Axisymmetric buckling of FGM shells under thermal shock

The paper deals with the problem of vibrations and dynamic buckling of FGM shells of revolution subjected to thermal shock. A finite-element algorithm is developed for studying geometrically nonlinear vibration of shells under unsteady heat conduction through the shell wall. A two-node shell finite element is formulated taking into account transverse shear deformation based on the first-order shear deformation theory by S. Timoshenko. To avoid the shear locking effect, the strains of the element are approximated using an exact analytical solution of the bending problem obtained in the framework of the Timoshenko beam theory. The reference surface used is the surface, passing through the centers of mass of the transverse fibers, which makes it possible to separate the inertial terms corresponding to translational motion and rotation. Equations of motion of the finite-element model are solved numerically using a step-by-step method based on the implicit Newmark integration scheme. At each time step of integration, temperature loads and stiffness characteristics of the shell are determined by numerically solving the unsteady heat conduction problem formulated with account of temperature-dependent properties of the material. Deformed configuration of the shell at each time step is determined using the Newton-Raphson iterative procedure. The proposed numerical algorithm is verified by comparing the numerical solution of test problems with the data available in the literature. Dynamic snap-through buckling of shallow spherical and conical shells, experiencing a sudden rise of temperature at the inner surface, is studied. The effect of material properties, geometrical parameters, and the rate of heating of the inner surface on the dynamic loss of stability is discussed.