ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 1, pp. 148-159.

Let ${F}^{\mathrm{p}}\in \mathrm{GL}\left(3\right)$ be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form $\mathrm{Curl}\left[{F}^{\mathrm{p}}\right]·{\left({F}^{\mathrm{p}}\right)}^{T}$ applied to rotations controls the gradient in the sense that pointwise $\forall R\in {C}^{1}\left({ℝ}^{3},\mathrm{SO}\left(3\right)\right):\parallel \mathrm{Curl}\left[R\right]·{R}^{T}{\parallel }_{{𝕄}^{3×3}}^{2}\ge \frac{1}{2}{\parallel \mathrm{D}R\parallel }_{{ℝ}^{27}}^{2}$. This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math. 55 (2002) 1461-1506; John, Comme Pure Appl. Math. 14 (1961) 391-413; Reshetnyak, Siberian Math. J. 8 (1967) 631-653)] as well as an associated linearized theorem saying that $\forall A\in {C}^{1}\left({ℝ}^{3},\mathrm{𝔰𝔬}\left(3\right)\right){:\parallel \mathrm{Curl}\left[A\right]\parallel }_{{𝕄}^{3×3}}^{2}\ge \frac{1}{2}{\parallel \mathrm{D}A\parallel }_{{ℝ}^{27}}^{2}={\parallel \nabla \mathrm{axl}\left[A\right]\parallel }_{{ℝ}^{9}}^{2}$.

DOI: 10.1051/cocv:2007050
Classification: 74A35,  74E15,  74G65,  74N15,  53AXX,  53B05
Keywords: rotations, polar-materials, microstructure, dislocation density, rigidity, differential geometry, structured continua
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author = {M\"unch, Ingo and Neff, Patrizio},
title = {Curl bounds grad on {SO(3)}},
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Münch, Ingo; Neff, Patrizio. Curl bounds grad on SO(3). ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 1, pp. 148-159. doi : 10.1051/cocv:2007050. http://archive.numdam.org/articles/10.1051/cocv:2007050/

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