Linearization and Homogenization of nonlinear elasticity close to stress-free joints

TL;DR

This paper investigates the homogenization and linearization of nonlinear hyperelastic composites with microstructure and prestrain near stress-free joints, revealing coupled effects and providing a new analytical estimate for Jones domains.

Contribution

It establishes convergence of limits to a unified linearized homogenized model, introduces a coupled homogenization of prestrain, and develops a new geometric rigidity estimate for Jones domains.

Findings

Converged homogenized model with prestrain for nonlinear elasticity.

Coupled homogenization of material and prestrain effects.

New geometric rigidity estimate for Jones domains.

Abstract

In this paper, we study a hyperelastic composite material with a periodic microstructure and a prestrain close to a stress-free joint. We consider two limits associated with linearization and homogenization. Unlike previous studies that focus on composites with a stress-free reference configuration, the minimizers of the elastic energy functional in the prestrained case are not explicitly known. Consequently, it is initially unclear at which deformation to perform the linearization. Our main result shows that both the consecutive and simultaneous limits converge to a single homogenized model of linearized elasticity. This model features a homogenized prestrain and provides first-order information about the minimizers of the original nonlinear model. We find that the homogenization of the material and the homogenization of the prestrain are generally coupled and cannot be considered separately. Additionally, we establish an asymptotic quadratic expansion of the homogenized stored energy function and present a detailed analysis of the effective model for laminate composite materials. A key analytical contribution of our paper is a new mixed-growth version of the geometric rigidity estimate for Jones domains. The proof of this result relies on the construction of an extension operator for Jones domains adapted to geometric rigidity.