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- W1540571830 abstract "In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field $F := nablavarphi: Omega to mathrm{GL}^+(n)$ and the microrotation field $R: Omega to mathrm{SO}(n)$, the shear-stretch energy is necessarily of the form begin{equation*} W_{mu,mu_c}(R,;F) := mu,leftlVert{mathrm{sym}(R^T F - boldsymbol{1})}rightrVert^2 + mu_c,leftlVert{mathrm{skew}(R^T F - boldsymbol{1})}rightrVert^2;, end{equation*} where $mu > 0$ is the Lame shear modulus and $mu_c geq 0$ is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations $mathrm{argmin}_{R,in,mathrm{SO}(n)}{W_{mu,mu_c}(R,;F)}$ as a function of $F in mathrm{GL}^+(n)$ and weights $mu > 0$ and $mu_c geq 0$. For $n geq 2$, we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: $(mu, mu_c) = (1,1)$ and $(mu,mu_c) = (1,0)$. In contrast to Grioli's theorem, we derive non-classical minimizers for the parameter range $mu > mu_c geq 0$ in dimension $n!=!2$. Currently, optimality results for $n geq 3$ are out of reach for us, but we contribute explicit representations for $n!=!2$ which we name $mathrm{rpolar}^{pm}_{mu,mu_c}(F) in mathrm{SO}(2)$ and which arise for $n!=!3$ by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non-classical optimal Cosserat rotations $mathrm{rpolar}^pm_{mu,mu_c}(F_gamma)$ for simple planar shear." @default.
- W1540571830 created "2016-06-24" @default.
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- W1540571830 date "2015-07-20" @default.
- W1540571830 modified "2023-09-27" @default.
- W1540571830 title "The geometrically nonlinear Cosserat micropolar shear-stretch energy. Part I: A general parameter reduction formula and energy-minimizing microrotations in 2D" @default.
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