*Mathematical Methods in the Applied Sciences*3D structure - 2D plate - 1D rod interaction problem

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 is a thin layer/quboid of thickness
$\eps$ of material whose stiffness is of order
$\frac{1}{\eps}$ that in the middle
contains another cuboid which is of width and thickness
$\eps$ that is made of material with elasticity
coefficients of order
$\frac{1}{\eps^q}$ for
$q>0$. We show that the family of solutions of linearized
elasticity problems, when $\eps$ 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 $q$
($q\in \langle 0,
2\rangle, \{2\},
\langle 2,
4\rangle,\{4\},\langle
4, \infty\rangle$) with different limit
problems. We further formulate a model posed on the unit cube that has
the same asymptotics when $\eps$ 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.

26 Aug 2022

27 Aug 2022

27 Aug 2022

09 Sep 2022

21 Dec 2022

22 Dec 2022