Asymptotic and quadratic failure criteria for anisotropic materials

Abstract

Abstract A criterion is proposed for describing the macroscopic states of stress which result in failure of brittle and ductile materials. For isotropic conditions, the criterion defines a failure surface in stress space where the deviatoric stress at failure is a nonlinear function of hydrostatic stress and asymptotically approaches a limiting value for extreme hydrostatic compression. The criterion involves three material parameters, and as special limiting cases, the criterion reduces to the isotropic von Mises and Drucker-Prager criteria. The anisotropic form of the criterion is based on a generalization of the distortion energy which is exponentially dependent upon the hydrostatic pressure. The criterion is especially suited for describing the failure envelope of materials which exhibit brittle deformation at low hydrostatic pressure and ductile deformation at high hydrostatic pressure. Unlike many criteria, the anisotropic criterion is capable of describing material failure due to the application of purely hydrostatic compression. The exponential yield function is then used in the application of plastic limit analysis. This function is used for isotropic materials which are idealized as rigid, perfectly plastic. Post yield deformations are assumed to obey the normality rule. The associated rate of energy dissipation is derived and proof of convexity is provided. It is demonstrated that post-yield material dilatancy and Poison's ratio can be accurately modelled. The upper and lower bound theorem can be applied in a straightforward manner without the need for using a tensil cutoff. It is also shown that for some materials, the generalization of the exponential criterion for anisotropic materials can be approximated by the quadratic yield function. The associated rate of energy dissipation is derived for the anisotropic quadratic criterion and the conditions for convexity are described. Example limit analyses are provided for the forces necessary to split a circular cylinder and for the edge pressures required to indent a semi-infinite sheet.