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Structural Domain

Following Hamilton's principle, the energy function for the structural domain may be written as

$\displaystyle \Pi_s$=$\displaystyle \int_{t_1}^{t_2} \left[ \frac{1}{2} \int_{\Omega_s}
...rac{1}{2} \int_{\Omega_s}\rho_s {\bf\dot{u}}^T {\bf\dot{u}}\; d\Omega_s
 -$\displaystyle \left. \int_{\Gamma_w} {\bf u}^T {\bf f}^I \; d\Gamma_w
- \int_{\Omega_s} {\bf u}^T {\bf f}^E \;d\Omega_s
\right] dt$(126)

where ${\bf E}$ is the stress-strain matrix, $\epsilon$ is the strain vector, ${\bf u}$ is the displacement vector, $\Omega_s$ is the structural domain, $\Gamma_w$ is the wet surface of the structure, ${\bf f}^I$ is the liquid pressure vector, ${\bf f}^E$ is the body force vector and $\rho_s$ is the mass density of the structure. Setting variations on $\Pi_s$ to zero gives the principle of virtual displacements
$\displaystyle \delta \Pi_s = \int_{\Omega_s} \delta \mbox{\boldmath$\epsilon$ }^T {\bf E} \;
\mbox{\boldmath$\epsilon$ } \; d\Omega_s$+$\displaystyle \int_{\Omega_s}\rho_s \delta {\bf u}^T { \bf\ddot{u} } \; d\Omega_s
- \int_{\Gamma_w} \delta {\bf u}^T {\bf f}^I \; d\Gamma_w$ 
 -$\displaystyle \int_{\Omega_s} \delta {\bf u}^T {\bf f}^E \; d\Omega_s =0$(127)

A. Zeiny