On the characterization of drilling rotation in the 6-parameter resultant shell theory

We analyze geometrically non-linear isotropic elastic shells and prove the existence of minimizers. In general, the model takes into account the effect of drilling rotations in shells. For the special case of shells without drilling rotations we present a representation theorem for the strain energy function

Counterexamples in the theory of coerciveness for linear elliptic systems related to generalizations of Korn's second inequality

We show that the following generalized version of Korn's second inequality with nonconstant measurable matrix valued coefficients P ||DuP+(DuP)^T||_q+||u||_q >= c ||Du||_q for u in W_0^{1,q}({\Omega};R^3), 1<q<{\infty} is in general false, even if P is in SO(3), while the Legendre-Hadamard condition and ellipticity on C^n for the quadratic form |Du P+(DuP)^T|^2 is satisfied. Thus Garding's inequality may be violated for formally positive quadratic forms

The geometrically nonlinear Cosserat micropolar shear-stretch energy. Part II: Non-classical energy-minimizing microrotations in 3D and their computational validation

In any geometrically nonlinear, isotropic and quadratic Cosserat micropolar extended continuum model formulated in the deformation gradient field $F = \nabla\varphi : \Omega \to GL^+(n)$ and the microrotation field $R: \Omega \to SO(n)$, the shear-stretch energy is necessarily of the form $$W_{\mu,\mu_c}(R;F) = \mu\, \| sym(R^T F - 1) \|^2 + \mu_c\, \| skew(R^T F - 1) \|^2 .$$ We aim at the derivation of closed form expressions for the minimizers of $W(R;F)$ in $SO(3)$, i.e., for the set of optimal Cosserat microrotations in dimension $n = 3$, as a function of $F \in GL^+(n)$. In a previous contribution (Part I), we have first shown that, for all $n \geq 2$, the full range of weights $\mu > 0$ and $\mu_c \geq 0$ can be reduced to either a classical or a non-classical limit case. We have then derived the associated closed form expressions for the optimal planar rotations in $SO(2)$ and proved their global optimality. In the present contribution (Part II), we characterize the non-classical optimal rotations in dimension n = 3. After a lift of the minimization problem to the unit quaternions, the Euler-Lagrange equations can be symbolically solved by the computer algebra system Mathematica. Among the symbolic expressions for the critical points, we single out two candidates $rpolar^{\pm}_{\mu,\mu_c}(F) \in SO(3)$ which we analyze and for which we can computationally validate their global optimality by Monte Carlo statistical sampling of $SO(3)$. Geometrically, our proposed optimal Cosserat rotations $rpolar^{\pm}_{\mu,\mu_c}(F)$ act in the "plane of maximal strain" and our previously obtained explicit formulae for planar optimal Cosserat rotations in $SO(2)$ reveal themselves as a simple special case. Further, we derive the associated reduced energy levels of the Cosserat shear--stretch energy and criteria for the existence of non-classical optimal rotations.Comment: 30 pages, 8 figure

The geometrically nonlinear Cosserat micropolar shear-stretch energy. Part I: A general parameter reduction formula and energy-minimizing microrotations in 2D

In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field $F := \nabla\varphi: \Omega \to \mathrm{GL}^+(n)$ 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\,\left\lVert{\mathrm{sym}(R^T F - \boldsymbol{1})}\right\rVert^2 + \mu_c\,\left\lVert{\mathrm{skew}(R^T F - \boldsymbol{1})}\right\rVert^2\;, \end{equation*} where $\mu > 0$ is the Lam\'e 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.Comment: 17 pages, 3 figure

Shells without drilling rotations: a representation theorem in the framework of the geometrically nonlinear 6-parameter resultant shell theory

In the framework of the geometrically nonlinear 6-parameter resultant shell theory we give a characterization of the shells without drilling rotations. These are shells for which the strain energy function $W$ is invariant under the superposition of drilling rotations, i.e. $W$ is insensible to the arbitrary local rotations about the third director $\boldsymbol{d}_3\,$. For this type of shells we show that the strain energy density $W$ can be represented as a function of certain combinations of the shell deformation gradient $\boldsymbol{F}$ and the surface gradient of $\boldsymbol{d}_3\,$, namely $W\big(\boldsymbol{F}^{ T}\boldsymbol{F} , \, \boldsymbol{F}^T \boldsymbol{d}_3 \,, \, \boldsymbol{F}^T\mathrm{Grad}_s\boldsymbol{d}_3 \big)$. For the case of isotropic shells we present explicit forms of the strain energy function $W$ having this property.Comment: 18 page

