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1. Simulations of multivariant Si I to Si II phase transformation in polycrystalline silicon with finite-strain scale-free phase-field approach

   https://doi.org/10.1016/j.actamat.2023.118996  

   Babaei, H; Pratoori, R; Levitas, V.I. (May 2023, Acta Materialia)

   Scale-free phase-field approach and corresponding finite element method simulations for multivariant martensitic phase transformation from cubic Si I to tetragonal Si II in a polycrystalline aggregate are presented. Important features of the model are large and very anisotropic transformation strain tensor 𝜺𝑡 = {0.1753; 0.1753; −0.447} and stress-tensor dependent athermal dissipative threshold for transformation, which produce essential challenges for computations. 3D polycrystals with stochastically oriented grains are subjected to uniaxial strain- and stress-controlled loadings under periodic boundary conditions and zero averaged lateral strains. Coupled evolution of discrete martensitic microstructure, volume fractions of martensitic variants and Si II, stress and transformation strain tensors, and texture are presented and analyzed. Macroscopic variables effectively representing multivariant transformational behavior are introduced. Macroscopic stress–strain and transformational behavior for 55 and 910 grains are close. Large transformation strains and grain boundaries lead to huge internal stresses of tens GPa, which affect microstructure evolution and macroscopic behavior. In contrast to a single crystal, the local mechanical instabilities due to phase transformation and negative local tangent modulus are stabilized at the macroscale by arresting/slowing the growth of Si II regions by the grain boundaries. This leads to increasing stress during transformation. The developed methodology can be used for studying similar phase transformations with large transformation strains and for further development by including plastic strain and strain-induced transformations. 

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2. An exact formulation for exponential-logarithmic transformation stretches in a multiphase phase field approach to martensitic transformations

   https://doi.org/10.1177/1081286520905352  

   Basak, Anup; Levitas, Valery I (January 2020, Mathematics and Mechanics of Solids)

   A general theoretical and computational procedure for dealing with an exponential-logarithmic kinematic model for transformation stretch tensor in a multiphase phase field approach to stress- and temperature-induced martensitic transformations with N martensitic variants is developed for transformations between all possible crystal lattices. This kinematic model, where the natural logarithm of transformation stretch tensor is a linear combination of natural logarithm of the Bain tensors, yields isochoric variant–variant transformations for the entire transformation path. Such a condition is plausible and cannot be satisfied by the widely used kinematic model where the transformation stretch tensor is linear in Bain tensors. Earlier general multiphase phase eld studies can handle commutative Bain tensors only. In the present treatment, the exact expressions for the first and second derivatives of the transformation stretch tensor with respect to the order parameters are obtained. Using these relations, the transformation work for austenite ↔ martensite and variant ↔ variant transformations is analyzed and the thermodynamic instability criteria for all homogeneous phases are expressed explicitly. The finite element procedure with an emphasis on the derivation of the tangent matrix for the phase field equations, which involves second derivatives of the transformation deformation gradients with respect to the order parameters, is developed. Change in anisotropic elastic properties during austenite–martensitic variants and variant–variant transformations is taken into account. The numerical results exhibiting twinned microstructures for cubic to orthorhombic and cubic to monoclinic-I transformations are presented. 

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3. Resolving puzzles of the phase-transformation-based mechanism of the deep-focus earthquake

   Levitas, Valery I. (October 2021, ArXivorg)

