Constitutive Relationship in a Composite Lamina

The constitutive relationship in a composite lamina is specially orthotropic in nature and is given by the following equation:

Where,

Q₁₁ = E₁ /(1- ? ₁₂ ? ₂₁ ),

Q₁₂ = ? ₁₂ E₂ /(1- ? ₁₂ ? ₂₁ ),

Q₂₂ = E₂ /(1- ?₁₂ ?₂₁ ),

Q₆₆ = G₁₂

E₁ for a composite is obtained by using rule of mix and fibre & matrix volume fraction as follows:

E₁ = E_1f V_f + E_1m V_m

However in the transverse direction, estimating the modulus of elasticity is more difficult. A simple model based on springs in series (matrix and reinforcing elements both modelled as springs connected in series). Thus,
1/E₂ = V_f /E_2f + V_m /E_2m
The same principle is used to estimate the shear modulus of the lamina.