Derivation of a rod theory for biphase materials with dislocations at the interface

TL;DR

This paper derives a one-dimensional rod theory for biphase materials with interface dislocations, using asymptotic analysis and $ ext{Gamma}$-convergence from three-dimensional elasticity, providing a rigorous mathematical foundation.

Contribution

It introduces a novel derivation of a rod model for biphase materials with interface dislocations, justified through rigorous asymptotic analysis.

Findings

Rigorous justification of interface dislocation assumptions

Asymptotic analysis of energy scaling as rod diameter shrinks

Development of a $ ext{Gamma}$-convergence framework for the model

Abstract

Starting from three-dimensional elasticity we derive a rod theory for biphase materials with a prescribed dislocation at the interface. The stored energy density is assumed to be non-negative and to vanish on a set consisting of two copies of SO(3). First, we rigorously justify the assumption of dislocations at the interface. Then, we consider the typical scaling of multiphase materials and we perform an asymptotic study of the rescaled energy, as the diameter of the rod goes to zero, in the framework of $\Gamma$-convergence.