A three dimensional field formulation, and isogeometric solutions to point and line defects using Toupin's theory of gradient elasticity at finite strains

TL;DR

This paper develops a unified field formulation for defects in nonlinear elasticity using distribution theory, enabling accurate 3D solutions for various defect types within Toupin's gradient elasticity framework, leveraging isogeometric analysis.

Contribution

It introduces a weak form approach for defects in nonlinear elasticity, extending solutions to complex defect configurations using isogeometric analysis within Toupin's theory.

Findings

Numerical solutions match classical linearized elasticity for small strains.

Extended to nonlinear elasticity and gradient elasticity at finite strains.

Successfully modeled complex dislocation structures like loops and grain boundaries.

Abstract

We present a field formulation for defects that draws from the classical representation of the cores as force dipoles. We write these dipoles as singular distributions. Exploiting the key insight that the variational setting is the only appropriate one for the theory of distributions, we arrive at universally applicable weak forms for defects in nonlinear elasticity. Remarkably, the standard, Galerkin finite element method yields numerical solutions for the elastic fields of defects that, when parameterized suitably, match very well with classical, linearized elasticity solutions. The true potential of our approach, however, lies in its easy extension to generate solutions to elastic fields of defects in the regime of nonlinear elasticity, and even more notably for Toupin's theory of gradient elasticity at finite strains(Arch. Rat. Mech. Anal., 11, 385, 1962). In computing these solutions we adopt recent numerical work on an isogeometric analytic framework that enabled the first three-dimensional solutions to general boundary value problems of Toupin's theory (Rudraraju et al. Comp. Meth. App. Mech. Engr., 278, 705, 2014). We first present exhaustive solutions to point defects, edge and screw dislocations, and a study on the energetics of interacting dislocations. Then, to demonstrate the generality and potential of our treatment, we apply it to other complex dislocation configurations, including loops and low-angle grain boundaries.