The Coupling of Pressure and Shear Stress in Compressible, Isotropic, Nonlinear Hyperlastic Solids

Michael J. Scheidler, U.S. Army Research Laboratory, U.S. Army Research Laboratory, Impact Physics Br., B 4600, Aberdeen Proving Ground, MD 21078

For compressible, isotropic, hyperelastic solids, we consider two effects which are absent in

the linear theory but necessarily present (to some extent) in the nonlinear theory, namely, the

dependence of pressure on shear strain and the dependence of shear stress on volumetric strain.

Typically, only limited experimental data is available on these nonlinear effects, so it is

common to make some simplifying assumptions on their functional form. However, some care must be

taken here since the dependence of pressure on shear strain and the dependence of shear stress

on volumetric strain are coupled through the strain energy function. For example, it is often

assumed that the pressure is independent of the shear strain or that the shear stress is

independent of the volumetric strain. While neither one of these assumptions individually

violates the existence of a strain energy function, both assumptions taken together are

inconsistent with the existence of a strain energy function. The problem that will be addressed

is to determine the most general form of the strain energy function compatible with various

assumed forms of the dependence of pressure on shear strain, and consequently to determine the

restrictions imposed on the dependence of shear stress on volumetric strain, or vice versa.

Since experimental data on the latter effect is often reported as a dependence of shear stress

on pressure, we also consider the consequences of various simplifying assumptions for this

dependence. Several known results are recovered as special cases.