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International Journal of Plasticity 96 (2017) 156e181

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International Journal of Plasticity

journal homepage: www.elsevier .com/locate/ i jp las

Coupled elastoplasticity and plastic strain-induced phasetransformation under high pressure and large strains:Formulation and application to BN sample compressed in adiamond anvil cell

Biao Feng a, b, Valery I. Levitas c, d, *

a Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USAb Los Alamos National Laboratory, Los Alamos, NM 87545, USAc Departments of Aerospace Engineering, Mechanical Engineering, and Material Science and Engineering, Iowa State University, Ames, IA50011, USAd Ames Laboratory, Division of Materials Science and Engineering, Ames, IA 50011, USA

a r t i c l e i n f o

Article history:Received 6 March 2017Received in revised form 8 May 2017Accepted 12 May 2017Available online 18 May 2017

Keywords:Large deformationPlasticityPlastic strain-induced phase transformationHigh pressureDiamond anvil cellComputational algorithm

* Corresponding author. Departments of AerospacAmes, IA 50011, USA.

E-mail addresses: [email protected] (B. Fen

http://dx.doi.org/10.1016/j.ijplas.2017.05.0020749-6419/© 2017 Elsevier Ltd. All rights reserved.

a b s t r a c t

In order to study high-pressure phase transformations (PTs), high static pressure is producedby compressing a thin sample within a high strength gasket in a diamond anvil cell (DAC).However, since a PT occurs during plastic flow, it is classified and treated here as a plasticstrain-induced PT. A thermodynamically consistent system of equations for combinedplastic flow and plastic strain-induced PTs is formulated for large elastic, plastic, andtransformation strains. TheMurnaghan elasticity law, pressure-dependent J2 plasticity (bothdependent of the concentration of a high-pressure phase), and plastic strain-induced andpressure-dependent PT kinetics are utilized. A computational algorithm within finiteelementmethod (FEM) is presented and implemented in a usermaterial subroutine (UMAT)in the FEM code ABAQUS. Combined plastic flow and strain-induced PT from the highly-disordered hexagonal boron nitride (hBN) sample to a superhard wurtzitic wBN is simu-lated within the rhenium gasket for pressures up to 50 GPa. The evolution of the fields ofstresses and plastic strains, aswell as the concentration of phases in a sample is obtained anddiscussed in detail. Stress-strain fields in a gasket and diamond are presented as well. Anunexpected shape of the deformed sample with almost complete PT in the external part ofthe sample that penetrated the gasket was found. Obtained results demonstrated the dif-ference between material and system behavior which are often confused by experimen-talists. Thus, while plastic strain-induced PT may start (and end) at plastic straining slightlyabove 6.7 GPa, it is not visible below 12 GPa. It becomes detectable at 21 GPa and is notcompleted everywhere in a sample even at a maximum pressure of 50 GPa. Due to a stronggasket the gradient of pressure is much smaller than the gradient of plastic strain, andtherefore the distribution of the high pressure phase is mostly determined by the plasticstrain field instead of the pressure field. Possible misinterpretation of the experimental dataand characterization of the PT is discussed. The developed model will allow computationaldesign of experiments for synthesis of high-pressure phases.

© 2017 Elsevier Ltd. All rights reserved.

e Engineering, Mechanical Engineering, and Material Science and Engineering, Iowa State University,

g), [email protected] (V.I. Levitas).

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B. Feng, V.I. Levitas / International Journal of Plasticity 96 (2017) 156e181 157

1. Introduction

A diamond anvil cell (DAC), see Fig.1, allows one an in-situ study of thematerial's physical andmechanical behavior as wellas PTs under high pressure, using advanced diagnostics such as optical, Raman, and X-ray techniques (Lazicki et al., 2012; Nisret al., 2012; Oganov et al., 2009; Zha et al., 2012). Within a liquidmedia, the sample in a DAC is under a hydrostatic pressure inwhich the PTs are pressure-induced. At some pressure, liquid freezes and the sample is subjected to a stress tensor, but mostlikely still below the yield strength. The highest solidification pressure of 11 GPa at room temperature is for helium. Note thatfreezing of a liquid transmitting media, even if not reported, exhibits itself in a drastic difference in PT pressures in differentmedia. For example, pressure for the beginning of a/ ε and reverse ε/ a PTs in iron varied for different transmitting mediain the range from 6 to 16 GPa (Bargen and Boehler, 1990). Thus, most PT studies are, in reality, performed under such non-hydrostatic (or quasi-hydrostatic) conditions, and PTs are stress-induced. Without a hydrostatic medium, or after the so-lidification of the transmitting medium if its yield strength is getting comparable with that for a sample, the sample un-dergoes plastic deformation. To produce high pressure, the thickness of a sample should be irreversibly reduced by hundredsof percent. Additional large plastic shearing occurs due to friction stress between the sample and diamond. Thus, in this case,PTs occur during large elastoplastic deformations, i.e., they are plastic strain-induced PTs. There are many applications forwhich plastic strain-induced PT under high pressure is of interest. They include: friction, wear, polishing, and cutting of somestrong semiconductors (silicon and germanium) and ceramics (e.g., SiC). Utilizing plastic strain-induced PTs from brittle low-to ductile high-pressure phases, one can introduce a ductile regime of machining of Si, Ge, and SiC, which minimizesmicrocracking (Patten et al., 2004). Various geophysical applications are based on analysis of combined plastic flow andplasticity. In particular, some of the mechanisms of deep earthquakes are caused by the instability of geological materials dueto shear strain-induced PTs (Green and Burnley, 1989). Penetration of a projectile in a target is also accompanied by largeplastic deformations and PTs (see, e.g., Chen et al., 2003). While these processes are dynamic, static experiments are stilldesirable to get initial information since it is easier to extract in situ data in DAC than in shock waves.

Recent discoveries of a new superconducting phase at megabar pressures (Drozdov et al., 2015; Troyan et al., 2016; Diasand Silvera, 2017) are also produced due to plastic strain-induced PTs without pressure transmitting media. The most excitingresults are obtained under application of large plastic shear in rotational DAC (Alexandrova et al., 1993; Blank et al., 1984,1993; Novikov et al., 1999; Levitas et al., 2006; Ji et al., 2012; Blank and Estrin, 2014), where plastic straining significantlyreduces the PT pressure by a factor of 2e10 in comparison to hydrostatic conditions. For example, the highly disorderednanocrystalline hBN-to-wBN transformation occurs at 6.7 GPa with plastic straining, whereas wBN has not been obtainedeven at pressures up to 52.8 GPa under quasi-hydrostatic conditions (Ji et al., 2012). Even without rotation, the irreversiblerhombohedral-to-cubic boron nitride (rBN-to-cBN) transformation under compression without hydrostatic media in DACoccurs at 5.6 GPa (Levitas and Shvedov, 2002), whereas it occurs at 55 GPa under the hydrostatic condition (Ueno et al., 1992).Similar results are obtained under lower pressures when the anvil is made of hard alloys (Bridgman, 1935; Zhilyaev et al.,2011; Srinivasarao et al., 2011; Edalati and Horita, 2016), e.g., for PTs in Zr and Ti. This popular processing approach iscalled high-pressure torsion. In addition, plastic deformation often substitutes a reversible PT with an irreversible PT (Blankand Estrin, 2014; Levitas et al., 2006), which allows one to use high-pressure phases in engineering applications. Thus, themain practical interest in PT under plastic straining is in the possibility to obtain both known and new phases under muchlower pressures while keeping them under normal pressure. This illustrates the transformation of discovery into technology.Lastly, plastic straining may lead to new phases that could not be obtained under hydrostatic conditions (Blank and Estrin,2014; Novikov et al., 1999; Levitas et al., 2012).

Thus, understanding of the interaction between PTs and plasticity is an important fundamental problem of high-pressurematerial physics and mechanics. DAC and rotational DAC are the only experimental tools that are used for these studies. Themain challenges in these studies are the following.

1. The high-pressure community does not clearly recognize and utilize the difference between pressure- or stress-inducedPTs and plastic strain-induced PTs under high pressure. Both pressure- and stress-induced PTs start by nucleating at pre-existing defects, e.g., various dislocation configurations or different boundaries (grain, subgrain, and twin). These disloca-tions and boundaries represent stress tensor concentrators. Plastic strain-induced PTs occur by nucleating at new defects thatare continuously generated during the plastic flow. In particular, dislocations are generated and densely piled up against grainboundaries or other obstacles during plastic deformation. They generate a strong concentrator of the stress tensor. Then, thelocal stresses may satisfy local PT (i.e., lattice instability) criterion and cause barrierless nucleation of the high-pressure phaseat a much lower applied pressure than that under hydrostatic conditions (Levitas, 2004a and b). Plastic strain-induced PTshave differentmechanisms and require completely different thermodynamic and kinetic descriptions as well as experimentalcharacterization than pressure- and stress-induced PTs. A multiscale theory (Levitas, 2004a and b) for high-pressuremechanochemistry was proposed, which claims that plastic strain-induced PTs could be characterized by a strain-controlled, pressure-dependent kinetic equation of type of Eq. (41). In this equation, the derivative of concentration of thehigh-pressure phase with respect to the accumulative plastic strain (rather than time) is determined.

2. Due to highly heterogeneous fields of stress and strain tensors, and complex distributions of phases, only the pressureand concentration of high pressure phases along the radius of the sample on a contact surface are experimentally available(Levitas et al., 2006; Blank and Estrin, 2014). In principle, there are currently methods that allow one to measure distributionof elastic lattice strain that may be converted into distribution of stresses (Wenk et al., 2007; Merkel et al., 2013; Nisr et al.,2012). However, the main problem is that distribution of plastic strain cannot be directly measured. The only way toward

B. Feng, V.I. Levitas / International Journal of Plasticity 96 (2017) 156e181158

quantitative high pressure science is to develop models of materials' behavior and FEM procedure. Then we can extractmaterial parameters and functions by comparing FEM modeling with available measurements (Levitas, 2004a,b; Levitas andZarechnyy, 2010a; Feng et al., 2013a, 2013b, 2014). Surprisingly, despite such an interesting, important, and urgent problem,we are the only group which works in this direction. FEM have been developed and applied for investigation of the evolutionof stresses, strains, and concentration of phases in the entire sample during plastic flow as well as PTs with external forcegrowth (Feng et al., 2013a, 2013b, 2014; Levitas and Zarechnyy, 2010a). In these papers, coupled problems of elastoplasticityand PT with large plastic deformation were solved. They are quite sophisticated in terms of nonlinear constitutive equationsand computational algorithms, as well as convergence of the FEM solutions. In exploratory studies (Feng et al., 2013a, 2013b,2014; Levitas and Zarechnyy, 2010a), the simplest system of equations was postulated without continuum thermodynamictreatment. In particular, the assumptions of small transformational and elastic strains, linear elasticity, and pressure-independent perfect plasticity have been used. Thus, pressure was limited to 0.1K (K is the bulk modulus), which is rela-tively low, and diamond anvils were considered as rigid. Transformation strain was pure volumetric and transformation-induced plasticity (TRIP) was neglected. Calculations have been performed for generic material parameters (i.e., withoutcalibration for any specific material), which have been varied in order to understand their effect on PTs and plastic flow.

The major goals of this paper are:

(a) to formulate a continuum thermodynamic framework for coupled elastoplasticity and plastic strain-induced PTs withlarge elastic, plastic, and tensorial transformation strains, nonlinear elasticity, strain hardening, pressure-dependentyield condition, and TRIP;

(b) to propose a computational algorithm for this theory with emphasis on the stress update and derivation of consistenttangent stiffness;

(c) to specify this model for plastic strain-induced PT in BN;(d) to model and simulate PT and plastic flow in the BN sample within a rhenium gasket in a DAC, for which all de-

formations are finite (including deformation of a diamond anvil, and(e) to interpret some experimental high-pressure phenomena observed in DAC, first of all, difference between material

behavior (constitutive equations) and system (sample/gasket/anvil) behavior as well as the difference betweenpressure-induced PTs and plastic strain-induced PTs at high pressure.

