On the Solution of Dynamic Stability Problem of Functionally Graded Viscoelastic Plates with Different Initial Conditions in Viscoelastic Media

The widespread use of structural elements consisting of functionally graded (FG) materials in advanced technologies has led to extensive research. Due to the difficulties encountered during modeling and problem solving, the number of studies on the dynamic behavior of structural elements made of FG viscoelastic materials is quite limited compared to the number examining FG elastic materials. This study is one of the first attempts to solve the dynamical problem by the mathematical modeling of functionally graded viscoelastic plates (FG-VE-Ps) and viscoelastic media together with different initial conditions. FG-VE-Ps on viscoelastic foundations (VE-Fs) are assumed to be under compressive edge load in the longitudinal direction. The governing equations for FG-VE-Ps on VE-Fs are derived using Boltzmann and Volterra concepts. The problem is reduced to the solution of integro-differential equation system using the Galerkin method. Then, by performing Laplace transforms, new analytical expressions for the time-dependent deflection function and critical time at different initial conditions are found. The loss of stability of FG-VE-Ps on VE-Fs is modeled to cover three time-varying ranges: the first is the range in which the deflection function decreases; the second is the transition interval; the third is the increase range of deflection function, which leads to the loss of stability. The time corresponding to the sharp increase of the deflection function is defined as the critical time, and is determined both theoretically and numerically. The results are compared with the results obtained by various methods to confirm their accuracy. Finally, the effects of VE-Fs, VE material properties, and FG profiles on the critical time behavior of plates are studied numerically.