Boundary value problems in plastic flow of random media

Effects of spatial random fluctuations in the yield condition of rigid-perfectly plastic continuous media are analyzed in cases of a Cauchy boundary value problem and a characteristic boundary value problem. Plastic material is assumed to be isotropic and hence it is described by a random field of plastic limit k, which is taken as space-homogeneous, ergodic, and having a high signal-to-noise ratio. The solution method is based on a stochastic generalization of the method of slip-lines, whose significant feature is that the deterministic characteristics are replaced by the forward evolution cones. Comparisons of response of this random medium and a deterministic homogeneous medium with a plastic limit equal to the average of k are carried out in specific examples of the two boundary value problems under study. It is found that the weak material randomness always leads to a relatively much stronger scatter in the position and field variables. In cases of inhomogeneous boundary data the sensitivity of field quantities to the strength of randomness of plastic limit k is very high and amplifies the differences between the average solution of a stochastic problem and the solution of the homogeneous medium problem.