Plastic strain-induced solid-solid PT from highly disordered nanograined hexagonal hBN to superhard wurtzitic wBN willbe studied in the paper. This PT does not occur even at 52.8 GPa under quasi-hydrostatic conditions but occurred at 6.7 GPa inrotational DAC with large plastic shear, demonstrating extremely strong effect of plastic straining on PT. During this PT, thebulk and shear moduli increase by a factor of 10 and 19 respectively, and the yield strength increases by a factor of 30. Thevolume is reduced by 39%, which along with geometrically and physically nonlinear elasticity rules for all materials (BNsample, rhenium gasket, and diamond anvil), as well as large plastic deformations and contact sliding, makes this problemquite challenging for convergence.

As the first step toward this goal, we developed a large deformation model for elastoplasticity without PTs (Feng et al.,2016), and solved the problem for compression of a rhenium sample in DAC with a maximum pressure up to 300 GPa. Re-sults were in good correspondence with experiments in terms of pressure distribution along the sample-diamond surface,and the change in shape (cupping) of this surface.

The paper is organized as follows. General equations for coupled plastic flow and plastic strain-induced PTs are presentedin Section 2. Specific models are described in Section 3. Complete system of equations is summarized in Section 4. Algorithmicaspects are discussed in Section 5. Material parameters used in simulations are described in Section 6. Nonlinear elasticequations and material properties for a single-crystal diamond anvil are given in Section 7. Results and discussion on plasticstrain-induced PT in the BN sample are described and analyzed in Section 8. Section 9 includes concluding remarks.

Contractions of the second-order tensors A ¼ fAijg and B ¼ fBijg over one and two indices are defined as A$B ¼ fAijBjkgand A : B ¼ fAijBjig, respectively. Similarly, we designate contractions of the forth-order tensor A and second-order tensor B,over one and two indices as A$B ¼ fAijkmBmng and A : B ¼ fAijkmBmkg. The subscripts s and amean symmetrization and skew-symmetrization, the superscripts t and �1 are the transposition and inverse of a tensor, the subscripts e, p and t designateelastic, plastic and transformational strain or deformation gradient. I is the second-order unit tensor,:¼ is equal by definition,and subscripts 1 and 2 are for material parameters of the low pressure and high pressure phases, respectively.

2. General equations for coupled plastic flow and plastic strain-induced phase transformations

The constitutive behavior of two-phase elastoplastic mixture with the variable concentration of phases during PT underlarge elastic, plastic and transformational strains is extremely complex. While it can be studied numerically for a repre-sentative volume using micromechanical approach for some typical loadings, we need analytical expressions to be imple-mented in the FEM code for the simulation of materials behavior in DAC. It is clear that significant simplifications are required.Also, the experimental data on the deformation and transformation behavior under high pressure are much more limitedthan under normal conditions, because of impossibility to produce homogeneous fields in a sample and difficulty inmeasuring stress and, especially, plastic strain tensors. In particular, there are no any experimental results on strain- and PT-


induced anisotropy of elastic and plastic properties. Under such circumstances, there is no sense to develop a complex model,which could not be calibrated nowor used to develop procedures for coupled experimental and computational determinationof all fields and extracting materials properties in the nearest future. It is necessary to start with a simple model, whichincludes the main effects and neglects the secondary ones. After the simulation of the known high-pressure experiments, onecan determine which experimental features could be reproduced and described and which not, and what should be addedinto themodel for the description of the later. Such an approachworkedwell at lower pressures, when isotropicmodels, smallelastic and transformation strains and linear elasticity were accepted. Both the simplest analytical treatment (Levitas,2004a,b, Levitas and Shvedov, 2002) and the FEM simulations (Feng et al., 2013a, 2013b, 2014; Levitas and Zarechnyy,2010a; 2010b) demonstrated to be very useful in explaining and predicting numerous effects, interpreting experimentaldata, and suggesting on experimental procedures that would allow extraction of elastoplastic and transformational propertiesfrom the heterogeneous fields. Examples include:

(a) predicting PT induced by rotational plastic instability for transformation of highly textured rhombohedral rBN to thecubic BN, which allowed us to reduce transformation pressure from 55 GPa at quasi-hydrostatic conditions to 5.6 GPaunder compression along the c axis (Levitas and Shvedov, 2002);

(b) reproducing plateaous (steps) with approximately constant pressure in otherwise very heterogeneous pressure dis-tribution (Levitas and Zarechnyy, 2010a and b);

(c) reproducing and interpretating of very irregular pressure distribution during some PTs in terms of smaller yieldstrength of the high-pressure phase and formation of multiple transformation-deformation bands (Feng et al., 2013a,2013b, 2014; Levitas and Zarechnyy, 2010a and b);

(d) explaining the pressure self-multiplication effect during PT to a stronger high-pressure phase (Levitas, 2004a,b, Levitasand Zarechnyy, 2010a,b, Feng and Levitas, 2013, 2016);

(e) finding possible misinterpretation of reported low PT pressures due to neglecting convective radial flow of particles(Levitas and Zarechnyy, 2010a,b, Feng and Levitas, 2013), and

(f) revealing an extrusion-based pseudoslip mechanism of contact sliding between sample, anvil, and gasket (Feng et al.,2014).

Also, these simulations allow us to study effects of variousmaterial and geometric parameters on the fields of stress, strain,and concentration of the high-pressure phase. That it is why below, we develop a simplified model, but muchmore advancedthan in Levitas and Zarechnyy (2010a), Feng et al. (2014), because it takes into account large elastic, plastic, and tensorialtransformation strains, physically nonlinear elasticity of a sample, gasket, and anvil, strain hardening, pressure-dependentyield condition, and TRIP. For simulations, additional simplifications will be made.

Sincewe consider initially isotropic polycrystallinematerials, themain assumption is isotropic elasticity and plasticity. Themain reason for this assumption is that there are no any data that allow one to quantify evolution of strain (and trans-formation) induced anisotropy of elastic and plastic properties under high pressure. It is not difficult to consider anisotropicmaterials andwrite down general equations for plastic strain-induced anisotropy of elastic and plastic properties. It is also notdifficult to take into account plastic strain-induced evolution of the mean grain size and its effect on the yield strength.However, we will not be able to quantify these dependences from experiments and use them in simulations. Also, someissues, like formulation of equations for plastic (and transformational) spin are not fully resolved even at normal pressure, andthis is not a goal of the current paper towork on this problem. Since this is to our best knowledge the first thermodynamicallyconsistent model and first simulations in DAC for plastic strain-induced PT at high pressure in fully geometrically andphysically nonlinear formulation, one needs to start it in any case with the simple but reasonable case.

In addition, there are some data at large accumulative plastic strain q > 0.6e1 and for a monotonous deformation (i.e., for adeformation path in the strain space without sharp changes in directions), which justify neglecting plastic anisotropy. Thus, itwas found (Levitas, 1996) for more than 60 materials belonging to different material classes (e.g., rocks, metals, oxides, alloys,and compacted powders) that when plastic strain is above some level the polycrystalline, initially-isotropic materials deformas the perfectly plastic and isotropic ones with a strain-history-independent limit surface of the perfect plasticity. Since grainsize depends on plastic strain history, independence of the yield surface of the plastic strain and strain history means that thegrain size reduction is saturated and its effect is effectively included in the value of the yield strength. Some additionalconfirmations are presented in Novikov et al. (1999) for alloyed steel and NaCl. Transition to perfect plasticity was strictlyproved under large uniform compression for six metals in Levitas et al. (1994). Saturation of the strain hardening is currentlygenerally accepted, see review (Edalati and Horita, 2016). Thus, our general theory will include strain hardening at relativelysmall q but neglect plastic strain-induced elastic and plastic anisotropy. In simulations, strain hardening will be neglected aswell due to relatively large strains and unknown hardening rules. Without PTs, such a model described well experimentalresults for rhenium for pressures up to 300 GPa (Feng et al., 2016). Note that all simulations for high pressure torsion are alsoperformed with isotropic plastic model, see Lee et al. (2014) and Edalati et al. (2016).

Effect of temperature will be neglected here due to our focus on PT at the room temperature without any external heating/cooling. Heating due to plastic flow and latent heat of PT can be neglected for actual loading in DAC due to the followingreasons: (a) The compression process is slow and is often interrupted for producing measurements of X-ray patterns, Ramanspectra, and fluorescence of the ruby particles used for measurement of the pressure or pressure distribution. (b) The sample


and gasket are thin (10e150 mm) and are compressed by two diamonds, the best heat conductor. We are not aware of anyworks measuring or including the effect of temperature under compression in DAC without external heating/cooling.

2.1. Kinematics

The motion of a material with large elastic, plastic, and transformational deformations is described by a vector functionr ¼ rðr0; tÞ, where r0 and r are the position vectors of material points in the reference (undeformed) configuration U0 at thetime instant t0 and in the actual (deformed) configuration U at the instant t. The deformation gradient F ¼ vr

vr0 is decomposedinto elastic Fe and inelastic F i contributions:

F ¼ Fe$F i ¼ Re$Ue$Ri$U i ¼�Re$Ri

�$�Rti$Ue$Ri

�$U i ¼ Re$Ue$U i ¼ Fe$U i ¼ Ve$Re$U i

with Re ¼ Re$Ri; Ue ¼ Rti $Ue$Ri; Fe ¼ Re$Ue ¼ Ve$Re

(1)

where F i is the inelastic deformation gradient in the locally stress-free intermediate configuration Ui, which includes theplastic and transformational deformations; Ue and U i are the symmetric elastic and inelastic right stretch tensors; Re and Riare the proper orthogonal elastic and inelastic rotation tensors. Rearrangements Re$Ri ¼ Re; Rt

i $Ue$Ri ¼ Ue in Eq. (1) areequivalent to considering motion with respect to the frame of reference rotating with the tensor Rt

i in the configuration Ui.Alternatively, this is equivalent to choosing the intermediate stress-free configuration Ui in which inelastic rotation iseliminated (i.e., Ri ¼ I). Such a choice of the intermediate configuration Ui allows one to automatically satisfy the invarianceof the kinematic and constitutive equations with respect to superposition of the rigid-body rotations inUi. All tensors withouta bar in Eq. (1) are defined when Ui is chosen as intermediate configuration. The velocity gradient l ¼ _F$F�1 ¼ W þ d isdecomposed into symmetric deformation rate d ¼ ðlÞs and skew-symmetric spin W ¼ ðlÞa. With the help of Eq. (1) wedecompose l and d into elastic and inelastic contributions:

l ¼ _Fe$F�1e þ Fe$ _U i$

_U�1i $F�1

e ¼ le þ li; d ¼�_Fe$F�1

e

�sþ�Fe$ _U i$U

�1i $F�1

e

�s: (2)

Sincewewill limit ourselves to isotropic materials, the Eulerian elastic strain tensor, Be ¼ 0:5ðFe$Fte � IÞ ¼ 0:5ðVe,Ve � IÞwill

be used. For isotropic materials, similar to the derivation process for Eq. (3.11) in (Levitas, 1996), the following kinematicdecomposition can be derived,

d ¼ BV

e � 2ðd$BeÞs þ Ve$Di$Ve; Di ¼ Re$�_U i$U

�1i

�s$Rt

e; BV

e¼ _Be � 2ðW$BeÞs; (3)

where BV

e is the Jaumann objective time derivative of Be, and Di is the inelastic deformation rate. The first of Eq. (3) can betransformed in the following steps

d ¼ BV

e � 2d$Be þ 2ðd$BeÞa þ Ve$Di$Ve/

dþ 2d$Be ¼ d$ðI þ 2BeÞ ¼ d,V2e ¼ B

V

eþ2ðd$BeÞa þ Di$V2e/

d ¼ BV

e$V�2e þ2ðd$BeÞa$V�2

e þ Di;

(4)

where we took into account that for isotropic materials, tensors s, Ve, Be, and Di have the same principal axes and can bearbitrarily permuted in the product of tensors. As we will see, such a form is convenient for thermodynamic treatment forisotropic materials. There are two main options on how to decompose the inelastic deformation gradient into plastic andtransformational parts. In (Levitas, 1998; Levitas and Javanbakht, 2015), a multiplicative decomposition is used, which isreasonably justified. For plastic strain-induced PTs, both direct and reverse PTs occur during plastic flow, and it is difficult tojustify the procedure in the actual or thought experiment to separate plastic and transformational deformation gradients forpolycrystalline materials. In this case, additive decomposition of the stress power and consequently additive decompositionof inelastic deformation rate into plastic Dp and transformational Dt components are more appropriate (see, Levitas, 1996;where, however, PTs were not included)

