Elastic solids under frictionless rigid contact and configurational force

TL;DR

This paper develops a nonlinear solid mechanics framework to analyze elastic solids under frictionless rigid contact, introducing a path-independent $J$-integral linked to configurational forces, with applications to buckling, motion, and biological cell migration.

Contribution

It introduces a new interpretation of configurational forces in elastic solids under frictionless contact, connecting them to energy release rates and force components, validated by finite element simulations.

Findings

A $J$-integral can be defined for frictionless contact problems.

Configurational forces are equal to the elastic solid's force on the constraint.

Applications include buckling, dynamic motion, and biological cell migration.

Abstract

A homogeneous elastic solid, bounded by a flat surface in its unstressed configuration, undergoes a finite strain when in frictionless contact against a rigid and rectilinear constraint, ending with a rounded or sharp corner, in a two-dimensional formulation. With a strong analogy to fracture mechanics, it is shown that (i.) a path-independent $J$--integral can be defined for frictionless contact problems, (ii.) which is equal to the energy release rate $G$ associated with an infinitesimal growth in the size of the frictionless constraint, and thus gives the value of the configurational force component along the sliding direction. Furthermore, it is found that (iii.) such a configurational sliding force is the Newtonian force component exerted by the elastic solid on the constraint at the frictionless contact. Assuming the kinematics of an Euler-Bernoulli rod for an elastic body of rectangular shape, the results (i.)--(iii.) lead to a new interpretation from a nonlinear solid mechanics perspective of the configurational forces recently disclosed for one-dimensional structures of variable length. Finally, approximate but closed-form solutions (validated with finite element simulations) are exploited to provide further insight into the effect of configurational forces. In particular, two applications are presented which show that a transverse compression can lead to Eulerian buckling or to longitudinal dynamic motion, both realizing novel examples of soft actuation mechanisms. As an application to biology, our results may provide a mechanical explanation for the observed phenomenon of negative durotaxis, where cells migrate from stiffer to softer environments.