Non-linear rapid heating of FGM beams

Geometrically non-linear thermally induced vibrations of functionally graded material (FGM) beams are analyzed in this research. All thermomechanical properties of the beam are assumed to be temperature and position dependent. Beam is subjected to thermal shock on the ceramic-rich surface whereas the metal-rich one is kept at reference temperature or thermally insulated. The one-dimensional transient heat conduction equation is established and solved via a hybrid iterative central finite difference method and Crank-Nicolson method. Total functional of the beam is obtained under the assumptions of uncoupled thermoelasticity laws, first order beam theory, and the von Kármán type geometrical non-linearity. The conventional multi-term p-Ritz method appropriate for arbitrary in-plane and out-of-plane boundary conditions is applied to the total functional of the system which results in the matrix representation of the equations of motion. Non-linear coupled equations of motion are solved via the iterative Newton-Raphson method accompanied with the β-Newmark time approximation technique. Numerical results are well validated with the available results for the case of isotropic homogeneous beams. Some parametric studies are conducted to examine the influences of beam geometry, material composition, temperature dependency, in-plane and out-of-plane mechanical and thermal boundary conditions. It is shown that, thermally induced vibrations indeed exist especially for the case of sufficiently thin beams.