Stress-Measure Dependence of Phase Transformation Criterion underFinite Strains: Hierarchy of Crystal Lattice Instabilities for Homogeneous and

Heterogeneous Transformations

Hamed Babaei1 and Valery I. Levitas 1,2,3

1Department of Aerospace Engineering, Iowa State University, Ames, Iowa 50011, USA2Departments of Mechanical Engineering, Iowa State University, Ames, Iowa 50011, USA

3Ames Laboratory, Division of Materials Science and Engineering, Ames, Iowa 50011, USA

(Received 6 November 2019; revised manuscript received 25 December 2019; accepted 17 January 2020;published 21 February 2020)

Hierarchy of crystal lattice instabilities leading to a first-order phase transformation (PT) is found, whichconsists of PT instability described by the order parameter and elastic instabilities under differentprescribed stress measures. After PT instability and prior to the elastic instability, an unexpected continuousthird-order PT was discovered, which is followed by a first-order PT after the elastic instability. Underprescribed compressive second Piola-Kirchhoff stress, PT is third order until completion; it occurs withouthysteresis and dissipation, properties that are ideal for various applications. For heterogeneousperturbations and PT, first-order PT occurs when the first elastic instability criterion (among criteriacorresponding to different stress measures) is met inside the volume, surprisingly independent of the stressmeasure prescribed at the boundary.

DOI: 10.1103/PhysRevLett.124.075701

Introduction.—Theoretical description of the mechanicalstability of a crystal lattice is one of the essential bases forunderstanding structural transformation in solids. The lossof stability of crystal lattice causes structural transforma-tions such as martensitic or displacive PTs, melting,amorphization, twinning, dislocation nucleation, cavita-tion, and fracture [1–7]. Therefore, crystal lattice instabilitycriteria have fundamental importance. However, despite thenumerous previous works in this direction, there are severaloutstanding problems to be resolved when instabilityoccurs at finite strains.The authors of the general elastic instability criteria for

finite strains [8,9] expressed them in terms of arbitrarymeasures of stress and work-conjugate strain and empha-sized that the instability criteria depend on the chosen(prescribed) stress or strain measure. Since for hetero-geneous solutions of a boundary-value problem stresstensors can only be prescribed at the external boundary,it is impossible to define which stress measure is prescribedat each material point, i.e., elastic instability is ambiguous.For applications, practically all instability criteria areformulated in terms of the Cauchy stress [1–3,10–12]without justification. Other approaches discussed in text-books, e.g., Ref. [13], define elastic instability based onloss of positive definiteness of some tensors of elasticmoduli (generalizing Born’s work [14] for finite strains) orof acoustic tensor [15] (generalizing Hadamard’s work[16]). They include, in particular, conditions for loss ofellipticity of the elastostatic equations [17]. Since elastic

moduli and acoustic tensors depend on the choice of strainand stress measures, this leads to a similar ambiguity.Different criteria are collected and compared, e.g., inRefs. [9,18]. In mathematical literature on martensiticPTs [7,17], loss of stability is necessary for developmentof martensitic and twinned microstructure.The alternative approach to the material instability was

based on the utilization of the order parameters describingPTs in the spirit of the Landau theory [4]. The PT instabilitycriterion for a homogeneous equilibrium phase underspontaneous variation of the order parameters was derivedin Refs. [19,20] using a phase-field approach (PFA) to thefirst-order PTs under large strains and applied stress tensor.This criterion is linear in the normal components of theCauchy stress tensor, which is confirmed by atomisticsimulations for PTs Si I ↔ Si II [10–12] and graphite-diamond [21].In this Letter we resolve some outstanding problems in

the crystal lattice instability. Initially, we consider homo-geneous perturbations and the PT process under a pre-scribed homogeneous stress measure. As it wasmathematically proven in Refs. [19,20], the PT instabilitycriteria are independent of the stress measures. We foundhere that they occur at the same strain. This is not the casefor elastic instability. That is why one has a hierarchy oflattice instabilities corresponding to PT instability andelastic instabilities under various prescribed stress mea-sures. When under the prescribed stress measure, the PTinstability occurs prior to the elastic instability; a new type

PHYSICAL REVIEW LETTERS 124, 075701 (2020)

0031-9007=20=124(7)=075701(6) 075701-1 © 2020 American Physical Society

of PT was discovered. There is no jump in strain and orderparameter, which occurs for first-order PT. The equilibriumvalues of the order parameter, corresponding to stableelastic equilibrium, continuously vary with varyingstresses, until elastic instability is reached. This PT is thirdorder in this stress range (see Supplemental Material [22]),with the equilibrium structure Siin corresponding to theintermediate structure along the initial part of the originaltransformation path for the first-order PT. When elasticinstability is reached, the first-order PT Siin ↔ Si II occurs.Since the second Piola-Kirchhoff stress (PK2S)-straincurve under compression does not possess an elasticinstability point, PT under the prescribed PK2S is athird-order PT until completion, and occurs without hys-teresis and dissipation under cyclic loading, properties thatare ideal for various applications. This opens the possibilityof controlling PT order and properties by controllingprescribed stress measures.For heterogeneous perturbations and the PT process,

stresses are prescribed at the boundaries only. After satisfy-ing the PT instability criterion and continuous third-orderPT, first-order PT to the product phase occurs when the firstelastic instability criterion (among criteria corresponding todifferent stress measures) is met inside the volume, inde-pendent of the prescribed stress measure at the boundary.We designate contractions of tensors A and B over one

and two indices as A · B ¼ fAijBjkg and A∶B ¼ AijBji; Iis the unit tensor; the transpose of A is AT and the inverse ofA is A−1. The deformation gradient F ¼ Fe · UtðηÞ, themapping crystal from an undeformed into a deformedconfiguration, is multiplicatively decomposed into elasticFe and transformational Ut parts; Ut maps the stress-freecrystal cell of the parent phase to that of the transformingphase; η is the order parameter which varies from 0 for theparent phase to 1 for the product phase. Lagrangian strainis E ¼ 0.5ðFT · F − IÞ.Phase transformation instability.—Instability of the

homogeneous equilibrium state of a phase under homo-geneous perturbations can only be analyzed when a par-ticular stress measure is prescribed at the boundary. It doesnot mean that the Cauchy (true) stress σ or the first Piola-Kirchhoff stress (PK1S) P ¼ Jσ · F−1T (force per unitundeformed area), which directly participate in the boun-dary conditions, can only be prescribed; here J ¼ detF. Anyother stress measure can be prescribed with the properfeedback and control of σ orP in the experiment or atomisticsimulations [22]. We will also use the Kirchhoff stressτ ¼ Jσ and the PK2S T ¼ F−1 · P ¼ JF−1 · σ · F−1T . Wewill start with prescribed P.PT instability for the thermodynamic equilibrium value