Di ¼ Dp þ Dt : (5)

For the given model, there is no need to define plastic Fp and transformational Ft deformation gradients and connect them toDp and Dt , because they do not participate in the further constitutive equations. However, we will need a volumetrictransformation strain. We further decompose Dp into volumetric and devitoric parts as

Dp ¼ _εp0I þ g; _εp0 ¼ I: Dp�3; I: g¼ 0: (6)


As usual, we will consider that the plastic flow is incompressible, i.e., _εp0 ¼ 0 and εp0 ¼ 0. Similar decomposition will beconsidered for the transformational deformation rate Dt ,

Dt ¼ _εt0I þ gt _c; _εt0 ¼ I: Dt=3; I: gt ¼ 0; εt0 ¼ 1=3ln detU i; (7)

where c is the concentration of high pressure phase per unit reference volume, 3εt0 is the volumetric transformation strain,and the deviatoric part of Dt is assumed to be proportional to the rate of concentration of the high pressure phase _c. We use U ifor the definition of volumetric transformation strain because plastic strain is incompressible. To define _εt0, we consider PTwith a pure volumetric transformation strain, e.g., like martensitic PT in cerium and its alloys. This is also the case forreconstructive phase transformationsdin particular, in BN considered below. For this case, only the volumetric trans-formation strain is known, and the transformational change in shape is undefined. Then for complete PTUt :¼ Utðc ¼ 1Þ ¼ ð1þ aÞI, where a is a component of the transformation strain Ut � I. For a two-phase material, neglectingsmall residual elastic strains in comparisonwith large transformational strains, we obtain according to an averaging theorem(Hill, 1972) Ut ¼ Utcþ Ið1� cÞ ¼ ðU � IÞcþ I ¼ ð1þ acÞI. Note that for reconstructive PT in BN considered below, theinterface is incoherent and it does not generate (or generate small) elastic residual strains. Then

εt0 ¼ 1=3ln detUt ¼ lnð1þ acÞ and _εt0 ¼ a1þ ac

_c: (8)

Usually, the volumetric transformation strain for complete PT, 3εt0 :¼ 3εt0ðc ¼ 1Þ, is known; then a ¼ expðεt0Þ � 1. If wesubstitute εt0 in Eq. (8) with the linear approximation εt0 ¼ εt0c, it gives a correct value for c ¼ 0 and c ¼ 1. The difference,normalized by εt0 for 0 < c < 1, does not exceed 1.25% for εt0 ¼ 0:13 accepted below for the PT in BN. That is why belowwewilluse

εt0 ¼ εt0c and Dt ¼ ðεt0I þ gtÞ _c: (9)

Note that transformation-induced plasticity (TRIP), for which deformation rate is proportional to _c, can also be effectivelyincluded in gt .

2.2. The dissipation rate and elasticity rule

The local dissipation inequality for isothermal processes is accepted in the form

D ¼ s: d� J�1 _J � 0; J ¼ r0=r¼ detF: (10)

Here D is the dissipation rate per unit current volume, r and r0 are the mass densities in the actual and undeformed con-
figurations respectively, s is the Cauchy stress, and J is the specific Helmholtz free energy per unit reference volume. Withthe help of Eq. (4) we transform the stress power
s: d ¼ s: BV

e$V�2e þ2s: ðd$BeÞa$V�2

e þ s: Di: (11)

V �2 �2V �2 _ �2 �2 �2
First, s: Be$Ve ¼ Ve $s: Be ¼ Ve $s: ðBe � 2ðW$BeÞsÞ and Ve $s: ðW$BeÞs ¼ Ve $s: W$Be ¼ Ve $Be$s: W ¼ 0, because
V�2e $s and V�2

e $Be$s are symmetric due to coaxiality of s, V�2e , and Be. Thus s: B

V

e$V�2e ¼ V�2

e $s: _Be. Also, s: ðd$BeÞa$V�2e ¼

V�2e $s: ðd$BeÞa ¼ 0 as a double contraction of symmetric and antisymmetric tensors. Substituting these equations in Eq. (10),

we obtain decomposition of the stress power into elastic and inelastic parts

s: d ¼ V�2e $s: _Be þ s: Di: (12)

Finally, using Eqs. (5), (6) and (9) as well as decomposition of the Cauchy stress s ¼ �pI þ s into a deviatoric part s and
hydrostatic pressure p ¼ �s : I=3, we obtain
s: d ¼ V�2e $s: _Be þ s : gþ ð � 3pεt0 þ s : gtÞ _c: (13)

We will assume the isotropic free energy J ¼ JðBe; cÞ, which does not depend on temperature since isothermal conditions
are assumed. Substituting JðBe; cÞ and Eq. (12) into Eq. (10), we have
D ¼ V�2e $s: _Beþs : gþ ð � 3pεt0 þ s : gtÞ _c� J�1 vJ

vBe: _Be�J�1vJ

vc_c ¼

�V�2

e $s� J�1 vJ

vBe

�: _Be þ s : gþ

�� 3pεt0 þ s : gt�J�1vJ

vc

�_c � 0:

(14)

Assuming the dissipation rate is independent of _Be, we obtain the nonlinear elastic rule and residual dissipation inequality


s ¼ J�1V2e$

vJ

vBe¼ J�1ð2Be þ IÞ$vJ

vBe; D ¼ s : gþ


vc

�_c � 0: (15)

2.3. The yield condition and plastic flow rule

For the pressure range in which plastic strain induced PTs do not occur and _c ¼ 0, the dissipation rate in Eq. (15) simplifiesto

Dp ¼ s : g � 0: (16)

We will assume that PTs affect plasticity only in terms of the dependence of yield strength on the concentration c. As it was
mentioned above, TRIP can be included in gt . Then inequality Eq. (16) is valid for simultaneous plastic flow and PTs as well. Inorder to satisfy it, the stress deviator s should be a function of g otherwise s and g can be chosen in a way such that s : g<0.For rate-independent plasticity, stress s depends on the direction n ¼ g=jgj of the plastic deformation rate and is independentof its magnitude jgj ¼ ðg : gÞ0:5, i.e., s is a homogeneous function of g of the zeroth degree:
s ¼ sðg; p; q; cÞ ¼ sðn; p; q; cÞ: (17)

In Eq. (17), the accumulated plastic strain q, defined by

_q ¼ ð2g: g=3Þ0:5 ¼ ð2=3Þ0:5jgj; (18)

is included along with pressure p and concentration c. Inverting the function in Eq. (17), n ¼ nðs; p; q; cÞ and considering that
the magnitude of n is equal by definition to unity, one receives the identity
4ðs; p; q; cÞ ¼ 4ðs; q; cÞ ¼ nðs; p; q; cÞ : nðs; p; q; cÞ � 1 ¼ 0: (19)

It is evident that 4ðs; q; cÞ¼ 0 is the yield condition, because it is met for any nonzero plastic deformation rate g. For jgj ¼ 0,
the tensor n and function sðg; p; q; cÞ are undetermined. Traditionally, we assume that for jgj ¼ 0 one has 4ðs; q; cÞ � 0.Various postulates (e.g., Ilyshin's or Drucker’ postulates (Lubarda, 2002), postulate of dissipation (Levitas, 1996)) lead to theassociated flow rule in the deviatoric space:
n ¼ v4

vs

v4vs g ¼

gnðs; p; q; cÞ: (20)

2.4. Plastic strain-induced kinetics of phase transformations

Generally, since for plastic strain-induced PTs _c ¼ 0 when g¼ 0, we accept that

_c ¼ Aðs; p; q; cÞ : g; (21)

where A is some second-rank tensor function. For an isotropic material, we assume a simpler relationship

_c ¼ Aðp; q; cÞ _q (22)

with a scalar function, A. Thus, the transformation and inelastic deformation rates in Eqs. (6) and (7) can be expressed as

Dt ¼ ðεt0I þ gtÞAðp; q; cÞ _q ¼ ðεt0I þ gtÞAðp; q; cÞffiffiffiffiffiffiffiffi2=3

p ijgj;

Di ¼ gþ ðεt0I þ gtÞAðp; q; cÞ _q ¼hnþ ðεt0I þ gtÞAðp; q; cÞ

ffiffiffiffiffiffiffiffi2=3

p ijgj: (23)

The dissipation inequality (15) transforms to

D ¼�s : nþ


vc

�A


p jgj � 0/

Xc: ¼ s : nþ�� 3pεt0 þ s : gt�J�1vJ

vc

�A


p� 0:

(24)

The term in parenthesis in the second Eq. (24) can be interpreted as the macroscopic driving force for PT if gt only includestransformational deformation rate without TRIP. One may speculate that due to the positive plastic dissipation rate (which is


proportional to s: n � 0), the macroscopic driving force for direct PTs (i.e., _c>0 and A>0) is allowed to be negative butsatisfying inequality (24), i.e., s: n � 0 is an additional driving force for plastic strain-induced PTs. Onemay also claim that thisis a thermodynamic mechanism of the promotion of PTs by plastic deformation, which explains why plastic strain-inducedPTs may occur at pressures, which are lower than the thermodynamic equilibrium pressures (Alexandrova et al., 1993; Blankand Estrin, 2014). However, this is not true, and it can be proved by a simple estimate. For simplicity, let us consider PT withεt0 ¼ 0 (e.g., in a shapememory alloy); collinearity of s, gt , and n; and jgt j ¼ 0:1. Then, without plasticity, transformationworkis s : gt ¼ jsjjgt j ¼ 0:1jsj. Infinitesimal plastic strainwill add the term s : n ¼ jsj to Xc, i.e., it increases the “driving force for PT”by an order of magnitude. However, this is nonsense from an experimental point of view.

Rather, the interaction between PT and plasticity occurs through corresponding nanoscale mechanisms, which will be dis-cussed below. In fact, in a detailed nanoscale phase field approach to the interaction between PT and dislocations (Levitas andJavanbakht, 2014, 2015; Javanbakht and Levitas, 2015, 2016), each of the dissipation rates, due to PT and dislocations, is nonneg-ative separately. It was found that the resultant phase interface equilibrium is described by a traditional driving force in terms oflocal stresses,which include stresses due to dislocations (Javanbakht and Levitas, 2015, 2016). Still, these approaches allowedus todescribe the reduction in PT pressure due to stress concentration at the tips of the dislocation pile up by an order of magnitude.

2.5. Consistency condition and stress rate e deformation rate relationship

According to Eq. (15), the Cauchy stress s is a function of both the strain tensor Be, and the concentration of high pressurephase c, i.e., s ¼ f ðBe; cÞ. Therefore

sV ¼ vf

vBe: Be

V

þ vfvc

_c ¼ vfvBe

$V2e :�d� Di � 2ðd$BeÞa

�þ vfvc

_c ¼ vfvBe

$V2e :

d�

"nþ ðεt0I þ gtÞA

ffiffiffi23

r #jgj!

þvfvc

A

ffiffiffi23

rjgj ¼ Y :

d�

"nþ ðεt0I þ gtÞA

ffiffiffi23

r #jgj!þ vf

vcA

ffiffiffi23

rjgj ¼ Y : dþ Zjgj;

(25)

where

Y ¼ vfvBe

$V2e and Z ¼ �Y :

hnþ ðεt0I þ gtÞA


p iþ vf

vcA

ffiffiffi23

r: (26)

We take into account that vfvBe

$V2e : ðd$BeÞa ¼ 0 because the fourth-rank tensor vf

vBe$V2

e is symmetric with respect to permu-

tation of indices (or basis vectors) 3 and 4. This can be proven bywriting the tensor vfvBe

$V2e in the principle axes of tensors vf

vBeor

V2e , which coincide vfij

vBe;klV2e;lm ¼ vfii

vBe;kkV2e;kk. The consistency condition for the yield surface 4ðs; q; cÞ ¼ 0 is

_4 ¼ v4

vs: sV þ v4

vq_qþ v4

vc_c ¼ v4

vs: ðY : dþ ZjgjÞ þ

�v4

vcAþ v4

vq

� ffiffiffi23

rjgj ¼ 0: (27)

Solving Eq. (27) for jgj yields

jgj ¼ �v4vs : Y

v4vs : Z þ

�v4vc Aþ v4

vq

� ffiffiffi23

q : d; (28)

and substituting Eq. (28) into Eq. (25), we obtain

sV ¼

8><>:Y þ Z

�v4vs : Y

v4vs : Z þ

�v4vc Aþ v4

vq

� ffiffiffi23

q9>=>; : d: (29)

This relationship will be used in Section 2.6 for derivation of the consistent tangent moduli.

