Two-dimensional model for the combined bending,stretching and transverse shearing of laminated plates derived from three-dimensional elasticity

Abstract

For laminated plates, the displacement field can be approximated in each layer by a third-order Taylor—Young expansion in thickness. These involve (sections 3.1 and 3.2) that the highest order term of transverse shear is of first-order in thickness. Then we are motivated to consider a simplified theory based on the thickness-wise expansion of the potential energy truncated at third order in thickness. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field in each layer. 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 transverse shearing energy and coupled bending—stretching energy. As a consequence this potential energy satisfies the stability condition of Legendre—Hadamard which is necessary for the existence of a minimizer.