On Saint-Venant compatibility and stress potentials in manifolds with boundary and constant sectional curvature

TL;DR

This paper extends the theory of elasticity to manifolds with boundary and constant curvature, establishing conditions for compatibility, solutions, and stress potentials using elliptic boundary-value problems, including non-Euclidean geometries.

Contribution

It introduces stress potentials in non-Euclidean geometries and generalizes the biharmonic equation for higher dimensions, expanding elasticity theory on curved manifolds.

Findings

Stress potentials are applicable in non-Euclidean geometries.

Elliptic boundary-value problems ensure solution existence and uniqueness.

Generalization of biharmonic equations for stress potentials in higher dimensions.

Abstract

We address three related problems in the theory of elasticity, formulated in the framework of double forms: the Saint-Venant compatibility condition, the existence and uniqueness of solutions for equations arising in incompatible elasticity, and the existence of stress potentials. The scope of this work is for manifolds with boundary of arbitrary dimension, having constant sectional curvature. The central analytical machinery is the regular ellipticity of a boundary-value problem for a bilaplacian operator, and its consequences, which were developed in [KL21]. One of the novelties of this work is that stress potentials can be used in non-Euclidean geometries, and that the gauge freedom can be exploited to obtain a generalization for the biharmonic equation for the stress potential in dimensions greater than two.