Curl bounds Grad on SO(3)

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$ .