and Temkin (1986), and Kaganova and Roitburd (1987). Two different approaches were used fordifferent problem formulations. The first one was based on the Eshelby driving force (Cherkaoui et al., 1998; Cherkaoui andBerveiller, 2000; Fischer and Reisner, 1998; Fischer et al., 2000), similar to that for elastic materials. In the second one, thedriving force for nucleation and interface propagation is the dissipation increment due to PT only, i.e. excluding plastic andall other types of dissipation (Levitas, 1998; Idesman et al., 1998, 1999, 2000; Levitas et al., 2002). Advantages of the secondapproach were demonstrated in Levitas (2002). This approach was applied to numerous problems, including nucleation andgrowth of martensitic unit (Levitas et al., 2002), nucleation at shear-band intersection (Levitas et al., 1999), and highpressure strain-induced PTs under compression (Levitas and Shvedov, 2002; Levitas, 2004a; Levitas and Zarechnyy, 2010a)in a diamond anvil cell and compression and shear in a rotational diamond anvil cell (Levitas, 2004a,b; Levitas and Zar-echnyy, 2010b). Large strain formulations has been developed in Levitas (1996a, 1998) and applied to numerical solutions ofvarious problems in Idesman et al. (1999, 2000) and Levitas et al. (2002, 1999). Crystallographic aspects for stress-inducedmartensitic PTs in elastoplastic materials were treated at the microscale in Bartel et al. (2011).

With the development of nano science and technology, PT and plasticity are studied in nano particles, films, wires, andfor smart nanosystems. That is why there is a need to study the interaction between PT and plasticity at the nanoscale,including discrete dislocations. In particular, nucleation of a product phase occurs at various dislocation configurations andnucleation is always a nanoscale problem even for large samples. Also, the phase interface loses its coherency through thenucleation of dislocations. PFA represents a perfect nanoscale and continuum tool to study PTs, dislocations, and theirinteraction. It is based on the concept of the order parameters that describe instabilities of the crystal lattice during PT anddislocation nucleation, as well the evolution of phase and dislocational structures in a continuous way by solving Ginzburg–Landau evolution equations for the order parameters. Typical solutions for these equations are propagating finite-widthphase interfaces that separate multiple phases and dislocation core regions that separate parts of sheared and non-sheared

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319290

perfect crystals, which describe the evolution of complex phase and dislocational structures. Thermodynamic potential andtransformation and plastic strains depend on two types of the order parameters: ηi that describes PT from the austenite A tomartensitic variant Mi and ξα that characterizes increase in the magnitude of the Burgers vector of a dislocation in the α slipsystem from zero for perfect lattice to an integer number characterizing the number of complete dislocations. Thermo-dynamic potential has as many minima in the space of the order parameters ηi and ξα as many phases (including variants)and complete dislocations that we have. These minima are separated by energy barriers. Also, the potential depends on thegradients of the order parameters, which are concentrated at the finite-width phase interfaces and dislocation cores; thistakes into account interface and dislocation core energies. In most of the previous papers on both PTs and dislocations (Jinet al., 2001a; Lookman et al., 2008; Wang and Khachaturyan, 1997; Artemev et al., 2001; Wang et al., 2001a,b; Hu and Chen,2001; Kundin et al., 2011a; Koslowski et al., 2002) these were the only conditions which thermodynamic potentials weresatisfied. However, there are several other important conditions, which were formulated in Levitas and Preston (2002a,b) forPTs and Levitas and Javanbakht (2012, 2015) for dislocations.

In this paper, we will start with the most advanced from a mechanics point of view PFAs to multivariant martensitic PTs(Levitas, 2013b) and dislocations (Levitas and Javanbakht, 2012, 2015), both at large strains, and will synergistically combinethem in order to develop coupled PFA to PT and dislocations. This is why weminimize repetition and a justification of resultsin Levitas (2013b) and Levitas and Javanbakht (2012, 2015). In the following review we will focus on the related papers andadvantages of these approaches.

PFA is broadly used for simulations of martensitic PTs and microstructure evolution (Jin et al., 2001a; Lookman et al.,2008; Levitas and Lee, 2007; Wang and Khachaturyan, 1997; Artemev et al., 2001), see also the reviews (Chen, 2002;Mamivand et al., 2013; Wang and Khachaturyan, 2006). Here we will focus on the theory, in which order parameters arerelated to the transformation strain (Jin et al., 2001a; Levitas and Lee, 2007; Wang and Khachaturyan, 1997; Artemev et al.,2001; Chen, 2002) because for total strain-related order parameters (Lookman et al., 2008; Falk, 1983) we cannot satisfy allthe desired conditions. These conditions were introduced in Levitas and Preston (2002a,b) in order to describe typicalfeatures of stress–strain curves that are conceptually consistent with known experimental data and to include all ther-momechanical properties of both A and Mi. These requirements are

(a)

The transformation strain tensor that followed from the thermodynamic equilibrium conditions for homogeneousphases should be constant (i.e., stress- and temperature-independent), in agreement with crystallographic theory(Wayman, 1964; Bhattacharya, 2003). Temperature dependence of the crystal lattice parameters of each phase is de-scribed with the help of a corresponding thermal expansion tensor. Stress hysteresis should be controlled, and could bechosen in particular, to be constant or weakly temperature-dependent.

(b)

PT criteria, which follow from the crystal lattice instability conditions, should have a desired form in terms of the stresstensor and should not involve zero elastic moduli.

(c)

All material properties of A and M could be included in the potential. The theory in Levitas and Preston (2002a,b) and Levitas et al. (2003) that satisfies these conditions has been generalized

for large strains in Levitas and Preston (2005), Levitas et al. (2009), and Levitas (2013b). It was applied for finite elementsolutions of various static (Levitas and Lee, 2007; Levitas et al., 2010) and dynamic (Idesman et al., 2008; Cho et al., 2012)problems at small strains, as well as at large strains in Levitas et al. (2009) and Levin et al. (2013).

PFA to dislocations is based on the ideas similar to those for martensitic PT and is broadly used for simulation of plasticflow (Wang et al., 2001a,b; Wang and Li, 2010; Hu and Chen, 2001, 2002; Jin and Khachaturyan, 2001; Kundin et al., 2011a;Koslowski et al., 2002; Hu et al., 2004; Hunter et al., 2010, 2011; Lei and Koslowski, 2011). The main idea comes from the PFAto martensitic PTs (Jin et al., 2001a; Wang and Khachaturyan, 1997). Transformation strain in this case is substituted byplastic strain, which represents combination of simple shears along slip systems, characterized by the Burgers vector andnormal to the slip planes. Order parameters for each slip system are related to the magnitude of the Burgers vector. Smallstrain formulation and linear elasticity are used in all theories. Spectral methods were utilized for numerical solution ofGinzburg–Landau equations for the order parameters, and the mechanical problem was solved using the Khachaturyanmicroelasticity theory (Wang et al. 2001a–c; Hu and Chen, 2001, 2002; Rodney et al., 2003). However, these theories sufferthe drawbacks, which are similar to those for PFA for martensitic PTs, and some additional ones. Some of these drawbackshave been eliminated in Levitas et al. (2003) and Levitas and Javanbakht (2012, 2015). Thus, additional thermodynamicequilibrium and lattice instability conditions for homogeneous states were suggested and the Helmholtz free energy andorder parameter dependence of the Burgers vector are found that satisfy these conditions. They allow us to obtain a stress-independent magnitude of the thermodynamically equilibrium Burgers vector and to avoid artificial dissipation duringelastic deformation, as well as to reproduce desired lattice instability conditions and a resolved shear stress – orderparameter dependence, in contrast to previous theories. Large strain kinematics is employed based on the multiplicativedecomposition of the deformation gradient into elastic and plastic parts. Instead of expression for plastic strain in terms oforder parameters in the previous theories, the relationship between the plastic part of the velocity gradient and the rate ofthe order parameters is formulated, which is consistent with traditional crystal plasticity. Instead of dislocation heightdetermined by computational mesh in the previous theories, dislocation height in Levitas and Javanbakht (2012, 2015) wasdetermined by equations. This theory is presented in a framework of nonlinear continuum thermodynamics.

There are several simplified PFA approaches to study the interaction between PT and dislocations. There are a number of

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 291

analytical treatments of M nucleation on dislocations based on PFA to PT (Korzhenevskii et al., 2003), followed by numerical(Reid et al., 1998; Wang and Khachaturyan, 2006) simulations. Dislocations are introduced through their stationary stressfield or are located at the moving phase interface only (Kundin et al., 2011a) and therefore do not require additional PFAequations. PFA for martensitic PT with continuum dislocation theory is presented in Kundin et al. (2011b), but we areinterested in discrete dislocations. Both approaches (Kundin et al., 2011a,b) have all drawbacks of the approaches to PT anddislocations mentioned above. Also, there are works (Malik et al., 2012a; Yeddu et al., 2012; Cottura et al., 2012; Yamanakaet al., 2008) in which PFA to martensitic PTs is coupled with classical isotropic plasticity. Note that there is a conceptualproblem in coupling PFAwith continuum dislocation theory and even bigger problemwith phenomenological plasticity. Thewidth of phase interface is on the order of 1 nm, which defines characteristic scale of the problem. Mean distance betweendislocations ld d

0.5ρ= − , where ρd is the dislocation density. For annealed materials 10 md10 2ρ = − , which leads to l 10 md

5= − .Even for heavily deformed materials 10 md

15 2ρ = − , which leads to l 32 nmd = . One has to have at least few dislocations in therepresentative volume to justify continuum description, which makes continuum theory of dislocations applicable at thescale two to five orders of magnitude larger than the interface width only. Thus, conceptually, nanoscale PFA to PT iscompatible with discrete dislocations only. Of course, any reasonable way to relax large elastic stresses is better than pureelastic formulation; but this cannot be considered as a strict theory.

In Levitas and Javanbakht (2012), solution to some problems on interactions between martensitic PT and dislocations wasobtained using a simplified version of PFA without presenting detailed equations. In Levitas and Javanbakht (2013, 2014) asimplified system of equations for the interaction between a single martensitic variant and dislocation evolution waspresented but without any derivations and justification. A number of important problems have been solved in Levitas andJavanbakht (2012, 2013, 2014), which include revealing scale-dependent athermal hysteresis for the semicoherent interfacemotion, pushing and inheriting dislocations by a moving interface, the generation of dislocations by a growing martensiticplate (including its arrest), and an order of magnitude reduction in PT pressure due to dislocations generated by appliedshear stresses.

Thus, general thermodynamically consistent PFA for interaction between PT and dislocations is still missing. In the paper,we attempt to develop such a theory as a combination of the most advanced (and the only available large strain) PFAs formultivariant martensitic PTs (Levitas, 2013b) and dislocation evolutions (Levitas and Javanbakht (2012, 2015)) with non-trivial couplings. Coupling between PT and dislocations includes multiplicative kinematic decomposition, dependence of allmaterial parameters for dislocations on the order parameters that describe PT (which in turn produces additional con-tributions to the driving force for PT due to the variation of the dislocations energy in the course of the PT), an inheritance ofdislocation during PT and their further evolution along the nontraditional slip systems. The main interaction between PT anddislocations occurs through their stress fields and is determined by a solution of the mechanical problem.

The paper is organized as follows. In Section 2, the key finding is the justification of the multiplicative decomposition ofthe deformation gradient into elastic, transformational, and plastic parts, in which the plastic part includes four mechan-isms: dislocation motion in M along slip systems of M and slip systems of A inherited during PT and dislocation motion in Aalong slip systems of A and slip systems of M inherited during reverse PT. Transformation rules for the parameters of the slipsystems during PT are presented. It is found that the definition of the plastic part of the velocity gradient (a) in M in terms ofcombination of shear rates along the slip system of M and (b) in M transformed back to A in terms of combination of shearrates along the transformed back to A slip system of M is the same. That means that one can describe plastic deformation ofM in the crystal lattice of A in the same way like in the crystal lattice of M, provided that crystallographic parameters of slipsystems of M transformed back to the A. This also allowed us to combine plastic deformation of A and M in a single plasticdeformation gradient. In Section 3, the global first and second laws of thermodynamics are applied to derive the localdissipative inequalities. Generalized forces related to gradients of the order parameters are introduced at the surface. InSection 4, explicit expressions for these forces, as well as for the stress tensor, entropy, and driving forces to change orderparameters related to PTs and dislocations have been derived. The structure of the Ginzburg–Landau equations is derived inSection 5. Main conditions for thermodynamic potentials, which include thermodynamic equilibrium conditions and latticeinstability conditions for homogeneous states are presented in Section 6. Explicit expressions for the gradient energies for PTand dislocations are analyzed in Section 7. In Section 8, an explicit expression for the local Helmholtz free energy is pre-sented. Boundary conditions for the order parameters for PTs and dislocations which include the variation of the surfaceenergy during PT and exit of dislocations are given in Section 9. Explicit expressions for Ginzburg–Landau equations for thechosen thermodynamic potential are presented in Section 10. Section 11 contains concluding remarks and future directions.The total system of equations for large and small strains are summarized in Appendix A. Application of the developed theoryto numerical simulations of the some important problems is presented in the accompanied paper (Javanbakht and Levitas,2015).

We designate vectors and tensors with boldface symbols and designate contractions of tensors A Aij= { } and B Bji= { }over one and two indices as A B A Bij jk⋅ = { } and A B A B: ij ji= .

2. Kinematics

As usual, the motion of the material is described by a continuous vector function r r r t,0= ( ), where r0 and r are thepositions of points in the reference (undeformed) 0Ω and the actual (deformed) Ω configurations, respectively; t is the time.

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319292

Material in the reference configuration at time t0 is assumed to be a virgin (i.e., plastically undeformed) and stress-freeaustenitic phase. The deformation gradient is defined as F r e er

rr

ri j

ij0 0

= = ∇ = ⊗∂∂

∂

∂, with ri and r j

0 for the components of theposition vectors r and r0 in the Cartesian unit basis vectors ei.

2.1. Kinematics of martensitic phase transformations

The multiplicative decomposition of the deformation gradient

F F U , 1e t= ⋅ ( )

into elastic and transformational parts will be utilized. Without loss of generality,Ut is considered to be symmetric (rotation-free) tensors and rotation is included in Fe. After local release of stresses, elastic deformation disappears (i.e., F Ie = ) and anunloaded configuration, characterized by Ut , is designated as tΩ . We assume that the reverse transformation does not occurduring an unloading; if this is not the case (for example, for pseudoelastic regime), unloading is considered as the thoughtexperiment under fixed phase state.