η ¼ 0 occurs and PT starts when the driving force X forchange in η in the Ginzburg-Landau equation (seeRef. [22]) is getting positive and η grows. The generalPT criterion that follows from this definition is derived inRef. [20]. For cubic-to-tetragonal PT under action of three

normal-to-cubic-faces Cauchy stresses σi, this criterion issimplified to

ðσ1 þ σ2Þεt1aε1 þ σ3εt3aε3 ≥ ðAþ 3ΔψθÞ=Je; ð1Þ

where εti are components of the transformation strainεt ¼ Utð1Þ − I, Je ¼ detFe, and aε, A, and Δψθ areinterpolation constants in expression for UtðηÞ, the magni-tude of the double-well barrier, and jump in the thermalenergy, respectively. This instability criterion was cali-brated and verified for Si I ↔ Si II PTs [23] using results ofatomistic simulations [11,12].Elastic instability is defined based on practical methods

used in experiments or simulations. If for the equilibriumstate under chosen fixed prescribed stresses at the boundaryfor some spontaneous perturbations of the deformationgradient ΔF the mechanical equilibrium cannot be kept,then such a state is unstable. For homogenous states andperturbation, such an instability criterion results in thecriterion presented in Ref. [8]. It is demonstrated that elasticinstability conditions depend on the prescribed stress (andwork-conjugate strain) measure [8,9]. We study the rela-tionship between PT and elastic instability conditions fordifferent prescribed stresses using PT between semicon-ducting Si I and metallic Si II using the PFA presented inRefs. [20,22,23]. To model homogeneous processes, weconsider the solution for a single cubic finite element.Elastic energy is ψe ¼ 0.5Ee∶CðηÞ∶Ee, where C is thefourth-order elastic moduli tensor, which leads to linearrelationship T ¼ CðηÞ∶Ee. Other stress measures are non-linearly related to Ee. To initiate PT, an initial valueη ¼ 0.01 is prescribed, and η cannot evolve below 0.01.We apply σ1 ¼ σ2 ¼ 1 GPa and perform slow strain-controlled compressive loading in the third direction,meaning that for each strain η reaches stationary value.Stress—strain E3 curves are shown in Fig. 1 for four stressmeasures. They are the primary information for the analysisof instability for heterogeneous processes as well as for thecase of stress-controlled loading. The strain-controlledhomogeneous loading does not allow the instability tooccur spontaneously.For all stress measures, PT instability occurs at the same

strain (Fig. 1). This explains how PT instability is inde-pendent of the prescribed stress measure.For small strains, analytical expression for the equilib-

rium stress-order parameter (obtained from conditionX ¼ 0 for η varying from 0 to 1) and the correspondingstress-strain curve describe reducing stress during PT [31].That is why when PT starts at fixed stress, it continues tocomplete until stress is equilibrated at the elastic branch ofthe product phase. In contrast, for finite strain in Fig. 1 afterPT instability, each stress measure continues to grow andreaches maximum (except T) corresponding to elasticinstability at corresponding prescribed stress and differentstrains for different stress measures. Indeed, at prescribed

PHYSICAL REVIEW LETTERS 124, 075701 (2020)

075701-2

stress corresponding to the maximum point positive per-turbation ΔE3 leads to reduction in elastic resistance andunstable deformation transformation until PT completionand equilibration of prescribed stress at the elastic branch ofthe product phase. This is further analyzed consideringstress-order parameter curves under three different pre-scribed stress measures (Fig. 2). The order parameter startsgrowing at stresses corresponding to PT instability inFig. 1. With increasing stresses, the order parameterevolves in a stable equilibrium and continuous way,describing smooth transition to intermediate structuresSiin along the pathway Si I → Si II. Since at PT instabilitystresses there is no jump in the order parameter and thecorresponding jump in strain and entropy, PT initially

occurs as a third-order PT [22]. When elastic instabilitystress is reached for the prescribed stress measure, the orderparameter grows in a nonequilibrium way to 1 and the PTproceeds until completion. This process is accompanied bya jump in the order parameter, strain, and entropy.Therefore, Siin → Si II PT is the first-order after elasticinstability. Thus, a hierarchy of the PTand elastic instabilitypoints is found under different prescribed stress measures.For the reverse, PT elastic instability and PT instability

coincide and occur at the same strain E3 ¼ 0.36 corre-sponding to the local stress minimum for any prescribedstress measure. The difference between stresses related toelastic instability for direct and reverse PTs constitutesstress hysteresis and energy dissipation during PT. Duringequilibrium, third-order PT between the PT instability pointand the elastic instability point, the PT or deformation isfully reversible without hysteresis for any prescribed stress.Since the equilibrium PK2S-strain curve in Fig. 1 does nothave a maximum related to elastic instability point, PTunder the prescribed increasing PK2S is a third-order PTuntil completion and occurs without hysteresis and dis-sipation. Reducing hysteresis and dissipation are importantfor various applications, e.g., for shape memory alloys[32–34] or caloric materials [34–36]; see also Ref. [10].PT and elastic instabilities under heterogeneous

perturbations in a finite volume.—During the solution ofboundary-value problems with heterogeneous fields,chosen stress measure can only be prescribed at theboundary, not for each material point within the bulk.This does not allow one to directly apply elastic instabilitycriteria obtained for homogenous states. Consequently, PTconditions under heterogeneous perturbations are notcurrently defined. To address this question, let us considera PT in a sample of sizes 20 × 60 × 5 nm3 under the sameloading as for the homogeneous field (Fig. 3). The left face,bottom face, and one of the faces in the thickness directionare fixed by zero normal-to-the-face displacement and aresymmetry planes. The right face and the other face in the

FIG. 2. Stress-order parameter curves for homogeneousdeformation in the third direction in which a stress-controlledcompressive loading for three different prescribed stress mea-sures is applied at σ1 ¼ σ2 ¼ 1 GPa.

Time (ps)0.0

0.07

0.3

0.75

1.5

3.0

30

0.0 0.01

0.0 0.01

0.0 0.07

0.0 0.22

FIG. 3. Nanostructure evolution for a triaxial compressive-tensile loading with initial random heterogeneous field η in therange (0;0.01]. Compressive PK1S is applied at the top face up tothe value P slightly above the peak for the Kirchhoff stress, butbelow the peak points for the Cauchy stress and PK1S in Fig. 1,along with σ1 ¼ σ2 ¼ 1 GPa. Presented solution is for the entiresample after mirroring with respect to the symmetry planes of thesimulation field.

