Nonlinear dynamic snap-through and vibrations of temperature-dependent FGM deep spherical shells under sudden thermal shock

In This article, nonlinear thermally induced vibrations of temperature-dependent functionally graded material (FGM) deep spherical shells are analyzed. One surface of the shell is kept at the reference temperature while the other one is subjected to rapid surface heating. Based on the theory of uncoupled thermoelasticity, the one-dimensional heat conduction equation across the thickness of the shell is developed and solved using the combined Crank–Nicolson method and the central finite difference method. The temperature profile obtained from the numerical solution of the heat conduction equation is used to calculate the thermal forces and the thermally induced bending moment to insert into the equations of motion of the shell. Under the assumptions of the first-order shear deformation theory (FSDT), uncoupled thermoelasticity laws, the von Kármán type of geometrical nonlinearity, and employing the Hamilton principle the axisymmetric equations of motion are obtained. The conventional multi-term polynomial Ritz method is used to discretize the governing nonlinear equations of motion. To convert the differential equations of motion into algebraic equations in each time step, the β-Newmark time marching scheme is used, and the obtained equations are solved with the help of the Newton–Raphson linearization method. In some cases, the dynamic buckling temperature is detected using the Budiansky criterion. Various numerical results are presented to analyze the effects of different parameters such as the opening angle of the shell, the thickness of the shell, and the power law index of the material composition rule.