Let us describe the meaning of the transformation deformation gradient Ut and corresponding transformation strainU It tε = − . For each martensitic variant Mi, the transformation–deformation gradient Uti transforms the crystal lattice of an

A into a lattice of a Mi, i n1, 2, ,= … , similar to the case in the crystallographic theory (Wayman, 1964). For an A Mi→transformation, the order parameter ηi is unambiguously related to the transformation strain through some monotonousfunction iφ η( ). Order parameter ηi varies between 0 for A (U It = ) and 1 for Mi (U Ut ti= ). For multivariant transformations, thetransformation deformation gradient Ut is some combination of all Uti and ηi. Since Uti are constants, U Ut t iη= ( ). We willdesignate the set of the arbitrary order parameters as , , , ,i n1η η η η˜ = ( … … ), with 0, , 00η = ( … ) for A and

0, , 1, , 0i iη η^ = ( … = … ) for Mi, and with 0, , , , 0i iη η¯ = ( … … ) for one nonzero parameter only. The requirement stating that inthermodynamic equilibrium 0iη = i∀ for A and 1iη = and 0kη = k i∀ ≠ for Mi for any temperature θ and stress tensor (Levitasand Preston, 2002a,b; Levitas, 2013b and Section 6) imposes the following conditions on function Ut iη( ):

U UU

j n i n; 0; 0, 1, 2, , ; 1, 2, , .2

t j tjt j

i

ηη

η(^) =

∂ (^)

∂= = … = …

( )

There are some additional limitations (Levitas and Preston, 2002a,b; Levitas, 2013b) related to obtaining the desired PTcriteria from thermodynamic instability conditions, including k i0

Ut j

i k

2

= ∀ ≠η

η η

∂ (^ )

∂ ∂– i.e., all mixed second derivatives of Ut

vanish for any equilibrium phase jη . The most popular expression for Ut , which satisfies all conditions is (Levitas, 2013b)

U I I L L

L

a

a a a a

, ;

3 3 ; , 1 4 3 ; 0 6. 3

t tk

n

tk ki

n

j i

n

i j i ij j ji

ji ti tj k k k k k

1 1

1

1

2 2

2 2 3 4

∑ ∑ ∑ε ε

ε ε

φ η η η η η

φ η η η η η

= + = + ( ) − ( + )

= ( − ) + ( ) = ( − ) + ( − ) < < ( )

= =

−

= +

For a single martensitic variant, Eqs. (3) simplify to U I a,t t1 1ε φ η= + ( ).

2.2. Kinematics of plastic deformation due to dislocations

Multiplicative decomposition of the deformation gradient

F F F , 4e p= ⋅ ( )

into elastic Fe and plastic Fp parts is used. An unloaded stress-free configuration characterized by plastic deformation gra-dient Fp, is designated as pΩ . Lattice rotation is included in Fe; Fp does not change an orientation of the crystal lattice andrepresents a combination of lattice-invariant shears. For a single slip at the slip plane with the unit normal nα in the slipdirection with the Burgers vector bα, Fp represents a simple shear

F I b n I m nH1

, 5p Φ ξ γ Φ ξ= + ⊗ ( ) = + ⊗ ( ) ( )αα α

α αα α

α

where b H/γ = | |αα α is the plastic shear strain per single dislocation in a dislocation band of the height Hα, mα is the unit vector

in the direction of bα, ξα is the order parameter for a dislocation in a slip system m n,( )α α , which varies between n�1 and nwhen n�1 complete dislocations exist and the nth dislocation appears. All parameters (nα, bα, H , …α ) and the gradientoperator ∇ are determined in the undeformed configuration. Function Φ satisfies the following conditions (see Levitas andJavanbakht (2012, 2015) and Section 6)

n nd n

dp; 0; 1, 2, , .

6Φ ξ

Φ ξξ

α( = ) =( = )

= = …( )α α α

α α

α

These conditions guarantee that the integer values of the order parameter nξ =α α for any slip system α satisfy the ther-modynamic equilibrium conditions for any stresses. The only existing expression for Φ, which satisfies conditions (6) and

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 293

lead to the desired lattice instability condition is (Levitas et al., 2003; Levitas and Javanbakht, 2012)

Int ; 3 2 . 72Φ ϕ ξ ξ ϕ ξ ξ ξ= ( ¯ ) + ( ) ( ¯ ) = ¯ ( − ¯ ) ( )α α α α α

It is expressed in terms of the integer part Int ξ( )α and the fractional part Intξ ξ ξ¯ ≔ − ( )α α α parts of ξα. It is evident that3,ϕ ξ φ ξ( ) = ( ). Thus, the order parameter is unambiguously connected to the magnitude of the plastic shear γ Φ ξ( )α α or the

Burgers vector b Φ ξ( )αα in the transitional state between n�1 and n dislocations: when ξα varies between n�1 and n, the

Burgers vector and plastic shear vary between their values for n�1 and n dislocations.For Eq. (5) the plastic velocity gradient is defined as

l F F b n m n FH1

, 8p p p p1 Φ ξ γ Φ ξ≔ · = ⊗ ( ) = ⊗ ( ) =

( )αα α

α αα α

α−

were we used F I b np H1 1 Φ ξ= − ⊗ ( )α α

α−

α and b n 0· =α α . For multiple slip systems, an additivity of the plastic velocity gradientsis usually assumed in crystal plasticity (Lubarda, 2002) and will be accepted here

l F F l b n m nH1

.9

p p p

p

p

p p1

1 1 1

∑ ∑ ∑Φ ξ γ Φ ξ≔ · ≔ = ⊗ ( ) = ⊗ ( )( )α

α

αα

α αα

αα

α αα

−

= = =

It is explicitly assumed in Eq. (9) that there is no interaction between different slip systems and partial dislocations and thefunction Φ is the same for each slip system. Introducing the Jacobian determinant FJ detp p≔ , the rate of volumetric plasticstrain is F F I lJ J J: : 0p p p p p p

1 = = =− for any values of the order parameters because m n 0· =α α . Thus, in contrast to the PFA formartensitic PTs and twinning at large strains (Levitas and Preston, 2005; Levitas et al., 2009; Clayton and Knap, 2011a) inwhich the finite expression (3) for the transformation strain in terms of the order parameters were assumed, the differentialequation (9) for Fp is formulated for dislocations. Eq. (9) is valid for A and Mi and all crystallographic parameters will bemarked with sub- or superscripts A and M, respectively: bA

α, nAα, Aγα , and HA

α for austenite and bMω , nM

ω , Mγω , and HMω for

martensitic variants. Since all martensitic variants are crystallographically equivalent, they have the same slip systems.

2.3. Combined kinematics of phase transformations and dislocations

The description of plasticity in the presence of martensitic PT is quite sophisticated due to the necessity of includingproper physics and satisfying certain conditions. In particular, dislocations in A are inherited by M during martensitic PT anddislocations in M are inherited by A during reverse PT. An experimentalist can obtain a sample of M with dislocations of Mand inherited dislocations of A, and further plastic deformation of M can be described without the knowledge of the initialundeformed state of austenite. Similarly, the description of plastic deformation of a sample of A transformed back fromplastically deformed M should be possible without the knowledge of preliminary plastic deformation in M. This should bealso true for multiple direct and reverse PTs. Note that in bulk (i.e., outside the interfaces), the material is either in A or in Mi

states. Thus, for material points in the M state, plastic strain in A and PT does not occur. Similarly, in an austenitic state,plastic strain in M and PT does not take place. It looks logical to split plastic deformation in A and M multiplicatively, but aswe will show below, it causes some problems. In principle, plastic deformation in any phase does not affect the crystallattice. That is why we will pursue a way to decompose the deformation gradient F F Fcl p= ⋅ into the deformation of thecrystal lattice and the deformation due to dislocations without changes in the crystal lattice. Deformation of the crystallattice F F Ucl e t= ⋅ includes elastic and transformational parts. However, the incorporation of plastic deformation within twodifferent lattices, A and M, is not trivial. Thus, our initial objective will be to demonstrate that the kinematics of plasticdeformation of M can be naturally performed within the crystal lattice of A. Then, the plastic deformation of both phases canbe described within the crystal lattice of A which resolves the problem of kinematic decomposition.

2.3.1. Parameters of the transformed slip systemsIf PT is not included, plastic deformation gradients in A, Fp

A, and in Mi, FpM , represent a combination of simple shears under

a fixed orientation of the crystal lattice of A and Mi, respectively, and are described by equations similar to those in Section2.2. However, with PT, there is an important difference related to the inheritance of dislocations of A during PT to Mi into thecrystal lattice of Mi and the inheritance of dislocations of Mi during reverse PT to A or transformation to Mj. In addition toslip systems of A, slip of dislocations along the slip systems of Mi transformed back to A is possible as well. Since thedislocations in Mi that are transformed back to A generally do not belong to the favorable slip system of A, they may bearrested or evolve (including appearance of new dislocations) under higher resolving shear stresses. Similarly in Mi, slip ofdislocations along the slip systems of A transformed to Mi is possible. While we could not find in known experiments(Brainin et al., 1981; Lovey and Torra, 1999; Lovey et al., 2004) direct proving that inherited dislocations can further evolvealong the unfavorable slip systems, our theory will take this into account, which hopefully will encourage correspondingexperiments. Possibility of inherited dislocations to evolve can be controlled by the ratio of the yield strength and kineticcoefficients along the favorable and unfavorable slip systems and their mutual orientation. Also, unfavorable dislocationsmay dissociate into partial dislocations along favorable slip systems, but we do not consider dislocation reactions here.While we consider general 3D theory, it is more vivid to illustrate inherited dislocations with simple 2D schematics for theparameters of the slip systems in A and M, including inherited slip systems in configurations Ω and Ωt, see Fig. 1. Examples

Fig. 1. Schematics for Burgers vectors and normals to the slip planes in austenite and martensite in different configurations. (a) 2D f.c.c. lattice of austenitewith two slip systems (characterized by bA

α and nAα) along the faces in the reference configuration Ω0. (b) 2D b.c.c. lattice of martensite with two slip systems

(characterized by bMω , nM

ω ) along the diagonals in the transformed configuration Ωt. Slip systems of austenite inherited by martensite (b U bAM t A= ·α α ,n n U n U/AM A t A t

1 1= · | · |α α α− − ) are also included. (c) Slip systems of martensite inherited by austenite (b U bMA t M1= ·ω ω− , n n U n U/MA M t M t= · | · |ω ω ω ) during reverse phase

transformations are included in the reference configuration Ω0.

Fig. 2. Schematics of the ideal crystal lattice of austenite (a) and crystal lattice of austenite shifted by b2 A2 in the horizontal plane and by bA

1 in the verticalplane (b); schematics of the ideal crystal lattice of martensite (c) and crystal lattice of martensite shifted by b2 M

1 (d).

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319294

of plastic deformation of crystal lattices are presented in Figs. 2 and 3.Parameters of the slip systems of A inherited by M are determined by the equations

b U b n n U n U

n U b U b n UH H H H

; / ;

/ ; / / . 10

AM t A AM A t A t

AM A A tAM

AM AM t A A t A

1 1

1 1γ

= · = · | · |

= | · | = | | = | · || · | ( )

α α α α α

α α αα

α α α α α

− −

− −

Thus, the Burgers vector transforms as a material vector embedded into the crystal lattice. Vector nAMα is defined as a unit

normal to the slip plane of A transformed to M and determined utilizing Nanson's relationship (Lubarda, 2002) betweenoriented area in the deformed n dS and undeformed n dS0 0 states with unit normals n and n0, respectively:

n F n n F n n F n FdS J dS dS dS J; / ; / . 11t10 0 0

10

10

1= · = | · | = · | · | ( )− − − −

Since Ut depends on the order parameters ηi, vectors b and n vary smoothly across phase interface from bAα and nA

α to bAMα and

nAMα , respectively. Distance between slip planes of A after transformation to M, HAM

α , is determined as a projection of thetransformed vector nHA A

α α, i.e., n UHA A t·α α , into the unit normal to the transformed plane nAMα

n U n n U U n n U n UH H H H/ / . 12AM A A t AM A A t t A A t A A t1 1 1= · · = · · · | · | = | · | ( )α α α α α α α α α α− − −

Similarly, parameters of the slip systems of M inherited by A during the reverse PT are determined by the same Eq. (10) afterexchanging A and M, as well as Ut with Ut

1−

Fig. 3. (a) Schematics of the crystal lattice of austenite from Fig. 2b shifted by b2 M1 along the slip system of martensite (in austenitic state or in martensitic

state and transformed back to the austenite); (b) schematics of the crystal lattice of martensite from Fig. 2d shifted by bA1 along the slip system of austenite

(in austenitic state and transformed to the martensite or in martensitic state).

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 295

b U b n n U n U

n U b U b n UH H H H

; / ;

/ ; / / . 13

MA t M MA M t M t

MA M M tMA

MA MA t M M t M

1

1γ

= · = · | · |

= | · | = | | = | · || · | ( )

ω ω ω ω ω

ω ω ωω

ω ω ω ω ω

−

−

We will also consider an important particular case when the slip systems of martensite (bMω , nM

ω ) coincide with the slipsystems of A transformed after martensitic PT (b U bAM t A= ·α α, n n U n U/AM A t M t

1 1= · | · |α α α− − ), i.e., b U bM t A= ·α α and n n U n U/M A t M t1 1= · | · |α α α− −

(we do not need to distinguish α and ω in this case). In this case, slip systems of M inherited by A (bMAα , nMA

α ) during reverse PTin the reference configuration Ω0 coincide with the slip systems of A in A (bA

α, nAα). This is always the case for slip systems of

different martensitic variants, which transform to each other during M Mi j↔ transformations (twinning in M).

2.3.2. Plastic deformation of martensite treated in crystal lattice of austeniteIn Appendixes A.1 and A.2, we discussed treatment of plastic deformation of M in crystal lattice of A and four-term

multiplicative decomposition, which as we found has essential drawbacks. To eliminate these problems, the followingmultiplicative decomposition will be used

F F U F . 14e t pMA= ⋅ ⋅ ( )

After A transforms to M and then it deforms plastically, stresses are reduced to zero and crystal lattice of M is returned to itsinitial orientation, determined by applying Ut to the lattice of the A. Then elastic deformation gradient disappears, i.e., F Ie = .After this, reverse PT (actual or in thought experiment) is caused by change in temperature, i.e., U It = and plastic de-formation remains only. Thus, Fp

MA represents plastic deformation gradient of M measured after reverse PT to A. For thedecomposition (14), one obtains

F F U F F U F F U F F F U F

l F F F F F U U F F U F F U F

; ;

. 15

e t pMA

e t pMA

e t pMA

pMA

t e

e e e t t e e t pMA

pMA

t e

1 1 1 1

1 1 1 1 1 1 1

= · ⋅ + · ⋅ + ⋅ ⋅ = · ·

≔ · = · + · · · + ⋅ ⋅ · · · ( )

− − − −

− − − − − − −

Plastic part of the velocity gradient in Eq. (15) should be the same as for physically correct definition in Eq. (97), i.e.,

l F F U F F U b nH1

.16

pM pM

pM

t pMA

pMA

t

m

MM M

M1 1 1

1

∑ Φ ξ≔ · = ⋅ · · = ⊗ ( )( )ω

ωω ω

ω− − −

=

Then from Eq. (16) and utilizing Eq. (13) we obtain

F F U b n U b nH H1 1

.17

pMA

pMA

m

Mt M M t

Mm

MAMA MA

M1

1

1

1

∑ ∑Φ ξ Φ ξ · = ⋅ ⊗ · ( ) = ⊗ ( )( )ω

ωω ω

ωω

ωω ω

ω−

=

−

=

Eq. (17) represents the key new kinematic result, which will allow us to develop noncontradictory kinematics of coupled PT

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319296

and dislocations: definition of plastic part of the velocity gradient (a) in M in terms of combination of shear rates along theslip systems of M and (b) in M transformed back to A in terms of combination of shear rates along the transformed back to Aslip systems of M is the same.

That means that one can use multiplicative decomposition (14) and describe plastic deformation of M in the crystallattice of A in the same way like in the crystal lattice of M, if crystallographic parameters of slip systems of M transformedback to the A. Substituting Eq. (17) into Eq. (15) one obtains

l F F F F F U U F F U b n U F

F F F U U F F b n F

H

H

1

1.