FIG. 1. Equilibrium stress-strain curves for homogeneousdeformation of Si in the third spacial direction in which astrain-controlled compressive loading is applied atσ1 ¼ σ2 ¼ 1 GPa. Hierarchy of PT and elastic instability pointsare shown by markers. The intermediate phase of Si betweenpoints of PT and elastic instability, which appears via third-orderPT, is designated as Siin. P is the prescribed stress for problemsdescribed below.

PHYSICAL REVIEW LETTERS 124, 075701 (2020)

075701-3

thickness direction are under 1 GPa tensile Cauchy stress.The PK1S is prescribed at the top face up to a value Pslightly above the first peak point for the Kirchhoff stress,but below the peak points for the Cauchy stress and PK1Sin Fig. 1. Weak heterogeneity is introduced by a randomdistribution of the initial values of η ⊆ ð0; 0.01. Thesolution is shown in Fig. 3.From the results for a homogeneous field discussed

above, we expect that for the prescribed PK1S, the PTshould not continue unless we reach the PK1S peak point.Surprisingly, although the PK1S peak point is not reachedyet, the exceeding strain for elastic instability for theKirchhoff stress is sufficient for the initiation and com-pletion of the first-order PT. Within the bulk close to theupper right corner, where the internal stresses due toheterogeneity are maximum, the critical condition forinitiation of the first-order PT for the prescribedKirchhoff stress is met locally and the region of thecomplete product phase is formed and grows, producingcomplex stationary nanostructure with significant amountof Si II. Residual austenite is stabilized by changes ingeometry of the sample.However, for different loadings, while the general

principle is the same, the different stress measure produceselastic instability prior to other stress measures. Forinstance, we apply σ3 ¼ −8 GPa and increase tensilestrains E1 ¼ E2. The tensile stress-strain E2 curves areshown in Fig. 4. In contrast to the previous compressiveloading, here the PK2S peaks first and PK1S, Kirchhoff,and Cauchy stresses peak afterward. This means that PK2Sshould be the first elastic instability point for tensile loadingunder heterogeneous perturbations. For the initial values ofη ⊆ ð0; 0.01 we apply σ3 ¼ −8 GPa and increase tensileσ1 ¼ σ2 up to σ ¼ 10 GPa, slightly above the peak strainfor the PK2S but lower than the peak strains for otherstresses. As shown in Fig. 5, this is sufficient for initiationof the first-order PT and its completion in the major part of

the sample. Since the general expressions for the instabilitycriteria in Refs. [8,9] are expressed in arbitrary work-conjugate stress and strain measures, one should justsubstitute stress measures responsible for instabilityobtained above for different loadings to derive specificelastic instability criteria.In the above treatment we neglected phonon instability,

which may occur before elastic instability and limit maxi-mum stress [37–42]. In particular for Si, phonon instabilityoccurs at hydrostatic pressure at 26 GPa [43] (well belowelastic instability pressure of 75.8 GPa [12]) and at a shearstrain of 0.22 for simple shear causing second-order PT[38]. These results do not affect our solutions in Fig. 1because for uniaxial compression at σ1 ¼ σ2 ¼ 0 GPaphonon instability was not found before elastic instability[44]; we are not aware of any published results on phononinstability for loading considered in Fig. 4. Generally, ifstress-strain curves include information about phononinstability (and other instabilities, like electronic, etc.)and the following equilibrium processes obtained withatomistic simulations (see, e.g., Refs. [24,38]), our approachcan be applied in the same way. In particular, it can beapplied to the PFA to PTwhen an order parameter describesphonon instability [45].Conclusions.—A number of basic long-term problems in

lattice instability under finite strains are resolved in thisLetter. Using PFA simulations for the Si I ↔ Si II PTs, it isshown that the PT instability is independent of theprescribed stress measures because it occurs at the samestrain for any prescribed stress. Prior to elastic instabilityand after PT instability, a continuous third-order PT isdiscovered, which is followed by a first-order PT afterelastic instability until PT completion. Thus the trans-formation path changes to Si I → Siin → Si II. Third-orderPTs are quite rarely discussed in literature (see, e.g.,Ref. [46] and references); we are not aware of any previousreports for elastic materials or in connection with the first-order PTs. Under prescribed compressive second Piola-Kirchhoff stress, PT is third order until completion; itoccurs without hysteresis and dissipation under cyclicloading, properties that are ideal for various applications.

FIG. 4. Equilibrium stress-strain curves for homogeneousdeformation under σ3 ¼ −8 GPa and increasing tensile strainsE1 ¼ E2.

Time (ps)0.0

0.25

0.5

1.5

3.5

3.9

6.0

0.0 0.01

0.0 0.01

0.0 0.82

0.0 0.20

FIG. 5. Nanostructure evolution for random distribution ofinitial η ⊆ ð0; 0.01, σ3 ¼ −8 GPa and increasing tensile σ1 ¼ σ2up to σ ¼ 10 GPa, slightly above the peak strain for the PK2S butlower than the peak strains for other stresses.

PHYSICAL REVIEW LETTERS 124, 075701 (2020)

075701-4

Since for heterogeneous perturbations stress tensors canonly be prescribed at the external boundary, it is impossibleto define which stress measure is prescribed at eachmaterial point, i.e., elastic instability is ambiguous. Afterthird-order PT, the first-order PT occurs when the firstelastic instability criterion (among criteria corresponding todifferent stress measures) is met inside the volume, sur-prisingly independent of the stress measure prescribed atthe boundary. For two considered loadings, the elasticinstability corresponds to the prescribed Kirchhoff stressand PK2S, respectively, even when the Cauchy stress wasprescribed. None of our results suggests Cauchy-stress-based criterion, in contrast to most of the previouspublications. This also means that finding the PT criterionfrom atomistic simulations [3,10–12] in terms of theCauchy stress should be reconsidered. The general latticeinstability criterion should be found for all possibleprescribed stress tensors, which involves the choice ofthe local prescribed stresses corresponding to the firstinstability being different for different stress states. Onemore problem that was not considered here is taking intoaccount the finite particle and lattice rotations. Thisproblem was solved for PT instability and homogeneousperturbations in Refs. [19,20], which may help, along withthe numerical procedure presented here, to treat elasticinstabilities under heterogeneous perturbations.

The support of NSF (CMMI-1536925 and MMN-1904830), ARO (W911NF-17-1-0225), and Iowa StateUniversity (Vance Coffman Faculty Chair Professorship)are gratefully acknowledged. The simulations were per-formed at the Extreme Science and Engineering DiscoveryEnvironment (XSEDE), allocation TG-MSS170015.

[1] F. Milstein and R. Hill, Divergences among the Born andClassical Stability Criteria for Cubic Crystals under Hydro-static Loading, Phys. Rev. Lett. 43, 1411 (1979).

