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arXiv · 2505.23348

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

Marco Caroccia·Nicolas Van Goethem
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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.

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Marco Caroccia, Nicolas Van Goethem. 2025-05-29. Rigidity and functional properties of $\mathrm{BD}_{dev}(\Omega)$. https://arxiv.org/abs/2505.23348

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