Torsion of Incompressible Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

Abstract

Torsion of solid cylinders in the context of nonlinear elasticity theory has been widely investigated with application to the behavior of rubber-like materials. More recently, this problem 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. A recent study in nonlinear elasticity was concerned specifically with the effects of strain-stiffening on the torsional response of solid circular cylinders. The cylinders are composed of incompressible isotropic nonlinearly elastic materials that undergo severe strain-stiffening in the stress-stretch response. Here we investigate similar issues for fiber-reinforced transversely-isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect stretch induced strain-stiffening of collagen fibers on loading and have been shown to model the mechanical behavior of many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The classic Poynting effect found for 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 large anisotropy and under certain conditions on the amount of twist, 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 tissues.

Rotation, inversion and perversion in anisotropic elastic cylindrical tubes and membranes

“…We further restrict our attention to simple forms for the isotropic and anisotropic responses in (2.14), the so-called standard fibre reinforcing model [42][43][44][45][47][48][49] 17) where the material parameters μ i > 0 have the dimension of a stress (pressure). The parameters ν i are of particular importance for our problem as they describe the effect of pre-compression (or pre-stretch) of the fibres in the matrix.…”

Extension, inflation and torsion of a residually stressed circular cylindrical tube

“…Recently Horgan and Murphy [13,14] considered the pure torsion and finite extension combined with torsion of incompressible solid cylinders reinforced with a single family of fibres arranged helically and locally in planes normal to the cylinder radius, with particular reference to axially aligned fibres.…”

On the Modeling of Extension-Torsion Experimental Data for Transversely Isotropic Biological Soft Tissues

“…However, if the inequality sign in (3.9) is reversed, then the leading term in (3.5) given by (3.7) is positive and so to within the order ofτ under consideration the axial force is tensile and in the absence of such a force, the cylinder would shorten on twisting. This reverse-Poynting effect has been discussed recently by Horgan and Murphy [7] for the exact theory of pure torsion for transversely isotropic materials modeled by (2.1).…”

“…An excellent discussion of the Poynting effect within the context of rubber elasticity can be found in [1] and within the context of the passive response of papillary muscles in [19]. It was shown recently by Horgan and Murphy [7] that the classic Poynting effect is not necessarily observed for transversely isotropic cylinders. In fact, for sufficiently large anisotropy, a reverse Poynting effect was predicted to occur where the cylinder shortens on twisting.…”