3D structure–2D plate–1D rod interaction problem

Abstract

In this paper, we consider the equilibrium problem of interaction of three elastic bodies of different elastic properties. The main body is the unit cube. On top of it, there is a thin layer/cuboid of thickness of material whose stiffness is of order that in the middle contains another cuboid which is of width and thickness that is made of material with elasticity coefficients of order for . We show that the family of solutions of linearized elasticity problems, when tends to zero, converges to a solution of a problem that is posed only on the unit cube with possibly additional elastic terms on the boundary related to the plate/rod energy of the thin elastic parts. It turns out that there are five different regimes related to different values of with different limit problems. We further formulate a model posed on the unit cube that has the same asymptotics when tends to zero as the full 3D problem posed on the union of the unit cube and thin cuboids. This model then can be used as the approximating model in all regimes.