Two-dimensional model of fifth order in thickness for homogeneous plates

摘要

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