Energy minimizing configurations for single-director Cosserat shells (Q6064114)

The authors consider a class of single-director Cosserat shell models accounting for both curvature and finite mid-plane strains.\ Let \(\Omega \subset \mathbb{R}^{2}\) be an open and bounded domain with a strongly locally Lipschitz boundary \(\partial \Omega\). \(\overline{\Omega }\) is a reference configuration for a material surface in a ``flat'' state, a configuration being specified by a deformation \(f:\overline{\Omega }\rightarrow \mathbb{R}^{3}\) and a director field \(d:\overline{\Omega }\rightarrow \mathbb{R}^{3}\). The gradient of \(f\) is denoted by \(F\) and that of \(d\) by \(G\), \(\{m_{l}(F,G)\}_{l=1}^{15}\) is the set of 15 independent \(2\times 2\) sub-determinants. A smooth configuration is required to satisfy the local orientation condition: \(J(d,\nabla f)=d\cdot (f_{,1}\times f_{,2})>0\) in \(\overline{\Omega }\). The surface is equipped with a stored-energy function \(W(x,d,F,G)\), \(W:\Phi \times \mathcal{ O}^{+}\times \mathbb{R}^{3\times 2}\rightarrow \lbrack 0,\infty)\), where \(\mathcal{O}^{+}=\{(d,F)\in \mathbb{R}^{3}\times \mathbb{R}^{3\times 2}:J>0\}\), satisfying the objectivity property: \(W(x,Qd,QF,QG)\equiv W(x,d,F,G)\) for all \(Q\in \mathrm{SO}(3)\), the existence of constants \(p,q,r>4/3\), \(s>1\), \(C_{1}>0\) and \(C_{2}\in \mathbb{R}\) such that \(W(x,d,F,G)\geq C_{1}\{\left\vert F\right\vert ^{p}+\left\vert G\right\vert ^{q}+\sum_{l=1}^{3}\left\vert m_{l}\right\vert ^{r}+\sum_{l=4}^{15}\left\vert m_{l}\right\vert ^{s}\}+C_{2}\), the existence of a \(C^{1}\) function \(\Phi :\overline{\Omega }\times \mathbb{R}^{3}\times \mathbb{R}^{3\times 2}\times \mathbb{R}^{3\times 2}\times (0,\infty)\times \mathbb{R}^{12}\rightarrow \lbrack 0,\infty)\), such that \((F,G,J,m_{4},\ldots,m_{15})\rightarrow \Phi (x,d,F,G,J,m_{4},\ldots,m_{15})\) is convex and \(W(x,d,F,G)\equiv \Phi (x,d,F,G,J,m_{4},\ldots,m_{15})\), and \(\Phi \rightarrow +\infty\) as \(J\rightarrow 0^{+}\). The total potential energy is given by \(E[f,d]=\int_{\Omega }W(x,d(x),\nabla f(x),\nabla d(x))dx-L(f,d)\), where \(L\) is a bounded linear functional on \(W^{1,p}(\Omega,\mathbb{R}^{3})\times W^{1,q}(\Omega,\mathbb{R}^{3})\) representing ``dead'' loading. The authors give an example of such functional \(L(f,d)\). The main result proves that if the admissible set \(\mathcal{A}=\{(f,d)\in W^{1,p}(\Omega,\mathbb{R}^{3})\times W^{1,q}(\Omega, \mathbb{R}^{3}):m_{l}\in L^{r}(\Omega)\), \(l=1,2,3\); \(m_{l}\in L^{s}(\Omega)\), \(l=4,\ldots,15\); \(J\in L^{1}(\Omega)\); \(J>0\) a.e. in \(\Omega\); \(f-f_{o}\in W_{\Gamma }^{1,p}(\Omega,\mathbb{R}^{3})\); \(d-d_{o}\in W_{\Gamma }^{1,q}(\Omega,\mathbb{R}^{3})\}\), where \((f_{o},d_{o})\in W^{1,p}(\Omega, \mathbb{R}^{3})\times W^{1,q}(\Omega,\mathbb{R}^{3})\) are prescribed and satisfy \(d_{o}\cdot (f_{o,1}\times f_{o,2})>0\) a.e. is non-empty with \(inf_{ \mathcal{A}}E[f,d]<\infty\), then there exists \((f^{\ast },d^{\ast })\in \mathcal{A}\) which minimizes the potential energy on \(\mathcal{A}\): \(E[f^{\ast },d^{\ast }]=\mathrm{inf}_{\mathcal{A}}E[f,d]\). The authors first prove that the potential energy functional is weakly lower semicontinuous on \(W^{1,p}(\Omega,\mathbb{R}^{3})\times W^{1,q}(\Omega,\mathbb{R}^{3})\). They then use Mazur's theorem, among other tools and computations. The paper ends with three examples of minimization problems with constraints: \(\left\vert d\right\vert =1\) a.e. in \(\Omega\), \(d\cdot f_{,\alpha }=0\) a.e. in \(\Omega\), \(\alpha =1,2\), and the combination of the preceding constraints. The authors briefly show how the preceding existence result may be adapted to these situations, through appropriate modifications of the admissible set.