Dynamic Stability of Viscoelastic Plates under Increasing Compressing Loads

Abstract

The dynamic stability problem of viscoelastic orthotropic and isotropic plates is considered in a geometrically nonlinear formulation using the generalized Timoshenko theory. The problem is solved by the Bubnov-Galerkin procedure combined with a numerical method based on quadrature formulas. The effect of viscoelastic and inhomogeneous properties of the material on the dynamic stability of a plate is discussed.Introduction. The use of new composite materials in the design and development of strong, light-weight, and reliable structures calls for improved mechanical models of deformable solids and mathematical models for structure calculations taking into account the actual properties of structural materials. Numerous experimental studies have shown that most composite materials possess pronounced viscoelastic properties [2-6] and are inhomogeneous [6,7].The classical Kirchhoff-Love model is effective in solving some applied problems, but in most cases it fails to give adequate solutions [8]. This is true primarily for calculations of the dynamic stability of viscoelastic shells made of composite materials with heterogeneous anisotropic structure [3,6,9]. This formulation of elastic problems was considered in [7,8,[10][11][12][13], where, however, only some properties of structural materials were taken into account.In a number of papers, the viscoelastic properties of materials were taken into account only in the shear directions (see, e.g., [3]). In previous calculations, exponential kernels were used as relaxation kernels but they cannot provide an adequate description of the real processes occurring in shells and plates at the initial time [14]. The choice of exponential kernels in the calculations is not random. By differentiation, the resulting systems of integrodifferential equations were reduced to high-order ordinary differential equations, which, in most cases, were solved by the Runge-Kutta numerical method. Previously existing methods were not applicable for solving these problems with weakly singular kernels of the type of Koltunov, Rzhanitsyn, Abel, and Rabotnov kernels.The numerical method developed in [1] using quadrature formulas has made it possible to solve systems of nonlinear integrodifferential equations with singular kernels. This method provides a high accuracy of calculations, is universal, and can be used to solve a wide class of dynamic problems of the theory of viscoelasticity. The results of [2,9] obtained by this method agree well with experimental data.It is worth noting that in contrast to the isotropic formulation of the dynamic problems of viscoelastic systems, where the integrodifferential equations contain only one relaxation kernel with three different rheological viscosity parameters, the orthotropic formulation includes seven different kernels and the number of different rheological parameters increases to 21, which leads to very intricate calculations.The objective of the present paper is to study the dynamic stability of viscoelastic isotropic and orthotropic plates using va...

Nonlinear vibrations of viscoelastic cylindrical shells taking into account shear deformation and rotatory inertia

“…The numerical results of Refs. [28][29][30][31][32][33][34][35][36] obtained by this method agree well with experimental data [3].…”

“…In accordance with Equations (1) and (3), the bending and twisting moments M x , M y , and H and the transverse shear forces Q x and Q y are given by [34]:…”

“…Writing the system (5) in terms of the deflection w = w (x, y, t), stress function F = F (x, y, t), and rotations ψ x = ψ x (x, y, t) and ψ y = ψ y (x, y, t), we obtain the system of equations of von Karman type [34] …”

Thermoviscoelastic dynamic response for a composite material thin narrow strip

“…In the same year, Khudayarov [9] determined the flutter velocities of viscoelastic plates and shown the viscoelastic characteristics reduce them. In the next year, the dynamic stability problem of viscoelastic orthotropic and isotropic plates was considered in a geometrically nonlinear formulation using the generalized Timoshenko theory by Eshmatov [10,11], the problem was solved by the Bubnov-Galerkin procedure combined with a numerical method based on quadrature formulas. At the same time, he modeled the nonlinear vibrations of viscoelastic orthotropic and isotropic shells mathematically using a geometrically nonlinear Timoshenko theory.…”