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Classical Laminate Theory

Consider an orthotropic material:

Lamina stiffness given by:
$\left[{\begin{array}{c}\sigma _{1}\\\sigma _{2}\\\sigma _{12}\\\end{array}}\right]=\left[{\begin{array}{c c c}Q_{11}&Q_{12}&0\\Q_{12}&Q_{2}2&0\\0&0&2Q_{66}\\\end{array}}\right]\left[{\begin{array}{c}\varepsilon _{1}\\\varepsilon _{2}\\{\frac {\varepsilon _{12}}{2}}\\\end{array}}\right]$ 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.