A Duality Principle and Related Convex Dual Approximate Formulation Applied to a Non-Linear Plate Model

1. Introduction

This article develops a duality principle applicable to a large class of models in the calculus of variations. Specifically in this text, we present applications to the non-linear Kirchhoff-Love plate model.

We emphasize the results on duality theory here addressed and developed are inspired mainly in the approaches of J.J.Telega, W.R. Bielski and co-workers presented in the articles [1,2,3,4]. Other main reference is the article by Toland,[5].

Moreover, details on the Sobolev spaces involved may be found in [6].

Similar results and models are addressed in [7,8,9,10,11].

Basic results on convex analysis are addressed in [12]. Other similar results and approaches may be found in [13,14,15].

Now we start to describe the primal variational formulation for the plate model in question.

Let $\Omega \subset {\mathbb{R}}^2$ be an open, bounded and connected set with a regular (Lipschitzian) boundary denoted by $\partial \Omega .$

We assume such a $\Omega$ set represents the middle surface of a thin plate with a constant thickness $h>0$.

Moreover, we suppose such a plate is subject to a external load $(P_{\alpha },P)\in L^2(\Omega ;{\mathbb{R}}^3)$ resulting a field of displacements denoted by

$$(u_{\alpha },w)=(u_1,u_2,w)\in W_0^{1,2}(\Omega ;{\mathbb{R}}^2)\times W_0^{2,2}(\Omega )=V.$$

Both the load and displacements fields refers to a cartesian system $(0,x_1,x_2,x_3)$ and related canonical basis in ${\mathbb{R}}^3.$

Finally, we denote $Y_1=Y_1^{*}=L^2(\Omega ;{\mathbb{R}}^4)$ and $Y_2=Y_2^{*}=L^2(\Omega ;{\mathbb{R}}^2).$

We also emphasize the boundary conditions in question refer to a clamped plate.

The strain tensors are defined by

$${\gamma }_{\alpha \beta }\left(u\right)={\displaystyle \frac{u_{\alpha ,\beta }+u_{\beta ,\alpha }}{2}}+{\displaystyle \frac{1}{2}}{w}_{,\alpha }{w}_{,\beta },$$

and

$${\kappa }_{\alpha \beta }\left(w\right)=-w_{,\alpha \beta }.$$

The plate total energy functional is defined by

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\displaystyle \frac{1}{2}}{\int }_{\Omega }{H}_{\alpha \beta \lambda \mu }{\gamma }_{\alpha \beta }\left(u\right){\gamma }_{\lambda \mu }\left(u\right)dx\hfill \\ & & +{\displaystyle \frac{1}{2}}{\int }_{\Omega }{h}_{\alpha \beta \lambda \mu }{\kappa }_{\alpha \beta }\left(u\right){\kappa }_{\lambda \mu }\left(u\right)dx\hfill \\ & & -{\langle w,P\rangle }_{L^2}-{\langle u_{\alpha },P_{\alpha }\rangle }_{L^2}.\hfill \end{array}$$

(1)

Here $\left\{H_{\alpha \beta \lambda \mu }\right\}$ is a fourth order positive definite symmetric constant tensor.

Moreover

$$\left\{h_{\alpha \beta \lambda \mu }\right\}={\displaystyle \frac{h^2}{12}}\left\{H_{\alpha \beta \lambda \mu }\right\}$$

and we denote

$$\left\{{\overline{H}}_{\alpha \beta \lambda \mu }\right\}={\left\{H_{\alpha \beta \lambda \mu }\right\}}^{-1}$$

and

$$\left\{{\overline{h}}_{\alpha \beta \lambda \mu }\right\}={\left\{h_{\alpha \beta \lambda \mu }\right\}}^{-1}$$

in an appropriate tensor sense.

2. The Main Duality Principle and Related Convex Dual Approximate Formulation

We start by defining the approximate functional $J_1:V\to \mathbb{R}$ by

$$J_1\left(u\right)=J\left(u\right)+\sum _{\alpha =1}^2{\displaystyle \frac{{\epsilon }_1}{2}}{\int }_{\Omega }{\left(u_{\alpha }\right)}^{2}dx,$$

and considering an appropriate real constant $K>0$, the functionals $F_1:V\to \mathbb{R}$, $F_2:V\times Y_1^{*}\to \mathbb{R}$, $F_3:V\to \mathbb{R}$ and $F_4:V\to \mathbb{R}$, by

