# A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates
## 5 Experimental Study  
The proposed method was then applied to the elastic constants of a symmetric graphite/epoxy [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\mathrm{s}}\) laminate using three measured strains of the laminate. First, the material constants of the graphite/epoxy lamina were determined using three types of the standard specimens ([0\({}^{\circ}\)]\({}_{\mathrm{d}}\), [90\({}^{\circ}\)]\({}_{\mathrm{s}}\), and [\(\pm\)45\({}^{\circ}\)]\({}_{\mathrm{s}}\)), in accordance with the ASTM standards of D3039 and D3518 [28]. The average values and coefficients of variation (indicated in the parentheses) of the material constants are given as follows:  
\[ \begin{array}{l}E_{1}=138.5\ GPa\ (0.96\%)\\ E_{2}=8.1\ GPa\ (1.64\%)\\ G_{12}=5.2\ GPa\ (0.38\%)\\ v_{12}=0.31(1.85\%)\\ \end{array}  \tag{21}\]  
In view of the small coefficients of variation (less than 2%), the average values in the above equation were then treated as the \"actual\" elastic constants of the composite material. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Actual Elastic Constant**} \\ \cline{3-6}  & & \(E_{1}\) (GPa) & \(E_{2}\) (GPa) & \(\
u_{12}\) & \(G_{12}\) (GPa) \\ \hline \multirow{3}{*}{Material type} & A. Carbon/epoxy & 146.50 & 9.22 & 0.30 & 6.84 \\  & B. Scotchly® 1002 E-glass/epoxy & 38.60 & 8.27 & 0.26 & 4.14 \\ \hline \multicolumn{6}{c}{Given Strain} \\ \cline{3-6}  & & \(\varepsilon_{x}\) (\(\mu\varepsilon\)) & \(\varepsilon_{y}\) (\(\mu\varepsilon\)) & \(\varepsilon_{45}\) (\(\mu\varepsilon\)) \\ \hline \([0^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\mathrm{s}}\) Laminate & A. Carbon/epoxy & 210.6170 & \(-\)149.9256 & 28.5322 \\ (F\({}_{\mathrm{x}}\) = 0.5 kN) & B. Scotchly® 1002 E-glass/epoxy & 579.4597 & \(-\)269.9758 & 154.7419 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Properties of different composite materials used for elastic constants identification. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material Type**} & & \multicolumn{4}{c}{**Material Constant**} \\ \cline{3-6}  & & \(E_{1}\) (GPa) & \(E_{2}\) (GPa) & \(\
u_{12}\) & \(G_{12}\) (GPa) \\ \hline \multirow{3}{*}{A. Carbon/epoxy} & Actual & 146.50 & 9.22 & 0.30 & 6.84 \\  & Level & & & & \\ \cline{2-6}  & \multirow{2}{*}{1 st} & Mean (Identified) & 146.49 & 9.507 & 0.293 & 6.818 \\  & Standard deviation & 0.89 & 0.67 & 0.07 & 0.22 \\ \cline{2-6}  & \multirow{2}{*}{2 nd} & Mean (Identified) & 146.500 (0\%) & 9.228 (0.09\%) & 0.300 (0\%) & 6.838 (\(-\)0.03\%) \\  & Standard deviation & 0.18 & 0.06 & 0.01 & 0.05 \\ \hline \multirow{3}{*}{B. Scotchly® 1002 E-glass/epoxy} & Actual & 38.60 & 8.27 & 0.26 & 4.14 \\  & Level & & & & \\ \cline{2-6}  & \multirow{2}{*}{1 st} & Mean (Identified) & 38.42 & 7.993 & 0.271 & 4.196 \\  & Standard deviation & 1.37 & 2.14 & 0.07 & 0.26 \\ \cline{2-6}  & \multirow{2}{*}{2 nd} & Mean (Identified) & 38.60 (0\%) & 8.161 (\(-\)1.3\%) & 0.264 (1.5\%) & 4.147 (0.17\%) \\  & Standard deviation & 0.21 & 0.60 & 0.02 & 0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Solution of optimization for [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\mathrm{s}}\) laminate. The dimensions of the symmetric graphite/epoxy [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\mathrm{s}}\) laminate are shown in Figure 4. The average thickness of each lamina in the laminate is 0.269 mm. The graphite/epoxy [0\({}^{\circ}\)/\(\pm\)45\({}^{\circ}\)]\({}_{\mathrm{s}}\) laminates were subjected to uniaxial tensile testing, in which a tri-axial strain gauge produced by TML, Japan (TML FRA-6-11-1L, gage length 6 mm and gauge factor \(2.11\pm 1\%\)) were used to measure the surface axial, lateral, and 45\({}^{\circ}\) strains (\(\varepsilon_{x}^{*}\), \(\varepsilon_{y}^{*}\), and \(\varepsilon_{45}^{*}\)) at the mid-span of the laminate. The uniaxial tensile test of the laminate was performed using the MTS 810, with test speeds less than 0.01 mm/s.