Geometrically non-linear equations in the theory of momentless shells with applications to problems on the non-classical forms of loss of stability of a cylinder

Geometrically non-linear and linearized equations in the theory of momentless shells are set up based on the kinematic relations in [Paimushin VN, Shalashilin VI. Relations of the theory of deformations in the quadratic approximation and problems of constructing refined versions of the geometrically non-linear theory of laminar structural components. Prikl Mat Mekh 2005; 69(5): 861–81]. The use of these equations, unlike in the case of the well-known equations, enables one to avoid the occurrence of spurious bifurcation points in solving real problems. Non-classical problems of the stability of cylindrical shells under an external pressure, axial compression and torsion are considered, which can be formulated on the basis of the derived equations of the theory of momentless shells. Their exact analytical solutions are found and enable one to estimate the quality of the previously obtained relations [Paimushin VN, Shalashilin VI. Relations of the theory of deformations in the quadratic approximation and problems of constructing refined versions of the geometrically non-linear theory of laminar structural components. Prikl Mat Mekh 2005; 69(5): 861–81] and the richness of content of the equations which have been constructed compared with well-known equations in the mechanics of thin shells. It is established that the majority of the new forms of loss of stability of cylindrical shells which are revealed relate to a number of shear forms, the onset of which is possible before the flexural forms which have been well studied up to now, in the case of small values of the shear modulus of a shell material with a very highly pronounced anisotropy in its properties.