A Reciprocal Theorem for Finite Deformations in Incompressible Bodies

TL;DR

This paper introduces a reciprocal theorem for large deformations in incompressible bodies, extending classical theorems to dynamic, finite deformation scenarios with complex geometries, enabling new experimental and computational approaches.

Contribution

It presents a novel reciprocal theorem applicable to large deformations in incompressible bodies, unifying static and dynamic cases with complex geometries and boundary conditions.

Findings

The theorem applies to both static and dynamic problems.

It accommodates large deformations and complex geometries.

It simplifies numerical implementation of boundary value problems.

Abstract

The reciprocal theorems of Maxwell and Betti are foundational in mechanics but have so far been restricted to infinitesimal deformations in elastic bodies. In this manuscript, we present a reciprocal theorem that relates solutions of a specific class of large deformation boundary value problems for incompressible bodies; these solutions are shown to identically satisfy the Maxwell-Betti theorem. The theorem has several potential applications such as development of alternative convenient experimental setups for the study of material failure through bulk and interfacial cavitation, and leveraging easier numerical implementation of equivalent auxiliary boundary value problems. The following salient features of the theorem are noted: (i) it applies to dynamics in addition to statics, (ii) it allows for large deformations, (iii) generic body shapes with several potential holes, and (iv) any general type of boundary conditions.