Curl bounds grad on SO(3)

Let F-p is an element of 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 Curl[F-p].(F-p)(T) applied to rotations controls the gradient in the sense that pointwise for all R is an element of C-1 (R-3, SO(3)) : parallel to Curl[R].R-T parallel to(2)(M3 x 3) >= 1/2 parallel to DR parallel to(2)(R27). This result complements rigidity results [Friesecke, James and Muller, 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 for all R is an element of C-1 (R-3, so(3)) : parallel to Curl[A].parallel to(2)(M3 x 3) >= 1/2 parallel to DA parallel to(2)(R27). = parallel to del axl[A]parallel to(2)(R9).