18

e e e t t e e t

m

MAMA MA

Mt e

e e e t t e e

m

MM M

Me

1 1 1 1

1

1 1

1 1 1

1

1

∑

∑

Φ ξ

Φ ξ

≔ · = · + · · · + ⋅ ⋅ ⊗ ( )· ·

= · + · · · + · ⊗ ( )·( )

ωω

ω ωω

ωω

ω ωω

− − − −

=

− −

− − −

=

−

Thus, with multiplicative decomposition (14), plastic part of the velocity gradient can be equivalently presented in terms ofboth crystallographic parameters of slip system of M or the same crystallographic parameters but transformed back to the A.

2.3.3. Optimal kinematic decomposition for the general caseFor the general case of multiple direct-reverse PTs and plastic deformation both in A and M, the following multiplicative

decomposition is postulated:

F F U F . 19e t p= ⋅ ⋅ ( )

Geometric interpretation of this decomposition is similar to that for decomposition (14). After reaching current state Ω by anthermomechanical loading, stresses are reduced to zero and crystal lattice of M is returned to its initial orientation. Thiseliminates (and defines) the elastic deformation gradient Fe and keeps transformational and plastic deformation gradients,producing intermediate configuration Ωt. After this, reverse PT (actual or in thought experiment) produced by change intemperature returns crystal lattice of M to that of A and leads to disappearance of Ut , forming intermediate configuration Ωp.Remaining plastic deformation gradient Fp includes both plastic deformation gradient of M (measured after reverse PT to A)and of A. Thus, the obtained result Eq. (17) about straightforward possibility of the description of plastic deformation of Mutilizing crystal lattice of A allows us in the general case to describe plastic deformation of both M and A in the crystal latticeof A with the help of single plastic deformation gradient Fp, formulate for plastic velocity gradient kinematic equation of thetype of Eq. (17), and keep terms responsible for deformation of crystal lattice, F Ue t⋅ , together. Eq. (19) results in

F F U F F U F F U F F F U F; ; 20e t p e t p e t p p t e1 1 1 1 = · ⋅ + · ⋅ + ⋅ ⋅ = · · ( )− − − −

l F F F F F U U F F U F F U F . 21e e e t t e e t p p t e1 1 1 1 1 1 1≔ · = · + · · · + ⋅ ⋅ · · · ( )− − − − − − −

d l F F F F F U U F F U F F U F . 22s s e e s e t t e s e t p p t e s1 1 1 1 1 1 1≔( ) = ( · ) = ( · ) + ( · · · ) + ( ⋅ ⋅ · · · ) ( )− − − − − − −

Plastic part of the velocity gradient, l F Fp p p1≔ · − , consists of four different contributions: plastic deformation in A along slip

systems of austenite (bAα, nA

α) and inherited slip systems of martensite (bMAω , nMA

ω ) and plastic deformation in M along slipsystems of martensite (bM

ω , nMω ) and inherited slip systems of austenite (bAM

α , nAMα ). However, when we develop theory in the

reference configuration, last two slip systems should be transformed back to A according to Eqs. (10) and (13). Since slipsystem bAM

α and nAMα is a transformed slip system bA

α and nAα from A to M with the help of tensorUt , its reverse transformation

from M to A with the help of inverse tensor Ut1− brings it back to bA

α and nAα. Similar, bM

ω and nMω transform to bMA

ω and nMAω ,

according to definition (13) of bMAω and nMA

ω . Thus, in the reference configuration only two slip systems in A are required todescribe all four contributions to lp: slip systems of A (bA

α, nAα) and inherited slip systems of M (bMA

ω , nMAω ). They are shown in

Fig. 1c. This is one more advantage of the justified kinematic decomposition (19). Utilizing Eq. (17), we obtain

l F F b n

b n

H

H

1

1,

23

p p p

p

AA A

AA AM

m

MAMA MA

MM MA

1

1

1

∑

∑

Φ ξ Φ ξ

Φ ξ Φ ξ

≔ · ≔ ⊗ ( ( ) + ( ))

+ ⊗ ( ( ) + ( ))( )

αα

α αα α

ωω

ω ωω ω

−

=

=

where the order parameter AAξα characterizes slip in A along slip systems of austenite, AMξα describes slip in M along slipsystems of A, MMξω characterizes slip in M along slip systems of M, and MAξω depicts slip in A along slip systems of M.

In fact, two sets of the order parameters, AAξα and AMξα can be substituted with the single set of the order parameters Aξα ,which characterizes slip along slip systems of A, independent whether it occurs in A or M. Change in material propertiesbetween A and M will be described with the help of the order parameters ηi for PTs. Similar, two order parameters, MMξω and

MAξω can be substituted with the single order parameters Mξω , which characterizes slip along slip systems of M, independentwhether it occurs in A or M. Then Eq. (23) reduces to two terms

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 297

l b n b n m n m nH H1 1

.24

p

p

AA A

Am

MAMA MA

Mp

AA A

Am

MAMA MA

M

1 1 1 1

∑ ∑ ∑ ∑Φ ξ Φ ξ γ Φ ξ γ Φ ξ≔ ⊗ ( ) + ⊗ ( ) = ⊗ ( ) + ⊗ ( )( )α

αα α

αω

ωω ω

ωα

αα α

αω

ωω ω

ω= = = =

This is the final kinematic equation for plastic part of the velocity gradient for the case of coupled plastic deformation of A andM and multiple direct and reverse PTs.

The rate of volumetric plastic strain is

I F U F F U F U F F U F F F F b n b nH H

0: : :1 1

,25

e t p p t e t e e t p p p p

p

AA A

Am

MAMA MA

M1 1 1 1 1 1 1

1 1

∑ ∑Φ ξ Φ ξ⋅ ⋅ · · · = · · ⋅ ⋅ = = · ( ) + · ( ) =( )α

αα α

αω

ωω ω

ω− − − − − − −

= =

where possibility of cyclic permutation of tensors in double contraction was used and orthogonality of the Burgers vectorsand normals to the corresponding slip plane was utilized. Thus, elastic and transformational strains do not violate conditionsof plastic incompressibility.

For an important particular case when the slip systems of M inherited by austenite (bMAω , nMA

ω ) during reverse PTs coincide inthe reference configuration Ω0 with the slip systems of A in austenite (bA

α , nAα), further simplifications are possible. In this case,

the only slip systems necessary for the kinematic description in Eq. (24) are slip systems of A in A in Ω0, i.e., (bAα, nA

α). Also,two sets of the order parameters, Aξα and Mξω can be substituted with the single set of the order parameters ξα, whichcharacterizes slip along any slip systems of A andM, independent whether it occurs in A or M. Change in material propertiesbetween A and M will be described with the help of the order parameters ηi for PTs. Then Eq. (24) simplifies to

l b nH1

.26

p

p

AA A

1

∑ Φ ξ≔ ⊗ ( )( )α

αα α

α=

Thus, novelty of the current approach is related to (a) justification of Eq. (17), which allows us to treat plastic deformation ofM in the crystal lattice of A; (b) justification of kinematic decomposition (19) (in contrast to e.g., four term decomposition(98)), which was done by demonstrating some drawbacks/contradictions in alternative approaches; (c) utilization ofmultiplicative decomposition (19) along with rules for change of the parameters of the slip system during direct and reversePTs (10) and (13) to derive the final decomposition (24) for the plastic velocity gradient with just two (rather than four)types of slip systems and order parameters, and (d) its further simplification equation (26).

2.3.4. Small strain and rotation approximationFor geometrically linear approximation

F I F I U I F I; ; ; , 27e e e t t t p p pε ω ε ω ε ω ε ω≃ + + ≃ + + ≃ + + ≃ + + ( )

where ε is the symmetric small strain tensor and its elastic, transformational, and plastic contributions, and ω is the skew-symmetric small rotation tensor and its elastic, transformation, and plastic parts. All components of ε and ω are small incomparison with unity. Then

F I I I I

; . 28

e e t t p p e e t t p p

e t p e t p

ε ω ε ω ε ω ε ω ε ω ε ω

ε ε ε ε ω ω ω ω

= ( + + )·( + + )·( + + ) ≃ + + + + + + →

= + + = + + ( )

Decomposition of the velocity gradient (21) simplifies to

; . 29e t p e t pε ε ε ε ω ω ω ω = + + = + + ( )

Also, since lp p pε ω≃ + , Eq. (24) can be integrated\:

b n b nH H1 1

.30

p p

p

AA A

Am

MAMA MA

M

1 1

∑ ∑ε ω Φ ξ Φ ξ+ = ⊗ ( ) + ⊗ ( )( )α

αα α

αω

ωω ω

ω= =

3. Application of laws of thermodynamics

We will consider an arbitrary volume V0 of material bounded by a surface S0 with unit external normal n0 in the un-deformed configuration Ω0. This volume can be cut from any actual body, thus S0 is not an external surface and surfaceenergy will not be included in the thermodynamics laws. The first and second laws of thermodynamics for the volume V0

can be presented in the form of the global energy balance equation

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319298

p r h n n Q n

f r v v

dS dS dS

r dVddt

U dV0.531

S S S i i

V V

0 0 0 0 0 0

0 0 0 0

0 0 0

0 0

∫ ∫ ∫

∫ ∫

ξ η

ρ ρ

Θ( ⋅ − ⋅ ) + ⋅ + ⋅

+ ( ⋅ + ) = ( + ⋅ )( )

α α

and combination of the global entropy balance equation with the Clausius–Duhem inequality

hn

ddt

s dVr

dV dS 0.32V V S

0 0 0 00 0 0

∫ ∫ ∫ρ ρθ θ

− + ⋅ ≥( )

Here n0 is the unit outer normal to S0, p P n0= · is the traction vector, P is the first nonsymmetric Piola–Kirchhoff stresstensor, U is the specific (per unit mass) internal energy, 0ρ is the mass density, s is the specific entropy, and 0θ ≥ is thetemperature. To make Eq. (32) applicable for an arbitrary volume, extra generalized thermodynamic forces n0Θ ⋅α conjugateto ξα and Q ni 0⋅ conjugate to iη are included at the surface in order to balance terms due to dependence of the thermodynamicpotential on ξ∇ α and iη∇ , respectively. Designation ξα includes both Aξα and Mξω and summation is assumed over the repeatedindices. Performing all the same transformations like in Levitas (2013b) but with Θα and utilizing the specific Helmholtz freeenergy U sψ θ= − , we obtain for the dissipation rate:

P F Qs: 0. 33T

i i0 0 0ρ ρ ψ ρ θ ξ η∇ Θ ∇= − − + ·( ) + ·( ) ≥ ( )α α

4. Structure of the constitutive equations

Assumption that F , , , , ,e i iψ ψ ξ ξ θ η η∇ ∇= ¯ ( )α α with traditional objectivity arguments leads to E , , , , ,e i iψ ψ ξ ξ θ η η∇ ∇= ( )α α ,where E F F I0.5e e

Te= ( · − ) is the elastic Lagrangian strain tensor. Transposing Eq. (20) for F , using

Q Q Q; , 34i i i i i iξ ξ ξ η η η∇ Θ ∇ Θ Θ ∇ ∇ ∇ ∇·( ) = ( · ) + · ·( ) = ( · ) + · ( )α α α α α α

ξ ξ∇ ∇ = α α and i iη η∇ ∇ = , and substituting these equations in Eq. (33), one evaluates

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

P F UF

F F PF

U

F P FU

Q

Q

D

s

: :

:

0.35

pT

te

eT

eT p

T

t

eT

pT t

ii

ii

ii

i

0 0 0

0 0 0

0

ρ ρ ψξ

ρ ψξ

ξ

ρ ψθ

θη

ρ ψη

η ρ ψξ

ξ

ρ ψη

η

∇ Θ

∇ Θ∇

∇

∇∇

= ⋅ ⋅ − ∂∂

+ ⋅∂∂

⋅ + · − ∂∂

− + ∂∂

+ ⋅ ⋅∂∂

+ · − ∂∂

+ − ∂∂

·

+ − ∂∂

· ≥( )

αα

αα

αα

α

Traditional assumption that the dissipation rate is independent of FeT , as well as of iη∇ , and ξ∇

α leads to the constitutiverelation for the Piola–Kirchhoff stress tensor (i.e., elasticity rule) and definition of the generalized thermodynamic forces Θαand Q i:

P F UF

Q; ; .36

pT

te

ii

0 0 0ρ ψ ρ ψξ

ρ ψη

Θ∇ ∇

⋅ ⋅ = ∂∂

= ∂∂

= ∂∂ ( )

αα

We can express the constitutive relationship for stresses also as

P FE

U F FE

F; ,37e

et p

Te

eeT

01 1

0σρ ψ ρ ψ= ⋅ ∂∂

⋅ ⋅ = ⋅ ∂∂

⋅( )

− −

where P FJ

T1σ = ⋅ is the true Cauchy stress and FJ det= is the Jacobian determinant. The residual dissipation inequalityreduces to

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

P FU

F

P F UF

D X X X

X

0; : ;

: ,38

i i iT

et

ip

i i

Te t

p

0 0 0 0 0 0

0 0 0

ρ ρ η ρ ξ ρη

ρ ψη

ρ ψη

ρξ

ρ ψξ

ρ ψξ

∇∇

∇∇

= + ≥ ≔ ⋅∂∂

⋅ + · ∂∂

− ∂∂

≔ ⋅ ⋅∂∂

+ · ∂∂

− ∂∂ ( )

ηαξ

αη

αξ

α α α

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 299

where Xiη and Xα

ξ are the dissipative forces conjugate to iη and ξα or the driving forces for change in ηi and ξα, respectively. Wepostulate that the dissipative processes described by generalized rates iη and ξα are thermodynamically independent (butinteract through the stress field). Then dissipative inequality (38) splits into two stronger inequalities

D X D X0; 0. 39i iη ξ= ≥ = ≥ ( )ηη

ξ αξ

α

It is straightforward to consider more general case and take into account kinetic coupling, i.e., dependence of iη on Xαξ and

dependence of ξα on Xiη. Next, we will elaborate the mechanical work term in Xα

ξ in Eq. (38) utilizing expression for Fp

ξ

∂∂ α

thatfollows from Eq. (24)

⎡⎣⎢⎢

⎤⎦⎥⎥

Fm n m n F .

