On the existence of solutions to geometrically nonlinear problems for shallow Timoshenko-type shells with free edges (Q465017)

The basic result of the article consists in the finding of necessary and sufficient conditions for solvability of the geometrically nonlinear equilibrium problem for shallow elastic Timoshenko-type shells with free edges. When these conditions are fulfilled the problem has the generalized solution \(a=(w_1,w_1,w_3,\psi_1,\psi_2)\in W^2_p(\Omega), 2<p<2/(1-\beta)\) up to the rigid body displacements \(a_*\) of the shell, where \(a\) is the vector of the generalized displacements, \(w\) is the vector of tangential and normal displacements of the shell middle surface, \(\psi_1,\psi_2\) are the rotation angles of its normal cross-section and \(0<\beta<1\). Additional conditions for the unique solvability of the problem are given. The author has marked that now the solvability of nonlinear problems for thin elastic shells has been studied with enough completeness only for the Kirchhoff-Love model while the results of solvability for more general models are scanty.