Exactly isochoric deformations of soft solids

TL;DR

This paper introduces an exact isochoric deformation theory for soft solids like gels and tissues, ensuring perfect volume conservation even at large strains, and demonstrates its application to nonlinear elasticity problems.

Contribution

It develops a linear elasticity framework that enforces exact incompressibility using a mixed coordinate transformation, extending validity to large strains.

Findings

Derived a nonlinear generalization of Kelvin's solution for soft solids.

Applied the theory to model muscular contraction in soft tissues.

Demonstrated the approach's applicability to various geometries and loadings.

Abstract

Many materials of contemporary interest, such as gels, biological tissues and elastomers, are easily deformed but essentially incompressible. Traditional linear theory of elasticity implements incompressibility only to first order and thus permits some volume changes, which become problematically large even at very small strains. Using a mixed coordinate transformation originally due to Gauss, we enforce the constraint of isochoric deformations exactly to develop a linear theory with perfect volume conservation that remains valid until strains become geometrically large. We demonstrate the utility of this approach by calculating the response of an infinite soft isochoric solid to a point force that leads to a nonlinear generalization of the Kelvin solution. Our approach naturally generalizes to a range of problems involving deformations of soft solids and interfaces in 2 dimensional and axisymmetric geometries, which we exemplify by determining the solution to a distributed load that mimics muscular contraction within the bulk of a soft solid.