$$\begin{array}{ccc}\hfill F_1\left(u\right)& =& {\displaystyle \frac{1}{2}}{\int }_{\Omega }{h}_{\alpha \beta \lambda \mu }{\kappa }_{\alpha \beta }\left(u\right){\kappa }_{\lambda \mu }\left(u\right)dx+\sum _{\alpha =1}^2{\displaystyle \frac{K}{2}}{\int }_{\Omega }{w}_{,\alpha }^{2}dx\hfill \\ & & +\sum _{\alpha =1}^2\epsilon {\int }_{\Omega }{w}_{,\alpha }^{2}dx-{\langle w,P\rangle }_{L^2}\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill F_2(u,N)& =& {\displaystyle \frac{1}{2}}{\int }_{\Omega }{N}_{\alpha \beta }{w}_{,\alpha }{w}_{,\beta }dx\hfill \\ & & -\sum _{\alpha =1}^2{\displaystyle \frac{K}{2}}{\int }_{\Omega }{w}_{,\alpha }^{2}dx\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill F_3\left(u\right)& =& \sum _{\alpha =1}^2{\displaystyle \frac{{\epsilon }_1}{2}}{\int }_{\Omega }{u}_{\alpha }^{2}dx\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill F_4\left(u\right)& =& \sum _{\alpha =1}^2\epsilon {\int }_{\Omega }{w}_{,\alpha }^{2}dx.\hfill \end{array}$$

(5)

Moreover, we define the polar functionals $F_1^{*}:{\left[Y_2^{*}\right]}^2\to \mathbb{R}$, $F_2^{*}:{\left[Y_2^{*}\right]}^2\times Y_1^{*}\to \mathbb{R},$ $F_3^{*}:Y_1^{*}\to \mathbb{R}$ and $F_4^{*}:{\left[Y_2^{*}\right]}^2\to \mathbb{R}$, by

$$\begin{array}{ccc}\hfill F_1^{*}(R,L)& =& \underset{u\in V}{sup}\{{\langle w_{,\alpha },R_{\alpha }-L_{\alpha }\rangle }_{L^2}-F_1\left(u\right)\},\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill F_2^{*}(Q,L)& =& \underset{v\in Y_2}{inf}\left\{{\langle v_{\alpha },Q_{\alpha }+L_{\alpha }\rangle }_{L^2}-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{N}_{\alpha \beta }{v}_{\alpha }{v}_{\beta }dx\right.\hfill \\ & & \left.+\sum _{\alpha =1}^2{\displaystyle \frac{K}{2}}{\int }_{\Omega }{v}_{\alpha }^{2}dx+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\overline{H}}_{\alpha \beta \lambda \mu }{N}_{\alpha \beta }{N}_{\lambda \mu }dx\right\}\hfill \\ & =& {\displaystyle \frac{1}{2}}{\int }_{\Omega }\overline{N_{\alpha \beta }^{(-K)}}(Q_{\alpha }+L_{\alpha })(Q_{\beta }+L_{\beta })dx\hfill \\ & & +{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\overline{H}}_{\alpha \beta \lambda \mu }{N}_{\alpha \beta }{N}_{\lambda \mu }dx,\hfill \end{array}$$

(7)

if $N=\left\{N_{\alpha \beta }\right\}\in B^{*}$, where

$$B^{*}=\{N\in Y_1^{*}:\parallel N_{\alpha \beta }{\parallel }_{\infty }\le K/8,\forall \alpha ,\beta \in \{1,2\}\},$$

$$\left\{N_{\alpha \beta }^{(-K)}\right\}=\{N_{\alpha \beta }-K{\delta }_{\alpha \beta }\},$$

and

$$\left\{\overline{N_{\alpha \beta }^{(-K)}}\right\}={\left\{N_{\alpha \beta }^{(-K)}\right\}}^{-1}.$$

Furthermore,

$$\begin{array}{ccc}\hfill F_3^{*}\left(N\right)& =& \underset{u\in V}{sup}\{{\langle u_{\alpha },N_{\alpha \beta ,\beta }+P_{\alpha }\rangle }_{L_2}-F_3\left(u\right)\}\hfill \\ & =& \sum _{\alpha =1}^2{\displaystyle \frac{1}{2{\epsilon }_1}}{\int }_{\Omega }{(N_{\alpha \beta ,\beta }+P_{\alpha })}^{2}dx,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill F_4^{*}(R,Q)& =& \underset{(v_1,v_2)\in Y_2^{*}}{sup}\left\{{\langle {\left(v_1\right)}_{\alpha },R_{\alpha }\rangle }_{L^2}-{\displaystyle \frac{\epsilon }{2}}{\int }_{\Omega }{\left(v_1\right)}_{\alpha }^{2}dx\right.\hfill \\ & & \left.+{\langle {\left(v_2\right)}_{\alpha },Q_{\alpha }\rangle }_{L^2}-{\displaystyle \frac{\epsilon }{2}}{\int }_{\Omega }{\left(v_2\right)}_{\alpha }^{2}dx\right\}\hfill \\ & =& \sum _{\alpha =1}^2\left({\displaystyle \frac{1}{2\epsilon }}{\int }_{\Omega }{R}_{\alpha }^{2}dx+{\displaystyle \frac{1}{2\epsilon }}{\int }_{\Omega }{Q}_{\alpha }^{2}dx\right).\hfill \end{array}$$