40

pp

AA A A

mMA

MA MA M p1 1

∑ ∑ξ

γ Φξ

γ Φξ

∂∂

= ⊗ ∂∂

+ ⊗ ∂∂

·( )α α

αα α

α ωω

ω ω

ω= =

Then

⎡⎣⎢⎢

⎤⎦⎥⎥P F U

FP F U m n m n F

n F P F U m n F P F U m

: :

,41

Te t

p Te t

pA

A A A

mMA

MA MA M p

p

A pT

e t AA

A

m

MA pT

e t MAMA

M

pA A

A

mMA MA

M

1 1

1 1

1 1

∑ ∑

∑ ∑

∑ ∑

ξγ Φ

ξγ Φ

ξ

γ Φξ

γ Φξ

τ γ Φξ

τ γ Φξ

⋅ ⋅∂∂

= ⋅ ⋅ ⊗ ∂∂

+ ⊗ ∂∂

·

= ⋅ ⋅ ⋅ ⋅ ⋅ ∂∂

+ ⋅ ⋅ ⋅ ⋅ ⋅ ∂∂

= ∂∂

+ ∂∂ ( )

α αα

α α

α ωω

ω ω

ω

α

α αα

α ω

ω ωω

ω

αα α

α ωω ω

ω

= =

= =

= =

where

n F P F U m n F P F U mand 42A

A pT

e t AMA

MA pT

e t MAτ τ≔ ⋅ ⋅ ⋅ ⋅ ⋅ ≔ ⋅ ⋅ ⋅ ⋅ ⋅ ( )αα α

ωω ω

are the resolved shear stress for slip systems in A and M transformed back to A. Eq. (41) allows us to present the dissipativeforce for dislocation evolution in more compact and specific form

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

D X X X

X

0; ;

.43

A A M M A A AA A A

M MA MAM M M

0 0 0

0 0 0

ξ ξ ρ τ γ Φξ

ρ ψξ

ρ ψξ

ρ τ γ Φξ

ρ ψξ

ρ ψξ

∇∇

∇∇

= + ≥ ≔ ∂∂

+ · ∂∂

− ∂∂

≔ ∂∂

+ · ∂∂

− ∂∂ ( )

α α ω ω α α αα α α

ω ω ωω ω ω

Expressing P F F U FJTp t e

1 1 1 1σ σ= · = · · ·− − − − in terms of the Cauchy stress, the resolved shear stresses can be presented in thealternative form:

n U F F U m n U F F U mJ J; . 44AA t e e t A

MAMA t e e t MA

1 1 1 1σ στ τ≔ ⋅ ⋅ · · ⋅ ⋅ ≔ ⋅ ⋅ · ⋅ ⋅ ⋅ ( )αα α

ωω ω− − − −

For small strains and rotations, P σ≃ , F U F Ip t e≃ ≃ ≃ , J 1≃ , and

n m n m; . 45AA A

MAMA MAσ στ τ≔ ⋅ ⋅ ≔ ⋅ ⋅ ( )α

α αω

ω ω

5. Structure of the Ginzburg–Landau equations

To satisfy dissipative inequalities (39) and (43), generalized rates have to be functions of the work-conjugate thermo-dynamic forces, i.e.,

f X q X X q X X; , , ; , , . 46i jA A A M

iM M A M

iη ξ η ξ η = ( ) = ( ) = ( ) ( )η

α β γ ω β γ

Otherwise, one can always chose independent thermodynamic forces and rates that violate the second law of thermo-dynamics. Explicit dependence of the kinetic equations for dislocation on ηi is included to underscore that kinetics of slipdepends on whether dislocation is in the A or M. In the framework of the linear irreversible thermodynamics, Eqs. (46)simplify to

L X L X L X L X L X

L L L L L L L L

; ; ;

; ; ; . 47

j ji iA A

jA

jM M M

jM

jA

ji ijA A M M

η ξ η η ξ η η = = ( ) + ( ) = ( ) + ( )

= = = = ( )

η ηα αβ β αγ γ ω ωγ γ ωβ β

η ηαβ βα ωγ γω αγ γα

⁎ ⁎

⁎ ⁎

We assume summation over the repeated indices. Symmetry conditions in Eq. (47) follow from the Onsager reciprocalrelationships. Also, both matrices connecting thermodynamics forces and rates for PT and dislocations are positive definite.

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319300

Kinetic coefficients for PT Liiη characterize the mobility of the A Mi− interfaces and Lji

η i j∀ ≠ characterize the mobility of theM Mi j− interfaces. If all martensitic variants are equivalent, then L L 0ii = >η η and L zLji =η η i j∀ ≠ , where Lη is the kineticcoefficient characterizing the mobility of the A Mi− interfaces and z is the parameter. While each A Mi− interface isdescribed by variation of a single order parameter ηi, each M Mi j− interface is described by variation of two orderparameters ηi and ηj. That is why assumption z¼0 (which is used in practically all simulations) does not mean neglectingcross-effects; it means some specific unjustified relationship between mobilities of the M Mi j− and A Mi− interfaces.

The anisotropy of the interface mobility can be described with the help of functions kL L j= ( )η η and kz z j= ( ), wherekj

j

j=

η

η

∇

∇| |is the unit normal to the finite-width interface.

Similar simplification of the structure of kinetic coefficients based on the equivalence of some slip systems of the A andM can be performed for dislocation evolution. In addition, neglecting kinetic cross-effects for dislocations is quite reason-able, because when each dislocation is single out, the main mechanism of the interaction of dislocations is related tointeraction through local stress field, and this is explicitly taken into account by solution of the coupled system of phase fieldand mechanics equations. Thus, Eq. (47) can be simplified as

L X L L L zL i j

L X L X

; 0; ;

; , 48

j ji i ii ji

A Aj

A M Mj

M

η

ξ η ξ η

= = > = ∀ ≠

= ( ) = ( ) ( )

η η η η η η

α αβ β ω ωγ γ

where all L Aαβ and LM

ωγ are the same for equivalent slip systems. Using the expressions for the thermodynamic forces, Eqs. (38)and (43), Ginzburg–Landau equations are given below:

⎛⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞⎠⎟⎟P F

UFL X L

1:

1;

49j ji i ji

Te

t

ip

i i0 00η

ρ η ρρ ψ

ηψη

∇∇

= = ⋅∂∂

⋅ + · ∂∂

− ∂∂ ( )

η η

⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞

⎠⎟⎟

L X L

L X L

1 1;

1 1.

50

A Aj

A Aj

A AA A A

M Mj

M Mj

MA MAM M M

0 00

0 00

ξ η ηρ

τ γ Φξ ρ

ρ ψξ

ψξ

ξ η ηρ

τ γ Φξ ρ

ρ ψξ

ψξ

∇∇

∇∇

= ( ) = ( ) ∂∂

+ · ∂∂

− ∂∂

= ( ) = ( ) ∂∂

+ · ∂∂

− ∂∂ ( )

α αβ β αβ β ββ β β

ω ωγ γ ωγ γ γγ γ γ

For initially homogeneous material, ρ0 disappears from the gradient-related terms in Eqs. (49) and (50). To describedependence of the kinetic coefficients (and later all other material parameters) on jη , we introduce the “equivalent” orderparameter

⎛⎝⎜⎜

⎞⎠⎟⎟p ,

51i

n

im

m

1

1/

∑ η=( )=

where m 2≥ is an integer, which in bulk varies from 0 for austenite to 1 for any martensitic variant Mi ( 1iη = , 0jη = j i∀ ≠ ).In Levitas and Javanbakht (2010, 2015) m¼2 was used. For M Mi j− interface or triple junction, when two or three orderparameters differ from zero, we cannot prove that p does not exceed 1 without knowledge of the entire thermodynamicpotential and corresponding solution to the Ginzburg–Landau equation. If it will exceed, one can increase m. There are noany physical limitations on functions L pA ( )αβ and L pM( )ωγ (in contrast to material parameters in the free energy, see Section 6). Itis reasonable to assume that

L p L L L q p q q dq dp0 1 0 ; 0 0; 1 1; / 0 52A A A A A( ) = ( ) + ( ( ) − ( )) ( ) ( ) = ( ) = ≥ ( )αβ αβ αβ αβ

and similarly for L pA ( )αβ . Alternatively to introducing the “equivalent” order parameter p, one can apply additive relationship

L L L L q q q0 1 0 ; 1; 1.53

Ai

A A A

j

nA

jj

nA

jj

nM

j1 1 1

∑ ∑ ∑η η η η( ) = ( ) + ( ( ) − ( )) ( ) ( ) ≤ ( ) ≤( )

αβ αβ αβ αβ= = =

6. Main conditions for thermodynamic potentials

Main conditions for thermodynamic potential for PT are formulated and satisfied in Levitas and Preston (2002a,b) forsmall strains and in Levitas (2013b) at large strains. Similar conditions are formulated for dislocations (Levitas and Ja-vanbakht, 2012, 2015). They are related to thermodynamic equilibrium and instability conditions for homogeneous states,i.e., when the gradient energy is zero. Here they will be shortly summarized.

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 301

6.1. Thermodynamic equilibrium conditions

The thermodynamic equilibrium is determined by condition X X X 0iA M= = =ηβ γ in Eqs. (49) and (50) for all i, ,β γ . When

we introduced the order parameters, it was stated that 0η η˜ = ^ for the austenite, iη η˜ = ^ for martensitic variant Mi, and that forn complete dislocations passed through the given material point nξ =α . Otherwise, the transformation strain for martensiticvariant Mi would not be equal to Uti and will depend on stresses and temperature, and the Burgers vector for n dislocationswould not be equal to bn . However, it is not enough to claim these conditions verbally; they should follow from thethermodynamic equilibrium conditions.

Now, we impose the following: Condition I: For the homogeneous states, the sets of constant order parameters for theaustenite 0η η˜ = ^ , for each martensitic variant iη η˜ = ^ , and for n dislocations nAξ =β β and nMξ =γ γ should satisfy the thermo-dynamic equilibrium conditions

P FU

FE

E

X i j

Xn n

:, ,

0 , ;

, , ,0 , ;

54

iT

et j

ip

e j

i

A A AA

A

Ae j

A

0 0

0 0

ρη

ηρ

ψ θ η

η

ρ τ γΦ ξ

ξρ

ψ ξ θ η

ξα β

= ⋅∂ (^)

∂⋅ −

∂ ( ^)

∂= ∀

=∂ ( = )

∂−

∂ ( = ^)

∂= ∀

( )

η

α α αα α

α

β β

α

for any stress P , temperature θ, and corresponding Fe. The same condition is true for X Mγ . Because the terms Ut

iη∂∂

,A

Φ

ξ

∂

∂ α, and

MΦ

ξ

∂

∂ ω

in each of Eqs. (54) are independent of stress P , temperature θ, and corresponding elastic deformation gradient Fe, but thesecond terms depend on these variables, Eqs. (54) can be satisfied when each of the term is equal to zero only:

U E

E E

i j

n n n ni j

0;, ,

0, , ;

0;, , , , , ,

0 , , , .55

t j

i

e j

i

A M

Ae j

A

Me j

M

η

η

ψ θ η

η

Φξ

Φξ

ψ ξ θ η

ξ

ψ ξ θ η

ξα ω

∂ (^)

∂=

∂ ( ^)

∂= ∀

∂ ( )∂

=∂ ( )

∂=

∂ ( = ^)

∂=

∂ ( = ^)

∂= ∀

( )α

α

ω

ω

β β

α

γ γ

ω

Conditions for transformation strain Ut and function Φ have been already discussed (Eqs. (2) and (6)) and functions thatsatisfied these conditions have been already presented in Eqs. (3) and (7). Conditions for the local free energy will besatisfied in Section 8.

6.2. Lattice instability conditions

In the PFA, PT and dislocation nucleation criteria are formulated as conditions for instability of the homogeneousequilibrium state of crystal lattice under spontaneous variation of the order parameters (Levitas, 2013b; Levitas and Ja-vanbakht, 2015). We will present instability conditions under prescribed nominal forces – i.e., for prescribed Piola–Kirchoffstress tensor P; it is shown in Levitas (2013b) that the same conditions are valid under any other stress tensor. Temperatureis also fixed and will not be shown for compactness.

Definition. If a spontaneous deviation of the order parameters from the thermodynamic equilibrium values is thermo-dynamically admissible under prescribed boundary conditions (i.e., corresponding dissipation rate is positive), then theequilibrium is unstable.

Thus, one has the following instability conditions for PT and dislocations for stress P const= and elastic deformationgradient Fe that varies due to spontaneous variation in η

P F F

P F F

X

X n n

, , 0 equilibrium of phase is unstable;

, , 0 equilibrium of state is unstable. 56

i e e j i j

e e

η η η η

ξ ξ ξ

( + Δ ^ + Δ ˜) ≥ → ^

( + Δ + Δ ) ≥ → = ( )

η

αξ

β β α β β

Wewill not here distinguish dislocations in A and M. Utilizing the Taylor expansion of Xi and Xα around equilibrium values ofthe order parameters and employing that P F P FX X n, , , , 0i e j e( )η = ( ) =η

αξ

β , one obtains from Eq. (56)

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟

P F P F

F

F P F

P F P F

F

F P F

X X X

X n X n X n

, , , ,:

, ,0;

, , , ,:

, ,0.

57

P

P

i e j

ki k

i e j

eT

e

k

i e j

ki k

e

kk

e

eT

e

k

e

kk

( )η

ηη η

η

η

η

ηη η

ξξ ξ

ξ ξξ ξ

∂ ( ^)

∂ =

∂ ( ^)

∂∂∂

+∂ ^

∂ ≥

∂ ( )∂

=∂ ( )

∂∂∂

+∂ ( )

∂ ≥

( )

η η η

αξ

βα

αξ

β αξ

βα

Thus, the lattice instability takes place when X / Pi kη∂ ∂ |η and/or X / Pkξ∂ ∂ |αξ first cease to be negative definite. After transfor-

mations similar to those in Levitas (2013b) and Levitas and Javanbakht (2015) and recognizing that Fp

ξ

∂∂ α

depends on ξα for the

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319302

same αonly, one obtains

P FP F

UF

E

P FP F U

F E

X

X n n n

, , 1:

,;

, , 1:

,.

58

P

P

i e j

k

Te

t j

i kp

le j

i k

e

k

Te t

pk

le

k

0

2 2

0

2

2

2

( ) ( )η

η ρ

η

η η

ψ η

η η

ξ ρ ξδ

ψξ ξ

∂ ^

∂= ⋅

∂ (^)

∂ ∂⋅ −

∂ ^

∂ ∂

∂ ( )∂

= ⋅ ⋅∂ ( )

∂−

∂ ( )∂ ∂ ( )

η

αξ

β α

αα

β

α

Note that in these derivations we fixed jη when considered dislocations and fixed ξα when considered PT. It is natural toassume that if the instability condition for PT in Eq. (57) is met for one specific i only, the PT from the phase jη will occurtoward this iη phase. In the sharp interface, PT conditions between any two phases are independent of any other phases. It isnatural to impose the same conditions in our PFA for PT. For dislocations, transformation of complete dislocations in one slipsystem nξ =β β into complete dislocations of any other slip system α does not occur and is not important for the description.The goal of the instability conditions is to describe the transition from n dislocations nξ =α α to n 1+ dislocations n 1ξ = +α αor n�1 dislocations n 1ξ = −α α for a fixed number of dislocations for any other slip system. The lack of cross-effects isalready reflected in Eq. (9) for plastic velocity gradient and Eq. (57) for instability condition. That is why we imposeconditions that completely eliminates cross-effects in instability conditions, similar to those for PT. Without this, one cannotobtain that lattice instability occurs when resolved shear stress reaches some critical value. Thus, we postulate thefollowing:

Condition II:

P F P FXk i

X nk

, ,0 ;

, ,0 ;

59P P

i e j

k

e

k

( )η

η ξα

∂ ^

∂= ∀ ≠

∂ ( )∂

= ∀ ≠

( )

ηαξ

β

which results in

U E Ek i

nk0;

,0 and

,0 .

60F E

t j

i k

e j

i k

e

k

2 2 2

e e

( )η

η η

ψ η

η ηψ

ξ ξα

∂ (^)

∂ ∂=

∂ ^

∂ ∂= ∀ ≠

∂ ( )∂ ∂

= ∀ ≠

( )

β

α

Consequently, all mixed second derivatives of Ut and ψ with respect to order parameters disappear for all equilibrium phases

jη any number of complete dislocations, which, however, does not constrain them for intermediate values of the orderparameters. In this case, the matrices in Eq. (57) become diagonal and the instability conditions (Eq. (57)) with thederivatives from Eq. (58) simplify to non-negativeness of any of the diagonal term:

P FP F

UF

E

P F E

X

X n d n

d

n

, , 1:

,0;

, , 1 ,0.