[2] F. Milstein, J. Marschall, and H. E. Fang, Theoreticalbcc ↔ fcc Transitions in Metals via Bifurcations underUniaxial Load, Phys. Rev. Lett. 74, 2977 (1995).

[3] J. Wang, J. Li, S. Yip, S. Phillpot, and D. Wolf, Mechanicalinstabilities of homogeneous crystals, Phys. Rev. B 52,12627 (1995).

[4] L. D. Landau and E. M. Lifshits, Statistical Physics(Pergamon, New York, 1980), Vol. 5.

[5] H. Hang-Sheng and R. Abeyaratne, Cavitation in elastic andelastic-plastic solids, J. Mech. Phys. Solids 40, 571 (1992).

[6] J. M. Ball, Discontinuous equilibrium solutions andcavitation in nonlinear elasticity, Phil. Trans. R. Soc. A306, 557 (1982).

[7] R. D. James, Finite deformation by mechanical twinning,Arch. Ration. Mech. Anal. 77, 143 (1981).

[8] R. Hill, On the elasticity and stability of perfect crystals atfinite strain, inMathematical Proceedings of the Cambridge

Philosophical Society (Cambridge University Press,Cambridge, England, 1975), Vol. 77, pp. 225–240.

[9] R. Hill and F. Milstein, Principles of stability analysis ofideal crystals, Phys. Rev. B 15, 3087 (1977).

[10] V. I. Levitas, H. Chen, and L. Xiong, Triaxial-Stress-Induced Homogeneous Hysteresis-Free First-Order PhaseTransformations with Stable Intermediate Phases, Phys.Rev. Lett. 118, 025701 (2017).

[11] V. I. Levitas, H. Chen, and L. Xiong, Lattice instabilityduring phase transformations under multiaxial stress: modi-fied transformation work criterion, Phys. Rev. B 96, 054118(2017).

[12] N. A. Zarkevich, H. Chen, V. I. Levitas, and D. D. Johnson,Lattice Instability during Solid-Solid Structural Transfor-mations under a General Applied Stress Tensor: Example ofSi I → Si II with Metallization, Phys. Rev. Lett. 121, 165701(2018).

[13] A. I. Lurie, Non-Linear Theory of Elasticity (Elsevier,New York, 1990), Vol. 36.

[14] M. Born, On the stability of crystal lattices. I, in Mathe-matical Proceedings of the Cambridge PhilosophicalSociety (Cambridge University Press, 1940), Vol. 36,pp. 160–172.

[15] R. Abeyaratne and J. K. Knowles, On the stability ofthermoelastic materials, J. Elast. 53, 199 (1998).

[16] J. Hadamard, Leçons sur la Propagation des Ondes et leseQuations de l’Hydrodynamique (A. Hermann, 1903).

[17] J. K. Knowles and E. Sternberg, On the failure of ellipticityand the emergence of discontinuous deformation gradientsin plane finite elastostatics, J. Elast. 8, 329 (1978).

[18] J. Clayton, Towards a nonlinear elastic representation offinite compression and instability of boron carbide ceramic,Philos. Mag. 92, 2860 (2012).

[19] V. I. Levitas, Phase-field theory for martensitic phase trans-formations at large strains, Int. J. Plast. 49, 85 (2013).

[20] V. I. Levitas, Phase field approach for stress- and temper-ature-induced phase transformations that satisfies latticeinstability conditions. Part I. General theory, Int. J. Plast.106, 164 (2018).

[21] Y. Peng and L. Xiong, Atomistic computational analysis ofthe loading orientation-dependent phase transformation ingraphite under compression, JOM 71, 3892 (2019).

[22] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.124.075701 for thesystem of equations for the phase field approach to phasetransformations in elastic materials, proof of the third-orderphase transformation at the phase transformation instabilitypoint before an elastic instability, and a heterogeneous fieldof internal stresses under triaxial compression, whichincludes Refs. [8–11,13,20,23–30].

[23] H. Babaei and V. I. Levitas, Phase-field approach for stress-and temperature-induced phase transformations that satis-fies lattice instability conditions. Part II. simulations ofphase transformations Si I ↔ Si II, Int. J. Plast. 107, 223(2018).

[24] R. S. Elliott, N. Triantafyllidis, and J. A. Shaw, Reversiblestress-induced martensitic phase transformations in a bi-atomic crystal, J. Mech. Phys. Solids 59, 216 (2011).

[25] S. G. Lekhnitskii, Theory of Elasticity of an AnisotropicElastic Body (1963).

PHYSICAL REVIEW LETTERS 124, 075701 (2020)

075701-5

[26] R. G. Hennig, A. Wadehra, K. P. Driver, W. D. Parker, C. J.Umrigar, and J. W. Wilkins, Phase transformation in Si fromsemiconducting diamond to metallic β-Sn phase in QMCand DFT under hydrostatic and anisotropic stress, Phys.Rev. B 82, 014101 (2010).

[27] V. I. Levitas, D. L. Preston, and D.-W. Lee, Three-dimen-sional Landau theory for multivariant stress-induced mar-tensitic phase transformations. III. Alternative potentials,critical nuclei, kink solutions, and dislocation theory, Phys.Rev. B 68, 134201 (2010).

[28] H. Babaei, A. Basak, and V. I. Levitas, Algorithmic aspectsand finite element solutions for advanced phase fieldapproach to martensitic phase transformation under largestrains, Comput. Mech. 64, 1177 (2019).

[29] M. Parrinello and A. Rahman, Polymorphic transitions insingle crystals: A new molecular dynamics method, J. Appl.Phys. 52, 7182 (1981).

[30] R. E. Miller, E. B. Tadmor, J. S. Gibson, N. Bernstein, andF. Pavia, Molecular dynamics at constant Cauchy stress,J. Chem. Phys. 144, 184107 (2016).

[31] V. I. Levitas and D. L. Preston, Three-dimensional Landautheory for multivariant stress-induced martensitic phasetransformations. I. austenite ↔ martensite, Phys. Rev. B66, 134206 (2002).

[32] J. Cui, Y. S. Chu, O. O. Famodu, Y. Furuya, J. Hattrick-Simpers, R. D. James, A. Ludwig, S. Thienhaus, M. Wuttig,Z. Zhang et al., Combinatorial search of thermoelasticshape-memory alloys with extremely small hysteresiswidth, Nat. Mater. 5, 286 (2006).

[33] C. Chluba, W. Ge, R. L. de Miranda, J. Strobel, L. Kienle, E.Quandt, and M. Wuttig, Ultralow-fatigue shape memoryalloy films, Science 348, 1004 (2015).

[34] Y. Song, X. Chen, V. Dabade, T.W. Shield, and R. D. James,Enhanced reversibility and unusual microstructure of a phase-transforming material, Nature (London) 502, 85 (2013).