(9)

At this point, denoting

$$D^{*}=\{Q=\left\{Q_{\alpha }\right\}\in Y_2^{*}:\parallel Q_{\alpha }\parallel \le 5,\forall \alpha \in \{1,2\}\},$$

we define $J_1^{*}:{\left(D^{*}\right)}^2\times B^{*}\times Y_2^{*}\to \mathbb{R}$ by

$$J_1^{*}(R,Q,N,L)=-F_1^{*}(R,L)-F_2^{*}(Q,L,N)-F_3^{*}\left(N\right)+F_4^{*}(R,Q),$$

and $J_2^{*}:{\left(D^{*}\right)}^2\times B^{*}\to \mathbb{R}$ by

$$J_2^{*}(R,Q,N)={\mathrm{sta}}_{L\in Y_2^{*}}{J}_1^{*}(R,Q,N,L)=J_1^{*}(R,Q,N,\widehat{L}(R,Q,N)),$$

where $\widehat{L}=\widehat{L}(R,Q,N)\in Y_2^{*}$ is the only solution of the linear equation in L

$${\displaystyle \frac{\partial J_1^{*}(R,Q,N,\widehat{L})}{\partial L}}=\mathbf{0}.$$

Moreover, we define $J_3^{*}:{\left(D^{*}\right)}^2\times B^{*}\to \mathbb{R}$, by

$$\begin{array}{ccc}& & J_3^{*}(R,Q,N)\hfill \\ & =& J_2^{*}(R,Q,N)\hfill \\ & & -\sum _{\alpha =1}^2\sum _{\beta =1}^2{\displaystyle \frac{K_1}{2}}{∥{\overline{H}}_{\alpha \beta \lambda \mu }{N}_{\lambda \mu }-{\displaystyle \frac{{(N_{\alpha \rho ,\rho }+P_{\alpha })}_{,\beta }+{(N_{\beta \rho ,\rho }+P_{\alpha })}_{,\alpha }}{2{\epsilon }_1}}-{\displaystyle \frac{1}{2}}{\tilde{v}}_{\alpha }{\tilde{v}}_{\beta }∥}_{0,2}^2,\hfill \end{array}$$

(10)

where

$${\tilde{v}}_{\alpha }=\overline{N_{\alpha \beta }^{(-K)}}(Q_{\beta }+L_{\beta }(R,Q,N)),\forall \alpha \in \{1,2\}.$$

Here, we assume

$$K_1\gg max\{1,K,max\{{\overline{h}}_{\alpha \beta \lambda \mu },\alpha ,\beta ,\lambda ,\mu \in \{1,2\}\}\},$$

$$0<\epsilon ,{\epsilon }_1\ll 1$$

and

$${\displaystyle \frac{1}{\epsilon }}\gg max\left\{K_1,{\displaystyle \frac{1}{{\epsilon }_1}}\right\}.$$

Observe that

$${\displaystyle \frac{{\partial }^2{J}_3^{*}(R,Q,N)}{\partial Q_{\alpha }^2}}=\mathcal{O}\left({\displaystyle \frac{1}{\epsilon }}\right)>\mathbf{0},$$

$${\displaystyle \frac{{\partial }^2{J}_3^{*}(R,Q,N)}{\partial R_{\alpha }^2}}=\mathcal{O}\left({\displaystyle \frac{1}{\epsilon }}\right)>\mathbf{0},$$

$\forall \alpha \in \{1,2\}.$

Thus, considering also the remaining mixed variations in $Q_{\alpha }$ and $R_{\alpha }$, we may infer that

$$det\left\{{\displaystyle \frac{{\partial }^2{J}_3^{*}(R,Q,N)}{\partial Q_{\alpha }\partial R_{\beta }}}\right\}>\mathbf{0},$$

in ${\left(D^{*}\right)}^2\times B^{*}$.

Moreover, by direct computation, clearly

$$det\left\{{\displaystyle \frac{{\partial }^2{J}_3^{*}(R,Q,N)}{\partial N_{\alpha \beta }\partial N_{\lambda \mu }}}\right\}<\mathbf{0},$$

in ${\left(D^{*}\right)}^2\times B^{*}$.

