Formal derivation of equations of a nonlinear hydroelastic structure, which is a volume of an ideal incompressible fluid covered by a shell, is proposed. The study is based on two assumptions. The first assumption implies that the energy stored in the shell is completely determined by the mean curvature and by the elementary area. In a three-dimensional case, the energy stored in the shell is chosen in the form of the Willmore functional. In a two-dimensional case, a more generic form of the functional can be considered. The second assumption implies that the equations of motionhave a Hamiltonian structure and can be obtained from the Lagrangian variational principle. In a two-dimensional case, a condition for the hydroelastic structure is derived, which relates the external pressure and the curvature of the elastic shell.
G. Friesecke, R. James, and S. Muller, “A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity,” Comm. Pure Appl. Math., 35, No. 11, 1461–1506 (2002).CrossRefMathSciNetGoogle Scholar
T. Willmore, Total Curvature in Riemannian Geometry, John Wiley and Sons, New York (1982).MATHGoogle Scholar
I. Ivanova-Karatopraklieva, P. E. Markov, and I. Kh. Sabitov, “Bending of surfaces. III,” Fundam. Prikl. Mat., 12, 3–56 (2006).Google Scholar