Question

# For orthotropic materials, the components of stiffness matrix Cij are related to the components of the compliance matrix Sij as follows: C_{11}=\frac{S_{22} S_{33}-S_{23}^{2}}{S} \quad C_{12}=\frac{S_{13} S_{23}-S_{12} S_{33}}{S} \quad C_{13}=\frac{S_{12} S_{23}-S_{13}

S_{22}}{S} \mathrm{C}_{22}=\frac{\mathrm{S}_{33} \mathrm{~S}_{11}-\mathrm{S}_{13}^{2}}{\mathrm{~S}} \quad \mathrm{C}_{23}=\frac{\mathrm{S}_{12} \mathrm{~S}_{13}-\mathrm{S}_{23} \mathrm{~S}_{11}}{\mathrm{~S}} \quad \mathrm{C}_{33}=\frac{\mathrm{S}_{11} \mathrm{~S}_{22}-\mathrm{S}_{12}^{2}}{\mathrm{~S}} \mathrm{C}_{44}=\frac{1}{\mathrm{~S}_{44}} \quad \mathrm{C}_{55}=\frac{1}{\mathrm{~S}_{55}} \quad \mathrm{C}_{66}=\frac{1}{\mathrm{~S}_{66}} where \mathrm{S}=\mathrm{S}_{11} \mathrm{~S}_{22} \mathrm{~S}_{33}-\mathrm{S}_{11} \mathrm{~S}_{23}^{2}-\mathrm{S}_{22} \mathrm{~S}_{13}^{2}-\mathrm{S}_{33} \mathrm{~S}_{12}^{2}+2 \mathrm{~S}_{12} \mathrm{~S}_{23} \mathrm{~S}_{13} Show (derive) that the components of stiffness matrix can also be represented in the following form using engineering constants E and v.

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