123 lines
3 KiB
Markdown
123 lines
3 KiB
Markdown
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.
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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.
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There is a unique displacement field per layer + interlaminar continuity of displacements (and sometimes of transverse stresses).
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-> very computationally expensive, since the number of degrees of freedom increase proportionally with the number of layers.
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([[Secondary articles descriptions#First order Zig-Zag plate Theory|source]])
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## Stiffness
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([[Videos descriptions#Composite materials course|video source]])
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When pulling on the material, strain is the same for every ply, but stress is not.
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You need to average it along the whole thickness
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$$
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\left[\begin{array}{c}
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\overline{\sigma_{xx}} \\
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\overline{\sigma_{yy}} \\
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\overline{\tau_{xy}} \\
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\end{array}\right]
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= \frac{1}{h}\int_{-h/2}^{h/2} \left[\begin{array}{c}
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\sigma_{xx} \\
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\sigma_{yy} \\
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\tau_{xy} \\
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\end{array}\right] dz \tag{1}$$
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$Q$ is the stiffness matrix :
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$$ \left[\begin{array}{c}
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\sigma_{xx} \\
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\sigma_{yy} \\
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\tau_{xy}
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\end{array}\right]
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=
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\left[\begin{array}{c}
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Q_{11} & Q_{12} & Q_{13} \\
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Q_{21} & Q_{22} & Q_{23} \\
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Q_{31} & Q_{32} & Q_{33} \\
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\end{array}\right]
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\left[\begin{array}{c}
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\varepsilon_{xx} \\
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\varepsilon_{yy} \\
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\gamma_{xy}
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\end{array}\right] \tag{2}$$
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Since the strain is the same for every ply, (1) & (2) gives us :
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$$
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\left[\begin{array}{c}
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\overline{\sigma_{xx}} \\
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\overline{\sigma_{yy}} \\
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\overline{\tau_{xy}} \\
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\end{array}\right]
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= \frac{1}{h}\int_{-h/2}^{h/2}
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\left[\begin{array}{c}
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Q_{11} & Q_{12} & Q_{13} \\
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Q_{21} & Q_{22} & Q_{23} \\
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Q_{31} & Q_{32} & Q_{33} \\
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\end{array}\right]dz
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\left[\begin{array}{c}
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\varepsilon_{xx} \\
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\varepsilon_{yy} \\
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\gamma_{xy}
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\end{array}\right]
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\tag{3}$$
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and we can give a name to the mean matrix of the stiffness :
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$$
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\left[\begin{array}{c}
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A \\
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\end{array}\right]
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= \frac{1}{h}\int_{-h/2}^{h/2}
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\left[\begin{array}{c}
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Q_{11} & Q_{12} & Q_{13} \\
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Q_{21} & Q_{22} & Q_{23} \\
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Q_{31} & Q_{32} & Q_{33} \\
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\end{array}\right]dz
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\tag{4}$$
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or when the ply are continuous :
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$$
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\left[\begin{array}{c}
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A \\
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\end{array}\right]
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= \frac{1}{h}\sum_{1}^{n\_ply}
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\left[\begin{array}{c}
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Q
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\end{array}\right]_i *h_i
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\tag{5}$$
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to rotate the stiffness matrix :
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$$[\overline{Q}]_\theta
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= [C]_{|\theta}^T
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\left[\begin{array}{c}
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Q_{11} & Q_{12} & 0 \\
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Q_{21} & Q_{22} & 0 \\
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0 & 0 & Q_{33} \\
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\end{array}\right]
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[C]_{|\theta}
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\tag{4}$$
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where $[C]$ is the transformation matrix :
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$$[C]_{|\theta} = \left[\begin{array}{c}
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c^2 & s^2 & cs \\
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s^2 & c^2 & -cs \\
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-2sc & 2sc & (c^2-s^2)
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\end{array}\right]$$
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## Tsai-Hill criterion
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$$
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F.O.S = \frac{T_x}{\sqrt{\sigma_{xx}^2 - \sigma_{xx}\sigma_{yy} + \frac{T_x^2\sigma_{yy}^2}{T_y^2} + \frac{T_x^2\tau_{xy}^2}{S_{xy}^2}}}$$
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