ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV)

ESAIM: COCV, Vol. 14, N°1, pp. 148-159
DOI: 10.1051/cocv:2007050

Curl bounds Grad on SO(3)

Patrizio Neff¹ and Ingo Münch²

¹ Department of Mathematics, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany; neff@mathematik.tu-darmstadt.de
² Institut für Baustatik, Universität Karlsruhe (TH), Kaiserstrasse 12, 76131 Karlsruhe, Germany; im@bs.uka.de

(Received May 19, 2006. Revised September 5, 2006. Published online September 21, 2007.)

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 $\forall R \in C^1(\mathbb{R} ^3, {\rm SO}(3)): \Arrowvert {\rm Curl}[R] \cdot ... ...{3\times3}}^2 \ge \frac \Arrowvert{\rm D}R\Arrowvert_{\mathbb}^2$ . 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 $\forall A \in C^1(\mathbb{R} ^3, \mathfrak{so} (3)): \Arrowvert {\rm Curl}[A]\... ...{\mathbb}^2 = \Arrowvert\nabla{\rm axl}[A]\Arrowvert_{\mathbb^2$ .

Mathematics Subject Classification. 74A35, 74E15, 74G65, 74N15, 53AXX, 53B05

Key words: Rotations, polar-materials, microstructure, dislocation density, rigidity, differential geometry, structured continua

© EDP Sciences, SMAI 2007