61

P

P

i e j

i

Te

t j

ip

e j

i

e e

0

2

2

2

2

0

2

2

2

2

( )η

η ρ

η

η

ψ η

η

ξ ρτ γ

Φξ

ψ

ξ

∂ ( ^)

∂= ⋅

∂ (^)

∂⋅ −

∂ ^

∂≥

∂ ( )∂

=( )

−∂ ( )

∂≥

( )

η

αξ

β

αα α

α

α

β

α

Thus, the transformation conditions between A and Mi as well as between M j and Mi are

P FP F

UF

E

P FP F

UF

E

X

X

A M :, , 1

:,

0;

M A:, , 1

:,

0.62

P

P

ii e

i

Te

t

ip

e

i

ii e i

i

Te

t i

ip

e i

i

0

0

20

2

20

2

0

2

2

2

2

( )

( )

ηη ρ

η

η

ψ η

η

ηη ρ

η

η

ψ η

η

→∂ ( ^ )

∂= ⋅

∂ (^ )∂

⋅ −∂ ^

∂≥

→∂ ( ^)

∂= ⋅

∂ (^)∂

⋅ −∂ ^

∂≥

( )

η

η

P FP F

UF

EXM M :

, , 1:

,0.

63P

j ii e j

i

Te

t j

ip

le j

i0

2

2

2

2

η

η ρ

η

η

ψ η

η→

∂ ( ^)

∂= ⋅

∂ (^)

∂⋅ −

∂ ( ^)

∂≥

( )

η

While Condition II is not mandatory, it makes instability conditions treatable, and allows us to find explicit expressions for ψand transformation and plastic deformation gradients.

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 303

7. Elaboration of the gradient energies

7.1. Gradient energy for phase transformation

Let us specify contribution to the Helmholtz free energy related to gradient iη∇ , ψη∇, which localizes at the finite-width

interface between phases and contributes to the interface energy. For 0.5 i ij jβψ η η∇ ∇= · ·ηη∇ , where ijβ η are positive definite

second-rank tensors, Eq. (49) for PT in initially homogeneous material specifies to

⎛⎝⎜⎜

⎞⎠⎟⎟P F

UFL

1: .

64j ji

Te

t

ip

iik k

0

βηρ η

ψη

η∇ ∇ = ⋅∂∂

⋅ − ∂∂

+ ·( · )( )

η η

Wewill focus on the case when tensors ikβ η are independent of ηi and coordinates; then the gradient related terms simplify to:ik kβ η∇ ∇⊗η . Anisotropy of the interface energy can be also described with the scalar coefficients ikβ η (i.e., Iik ikβ β=η η ), which

depend on interface orientation kj. If all martensitic variants are crystallographically equivalent, then 0kkβ β= >η η (or fortensorial coefficients kkβ β=η η) characterizes contribution to the A Mk− interfaces and bkjβ β=η η (or for tensorial coefficients

kj 1β β=η η) k j∀ ≠ characterize contribution to the M Mk j− interfaces. For these cases Ginzburg–Landau equation (64) takesthe forms

⎛⎝⎜⎜

⎞⎠⎟⎟P F

UFL

1: : : ;

65j ji

Te

t

ip

ii

k k i

n

k0

11,

∑β βηρ η

ψη

η η∇ ∇ ∇ ∇ = ⋅∂∂

⋅ − ∂∂

+ ⊗ + ⊗( )

η η η

= ≠

⎛⎝⎜⎜

⎞⎠⎟⎟P F

UFL b

1: .

66j ji

Te

t

ip

ii

k k i

n

k0

2

1,

2∑ηρ η

ψη

β η β η∇ ∇ = ⋅∂∂

⋅ − ∂∂

+ +( )

η η η

= ≠

7.2. Gradient energy for dislocations

To present the gradient energy for dislocations, ψξ∇, we first decompose the gradient of the order parameters (both Aξα and

Mξ )ω

m n m n; ; , 67m n m nξ ξ ξ ξ ξ ξ ξ∇ ∇ ∇= ∇ + ∇ ∇ ≔ · ∇ ≔ · ( )α αα

αα

α αα

α αα

into two parts: one along the slip direction and another along the normal to the slip plane. The following quadratic ex-pression is justified in Levitas and Javanbakht (2012, 2015):

Z0.5 1 ,68

km m

kn n2∑ ∑ψ β η ξ ξ ξ ν η ξ ξ= ( ( )∇ ∇ + ( − ¯ ) ( )∇ ∇ )

( )ξ

α ςας

α ς α ας

α ς∇

where βας , να

ς, and Z are the coefficients that vary during PTs. Gradient energy for dislocation should localize at the dislocationcore region (in order to describe the dislocation core energy) and must be independent of the gradient nξ∇ α along thenormal to the slip plane. Indeed, slip planes represent a perfect lattice and do not possess any dislocation-related energyaway from the dislocation cores. Thus, in all previous papers (e.g., Shen and Wang, 2004; Wang et al., 2001b; Jin andKhachaturyan, 2001) ψξ

∇ depends on mξ∇ α only, which corresponds to Z¼0 in Eq. (68). However, as it was shown in Levitasand Javanbakht (2012, 2015), this eliminates a characteristic length in the directions nα and leads to an ill-posed system ofequations for ξα. This results in localization of the order parameter and plastic shear in a layer of the width of a single finiteelement and strong mesh-dependence of the solution. That it why regularizing term scaled with a small parameter Z isincluded in Eq. (68), which introduces a characteristic length in the directions nα and results in mesh-independentpropagation of a dislocation within the entire band of the prescribed height Hα. This term disappears when dislocation iscompleted ( 1ξ =α ), and does not produce artificial surface energy between shifted and initial crystal. If all slip systems of Aare equivalent and all slip systems of M are equivalent, and all cross-effects are neglected, Eq. (68) simplifies to

Z

Z

0.5 1

0.5 1 .69

Ak

pm A

AA n A

Mk

mm M

MM n M

1

2 2 2

1

2 2 2

∑

∑

ψ β η ξ ξ ξ

β η ξ ξ ξ

= ( ) ((∇ ) + ( − ¯ ) (∇ ) )

+ ( ) ((∇ ) + ( − ¯ ) (∇ ) )( )

ξ ξα

α α α

ξω

ω ω ω

∇

=

=

Note that dependence of the coefficients in ψ ξ on ηi produces corresponding contribution to the Ginzburg–Landau equations(50) for PT. Since constraints related to satisfaction of the thermodynamic equilibrium and instability conditions areformulated for homogeneous states, they do not impose any limitations on dependence of ψξ

∇ on ηi, and it can beapproximated in the same way as for kinetic coefficients in Eqs. (52) and (53). In fact, near such a strong stress heterogeneitylike dislocations, barrierless M nucleation may occur and for the small driving force an embryo is in an intermediate state,

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319304

i.e., 0 1iη≤ ≤ (Reid et al., 1998; Wang and Khachaturyan, 2006; Levitas and Javanbakht, 2013, 2014). Thus, one cannotrequire that for heterogeneous state condition for thermodynamic equilibrium Xi¼0 must contain solutions 0η η˜ = ^ and iη η˜ = ^for any stresses and temperature. If function q does not have zero derivatives at 0η η˜ = ^ and iη η˜ = ^ , then an additional (to thestress-induced) driving force for heterogeneous phase nucleation appears near dislocations in homogeneous A and M. Iffunction a, kφ η( ) from Eq. (3) or its particular case 3,k kϕ η φ η( ) = ( ) (Eq. (7)), which has zero derivative at 0η η˜ = ^ and iη η˜ = ^ , isused as q, then such an additional driving force is absent. Since ZA and ZM are regularizing parameters, we will not includetheir dependence on ηk for simplicity.

8. Expression for the Helmholtz free energy

We accept the expression for the Helmholtz free energy per unit mass as the sum of elastic ψe, thermal energy for PT ψηθ ,

crystalline energy for dislocations cψξ , the energy of interaction of dislocation cores belonging to different slip systems intψξ ,and gradient energies related to multivariant martensitic PT ψη

∇ and dislocations ψξ∇:

J , 70te c intψ ψ ψ ψ ψ ψ ψ= + + + + + ( )η

θξ ξ η ξ

∇ ∇

where UJ dett t= is the ratio of elementary volumes of stress-free M and A.Elastic energy per unit volume in the unloaded configuration Ωt is accepted as

E E C E E C E E E E C E E

C C C C a

,12

: :13

: : :14

: : : : ;

, ,71

ee i e i e e i e e e e i e e

ki A

k

i

n

ik

Ak

Ck i

02 3 4

1

∑

ρ ψ η η η η

η φ η

( ) = ( ) +!( ( ) ) +

!( ( ) ) + ⋯

( ) = + ( − ) ( )( )=

where Ck are the elastic moduli tensors of the kth rank. We assume that the elastic energy is independent of ξα, i.e., elasticconstants in the region of the dislocation core are the same as in perfect lattice. Should such an information be available, itcan easily be included in elastic constants. The reason why E ,e

e i0ρ ψ η( ) is defined per unit volume in Ωt is described in Levitasand Attariani (2013a,b) and Levitas (2014b).

Expression for thermal energy

E C C C

A G F F B D

A A K K G s

K a a A A A

1 4 3 , ; 1

; ;

, , 3 3 ; ; 0, 72

k

n

k k k ki

n

j i

n

ij i j ij i j i j i j i j i j

i j i j i j i ij j ji e

jie

e Ck ik

Ck Ak

jk

c

1

2 2 3 4

1

1

1

2

2 2 2 2

0 0

∑ ∑ ∑ψ η η η η η η η η η η η η η η η η

η η η η η η η η θ θ

ψ θ θ θ

= ( ( − ) + Δ ( − )) + ˜ ( ) ˜ = ( − − )[ (( − ) − − ) + ]

+ ( ¯ − ) ( + ) + ( + ) Δ = − Δ ( − )

= ( ( − ) + − ) = ( − ) > ( )

ηθ θ

θ

= =

−

= +

includes the thermal driving force for PT, double-well barriers between all phases and cross terms that allow us to satisfyCondition I and instability conditions (62)–(63). This expression was derived in Levitas and Preston (2002b) and used in allour simulations (Levitas and Javanbakht, 2012, 2013, 2014, 2015). Here GΔ θ and sΔ are the differences between the thermalpart of free energy and entropy of M and A, respectively; A and A are the double-well barrier between A and M and betweenmartensitic variants, respectively; parameters B and D control energy away from both the A and Mi minima and do not affectthe phase equilibrium or PT conditions; functions Kji are related to the jump in elastic moduli during PTs; eθ is theequilibrium temperature for stress-free A and M; A0 is a parameter, and cθ is the critical temperature at which stress-free Aloses its thermodynamic stability.

Crystalline energy for dislocations is accepted like in Levitas and Javanbakht (2012, 2015)

A y A y

A A A A a A A A A a

, 1 , 1 ;

, ; , ,73

cp

Ai A

A Am

Mi M

M M

A AA MA AA

i

n

A iM AM MM AM

i

n

M i

1

2 2

1

2 2

1 1

∑ ∑

∑ ∑

ψ η ξ ξ η ξ ξ

φ η φ η

= ( ¯ )( ¯ ) ( − ¯ ) + ( ¯ )( ¯ ) ( − ¯ )

= + ( − ) ( ) = + ( − ) ( )( )

ξα

αα

α αω

ωω

ω ω

α α α α ω ω ω ω

= =

= =

but with taking into account PTs and inheritance of slip systems during PTs. The first sum is related to slip along the slipsystems of the A, and the hight of multi-well barrier A A

α characterizes the theoretical yield strength in shear in A (for 0η η˜ = ^ ,A AA AA=α α ) and martensitic variants (for iη η˜ = ^ , A AA MA=α α ). Similarly, the second sum is related to slip along the slip systems ofthe M, and AM

ω characterizes the theoretical yield strength in shear in austenite (A AM AM0η(^ ) =ω ω ) and martensite

(A AMi

MMη(^) =ω ω ). Parameters Aα are the same for all equivalent slip systems. In order to define the height Hα of dislocationbands via equations rather than via computational mesh, the parameters Aα are assumed to be a periodic step-wise functionof the coordinate yα along the normal to the corresponding slip plane nα:

Fig. 4. (a) Schematics of the distribution of the multiplier k for the crystalline energy barrier kAα along the normal to each slip plane. It represents periodicstep-wise function with k¼1 within slip band and k¼100 in a thin boundary layer between dislocations of the width w bH=α

α (b 1⪡ ). (b) An example of thedistribution of the dislocation order parameter ξα for the corresponding distribution of the multiplier k.

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 305

⎪

⎪⎧⎨⎩

⎛⎝⎜⎜

⎞⎠⎟⎟A y

A y H

kA y Hy y Int

yH w

H w;

;.

74

A M, ( ) =¯ ¯ ≤¯ ¯ >

¯ = −+

( + )( )

αα α

α α

αα α

α αα

αα

αα

Consequently, the parameter Aα is equal to its normal value Aα within each dislocation band of the height Hα and kAα (k 1⪢ ) ina thin boundary layer between dislocations of the width w bH=α

α (b 1⪡ ) (Fig. 4). Large values of Aα exclude the spreading ofthe dislocation outside the desired dislocation band. Eq. (74) should be applied both to A A

α and AMω .

The energy of interaction of dislocation cores belonging to different slip systems is

A

A A A

1 1

1 1 ; 0.75

int

k

p

kA

iA A

kA

kA

l

m

lM

iM M

lM

lM A M

, 1

2 2 2 2

, 1

2 2 2 2

∑

∑

ψ η ξ ξ ξ ξ

η ξ ξ ξ ξ

= ( )( ¯ ) ( − ¯ ) ( ¯ ) ( − ¯ )

+ ( )( ¯ ) ( − ¯ ) ( ¯ ) ( − ¯ ) = =( )

ξα

α α α

ωω ω ω αα ωω

=

=

Since after a complete dislocation has passed through the material point under study, it does not affect the crystal structureat this point, intψξ depends on Aξα and Mξω rather than Aξα and Mξω , respectively. Self-interaction of cores is excluded with thehelp of the last equation in (75). Thus, when just one incomplete dislocation is present in the given point, this termdisappears. The interaction energy must be invariant with respect to exchange 1A Aξ ξ¯ ↔ − ¯

α α and subscripts kα ↔ for A aswell as 1M Mξ ξ¯ ↔ − ¯

ω ω and subscripts lω ↔ for M, which is met by using in Eq. (75) the same exponents for all multipliers.Because without intψξ Condition I and instability conditions for dislocations are met, ψint is chosen as the minimum degreepolynomial that does not violate any of them. Indeed, exponent 2 is the minimal integer for which interaction energy and itsfirst and second derivatives disappear for complete dislocations. The coefficients A k

Aα as well as A l

Mω are the same for

equivalent dislocation pairs. Parameters A kAα and A l

Mω can be described as a function of ηi by equations similar to Eq. (74). Note

that in the previous theories intψξ was neglected. However, it was demonstrated in Levitas and Javanbakht (2015) that itmakes essential difference for dislocation evolution along the intersecting slip systems.

Gradient energy for PTs is taken as the simplest form discussed in Section 7.1:

⎛⎝⎜⎜

⎞⎠⎟⎟b

22 .

