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 𝜀 of material whose stiffness is of order 1 𝜀 that in the middle contains another cuboid which is of width and thickness 𝜀 that is made of material with elasticity coefficients of order 1 𝜀 q for q > 0. We show that the family of solutions of linearized elasticity problems, when 𝜀 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 q (q ∈ ⟨0, 2⟩, {2}, ⟨2, 4⟩, {4}, ⟨4, ∞⟩) with different limit problems. We further formulate a model posed on the unit cube that has the same asymptotics when 𝜀 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.

A quasi-static model of a thermoelastic body reinforced by a thin thermoelastic inclusion

“…Using this notation, re-denoting x 1 := z 1 in the integrals over m and, accordingly, noticing that γ = {0 < x 1 < l * }, we reduce equation (20) to the form:…”

“…Finalizing the introduction, we give a short review of the works related to thin inclusion (thin inhomogeneity) problems in the theory of elasticity. Asymptotic analysis was performed for different models of elasticity with a wide range of constitutive relations, such as elastic models [18][19][20][21][22], multiphysics models [23,24], and models of plates [25][26][27][28][29]. In this line, let us mention also the recent articles [25,[29][30][31][32][33][34][35][36][37].…”

The homogenized dynamical model of a thermoelastic composite stitched with reinforcing filaments

“…See, e.g. papers [2][3][4][5][6][7][8], in which static and dynamical problems are analysed for different models of elastic bodies, and also papers [9][10][11][12][13][14][15][16][17][18] that deal with thermoelasticity. Owing to the similarity of such an asymptotic analysis, we do not perform it in the present paper.…”