3 KiB
To Composite laminate models there are a few options, the most important are the Equivalent Single Layer Theories 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. (Secondary articles descriptions#First order Zig-Zag plate Theory)
Stiffness
(Videos descriptions#Composite materials course)
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}
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}}}