2.6. Consistent tangent moduli

The consistent tangentmoduli C are determined from the following equation inABAQUS 6.11Manual (see ABAQUS 6.11, 2011)

ðJsÞ� ¼ JðC : dþW$s� s$WÞ: (30)


Here, ðÞ� means the material time derivative of the expression in the parenthesis. C will be evaluated in the user-definedsubroutine UMAT. The fourth-rank tensor C determines the convergence rate but does not influence the accuracy of thefinal results. For the problems with large elastoplasticity along with large transformation strain and multiple physical non-linearities, the exact expression for C should be used in the simulations; otherwise the convergence cannot be reached. FromEq. (30) and by using the derivation in (Feng et al., 2016), we can obtain

C : d ¼_JJsþ ð _sþ s$W �W$sÞ/C ¼ s

1Jv _Jvd

þ vsV

vd: (31)

Since _J ¼ J _F : F�1 ¼ JI: d (see, e.g., Lurie (1990), Lubarda (2002), Levitas (1996)), then v _Jvd ¼ JI. Substituting this expression in

Eq. (31) and inserting Eq. (29), we obtain the consistent tangent moduli as

C ¼ sI þ Y þ Z�v4

vs : Y

v4vs : Z þ

ffiffiffi23

q �v4vc Aþ v4

vq

� : (32)

The components of fourth-rank tensor sI can bewritten as sijdkl, where dkl is the Kronecker delta. The expressions forvfvc,

vfvBe

, v4vs,and v4

vc in Eqs. (26) and (32) can be found in the Appendix.

3. Specific models

The simplest mixture rule will be used for all elastic and plastic constants. Due to a large ratio of elastic and plasticproperties of phases (see Section 6), actual properties strongly depend on morphology of a mixture, which evolves and isunknown. Also, scatter of the pressure dependence of the yield strength (e.g., for Rhenium, see Feng et al. (2016)) is quitelarge. Under these circumstances, there is no sense to use a more complex model than the linear mixture rule.

3.1. Nonlinear isotropic elasticity

The third-order Murnaghan potential (Murnaghan, 1951) is traditionally used for high pressure

JðBeÞ ¼ le þ 2G2

I21 � 2GI2 þ�lþ 2m

3I31 � 2mI1I2 þ nI3

�; (33)

where le, G, m, l, and n are elastic material parameters which depend on the concentration c of the high pressure phase. Ii isthe ith invariant of the strain tensor Be

I1 ¼ Be11 þ Be22 þ Be33 ; I2 ¼ Be22Be33 � B2e23 þ Be11Be33 � B2e13 þ Be22Be11 � B2e12 ; I3 ¼ det Be: (34)

The simplest linear dependence of elastic material parameters on c is accepted here as

le ¼ ð1� cÞle1 þ cle2; G ¼ ð1� cÞG1 þ cG2; m ¼ ð1� cÞm1 þ cm2;l ¼ ð1� cÞl1 þ cl2 and n ¼ ð1� cÞn1 þ cn2:

(35)

Substituting J from Eq. (33) in Eq. (15), we obtain the Cauchy stress as

s ¼ J�1ð2Be þ IÞ$�leI1I þ 2GBe þ

�lI21 � 2mI2

�I þ n

vI3vBe

þ 2mI1Be

�; (36)

0Be22Be33 � Be23Be32 Be23Be31 � Be33Be21 Be21Be32 � Be22Be31

1
in which
vI3vBe

¼ @Be23Be31 � Be33Be21 Be11Be33 � Be13Be31 Be12Be31 � Be11Be32Be21Be32 � Be22Be31 Be12Be31 � Be11Be32 Be11Be22 � Be12Be21

A:

3.2. Yield condition and flow rule

The J2 flow theory with pressure-, accumulated plastic strain-, and concentration-dependent yield strength, sy is used(e.g., Levitas, 1996):

4 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=2s : s

p� syðp; q; cÞ ¼ 0: (37)

Then the plastic flow rule is


g ¼ l s� ffiffiffiffiffiffiffiffis: s

p ¼gs� ffiffiffiffiffiffiffiffi

s: sp ; (38)

where l ¼ jgj � 0 is a scalar function determined from the consistency condition _4¼ 0, see Eq. (28).As it was mentioned in Section 2, when the plastic strain is above some level, for a monotonous deformation dozens of

polycrystalline initially-isotropicmaterials belonging to differentmaterial classes deform as the perfectly plastic and isotropicwith a strain-history-independent limit surface of the perfect plasticity (Levitas, 1996). While BN and rhenium were notstudied in the above references, we assume that they also exhibit this type of behavior. We also assume that deformation inboth the BN sample and rhenium gasket exceeds this critical value of plastic strain at early stages of compression. Therefore,we can neglect strain hardening, i.e., sy ¼ syðc; pÞ. Note that for rhenium, such a perfectly plastic isotropic model was used inFeng et al. (2016) and gave a good description of experiments in DAC up to 300 GPa. Assuming linear dependence of yieldstrength on pressure, we obtain

sy ¼ sy0 þ bp; (39)

where b is the material parameter and sy0 is the initial yield strength at zero pressure. We assume also the simplest mixture
rule for b and sy:
b ¼ ð1� cÞb1 þ cb2 and sy0 ¼ ð1� cÞsy01 þ csy02: (40)

3.3. Strain-controlled pressure-dependent PT kinetics

There is extended literature on the modeling of plastic strain-induced PT kinetics in TRIP steels under normal pressure.Based on the mechanism of plastic strain-induced nucleation at the shear-band intersection (Olson and Cohen, 1972),various simple (Olson and Cohen, 1975; Stringfellow et al., 1992) and quite sophisticated (Cherkaoui et al., 2000; Dianiand Parks, 1998; Fischer et al., 2000; Garion et al., 2006; Kouznetsova and Geers, 2008; Lebedev and Kosarchuk, 2000;Ma and Hartmaier, 2015; Turteltaub and Suiker, 2005) models have been suggested. They can be calibrated and veri-fied using experiments under different loadings and moderate strains (e.g., (Lebedev and Kosarchuk, 2000)).

However, there is no experimental data on the kinetics of plastic strain-induced PTs under high pressure and large plasticstrain for any material. The reason is that all fields in a sample compressed in DAC are very heterogeneous (Feng et al., 2013a,2013b, 2014; Levitas and Zarechnyy, 2010a and b), and it is difficult to measure them. In particular, the most detailed mea-surements include pressure distribution at the contact surface (Blank et al., 1984; Hemley et al., 1997; Novikov et al., 1999;Levitas et al., 2006; Blank and Estrin, 2014) and concentration distribution averaged over the sample thickness (Levitaset al., 2006). Plastic strain field was not measured because most experimentalists do not distinguish between pressure-and plastic strain-induced PTs and still characterize plastic strain-induced PT in terms of pressure without any interest inplastic strain. The only way that we see them is by the simulation of all the fields in a sample coupled to experiments thatmeasure pressure, displacement, and concentration distributions with iterative improvement, calibration, and verification ofthe models. Some suggestions on possible experimental procedures and extraction of information from heterogeneous fieldscan be found in (Feng and Levitas, 2016; Feng et al., 2014). This procedure should start with the simplest model.

Such a model was developed in (Levitas, 2004a,b, Levitas and Zarechnyy, 2006). The starting point was a simplenanoscale model for plastic strain-induced nucleation at the tip of the dislocation pile up (Levitas, 2004a and b). Stressand pressure concentration at the tip of the dislocation pile up is proportional to the number of dislocations in a pile upand can be very strong. It reduces the nucleation barrier due to surface energy. Also, for some critical number of dis-locations it substitutes thermally activated nucleation with a barrierless nucleation, which does not require thermalfluctuations. This explains the strain-controlled kinetics, for which time is not a parameter and is substituted with theplastic strain. The same is qualitatively true for other types of defects (various dislocation configurations, grain, subgrain,tilt, and twin, boundaries, shear-band intersections, and stacking faults). Under the prescribed strain increment, defectsare generated with fast barrierless nucleation for the high-pressure phase in the regions of high stress concentration.

Since stress decreases away from the tip of a defect, nuclei grow to some size and shape corresponding to thethermodynamic equilibrium interface. This analytical prediction in (Levitas, 2004a and b) was confirmed by detailedphase field simulations (Levitas and Javanbakht, 2014; Javanbakht and Levitas, 2015, 2016). For the typical observationtime (e.g., few seconds), this corresponds to instantaneous PT. Without straining, new defects and nuclei do not appear,and essential growth of the already existing nuclei is thermodynamically impossible. Note that plastic straining (incontrast to shear stresses) promotes PT with pure volumetric strain (e.g., like reconstructive PTs, in particular, in BN,considered below) as well, because dislocation pile up produces concentration of all components of the stress tensor, and,consequently, pressure. Similar conclusions follow from modeling the nucleation of martensite at the shear-bandintersection (Levitas et al., 1998). An important point in (Levitas, 2004a,b, Levitas and Zarechnyy, 2006), which doesnot have counterpart in studying of the plastic strain-induced PT in TRIP steels, is that defects like dislocation pile upsgenerate both compressive and tensile stresses of the same magnitude. Consequently, defects nucleated in high pressurephase facilitate the reverse PT. Thus, plastic strain-induced defects simultaneously promote both direct and reverse PTs


but in different regions. In addition, due to different yield strengths at different phases, plastic strain is larger in the phasewith a smaller yield strength, which promotes PT from the weaker to stronger phase.

All these results have been conceptually incorporated in the microscale model along with a simplified nano-to micro-transition in the thermodynamic derivations (Levitas, 2004a,b, Levitas and Zarechnyy, 2006). The resulting pressure-dependent, strain-controlled kinetic equation additively combines kinetics for direct and reverse PTs:

dcdq

¼ A ¼ kð1� cÞpdHðpdÞ sy2

sy1� cprHðprÞ

cþ ð1� cÞ sy2sy1

: (41)

Here, k is the kinetic parameter which scales the rate of PTs, pd ¼ p�pdε

pdh�pd

ε

and pr ¼ p�prε

prh�pr

ε

are dimensionless characteristic

pressures for direct and reverse PT, and pdεis the minimum pressure below which a direct plastic strain-induced PT to high

pressure phase cannot occur. Similarly, prεis the maximum pressure above which a reverse plastic strain-induced PT to low

pressure phase does not take place. pdh and prh are the pressures for direct and reverse PTs under hydrostatic loading,respectively. Yield strengths of phases are sy1 ¼ sy01 þ b1p and sy2 ¼ sy02 þ b2p. Note that the pressure dependence of theyield strength in the kinetic equation (41) is included here for the first time. The Heaviside function H (HðxÞ ¼ 1 for x � 0 andHðxÞ ¼ 0 for x<0) ensures that the contribution to Eq. (41) for a direct PT to high pressure phase occurs for p> pd

ε, and the

contribution for the reverse PT takes place for p<prεonly. Eq. (41) specifies the expression for the parameter A from Eq. (22).

The deviatoric stresses are not visible in Eq. (41) because plastic strain-induced PToccurs during plastic flowand their norm ineach phase is proportional to the corresponding yield strength, i.e., it cannot be varied independently. However, characteristicpressures pd

εand pr

εare determined under fulfilment of the yield condition, i.e., the effect of deviatoric stresses is included.Eq.

(41) was generalized in (Levitas and Zarechnyy, 2006) for multiphase materials. Stationary and nonstationary solutions to Eq.(41) were studied in detail in (Levitas, 2004a,b, Levitas and Zarechnyy, 2006). While all parameters in the kinetic equationswere not determined experimentally, some nontrivial experimental phenomena were explained and described:

(1) Incomplete PT and existence of the stationary concentration (Straumal et al., 2015) for q/∞ if pdε<pr

εand pd

ε< p<pr

ε.

(2) Zero pressure hysteresis observed in (Blank et al., 1984; Blank and Estrin, 2014) for B1 to B2 PT in KCl. However, this doesnot mean the PT pressure in this experiment coincides with the phase equilibrium pressure. The phase equilibriumpressure cannot be determined from amacroscopic plastic strain-induced experiment because it does not appear in themicroscale kinetic Eq. (41).

