Mass moments of functionally graded 2D domains and axisymmetric solids

We present a general methodology to evaluate the mass moments of two-dimensional domains and axisymmetric solids made of functionally graded materials. The approach developed in the paper is based on the sequence of two steps. First, the original domain integrals are converted to integrals extended to the relevant boundary by exploiting Gauss theorem. Second, for domains having a polygonal or circular shape, the boundary integrals are evaluated analytically by providing algebraic expressions that depend upon the parameters defining the density distribution, the position vectors of the vertices of the polygonal domain or the initial and ending points of an arbitrary circular sector, respectively. While the first step refers to moments of arbitrary order, the second step is limited to the most useful quantities for engineering applications, i.e. generalised mass, static moment and inertia tensor. The formulas derived in the paper are validated by means of examples retrieved from the specialised literature for which analytical results are available or have been specifically derived by the authors. Finally, in order to ascertain the computational savings entailed by the use of the proposed analytical formulas with respect to numerical techniques, the mass moments of a longitudinal section of a human femure, made of a functionally graded material and characterised by a linear density distribution, have been computed.