Curl bounds grad on SO(3)
ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 1, pp. 148-159.

Let F p 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 RC 1 ( 3 , SO (3)): Curl [R]·R T 𝕄 3×3 2 1 2DR 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 AC 1 ( 3 ,𝔰𝔬(3)): Curl [A] 𝕄 3×3 2 1 2DA 27 2 = axl [A] 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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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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