(3) Hard (soft) inert particles promote (decelerate) plastic strain-induced chemical reactions (Zharov, 1984, 1994), whichare described by Eq. (41) similar to PTs. However, the acceleration (deceleration) of the reaction occurs at the initialstage only, and the stationary solution is independent of the inert particles.

(4) Stationary concentration of the high-pressure phase increases with the ratio sy2=sy1, i.e., plastic strain-induced syn-thesis is more suitable for strong high pressure phases. A number of useful conclusions and suggestions for experi-mentalists were derived from the analysis of Eq. (41).

4. Complete system of equations

Box 1 summarizes all equations in the form used in our simulations.

Box. 1

Complete system of equations

Decomposition of the deformation gradient F into elastic Fe and inelastic F i contributions

F ¼ vr=vr0¼ Ve$Re$U i ¼ Ve$F i; F i ¼ Re$U i; Be ¼ 0:5

�Fe$Ft

e � I� ¼ 0:5

�V2

e � I�: (42)

Decomposition of the deformation rate into elastic, plastic, and transformation parts

d ¼ BV

e,V�2e þ2ðd$BeÞa,V�2

e þ gþ εt0 _cI; BV

e¼ _Be � 2ðW$BeÞs: (43)

The third-order Murnaghan potential

JðBeÞ ¼ le þ 2G2

I21 � 2GI2 þ�lþ 2m

3I31 � 2mI1I2 þ nI3

�: (44)


0@ I1 ¼ Be11 þ Be22 þ Be33 ; I2 ¼ Be22Be33 � B2e23 þ Be11Be33 � B2e13 þ Be22Be11 � B2e12 ; I3 ¼ detBe:

le ¼ ð1� cÞle1 þ cle2; G ¼ ð1� cÞG1 þ cG2; m ¼ ð1� cÞm1 þ cm2; l ¼ ð1� cÞl1 þ cl2;n ¼ ð1� cÞn1 þ cn2:

1A

Elasticity rule

s ¼ J�1ð2Be þ IÞ$vJvBe

¼ J�1ð2Be þ IÞ$�leI1I þ 2GBe þ

�lI21 � 2mI2

�I þ n

vI3vBe

þ 2mI1Be

�: (45)

Yield surface

4 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=2s : s

p� �sy0 þ bp

� ¼ 0; sy0 ¼ ð1� cÞsy01 þ csy02; b ¼ ð1� cÞb1 þ cb2: (46)

Plastic flow rule

g ¼gs� ffiffiffiffiffiffiffiffi

s: sp ; (47)

jgj¼ 0 in the elastic region (4ðs; p; cÞ<0 or 4ðs; p; cÞ ¼ 0 but _4ðs; p; cÞ � 0), and jgj is determined from the consistency
condition _4ðs; p; cÞ ¼ 0 in the elastoplastic region ð4ðs;p; cÞ ¼ 0 and _4ðs;p; cÞ>0):
jgj ¼� ffiffiffiffiffi

1:5p

sffiffiffiffiffis:s

p þ b3 I

!: Y

ffiffiffiffiffi1:5

psffiffiffiffiffi

s:sp þ b

3 I

!: Z þ v4

vc Affiffi23

q : d; (48)

where Y :¼ vsvBe

$V2e and Z :¼ �Y : ½nþ ðεt0I þ gtÞA


p � þ vfvc A

ffiffiffi23

q. Accumulated plastic strain

q: ¼ ð2g: g=3Þ0:5 ¼


p g: (49)

Strain-controlled kinetic equation for phase transformation

dcdq

¼ A ¼ kð1� cÞpdHðpdÞ sy2

sy1� cprHðprÞ

cþ ð1� cÞ sy2

sy1

; pd ¼ p� pdε

pdh � pdε

and pr ¼p� pr

ε

prh � prε

: (50)

Stress rate e deformation rate relationship

sV ¼

8><>:Y þ Z

� ffiffiffiffiffi

1:5p

sffiffiffiffiffis:s

p þ b3 I

!: Y

v4vs : Z þ

�v4vc Aþ v4

vq

� ffiffi23

q9>=>; : d (51)

Consistent tangent moduli !
C ¼ s
1JvJ:

vdþ vs

V

vd¼ sI þ Y þ Z

�ffiffiffiffiffi1:5

psffiffiffiffiffi

s:sp þ b

3 I : Y

v4vs : Z þ

ffiffi23

q �v4vc Aþ v4

vq

� : (52)

Equilibrium equation

V$s ¼ 0: (53)

In simulations, we assumed that materials reached a perfectly plastic state and that transformation strain was purelyvolumetric. The later assumption is justified for BN at the end of Section 6 based on reconstructive mechanism of PT.We note that in the final simplified model the inelastic volumetric strain is solely due to PT and change in shape issolely due to plasticity. In this case a multiplicative decomposition of the inelastic deformation gradient F i into thevolumetric part for PT and the isochoric contribution for plasticity can be used, see Clayton (2009).

5. Update of Cauchy stress for an elastoplastic material with a phase transformation

We will formulate the computational algorithm for updating Cauchy stress for elastoplastic materials coupled withplastic strain-induced PTs, which is similar to the radial return algorithm in the book (Simo and Hughes, 1998). However,


due to multiple nonlinearities in our constitutive equations, the return direction is not exactly along the radial directionas it is in Simo and Hughes' book for small elastoplastic deformation problems. At time instant t ¼ tn (n ¼ 0, 1, 2, …) allstate variables are known. All unknowns have to be found for time t ¼ tnþ1 ¼ tn þ Dt during the incremental step nþ1.Note for n ¼ 0, we need to define the initial values of the state variables, which are q ¼ c ¼ 0, and U i ¼ I at t ¼ 0. Inaddition, the deformation gradient Fnþ1 at tnþ1 is assumed to be known. The goal of the computational algorithm is toupdate the Cauchy stress and state variables at tnþ1 by using the known time increment Dt ¼ tnþ1 � tn, state variables attn, and deformation gradient Fnþ1.

As the usual first step (corresponding to the iteration time i ¼ 0) for the radial return algorithm, we consider the elasticpredictor, i.e., we assume there is no change in inelastic strain during the time increment Dt. Then,

Fnþ1 ¼ F0e$U i n/F0

e ¼ Fnþ1$U�1i n ¼ V0

e$R0e ; (54)

where F0e is the elastic deformation predictor at time tnþ1 (superscript 0 means “trial”), V0

e is the trial elastic leftstretch tensor for the elastic predictor at tnþ1, and the known state variable U i n is the inelastic right stretch tensor U iat the time tn. In Eq. (54), Fnþ1 and U i n are known so that F0

e and V0e can be found. Substituting B0

e ¼ 0:5ðV0e$V

0e � IÞ

into elasticity rule (45), where the concentration c in elastic constants (e.g., le ¼ ð1� cÞle1 þ cle2) is taken as cn attime tn, we obtain the trial Cauchy stress s0 at tnþ1 and its deviatoric s0 and spherical ep0I parts. If4 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32s

0 : s0q

� syðp0; cnÞ<0, then we update U i nþ1 ¼ U i n, and cnþ1 ¼ cn, and this completes the determination of allparameters at time tnþ1. If 4 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32s

0 : s0q

� syðp0; cnÞ � 0, inelastic deformations occur during time increment Dt, andthe stress should be iteratively reduced to satisfy the yield condition 4ðsnþ1; pnþ1; cnþ1Þ ¼ 0. In this stage (which iscalled the return stage or inelastic corrector stage), the deformation gradient F ic

nþ1 is fixed, and consequentlydicnþ1 ¼ W ic

nþ1 ¼ 0, where superscript ic is for the inelastic corrector. Then, kinematic Eq. (3) reduces to

_Be þ V2e$Di ¼ 0: (55)

Substituting Eqs. (5), (47), (23), and (18) into the plastic flow rule (55), we obtain for the inelastic corrector stage

_Be¼ � Ve$�Dp þ Dt

�$Ve ¼ �jgjV2

e$

sffiffiffiffiffiffiffiffis: s

p þ εt0A�23

�0:5I

!: (56)

Integrating Eq. (56) for Be from iteration step i to iþ1, we derive

Bðiþ1Þeðnþ1Þ ¼ BeðnÞ � jgjðiþ1ÞV2ðiÞ

eðnþ1Þ$

0B@ sðiÞnþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sðiÞnþ1: sðiÞnþ1

q þ εt0AðiÞðnþ1Þ

�23

�0:5I

1CADt; (57)

where jgj is a scalar to be determined. Substituting elastic strain Bðiþ1Þeðnþ1Þ in the elasticity rule (36), we obtain the expression for

the Cauchy stress

sðiþ1Þðnþ1Þ ¼ J�1

�2Bðiþ1Þ

eðnþ1Þ þ I�

$

0@leI1I þ 2GBðiþ1Þ

eðnþ1Þ þ�lI21 � 2mI2

�I þ n

vI3vBðiþ1Þ

eðnþ1Þþ 2mI1B

ðiþ1Þeðnþ1Þ

1A: (58)

Here, concentration c in the elastic constants (e.g. le ¼ ð1� cÞle1 þ cle2) in Eq. (44), and the concentration in the yieldstrength in Eq. (50) should also be written as a function of jgj :

cðiþ1Þnþ1 ¼ cn þ AðiÞ

ðnþ1Þffiffiffiffiffiffiffiffi2=3

pjgjDt: (59)

The parameter jgj can be found from the yield condition in Eq. (37)

4ðsnþ1; pnþ1; cnþ1Þ ¼ 4ðsnþ1ðjgjÞ; cnþ1ðjgjÞÞ (60)

where all arguments (snþ1 from Eqs. (57)e(59) and cnþ1 from Eq. (59)) depend on a single parameter jgj. This nonlinearequationwas iteratively solved using a Newton-Raphson method with the help of a developed Fortran subroutine. It involves

the derivative 40ðjgjðiÞÞ at the i-th iteration, which can be found in the Appendix. Due to geometrical and physical nonlinearityof the elasticity rule, pressure piþ1

nþ1 also depends on jgj despite the incompressibility of plastic flow. After finding jgj from Eq.

(60), sðiþ1Þnþ1 , Bðiþ1Þ

eðnþ1Þ, and cðiþ1Þnþ1 can be updated.

The elastic left stretch tensor V ðiþ1Þeðnþ1Þ can be updated by known Bðiþ1Þ

eðnþ1Þ either from the nonlinear equationV2ðiþ1Þ

eðnþ1Þ¼ 2Bðiþ1Þeðnþ1ÞþI written in principle axes or from its linearized version as


2�Bðiþ1Þeðnþ1Þ � BðiÞ

eðnþ1Þ�¼�V ðiþ1Þ

eðnþ1Þ � V ðiÞeðnþ1Þ

�$V ðiÞ

eðnþ1Þ þ V ðiÞeðnþ1Þ$

�V ðiþ1Þ

eðnþ1Þ � V ðiÞeðnþ1Þ

�: (61)

Fortran subroutines have been written for these purposes. Then, the symmetric tensor Uðiþ1Þi nþ1 and orthogonal Rðiþ1Þ

e nþ1 parts
can be found from
V�1ðiþ1Þe ðnþ1Þ $Fnþ1 ¼ Rðiþ1Þ

e nþ1$Uðiþ1Þi nþ1; (62)

where another Fortran subroutine is written for the polar decomposition of the tensor V�1ðiþ1Þeðnþ1Þ $Fnþ1.

Flowchart for the computational algorithm


6. Material properties for the sample and gasket

6.1. Rhenium

Due to the low yield strength of hBN, the sample gets very thin under high pressure if compressed without gasket. Thesample is placed in a gasket made of a strong material to reduce sample radial flow and increase the volume of a high-pressure region in the sample, as well as reduce the radial pressure gradient in the sample, rhenium is often used as agasket material (Jeanloz et al., 1991; Manghnan et al., 1974; Vohra et al., 1987; Dubrovinsky et al., 2012, 2015) due to its highcompressive strength, large bulk (K) and shear (G) moduli (Jeanloz et al., 1991; Manghnan et al., 1974; Vohra et al., 1987), andexcellent chemical stability. Rhenium is a heavy group-VII transition metal, which crystallizes in the hexagonal-close-packedstructure. Usually, a rhenium gasket is either polycrystalline or compacted powder. The grain size at a pressure of 250 GPa is10e20 nm (Singh et al., 2012), which is 20e40 times smaller than the sample thickness at the same pressure. The equations inSection 4 will be used for Re at concentration c ¼ 0.

