Stochastic homogenization of the first gradient-strain modelling of elasticity

Stochastic $\Gamma$-convergence concept of the mathematical theory of ho-mogenization in a version is applied to calculation of effective energy of non-simple elastic body with a microinhomogeneous random structure. The problem of homogenization in the case considered is solved by the application and generalization of the idea introduced by Dal Maso and Modica (1986). The authors proved the theorem of convergence for a special class of stochastic integral functionals. The form of homogenized, non-random integral functional was given. The first gradient-strain modelling of elasticity leads to integral functionals of internal energy depending on the second gradients of displacements. The main theorem formulated in the paper is a generalization of the result of Dal Maso and Modica (1986) for integrands of functionals depending on the second gradients of displacement limited, however the, to linear constitutive relations. The form of effective non-random, integral functional is given. It is interpreted as a internal energy of homogenized, effective non-simple material body with well defined effective, constant properties. An example of the Kirchhoff plate with thickness randomly changing in one direction is considered and effective stiffness coefficients are explicitly calculated. They depend on a volume fraction of stiffeners in a matrix material.