MSR/ressources/Layerwise Theories.md

To [[Composite laminate models|model a composite laminate]] there are a few options, the most important are the [[Equivalent Single Layer Theories|equivalent single layer]] and the layerwise.

Those theories describe kinetically each layer. They can be used for very thick laminate, and useful to compute delamination. They predicts correct inter-laminar stresses and there is no need of shear correction factor.

There is a unique displacement field per layer + interlaminar continuity of displacements (and sometimes of transverse stresses).
-> very computationally expensive, since the number of degrees of freedom increase proportionally with the number of layers.
([[Context articles descriptions#First order Zig-Zag plate Theory|source]])

## Stiffness
([[Videos descriptions#Composite materials course|video source]])

When pulling on the material, strain is the same for every ply, but stress is not.

You need to average it along the whole thickness
$$

\left[\begin{array}{c}
\overline{\sigma_{xx}} \\
\overline{\sigma_{yy}} \\
\overline{\tau_{xy}} \\
\end{array}\right]

 = \frac{1}{h}\int_{-h/2}^{h/2} \left[\begin{array}{c}
\sigma_{xx} \\
\sigma_{yy} \\
\tau_{xy} \\
\end{array}\right] dz \tag{1}$$

$Q$ is the stiffness matrix : 
$$ \left[\begin{array}{c} 
\sigma_{xx} \\ 
\sigma_{yy} \\ 
\tau_{xy} 
\end{array}\right] 
=
\left[\begin{array}{c} 
Q_{11} & Q_{12} & Q_{13} \\
Q_{21} & Q_{22} & Q_{23} \\
Q_{31} & Q_{32} & Q_{33} \\
\end{array}\right]
\left[\begin{array}{c} 
\varepsilon_{xx} \\ 
\varepsilon_{yy} \\ 
\gamma_{xy} 
\end{array}\right] \tag{2}$$

Since the strain is the same for every ply, (1) & (2) gives us :
$$

\left[\begin{array}{c}
\overline{\sigma_{xx}} \\
\overline{\sigma_{yy}} \\
\overline{\tau_{xy}} \\
\end{array}\right]

 = \frac{1}{h}\int_{-h/2}^{h/2} 
 \left[\begin{array}{c} 
Q_{11} & Q_{12} & Q_{13} \\
Q_{21} & Q_{22} & Q_{23} \\
Q_{31} & Q_{32} & Q_{33} \\
\end{array}\right]dz 

\left[\begin{array}{c} 
\varepsilon_{xx} \\ 
\varepsilon_{yy} \\ 
\gamma_{xy} 
\end{array}\right]

\tag{3}$$

and we can give a name to the mean matrix of the stiffness :
$$
\left[\begin{array}{c}
A \\
\end{array}\right]

 = \frac{1}{h}\int_{-h/2}^{h/2} 
 \left[\begin{array}{c} 
Q_{11} & Q_{12} & Q_{13} \\
Q_{21} & Q_{22} & Q_{23} \\
Q_{31} & Q_{32} & Q_{33} \\
\end{array}\right]dz 

\tag{4}$$

or when the ply are continuous :
$$
\left[\begin{array}{c}
A \\
\end{array}\right]

 = \frac{1}{h}\sum_{1}^{n\_ply} 
 \left[\begin{array}{c} 
Q
\end{array}\right]_i *h_i 

\tag{5}$$

to rotate the stiffness matrix :
$$[\overline{Q}]_\theta

 = [C]_{|\theta}^T
 \left[\begin{array}{c} 
Q_{11} & Q_{12} & 0 \\
Q_{21} & Q_{22} & 0 \\
0 & 0 & Q_{33} \\
\end{array}\right]
[C]_{|\theta}

\tag{4}$$

where $[C]$ is the transformation matrix :
$$[C]_{|\theta} = \left[\begin{array}{c} 
c^2 & s^2 & cs \\
s^2 & c^2 & -cs \\
-2sc & 2sc & (c^2-s^2)
\end{array}\right]$$

## Tsai-Hill criterion

$$
F.O.S = \frac{T_x}{\sqrt{\sigma_{xx}^2 - \sigma_{xx}\sigma_{yy} + \frac{T_x^2\sigma_{yy}^2}{T_y^2} + \frac{T_x^2\tau_{xy}^2}{S_{xy}^2}}}$$