76i

n

ii

n

j i j

n

i j1

2

1 1,

∑ ∑ ∑ψ β η η η∇ ∇ ∇= | | + ·( )

η

η∇

= = = ≠

Gradient energy for dislocation is presented in Eq. (69). All parameters in the equations for dislocations depend on ηaccording to the rule B B B BA M A ϕ η= + ( − ) ( ), where BA and BM are the values of a parameter B in A and M. This leads tocontributions of the dislocation-related terms in Ginzburg–Landau equation (49) for PT. In addition, both processes arecoupled through the mechanical driving force (stress power) in Eqs. (49) and (50) and the evolving stress field.

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319306

9. Boundary conditions for order parameters

9.1. Boundary conditions for phase transformations

The generalized vector-forces Q i is introduced at the external surface (see Eq. (36)), thus, their normal component at theboundary can be prescribed similar to the heat flux as

n Q n W ,77i

ii0 0 0ρ ψ

η∇· = · ∂

∂=

( )

where Wi are some given functions. According to Lipowsky (1982), Pluis et al. (1990), Levitas et al. (2006), Levitas andJavanbakht (2010), and Levitas and Samani (2011a,b), the most important physically based option for functions Wi is

n ,78i

s i

i0 0ρ ψ

ηγ η

η∇· ∂

∂= −

∂ ( )∂ ( )

where s iγ η( ) is the surface energy per unit reference area. It was suggested in Levitas and Javanbakht (2010) that

⎛⎝⎜⎜

⎞⎠⎟⎟a p a p a p p4 2 3 ; ,

79s s

AsM

sA

s s si

n

i2 3 4

1

2

0.5

∑γ η γ γ γ η( ) = + ( − )( + ( − ) + ( − ) ) =( )=

where p is the “equivalent” order parameter. Also, for equal surface energy of the phases, Wi¼0. In Levitas and Javanbakht(2011b), one more contradiction in the PFA was stated: while the finite width of the phase interfaces and the surfacetransformed layer are resolved, the external surface is treated as the sharp surface with boundary conditions (i.e. similar toEq. (79)). In order to resolve this contradiction, a finite-width external surface was introduced in Levitas and Javanbakht(2011b) with the help of an additional order parameter, which led to several interesting scale and mechanics effects andmorphological transitions in surface-induced PTs and various surprising phenomena on barrierless and thermally activatedtransformations (Levitas and Samani, 2014) for melting of Al nanoparticles. After substituting the Helmholtz energyequation (70) into Eq. (79), the boundary conditions for PTs simplify to

⎛⎝⎜⎜

⎞⎠⎟⎟n nb a p a p a p2 3 4 2 4 3 .

80i

j i j

n

j sM

sA

i s s s0 0 01,

2 3∑ρ β η η γ γ η∇ ∇· + · = ( − ) ( + ( − ) + ( − ) )( )

η

= ≠

Alternative to Eq. (81), periodic boundary conditions for ηi, or constant consti iη = (in particular, 0 or 1), or mixed boundaryconditions are possible.

9.2. Boundary conditions for dislocations

Periodic boundary conditions for dislocation order parameters ξα have been applied in majority of papers on the phasefield simulations of dislocation evolution. Similar to that for PTs, this is always the case when spectral methods of solutionsof the boundary value problems are utilized (Wang et al. 2001a–c, 2003; Jin and Khachaturyan, 2001; Wang and Li, 2010).Applying conditions nξ =α can fix a number of dislocations at the surface, if there are physical reasons for this. Similar to Eq.(78) one can prescribe

n n ,81

s0 0 0ρ ψ

ξγ ξ

ξΘ

∇· = · ∂

∂= −

∂ ( )∂ ( )α

α

α

α

where sγ ξ( )α is the surface energy per unit reference area. When dislocations leave the crystal, they create steps at the surfaceand a new surface appears. Therefore, the surface energy for dislocations should also depend on the order parameters ξα. Infact, the dislocation surface energy is proportional to the number of exiting dislocations and depends on the orientation ofnew area. At the same time dislocation core energy disappears. Here, we can assume that

n nsA M

0γ ξ γ Λ ξ Λ ξ( ) = + ∑ ( ) + ∑ ( )α αα

α ωω

ω , where nΛ( )α characterizes energetic changes associated with the exit of a single dis-location at the surface. Substituting the Helmholtz free energy from Eq. (70) into Eq. (81) and using the decompositionequation (67), we obtain

n n n

n m n n n

Z

Z

1 1

1 82

Ai

AA

A A

Ai

m AA

A n A

0 02

0 02

0

ρ β η ξ ξ ξ

ρ β η ξ ξ ξ Λ

∇ ∇· ( )[ ¯ + [ ( − ¯ ) − ]( ¯ · ) ]

= ( )[ · ∇ ¯ + ( − ¯ ) ∇ ¯ · ] = − ( ) ( )

ξ α α αα α

ξα

α α αα α

and similar equation for Mξω . The term with Z disappears when dislocation completes ( 1ξ →α ), which is expected for thestationary state. Note that this term was introduced to regularize differential equation and can be omitted in the boundaryconditions. Thus

n m n n m n; . 83A

im A M

im M

0 0 0 0ρ β η ξ Λ ρ β η ξ Λ( ) · ∇ ¯ = − ( ) ( ) · ∇ ¯ = − ( ) ( )ξα

αα

ξω

ωω

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 307

In the simplest case when the surface energy is independent of ξα and Λ¼0, one obtains

b b0; 0. 84m A A m M Mξ ξ ξ ξ∇ ∇∇ ¯ = ¯ · = ∇ ¯ = ¯ · = ( )α αα

ω ωω

Eq. (84) is independent of the direction of the normal to the surface. This boundary condition does not change the slipdirection when dislocation exits a crystal because within a volume and away from the dislocation core, ξ∇ α and bα areorthogonal and Eq. (84) is satisfied.

10. Explicit expressions for Ginzburg–Landau equations

Substituting the energy terms from Eqs. (69), (71)–(73), (75), and (76) into the Helmholtz free energy equation (70), andthen Eq. (70) into Eq. (66) after some transformation, the Ginzburg–Landau equations for PTs specify to

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

P FU

F EE

LJ

J

A GF

A y A y

A A

Z

Z b

1: ,

,

2 1 1 2 12 1,

,1

,1

1 1 1 1

0.5 1

0.5 1 .85

j jiT

et

ip

t

i

ee i t

ee i

i

i i i i ij

n

j i

nij i j

i

p Ai A

i

A Am M

i M

i

M M

k

pk

Ai

i

A AkA

kA

l

ml

Mi

i

M MlM

lM

Ai

i

pm A

AA n A

Mi

i

mm M

MM n M

ik k i

n

k

0

2

1

1

1

1

2 2

1

2 2

, 1

2 2 2 2

, 1

2 2 2 2

1

2 2 2

1

2 2 2 2

1,

2

∑ ∑

∑ ∑

∑ ∑

∑

∑ ∑

ηρ η η

ψ ηψ η

η

η η η η ηη η

η

ηη

ξ ξηη

ξ ξ

ηη

ξ ξ ξ ξη

ηξ ξ ξ ξ

β η

ηξ ξ ξ

β η

ηξ ξ ξ β η β η

= ⋅∂∂

⋅ −∂∂

( ) −∂ ( )

∂

− [ ( − )( − ) + Δ ( − )] −∂ ˜ ( )

∂

−∂ ( ¯ )

∂( ¯ ) ( − ¯ ) −

∂ ( ¯ )∂

( ¯ ) ( − ¯ )

−∂ ( )

∂( ¯ ) ( − ¯ ) ( ¯ ) ( − ¯ ) −

∂ ( )∂

( ¯ ) ( − ¯ ) ( ¯ ) ( − ¯ )

−∂ ( )

∂((∇ ) + ( − ¯ ) (∇ ) )

−∂ ( )

∂((∇ ) + ( − ¯ ) (∇ ) ) + ∇ + ∇

( )

η

θ

α

αα

α αω

ωω

ω ω

α

αα α

ω

ωω ω

ξ

αα α α

ξ

ωω ω ω

η η

=

−

= +

= =

= =

=

= = ≠

In this equation

UU

UUJ dJ

dJ: : .

86t

i

t

t

t

it t

t

i

1

η η η∂∂

=∂∂

=∂∂ ( )

−

For small distortions and linear elasticity, the strain-related terms in Eq. (85) simplify to

⎛⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞⎠⎟⎟I C

CL

J J1:

2: : :

2: : .

87j ji

t

i

t t

ie i e

te

i

ie

0 0 0

σε ε

ε ε ε εηρ η ρ η

ηρ

ηη

=∂∂

−∂∂

( ) −∂ ( )

∂+ ⋯

( )η

Substituting the energy terms from Eqs. (71)–(73), (75), (76), and (69) into the Helmholtz free energy equation (70), andthen Eq. (70) into Eq. (50) after some transformation, the Ginzburg–Landau equation for dislocations specify to

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319308

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

n n

n n

n n n

L Z

Z

Z Z

A y A

Z

61

12

1 1

12

12

1 1

2 1 1 1

2 , 1 1 2 2 1 1 2 1

1 .88

A Aj

A A A AA

A A Aj

Aj

A Aj

AA

A A

AA A

AA A

Aj A

A A Ak

Aj

A A AkA

kA

Aj A

A n A

0

2

2 2

2 2

2 2

2

ξ ηρ

τ γ ξ ξ ξ ξ β η

β η ξ β η ξ ξ ξ

ξ ξ ξ ξ

η ξ ξ ξ η ξ ξ ξ ξ ξ

β η ξ ξ

∇ ∇

∇ ∇ ∇ ∇

∇ ∇ ∇

= ( ) ¯ ( − ¯ ) + [ ( − ¯ ) − ]( ¯ ⋅ )( ( )⋅ )

+ ( )⋅ ¯ + ( )[∇ ¯ + ( ( − ¯ ) − )( ⋅ )( ¯ ⋅ )]

− ( − ¯ )( ¯ ⋅ ) + [ ( − ¯ ) − ] ( ¯ ⋅ )·

− ( ¯ ) ¯ ( − ¯ )( − ¯ ) − ( ) ¯ ( − ¯ )( − ¯ )( ¯ ) ( − ¯ )

+ ( ) ( − ¯ )(∇ )( )

α α α α α α α αα

ξα

ξ α ξ α αα

αα

α αα

α αα α

αα

α α α α α α α

ξ α α

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

n n

n n

n n n

L Z

Z

Z Z

A y A

Z

61

12

1 1

12

12

1 1

2 1 1 1

2 , 1 1 2 2 1 1 2 1

1 .89

M Mj

MA MA M MM

M M Mj

Mj

M Mj

MM

M M

MM M

MM M

Mj M

M M Ml

Mj

M M MlM

lM

Mj M

M n M

0

2

2 2

2 2

2 2

2

ξ ηρ

τ γ ξ ξ ξ ξ β η

β η ξ β η ξ ξ ξ

ξ ξ ξ ξ

η ξ ξ ξ η ξ ξ ξ ξ ξ

β η ξ ξ

∇ ∇

∇ ∇ ∇ ∇

∇ ∇ ∇

= ( ) ¯ ( − ¯ ) [ ( − ¯ ) − ]( ¯ ⋅ )( ( )⋅ )

+ ( )⋅ ¯ ++ ( )[∇ ¯ + ( ( − ¯ ) ) − )( ⋅ )( ¯ ⋅ )]

− ( − ¯ )( ¯ ⋅ ) + [ ( − ¯ ) − ] ( ¯ ⋅ )·

− ( ¯ ) ¯ ( − ¯ )( − ¯ ) − ( ) ¯ ( − ¯ )( − ¯ )( ¯ ) ( − ¯ )

+ ( ) ( − ¯ )(∇ )( )

ω ω ω ω ω ω ω ωω

ξω

ξ ω ξ ω ωω

ωω

ω ωω

ω ωω ω

ωω

ω ω ω ω ω ω ω

ξ ω ω

For small strains and rotations, Eqs. (88) and (89) do not change their appearance.Instability conditions for homogeneous state. Using the Helmholtz free energy (Eq. 70), the instability conditions between

A and Mi as well as between M j and Mi (Eqs. (62) and (63)) are simplified to

P F F E C C I E C

P F F E C C

U E C

aA J a aJ

aG A J a

a J

A M : : , : , ;

M A:6

: 6 , 6

6 : , ; 90

iT

e ti p te

e ck ik

Ak

t tie

e Ak

iT

e ti p te

e ck ik

Ak

t t tie

e ik

0

01

ε ε

ε

ε

ρψ ψ

ρψ

ψ

→ ⋅ ⋅ ≥ + ( ( − )) + ( ) ( )

→ − ⋅ ⋅ ≤ Δ − + ( ( − )( − ))

+ ( − ) ( ) ( ) ( )

θ

−

P F U U F E C CA

J aM M :1

:3

, .91

i jT

e ti tj p te

e ck ik

jk

0ρψ→ ⋅ ( − )⋅ ≥

¯+ ( ( − ))

( )

Here, E C Ca,ee ck i

kAkψ ( ( − )), E C Ca, 6e

e ck ik

Akψ ( ( − )( − )), and E C C,e

e ik

jkψ ( − ) are obtained from the expression for elastic energy

E C,ee

kiψ η( ( )) (Eq. (71)) by substituting elastic moduli of the rank k, Ck

iη( ), with C Cack ik

Ak( − ), C Ca6 ck i

kAk( − )( − ), and C Ci

kjk− ,

respectively.Substituting the Helmholtz free energy equation (70) in the thermodynamic equilibrium condition X X 0A M= =α ω for

homogeneous states, and using Eq. (7) for ϕ ξ( ¯ )α function, we obtain (in addition to the trivial solution ξ¼n)

A

31 2 .

920τ

ργ

ξ= ( − ¯ )( )

αα

αα

Placing in Eq. (92) 0ξ =α , we obtain the critical shear stress to nucleate a dislocation:A

30τ =α

ργ

α

α. Substituting in Eq. (92) 1ξ =α ,

one obtains the critical shear stress to nucleate a dislocation in the opposite direction:A

30τ = −α

ργ

α

α. Also, Aα depends on the

phase where the dislocation instability occurs, see Eq. (73). Thus, the instability conditions are

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 309

n nA

n nA

n nA

n nA

n nA

n nA

n nA

n nA

AA: 1:3

; 1:3

;

AM: 1:3

; 1:3

;

MA: 1:3

; 1:3

;

MM: 1:3

; 1:3

,93

AAAA

AAA

AA

A

AMAM

MAAM

AM

MA

MAMA

MMA

MA

M

MMMM

MAMM

MM

MA

0 0

0 0

0 0

0 0

τρ

γτ

ρ

γ

τρ

γτ

ρ

γ

τρ

γτ

ρ

γ

τρ

γτ

ρ

γ

→ + ≥ → − ≤ −

→ + ≥ → − ≤ −

→ + ≥ → − ≤ −

→ + ≥ → − ≤ −( )

αα

αα

α

α

αα

αα

α

α

ωω

ωω

ω

ω

ωω

ωω

ω

ω

where AA and MA , AM and MM correspond to the slips along the slip system of A in A and in M, and along the slip system ofM in A and in M, respectively.

Remark. While both PT and plastic flow occur via nucleation and growth processes, i.e., heterogeneously, to develop ki-nematics and constitutive equations, we have to use a concept of a homogeneously deformed small material volume.Homogeneous deformation of a material during PT can be provided by applying displacement controlled deformation thatsuppresses heterogeneous strain localization due to mechanical instability. The same can be done for dislocation by pre-scribing simple-shear displacement to the crystal slab of the size of dislocation height along the normal to the slip system.This allows us to determine kinematic and constitutive relationships for homogenous states for each of the processes. Forcoupled PT and plasticity, we cannot consider dislocation within a crystal, because it creates heterogeneity. We can considercombined homogeneous deformation during PT and plastic shears at the end of which dislocations appear at the surface of asample only.

