DEFORMATION OF COMPOSITE SHELLS DILATING MATERIALS BEYOND ELASTIC LIMITS

A mathematical model is proposed for the deformation of a thin flat shell of positive Gaussian curvature with a rectangular contour loaded with a transverse load. The model takes into account the deformation of the shell both at the elastic stage of the material’s operation and at the plastic stage. The model is developed for shells made of initially isotropic composite and polymer materials, for which physical linearity corresponding to the generalized Hooke’s law is valid in the elastic stage of deformation. During the transition to the plastic region of deformation for these materials, the manifestation of dilatation and the dependence of the yield strength on the type of stress state are taken into account. The model describes all stages of deformation of the elastic shell, the stage of the initial appearance of plastic zones up to the formation of plastic hinges. The model is applicable to dilating materials whose plastic properties can be interpreted as ideal without hardening. In this regard, two plasticity conditions, specially developed for isotropic composites, are considered. The first of them was proposed in the works of E.V. Lomakin, and the second - in the publications of the author of the presented study. At the same time, the disadvantages of the first version of the condition are noted. In addition, the developed model can naturally function with the classical von Mises plasticity condition. Therefore, for quantitative and qualitative comparison of the results obtained, calculations were carried out using three variants of plasticity conditions. To construct the model, technical hypotheses traditional for thin shells were used within the framework of geometrically nonlinear representations of the Karman deformation components. Nonlinear resolving differential equations for the bending of shallow shells were obtained, which were processed by V.V. Petrov’s two-step method of successive perturbations of parameters with subsequent numerical implementation. For specific calculations, a shell with a square plan, hinged along the entire contour and loaded with a distributed transverse load of constant intensity, was adopted. As a result of applying the developed mathematical model, the dependences of the calculated maximum deflections of a shell made of polymethyl methacrylate on the load intensity are presented. The fields of propagation of plasticity over the surfaces and through the thickness of the shell are constructed at individual load values. In addition, the values of the loads at which plasticity first develops on the surfaces of the shell and the ultimate loads corresponding to the formation of plastic hinges are given in tabular form. It is shown that when dilating materials go beyond the elastic limits, traditional theories of plasticity lead to significant errors in determining the states and ultimate loads for shallow shells. Considering the obvious advantages for the calculation of such structures, it is advisable to apply the plasticity conditions proposed by the author.