Publication Date:
2016-12-19
Description:
In this note, we extend integrability conditions for the symmetric stretch tensor U in the polar decomposition of the deformation gradient \(\nabla \varphi =F=R\,U\) to the nonsymmetric case. In doing so, we recover integrability conditions for the first Cosserat deformation tensor. Let \(F=\overline{R}\,\overline{U}\) with \(\overline{R}:\varOmega \subset \mathbb {R}^3\longrightarrow {{\mathrm{\mathrm {SO}}}}(3)\) and \(\overline{U}:\varOmega \subset \mathbb {R}^3\longrightarrow \mathrm {GL}(3)\) . Then, $$\begin{aligned} \mathfrak {K}:=\overline{R}^T\mathrm {Grad}\,\overline{R}={{\mathrm{\mathrm {Anti}}}}\left( \frac{\Big [\overline{U}({{\mathrm{\mathrm {Curl}}}}\overline{U})^T-\frac{1}{2}{{\mathrm{tr}}}(\overline{U}({{\mathrm{\mathrm {Curl}}}}\overline{U})^T){{\mathbbm {1}}}\Big ]\overline{U}}{\det \overline{U}}\right) , \end{aligned}$$ giving a connection between the first Cosserat deformation tensor \(\overline{U}\) and the second Cosserat tensor \({\mathfrak {K}}\) . (Here, \({{\mathrm{\mathrm {Anti}}}}\) denotes an isomorphism between \(\mathbb {R}^{3\times 3}\) and \(\mathfrak {So}(3):=\{\,\mathfrak {A}\in \mathbb {R}^{3\times 3\times 3}\,|\,\mathfrak {A}.u\in \mathfrak {so}(3)\;\forall u\in \mathbb {R}^3\}\) ). The formula shows that it is not possible to prescribe \(\overline{U}\) and \(\mathfrak {K}\) independent from each other. We also propose a new energy formulation of geometrically nonlinear Cosserat models which completely separate the effects of nonsymmetric straining and curvature. For very weak constitutive assumptions (no direct boundary condition on rotations, zero Cosserat couple modulus, quadratic curvature energy), we show existence of minimizers in Sobolev spaces.
Print ISSN:
0044-2275
Electronic ISSN:
1420-9039
Topics:
Mathematics
,
Physics
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