   Deep-focus earthquakes that occur at 350–660 km, where pressures p =12-23 GPa and temperature T =1800-2000 K, are generally assumed to be caused by olivine→spinel phase transformation, see pioneering works [1–10]. However, there are many existing puzzles: (a) What are the mechanisms for jump from geological 10−17−10−15 s−1 to seismic 10−103s−1(see [3]) strain rates? Is it possible without phase transformation? (b) How does metastable olivine, which does not completely transform to spinel at high temperature and deeply in the region of stability of spinel for over the million years, suddenly transforms during seconds and generates seismic strain rates 10−103s−1 that produce strong seismic waves? (c) How to connect deviatorically dominated seismic signals with volume-change dominated transformation strain during phase transformations [9,11]? Here we introduce a combination of several novel concepts that allow us to resolve the above puzzles quantitatively. We treat the transformation in olivine like plastic strain-induced (instead of pressure/stress-induced) and find an analytical 3D solution for coupled deformation-transformation-heating processes in a shear band. This solution predicts conditions for severe (singular) transformation-induced plasticity (TRIP) and self-blown-up deformation-transformation-heating process due to positive thermomechanochemical feedback between TRIP and strain-induced transformation. In nature, this process leads to temperature in a band exceeding the unstable stationary temperature, above which the self-blown-up shear-heating process in the shear band occurs after finishing the phase transformation. Without phase transformation and TRIP, significant temperature and strain rate increase is impossible. Due to the much smaller band thickness in the laboratory, heating within the band does not occur, and plastic flow after the transformation is very limited. Our findings change the main concepts in studying the initiation of the deep-focus earthquakes and phase transformations during plastic flow in geophysics in general. The latter may change the interpretation of different geological phenomena, e.g., the possibility of the appearance of microdiamond directly in the cold Earth crust within shear-bands [12] during tectonic activities without subduction to the mantle and uplifting. Developed theory of the self-blown-up transformation-TRIP-heating process is applicable outside geophysics for various processes in materials under pressure and shear, e.g., for new routes of material synthesis [12,13], friction and wear, surface treatment, penetration of the projectiles and meteorites, and severe plastic deformation and mechanochemical technologies. 

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4. Fifth-degree elastic energy for predictive continuum stress–strain relations and elastic instabilities under large strain and complex loading in silicon

   https://doi.org/10.1038/s41524-020-00382-8  

   Chen, Hao; Zarkevich, Nikolai A.; Levitas, Valery I.; Johnson, Duane D.; Zhang, Xiancheng (December 2020, npj Computational Materials)

   null (Ed.)

   Abstract Materials under complex loading develop large strains and often phase transformation via an elastic instability, as observed in both simple and complex systems. Here, we represent a material (exemplified for Si I) under large Lagrangian strains within a continuum description by a 5 th -order elastic energy found by minimizing error relative to density functional theory (DFT) results. The Cauchy stress—Lagrangian strain curves for arbitrary complex loadings are in excellent correspondence with DFT results, including the elastic instability driving the Si I → II phase transformation (PT) and the shear instabilities. PT conditions for Si I → II under action of cubic axial stresses are linear in Cauchy stresses in agreement with DFT predictions. Such continuum elastic energy permits study of elastic instabilities and orientational dependence leading to different PTs, slip, twinning, or fracture, providing a fundamental basis for continuum physics simulations of crystal behavior under extreme loading. 

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5. Nontrivial nanostructure, stress relaxation mechanisms, and crystallography for pressure-induced Si-I → Si-II phase transformation

   https://doi.org/10.1038/s41467-022-28604-1  

   Chen, Hao; Levitas, Valery I.; Popov, Dmitry; Velisavljevic, Nenad (December 2022, Nature Communications)

   Abstract Crystallographic theory based on energy minimization suggests austenite-twinned martensite interfaces with specific orientation, which are confirmed experimentally for various materials. Pressure-induced phase transformation (PT) from semiconducting Si-I to metallic Si-II, due to very large and anisotropic transformation strain, may challenge this theory. Here, unexpected nanostructure evolution during Si-I → Si-II PT is revealed by combining molecular dynamics (MD), crystallographic theory, generalized for strained crystals, and in situ real-time Laue X-ray diffraction (XRD). Twinned Si-II, consisting of two martensitic variants, and unexpected nanobands, consisting of alternating strongly deformed and rotated residual Si-I and third variant of Si-II, form $$\{111\}$$ { 111 } interface with Si-I and produce almost self-accommodated nanostructure despite the large transformation volumetric strain of $$-0.237$$ − 0.237 . The interfacial bands arrest the $$\{111\}$$ { 111 } interfaces, leading to repeating nucleation-growth-arrest process and to growth by propagating $$\{110\}$$ { 110 } interface, which (as well as $$\{111\}$$ { 111 } interface) do not appear in traditional crystallographic theory. 

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