Modes of loss of stability and critical loads of a three-layer spherical shell under a uniform external pressure

Problems of the modes of loss of stability of a three-layer spherical shell, consisting of thin external layers and a transversely soft filler of arbitrary thickness, which is under conditions of a uniform external pressure, are considered. The two-dimensional equations of the Kirchhoff-Love theory of the moderate flexure of thin shells are used. These equations are set up for the external layers, taking account of the interaction with the filler and, in the case of the filler, using the geometrically non-linear equations of the theory of elasticity, which correspond to the introduction of the assumption that the stretching deformations are small and the shear deformation are finites, which enables the purely shear modes of loss of stability in the filler to be described correctly. An exact analytical solution is found for the problem of an initial centro-symmetric deformation of a shell, which depends linearly on the external pressure. It is shown that the three-dimensional equations for the filler, which have been linearized in the neighbourhood of this solution, can be integrated with respect to the radial coordinate, and reduce to two two-dimensional differential equations, in addition to the six equations by which the neutral equilibrium of the external layers is described. It is established that the system of eight differential equations of stability, constructed for a shell with isotropic layers, when new unknowns in the form of scalar and vortex potentials are introduced, decomposes into two unconnected systems of equations. The first of these systems has two forms of solutions by which the shear modes of loss of stability are described for the same value of the critical load. A mixed flexural mode, the realization of which is possible for certain combinations of the governing parameters of the shell for high values of the external pressure compared with the shear modes, is described by the second system.