Existence and smoothness for a class of $n$D models in elasticity theory of small deformations

TL;DR

This paper proves existence and interior smoothness of solutions for a class of elasticity models with nonlinear bulk modulus in any space dimension, including the physically relevant 3D case, covering materials like rubber and concrete.

Contribution

It establishes the first regularity result for elasticity problems with nonlinear bulk modulus in arbitrary dimensions, including the critical 3D case.

Findings

Unique solutions exist for the PDE model in all dimensions

Solutions are smooth inside the domain

Applicable to common elastic materials like rubber and concrete

Abstract

We consider a model for deformations of a homogeneous isotropic body, whose shear modulus remains constant, but its bulk modulus can be a highly nonlinear function. We show that for a general class of such models, in an arbitrary space dimension, the respective PDE problem has a unique solution. Moreover, this solution enjoys interior smoothness. This is the first regularity result for elasticity problems that covers the most natural space dimension $3$ and that captures behaviour of many typical elastic materials (considered in the small deformations) like rubber, polymer gels or concrete.