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Classical Laminate Theory

Consider an orthotropic material:

Lamina stiffness given by:

$$\left[ \begin{array}{c} \sigma_1 \\ \sigma_2 \\ \sigma_{12} \\ \end{array}\right] = \left[ \begin{array}{c c c} Q_{11}&Q_{12}&0\\ Q_{12}& Q_22 & 0 \\ 0 & 0 & 2Q_{66} \\ \end{array}\right]\left[ \begin{array}{c} \varepsilon_1 \\ \varepsilon_2 \\ \frac{\varepsilon_{12}}{2} \\ \end{array}\right]$$ where:

$$Q_{11}=\frac{E_{11}}{1-\nu_{12}\nu_{21}}$$

$$Q_{22}=\frac{E_{22}}{1-\nu_{12}\nu_{21}}$$

$$Q_{12}=\frac{\nu_{21}E_{11}}{1-\nu_{12}\nu_{21}}=\frac{\nu_{12}E_{22}}{1-\nu_{12}\nu_{21}}$$

$$Q_{66}=E_{12}$$

and: $$E_{11}=$$Modulus in material's 1-direction

$$E_{11}=$$Modulus in material's 2-direction

$$E_{12}=$$Shear modulus

$$\nu_{11}=$$Poisson's effect of 2-direction on 1

$$\nu_{11}=$$Poisson's effect of 1-direction on 2

in vector form: $$\sigma_1 = Q_{11}\varepsilon_1+Q_{12}\varepsilon_2$$ Consider the transformation matrix:

$$T=\left[ \begin{array}{c c c} m^2 & n^2 & +2nm \\ n^2 & m^2 & -2nm \\ -nm &+nm & m^2 - n^2 \end{array}\right] $$ where:

$$ m = \cos \theta $$

$$ n = \sin \theta$$

Note also the inverse of T: $$T^{-1}=\left[ \begin{array}{c c c} m^2 & n^2 & -2nm \\ n^2 & m^2 & +2nm \\ +nm &-nm & m^2 - n^2 \end{array}\right] $$

Hence,

$$\vec{\sigma}_{1,2} = \mathbf{T}\vec{\sigma}_{x,y}$$

$$\vec{\varepsilon}_{1,2} = \mathbf{T}\vec{\varepsilon}_{x,y}$$

Premultiplying by $$\mathbf{T}^{-1}$$:

$$\mathbf{T}^{-1}\vec{\sigma}_{1,2} = \mathbf{T}^{-1}\mathbf{T}\vec{\sigma}_{x,y} = \vec{\sigma}_{x,y}$$

$$\mathbf{T}^{-1}\vec{\varepsilon}_{1,2} = \mathbf{T}^{-1}\mathbf{T}\vec{\varepsilon}_{x,y} = \vec{\varepsilon}_{x,y}$$

$$\therefore \vec{\sigma}_{x,y}= \mathbf{T}^{-1}\mathbf{Q}\vec{\sigma}_{1,2} = \mathbf{T}^{-1}\mathbf{Q}\vec{\varepsilon}_{1,2}$$

From above, $$\vec{\sigma}_{x,y} = \mathbf{T}^{-1}\mathbf{Q}\vec{\varepsilon}_{1,2}=\mathbf{T}^{-1}\mathbf{QT}\vec{\varepsilon}_{x,y}$$

$$\vec{\sigma}_{x,y}=\bar{\mathbf{Q}}\vec{\varepsilon}_{x,y}$$

Where $$\bar{\mathbf{Q}}$$ is the Global Stiffness matrix.

In long hand: $$\left[ \begin{array}{c} \sigma_x \\ \sigma_y \\ \sigma_{xy} \\ \end{array}\right] = \left[ \begin{array}{c c c} \bar{Q}_{11}&\bar{Q}_{12}&\bar{Q}_{16}\\ \bar{Q}_{12}& \bar{Q}_{22} & \bar{Q}_{26} \\ \bar{Q}_{16} & \bar{Q}_{26} & \bar{Q}_{66} \\ \end{array}\right]\left[ \begin{array}{c} \varepsilon_x \\ \varepsilon_y \\ \varepsilon_{xy} \\ \end{array}\right]$$

Strain in this layer given by:

$$ \varepsilon_{x,y} = \varepsilon_{x,y}^0 + z_k \kappa_{x,y}$$

where k is the curvature.