Plastic flow of random media. Micromechanics, Markov property and slip-lines

The classical method of slip-lines (characteristics) of planar flow of perfectly-plastic media is generalized to a stochastic setting. The media are characterized by space-homogeneous statistics of the yield limit k, whose derivation is outlined on the basis of micromechanics. The field equations of the random continuum approximation lead to a stochastic hyperbolic system. This system, when stated in a finite difference form, displays a Markov property for the forward evolution. On that basis, two methods of solution of boundary value problems - an exact one and a mean-field one - are outlined through an example of a Cauchy problem. The principal observation is that even for a weak material randomness the stochastic solution may differ qualitatively from that of a homogeneous deterministic medium and have a strong scatter.