[35] I. Takeuchi and K. Sandeman, Solid-state cooling withcaloric materials, Phys. Today 68, No. 12, 48 (2015).

[36] H. Hou, E. Simsek, T. Ma, N. S. Johnson, S. Qian, C. Cisse,D. Stasak, N. A. Hasan, L. Zhou, Y. Hwang, R.Radermacher, V. I. Levitas, M. J. Kramer, M. Asle Zaeem,A. P. Stebner, R. T. Ott, J. Cui, and I. Takeuchi, Fatigue-resistant high-performance elastocaloric materials via addi-tive manufacturing, Science 366, 1116 (2019).

[37] D. M. Clatterbuck, C. R. Krenn, M. L. Cohen, and J. L.Morris, Jr., Phonon Instabilities and the Ideal Strength ofAluminum, Phys. Rev. Lett. 91, 135501 (2003).

[38] S. M. M. Dubois, G. M. Rignanese, T. Pardoen, and J. C.Charlier, Ideal strength of silicon: An ab initio study, Phys.Rev. B 74, 235203 (2006).

[39] P. Řehák, M. Černý, and M. Šob, Mechanical stability of niand ir under hydrostatic and uniaxial loading, Model. Simul.Mater. Sci. Eng. 23, 055010 (2015).

[40] M. Černý, P. Řehák, and J. Pokluda, The origin of latticeinstability in bcc tungsten under triaxial loading, Philos.Mag. 97, 2971 (2017).

[41] M. Friák, D. Holec, and M. Šob, An ab initio study ofmechanical and dynamical stability of mosi2, J. AlloysCompd. 746, 720 (2018).

[42] J.-H. Wang, Y. Lu, X.-L. Zhang, and X.-h. Shao, Theelastic behaviors and theoretical tensile strength of γ-tialalloy from the first principles calculations, Intermetallics101, 1 (2018).

[43] K. Gaál-Nagy and D. Strauch, Phonons in the β-tin, i m m a,and sh phases of silicon from ab initio calculations, Phys.Rev. B 73, 014117 (2006).

[44] N. A. Zarkevich, V. I. Levitas, H. Chen, and D. D. Johnson,Phonon spectra and instabilities in Si I and II under triaxialloadings (to be published).

[45] P. Toledano and V. Dmitriev, Reconstructive Phase Tran-sitions: In Crystals and Quasicrystals (World Scientific,Singapore, 1996).

[46] F. D. Cunden, P. Facchi, M. Ligabò, and P. Vivo, Univer-sality of the third-order phase transition in the constrainedcoulomb gas, J. Stat. Mech. (2017) 053303.

PHYSICAL REVIEW LETTERS 124, 075701 (2020)

075701-6

Supplemental Material: Stress-measure dependence of phase transformation criterionunder finite strains: Hierarchy of crystal lattice instabilities for homogeneous and

heterogeneous transformations

Hamed Babaei1 and Valery I. Levitas1, 2, 3

1Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA2Departments of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA

3Ames Laboratory, Division of Materials Science and Engineering, Ames, IA 50011, USA(Dated: March 1, 2020)

SYSTEM OF EQUATIONS FOR PHASE FIELDAPPROACH TO PHASE TRANSFORMATIONS

IN ELASTIC MATERIALS

The formulations in this work are based on the generaladvanced phase field approach developed in [1], whichwas specified for cubic-to-tetragonal Si I↔Si II phasetransformations (PTs) in [2].

Some designations. We designate contractions of ten-sors AAA = Aij and BBB = Bij over one and two indicesas AAA···BBB = AikBkj and AAA:::BBB = AijBji, respectively; Aijand Bij are the components of tensors in Cartesian unitbasis, and summation is assumed over the repeated in-dices, unless otherwise stated. The superscripts t and −1indicate the transpose and inverse of the tensor; and III isthe second-rank unit tensor.

Kinematics

Let a vector function rrr = rrr(rrr0, t) describe the motionof the elastic material with PTs. Here rrr0 and rrr are thepositions of points in the reference (undeformed) Ω0 andthe actual (deformed) Ω configurations, respectively, andt is the time. Multiplicative decomposition of the de-formation gradient FFF = ∂rrr

∂rrr0= ∇∇∇0rrr into elastic FFF e and

transformation UUU t parts is utilized:

FFF = FFF e···UUU t; UUU t = UUU tt. (S1)

The Jacobian determinants describing the ratio of vol-umes in the corresponding configurations are

J =dV

dV0= detFFF ; Je =

dV

dVt= detFFF e;

Jt =dVtdV0

= detUUU t; J = JeJt, (S2)

where V0, Vt, and V are the volumes in the reference Ω0,stress-free intermediate Ωt (after elastic unloading), andactual Ω configurations, respectively. The Lagrangiantotal, elastic, and transformation strains are given re-spectively by

EEE =1

2(FFF t···FFF − III); EEEe =

1

2(FFF te···FFF e − III);

EEEt =1

2(UUU t···UUU t − III); EEE = UUU t···EEEe···UUU t +EEEt. (S3)

Phase transformation is described by the order param-eter η with η = 0 for the parent phase P0 and η = 1 forthe product phase P1. The transformation deformationgradient is expressed as

UUU t(η) = III + εεεt ϕϕϕ(aaaε,wwwε, η);

ϕϕϕ := aaaεη2 + (10ιιι− 3aaaε +wwwε) η

3 + (3aaaε − 2wwwε − 15ιιι)η4

+(6ιιι− aaaε +wwwε)η5, (S4)

where εεεt = UUU t(1) − III is the transformation strain af-ter complete transformation from the parent phase P0 tothe product phase P1; ϕϕϕ (and consequently, aaaε,wwwε, ιιι) arematrices (not second-rank tensors), which have the samenon-zero components and symmetry as εεεt in the coor-dinate system of the crystal lattice of P0; all non-zerocomponents of matrix ιιι are equal to one. The Hadamardproduct is defined as εεεt ϕϕϕ := εijt ϕij with no sum-mation over i and j. For cubic-to-tetragonal PT, thediagonal terms of transformation strain and all matricesare nonzero only, and two of three terms are the same forall matrices.

Helmholtz free energy

We accept the expression for the Helmholtz free energyper unit reference volume as

ψ(FFF , η, θ,∇∇∇0η) = Jtψe + ψθ + ψ∇, (S5)

where ψe is the elastic energy per unit volume in the in-termediate configuration Ωt, which is the reference con-figuration for the elasticity rule; ψθ is the thermal energy,which includes the double-well barrier between phases aswell as the thermal driving force for phase transforma-tion; and ψ∇ is the gradient energy which penalizes in-terfaces.

