A higher order theory for functionally graded shells

New theory of higher order for functionally graded (FG) shells, which is based on the expansion of the three-dimensional (3D) equations of elasticity for functionally graded materials (FGMs) into Legendre's polynomials series is developed here. The stress and strain tensors, the displacement, traction and body force vectors of the 3D equations of elasticity, are expanded into Legendre's polynomials series in terms of in the thickness coordinate. The mechanical parameters that describe the functionally graded material properties are also represented in the form of Legendre's polynomials series expansion. As result the equations of the 3D elasticity are turned into the infinite number of two-dimensional (2D) equations for the Legendre's polynomials series expansion coefficients. Considering finite number of the Legendre's polynomials series coefficients and substituting kinematic relations into generalized Hooke's law and the obtained result into the equations of motion the differential equations of motion the equations of motion in form of displacements have been obtained. The first order equations for the FG axisymmetric cylindrical plate and spherical shell are considered in more details. Corresponding boundary-value problems are solved using the finite element method (FEM) implemented in the MATEMATICA software. The numerical results are presented and discussed.