Geometrical super-elements for elasto-plastic shells with large deformation

This paper presents the application of the so-called Geometrical Elements Method (Lukasiewicz and Szyszkowski, 1974; Pogorelov, 1967) to the solution of elasto-plastic problems of shells. The approach is based on the observation that, during large deformations, the shell structure deforms in a nearly isometrical manner. Therefore, its deformed shape can be determined and analysed making use of the Gauss theorem according to which the Gaussian curvature of the isometrically deformed surface remains unchanged. The shell structure is subdivided into elements of two kinds: purely-isometrically deformed elements and quasi-isometrically deformed elements. The equilibrium of the whole structure is defined by the stationary value of the Hamiltonian function which requires the calculation of the strain energy in the elements. This can easily be obtained if we recognize that the isometrically deformed elements contain only bending energy. Using the method described, we are able to significantly the number of unknown values defining the shape of the deformed structure. The problem is reduced to the numerical evaluation of the minimum of a function of many variables. The elasto-plastic state of stress of the plastic material in the structure canbe determined by using the deformation theory of plasticity or the theory of plastic flow. Also, the strains and stresses in the plastic regions are the only functions of the assumed displacements field. The corresponding energy of the plastic deformation can easily be evaluated and added to the minimized functionals. For example, the elasto-plastic behaviour of a spherical shell under a concentrated load is studied. The solution obtained defines the large deformation behaviour and the motion of the plastic zones on the surface of the shell.