Formulation
Consider a shell that represents a thin domain \(\Omega\):
\[\Omega \triangleq \left\{\left(x_1, x_2, z\right) \in \mathbb{R}^3 \text { such that } z \in\left[-\frac{h}{2}, \frac{h}{2}\right] \text { and }\left(x_1, x_2\right) \in A \subset \mathbb{R}^2\right\} .\]
The linearized three dimensional displacement field \(u_i\) has the representation:
\[\begin{split}\left.\begin{array}{rl}
u_1 & \triangleq -z \theta_1\left(x_1, x_2\right)+\bar{u}_1\left(x_1, x_2\right) \\
u_2 & \triangleq -z \theta_2\left(x_1, x_2\right)+\bar{u}_2\left(x_1, x_2\right) \\
u_3 & \triangleq \bar{u}_3\left(x_1, x_2\right)
\end{array}\right\}\end{split}\]
where \(\bar{u}_i\) is the translation of the plate mid-surface and
\(\theta_i\) are rotations of fibers initially normal to the
mid-surface of the plate.
Strain
A ShellFiber section takes the shell strains \(\bar{\epsilon}_{\alpha \beta}\), \(\kappa_{\alpha \beta}\), and \(\gamma_{\alpha}\),
and passes a strain tensor to material points with the form:
\[\epsilon_{\alpha \beta}=\frac{1}{2}\left(u_{\alpha, \beta}+u_{\beta, \alpha}\right)= \bar{\epsilon}_{\alpha \beta} - z \kappa_{\alpha \beta}\]
and
\[\epsilon_{\alpha 3}=\epsilon_{3 \alpha}=\frac{1}{2}\left(u_{\alpha, 3}+u_{3, \alpha}\right)=\frac{1}{2} \gamma_\alpha .\]
Stress
Definte the membrane stress resultant as
\[p_{\alpha \beta}\triangleq \int_{-\frac{h}{2}}^{\frac{h}{2}} \sigma_{\alpha \beta} \mathrm{d} z\]
Define the moment tensor as
\[m_{\alpha \beta}\triangleq \int_{-\frac{\hbar}{2}}^{\frac{h}{2}} z \sigma_{\alpha \beta} \mathrm{d} z\]
Define the transverse shear forces as
\[q_\alpha\triangleq \int_{-\frac{h}{2}}^{\frac{h}{2}} \sigma_{\alpha 3} \mathrm{~d} z\]
Material
The material model is constrained by the plane stress hypothesis: \(\sigma_{33}=0\).
\[\sigma_{i j}=\mathfrak{C}_{i j k l} \cdot \epsilon_{k l}\]
where \(\mathfrak{C}\) is the symmetric (major and minor) rank-four
elasticity tensor. Enforcement of the plane stress condition
\(\sigma_{33}=0\) yields a condensed elasticity tensor
\(\mathbb{C}\) such that
\[\begin{split}\left.\begin{array}{rl}
\sigma_{i j} & =\mathbb{C}_{i j k l} \cdot \epsilon_{k l} \\
\mathbb{C}_{i j k l} & =\mathfrak{C}_{i j k l}-\mathfrak{C}_{i j 33}\left(\mathfrak{C}_{3333}\right)^{-1} \mathfrak{C}_{33 k l}
\end{array}\right\}\end{split}\]
The modified tensor \(\mathbb{C}\) is now appropriate for plate
analysis.
Integration through the thickness yields the stress resultant
constitutive response parameters.
