Some general results in the theory of large elastic deformations

Abstract

When the strain-energy function for an elastic body is expressed as a function of the six components of strain, the solution of a given problem for different types of material may assume very different forms. In the present paper, by regarding the strain-energy function as a function of the parameters defining the deformation, results are obtained which are valid for a wide range of materials. The analysis for each problem is performed initially for bodies possessing a suitable type of curvilinear aeolotropy, and results are derived which are independent of symmetries in the elastic material. These results are therefore valid, not only for the general type of material initially considered, but also for isotropic bodies and for materials which are orthotropic or transversely isotropic with respect to the curvilinear co-ordinate system which defines the aeolotropy. Both compressible and incompressible bodies are considered. From this point of view, a general type of cylindrically symmetrical deformation is examined which includes as special cases the problem of flexure, the inflation, extension and torsion of a cylindrical tube, and the shear of a cylindrical annulus. Particular results for these special cases are considered separately, and for the flexure and torsion problems, expressions are found for the resultant forces and couples required to maintain the deformation. A brief analysis is also given for the corresponding types of deformation for a cuboid. In the final section of the paper, a generalized shear problem is considered in which, during deformation, each point of the elastic body moves parallel to a given axis through a distance which is a general function of position in a plane normal to that axis.