Derivation of the Inelastic Behaviour of Plates and Shells From the Three Dimensional Models and Extensions

Summary

The elastoplastic behaviour of plates and shells is usually obtained in two different ways. First, the so-called global method assumes that the yield function can be expressed in terms of resultant in-plane stresses and bending moments. The success of such a formulation is due principally to its simplicity as a natural generalization of the elastic theory of plates and shells. Besides the well-known difficulty on an adequate expression of the adopted criterion, such a model leads to a linear variation of in-plane stresses through the thickness of the structure. If this description is widely used and seems to give entire satisfaction in limit analysis, on the contrary, in the context of elastoplastic analysis, a more refined model is sometimes necessary. The second possibility to derive the inelastic behaviour is to take account of the distribution of plastic strains along the thickness. This more complicated formulation is often preferred to the first one in computation by finite-element programs. The goal of our communication is to give a mathematical justification of the second formulation by asymptotic analysis. It consists in considering the thickness of the shell as a small parameter governing equations for plates and shells are then derived exactly from the three-dimensional description by asymptotic development with respect to the thickness. In the case of elastoplasticity, this leads to a two dimensional model with respect to the displacements of the middle surface. The plastic behaviour is characterized by multiple plastic potentials in the sense of Koiter-Mandel, with hardening parameters which are residual stresses along the thickness. The in-plane stresses are no more linear through the thickness. The derivation of the bidimensional behaviour is presented in detail in the case of standard materials for both deformation theory and incremental theory of plasticity. A comparison-between global models and the ones obtained in this paper is given for particular choice of yield criteria. A section is devoted to composite materials. It is assumed that a plasticity phenomenon can appear in the epoxy layers sticking together two layers of fibers. Then using asymptotic methods we deduce a plate model which takes into account the effect of a gliding between two layers. Finally a buckling model is suggested.