The properties of rhenium used in simulations are as follows: elastic constants (Jeanloz et al., 1991; Manghnan et al., 1974)G ¼ 200 GPa, le ¼ 247 GPa, l ¼ �291 GPa, m ¼ �662 GPa and n ¼ 0 as well as plastic constants (Jeanloz et al., 1991)sy0 ¼ 8:00 GPa and b ¼ 0:04. These properties were used in our recent paper (Feng et al., 2016) and gave good corre-spondence with experimental data (Hemley et al., 1997) at pressures up to 300 GPa.

6.2. Hexagonal and superhard wurtzitic boron nitride

BN exists in several crystalline forms: two soft graphite-like forms as hexagonal hBN and rhombohedral rBN, and twodense forms as cubic cBN andwurtzitic wBN. Hexagonal hBN is themost stable and soft at ambient conditions, and it is widelyused as a lubricant and an additive to cosmetic products. Due to high dynamic strength, hardness, and high wear resistance,superhard wBN and cBN are of great interest as tool material for various applications (drilling, cutting, etc.) and their syn-thesis under extreme conditions has been extensively studied. The crystalline structure of wBN is the same as lonsdaleite(hexagonal diamond), and the solid-solid PT from hBN to wBN can occur under high pressures. Transformation pressure andmechanism strongly depend on the degree of the two-dimensional disorder, which is characterized by the concentration ofthe turbostratic stacking faults (Britun and Kurdyumov, 2000). Turbostratic stacking faults are formed in hexagonal lattices byrelative displacement or rotation of two parts of a lattice in (001) planes to arbitrary positions (Britun and Kurdyumov, 2000).A highly ordered hBN transforms to wBN at pressures ranging from 8.1 to 13 GPa at either room or high temperatures (Bundyand Wentorf, 1963; Corrigan and Bundy, 1975; Solozhenko and Elf, 1998; Taniguchi et al., 1997). The lowest pressure in thehydrostatic condition, at which highly ordered hBN-to-wBN transformation starts at room temperature, is 8.1 GPa; PT be-comes irreversible above 10 GPa and transformation does not complete up to 25 GPa (Solozhenko and Elf, 1998). In (Taniguchiet al., 1997), highly ordered hBN starts to transform at a pressure of 10 GPa under nonhydrostatic conditions; after pres-surizing to 12 GPa, wBN becomes quenchable. In (Levitas et al., 2006), initially highly ordered hBN was transformed to hBNunder the same pressure of ~10 GPa under uniaxial compression and compressionwith torsion in rotational DAC. The unusuallack of effect of plastic shear on this PT was rationalized by a two-fold effect of the plastic shear: (a) it promotes PT byproducing new defects with strong stress concentrators and (b) it suppresses PT by increasing disordering. It happened thatboth effects compensated each other.

A model, which considers evolution of the degree of disordering during plastic straining and its effect on phase trans-formation pressure, was developed and calibrated with the help of in situ X-ray measurements in Levitas et al. (2006). Here,we consider completely two-dimensional disordered nanograined hBN, i.e., degree of disordering does not evolve duringplastic deformation, which simplifies the problem. It was found in (Ji et al., 2012) that such a hBN does not transform to wBNeven at a pressure as high as 52.8 GPa under hydrostatic conditions. Thus, we assume pdh ¼ 55 GPa. Under compression andtorsion in RDAC, this completely disordered hBN transforms to wBN at 6.7 GPa; we assume pd

ε¼ 6:7 GPa. Since this is the

lowest pressure for hBN-to-wBN transition and very large (almost by an order of magnitude) reduction in transformationpressure due to plastic straining, this is the most interesting case. Thus, plastic strain-induced PT for the highly disorderedhBN-to-wBN transformation will be studied and discussed, utilizing the developed model and FEM.

Wurtzitic BN obtained by a plastic strain-induced PT from disordered hBN is stable at atmospheric pressure. We alsoassume that reverse plastic strain-induced PT is impossible even at atmospheric pressure, thus, skipping the second term inkinetic equation (41). The yield strengths of hBN and wBN vary depending onmaterial defects, grain sizes, purities, etc. In thispaper, reasonable values sy02 ¼ 30sy01 ¼ 9GPa (Wills, 1985) and b1 ¼ b2 ¼ 0:01 will be used. Elastic constants for hBN

(Solozhenko and Elf, 1998) are G1 ¼ 20 GPa, dG1dp ¼ 2:5, B1 ¼ 37GPa, and dB1

dp ¼ 5:6. For wBN (Kim et al., 1996; Saib and

Bouarissa, 2009) they are G2 ¼ 384GPa, dG2dp ¼ 1:0, B2 ¼ 397GPa, and dB2

dp ¼ 3:7. Similar to the reference (Feng et al., 2016)

and by assuming n1 ¼ n2 ¼ 0, we can calculate le1 ¼ 24GPa, l1 ¼ �35GPa, m1 ¼ �75GPa for hBN and le2 ¼ 141GPa,l2 ¼ �245GPa, and m2 ¼ �657GPa for wBN. Both the elastic and plastic parameters for hBN are significantly smaller thanwBN. After PT, the bulk and shear moduli increase by about 10 and 20 times respectively, and the yield strength increases by


30 times. This leads to results which are very distinct from those in our previous simulations (Feng et al., 2013a, 2013b; 2014;Levitas and Zarechnyy, 2010b), where the changes in material properties during PT were much smaller. The volumetrictransformation strain is 3εt0 ¼ �0:39 (Nagakubo et al., 2013; Rumyantsev et al., 2001), which is also larger by a factor of 4 thanin previous simulations (Feng et al., 2013a, 2013b; 2014; Levitas and Zarechnyy, 2010b). The kinetic factor k is the kineticparameter which scales the rate of PT. The discussion of effects of k on PT into a stronger high pressure phase can be found indetail in (Levitas, 2004a,b; Levitas and Zarechnyy, 2010a; Feng et al., 2013a, 2013b, 2014). Based on our experience andpreliminary calculations we accepted that k ¼ 3 is a reasonable value for the rate of PT in BN, and it is utilized in the currentpaper.

In contrast to martensitic PTs in highly ordered hBN, highly disordered hBN transforms through a reconstructive mech-anism (Britun and Kurdyumov, 2000), or maybe even through an intermediate amorphous phase (Ji et al., 2012). That is thereasonwhy only the volumetric transformation strain 3εt0 is known and the deviatoric part of the transformational strain rateis undefined and can be neglected. TRIP in the constitutive equations will be neglected as well because its parameters areunknown. However, due to heterogeneous distribution of concentration of wBN during PT, macroscopic TRIP within a sampleappears as a result of the solution to the boundary-value problem. This is due to the large volumetric transformation strainunder nonhydrostatic loading (Greenwood-Johnson mechanism, see, e.g., Fischer et al. (2000)). Also, reconstructive PTproduces the incoherent interface, which significantly reduces internal (residual) stresses and elastic strains. This simplifiestheory and, in particular, gives additional justification for neglecting residual elastic strains in Eq. (8). In addition, despite thepure volumetric transformation strain, shear stresses strongly affect PT by producing dislocation pile up with strong con-centration of pressure (Levitas, 2004a,b, Levitas and Javanbakht, 2014; Javanbakht and Levitas, 2015), which is taken intoaccount in Eq. (41).

7. Nonlinear elastic equations and material properties for single-crystal diamond anvil

The elasticity rule for anisotropic materials (Feng et al., 2016; Feng and Levitas, 2017) has the form

s ¼ F$~TðEÞ$Ft.detF; T ¼ sdetF ¼ F$~TðEÞ$Ft ; ~T ¼ vJ=vE ; (63)

where T is the Kirchhoff stress, ~T is the second Piola-Kirchhoff stress, and since there is no plastic deformation in a diamond,the subscript e is dropped. Under megabar pressures, it is necessary to consider at least the third-order potentialJwith cubicsymmetry:

J ¼ 0:5c11�h21 þ h22 þ h23

�þ c12ðh1h2 þ h1h3 þ h2h3Þ þ 0:5c44

�h24 þ h25 þ h26

�þ c111

�h31 þ h32 þ h33

�.6

þ0:5c112hh21ðh2 þ h3Þ þ h22ðh1 þ h3Þ þ h23ðh1 þ h2Þ

iþ c123h1h2h3 þ 0:5c144

�h1h

24 þ h2h

25 þ h3h

26

�þ0:5c166

hðh2 þ h3Þh24 þ ðh1 þ h3Þh25 þ ðh1 þ h2Þh26

iþ c456h4h5h6;

(64)

where h1 ¼ E11, h2 ¼ E22, h3 ¼ E33, h4 ¼ 2E23, h5 ¼ 2E31, and h6 ¼ 2E12.The following equations give two explicit examples for the components of the second Piola-Kirchhoff stress:

~T11 ¼ vJ

vh1¼ c11h1 þ c12ðh2 þ h3Þ þ c111h

21

.2þ c112

h2h1ðh2 þ h3Þ þ h22 þ h23

i.2þ c123h2h3

þc144h24

.2þ c166

�h25 þ h26

�.2; ~T12 ¼ vJ

vh6¼ c44h6 þ c144h3h6 þ c166ðh1 þ h2Þh6 þ c456h4h5:

(65)

The consistent stiffness matrix is defined as follows (Feng et al., 2016) as

C ¼ J�1ðLþ2KÞ; (66)

where the forth order tensors L and K are

L ¼ F$F$

v2J

vEvE$Ft$Ft

!; Kijkl ¼

14

�Tikdjl þ Tjkdil þ Tildjk þ Tjldik

�: (67)

All components in Eq. (56) are either known or can be easily calculated. For example,

v2J

vh1vh1¼ c11 þ c111h1 þ c112ðh2 þ h3Þ;

v2J

vh1vh2¼ c12 þ c112ðh1 þ h2Þ þ c123h3: (68)


In this paper, the second-order elastic constants are taken from (Nielsen, 1986): c11 ¼ 1050GPa, c12 ¼ 127GPa, andc44 ¼ 550GPa The third-order elastic constants are accepted from (Lang and Gupta, 2011): c111 ¼ �7603GPa,c112 ¼ �1909GPa, c123 ¼ �835GPa, c166 ¼ �3938GPa, c144 ¼ 1438GPa, and c456 ¼ �2316GPa.

8. Results and discussion on plastic strain-induced phase transformation in the BN sample

Due to the complexity of the problem, it is impossible to guess the character of the solution. For example, large volu-metric transformation strain should lead to pressure drop, but drastic increase of elastic and, especially, plastic propertiesduring PT should lead to pressure growth. Interplay of these effects determines the actual distribution of all fields and PTkinetics. Thus, if the reduction in volume due to PT dominates and the material flows toward the center, then maximumpressure may be from any side of the sample/gasket boundary (but will be away from the center of the sample). In contrast,if material will flow from the center, then the pressure gradient and, consequently, pressure will strongly increase towardthe center due to increasing yield strength (Levitas and Zarechnyy, 2010a; Feng et al., 2014). In some cases, pressure growsalmost homogeneously (Levitas et al., 2006; Feng et al., 2014). The shape of the sample/gasket boundary may remain cy-lindrical, or plastic flow at the symmetry plane from or to the center may be much faster than at the sample/diamondinterface. Maximum stresses in the gasket can also be smaller or larger than in the sample. Due to heterogeneity of thepressure and, especially, plastic strain, it is not clear which of these two fields will more strongly affect the evolution andheterogeneity of concentration of the high-pressure phase. In addition, change in initial geometric parameters of a sample/gasket system and material of a gasket may drastically change all fields (Feng et al., 2014). At pressures of 40 GPa and higher,deformation of diamond anvils is getting essential and produces addition changes. The key interest is in finding shape of asample, heterogeneity of pressure, plastic strain, and, especially, concentration of high pressure phase fields and under-standing what controls them. Heterogeneity along the thickness is especially important since it was never measured for anyfield and was assumed to be negligible. While there are no experimental results for plastic strain-induced PT in highlydisordered hBN, our analysis should help in interpretation of future experimental data. However, there are experimentaldata (Ji et al., 2012) for this PT in rotational DAC, showing drastic reduction of PT pressure due to rotation of an anvil. Ourpreliminary results show that we can reproduce such a pressure reduction quantitatively within the current model (Fengand Levitas, in preparation).

