Nonlinear Thermoelastic Vibrations of Fractional Viscoelastic FGM Beams With Generalized Fractional Order Heat Conduction Model

This paper presents a fractional thermo‐viscoelastic formulation for studying the nonlinear vibrational behavior caused by thermal loads in a functionally graded (FGM) Timoshenko‐type beam exposed to a sudden thermal shock. It is presumed that the beam consists of a ceramic‐metal mixture whose effective properties vary continuously across the thickness following a power‐law gradation. The transient heat transfer is described using the Lord–Shulman generalized thermoelasticity theory extended to incorporate a fractional‐order time derivative of order α . Subsequently, the mechanical behavior follows the fractional‐order Kelvin‐Voigt viscoelastic constitutive law, where the stress–strain relation contains a fractional derivative of order γ . Hamilton's variational principle is employed to derive the weak form of the governing equations, with geometric nonlinearity incorporated through the von Kármán strain relations. Spatial discretization is performed by a combination of the variational differential quadrature (VDQ) and generalized differential quadrature (GDQ) methods. The fractional time derivatives are treated by the Grünwald‐Letnikov (GL) scheme, and the resulting nonlinear system is integrated in time using the Newmark‐β algorithm together with Newton–Raphson iterations. A detailed parametric study investigates the effects of the fractional‐order parameters and , the viscoelastic coefficient, the relaxation time, the power‐law index, and different thermal loading conditions on the transient displacement and temperature fields. The numerical results demonstrate that lower values of increase damping and reduce oscillation amplitudes. Regarding the parameter , increasing it leads to a higher peak temperature and larger oscillation amplitudes. The presented computational approach offers a reliable and accurate tool for designing FGM structures under extreme thermal environments.