After completing determination of the constitutive equations for homogeneous states, we should take into accountgradient energy terms and obtain analytical or numerical solutions for each phase interface and single dislocation for eachslip system (separately in each phase). Calibrating these solutions using values of interface and dislocation core widths andenergies, and interface and dislocation velocities, one can determine the coefficients of the gradient energy, the height of theenergy barrier, and the kinetic coefficient of the Ginzburg–Landau equations. Since after this we know all properties of thedislocation in each phase, we can interpolate these properties for intermediate values of the order parameters describingPTs.

Note that we did not use any information about the interaction between the finite-width interface and dislocation core,where both order parameters have intermediate values. That means that in the first approximation this interaction iscompletely determined by a solution of the coupled Ginzburg–Landau and mechanics equations. In the future, cross termswith i

a bη ξα and similar cross-gradient terms can be introduced and analyzed, if proper information will be available.

11. Concluding remarks and future directions

In the paper, thermodynamically consistent large-strain PFA for coupled multivariant martensitic PTs and dislocationevolution is developed. It is based on a combination and generalization of the most advanced from a mechanics point ofview PFA to multivariant martensitic PT (Levitas, 2013b) and dislocations (Levitas and Javanbakht, 2012, 2015), both at largestrains. The main problem and development is in the kinematics. Several possible and quite logical versions of multiplicativedecomposition of the deformation gradient have been analyzed that finally appeared to have undesired features. Con-sideration included cyclic A–M transformations and plastic deformation of A and M after it, with emphases on the in-heritance and further evolution of dislocations during transformations along the slip systems that are not traditional for thegiven crystal lattice. Transformation rules for the parameters of the slip systems inherited by the crystal lattice during PT arepresented. The multiplicative decomposition of the deformation gradient into elastic, transformational, and plastic parts issubstantiated, in which the plastic part includes four mechanisms: dislocation motion in M along slip systems of M and slipsystems of A inherited during PT and dislocation motion in A along slip systems of A and slip systems of M inherited duringreverse PT. It is found that the definition of the plastic part of the velocity gradient (a) in M in terms of a combination ofshear rates along the slip system of M and (b) in M transformed back to A in terms of a combination of shear rates along thetransformed back to A slip system of M are the same. This finding allowed us to describe plastic deformation of M in thecrystal lattice of A in the same way as in the crystal lattice of M, but with transformed crystallographic parameters of the slipsystems of M back to the A. This also allowed us to additively combine the plastic parts of the velocity gradients for all fourmechanisms using crystal lattice of the A. The plastic part of the velocity gradient for all these mechanisms is defined in thecrystal lattice of the A utilizing just slip systems of A and inherited slip systems of M and just two corresponding types of theorder parameters. This offers a noncontradictory solution to the kinematic problem, in contrast to, for example, multi-plicative decomposition of the plastic deformation gradient into plastic deformation gradients in the M and A.

Applications of the global thermodynamics laws for coupled PT and plasticity do not differ essentially from the appli-cations for each process separately. General expressions for the driving forces for PT and dislocation evolution have a slightly

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319310

more general form of the mechanical work than for each process separately because of the effect of plastic deformationgradient in the transformation work and transformational deformation gradient in the expression for resolved shearstresses. In small strain approximation this coupling disappears. Explicit Ginzburg–Landau equations for a specific ther-modynamic potential possess more new terms due to the interaction between PTs and plasticity. Ginzburg–Landau equa-tions for dislocations include the dependence of the material parameters in the local and gradient energy on the phase state(order parameters ηi). Because of this, corresponding contributions from dislocations appear in the Ginzburg–Landauequations for PTs as well. Boundary conditions for the order parameters include variation of the surface energy during PTand exit of dislocations. Note that previous theories utilizing spectral methods employ periodic boundary conditions andhave problems in incorporating surface effects. Complete system of coupled phase field and mechanics equations is pre-sented for a general case and for small strain approximation. Simplified cases of the developed theory have already beenused for the finite element solution of various problems on the interaction between PTs and plasticity (Levitas and Ja-vanbakht, 2012, 2013, 2014). Application of the current theory to solution of the some physically important problems ispresented in the accompanied paper (Javanbakht and Levitas, 2015).

The developed theory can be further advanced in the following directions:

1.

Thermodynamic potential for PTs for multiple variants describes A Mi− PT with help of a single order parameter, butM Mi j− transformation with two order parameters. While for single order parameter analytical solutions for static andnonequilibrium interfaces are available (Levitas et al., 2003, 2010; Levitas, 2013c, 2014a,b), which allowed us to calibratematerial parameters in Ginzburg–Landau equation using interface width, energy, and mobility, for two order parametersnumerical studies are only possible. In this case properties of a M Mi j− interface depend on temperature and stresses incomplex and not well-controlled way. This problem was overcame (at least for two variants) in the theory by Levitaset al. (2003, 2013c) utilizing hyperspherical order parameters. Latter (Levitas and Roy, 2015) it was resolved also in termsof ηi for an arbitrary number of variants. These advances can be used in the current coupled theory for PT and plasticity.

2.

Here and in Levitas et al. (2013c), tensorial properties of Mi are symmetry related and scalar properties are the same. Inthe theories by Levitas and Momeni (2014), Momeni (2014), and Levitas and Roy (2015) these properties are different, i.e.,theory describes PTs between different phases. They can be used instead of the theory in terms of ηi utilized here.However, in this case, because crystal lattices of phases are not symmetry-related, one needs to transform slip systems oftwo product phases to the crystal lattice of the parent phase and consider plasticity of all three types of slip systems.

3.

Here for simplicity we neglected interface stresses and line tension for dislocations, which may be quite significant at thenanoscale. In particular, interface stresses at the plane external surface caused PT in nanowires in Diao et al. (2003).Interface stresses have been introduced in the PFA for martensitic PTs for static interfaces in Levitas and Javanbakht(2010, 2011a)and for propagating interfaces in Levitas (2013a,c, 2014a,b). Their importance for modeling nanostructureformations was demonstrated in simulations in Levitas and Javanbakht (2010, 2011a). Since the main principles for thecorrect introduction of the interface stresses are now clear, this can be done for dislocations and for interaction betweenPTs and dislocations.

4.

Since all martensitic variants are in twin relation to each other for most crystal lattices, our theory automatically includestwinning as a mechanism of plastic deformation. Consequently, it allows us to study the interaction of twinning anddislocation plasticity, twinning-slip transition as the mechanism of plastic accommodation during martensitic PT and ingeneral.

5.

A similar theory can be developed for plastic deformation in the polycrystalline aggregate by evolving grain structure(instead of PT), for which one can use one of the advanced PFAs for evolution of the grain structure (see, e.g., Kobayashiet al., 1998). Our approach is applicable if dislocations are inherited when one grain transforms to another due to motionof the low-angle grain boundary. In contrast, for high-angle disordered grain boundaries, dislocations disappears at grainboundary, which should be taken into account in the theory. The same is true for phase boundaries for PTs between solidphases that occur via virtual or intermediate melt, which is present as a few nm thick liquid layer within moving in-terface between to solids (Levitas, 2005; Levitas et al., 2006, 2012; Luo and Chiang, 2008; Levitas and Momeni, 2014;Momeni, 2014; Momeni et al., 2015).

6.

Dislocation reactions should be included because dislocations inherited during PTs may belong to a nontraditionalproduct phase slip systems and they may dissociate into partial dislocations (Brainin et al., 1981). As the first step this canbe done utilizing PFAs from Shen and Wang (2004), Wang and Li (2010), and Hunter et al. (2011) but imposing ourconditions.

7.

Inheritance rules for parameters of the slip systems and developed kinematics are not limited to the nanoscale PFA onlybut can be used (adjusted) for any approach at any spatial scale. That is why they can be a basis for scaling up the currenttheory. Thus, microscale PFA for martensitic PTs was suggested in Levitas et al. (2004), Idesman et al. (2005), and Levitasand Ozsoy (2009a,b), and is applicable for a sample size greater than 100 and without upper limits. In this case, it can becombined with continuum dislocation theory in terms of dislocation density, or traditional crystal plasticity. At the scaleof polycrystalline aggregate, a similar approach to PTs can be combined with phenomenological plasticity.

8.

A similar theory can be developed for PFA to dislocations and other types of PTs, like reconstructive PTs and diffusive PTsdescribed by the Cahn–Hilliard equation.

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 311

Acknowledgment

The support of NSF, ARO (W911NF-12-1-0340), DARPA (W31P4Q-13-1-0010), ONR (N00014-12-1-0525), and Iowa StateUniversity are gratefully acknowledged.

Appendix A

A.1. Plastic deformation of martensite treated in crystal lattice of martensite

For elastoplastic deformation of M, when the undeformed lattice of M is taken as a reference configuration, one hasmultiplicative decomposition of the deformation gradient into elastic and plastic parts F F FM e p

M= · . The same deformationbut when the undeformed lattice of A is taken as the reference configuration is described as

F F F U . 94e pM

t= ⋅ ⋅ ( )

For the first decomposition, one obtains

F F F F F F F F

l F F F F F F F F

; ;

. 95

M e pM

e pM

M pM

e

M M M e e e pM

pM

e

1 1 1

1 1 1 1

= · + · = ·

≔ · = · + · · · ( )

− − −

− − − −

Similarly, for decomposition (94) we derive

F F F U F F U F F U F U F F

l F F F F F F F F F F U U F F

; ;

. 96

e pM

t e pM

t e pM

t t pM

e

e e e pM

pM

e e pM

t t pM

e

1 1 1 1

1 1 1 1 1 1 1

= · ⋅ + · ⋅ + ⋅ ⋅ = · ·

≔ · = · + · · · + ⋅ ⋅ · · · ( )

− − − −

− − − − − − −

It is clear that elastic and plastic contributions to the velocity gradients in Eqs. (95) and (96) are the same; if U constt = ,these equations are equivalent. For the plastic part of the velocity gradients we define as in Eq. (9)

l F F b nH1

.97

pM pM

pM

m

MM M

M1

1

∑ Φ ξ≔ · = ⊗ ( )( )ω

ωω ω

ω−

=

A.2. Four-term multiplicative decomposition

Natural generalization of decomposition (94) for the case with plastic deformation of A followed by PT is

F F F U F . 98e pM

t pA= ⋅ ⋅ ⋅ ( )

Such a decomposition (even with and additional plastic strain during PT) have been in some way justified in Levitas (1998)but was not used in the PFA. Undesirable points of this decomposition are

�

From any current state, elastic and transformational parts of the deformation gradient can be determined by reducingstresses to zero and performing reverse PT without plastic deformation. However, there is no experimental way toseparate the remaining inelastic deformation gradient of F F Fin p

MpA= · into Fp

M and FpA. Of course, if one of these con-

tributions is defined through constitutive equations, then another one could be determined as well.

� If after thermomechanical processes, including direct and reverse PTs and plastic deformation of A and M, an unloaded

austenitic sample has dislocations (both along slip systems of A and inherited slip systems of M), then there is no way todistinguish whether the current position of the dislocations along inherited slip systems of M was achieved by theirevolution inM, A, or in both in some sequence. Also, if such a dislocation configuration is taken as the initial condition forthe further deformation/transformation problem, the solution will not be affected by the way this initial condition wasproduced and how Fin was split into Fp

M and FpA. In this sense such a splitting is unphysical and unnecessary.

M

� Two types of deformation of the crystal lattice, Fe and Ut are separated by Fp . In some theories (Lookman et al., 2008;Falk, 1983; Reid et al., 1998), deformation of crystal lattice during PT is not divided into Fe and Ut , which is impossible torealize with Eq. (98).

�

Two types of plastic deformation are separated by Ut . This complicates the definition of the plastic part of the velocitygradients for A by an equation of the type of Eq. (97), see below.First, we consider an auxiliary problem. Let plastic strain occur first along some set of slip systems 1 producing Fp1, then

along another set of slip systems 2 at fixed Fp1 producing Fp2 on the top of Fp1 (i.e., keeping configuration after Fp1 as thereference one). Then Eq. (9) can be integrated as

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319312

F F F . 99p p p2 1= · ( )

Now, let both sets of slip system be activated in turn one after another while keeping decomposition Eq. (99). Then

F F F F F F F F

l F F F F F F

; ;

. 100

p p p p p p p p

p p p p p p p

2 1 2 11

11

21

2 21

2 1 11

21

= · + · = ·

≔ · + · · · ( )

− − −

− − −

Comparing with Eq. (9) leads to the following definition of the plastic velocity gradient for the decomposition (99):

l F F b n

l F F F F b n

H

H

1;

1.

101

p p p

p p p p p

2 2 21

1 2 1 11

21

∑

∑

Φ ξ

Φ ξ

≔ · ≔ ⊗ ( )

≔ · · · ≔ ⊗ ( )( )

ωω

ω ωω

αα

α αα

−

− −

Thus, while definition for lp2 is the same as in the general case (Eq. (9)), the definition of the plastic velocity gradient lp1 ismodified by the presence of Fp2. This definitions will be applied below when sets 1 and 2 correspond to slipping in the A andM, respectively.

Decomposition of the velocity gradient that follows from Eq. (98) is

l F F F F F F F F F F U U F F

F F U F F U F F . 102

e e e pM

pM

e e pM

t t pM

e

e pM

t pA

pA

t pM

e

1 1 1 1 1 1 1

1 1 1 1

≔ · = · + · · · + ⋅ ⋅ · · ·

+ ⋅ ⋅ · · · · · ( )

− − − − − − −

− − − −

After reverse PT, inelastic deformation gradient is F F Fin pM

pA= · , which is similar to decomposition Eq. (99). Then one has to

use the definitions Eq. (101) for the plastic velocity gradients

l F F b n

l F F F F b n

H

H

1;

1.

103

pM pM

pM

m

MM M

M

pA pM

pA

pA

pM

p

AA A

A

1

1

1 1

∑

∑

Φ ξ

Φ ξ

≔ · = ⊗ ( )

≔ · · · = ⊗ ( )( )

ωω

ω ωω

αα

α αα

−

=

− −

Thus, the necessity to properly describe plastic deformation of A after plastic deformation of M and reverse PT complicatesthe definition of the plastic velocity gradients for A. Resolving Eq. (103) for F Fp

ApA 1 · −

F F F b n FH1

,104

pA

pA

pM

p

AA A

ApM1 1

1

∑ Φ ξ · = · ⊗ ( )·( )α

αα α

α− −

=

and substituting into the last term in Eq. (102) results in

F F U F F U F F

F F U F b n F U F FH1

.105

e pM

t pA

pA

t pM

e

e pM

t pM

p

AA A

ApM

t pM

e

1 1 1 1

1

1

1 1 1∑ Φ ξ

⋅ ⋅ · · · · ·

= ⋅ ⋅ · · ⊗ ( )· · · ·( )α

αα α

α

− − − −

−

=

− − −

The complexity of this term gives an impression that multiplicative decomposition (98) and its consequences are far frombeing the best option.

A.3. Complete system of equations

1.