From such results, we may infer that $J_3^{*}$ is convex in $(R,Q)$ and concave in N in ${\left(D^{*}\right)}^2\times B^{*}$.

Let $(\widehat{R},\widehat{Q},\widehat{N},\widehat{L})\in {\left(D^{*}\right)}^2\times B^{*}\times Y_2^{*}$ be such that

$$\delta J_1^{*}(\widehat{R},\widehat{Q},\widehat{N},\widehat{L})=\mathbf{0}.$$

Let $u_0=({\left(u_0\right)}_{\alpha },w_0)\in V$ be such that

$${\left(u_0\right)}_{\alpha }={\displaystyle \frac{{\widehat{N}}_{\alpha \beta ,\beta }+P_{\alpha }}{{\epsilon }_1}},$$

and

$${\left(w_0\right)}_{,\alpha }={\displaystyle \frac{{\widehat{Q}}_{\alpha }}{\epsilon }},$$

$\forall \alpha \in \{1,2\}.$

From standard results in Duality Theory and the Legendre Transform properties, we may obtain

$$\delta J_1\left(u_0\right)=\mathbf{0},$$

$$\delta J_2^{*}(\widehat{R},\widehat{Q},\widehat{N})=\mathbf{0}$$

,

$$\delta J_3^{*}(\widehat{R},\widehat{Q},\widehat{N})=\mathbf{0}$$

and

$$\begin{array}{ccc}\hfill J_1\left(u_0\right)& =& J_1^{*}(\widehat{R},\widehat{Q},\widehat{N},\widehat{L})\hfill \\ & =& J_2^{*}(\widehat{R},\widehat{Q},\widehat{N})\hfill \\ & =& J_3^{*}(\widehat{R},\widehat{Q},\widehat{N}).\hfill \end{array}$$

(11)

From such results and the Min-Max Theorem, we have

$$J_3^{*}(\widehat{R},\widehat{Q},\widehat{N})=\underset{(R,Q)\in {\left(D^{*}\right)}^2}{inf}\left\{\underset{N\in B^{*}}{sup}{J}_3^{*}(R,Q,N)\right\}.$$

Joining the pieces, we have got

$$\begin{array}{ccc}\hfill J_1\left(u_0\right)& =& J_1^{*}(\widehat{R},\widehat{Q},\widehat{N},\widehat{L})\hfill \\ & =& J_2^{*}(\widehat{R},\widehat{Q},\widehat{N})\hfill \\ & =& \underset{(R,Q)\in {\left(D^{*}\right)}^2}{inf}\left\{\underset{N\in B^{*}}{sup}{J}_3^{*}(R,Q,N)\right\}\hfill \\ & =& J_3^{*}(\widehat{R},\widehat{Q},\widehat{N}).\hfill \end{array}$$

(12)

Remark 2.1.

Defining $J_5^{*}:{\left(D^{*}\right)}^2\to \mathbb{R}$ by

$$J_5^{*}(R,Q)=\underset{N\in B^{*}}{sup}{J}_3^{*}(R,Q,N),$$

we have that such a functional $J_5^{*}$ is convex in ${\left(D^{*}\right)}^2$ as the supremum in $N\in B^{*}$ of a family of convex functionals in $(R,Q).$

In such a case, we have also obtained

$$\begin{array}{ccc}\hfill J_1\left(u_0\right)& =& J_1^{*}(\widehat{R},\widehat{Q},\widehat{N},\widehat{L})\hfill \\ & =& J_2^{*}(\widehat{R},\widehat{Q},\widehat{N})\hfill \\ & =& \underset{(R,Q)\in {\left(D^{*}\right)}^2}{inf}\left\{\underset{N\in B^{*}}{sup}{J}_3^{*}(R,Q,N)\right\}\hfill \\ & =& J_3^{*}(\widehat{R},\widehat{Q},\widehat{N})\hfill \\ & =& \underset{(R,Q)\in {\left(D^{*}\right)}^2}{inf}{J}_5^{*}(R,Q)\hfill \\ & =& J_5^{*}(\widehat{R},\widehat{Q}).\hfill \end{array}$$

(13)

3. Conclusion

In this article, we have developed a duality principle and related convex dual approximate variational formulation for an originally non-convex primal one.

We highlight the results here obtained are applicable to a large class of models in the calculus of variations, including other plate and shell non-linear theories, models in superconductivity, phase transition and micro-magnetism, among many others.

In a near future research we intend to apply such results to some of these mentioned related models.