8.1. Geometry and boundary conditions

The schematic of a DAC is shown in Fig. 1a, where the sample in blue is placed inside a gasket and both are compressedbetween two diamond anvils with an increasing normal stress sn. The loading and geometry of the DAC are assumed to beaxisymmetric. As in (Feng et al., 2016), it is reasonable to neglect the material anisotropy of diamond anvils along thecircumferential direction but keep it within rz plane. The cubic lattice axis of anvils is along the z axis. Typically, at thepressure of 50 GPa, the bottom contact surface of the anvil used is flat, as in the cases for BN in (Ji et al., 2012). This contrastswith the preceding FEM studies under megabar pressures (Feng et al., 2016), where the bottom contact surface of thediamond anvil has a bevel angle of 8.5�, as in experiments in Hemley et al. (1997). Due to symmetry, the simulations isperformed for one-quarter of the system (Fig. 1b). The geometric parameters in the reference state are given in Fig. 1b for theanvil and in Fig. 1c for the sample and gasket. We notice that if there is a sharp corner at point K, the intersection pointbetween lines FK and KG, the penetration of the anvil finite elements into the sample elements may occur. Consequently, toavoid this and divergence in computations due to such a penetration, a smooth transition at a point K with a 45� arc is used(see Fig. 1c).

The boundary conditions for one-quarter of the DAC in Fig. 1b are summarized as follows:

(1) The normal compressive stress sn is applied at the top of anvil surface AB.(2) The radial displacement ur and shear stress trz are zero at the axis r ¼ 0 (line AC is for the anvil and line CD is for the

sample).(3) At the contact surface of diamond with the sample or gasket (the surface CFKG), the Coulomb friction model is applied

with the Coulomb friction coefficient m ¼ 0:15. Thus, if the friction stress t<msc , then relative sliding is absent.Sliding occurs at t ¼ msc, where sc is the contact stress normal to the contact surface.

(4) The cohesion condition is applied at the contact surface between the sample and gasket (the surface FE).(5) At the symmetry plane z ¼ 0 (the plane DEI), the radial shear stress trz ¼ 0, and the axial displacement uz ¼ 0.(6) Other surfaces not mentioned above are stress-free.

The mesh consisting of CGAX4 type elements (4-node generalized bilinear axisymmetric quadrilateral element with atwist) is shown in Fig. 1 (d). There are more than 25 elements over the half of a sample thickness, which is fine enough toobtain mesh-independent results. We performed the simulations by both reducing the number of elements along the halfsample thickness by 25% and increasing by 50%, and we obtained practically the same results as our current results. The

Fig. 1. (a) Diamond anvil cell scheme; (b) one-quarter of the sample, gasket, and anvil in the initial undeformed state, and the geometric parameter of the anvil;(c) the geometry of one-quarter of the sample (CDEF) and gasket (FKGHIE)); (d) the mesh used in the simulation.


results are mesh-independent for around 20 elements along the half of a thickness. Also, a similar mesh was used in Fenget al., (2016) to solve an elastoplastic problem with much larger plastic deformations (q � 23:5) and pressure gradient,leading to a pressure up to 300 GPa. The loading step was 10�4smax

n , where smaxn is the final stress applied to the surface

AB.

8.2. Plastic flow and phase transformations in hBN

Change in geometry of the sample and gasket as well as the evolution of distribution of concentration of wBN, c, pressure pand accumulative plastic strain q are presented in Fig. 2. Evolution of shear stress trz in the sample is given in Fig. 3. Evolutionof the pressure distribution at the contact surface is shown in Fig. 4.

Zero radial displacement of the sample at the contact surface of the anvil is caused by high contact stresses and relativelylow shear stresses, which are limited by the yield strength in shear, both in the sample and in a gasket near the sample. Thus,the cohesion condition at the contact surface is met for the sample and diamond contact surface since the pressure is high.Because of this, radial plastic flow of the material during compression has significant gradient along the z axis and is the mostintense at the symmetry plane. While pressure in this part of the sample, which penetrates into the gasket, is smaller thananywhere else in the sample, plastic strain is significantly larger, especially along the entire contact surfaces of the sample andgasket as well as the contact plane between sample and anvil. Plastic strain-induced PT starts in the penetrating part andtransformation rate ismaximal here during the entire compression until complete transformation occurs.We are not aware ofany experiments which check the cross section of a deformed sample. Presence of such a zone with PT in it should beexperimentally checked for any material because it allows a better understanding of the deformation process and gives thebest chance to obtain plastic strain-induced phase in a large concentration at relatively low pressure. Note, for very differentmaterial parameters that allowed relative sliding of the gasket with respect to the sample along the surface EF in (Feng et al.,2014), the initially cylindrical surface EF did not change its shape significantly.

The region with the smallest plastic deformation is near the contact surface and symmetry axis, so-called stagnationregion in upsetting of a thin sample. While pressure in this region is close to maximum, due to the smallest plastic strains,concentration of wBN is also minimal. Near the symmetry axis, due to small plastic strain near the contact surface, largeplastic straining occurs near the symmetry plane to accommodate the reduction in sample thickness. This, in combinationwith relatively high pressure causes intense PT in this region. Thus, for the given geometry, properties, and loading, PT is moreaffected by plastic strain fields than by pressure fields.

First visible traces of wBN appear near local plastic strain concentrators near the contact surface with a gasket at a localpressure of 12 GPa, well above pd

ε¼ 6:7GPa. The pressure in the sample (see Figs. 2 and 4) is quite homogeneous throughout

the entire sample for p<25 GPa at the applied normal stress sn � 0:6 GPa. This is caused by both a relatively thick sample anda small friction stress t due to its limitation by the yield strength in shear of a weak hBN, according to the simplified equi-

librium equation dpdr ¼ �2t

h . With an increase in sn, the pressure gradient in the sample rises as well, starting from the sample-

Fig. 2. Distributions of (a) concentration of high-pressure phase c, (b) pressure p, and (c) accumulated plastic strain q in the sample. The applied normal contactstress sn in GPa is 0.6 (1), 0.72 (2), 0.84 (3), 0.96 (4). The arrows in white in (a) are pointing to the boundary of the sample and gasket.

Fig. 3. Distribution of shear stress trz in the sample. The applied normal contact stress sn in GPa is 0.6 (1), 0.72 (2), 0.84 (3), and 0.96 (4).


0 30 60 90 120

10

20

30

40

505

3

2

1

r (μm)

1. σ =0.48 , 2. σ =0.6 3. σ =0.72, 4. σ =0.84 5. σ =0.96

4 p(GPa)

Fig. 4. Distribution of pressure p at the upper surface of the sample and gasket contacting with diamond. The applied normal contact stress sn in GPa is 0.48 (1),0.6 (2), 0.72 (3), 0.84 (4) and 0.96 (5).


gasket boundary. Note, while near the symmetry axis it is still homogeneous (Figs. 2 and 4). The pressure heterogeneity iscaused by the reduction of thickness and rise of the friction stress due to transformation hardening during PT to significantlystronger wBN. Shear stress (Fig. 3) increases away from the symmetry plane and axis towards the contact line between thesample, gasket, and anvil which is stimulated by the increase in yield strength in shear with increasing concentration of wBN.

The pressure distributions at the flat contact surface between diamond and the sample or gasket (Fig. 4) gives an addi-tional insight. For stress sn ¼ 0:48 GPa, the pressure in the gasket is larger than in the sample. Also, the pressure in thesample/anvil contact is homogeneous. Note, for a weaker gasket or without a gasket, pressure always reaches the maximumat the central part of the sample. With increasing applied stress, the pressure grows faster in the sample than in the gasket.Thus, it is getting approximately the same for gasket and sample under sn ¼ 0:6, and then pressure is getting larger in thesample than in the gasket.

Pressure at the central region is homogeneous at any load, but the radius of this region reduces with increasing load.Maximum of the pressure is not at the center but shifted to 25e30 mm. Homogeneity of pressure in the sample is one of thegoals in some of the experiments, in particular in (Levitas et al., 2006). Conditions for homogeneous pressurewas estimated in(Levitas et al., 2006) analytically and confirmed experimentally. One of the main points was to choose geometric parametersin such a way that sliding in the sample at the contact surface does not occur. Even for this case, we demonstrated here thatstartingwith some load, the pressure gradient appears at the periphery of the sample. Also, even in the regionwhere pressureis homogeneous along the radius, there is essential pressure heterogeneity along the sample thickness as well. There is asignificant pressure drop near the sample-gasket interface at the contact plane with an anvil. It may be caused by complexflow and jump in properties near this region, as well as relatively fast PT and volume reduction due to a large plastic strainrate.

Fig. 5 plots the variations of concentration of high-pressure phase c0 and accumulated plastic strain q0 averaged over thedeformed sample volume V (a0 :¼ V�1R

VadV) versus themaximumpressure pmax in the sample. The plastic strain required for

a plastic strain-induced PT can only be obtained during compression along with a pressure increase. Visible c0 starts aroundpmax ¼ 12:5 GPa, which is much larger than the critical pressure for a direct strain-induced PT, pd

ε¼ 6:7GPa. For comparison,

without a gasket (Feng et al., 2013a), PT is observable very close to pdε. The reason for such strong deviation is that with a

0 12 24 36 480.0

0.2

0.4

0.6

0.8

1.0

0.0

0.3

0.6

0.9

1.2

c0

q0

q0c

0

pmax

Fig. 5. Variations of concentration of high-pressure phase c0 and accumulated plastic strain q0 averaged over the deformed sample versus maximum pressurepmax in the sample. The subscript 0 means averaged values over the sample, which is used to distinguish from the local concentration c and plastic strain q.


strong gasket, the plastic flow in the sample is more limited, but the pressure grows faster than without a gasket. If thethreshold for detection of the high-pressure phase concentration is 0.1, then corresponding “PT pressure” is around 21 GPa. PTis not completed everywhere in a sample even at the maximum pressure of 50 GPa. This is due to strong heterogeneity ofpressure and especially plastic strain, which does not exceed 1.1 in themajor part of the sample, whilemaximum q¼ 2.48. Ourresults demonstrate that 12 or 21 GPa are not a fundamental characteristic of PT but rather a result of the behavior of thesample-gasket system under certain types of the loading. It is practically impossible to extract the basic parameter pd

εfrom an

experiment with a strong gasket under compression in DAC. At the same time, for a high-pressure phase stronger than a low-pressure phase, characteristic pressure pd

εcan be localized during compression of a sample without a gasket when the large

pressure gradient in the radial direction is observed in the experiment. Experimental pressure distribution contains steps orplateaus with almost constant pressure, located at the two-phase region's separated (almost completely) transformed andnon-transformed regions (Blank et al., 1984; Novikov et al., 1999; Blank and Estrin, 2014). While it is assumed in (Blank et al.,1984; Blank and Estrin, 2014) that pressure at the plateau corresponds to PT pressure and phase equilibrium pressure, wefound in FEM simulations that it corresponds to the value close to pd

ε(Feng et al., 2013a).

However, at relatively slow kinetics this plateau may not be observed in simulations (Levitas and Zarechnyy, 2010b). Evenin this case it can be observed in simulations for torsion under fixed loading in rotational DAC (Levitas and Zarechnyy, 2010a)because of more intense plastic deformation. One should be careful and not connect pd

εwith the pressure in the regionwhere

a small amount of high pressure phase is found. PTmay occur at amuch higher pressure, and then the transformed phasemaybe moved in the low-pressure region by convective radial flow, thus leading to misinterpretation of experiments. Also, as itwas shown in (Levitas and Zarechnyy, 2010a, 2010b), similar multiple plateaus in the pressure distribution for a low strengthhigh pressure phase do not correspond to any characteristic pressure but just exhibit complex mechanical behavior of theheterogeneously deformed material during PT. Evenwith a gasket, when the pressure gradient in a sample is relatively smalland plateaus are not observed, torsion in rotational DAC at the constant force allows an increase in plastic straining withoutessential pressure growth. This allows one to obtain PT at a pressure close to pd

ε(Feng and Levitas, 2016).

