HOMOGENIZATION OF A LAYERED COMPOSITE REINFORCED WITH THIN GRADIENT-ELASTIC INCLUSIONS
E I. Leonova1,2, E.M. Rudoi1, S.A. Sazhenkov1,2
1Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
2Altai State University, Barnaul, Russia
Keywords: boundary value problem, linear elasticity, gradient theory of elasticity, composite material, thin inclusion, homogenization, two-scale convergence
Abstract
Within the framework of antiplane shear theory, the equilibrium problem of an elastic composite body containing many thin gradient-elastic inclusions is considered. Each inclusion is geometrically represented by a straight-line segment with endpoints on the edges of the body. The inclusions-fibers or rods-are parallel to one another, and the problem contains a small positive parameter ε describing the distance between two adjacent inclusions. Using a variational formulation of the problem, the behavior of a family of its solutions as ε tends to zero is investigated. The limiting passage as ε → 0+ is justified using the two-scale convergence method. As a result, a correct averaged model of an anisotropic gradient-elastic composite body is constructed. This model describes the effective behavior on a macroscopic scale: on a scale at which it is not necessary to take each individual inclusion into account. Additional smoothness of the solution to the averaged model is proven, and an equivalent mixed formulation is derived on the basis of this proof. Finally, for the averaged problem in the mixed formulation, a numerical algorithm is proposed, and a series of numerical experiments is carried out to confirm the validity of the theoretical conclusions.
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