Two-dimensional model of order h 5 for the combined bending,stretching,transverse shearing and transverse normal stress effect of homogeneous plates derived from three-dimensional elasticity

Abstract

For homogeneous plates, the highest order term of transverse shear and normal stresses is of second order in thickness. To take this effect into account, we show that the thickness-wise expansion of the potential energy must be truncated at least from fifth order in thickness. The equilibrium equations imply local constraints on the through-thickness derivatives of the zeroth-order displacement field. These lead to an analytical expression for two-dimensional potential energy in terms of the zeroth-order displacement field and its derivatives, which include non-standard shearing and transverse normal energies and coupled stretching–shearing, bending–shearing and stretching–transverse normal energies. As a consequence, this potential energy satisfies the stability condition of Legendre–Hadamard, which is necessary for the existence of a minimizer.