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Curl bounds Grad on SO(3)

Published online by Cambridge University Press:  21 September 2007

Patrizio Neff  and
Ingo Münch
Patrizio Neff
Affiliation:
Department of Mathematics, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany; neff@mathematik.tu-darmstadt.de
Ingo Münch
Affiliation:
Institut für Baustatik, Universität Karlsruhe (TH), Kaiserstrasse 12, 76131 Karlsruhe, Germany; im@bs.uka.de
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Abstract

Let $F^{\rm p} \in {\rm GL}(3)$ be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form ${\rm Curl}[{F^{\rm p}}]\cdot (F^{\rm p})^T$ applied to rotations controls the gradient in the sense that pointwise $ \forall R \in C^1(\mathbb{R}^3, {\rm SO}(3)): \Arrowvert {\rm Curl}[R] \cdot R^T \Arrowvert_{\mathbb{M}^{3\times3}}^2 \ge \frac{1}{2} \Arrowvert{\rm D}R\Arrowvert_{\mathbb{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(\mathbb{R}^3, \mathfrak{so}(3)): \Arrowvert {\rm Curl}[A]\Arrowvert_{\mathbb{M}^{3\times3}}^2 \ge \frac{1}{2} \Arrowvert{\rm D}A\Arrowvert_{\mathbb{R}^{27}}^2 = \Arrowvert\nabla{\rm axl}[A]\Arrowvert_{\mathbb{R}^9}^2$.

Keywords

Rotationspolar-materialsmicrostructuredislocation densityrigiditydifferential geometrystructured continua
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 1 , January 2008 , pp. 148 - 159
Copyright
© EDP Sciences, SMAI, 2007

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