Existence of minimizers in the geometrically non-linear 6-parameter resultant shell theory with drilling rotations

The paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells with drilling degrees of freedom. This theory establishes a general model for shells, which is characterized by two independent kinematic fields: the translation vector and the rotation tensor. Thus, the kinematical structure of 6-parameter shells is identical to that of Cosserat shells. We show the existence of global minimizers for the geometrically non-linear 2D equations of elastic shells. The proof of the existence theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze separately the particular cases of isotropic shells, orthotropic shells, and composite shells

On the dislocation density tensor in the Cosserat theory of elastic shells

We consider the Cosserat continuum in its finite strain setting and discuss the dislocation density tensor as a possible alternative curvature strain measure in three-dimensional Cosserat models and in Cosserat shell models. We establish a close relationship (one-to-one correspondence) between the new shell dislocation density tensor and the bending-curvature tensor of 6-parameter shells

The exponentiated Hencky-logarithmic strain energy. Part III: Coupling with idealized isotropic finite strain plasticity

We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric-isochoric decoupled strain energies \begin{align*} F\mapsto W_{_{\rm eH}}(F):=\hat{W}_{_{\rm eH}}(U):=\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\frac{\kappa}{{\text{}}{2\, {\hat{k}}}}\,e^{\hat{k}\,[{\rm tr}(\log U)]^2}&\text{if}& {\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0, \end{array}.\quad \end{align*} based on the Hencky-logarithmic (true, natural) strain tensor $\log U$. Here, $\mu>0$ is the infinitesimal shear modulus, $\kappa=\frac{2\mu+3\lambda}{3}>0$ is the infinitesimal bulk modulus with $\lambda$ the first Lam\'{e} constant, $k,\hat{k}$ are dimensionless fitting parameters, $F=\nabla \varphi$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the strain tensor $\log U$. Based on the multiplicative decomposition $F=F_e\, F_p$, we couple these energies with some isotropic elasto-plastic flow rules $F_p\,\frac{\rm d}{{\rm d} t}[F_p^{-1}]\in-\partial \chi({\rm dev}_3 \Sigma_{e})$ defined in the plastic distortion $F_p$, where $\partial \chi$ is the subdifferential of the indicator function $\chi$ of the convex elastic domain $\mathcal{E}_{\rm e}(W_{\rm iso},{\Sigma_{e}},\frac{1}{3}{\boldsymbol{\sigma}}_{\!\mathbf{y}}^2)$ in the mixed-variant $\Sigma_{e}$-stress space and $\Sigma_{e}=F_e^T D_{F_e} W_{\rm iso}(F_e)$. While $W_{_{\rm eH}}$ may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of $W_{_{\rm eH}}$ on the one hand, and the elastic domain in $\Sigma_{e}$-stress space on the other hand, are closely related

Comparison of isotropic elasto-plastic models for the plastic metric tensor Cp=FpT FpC_p=F_p^T\, F_p

We discuss in detail existing isotropic elasto-plastic models based on 6-dimensional flow rules for the positive definite plastic metric tensor $C_p=F_p^T\, F_p$ and highlight their properties and interconnections. We show that seemingly different models are equivalent in the isotropic case

Loss of ellipticity for non-coaxial plastic deformations in additive logarithmic finite strain plasticity

In this paper we consider the additive logarithmic finite strain plasticity formulation from the view point of loss of ellipticity in elastic unloading. We prove that even if an elastic energy $F\mapsto W(F)=\hat{W}(\log U)$ defined in terms of logarithmic strain $\log U$, where $U=\sqrt{F^T\, F}$, is everywhere rank-one convex as a function of $F$, the new function $F\mapsto \widetilde{W}(F)=\hat{W}(\log U-\log U_p)$ need not remain rank-one convex at some given plastic stretch $U_p$ (viz. $E_p^{\log}:=\log U_p$). This is in complete contrast to multiplicative plasticity in which $F\mapsto W(F\, F_p^{-1})$ remains rank-one convex at every plastic distortion $F_p$ if $F\mapsto W(F)$ is rank-one convex. We show this disturbing feature with the help of a recently considered family of exponentiated Hencky energies.Comment: arXiv admin note: text overlap with arXiv:1409.755