On the deformation of an elastoplastic body when placed between flat plates rotating about non-coincident axes

Abstract

While several boundary value problems concerning the inelastic response of bodies have been studied analytically within the context of small strains, there are few studies that have been dedicated to such bodies undergoing large deformations despite the fact that numerous theories have been advanced to describe large inelastic deformations. Here, we discuss a solution to a boundary value problem (using an elastoplastic model for a polymeric material) wherein a layer is subject to a large deformation due to the rotation of two bounding planes of the layer. While the amount of rotation of the two planes is the same, the axes of rotation of the planes, while being parallel to one another, are non-coincident. The deformation under consideration amounts to a shear followed by a large rotation. It transpires that although the body is subject to a cyclic deformation, there is no unloading, i.e., the body, once it yields, continues to do so under deformation. The problem also highlights the shortcoming of a purely Eulerian approach to studying the inelastic response of bodies as the resulting permanent deformation and change of shape can be understood only in terms of a referential (or Lagrangian) approach. The solution also highlights the development of forces in a direction perpendicular to the plane of shear, namely the Poynting effect that is characteristic of the response of non-linear materials.