Homogenization in the Problem of Long-Term Loading of a Layered Elastic-Creeping Composite

The paper presents analytical estimates of the proximity of solutions to boundary value problems for elastic-creeping layered composite materials, which are widely used in construction, and must withstand loads for a long time, and the corresponding averaged model for such a material. The estimates show the possibility of using the averaged model over a long time interval for the problem of loading by a constantly acting force action. Previously, this statement was substantiated by numerical experiments comparing the solutions of boundary value problems for the effective (averaged) model and direct numerical calculation using the original model for a highly inhomogeneous layered material. Analytical estimates are based on previously obtained estimates of the proximity of solutions to stationary problems of elasticity theory. The problem under consideration is reduced to such problems using the Laplace transform in time. Next, we analyze analytically the dependence of the estimates for stationary problems with a complex parameter of the Laplace transform on this complex parameter, and the reverse transition to the original variables (time and spatial coordinate) is performed. The method used in this work for estimating the proximity of solutions for the averaged and original boundary value problems can also be used in the study of dynamic problems of viscoelasticity. It should also be noted that for the one-dimensional model considered in this work, an interesting property has been established: if the constitutive relations for various phases are written as dependences of deformations on stresses, then the coefficients for the same form of writing the constitutive relation of the averaged model are obtained as simple weighted averages of similar coefficients for individual phases.