# Motion of a finite composite cylindrical annulus comprised of nonlinear elastic solids subject to periodic shear

## TL;DR

This study analyzes the stress and strain response of a finite composite cylindrical annulus made of generalized neo-Hookean solids under periodic shear, revealing boundary layer effects and the influence of annular thickness.

## Contribution

It provides analytical solutions for the nonlinear elastic behavior of composite cylinders under shear, including boundary layer phenomena and the effect of annular thickness.

## Key findings

- Boundary layer structures appear in strain distribution for thick annuli.
- Solutions for infinite and finite cylinders converge beyond twice the wall thickness.
- Stress response varies with strain hardening and softening cases.

## Abstract

In this paper we study the motion of a finite composite cylindrical annulus made of generalized neo-Hookean solids that is subject to periodic shear loading on the inner boundary. Such a problem has relevance to several problems of technological significance, for example blood vessels can be idealized as finite anisotropic composite cylinders. Here, we consider the annulus to be comprised of an isotropic material, namely a generalized neo-Hookean solid and study the effects of annular thickness on the stress distribution within the annulus. We solve the governing partial differential equations and examine the stress response for the strain hardening and strain softening cases of the generalized neo-Hookean model. We also solve the problem of the annular region being infinite in length, that reduces the problem to a partial differential equation in only time and one spatial dimension. When the thickness of the annulus is sufficiently large, the solutions to the problems exhibit very interesting boundary layer structure in that the norm of the strain has a large gradient in a narrow region adjacent to one of the boundaries, with the strain being relatively uniform outside the narrow region. We also find that beyond a distance of two times the annular wall thickness from the ends of the cylinder the solutions for the infinite length cylinder match solutions for the finite length cylinder implying that end effects are not felt in most of the length of a sufficiently long annulus.