Unconditionally stable, second-order schemes for gradient-regularized, non-convex, finite-strain elasticity modeling martensitic phase transformations

Abstract

In the setting of continuum elasticity martensitic phase transformations are characterized by a non-convex free energy density function that possesses multiple wells in strain space and includes higher-order gradient terms for regularization. Metastable martensitic microstructures, defined as solutions that are local minimizers of the total free energy, are of interest and are obtained as steady state solutions to the resulting transient formulation of Toupin's gradient elasticity at finite strain. This type of problem poses several numerical challenges including stiffness, the need for fine discretization to resolve microstructures, and following solution branches. Accurate time-integration schemes are essential to obtain meaningful solutions at reasonable computational cost. In this work we introduce two classes of unconditionally stable second-order time-integration schemes for gradient elasticity, each having relative advantages over the other. Numerical examples are shown highlighting these features.

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.…”

Machine learning materials physics: Multi-resolution neural networks learn the free energy and nonlinear elastic response of evolving microstructures

“…Mechanical equilibrium in the setting of strain gradient elasticity is governed by [1,2,[32][33][34][35] (most transparently written in coordinate notation):…”

A graph theoretic framework for representation, exploration and analysis on computed states of physical systems

“…The three tetragonal crystal structures in Figure 4 occur in the martensitic microstructures of Figure 5, with each tetragonal variant being represented by the same color in the two figures. A more detailed exposition of this problem from the mathematical and computational points of view has been laid out elsewhere [20,15,21,17]. Using isogeometric analytic methods, described in the preceding references, for the ease of representing finite-dimensional functions of high-order continuity, we have obtained numerical solutions to boundary value problems posed on the system of equations (9-12e).…”