Elastic bodies with kinematic constraints on many small regions

TL;DR

This paper analyzes the equilibrium of hyperelastic bodies with small perforations under various kinematic constraints, deriving a limit model that captures the effects of these constraints through a penalization term.

Contribution

It introduces a novel asymptotic analysis for hyperelastic solids with multiple small constrained regions, deriving a $ ext{Gamma}$-limit that incorporates the effects of perforations.

Findings

Identifies the critical scale where perforation effects are non-trivial.

Derives a $ ext{Gamma}$-limit with a penalization term for constraints.

Provides numerical studies for specific geometries.

Abstract

We study the equilibrium of hyperelastic solids subjected to kinematic constraints on many small regions, which we call perforations. Such constraints on the displacement $u$ are given in the quite general form $u(x) \in F_x$, where $F_x$ is a closed set, and thus comprise confinement conditions, unilateral constraints, controlled displacement conditions, etc., both in the bulk and on the boundary of the body. The regions in which such conditions are active are assumed to be so small that they do not produce an overall rigid constraint, but still large enough so as to produce a non-trivial effect on the behaviour of the body. Mathematically, this is translated in an asymptotic analysis by introducing two small parameters: $\varepsilon$, describing the distance between the elements of the perforation, and $\delta$, the size of the element of the perforation. We find the critical scale at which the effect of the perforation is non-trivial and express it in terms of a $\Gamma$-limit in which the constraints are relaxed so that, in their place, a penalization term appears in the form of an integral of a function $\varphi(x,u)$. This function is determined by a blow-up procedure close to the perforation and depends on the shape of the perforation, the constraint $F_x$, and the asymptotic behaviour at infinity of the strain energy density $\sigma$. We give a concise proof of the mathematical result and numerical studies for some simple yet meaningful geometries.