Nonlinear dynamics of heterogeneous shells Part 1. Statics and dynamics of heterogeneous variable stiffness shells

The increasing complexity of the constructive forms and shell elements structure leads to the need to develop both the theory and methods for solving static and dynamic problems for non-homogeneous (heterogeneous) shells. By the shell heterogeneity, we mean heterogeneity in a broad sense: these are inclusions in the shell body of the different rigidity elements and, as a special case, these are holes; the material inhomogeneity caused by a change in the stress-strain state under the influence of both static and dynamic loads; stiffeners; and by taking into account the physical nonlinearity and different modulus of the shell material. This work is devoted to the mathematical model creation of the statics and dynamics for non-homogeneous shells in the above sense and consists of two parts. In the first part, a mathematical model of statics and dynamics for rectangular shells described by the Kirchhoff-Love kinematic model is constructed. Geometric nonlinearity is taken into account on the basis of T. von Karman's geometric model, physical nonlinearity - according to the deformation theory of plasticity, based on the elasticity variable parameters method. Stiffness heterogeneity is taken into account using the Heaviside function. The original equations were obtained from Hamilton's variational principle. A numerical experiment is performed using the Faedo-Galerkin method in higher approximations. The convergence of this method is investigated. In the second part, the nonlinear dynamics of axisymmetric elastic variable thickness shells is investigated. In contrast to the first part of this work, where the Faedo-Galerkin method in higher approximations was used to reduce partial differential equations to the Cauchy problem, in this part the Ritz method in higher approximations is employed. For differential operators used in two parts of this work, according to the research of S.G. Mikhlin, the Ritz and Faedo-Galerkin methods are equivalent. The convergence of the applied numerical method is investigated. An approach is proposed for the zones study of oscillation types for variable thickness shells using dynamic modes maps.