ON THE ASYMPTOTIC MEMBRANE THEORY OF THIN HYPERELASTIC PLATES

Applying the asymptotic expansion technique to the three-dimensional e quations of non-linear elasticity, a non-linear asymptotic membrane th eory considering large deflections and strains is obtained for thin hy perelastic plates. To this end, the displacement vector and stress ten sor components are scaled via an appropriate thickness parameter such that the present approximation takes into account larger deflections c ompared with those of the von Karman plate theory. Later, for an arbit rary form of the strain energy function, the hierarchy of the held equ ations is obtained by expanding the displacement vector and the stress tensor in terms of powers of the square root of the thickness paramet er. The equations belonging to the first three orders of this hierarch y are studied in detail. It is shown that the zeroth order approximati on corresponds to the well-known Foppl membrane theory, the first orde r approximation includes bending effects, and the effect of material n on-linearity appears in the second order approximation. Solving the pr oblem of an infinitely long strip under uniform load for clamped edge conditions, the effect of material non-linearity is discussed numerica lly for both compressible and incompressible hyperelastic solids. The results are also compared with the solutions of the asymptotic approxi mation which gives the von Karman plate equations in the zeroth order approximation. Copyright (C) 1997 Elsevier Science Ltd.