Reverse Poynting Effects in the Torsion of Soft Biomaterials

The torsion of solid cylindrical bodies has been widely investigated in the context of isotropic nonlinear elasticity theory with application to the behavior of rubber-like materials. More recently, this problem for anisotropic materials has attracted attention in investigations of the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. Here we consider the torsion of a solid circular cylinder composed of a transversely isotropic incompressible material described by a strain-energy function that depends on the full set of relevant invariants. Three specific strain-energy density functions modeling soft tissues are considered in detail. These models are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and are consistent with the linear theory. The classic Poynting effect found for isotropic rubber-like materials where torsion induces elongation of the cylinder is shown to be significantly different for the transversely isotropic materials considered here. For sufficiently small angles of twist that are consistent with the physiological strain range, a reverse Poynting effect is demonstrated where the cylinder tends to shorten on twisting. The results obtained here have important implications for the development of accurate torsion test protocols for determination of material properties of soft biomaterials.