Kinematics

I. Large strains1.1. Multiplicative decomposition of the deformation gradient F into elastic Fe, transformation Ut , and plastic Fp

contributions

F F U F . 106e t p= ⋅ ⋅ ( )

1.2. Jacobian determinants

F F U FJ det J det det J J det J J J; ; ; 1; . 107e e t t p p e t≔ ≔ = ≔ = = ( )

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 313

1.3. Transformation–deformation gradient

U I I L L

L

a

a a a a

, ;

3 3 ; , 1 4 3 ; 0 6. 108

t tk

n

tk ki

n

j i

n

i j i ij j ji

ji ti tj k k k k k

1 1

1

1

2 2

2 2 3 4

∑ ∑ ∑ε ε

ε ε

φ η η η η η

φ η η η η η

= + = + ( ) − ( + )

= ( − ) + ( ) = ( − ) + ( − ) < < ( )

= =

−

= +

1.4. Plastic part of the velocity gradient

l b n b n m n m nH H1 1

;109

p

p

AA A

Am

MAMA MA

Mp

AA A

Am

MAMA MA

M

1 1 1 1

∑ ∑ ∑ ∑Φ ξ Φ ξ γ Φ ξ γ Φ ξ≔ ⊗ ( ) + ⊗ ( ) = ⊗ ( ) + ⊗ ( )( )α

αα α

αω

ωω ω

ωα

αα α

αω

ωω ω

ω= = = =

Int ; 3 2 . 1102Φ ξ ϕ ξ ξ ϕ ξ ξ ξ( ) = ( ¯ ) + ( ) ( ¯ ) = ( ¯ ) ( − ¯ ) ( )α α α α α α

1.5. Parameters of the slip systems of M inherited by A during the reverse transformation

b U b n n U n U

n U b U b n UH H H H

; / ;

/ ; / / . 111

MA t M MA M t M t

MA M M tMA

MA MA t M M t M

1

1γ

= · = · | · |

= | · | = | | = | · || · | ( )

ω ω ω ω ω

ω ω ωω

ω ω ω ω ω

−

−

II. Small strains

u

L L

b n b n

a

H H

; ;

, ;

1 1.

112

e t p e t p

tk

n

tk ki

n

j i

n

i j i ij j ji

p p

p

AA A

Am

MAMA MA

M

s

1 1

1

1

2 2

1 1

∑ ∑ ∑

∑ ∑

ε ε ε ε ω ω ω ω

ε ε

ε ω

φ η η η η η

Φ ξ Φ ξ

∇= ( ) = + + = + +

= ( ) − ( + )

+ = ⊗ ( ) + ⊗ ( )( )α

αα α

αω

ωω ω

ω

= =

−

= +

= =

2.

Helmholtz free energy

J . 113te c intψ ψ ψ ψ ψ ψ ψ= + + + + + ( )η

θξ ξ η ξ

∇ ∇

2.1. Elastic energy

E E C E E C E E E E C E E

C C C C E F F Ia

,12

: :13

: : :14

: : : : ;

, ; 0.5 .114

ee i e i e e i e e e e i e e

ki A

k

i

n

ik

Ak

Ck i e eT

e

02 3 4

1

1

∑

ρ ψ η η η η

η φ η

( ) = ( ) +!( ( ) ) +

!( ( ) ) + ⋯

( ) = + ( − ) ( ) = ( · − )( )=

−

2.2. Thermal energy

E C C C

A G F F B D

A A K K G s

K a a A A A

1 4 3 , ; 1

; ;

, , 3 3 ; ; 0. 115

k

n

k k k ki

n

j i

n

ij i j ij i j i j i j i j i j

i j i j i j i ij j ji e

jie

e Ck ik

Ck Ak

jk

c

1

2 2 3 4

1

1

1

2

2 2 2 2

0 0

∑ ∑ ∑ψ η η η η η η η η η η η η η η η η

η η η η η η η η θ θ

ψ θ θ θ

= ( ( − ) + Δ ( − )) + ˜ ( ) ˜ = ( − − )[ (( − ) − − ) + ]

+ ( ¯ − ) ( + ) + ( + ) Δ = − Δ ( − )

= ( ( − ) + − ) = ( − ) > ( )

ηθ θ

θ

= =

−

= +

2.3. Crystalline energy

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319314

⎧⎨⎪⎩⎪

⎛

⎝⎜⎜

⎞

⎠⎟⎟

A y A y

A A A A a A A A A a

A yA y H

kA y Hy y Int

yH w

H w

, 1 , 1 ;

, ; , ;

;

;.

116

cp

Ai A

A Am

Mi M

M M

A AA MA AA

i

n

A iM AM MM AM

i

n

M i

A MA M

A M

1

2 2

1

2 2

1 1

,,

,

∑ ∑

∑ ∑

ψ η ξ ξ η ξ ξ

φ η φ η

= ( ¯ )( ¯ ) ( − ¯ ) + ( ¯ )( ¯ ) ( − ¯ )

¯ = + ( − ) ( ) ¯ = + ( − ) ( )

( ) =¯ ¯ ≤

¯ ¯ >¯ = −

+( + )

( )

ξα

αα

α αω

ωω

ω ω

α α α α ω ω ω ω

αα α

α α

αα α

α αα

αα

αα

= =

= =

2.4. Energy of interaction of dislocation cores belonging to different slip systems

A

A A A

1 1

1 1 ; 0.117

int

k

p

kA

iA A

kA

kA

l

m

lM

iM M

lM

lM A M

, 1

2 2 2 2

, 1

2 2 2 2

∑

∑

ψ η ξ ξ ξ ξ

η ξ ξ ξ ξ

= ( )( ¯ ) ( − ¯ ) ( ¯ ) ( − ¯ )

+ ( )( ¯ ) ( − ¯ ) ( ¯ ) ( − ¯ ) = =( )

ξα

α α α

ωω ω ω αα ωω

=

=

2.5. Gradient energies for PTs and dislocations

⎛⎝⎜⎜

⎞⎠⎟⎟b

22 ;

118i

n

ii

n

j i j

n

i j1

2

1 1,

∑ ∑ ∑ψ β η η η∇ ∇ ∇= | | + ·( )

η

η∇

= = = ≠

Z

Z

0.5 1

0.5 1 .119

Ai

pm A

AA n A

Mi

mm M

MM n M

1

2 2 2

1

2 2 2

∑

∑

ψ β η ξ ξ ξ

β η ξ ξ ξ

= ( ) ((∇ ) + ( − ¯ ) (∇ ) )

+ ( ) ((∇ ) + ( − ¯ ) (∇ ) )( )

ξ ξα

α α α

ξω

ω ω ω

∇

=

=

3.

Stresses

I. Large strains

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

P FE

U F

F C E E C E E C E E U F

FE

F

F C E E C E E C E E F

J

J

J

J

:12

: :13

: : : ;

1:

12

: :13

: : : .120

t e

e

et p

T

t e i e e i e e i e e t pT

t e

e

eeT

ee i e e i e e i e e e

T

01 1

2 3 4 1 1

3 4

σ

ρ ψ

η η η

ρ ψ

η η η

= ⋅∂∂

⋅ ⋅

= ⋅ ( ) + ( ( ) ) +!( ( ) ) + ⋯ ⋅ ⋅

= ⋅∂∂

⋅

= ⋅ ( ) + ( ( ) ) +!( ( ) ) + ⋯ ⋅

( )

− −

− −

II. Small strains and linear elasticity

C : .121e

i e2σ

εερ ψ η= ∂

∂= ( )

( )

4.

Ginzburg–Landau equations

4.1. Compact form in the reference configuration at large strains

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319 315

⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞

⎠⎟⎟

P FU

F

n F P F U m n F P F U m

L X L

L X L

L X L

L L L L q

L L L L q q q

1: ;

1;

1;

0 1 0 ;

0 1 0 ; 3 2 ;

and . 122

j ji i jiT

et

ip

i i

A Aj

A Aj

A AA A A

M Mj

M Mj

MA MAM M M

Aj

A A A

i

nA

i

Mj

M M M

i

nM

iA

iM

i i i

AA p

Te t A

MAMA p

Te t MA

0

0

0

1

1

2

∑

∑

ηρ η

ψη

ψη

ξ η ηρ

τ γ Φξ

ψξ

ψξ

ξ η ηρ

τ γ Φξ

ψξ

ψξ

η η

η η η η η η

τ τ

∇∇

∇∇

∇∇

= = ⋅∂∂

⋅ + · ∂∂

− ∂∂

= ( ) = ( ) ∂∂

+ · ∂∂

− ∂∂

= ( ) = ( ) ∂∂

+ · ∂∂

− ∂∂

( ) = ( ) + ( ( ) − ( )) ( )

( ) = ( ) + ( ( ) − ( )) ( ) ( ) = ( ) = ( − )

≔ ⋅ ⋅ ⋅ ⋅ ⋅ ≔ ⋅ ⋅ ⋅ ⋅ ⋅ ( )

η η η

α αβ β αβ β ββ β β

ω ωγ γ ωγ γ γγ γ γ

αβ αβ αβ αβ

ωγ ωγ ωγ ωγ

ββ β

γγ γ

=

=

4.2. Detailed form at large strains

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

P FU

F UU

EE

L J J

A GF

A y A y

A

A

Z

Z b

1: : ,

,

2 1 1 2 12 1,

,1

,1

1 1

1 1

0.5 1

0.5 1 .123

j jiT

et

ip t t

t

i

ee i t

ee i

i

i i i i ij

n

j i

nij i j

i

p Ai A

i

A Am M

i M

i

M M

k

pk

Ai

i

A AkA

kA

l

ml

Mi

i

M MlM

lM

Ai

i

pm A

AA n A

Mi

i

mm M

MM n M

ik k i

n

k

0

1

2

1

1

1

1

2 2

1

2 2

, 1

2 2 2 2

, 1

2 2 2 2

1

2 2 2

1

2 2 2 2

1,

2

∑ ∑

∑ ∑

∑

∑

∑

∑ ∑

ηρ η η

ψ ηψ η

η

η η η η ηη η

η

ηη

ξ ξηη

ξ ξ

ηη

ξ ξ ξ ξ

ηη

ξ ξ ξ ξ

β η

ηξ ξ ξ

β η

ηξ ξ ξ β η β η

= ⋅∂∂

⋅ −∂∂

( ) −∂ ( )

∂

− [ ( − )( − ) + Δ ( − )] −∂ ˜ ( )

∂

−∂ ( ¯ )

∂( ¯ ) ( − ¯ ) −

∂ ( ¯ )∂

( ¯ ) ( − ¯ )

−∂ ( )

∂( ¯ ) ( − ¯ ) ( ¯ ) ( − ¯ )

−∂ ( )

∂( ¯ ) ( − ¯ ) ( ¯ ) ( − ¯ )

−∂ ( )

∂((∇ ) + ( − ¯ ) (∇ ) )

−∂ ( )

∂((∇ ) + ( − ¯ ) (∇ ) ) + ∇ + ∇

( )

η

θ

α

αα

α αω

ωω

ω ω

α

αα α

ω

ωω ω

ξ

αα α α

ξ

ωω ω ω

η η

−

=

−

= +

= =

=

=

=

= = ≠

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

n n

n n

n n n

L Z

Z

Z Z

A y A

Z

61

12

1 1

12

12

1 1

2 1 1 1

2 , 1 1 2 2 1 1 2 1

1 .124

A Aj

A A A AA

A A Aj

Aj

A Aj

AA

A A

AA A

AA A

Aj A

A A Ak

Aj

A A AkA

kA

Aj A

A n A

0

2

2 2

2 2

2 2

2

ξ ηρ

τ γ ξ ξ ξ ξ β η

β η ξ β η ξ ξ ξ

ξ ξ ξ ξ

η ξ ξ ξ η ξ ξ ξ ξ ξ

β η ξ ξ

∇ ∇

∇ ∇ ∇ ∇

∇ ∇ ∇

= ( ) ¯ ( − ¯ ) + [ ( − ¯ ) − ]( ¯ ⋅ )( ( )⋅ )

+ ( )⋅ ¯ + ( )[∇ ¯ + ( ( − ¯ ) − )( ⋅ )( ¯ ⋅ )]

− ( − ¯ )( ¯ ⋅ ) + [ ( − ¯ ) − ] ( ¯ ⋅ )·

− ( ¯ ) ¯ ( − ¯ )( − ¯ ) − ( ) ¯ ( − ¯ )( − ¯ )( ¯ ) ( − ¯ )

+ ( ) ( − ¯ )(∇ )( )

α α α α α α α αα

ξα

ξ α ξ α αα

αα

α αα

α αα α

αα

α α α α α α α

ξ α α

V.I. Levitas, M. Javanbakht / J. Mech. Phys. Solids 82 (2015) 287–319316

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

n n

n n

n n n

L Z

Z

Z Z

A y A

Z

61

12

1 1

12

12

1 1

2 1 1 1

2 , 1 1 2 2 1 1 2 1

1 .125

M Mj

MA MA M MM

M M Mj

Mj

M Mj

MM

M M

MM M

MM M

Mj M

M M Ml

Mj

M M MlM

lM

Mj M

M n M

0

2

2 2

2 2

2 2

2

ξ ηρ

τ γ ξ ξ ξ ξ β η

β η ξ β η ξ ξ ξ

ξ ξ ξ ξ

η ξ ξ ξ η ξ ξ ξ ξ ξ

β η ξ ξ

∇ ∇

∇ ∇ ∇ ∇

∇ ∇ ∇

= ( ) ¯ ( − ¯ ) [ ( − ¯ ) − ]( ¯ ⋅ )( ( )⋅ )

+ ( )⋅ ¯ ++ ( )[∇ ¯ + ( ( − ¯ ) ) − )( ⋅ )( ¯ ⋅ )]

− ( − ¯ )( ¯ ⋅ ) + [ ( − ¯ ) − ] ( ¯ ⋅ )·

− ( ¯ ) ¯ ( − ¯ )( − ¯ ) − ( ) ¯ ( − ¯ )( − ¯ )( ¯ ) ( − ¯ )

+ ( ) ( − ¯ )(∇ )( )

ω ω ω ω ω ω ω ωω

ξω

ξ ω ξ ω ωω

ωω

ω ωω

ω ωω ω

ωω

ω ω ω ω ω ω ω

ξ ω ω

4.3. Small strains, linear elasticity

⎛⎝⎜⎜

⎞⎠⎟⎟I C

CL

J J1:

2: : :

2: : ,j ji

t

i

t t

ie i e

te

i

ie

0 0

2

0

σε ε

ε ε ε εηρ η ρ η

ηρ

ηη

=∂∂

− (∂∂

) ( ) −∂ ( )

∂+ ⋯η

where strain-unrelated terms remain the same. For dislocations, Eqs. (124) and (125) are the same for smallstrains and rotations.

5.

Momentum balance equation

I. Large strains

P f v. 1260 0ρ ρ∇⋅ + = ( )

II. Small strains

f v. 127σ ρ ρ∇⋅ + = ( )

6.

Boundary conditions for the order parameters

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

n nb

a p a p a p p2 3 4 2 4 3 ; .128

ij i j

n

j

sM

sA

i s s si

n

i

0 0 01,

2 3

1

2

0.5

∑

∑

ρ β η η

γ γ η η

∇ ∇· + ·

= ( − ) ( + ( − ) + ( − ) ) =( )

η

= ≠

=

n m n n m n

n n

; ;

.129

Ai

m A Mi

m M

s

pA

mM

0 0 0 0

01 1

∑ ∑

ρ β η ξ Λ ρ β η ξ Λ

γ ξ γ Λ ξ Λ ξ

( ) · ∇ ¯ = − ( ) ( ) · ∇ ¯ = − ( )

( ) = + ( ) + ( )( )

ξα

αα

ξω

ωω

αα

αα

ω

ωω

= =

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