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

Abstract

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.