The numerical intcgration of the rate equation of an elastic^ plastic material is considered. Special attention is focuscd on the discrctization via the fully implicit backward Eulcr method in the sinall strain casc with linear elasticity and the yield function :I general quadratic in stress spacc. Here the calculation of the plastic (Lagrange) multiplier reduces to the computation of the smallcst positive root of a polynomial in one variable. Explicit formulae are given for some special cases.
INTRODU CTION
The decomposition method we want to consider is applicable only to an elastic-plastic material with a quadratic complemcntary cnergy function and the yield function a general quadratic function in stress space, the so-called Tsai-Wu yield function.' This encompasses formulations like thc Huber-von Mises or, with some modifications, the Drucker-Prager yield function. The incentive for the present paper arose from modelling2 sea-ice as an eiastic-plastic material with a Pariseau yield function, which is again a special instance of the Tsai-Wu criterion. Owing to the already mentioned rcstriction on the numerical algorithm, we limit ourselves to the small strain case. After formulating the ratc problem as a variational inequality equivalent t o Koiter's3 minimum principles, the same pattern is repeated for the time discrete problem, thus resulting in a gcneralized radial return or projection method."-6 The numerical solution of the resulting minimum problem via the Lagrange multiplier technique is closely related to a generalized eigenvalue problem. By exploiting this fact, the calculation of the final stress and the plastic (Lagrangc) multiplier is decomposed into two steps: solution of the generalized eigenvalue problem, which in many cases may be pcrformcd before the time stepping starts or may be given by explicit formulae, and computation of the plastic multiplier. The plastic multiplier is shown to be the smallest positive root of a function in one variable. For some special but frequently encountcred cases explicit formulae are given. It is seen that for three-dimensional isotropic elasticity and the Huber-von Mises yield function the result is idcntical to that of the radial return m ~t h o d . ~.
* THE RATE PROBLEM
We assume that the elastic domain B in stress space is a closed convex set, such that the material is clastic whenever the current stress cr is in the interior of B. If on the other hand the stress point is at the boundary of B, the yield surface, plastic strains may develop. We shall limit ourselves to the