Thus, the obtained result illustrates the main reason of large scatter in experimentally reported pressures for different PTsand spreading transformation over large pressure range. Concentration of high-pressure phase for p>pd

εis determined

mostly not by pressure but by the increment of plastic strain q, which was never measured. While an increase in pressure isnot required for PT, it occurs because of further compression of the sample, which is necessary to increase plastic strain. Theonly way to obtain PT at pressure close to pd

εis to increase plastic strain by torsion at the fixed applied force in rotational DAC,

when pressure does not grow. Also, obtained results explain the difference in PT pressure observed under compression in DACand torsion in rotational DAC, despite the same mechanism, thermodynamics and kinetics of plastic strain-induced PTs. Thekey point is that one has to distinguish material behavior, i.e., constitutive equations, from the behavior of the system sample/gasket/anvil.

8.3. Stress-strain fields in diamond

Knowledge of stress and strain fields in diamond anvil is required for estimation of strength of the anvil and optimizationof its geometry and loading conditions. These fields have been determined in diamonds compressing the Re sample up to285 GPa (Feng et al., 2016). Here, we will present just evolution of the pressure field (Fig. 6) and deformation of the initialplane contact surface with the sample and gasket. As follows from (Lang and Gupta, 2011), allowing for physical nonlinearity,i.e., including the third-order elastic potential, is required at pressures over 30 GPa. In contrast to the pressure gradient in asample, which is relatively small in the thickness direction and much larger in the radial direction, Fig. 6 shows that the

Fig. 6. Distribution of pressure p in the bottom of the diamond anvil. The geometry of the undeformed state is shown in (0), and the applied normal contact stresssn in GPa is 0.6 (1), 0.72 (2), 0.84 (3), and 0.96 (4).


pressure gradient in the diamond is much smaller in the radial direction than in the thickness direction. This is because thecross-sectional area significantly increases along the z axis and total vertical force does not change.

For pressures smaller than 28.7 GPa, the maximum pressure is located near the corner point rather than center of thecontact surface because of the stress concentration due to a sudden change in geometry. Although the maximum pressure isonly 31.3 GPa in the diamond, obvious bending of the diamond contact surface can be found in Figs. 6 and 7. Fig. 7 plots half ofthe thickness of the sample and gasket along the radial coordinate. At applied normal stress sn ¼ 0:6 GPa, the thicknessdifference at r ¼ 0 and r ¼ 125mm is only 1.6 mm, which is 2.0% of the sample thickness at r ¼ 0. With an increasing force, thedifference between the center and periphery grows. For sn ¼ 0:96 GPa, the thickness difference at r ¼ 0 and r ¼ 125 mm is4.6mm,which is 8.8% of the thickness at r¼ 0. Therefore, the deformation of diamond cannot be ignored under such an appliedload.

9. Concluding remarks

In this paper, coupled plastic strain-induced PT and plastic flow in a sample compressed in a diamond anvil cell areinvestigated in the framework of fully geometrically nonlinear formulation and by utilizing FEM. A thermodynamicallyconsistent system of equations is formulated. It includes the Murnaghan elasticity law and pressure-dependent J2 plasticityfor both low and high pressure phases, linear mixture rule, and plastic strain-induced and pressure-controlled PT kinetics.Elastic, plastic, and transformation strains can be large. The relationship between the Jaumann derivative of the Cauchy stressand deformation rate as well as the explicit expression for the consistent tangent moduli are derived. A computational al-gorithm is presented and implemented as a user material subroutine UMAT in the FEM code ABAQUS. The material pa-rameters for the BN sample, rhenium gasket, and diamond anvils are calibrated based on the published experimental andatomistic simulation results in the literature.

Plastic strain-induced transformation in the highly-disordered hexagonal hBN sample to a superhard wurtzitic wBN issimulated under high pressure in DAC. Complexity of the problem is increased due to large volumetric transformation strain(3εt0 ¼ �0:39), and a much higher elastic modulus (10e20 times) and yield strength (30 times) of wBN in comparison withhBN.

The evolutions of the stress, strain, and concentration of the high-pressure phase fields in the sample are obtained anddiscussed in detail. Strain and stress fields were also presented for the gasket and diamond anvil. In particular, for pressure at48 GPa, deformation of the diamond anvil changes the sample's profile which affects PT in the sample as well. Thus, a rigiddiamond model (Feng et al., 2013a, 2013b, 2014; Levitas and Zarechnyy, 2010a) cannot be used in the current study.

Unexpected shape of the deformed sample is predicted with almost complete PT in the external part of the sample thatpenetrated into the gasket. This prediction should be checked experimentally and if confirmed, utilized in measurements.

Pressure heterogeneity is another concern for experimentalists because it significantly complicates characterization of PTs.It was found while at low pressure, heterogeneities are small, but with a pressure increase heterogeneities grow in the radialdirection and especially along the thickness. Accumulated plastic strain is very heterogeneous in the sample, and its evolutionis the major factor that determines the distribution of concentration of wBN rather than pressure heterogeneity. Thus, theonly way to determine PT kinetics experimentally is to find the best fit between FEM simulations and available experimentalmeasurements (displacements, pressure and concentration of wBN distributions at the contact surface or averaged over thesample thickness). An important concept is that one has to follow moving material points rather than focus on the evolutionof fields in selected spatial points.

Due to a broad variety of pressure and plastic strain paths within a sample, this method will represent a high-throughputcharacterization. Thus, one of our main result is that we produce a computational tool that can be utilized for the iterativecomparisonwith experimentally measurable parameters and extraction ofmaterial parameters and functions, in particular, inthe kinetic equation.

0 40 80 120

24

30

36

42

48

4

3

2

σ =0.6 (1), σ =0.72 (2),

σ =0.84 (3), σ =0.96 (4)

h/2

(μm

)

r (μm)

1

Fig. 7. Half the thickness of the sample and gasket (or the z coordinate at the contact surface of diamond) vs r. The applied normal contact stress sn in GPa is 0.6(1), 0.72 (2), 0.84 (3), and 0.96 (4).


One more main point of our simulations is to demonstrate the difference between material and system behavior, which isoften confused by experimentalists. Thus, while plastic strain-induced PTmay start at plastic straining close to pd

ε¼ 6:7 GPa, it

is not visible below 12 GPa. If the threshold for detection of the wBN concentration is 0.1, then the experimentally reported“PT pressure” would be around 21 GPa. Also, PT is not completed everywhere in a sample even at a maximum pressure of50 GPa. Such a scatter in measured pressures may lead to a claim of large scatter in “transformation pressure.” One of thereasons for such a misunderstanding is the confusion surrounding pressure-induced PTs and plastic strain-induced PTs. Thelatter cannot be characterized in terms of transformation pressure but rather in terms of Eq. (41) which is the plastic strain-controlled and pressure dependent kinetic equation. Thus, PT can occur at any pressure above pd

εin the presence of plastic

flow.In the framework of plastic strain-induced PT, significant deviation of “transformation pressure” from pd

εis because the

increase in plastic strain in a sample under compression in DAC causing growth in pressure, even if we do not need it. Incontrast, under torsion at a fixed load in a rotational DAC, plastic straining may occur without increasing pressure whichallows one to obtain high pressure phases just slightly above pd

ε. This does not mean plastic shear promotes plastic strain-

induced PTs more than plastic compression because our kinetic Eq. (41) does not include the effect of different strainstates. This just means the sample-gasket system behaves differently under different types of loading.

Possible misinterpretation of the experimental data and characterization of the PT is discussed. Obtained results arebeneficial to the understanding of plastic strain-induced PTs in BN and other superhard phases under compression in the DAC.The developedmodel will allow computational design of experiments for synthesis of high-pressure phases. Study of PT in BNunder compression and torsion in RDAC will be performed in future work, similar to that in (Feng and Levitas, 2013, 2016;Levitas and Zarechnyy, 2010a) but with a more sophisticated model presented here. There are experimental data for thiscase that PT occurs at 6.7 GPa (Ji et al., 2012), which will be used for comparison.

Acknowledgements

The support of NSF (DMR-1434613), ARO (W911NF-17-1-0225), and Iowa State University (Schafer 2050 Challenge Pro-fessorship) is gratefully acknowledged.

Appendix. Explicit expressions for vfvc,

vfvBe

, v4vs,v4vc , and 40ð jgjÞ

1. We can directly evaluate vfvc from the constitutive law s ¼ f ðBe; cÞ in Eq. (36) as

vfvc

¼hJ�1ð2Be þ IÞ

i$

�ðle2 � le1ÞI1I þ 2ðG2 � G1ÞBe

þ�ðl2 � l1ÞI21 � 2ðm2 �m1ÞI2

�I þ ðn2 � n1Þ

vI3vBe

þ 2ðm2 �m1ÞI1Be

: (A1)

2. vfvBe

can be evaluated as follows from Eq. (36) as

vfijvBekf

¼ vð1=detFÞvBekf

sijdetF þ 2dikdetF

leI1dfj þ 2mBefj þ

�lI21 � 2mI2

�dfj þ n

vI3vBefj

þ 2mI1Befj

!

þð1=detFÞ�2Beij þ dij�hledkf þ 2ldkf I1 � 2m

�I1dkf � Bekf

�iþ

ð1=detFÞ�2Beik þ dik��

2mdjf þ 2mdjf I1�þ ð1=detFÞð2Beim þ dimÞ

n

v2I3vBemjvBekf

þ 2mdkf Bemj

!:

(A2)

Tensor vI3vBe

is given in Eq. (36), and it is easy to obtain v2I3vBe Be

by further differentiation. Tensor vð1=det FÞvBe

can be obtained from
mj kf kf
the following procedures. First, we evaluate�1=detFÞ ¼ �detFtdetF

��0:5 ¼ �det�Ft$F��0:5 ¼ ½detðI þ 2BeÞ��0:5

¼ ½ð1þ 2Be1Þð1þ 2Be2Þð1þ 2Be3Þ��0:5 ¼ ð1þ 2I1 þ 4I2 þ 8I3Þ�0:5;(A3)

in which Be1, Be2, and Be3 are the three principle values of tensor Be, and Ii are the three invariants of the strain tensor Be, see
Eq. (34). Then we calculate


vð1=detFÞvBe

¼ vð1þ 2I1 þ 4I2 þ 8I3Þ�0:5

vBe¼ �1

detF3

�Iþ2ðI1I � BeÞ þ 4

vI3vBe

�; (A4)

which concludes evaluation of vsvBe

.

3. For linear pressure dependence of the yield surface, Eq. (14), we obtain

v4

vs¼

ffiffiffiffiffiffiffi1:5

psffiffiffiffiffiffiffiffiffi

s : sp þ b

3I;

v4

vc¼ sy01 � sy02: (A5)

4.The derivative 40ð jgjðiÞÞ at the ith iteration at the time step n þ 1 in the Newton-Raphson method is evaluated by usingEq. (60):

40�jgjðiÞ

�¼ v4

vsðiÞnþ1

:vs

ðiÞnþ1

vjgjðiÞþ v4

vcðiÞnþ1

vcðiÞnþ1

vjgjðiÞ¼ v4

vsðiÞnþ1

:

0@vf ðiÞnþ1

vcðiÞnþ1

vcðiÞnþ1

vjgjðiÞþ vf ðiÞnþ1

vBðiÞeðnþ1Þ

:vBðiÞ

eðnþ1ÞvjgjðiÞ

1Aþ v4

vcðiÞnþ1

vcðiÞnþ1

vjgjðiÞ: (A6)

Due to Eq. (A5), we have

v4

vsðiÞnþ1

¼ffiffiffiffiffiffiffi1:5

psðiÞnþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sðiÞnþ1 : sðiÞnþ1

q þ b3I and

v4

vcðiÞnþ1

¼ sy01 � sy02: (A7)

To explicitly express each term in Eq. (A6), we use Eq. (A1), (A2), (A5), (57) and (59), and obtain

vBðiÞeðnþ1Þvjgj ¼ �V2ði�1Þ

eðnþ1Þ$

0B@ sði�1Þ

nþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisði�1Þnþ1 : sði�1Þ

nþ1

q þ εt0Aði�1Þnþ1

�23

�0:5I

1CADt: (A8)

vcðiÞ ffiffiffiffiffiffiffiffip
nþ1
vjgjðiÞ¼ 3=2Aði�1Þ

nþ1 Dt: (A9)

The elastic strain, stress, and concentration in (A6)e(A9) at step n þ 1 at each iteration are used from Eqs. (57)e(59) aftersubstituting the known value jgjðiÞ.

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