On the asymptotic membrane theory of thin hyperelastic plates

Abstract

Applying the asymptotic expansion technique to the three-dimensional equations of non-linear elasticity, a non-linear asymptotic membrane theory considering large deflections and strains is obtained for thin hyperelastic plates. To this end, the displacement vector and stress tensor components are scaled via an appropriate thickness parameter such that the present approximation takes into account larger deflections compared with those of the von Kárman plate theory. Later, for an arbitrary form of the strain energy function, the hierarchy of the field equations is obtained by expanding the displacement vector and the stress tensor in terms of powers of the square root of the thickness parameter. The equations belonging to the first three orders of this hierarchy are studied in detail. It is shown that the zeroth order approximation corresponds to the well-known Föppl membrane theory, the first order approximation includes bending effects, and the effect of material non-linearity appears in the second order approximation. Solving the problem of an infinitely long strip under uniform load for clamped edge conditions, the effect of material non-linearity is discussed numerically for both compressible and incompressible hyperelastic solids. The results are also compared with the solutions of the asymptotic approximation which gives the von Kármán plate equations in the zeroth order approximation.