A quadratic elastic energy for an orthotropic materialis given by [3]

ψe =1

2EEEe:::CCC:::EEEe =

1

2CijklEije E

kle (S6)

=1

2

[λn(Ee

nn)2 + 2µnEnne Ekke + 4νnEnke Ekne].

The components of the fourth-rank elastic moduli tensor

2

CCC are expressed as

Cijkl = λnδinδjnδknδln + µn(δinδjnδkl + δijδknδln)

+νn(δinδjkδln + δjnδikδln + δinδjlδkn + δjnδilδkn),

(S7)

where λn, µn, and νn are nine independent elastic con-stants, with three for each of the three orthogonal di-rections and δjl is the Kronecker delta or components ofthe unit tensor. For materials with a tetragonal crystallattice, the constants for n = 2 and 3 are the same; forcubic lattices, constants for all n are the same; see [2] forexplicit expressions of elastic constants. The constantsλn, µn and νn are interpolated during the PT as

λn = λn0 + (λn1 − λn0 )ϕe(η); µn = µn0 + (µn1 − µn0 )ϕe(η);

νn = νn0 + (νn1 − νn0 )ϕe(η), (S8)

with λn0 , µn0 , νn0 and λn1 , µn1 , νn1 being the elastic constantsof P0 and P1, respectively. The corresponding interpola-tion function is [1]:

ϕe(η) = η3(10− 15η + 6η2). (S9)

The thermal part of the free energy can be expressedas

ψθ = Aη2(1− η)2 + ∆ψθ(3η2 − 2η3), (S10)

where A is the magnitude of the double-well barrier be-tween P0 and P1 and ∆ψθ is the difference between thethermal free energy of P0 and P1.

The gradient part of the free energy is presented in astandard form:

ψ∇ =β

2|∇∇∇0η|2, (S11)

where β is the gradient energy coefficient.

Ginzburg-Landau equation

The evolution of the order parameter and correspond-ing martensitic nanostructure can be described by theGinzburg-Landau equation, which represents a linear re-lationship between the rate of change of the order param-eter, η, and the conjugate generalized thermodynamicforce, X:

η = LX = L

(−∂ψ∂η

∣∣∣EEE

+∇∇∇0···(

∂ψ

∂∇∇∇0η

)), (S12)

where L > 0 is the kinetic coefficient. Substituting thefree energy Eq. (S5) into Eq. (S12) results in the moreexplicit but still compact form of the Ginzburg-Landauequation in the reference configuration

η = LX = L

(PPPT ···FFF e:::

∂UUU t∂η− Jt

∂ψe

∂η

∣∣∣EEEe

−

JtψeUUU−1t :::

∂UUU t∂η− ∂ψθ

∂η+ β∇2

0η

), (S13)

where PPP is the first Piola-Kirchhoff stress (PK1S).

Equilibrium equation in the reference configuration

∇∇∇0···PPP = 000. (S14)

Boundary conditions for the order parameter

Assuming that the surface energy does change duringthe PT, one obtains boundary conditions for the evolu-tion of the order parameter as

nnn0···∇∇∇0η = 0, (S15)

where nnn0 is the normal to the surface in Ω0.

Properties of the interpolation functions

For any material parameter m, interpolation functionϕm(η) satisfy the following properties:

ϕm(0) = 0, ϕm(1) = 1; (S16)

dϕm(0)

dη=dϕm(1)

dη= 0. (S17)

They allow one to reproduce parameters of the parentand product phases at equilibrium state X = 0 for anystress and temperature. In addition, for elastic param-eters the interpolation function ϕe in Eq. (S9) satisfiesthe condition

d2ϕe(0)

dη2=d2ϕe(1)

dη2= 0. (S18)

This condition allows one to ensure that the lattice insta-bility conditions Eq. (S19) are linear in normal stresses,in accordance with molecular dynamics results [4, 5].

Calibration of the crystal lattice instability criteriafor Si I-Si II PTs using results of molecular

dynamics simulations

The material parameters can be calibrated based onthe results of molecular dynamics simulations in [4, 5]. Aselaborated in [2], the crystal lattice instability conditionsfor cubic-to-tetragonal PTs can be presented as

P0 → P1 : (σ1 + σ2)εt1aε1 + σ3εt3aε3 ≥1

Je(A+ 3∆ψθ);

P1 → P0 : (σ1 + σ2)εt1wε11 + εt1

+σ3εt3wε31 + εt3

≥ 1

J(A− 3∆ψθ)

(S19)

where the components of the transformation strain ten-sor for Si I→Si II PT are εt1 = εt2 = 0.1753 and

3

εt3 = −0.447. Other calibrated parameters, includingthe jump in thermal energy ∆ψθ = 6.35 GPa, interpola-tion constants aaaε and wwwε for transformation strain, anddouble-well barrier magnitude A, are (see [2] for moredetails)

if − σ3 ≥ 6.23 GPa→ A = 0.75 GPa, aε1 = 3.31,

aε3 = 3.60, wε1 = −2.48, wε3 = −2.39,

otherwise→ A = −9.48 GPa, aε1 = 1.10,

aε3 = 2.26, wε1 = −3.88, wε3 = −3.73. (S20)

In addition, the following material parameters are used[6, 7]: L = 2600(Pa.s)−1, β = 2.59 × 10−10N, C11

0 =C22

0 = C330 = 167.5GPa, C44

0 = C550 = C66

0 =80.1GPa, C11

1 = C221 = 174.76GPa, C33

1 =136.68GPa, C44

1 = C551 = 60.24GPa, C66

1 =42.22GPa, C12

1 = 102GPa, C131 = C23

1 = 68GPa.Finite element algorithm for solution of boundary-

value problems was presented in [2, 8].

