# A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates
## 4 Elastic Constants Identification of Composite Laminate
### Strain Analysis of Symmetric Composite Laminate under Axial Load  
The symmetric laminate of size a(length) \(\times\) b(width) \(\times\) h(thickness) subjected to in-plane axial stress resultant \(N_{x}\) in the x-direction should be considered, as shown in Figure 3. The relations between the axial stress resultant and strains expressed in matrix form are [27]  
\[\left\{ \begin{array}{c}N_{x}\\ 0\\ 0\end{array} \right\}=\left[ \begin{array}{ccc}A_{xx}&A_{xy}&0\\ A_{xy}&A_{yy}&0\\ 0&0&A_{\rm ss}\end{array} \right]\left\{ \begin{array}{c}\varepsilon_{x}\\ \varepsilon_{y}\\ \gamma_{xy}\end{array} \right\} \tag{11}\]  
where \(\varepsilon_{x}\), \(\varepsilon_{y}\), and \(\gamma_{xy}\) are axial, lateral, and engineering shear strains, respectively; \(A_{xx}\), \(A_{yy}\), \(A_{xy}\), and \(A_{\rm ss}\) are in-plane axial, lateral, axial-lateral coupling, and shear stiffness coefficients, respectively. The total force in the x-direction is defined as \(F_{x}=bN_{x}\). The laminate in-plane stiffness coefficients are  
\[A_{ij}=\int_{-\frac{h}{2}}^{\frac{h}{2}}\overline{Q}_{ij}^{(m)}dz(i,j=x,y,s) \tag{12}\]  
where \(h\) is laminate thickness and \(\overline{Q}_{ij}^{(m)}(i,j=x,y,s)\) are the transformed lamina stiffness coefficients of the \(m\)th layer with fiber angle \(\theta_{m}\). For an orthotropic lamina, the lamina stiffness coefficients are expressed as [27]  
\[\underline{Q}=\left[ \begin{array}{ccc}Q_{11}&Q_{12}&0\\ Q_{12}&Q_{22}&0\\ 0&0&Q_{66}\end{array} \right] \tag{13}\]  
with  
\[ \begin{array}{c}Q_{11}=\frac{E_{1}}{1-\
u_{12}\
u_{12}},Q_{22}=\frac{E_{2}}{ 1-\
u_{12}\
u_{12}},\\ Q_{12}=Q_{21}=\frac{\
u_{21}E_{1}}{1-\
u_{12}\
u_{21}}=\frac{\
u_{12}E_{2}}{1- \
u_{12}\
u_{21}},Q_{66}=G_{12}\end{array}  \tag{14}\]  
where \(E_{1}\) and \(E_{2}\) are [PERSON]'s moduli in fiber and transverse directions, respectively; \(\
u_{ij}\) is [PERSON]'s ratio for transverse strain in the j-direction when stressed in the \(i\)-direction; and \(G_{12}\) is shear modulus in the 1-2 plane. The relations between \(\overline{Q}_{ij}\) and \(Q_{ij}\) can be found in the literature. Figure 3: Geometry and loading condition of composite laminate. The inversion of Equation (11) gives  
\[\left\{ \begin{array}{c}\varepsilon_{x}\\ \varepsilon_{y}\\ \gamma_{xy}\end{array} \right\}=\left[ \begin{array}{ccc}a_{xx}&a_{xy}&0\\ a_{xy}&a_{yy}&0\\ 0&0&a_{ss}\end{array} \right]\left\{ \begin{array}{c}N_{x}\\ 0\\ 0\end{array} \right\} \tag{15}\]  
where \(a_{ij}(i,j=x,y,s)\) are the in-plane compliance coefficients, which are functions of the elastic constants (\(E_{1}\), \(E_{2}\), \(G_{12}\), and \(\
u_{12}\)) and the layer fiber angles of the laminate. Hence, from the above equation, the axial and lateral strains can be obtained, respectively, as  
\[\varepsilon_{x}=a_{xx}N_{x} \tag{16}\]  
and  
\[\varepsilon_{y}=a_{xy}N_{x} \tag{17}\]  
It is noted that the strains (\(\varepsilon_{x}\) and \(\varepsilon_{y}\)) can be determined directly from Equations (16) and (17), provided that the elastic constants of the composite laminate are available. However, the strains of a composite laminate can generally be determined experimentally without knowing the values of the elastic constants. It is a premise that through an optimization approach, the elastic constants of a laminate may be extracted from the strains of the laminate, even though the number of strains was less than the number of elastic constants. Herein, as an example, the extraction of the elastic constants (\(E_{1}\), \(E_{2}\), \(\
u_{12}\), and \(G_{12}\)) from given strains of a symmetric [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{8}\) laminate will be achieved via an optimization approach. Traditionally, the given or measured axial and lateral strains (\(\varepsilon_{x}^{*}\) and \(\varepsilon_{y}^{*}\)), together with the theoretically predicted (\(\varepsilon_{x}\) and \(\varepsilon_{y}\)), can be used to construct the objective function to formulate the optimization based elastic constants identification problem.