A degenerating Robin-type traction problem in a periodic domain

TL;DR

This paper analyzes the asymptotic behavior of elastic displacement solutions in a periodic domain with voids under Robin-type traction conditions, as these conditions degenerate into pure traction, using a functional analytic approach.

Contribution

It introduces a power series expansion for the displacement solution in terms of a parameter controlling the Robin condition, demonstrating convergence near zero.

Findings

Displacement solutions can be expressed as convergent power series in the parameter.

The analysis reveals the asymptotic transition from Robin to pure traction conditions.

The approach provides a rigorous framework for understanding degenerating boundary conditions.

Abstract

We consider a linearly elastic material with a periodic set of voids. On the boundaries of the voids we set a Robin-type traction condition. Then we investigate the asymptotic behavior of the displacement solution as the Robin condition turns into a pure traction one. To wit, there will be a matrix function {$b[k](\cdot)$ that depends analytically on a real parameter $k$ and vanishes for $k=0$ and we multiply the Dirichlet-like part of the Robin condition by $b[k](\cdot)$}. We show that the displacement solution can be written in terms of power series of $k$ that converge for $k$ in a whole neighborhood of $0$. For our analysis we use the Functional Analytic Approach.