Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes

Abstract

Abstract. The axial shear problem for a hollow circular cylinder, composed of homogeneous isotropic compressible nonlinearly elastic material, is described. The inner surf ice of the tube is bonded to a rigid cylinder while the outer surface is subjected to a uniformly distributed axial shear traction and the radial traction is zero. For an arbitrary compressible material, the cylinder will undergo both a radial and axial deformation. These axisymmetric fields are governed by a coupled pair of nonlinear ordinary differential equations, one of which is second-order and the other first-order. The class of materials for which axisymmetric anti-plane shear (i.e., a deformation with zero radial displacement) is possible is described. The corresponding axial displacement and stresses are determined explicitly. Specific material models are used to illustrate the results.

Pure torsion of compressible non-linearly elastic circular cylinders

“…Issues similar to those addressed here have been examined recently by the present authors [28] for finite axial shear deformations of compressible nonlinearly elastic hollow cylinders. The inner surface of the tube is bonded to a rigid cylinder while the outer surface is subjected to a uniformly distributed axial shear traction and the radial traction is zero.…”

“…The inner surface of the tube is bonded to a rigid cylinder while the outer surface is subjected to a uniformly distributed axial shear traction and the radial traction is zero. The class of compressible materials for which axisymmetric antiplane shear (i.e., a deformation with zero radial displacement and axial displacement depending only on the radial coordinate) is possible is identified in [28], The results in [28] and the present paper complement one another in the sense that many of the materials described in [28] for which finite axisymmetric anti-plane shear is possible do not sustain pure torsion while, conversely, several of the admissible materials identified here do not sustain axisymmetric anti-plane shear.…”

Solutions for an infinite compressible nonlinearly elastic body under a line load

“…If a is then replaced by I\ and I2 in the resulting equations for M and N respectively, then w = | M{h)N{h)P{h) (4.16) DEBRA A. POLIGNONE determines a class of strain-energy densities that will have (3.13) and (3.14) as a solution, where P(h) is arbitrary except that P ( 1) will have (3.13) and (3.14) as a solution to the line load problem. A variety of materials proposed in the literature are of the form (4.28) (see [9] for a partial list). We remark that this process could be continued by choosing other forms for the strain-energy density and then obtaining restrictions on these forms.…”

“…We remark that this process could be continued by choosing other forms for the strain-energy density and then obtaining restrictions on these forms. We conclude by identifying an interesting comparison between the line load problem and the problem involving axisymmetric anti-plane shear of isotropic compressible nonlinearly elastic circular tubes studied in [9]. In [9], a circular tube centered on the z-axis and bonded on its inner surface to a rigid cylinder is subjected to a uniform axial shear traction on its outer surface.…”

“…We conclude by identifying an interesting comparison between the line load problem and the problem involving axisymmetric anti-plane shear of isotropic compressible nonlinearly elastic circular tubes studied in [9]. In [9], a circular tube centered on the z-axis and bonded on its inner surface to a rigid cylinder is subjected to a uniform axial shear traction on its outer surface. First, the invariants associated with both the axial shear problem (radial deformation possible) and the anti-plane shear problem (no radial deformation) take on the same form as (2.6)-(2.8) and (4.5)-(4.6), respectively (see Eqs.…”