## 2 Classical Laminate Theory and Lamination Parameters  
### Lamination Parameters and Feasible Region  
The laminate constitutive equation of strain-load intensity relationships according to classical laminate theory [21] is as follows:  
\[\left\{\begin{array}{c}N\\ M\end{array}\right\}=\left[\begin{array}{c}A_{ij}B_{ij}\\ B_{ij}D_{ij}\end{array}\right]\left\{\begin{array}{c}\varepsilon\\ k\end{array}\right\} \tag{1}\]  
in which \(N\) and \(M\) denote the force and moment resultants, respectively. \(\varepsilon\) and \(k\) denote normal and shear strains. \(A_{ij}\), \(B_{ij}\), and \(D_{ij}\) are the elements of the membrane stiffness matrix, the membrane-bending coupling matrix, and the bending matrix of the laminate, respectively. \(A_{ij}\), \(B_{ij}\), and \(D_{ij}\) can be expressed in terms of 12 lamination parameters and material stiffness invariants \(U\).  
\[\left\{\begin{array}{c}A_{11}\\ A_{22}\\ A_{12}\\ A_{66}\\ A_{16}\\ A_{26}\end{array}\right\}=h\left[\begin{array}{ccc}U_{1}&V_{1}^{A}&V_{2}^{A }\\ U_{1}&  
\[\left\{\begin{array}{l}D_{11}\\ D_{22}\\ D_{12}\\ D_{66}\\ D_{16}\\ D_{26}\end{array}\right\}=\frac{h^{3}}{12}\left[\begin{array}{ccc}U_{1}&V_{1}^{ D}&V_{2}^{D}\\ U_{1}&-V_{1}^{D}&V_{2}^{D}\\ U_{4}&0&-V_{2}^{D}\\ U_{5}&0&-V_{2}^{D}\\ 0&\frac{V_{2}^{D}}{2}&V_{4}^{D}\\ 0&\frac{V_{2}^{D}}{2}&-V_{4}^{D}\end{array}\right]\left\{\begin{array}{l}1\\ U_{2}\\ U_{3}\end{array}\right\} \tag{4}\]  
The material stiffness invariants \(U\) are calculated as follows:  
\[\left\{\begin{array}{l}U_{1}\\ U_{2}\\ U_{3}\\ U_{4}\\ U_{5}\end{array}\right\}=\frac{1}{8}\left[\begin{array}{cccc}3&2&3&4\\ 4&0&-4&0\\ 1&-2&1&-4\\ 1&6&1&-4\\ 1&-2&1&4\end{array}\right]\left\{\begin{array}{l}Q_{11}\\ Q_{12}\\ Q_{22}\\ Q_{66}\end{array}\right\} \tag{5}\]  
where \(Q_{11}=\frac{E_{11}}{1-v_{12}v_{21}}\), \(Q_{22}=\frac{E_{22}}{1-v_{12}v_{21}}\), \(Q_{21}=Q_{12}=\frac{v_{12}E_{2}}{1-v_{12}v_{21}}\), \(Q_{66}=G_{12}\), and \(v_{21}=v_{12}\frac{E_{22}}{v_{11}}\).  
The membrane, coupling, and bending lamination parameters are given, respectively, by the following integrals:  
\[V_{[1~{}2~{}3~{}4]}^{\text{A}}=\frac{1}{h}\int_{-h/2}^{h/2}\left[\cos 2\theta~{} \cos 4\theta~{}\sin 2\theta~{}\sin 4\theta\right]dz \tag{6}\]  
\[V_{[1~{}2~{}3~{}4]}^{\text{B}}=\frac{4}{h^{2}}\int_{-h/2}^{h/2}\left[\cos 2 \theta~{}\cos 4\theta~{}\sin 2\theta~{}\sin 4\theta\right]zdz \tag{7}\]  
\[V_{[1~{}2~{}3~{}4]}^{\text{D}}=\frac{12}{h^{3}}\int_{-h/2}^{h/2}\left[\cos 2 \theta~{}\cos 4\theta~{}\sin 2\theta~{}\sin 4\theta\right]z^{2}dz \tag{8}\]  
Lamination parameters cannot be arbitrarily prescribed since the trigonometric functions entering their definition are related. For a symmetric laminate, the number of independent lamination parameters can be reduced to eight. The feasible domain for membrane and bending lamination parameters for symmetric laminate is known to be convex and defined by [21; 22; 23].  
\[2(V_{1}^{X})^{2}-V_{2}^{X}-1\leq 0,\quad X=A,D \tag{9a}\] \[2(V_{1}^{X})^{2}-V_{3}^{X}-1\leq 0,\quad X=A,D\] (9b) \[2(1+V_{2}^{X})^{2}V_{3}^{X}-4V_{1}^{X}V_{3}^{X}V_{4}^{X}+{(V_{4}^{X})}^{2}-{(V_ {2}^{X}-2{(V_{1}^{X})}^{2}+1)}(1-V_{2}^{X})\leq 0,\quad X=A,D\] (9c) \[4(V_{i}^{D}-1)(V_{i}^{A}-1)\leq{(V_{i}^{A}-1)}^{4},\quad i=1,2,3\] (9d) \[4(V_{i}^{D}+1)(V_{i}^{A}+1)\leq{(V_{i}^{A}+1)}^{4},\quad i=1,2,3 \tag{9e}\]