Generalized Hashin-Shtrikman variational principle for boundary-value problem of linear and non-linear heterogeneous body

When the in-situ measurement of the effective properties is difficult, a ground or a crust of large dimensions is modeled as a body with probabilistically varying materials. As the variance of heterogeneity is large, conventional analysis methods require enormous numerical computation. As an alternative, this paper proposes the generalized Hashin-Shtrikman variational principle which provides upper and lower bounds for the expectation of the behavior of such a probabilistically varying body. The bounds are obtained by analyzing two fictitious bodies which are rigorously defined when probabilistic distributions of material properties are given. The generalized Hashin-Shtrikman principle can be applied to non-linear initial boundary-value problems. The fault formation process in surface ground layers is solved as an illustrative example. The surface layers are modeled as a probabilistically varying elasto-plastic body, and it is shown that the upper and lower bounds for the expectation actually bound the average behavior which is computed by the Monte-Carlo simulation. Discussions are made on these numerical results. (C) 1999 Elsevier Science Ltd. All rights reserved.