###
Reduction of the equation of elasticity theory with random coefficients in a domain with a periodic structure to the average equation of elasticity theory with constant coefficients. effective stiffness tensor

Vlasov A.N.

#### Abstract:

The paper presents a method of averaging the equations of the elasticity theory of with random coefficients on a periodic structure to the averaged equations of the elasticity theory with constant coefficients. The averaging procedure is reduced to averaging the equation over 310 the distribution functions of independent random variables of the coefficients of the equations that determine the stiffness tensor, with the subsequent construction of an asymptotic solution to the problem of elasticity theory in the form of a series in a small structural parameter. This parameter is the ratio of the periodicity cell size to the size of the computational domain. This approach makes it possible to reduce the original equation to the equation of the theory of elasticity with constant coefficients that determine the effective stiffness tensor, which is found from the solution of the cell problem, as in the Bakhvalov asymptotic averaging method. The only difference is that in the problem on a cell, the corresponding averaged functions of random variables, components of the stiffness tensor, depending on the fast variable, are used. It is shown that for layered media an analytical dependence is obtained that determines the effective stiffness tensor, which is similar to the dependence obtained by Bakhvalov’s asymptotic averaging method.

0