Knowledge Bank

A functionally graded material (FGM) is a kind of composite in which material properties vary continuously as a function of position. FGMs were first proposed in 1984 and have been in use ever since due to advances in manufacturing to suit specific applications in structural mechanics and materials science. The case of a composite cylinder with a radially graded core imperfectly bonded to a homogeneous cladding is the subject of the present work. The goal of this project is to calculate the dispersion relations of this type of composite cylinder subject to guided time-harmonic torsional waves. The methods include determination of proper equation forms for the circumferential displacements in both the core and cladding. Relevant shear stresses are then calculated from these displacements. Boundary conditions are applied so that the stresses in the core and cladding are continuous across the interface. It is also assumed that the outer surface of the cladding is stress free. Imperfect interface conditions are modeled by setting the circumferential displacement jump at the interface radius proportional to the shear stress through a parameter which represents the compliance of the interface. Placing the resulting expressions in matrix form leads to a dimensionless frequency equation that is solved numerically using Maple. Dispersion relations for three cases are plotted in MATLAB and discussed. The effect of interface damage is most easily distinguished in cases of high radial grading of the core, at low wavenumbers with frequencies associated with the third torsional mode.