A variational treatment of material configurations with application to interface motion and microstructural evolution

Abstract

We present a unified variational treatment of evolving configurations in crystalline solids with microstructure. The crux of our treatment lies in the introduction of a vector configurational field. This field lies in the material, or configurational, manifold, in contrast with the traditional displacement field, which we regard as lying in the spatial manifold. We identify two distinct cases which describe (a) problems in which the configurational field's evolution is localized to a mathematically sharp interface, and (b) those in which the configurational field's evolution can extend throughout the volume. The first case is suitable for describing incoherent phase interfaces in polycrystalline solids, and the latter is useful for describing smooth changes in crystal structure and naturally incorporates coherent (diffuse) phase interfaces. These descriptions also lead to parameterizations of the free energies for the two cases, from which variational treatments can be developed and equilibrium conditions obtained. For sharp interfaces that are out-of-equilibrium, the second law of thermodynamics furnishes restrictions on the kinetic law for the interface velocity. The class of problems in which the material undergoes configurational changes between distinct, stable crystal structures are characterized by free energy density functions that are non-convex with * Corresponding Author arXiv:1608.05355v2 [cond-mat.soft] 18 Nov 2016 respect to configurational strain. For physically meaningful solutions and mathematical well-posedness, it becomes necessary to incorporate interfacial energy. This we have done by introducing a configurational strain gradient dependence in the free energy density function following ideas laid out by Toupin (Arch. Rat. Mech. Anal., 11, 1962, 385-414). The variational treatment leads to a system of partial differential equations governing the configuration that is coupled with the traditional equations of nonlinear elasticity. The coupled system of equations governs the configurational change in crystal structure, and elastic deformation driven by elastic, Eshelbian, and configurational stresses. Numerical examples are presented to demonstrate interface motion as well as evolving microstructures of crystal structures.

PRISMS: An Integrated, Open-Source Framework for Accelerating Predictive Structural Materials Science

“…This code suite (v0.1) was publicly released on GitHub in September 2016, and is now under continuous integration and development by the Garikipati Group. These models are applied to initial and boundary value problems involving structural alloys, energy storage materials and other systems of current and future interest [75][76][77][78][79][80][81] at length scales of 100 nm to 1 cm. The mechanoChem models and codes have been developed to take as inputs quantities such as free energy densities, elasticity tensors, reaction rates, and mobility and conductivity tensors.…”

Numerical analysis of non-local calculus on finite weighted graphs, with application to reduced-order modelling of dynamical systems

“…Our treatment follows Toupin. 15 and has appeared as decoupled, gradient-regularization of non-convex, non-linear elasticity 16,17 , as well as mechano-chemical spinodal decomposition 18,19,20 The free energy density functional ψ depends on the fields {c, E, ∇c, ∇E} where c is the composition and E is the Green-Lagrange strain tensor.…”

Reduced order models from computed states of physical systems using non-local calculus on finite weighted graphs

“…Our treatment follows Toupin 12 and has appeared as decoupled, gradient-regularization of non-convex, non-linear elasticity 13,14 , as well as mechanochemical spinodal decomposition 15,16,17 . The free energy density functional is ψ = ψ tot , with a dependence on the fields {c, E, ∇c, ∇E} where c is the composition and E is the Green-Lagrange strain tensor.…”