Finite strain solutions for a compressible elastic solid

Abstract

Abstract. Several closed form finite strain equilibrium solutions are presented for a special compressible isotropic elastic material which was proposed as a model for foam rubber by Blatz and Ko. These solutions include bending of a cylindrical sector into another sector or a rectangular block, bending of a block into a sector, expansion, compaction or eversion of cylinders or spheres, and torsion and extension of circular cylinders or tubes.1. Introduction. Ericksen has examined the problems of finding all of the deformations which can be supported, in the absence of body force, in all homogeneous, isotropic, incompressible elastic solids [1], or in all homogeneous, isotropic, compressible elastic solids [2], The first category consists of all isochoric homogeneous deformations and five families of nonhomogeneous deformations, the so-called controllable or universal deformations ([1] and Singh and Pipkin [3]). The second category consists of homogeneous deformations only, i.e., there is no nonhomogeneous finite deformation which can be supported in every compressible isotropic elastic solid material without applying a body force.This latter result implies that nonhomogeneous deformations can be discussed, for compressible solids, only in the context of a particular strain energy function, or class of strain energy functions. Currie and Hayes [4] have suggested that Ericksen's results have had an unduly inhibiting effect on the study of nonhomogeneous finite deformations. For compressible solids, in particular, the analysis of such deformations is usually very complicated even for simple forms of the strain energy function and very few closed form solutions have been obtained.John [5] introduced the class of harmonic materials, for which the problem of finite plane strain simplifies considerably, and solutions of such problems have been presented by several authors [6][7][8][9][10], Abeyaratne and Horgan [11] obtained an exact solution of the problem of pressurization of a hollow sphere of harmonic material; see also Ogden [10]. Wheeler [12] has examined the deformation of a harmonic material containing an ellipsoidal cavity. Chung, Horgan, and Abeyaratne [13] obtained

Pure torsion of compressible non-linearly elastic circular cylinders

“…(2.9) of [18]. This was also observed in [16]. The stresses (2.14)-(2.18), for the material Observe that the stresses (3.4) are independent of r. In fact, these stresses are also identical to the corresponding stresses in the plane strain axisymmetric problem (see Eq.…”

“…3, we consider the special case of the Blatz-Ko material [14] for which pure torsion, in the absence of lateral surface tractions, is possible. This was recently observed by Beatty in [15], and independently by Carroll and Horgan in [16]. The stress distribution in torsion for the Blatz-Ko material is remarkably simple-simpler even than the corresponding stress field for a neo-Hookean incompressible material.…”

“…The Blatz-Ko material. It has been shown recently by Beatty [15] and independently by Carroll and Horgan [16] that pure torsion is possible for a special compressible material proposed by Blatz and Ko [14] as a model for foam rubber material. In this section, we examine this special case in detail.…”

“…5, a subclass of materials for which pure torsion is possible is identified. This includes the Hadamard material discussed in [13] and a family of Blatz-Ko materials previously investigated in [15,16]. In Sec.…”

“…Nevertheless, the material model (3.1) has recently attracted considerable attention. In particular, the analytic solution of a wide variety of problems in finite elastostatics for compressible materials has been possible for this material (see [15,16,[18][19][20][21]). When W is given by (3.1), the differential equation ( where r = dr/dR and r = d r/dR .…”