PROVE OF THE THIRD-ORDER PHASETRANSFORMATION AT PHASE

TRANSFORMATION INSTABILITY POINTBEFORE AN ELASTIC INSTABILITY

Here, we aim to prove that the phase transformationis the third-order at PT instability point if it occurs be-fore an elastic instability. To this end, we show that thefirst and second derivatives of the Gibbs energy with re-spect to stress is continuous at the PT instability point,but the third derivative is discontinuous. Consideringthe second Piola-Kirchhoff stress (PK2S) tensor TTT as anindependent parameter, the Gibbs potential per unit un-deformed volume is defined as

G(TTT , θ, η) := ψ(EEEe, θ, η)−TTT :::EEE, (S21)

where θ is temperature, which is assumed to be constantduring this derivation. The two terms in this definition,namely, free energy and stress work can be explicitly ex-pressed with the help of Eqs. (S3) and (S5) as

ψ(EEEe, θ, η) =Jt2EEEe:::CCC(η):::EEEe + ψθ(θ, η); (S22)

TTT :::EEE = TTT :::(UUU t ·EEEe ·UUU t +EEEt). (S23)

Next, we determine PK2S and the elastic strain:

TTT = JtUUU−1t ···

∂ψ

∂EEEe···UUU−1t = JtUUU

−1t ···(CCC:::EEEe)···UUU−1t ⇒

EEEe =1

Jtλλλ:::(UUU t···TTT ···UUU t); λλλ = CCC−1, (S24)

where λλλ is the elastic compliance tensor; both CCC and λλλare symmetric with respect to exchange of components

1 ↔ 2, 3 ↔ 4, and pairs 1, 2 ↔ 3, 4. Defining the PK2STTT in the intermediate stress-free configuration leads to

TTT = J−1t UUU t···TTT ···UUU t ⇒ EEEe = λλλ:::TTT . (S25)

Then the Gibbs potential is expressed in terms of PK2STTT as

G(TTT , θ, η) = −1

2TTT :::λλλ:::TTT + ψθ(θ, η)− TTT :::EEEt (S26)

with

EEEt :=Jt2

(III −UUU−1t ···UUU−1t ). (S27)

The thermodynamic equilibrium condition

X = −∂G∂η

= 0 → η = 0; η = 1; ηe = ηe(TTT , θ) (S28)

has two roots η = 0 and η = 1 because of Eqs. (S17) and

the third root, ηe(TTT , θ), which determines the equilibriumevolution of the order parameter during PT. Since whenPT instability precedes the elastic instability jump tothe product phase η = 1 is impossible, we will studytransformation behavior in the vicinity of the PT pointη = 0. Thus, along the equilibrium path ηe(TTT , θ) onwhich ∂G

∂η |ηe = 0, we have

dG

dTTT=∂G

∂TTT+∂G

∂η

∂ηe

∂TTT=∂G

∂TTT= −λλλ:::TTT − EEEt. (S29)

Therefore, the second and third derivatives of the Gibbsenergy along the equilibrium path ηe(TTT , θ) are

d2G

dTTT2 =

∂

∂TTT

(dG

dTTT

)+

∂

∂η

(dG

dTTT

)∂ηe

∂TTT

= −λλλ−

(dλλλ

dη:::TTT +

dEEEtdη

)∂ηe

∂TTT; (S30)

d3G

dTTT3 =

∂

∂TTT

(d2G

dTTT2

)+

∂

∂η

(d2G

dTTT2

)∂ηe

∂TTT

= −2dλλλ

dη

∂ηe

∂TTT−

(dλλλ

dη:::TTT +

dEEEtdη

)∂2ηe

∂TTT∂TTT

−

(d2λλλ

dη2:::TTT +

d2EEEtdη2

)∂ηe

∂TTT

∂ηe

∂TTT(S31)

in which

dEEEtdη

=dEEEtdUUU t

:::dUUU tdη

; (S32)

d2EEEtdη2

=dEEEtdUUU t

:::d2UUU tdη2

+d

dUUU t

(dEEEtdUUU t

:::dUUU tdη

):::dUUU tdη

.(S33)

To investigate the continuity of the first, second, andthird derivatives of the Gibbs energy at the PT instabilitypoint, we evaluate the equality of the values of derivativesin Eqs. (S29)-(S31) before the PT instability (when η ≡

4

0 and all η-derivatives are 0) with those at η = 0 butkeeping all η-derivatives). Before PT

dG

dTTT= −λλλ0:::TTT ;

d2G

dTTT2 = −λλλ0;

d3G

dTTT3 = 0. (S34)

During PT, based on Eqs. (S16) and (S17), we obtain

dUUU t(0)

dη=dλλλ(0)

dη= 0, (S35)

and

dG

dTTT

∣∣∣∣∣η=0

= −λλλ0:::TTT ;d2G

dTTT2

∣∣∣∣∣η=0

= −λλλ0, (S36)

and

d3G

dTTT3

∣∣∣∣∣η=0

= −

(d2λλλ(0)

dη2:::TTT +

dEEEt(0)

dUUU t:::d2UUU t(0)

dη2

)∂ηe(0)

∂TTT

∂ηe(0)

∂TTT. (S37)

The first and second derivatives are continuous at thePT point. However, the third derivative is discontinuous

because it is not zero in Eq. (S37). Indeed, ∂ηe(0)

∂˜TTT6= 0;

next, d2UUU t(0)dη2 6= 0 for model under study Eq. (S4) and in

general, because otherwise transformation strain wouldnot contribute to the PT condition Eq. (S19), which is

unrealistic. Note that for the current model d2λλλ(0)dη2 = 0

according to Eq. (S18), while this was not the case forprevious models.

Thus, it was proved that the PT is the third-order atthe PT instability when it precedes the elastic instability.

Prescribing boundary conditions is terms ofdifferent stress measures

In experiments or simulations, either displacement orforces are prescribed at the surface. For the prescribedelemental force dttt

dttt = pppdS = ppp0dS0, (S38)

two types of the boundary conditions are used in thereference Ω0 and deformed Ω configurations, respectively:

ppp0 = PPP ·nnn0; ppp = σσσ ·nnn. (S39)

Here, dS is infinitesimal surface element with the unitnormal nnn in the deformed configuration and dS0 andunit normal nnn0 are corresponding surface and normal inthe undeformed configuration; ppp and ppp0 are the tractionvectors in the deformed and undeformed states, respec-tively. Since in the reference state dS0 and nnn0 are fixedand known, controlling force dttt gives immediate controlover the components of the PK1S PPP contributing to the

traction. In particular, if we want to study processesat fixed PPP , we should just fix the force dttt. In contrast,if we want to study deformation at fixed (or controlled)Cauchy stress σσσ, since dS and nnn vary during deformation,force dttt should be adjusted according to equation

dttt = pppdS = σσσ ·nnndS = σσσ · JFFF−1T ·nnn0dS0. (S40)

In Eq. (S40) we utilized Eqs. (S38), (S39), and theNanson’s relationship [9] between oriented area in thedeformed and undeformed states:

nnndS = JFFF−1T ·nnn0dS0. (S41)

Thus, in experiment one has to measure continuously thecomponents of the deformation FFF participating in chang-ing nnndS at the deformed surface and use this feedback tochange force dt at fixed (controlled) σσσ according to Eq.(S40). This may be problematic in experiment. In atom-istic and continuum simulations, adjustment of force canbe easily done iteratively since FFF is known at each itera-tion or time step of the solution.

