Simple Shearing of Incompressible and Slightly Compressible Isotropic Nonlinearly Elastic Materials

2010

Proc. R. Soc. A.

Shearing is induced in soft tissues in numerous physiological settings. The limited experimental data available suggest that a severe strain-stiffening effect occurs in the shear stress when soft biological tissues are subjected to simple shear in certain directions. This occurs at relatively small amounts of shear (when compared with the simple shear of rubbers). This effect is modelled within the framework of nonlinear elasticity by consideration of a class of incompressible anisotropic materials. Owing to the large stresses generated for relatively small amounts of shear, particular care must be exercised in order to maintain a homogeneous deformation state in the bulk of the specimen. The results obtained are relevant to the development of accurate shear test protocols for the determination of constitutive properties of soft tissues. It is also demonstrated that there is a fundamental ambiguity in determining the normal stresses in simple shear when soft tissues are modelled as incompressible hyperelastic materials owing to the arbitrary nature of the hydrostatic pressure term. Two physically well-motivated approaches to determining the pressure are presented here, and the resulting hydrostatic stresses are compared and contrasted. The possible generation of cavitational damage owing to critical hydrostatic stress levels is briefly discussed.

Section: Introductionmentioning

confidence: 99%

Simple shearing of soft biological tissues

2010

Proc. R. Soc. A.

“…1) in order to maintain this deformation. As emphasized in our previous papers (Horgan and Murphy [11,12]), this implies that the experimental implementation of simple shear is indeed a complex process. On resolving these tractions into components tangential and normal to the surfaces on which they act in their deformed state (denoted by S and N respectively) it was shown in (2.12) of [12] that, irrespective of the constitutive law, one has…”

Section: The Inclined Facesmentioning

confidence: 71%

“…Some alternative assumptions are described in Horgan and Murphy [11]. On substitution of this value of p in (3.1), we find that there are two non-zero normal stresses given by…”

Section: Simple Shear Of Isotropic Materialsmentioning

confidence: 87%

See 1 more Smart Citation

On the Normal Stresses in Simple Shearing of Fiber-Reinforced Nonlinearly Elastic Materials

2011

J Elast

We are concerned with a particular aspect of the simple shear problem within the framework of nonlinear elasticity for a class of incompressible transversely-isotropic fiberreinforced materials. It is well known that, for isotropic hyperelastic materials, the normal stress effect characteristic of nonlinear elasticity is crucial in order to maintain a homogeneous deformation state in the bulk of the specimen. For the fiber-reinforced materials of concern here, we show that the confining traction that needs to be applied to the top and bottom faces of a block in order to maintain simple shear can be compressive or tensile depending on the degree of anisotropy and on the angle of orientation of the fibers. Inclusion of the second invariant in the isotropic part of the strain-energy used is shown to be of crucial importance in assessing the nature of the confining traction. In the absence of such an applied traction, an unconfined sample tends to bulge outwards or contract inwards perpendicular to the direction of shear. The character of the normal component of traction on the inclined faces is also investigated. The results are relevant to the development of accurate shear test protocols for the determination of constitutive properties of fiber-reinforced rubber-like materials and fibrous biological soft tissues.Keywords Simple shear · Nonlinear elasticity · Fiber-reinforced materials · Transversely isotropic materials · Normal stress effect · Second invariant for isotropic matrix material Mathematics Subject Classification (2000) 74B20 · 74G55Dedicated to the memory of Don Carlson.

“…For a given fiber extensibility parameter J , the condition (4.2) constrains the total angle of twist τ A in a rather complex way depending on the pitch angle. We remark that the constraint (4.2) is formally equivalent to that arising in simple shear for classes of fiberreinforced materials (see Horgan and Murphy [6,7]). An analysis of the dependence of this constraint on the fiber orientation angle carried out in Horgan and Murphy [7] is applicable to the present problem.…”

Section: Results For the Models (24) And (25)mentioning

confidence: 98%

Torsion of Incompressible Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

2010

J Elast

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.