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1. A multiphase phase-field study of three-dimensional martensitic twinned microstructures at large strains

   Basak, Anup; Levitas, Valery I. (June 2022, ArXivorg)

   A thermodynamically consistent multiphase phase-field approach for stress and temperature-induced martensitic phase transformation at the nanoscale and under large strains is developed. A total of N independent order parameters are considered for materials with N variants, where one of the order parameters describes A ↔ M transformations and the remaining N − 1 independent order parameters describe the transformations between the variants. A non-contradictory gradient energy is used within the free energy of the system to account for the energies of the interfaces. In addition, a non-contradictory kinetic relationships for the rate of the order parameters versus thermodynamic driving forces is suggested. As a result, a system of consistent coupled Ginzburg-Landau equations for the order parameters are derived. The crystallographic solution for twins within twins is presented for the cubic to tetragonal transformations. A 3D complex twins within twins microstructure is simulated using the developed phase-field approach and a large-strain-based nonlinear finite element method. A comparative study between the crystallographic solution and the simulation result is presented. 

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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 (February 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 field 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. Phase field theory for fracture at large strains including surface stresses

   Jafarzadeh, Hossein; Farrahi, Gholam Hossein; Levitas, Valery I.; Javanbakht, Mahdi (November 2020, ArXivorg)

   null (Ed.)

   Phase field theory for fracture is developed at large strains with an emphasis on a correct introduction of surface stresses. This is achieved by multiplying the cohesion and gradient energies by the local ratio of the crack surface areas in the deformed and undeformed configurations and with the gradient energy in terms of the gradient of the order parameter in the reference configuration. This results in an expression for the surface stresses which is consistent with the sharp surface approach. Namely, the structural part of the Cauchy surface stress represents an isotropic biaxial tension, with the magnitude of a force per unit length equal to the surface energy. The surface stresses are a result of the geometric nonlinearities, even when strains are infinitesimal. They make multiple contributions to the Ginzburg-Landau equation for damage evolution, both in the deformed and undeformed configurations. Important connections between material parameters are obtained using an analytical solution for two separating surfaces, as well as an analysis of the stress-strain curves for homogeneous tension for different degradation and interpolation functions. A complete system of equations is presented in the undeformed and deformed configurations. All the phase field parameters are obtained utilizing the existing first principle simulations for the uniaxial tension of Si crystal in the [100] and [111] directions. 

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4. A unified theory of free energy functionals and applications to diffusion

   https://doi.org/10.1073/pnas.2203399119  

   Li, Andrew B.; Miroshnik, Leonid; Rummel, Brian D.; Balakrishnan, Ganesh; Han, Sang M.; Sinno, Talid (June 2022, Proceedings of the National Academy of Sciences)

   Free energy functionals of the Ginzburg–Landau type lie at the heart of a broad class of continuum dynamical models, such as the Cahn–Hilliard and Swift–Hohenberg equations. Despite the wide use of such models, the assumptions embodied in the free energy functionals frequently either are poorly justified or lead to physically opaque parameters. Here, we introduce a mathematically rigorous pathway for constructing free energy functionals that generalizes beyond the constraints of Ginzburg–Landau gradient expansions. We show that the formalism unifies existing free energetic descriptions under a single umbrella by establishing the criteria under which the generalized free energy reduces to gradient-based representations. Consequently, we derive a precise physical interpretation of the gradient energy parameter in the Cahn–Hilliard model as the product of an interaction length scale and the free energy curvature. The practical impact of our approach is demonstrated using both a model free energy function and the silicon–germanium alloy system. 

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5. Phase-field description of fracture in NiTi single crystals

   https://doi.org/10.1016/j.cma.2023.116677  

   Kavvadias, D.; Baxevanis, Th. (February 2024, Computer Methods in Applied Mechanics and Engineering)

   A phase-field model for thermomechanically-induced fracture in NiTi at the single crystal level, i.e., fracture under loading paths that may take advantage of either of the functional properties of NiTi–superelasticity or shape memory effect–, is presented, formulated within the kinematically linear regime. The model accounts for reversible phase transformation from austenite to martensite habit plane variants and plastic deformation in the austenite phase. Transformation-induced plastic deformation is viewed as a mechanism for accommodation of the local deformation incompatibility at the austenite–martensite interfaces and is accounted for by introducing an interaction term in the free energy derived based on the Mori–Tanaka and Kröner micromechanical assumptions and the hypothesis of martensite instantaneous growth within austenite. Based on experimental observations suggesting that NiTi fractures in a stress-controlled manner, damage is assumed to be driven by the elastic energy, i.e., phase transformation and plastic deformation are assumed to contribute in crack formation and growth indirectly through stress redistribution. The model is restricted to quasistatic mechanical loading (no latent heat effects), thermal loading sufficiently slow with respect to the time rate of heat transfer by conduction (no thermal gradients), and a temperature range below 𝑀𝑑, which is the temperature above which the austenite phase is stable, i.e., stress-induced martensitic transformation is suppressed. The numerical implementation of the model is based on an efficient scheme of viscous regularization in both phase transformation and plastic deformation, an explicit numerical integration via a tangent modulus method, and a staggered scheme for the coupling of the unknown fields. The model is shown able to capture transformation-induced toughening, i.e., stable crack advance attributed to the shielding effect of inelastic deformation left in the wake of the growing crack under nominal isothermal loading, actuation-induced fracture under a constant bias load, and crystallographic dependence on crack pattern. 

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