Curl bounds Grad on SO(3)

Patrizio Neff; Ingo Münch

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 14, Issue: 1, page 148-159
  • ISSN: 1292-8119

Abstract

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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 ∀ R ∈ C 1 ( ℝ 3 , SO ( 3 ) ) : ∥ Curl [ R ] · R T ∥ 𝕄 3 × 3 2 ≥ 1 2 ∥ D R ∥ 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 ∀ A ∈ C 1 ( ℝ 3 , 𝔰𝔬 ( 3 ) ) : ∥ Curl [ A ] ∥ 𝕄 3 × 3 2 ≥ 1 2 ∥ D A ∥ ℝ 27 2 = ∥ ∇ axl [ A ] ∥ ℝ 9 2 .

How to cite

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Neff, Patrizio, and Münch, Ingo. "Curl bounds Grad on SO(3)." ESAIM: Control, Optimisation and Calculus of Variations 14.1 (2010): 148-159. <http://eudml.org/doc/90861>.

@article{Neff2010,
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$. },
author = {Neff, Patrizio, Münch, Ingo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Rotations; polar-materials; microstructure; dislocation density; rigidity; differential geometry; structured continua; multiplicative decomposition; elasto-plasticity; geometric dislocation density tensor},
language = {eng},
month = {3},
number = {1},
pages = {148-159},
publisher = {EDP Sciences},
title = {Curl bounds Grad on SO(3)},
url = {http://eudml.org/doc/90861},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Neff, Patrizio
AU - Münch, Ingo
TI - Curl bounds Grad on SO(3)
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 14
IS - 1
SP - 148
EP - 159
AB - 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$.
LA - eng
KW - Rotations; polar-materials; microstructure; dislocation density; rigidity; differential geometry; structured continua; multiplicative decomposition; elasto-plasticity; geometric dislocation density tensor
UR - http://eudml.org/doc/90861
ER -

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