Structural Domain

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Structural Domain

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

$\displaystyle \Pi_s$	=	$\displaystyle \int_{t_1}^{t_2} \left[ \frac{1}{2} \int_{\Omega_s} \mbox{\boldma... ...rac{1}{2} \int_{\Omega_s}\rho_s {\bf\dot{u}}^T {\bf\dot{u}}\; d\Omega_s \right.$
	-	$\displaystyle \left. \int_{\Gamma_w} {\bf u}^T {\bf f}^I \; d\Gamma_w - \int_{\Omega_s} {\bf u}^T {\bf f}^E \;d\Omega_s \right] dt$	(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
2000-09-06