Dilatationally Nonlinear Elastic Materials. 2. An Example Illustrating Stress Concentration Reduction

摘要

This paper, which is the second in a two part study, uses a specific boundary value problem to illustrate some of the features of the theory discussed in the first part. Here, the spherically symmetric deformation of a hollow sphere which has a traction-free inner wall and a prescribed radial displacement delta at its outer wall is studied. The analysis is carried out within the small-strain theory of nonlinear elasticity and the body is assumed to be composed of an elastic material which is homogeneous and isotropic, and which has a linear response in shear and a tri-linear response in dilatation. For a certain range of values of the applied displacement delta the problem has an infinity of solutions and these describe configurations which involve a phase boundary; the strain field is continuous on either side of the phase boundary but suffers a jump discontinuity across it. A kinetic law , which is a supplementary constitutive law pertaining to particles located on the phase boundary and relating the driving traction on the phase boundary to its velocity, is then imposed, leading to a unique response in all quasi-static motions. (MJM)