Rigidity and functional properties of $\mathrm{BD}_{dev}(\Omega)$

TL;DR

This paper analyzes the structure of the space of functions with bounded deviatoric deformation, providing new rigidity results and tools for relaxation and homogenization in plasticity and fluid mechanics.

Contribution

It identifies the annihilator and rigidity properties of $ ext{BD}_{dev}$-maps with constant polar vector, enabling advanced analysis of singularities and applications in material science.

Findings

Established the structure of singularities in $ ext{BD}_{dev}$

Developed an explicit kernel projection operator

Enabled iterative blow-up procedures for relaxation and homogenization

Abstract

We provide a structural analysis of the space of functions of bounded deviatoric deformation, $\mathrm{BD}_{dev}$, which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for $\mathrm{BD}_{dev}$-maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation and homogenization problems, allowing for integrands with explicit dependence on $u$ as well as $\mathcal{E}_d u$. Our approach overcomes several difficulties as compared to the $\mathrm{BD}$ case, in particular due to the lack of invariance of $\mathcal{E}_d$ under orthogonalization of the polar directions. Applications to integral representation and Material science are discussed.