\[\begin{split}\begin{aligned}
p_{\alpha \beta} & =\int_{-\frac{h}{2}}^{\frac{h}{2}} \sigma_{\alpha \beta} \mathrm{d} z \\
& =\int_{-\frac{h}{2}}^{\frac{h}{2}} \mathbb{C}_{\alpha \beta k l} \cdot \epsilon_{k l} \mathrm{~d} z \\
& =\int_{-\frac{h}{2}}^{\frac{h}{2}}\left[\mathbb{C}_{\alpha \beta \delta \gamma} \cdot \epsilon_{\delta \gamma}+\mathbb{C}_{\alpha \beta \delta 3} \cdot 2 \epsilon_{\delta 3}\right] \mathrm{d} z \\
& =\int_{-\frac{h}{2}}^{\frac{h}{2}}\left[\mathbb{C}_{\alpha \beta \delta \gamma} \cdot\left(-z \kappa_{\delta \gamma}+\bar{\epsilon}_{\delta \gamma}\right)+\mathbb{C}_{\alpha \beta \delta 3} \cdot 2 \epsilon_{\delta 3}\right] \mathrm{d} z \\
& =\left[\int_{-\frac{h}{2}}^{\frac{h}{2}}-z \mathbb{C}_{\alpha \beta \delta \gamma} \mathrm{d} z\right] \cdot \kappa_{\delta \gamma}+\left[\int_{-\frac{h}{2}}^{\frac{h}{2}} \mathbb{C}_{\alpha \beta \delta \gamma} \mathrm{d} z\right] \cdot \bar{\epsilon}_{\delta \gamma}+\left[\int_{-\frac{h}{2}}^{\frac{h}{2}} \mathbb{C}_{\alpha \beta \delta 3} \mathrm{~d} z\right] \cdot \gamma_\delta
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
m_{\alpha \beta} & =\int_{-\frac{h}{2}}^{\frac{h}{2}} z \sigma_{\alpha \beta} \mathrm{d} z \\
& =\int_{-\frac{h}{2}}^{\frac{h}{2}} z \mathbb{C}_{\alpha \beta k l} \cdot \epsilon_{k l} \mathrm{~d} z \\
& =\int_{-\frac{h}{2}}^{\frac{h}{2}} z\left[\mathbb{C}_{\alpha \beta \delta \gamma} \cdot \epsilon_{\delta \gamma}+\mathbb{C}_{\alpha \beta \delta 3} \cdot 2 \epsilon_{\delta 3}\right] \mathrm{d} z \\
& =\int_{-\frac{h}{2}}^{\frac{h}{2}} z\left[\mathbb{C}_{\alpha \beta \delta \gamma} \cdot\left(-z \kappa_{\delta \gamma}+\bar{\epsilon}_{\delta \gamma}\right)+\mathbb{C}_{\alpha \beta \delta 3} \cdot 2 \epsilon_{\delta 3}\right] \mathrm{d} z \\
& =\left[\int_{-\frac{h}{2}}^{\frac{h}{2}}-z^2 \mathbb{C}_{\alpha \beta \delta \gamma} \mathrm{d} z\right] \cdot \kappa_{\delta \gamma}+\left[\int_{-\frac{h}{2}}^{\frac{h}{2}} z \mathbb{C}_{\alpha \beta \delta \gamma} \mathrm{d} z\right] \cdot \bar{\epsilon}_{\delta \gamma}+\left[\int_{-\frac{h}{2}}^{\frac{h}{2}} z \mathbb{C}_{\alpha \beta \delta 3} \mathrm{~d} z\right] \cdot \gamma_\delta
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
q_\alpha & =\int_{-\frac{h}{2}}^{\frac{h}{2}} \sigma_{\alpha 3} \mathrm{~d} z \\
& =\int_{-\frac{h}{2}}^{\frac{h}{2}} \mathbb{C}_{\alpha 3 k l} \cdot \epsilon_{k l} \mathrm{~d} z \\
& =\int_{-\frac{h}{2}}^{\frac{h}{2}}\left[\mathbb{C}_{\alpha 3 \delta \gamma} \cdot \epsilon_{\delta \gamma}+\mathbb{C}_{\alpha 3 \delta 3} \cdot 2 \epsilon_{\delta 3}\right] \mathrm{d} z \\
& =\int_{-\frac{h}{2}}^{\frac{h}{2}}\left[\mathbb{C}_{\alpha 3 \delta \gamma} \cdot\left(-z \kappa_{\delta \gamma}+\bar{\epsilon}_{\delta \gamma}\right)+\mathbb{C}_{\alpha 3 \delta 3} \cdot 2 \epsilon_{\delta 3}\right] \mathrm{d} z \\
& =\left[\int_{-\frac{h}{2}}^{\frac{h}{2}}-z \mathbb{C}_{\alpha 3 \delta \gamma} \mathrm{~d} z\right] \cdot \kappa_{\delta \gamma}+\left[\int_{-\frac{h}{2}}^{\frac{h}{2}} \mathbb{C}_{\alpha 3 \delta \gamma} \mathrm{~d} z\right] \cdot \bar{\epsilon}_{\delta \gamma}+\left[\int_{-\frac{h}{2}}^{\frac{h}{2}} \mathbb{C}_{\alpha 3 \delta 3} \mathrm{~d} z\right] \cdot \gamma_\delta
\end{aligned}\end{split}\]