If one wants to study processes at prescribed otherstress measures, like the Kirchhoff stress or PK2S, theCauchy stress in Eq. (S40) should be expressed in termsof these stresses:

dttt = σσσ · JFFF−1T ·nnn0dS0 = τττ ·FFF−1T ·nnn0dS0

= FFF ·TTT ·nnn0dS0. (S42)

Thus, if one can control (fix) in experiment the Cauchystress, controlling (fixing) other stress measures is notmore complex. In molecular dynamics, the first Piola-Kirchhoff stress is controlled by controlling force. TheParinello-Rahman algorithm [10] controls the secondPiola-Kirchhoff stress [11]. Modification of this algorithmmade in [11] allowed to control Cauchy stress. In ad-dition, in [12] the Biot stress is prescribed for uniaxialloading, which excludes the effect of finite rotations fromthe first Piola-Kirchhoff stress. For our rotation-free de-formations, both stresses coincide.

If someone is still uncomfortable with prescription ofstresses other than the first Piola-Kirchhoff stress andthe Cauchy stress at the boundary, we can avoid thiswithout changing the main results of the paper. As it isdescribed in the main text, the strain-controlled homo-geneous loading provides the primary information for theanalysis of instability for heterogeneous processes as wellas for the case of stress-controlled loading. Indeed, straincontrolled loading allows to obtain stress-strain curves forany chosen stress (and strain) measure (Fig. 1). Then forheterogeneous processes, independent of the prescribedstress measure at the boundary, elastic instability occurswhen the first peak stress (among peak stresses corre-sponding to different stress measures) is reached. Also,since general expressions for the instability criteria in[13, 14] are expressed in arbitrary work-conjugate stressand strain measures, one should just substitute stress

5

measures responsible for instability obtained in the maintext for different loadings to derive specific elastic insta-bility criteria.

HETEROGENEOUS FIELD OF INTERNALSTRESSES UNDER TRIAXIAL COMPRESSION

To further investigate the internal processes leading tothe elastic instability for heterogeneous fields, we analyzethe solution presented in Fig. 3 for compressive PK1S atthe top face along with σ1 = σ2 = 1 GPa, see Figs. S1and S2. Solutions are presented for three time instants:(a) at PT instability; (b) and (c) at P3 = P slightlyabove the peak for the Kirchhoff stress, but below thepeak points for the Cauchy stress and PK1S.

The fields of the order parameter and all normalCauchy stresses are shown in Fig. S1. In addition, inFig. S2 the deviation of the normal Cauchy stresses fromthe averaged stress in direction 3 and external prescribedstresses σ1 = σ2 = 1 GPa are plotted along the horizontalline in the middle of the sample shown in Fig. S1.

Heterogeneity of stresses are caused by heterogeneityof the order parameter. In addition, at the top surface,even for homogeneous traction for the PK1S heterogene-ity in the deformation gradient causes heterogeneity ofthe traction for the Cauchy and other stresses. It can beobserved in both Figs. S1 and S2 that at the PT insta-bility point, the initial perturbation of order parameterbetween 0-0.01 produces a deviation of stresses from theaveraged values smaller than 10−5 GPa for the lateraltensile stresses σ11 = σ22 =1 GPa and smaller than 10−3

GPa for the compressive σ33= -11.81 GPa. At point Pwhen the maximum order parameter slightly increases to0.075 during the increase of an averaged Cauchy com-pressive stress to -12.75 GPa, the maximum deviationfrom the external stresses is still smaller than 5 × 10−3

GPa. This proves that despite the initial heterogene-ity and small changes in geometry, the stresses in thebulk remains below the peak point of Cauchy stress andPK1S. Yet, passing the first elastic instability point forKirchhoff stress causes heterogeneous elastic instabilityprocess and continuation of PT as it can be seen in Figs.S1 and S2 (c), where the maximum order parameter in-

creases to 0.25 while the solution localizes. This leadsto an increase in the deviation of stresses from averagedvalues up to 0.25 GPa for σ11 and σ22 and 0.4 GPa forσ33 (leading to reduction in local compressive σ33).

Thus, for heterogeneous fields under compressive load-ing, for which the Kirchhoff stress peak is at lower strainscompared to the Cauchy and PK1S, passing the elasticinstability for the Kirchhoff stress in terms of prescribed(averaged) stresses is sufficient for occurring progressingelastic instability leading to the first-order PT even ifother stress measures such as Cauchy or PK1S are pre-scribed at the boundary and do not reach correspond-ing instability peaks in terms of prescribed (averaged)stresses.

[1] V. I. Levitas, International Journal of Plasticity 106, 164(2018).

[2] H. Babaei and V. I. Levitas, International Journal ofPlasticity 107, 223 (2018).

[3] S. Lekhnitskii, Theory of Elasticity of an AnisotropicElastic Body (1963).

[4] V. I. Levitas, H. Chen, and L. Xiong, Physical ReviewLetters 118, 025701 (2017).

[5] V. I. Levitas, H. Chen, and L. Xiong, Physical ReviewB 96, 054118 (2017).

[6] R. Hennig, A. Wadehra, K. Driver, W. Parker, C. Um-rigar, and J. Wilkins, Physical Review B 82, 014101(2010).

[7] V. I. Levitas, D. L. Preston, and D.-W. Lee, PhysicalReview B 68, 134201 (2003).

[8] H. Babaei, A. Basak, and V. I. Levitas, ComputationalMechanics 64, 1177 (2019).

[9] A. I. Lurie, Non-linear theory of elasticity, Vol. 36 (Else-vier, 1990).

[10] M. Parrinello and A. Rahman, Journal of Applied physics52, 7182 (1981).

[11] R. E. Miller, E. B. Tadmor, J. S. Gibson, N. Bern-stein, and F. Pavia, The Journal of chemical physics144, 184107 (2016).

[12] R. S. Elliott, N. Triantafyllidis, and J. A. Shaw, Journalof the Mechanics and Physics of Solids 59, 216 (2011).

[13] R. Hill, in Mathematical Proceedings of the CambridgePhilosophical Society, Vol. 77 (Cambridge UniversityPress, 1975) pp. 225–240.

[14] R. Hill and F. Milstein, Physical Review B 15, 3087(1977).

6

FIG. S1. Field of the order parameter and normal Cauchy stresses for solution presented in Fig. 3 for compressive PK1S atthe top face along with σ1 = σ2 = 1 GPa. Solutions are presented for three time instances: (a) at PT instability; (b) and (c)at P3 slightly above the peak for the Kirchhoff stress, but below the peak points for the Cauchy stress and PK1S.

(a) (b) (c)

FIG. S2. Deviation of the normal Cauchy stresses from the averaged stress in direction 3 and external prescribed stressesσ1 = σ2 = 1 GPa for the three different instances considered in Fig. S1. The stresses are plotted along a horizontal line in themiddle of simulation domain shown in Fig. S1.