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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 ( 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. 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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4. 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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5. Ginzburg-Landau approach to the vortex–domain wall interaction in superconductors with nematic order

   https://doi.org/10.1103/PhysRevB.109.094513  

   Severino, R. S. ; Mininni, P. D. ; Fradkin, E. ; Bekeris, V. ; Pasquini, G. ; Lozano, G. S. ( March 2024 , Physical Review B)

   In this work, we study the interaction between vortices and nematic domain walls within the framework of a Ginzburg-Landau approach. The free energy of the system is written in terms of a complex order parameter characteristic of s-wave superconductivity and a real (Ising-type) order parameter associated with nematicity. The interaction between both order parameters is described by a biquadratic and a trilinear derivative term. To study the effects of these interactions, we solve the time-dependent dissipative Ginzburg-Landau equations using a highly effective pseudospectral method by which we calculate the trajectories of a vortex that, for different coupling parameters, is either attracted or repelled by a wall, as well as of the wall dynamics. We show that despite its simplicity, this theory displays many phenomena observed experimentally in Fe-based superconductors. In particular, we find that the sign of the biquadratic term determines the attractive (pinning) or repulsive (antipinning) character of the interaction, as observed in FeSe and BaFeCoAs compounds, respectively. The trilinear term is responsible for the elliptical shape of vortex cores as well as the orientation of the axes of the ellipses and vortex trajectories with respect to the axes of the structural lattice. For the case of pinning, we show that the vortex core is well described by a heart-shaped structure in agreement with scanning tunneling microscopy experiments performed in FeSe. 

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