## 2 Classical Laminate Theory and Lamination Parameters
### Poisson's Ratio of the Laminate [24]  
According to Equation (1):  
\[\left\{\begin{array}{l}N_{x}\\ N_{y}\\ N_{xy}\end{array}\right\}=[A]\left\{\begin{array}{l}\varepsilon_{x}\\ \varepsilon_{y}\\ \gamma_{xy}\end{array}\right\} \tag{10}\]  
in which \(N_{x}\), \(N_{y}\), and \(N_{xy}\) denote the inner forces in the longitudinal (\(x\)) direction, transverse (\(y\)) direction, and in-plane shear force resultants per unit length (load intensities), respectively. We can express the strains as functions of the applied load intensities by:  
\[\left\{\begin{array}{l}\varepsilon_{x}\\ \varepsilon_{y}\\ \gamma_{xy}\end{array}\right\}=\left[A^{-1}\right]\left\{\begin{array}{l}N_ {x}\\ N_{y}\\ N_{xy}\end{array}\right\} \tag{11}\]where \([A^{-1}]\) is the inverse of the matrix \([A]\). For a symmetric laminate, it may be shown that the in-plane load intensities do not produce out-of-plane curvature deformations. The average direct stresses across the thickness of the laminate as:  
\[\overline{\sigma}_{x}=\frac{N_{x}}{t},\,\,\overline{\sigma}_{y}=\frac{N_{y}}{t},\,\,\overline{\tau}_{xy}=\frac{N_{xy}}{t} \tag{12}\]  
Equation (10) then becomes:  
\[\left\{\begin{array}{c}\varepsilon_{x}\\ \varepsilon_{y}\\ \gamma_{xy}\end{array}\right\}=t\Big{[}A^{-1}\Big{]}\left\{\begin{array}{c} \overline{\sigma}_{x}\\ \overline{\sigma}_{y}\\ \overline{\tau}_{xy}\end{array}\right\} \tag{13}\]  
Let the terms in \([A^{-1}]\) be as follows:  
\[\Big{[}A^{-1}\Big{]}=\left[\begin{array}{ccc}a_{11}&a_{12}&a_{16}\\ a_{12}&a_{22}&a_{26}\\ a_{16}&a_{26}&a_{66}\end{array}\right] \tag{14}\]  
Suppose, now, that a symmetric laminate is subjected to a single load intensity, \(N_{x}\). Then:  
\[\overline{\sigma}_{x}=\frac{N_{x}}{t},\,\,\overline{\sigma}_{y}=0,\,\, \overline{\tau}_{xy}=0 \tag{15}\]  
From Equations (13) and (14), [PERSON]'s modulus for the laminate in the \(x\) direction:  
\[E_{x}=\overline{\sigma}_{x}/\varepsilon_{x}=\frac{1}{ta_{11}} \tag{16}\]  
\[\varepsilon_{y}=ta_{12}\overline{\sigma}_{x}=ta_{12}\varepsilon_{x}/ta_{11}= (a_{12}/a_{11})\varepsilon_{x} \tag{17}\]  
Equation (17) gives the strain in the \(y\) direction due to a load in the \(x\) direction, that is the [PERSON]'s effect. Then, [PERSON]'s ratio in the \(xy\) plane of the laminate is given by:  
\[v_{xy}=\frac{-\varepsilon_{y}}{\varepsilon_{x}}=\frac{-a_{12}}{a_{11}}=\frac {A_{16}A_{26}-A_{12}A_{66}}{A_{22}A_{66}-A_{26}A_{26}} \tag{18}\]  
The above arguments may be applied to a symmetric laminate subjected to a single load intensity, \(N_{y}\). Then, [PERSON]'s modulus for the laminate in the \(y\) direction and [PERSON]'s ratio in the \(xy\) plane are:  
\[E_{y}=\overline{\sigma}_{y}/\varepsilon_{y}=\frac{1}{ta_{22}} \tag{19}\]  
\[v_{yx}=\frac{-a_{12}}{a_{22}}=\frac{A_{16}A_{26}-A_{12}A_{66}}{A_{11}A_{66}-A_ {16}A_{16}} \tag{20}\]  
According to Equations (2), (18) and (20), [PERSON]'s ratios of \(v_{xy}\) and \(v_{yx}\) are not dependent on the stacking sequences of laminated composites and just functions of membrane stiffness matrix elements or membrane lamination parameters. [PERSON]'s ratio envelope can be studied by using membrane lamination parameters as design variables.