Inhomogeneous shear of orthotropic incompressible non-linearly elastic solids: singular solutions and biomechanical interpretation

TL;DR

This paper investigates shear deformation in orthotropic incompressible nonlinear elastic solids, revealing conditions for smooth and singular solutions, with implications for biomechanical modeling of fiber-reinforced materials.

Contribution

It identifies conditions under which strain singularities occur in orthotropic nonlinear elastic solids and extends these findings to biomechanical models.

Findings

Singular solutions develop when fiber families are mechanically different.

Smooth solutions occur when fiber families are mechanically equivalent.

Conditions for the existence of singular solutions are precisely determined.

Abstract

We present a detailed study of rectilinear shear deformation in the framework of orthotropic nonlinear elasticity, under Dirichlet and mixed-boundary conditions. We take a slab made of a soft matrix, reinforced with two families of extensible fibers. We consider the case where the shear occurs along the bissectrix of the angle between the two privileged directions aligned with the fibers. We show that if the two families of parallel fibers are mechanically equivalent, then only smooth solutions are possible, whereas if the mechanical differences among the two families of fibers is pronounced, then strain singularities may develop. We determine the precise conditions for the existence of such singular solutions for the standard reinforcing orthotropic model. We then extend our findings to some orthotropic models of interest in biomechanical applications, and we discuss the possible relevance of the singular solutions to biomechanics.