The Unexpected Kinematics of Simple Extension and Contraction of Incompressible Materials

Abstract

Simple extension and contraction of incompressible non-linearly elastic materials is considered here, with cuboid specimens stretched or contracted in one of the principal directions. Although the average stretch of infinitesimal line elements in such deformations is always extensile, the local response is much more nuanced. We examine the variation of the squared stretch with applied stretch and line element orientation in the undeformed configuration. The response of interest here is the percentage of directions at a point for which the material is in compression which we call the amount of material in compression. It is shown that the maximum amount of compression attainable is approximately $61\%$ which occurs for infinitesimal contractions. Further contraction results in a decrease in the amount of material in compression, with the material being extensile almost everywhere in the limit of zero axial stretch. On the other hand, for simple extension, the amount of material in compression is a monotonically decreasing function of axial stretch, with $39\%$ in compression for infinitesimal strains. The amount of material in compression therefore exhibits a discontinuity in the reference configuration. The amount of material in compression is still significant for moderate extensions, with, for example, $25\%$ of the material still in compression for $100\%$ stretch. Suspecting that this somewhat surprising response is a result of assuming perfect incompressibility, the effect of compressibility on the amount of material in compression is examined within the context of the linear theory for isotropic elastic materials.