Plastic limit analysis with an anisotropic, parabolic yield function

This paper applies the theory of plastic limit analysis to an anisotropic parabolic yield function. The methods of plastic limit analysis have been widely used to solve practical engineering problems of ultimate strength in such fields as metal-deforming processes and soil, rock, and ice mechanics. To date, the material idealizations used in these analyses have been isotropic and thus fail to describe any anisotropy in strength. This anisotropic, parabolic yield function describes an anisotropic material that has different tensile strengths and compressive strengths in three mutually orthogonal directions. However, the material is homogeneous in the sense that the strengths in these directions are uniform throughout the material. With increasing hydrostatic stress, the compressive strengths increase non-linearly. This model can therefore be regarded as the anisotropic generalization of the Torre material model. The implications of plasticity analysis with the anisotropic, parabolic yield function and the theoretical considerations required for the construction of upper and lower bound solutions are presented in this paper. The derivation of explicit expressions for the rate of energy dissipation for continuous and discontinuous velocity fields is the principal contribution of this work. The use of these theoretical developments is illustrated in a two-dimensional example problem where both upper and lower bounds on the punch pressure of a two-dimensional flat punch are obtained.