## 5 Formulation of optimal design problems for laminates  
As said in section 1, the design of laminates must take into account, besides the objective itself (for instance, a sufficient stiffness in one or more directions), also the general elastic properties of the final laminate. The general constitutive law of an elastic laminate is  
\[ \begin{cases}\mathbf{N}\\ \mathbf{M}\end{cases} = \begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{B}&\mathbf{D}\end{bmatrix} \begin{bmatrix}\mathbf{\varepsilon}^{o}\\ \boldsymbol{\chi}\end{cases}, \tag{15}\]  
where \(\mathbf{N}\) and \(\mathbf{M}\) are the tensors of in-plane tractions and of bending moments, \(\mathbf{\varepsilon}^{0}\) is the tensor of in-plane strains for the middle plane, \(\boldsymbol{\chi}\) is the tensor of curvatures. The fourth order tensors \(\mathbf{A}\) and \(\mathbf{D}\) describe respectively the in- and out of-plane behaviour of the plate, while \(\mathbf{B}\) take into account coupling between them. Usually, designers look for laminates having an orthotropic behaviour, at least in extension, and uncoupled, _i.e._\(\mathbf{B}\)=\(\mathbf{O}\), but other elastic properties can also be of interest. All these properties can be taken into account by the following minimization procedure, [13]: consider the quadratic form \(I(\mathbf{P})\) of the symmetric matrix \(\mathbf{H}\), defined on the 18-dimensional space of the variables \(P_{i}\):  
\[I(\mathbf{P})=\mathbf{P}\cdot\mathbf{H}\ \mathbf{P}=H_{ij}P_{i}P_{j},\ \ \ i,j=1,...,18 \tag{16}\]  
with the components of vector \(\mathbf{P}\) given by  
\[ \begin{split}& P_{1}=\frac{\widetilde{T}_{0}}{h\ M},\ \ P_{2}=\frac{ \widetilde{T}_{1}}{h\ M},\ \ P_{3}=\frac{\widetilde{R}_{0}}{h\ M},\ \ P_{4}=\frac{ \widetilde{R}_{1}}{h\ M},\ \ P_{5}=\overline{\mathcal{O}}_{0},\ \ P_{6}=\overline{ \mathcal{O}}_{1},\\ & P_{7}=\frac{2\ \hat{T}_{0}}{h^{2}\ M},\ \ P_{8}=\frac{2\ \hat{T}_{1}}{h^{2}\ M},\ \ P_{9}=\frac{2\ \hat{R}_{0}}{h^{2}\ M},\ \ P_{10}=\frac{2\ \hat{R}_{1}}{h^{2}\ M},\ \ P_{11}=\hat{ \mathcal{O}}_{0},\ \ P_{12}=\hat{\mathcal{O}}_{1},\\ & P_{13}=\frac{12\ \widetilde{T}_{0}}{h^{3}\ M},\ \ P_{14}=\frac{12\ \widetilde{T}_{1}}{h^{3}\ M},\ \ P_{15}=\frac{12\ \widetilde{R}_{0}}{h^{3}\ M},\ \ P_{16}=\frac{12\ \widetilde{R}_{1}}{h^{3}\ M},\ \ P_{17}= \widetilde{\mathcal{O}}_{0},\ \ P_{18}=\widetilde{\mathcal{O}}_{1}.\end{split}  \tag{17}\]  
Here, \(h\) is the total laminate thickness while \(M\) is a factor, whose dimensions are those of an elastic modulus, used to obtain non-dimensional variables, that can be chosen in various ways. In addition,\[ \begin{array}{l}\overline{T}_{0},\widehat{T}_{0},\widetilde{T}_{0}=\frac{1}{\mu} \sum\limits_{k=1}^{np}T_{0_{k}}\left(z_{k}^{\mu}-z_{k-1}^{\mu}\right),\\ \overline{T}_{1},\widehat{T}_{1},\widetilde{T}_{1}=\frac{1}{\mu} \sum\limits_{k=1}^{np}T_{1_{k}}\left(z_{k}^{\mu}-z_{k-1}^{\mu}\right),\\ \overline{R}_{0}e^{4i\overline{\phi}_{0}},\hat{R}_{0}e^{4i\hat{\phi}_{0}}, \widetilde{R}_{0}e^{4i\tilde{\phi}_{0}}=\frac{1}{\mu}\sum\limits_{k=1}^{np}R _{0_{k}}e^{4i(\phi_{0_{k}}+\delta_{k})}\left(z_{k}^{\mu}-z_{k-1}^{\mu}\right), \\ \overline{R}_{1}e^{4i\overline{\phi}_{1}},\hat{R}_{1}e^{4i\hat{\phi}_{1}}, \widetilde{R}_{1}e^{4i\tilde{\phi}_{1}}=\frac{1}{\mu}\sum\limits_{k=1}^{np}R _{1_{k}}e^{4i(\phi_{1_{k}}+\delta_{k})}\left(z_{k}^{\mu}-z_{k-1}^{\mu}\right), \end{array}  \tag{18}\]  
are, in complex form, the 18 polar components of, respectively, tensor **A**, **B** and **D**, with \(\mu\)=1 for **A**, \(\mu\)=2 for **B** and \(\mu\)=3 for **D**; \(np\) is the number of plies in the stack and \(\delta_{k}\) is the orientation angle of the \(k\)-th ply; \(T_{0k}\), \(T_{1k}\), \(R_{0k}\), \(R_{1k}\), \(\Phi_{0k}\) and \(\Phi_  
**H** is a symmetric matrix of real coefficients: the choice of its components \(H_{ij}\) determines the kind of problem to be treated. Several choices are possible, so several different problems can be stated in the same way, simply changing the \(H_{ij}\).