Stability of a viscoelastic plate under dynamic loading

Badalov, F. B.; Éshmatov, Kh.; Akbarov, U.

doi:10.1007/bf00887982

Cited by 14 publications

(12 citation statements)

References 3 publications

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“…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 various theories and to determine the applicability limits of these hypotheses in solving applied problems. …”

supporting

confidence: 90%

supporting

confidence: 90%

“…The relationship among the strains ε z x , ε z y , and γ z xy and the rotations ψ x and ψ y is given by (2). Substitution of (2) and (8) into the equations of motion of the plate yields the following system of integrodifferential equations for the displacements u and v, the deflection w, and the rotations ψ x and ψ y :…”

Section: Analysis Of Deformation Of a Viscoelastic Isotropic Rectangumentioning

confidence: 99%

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Dynamic Stability of Viscoelastic Plates under Increasing Compressing Loads

Éshmatov

2006

J Appl Mech Tech Phys

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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...

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supporting

confidence: 90%

supporting

confidence: 90%

Section: Analysis Of Deformation Of a Viscoelastic Isotropic Rectangumentioning

confidence: 99%

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Dynamic Stability of Viscoelastic Plates under Increasing Compressing Loads

Éshmatov

2006

J Appl Mech Tech Phys

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

supporting

confidence: 87%

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

Éshmatov

2007

Nonlinear Dyn

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The vibration problem of a viscoelastic cylindrical shell is studied in a geometrically nonlinear formulation using the refined Timoshenko theory. The problem is solved by the Bubnov-Galerkin procedure combined with a numerical method based on quadrature formulas. The choice of relaxation kernels is substantiated for solving dynamic problems of viscoelastic systems. The numerical convergence of the Bubnov-Galerkin procedure is examined. The effect of viscoelastic properties of the material on the response of the cylindrical shell is discussed. The results obtained by various theories are compared.

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“…Fortunately, it can be used for reference from the study on nonlinear dynamical analysis for thin un-corrugated plates. Badalov et al [13], Touati and Cederbaum [14] investigated the dynamic stability of nonlinear viscoelastic plates. Zhu and Cheng [15] analyzed the stability of viscoelastic rectangular plates from the point of view of dynamical system with ignoring the inertia terms.…”

Section: Introductionmentioning

confidence: 99%

A dynamical model for nonlinear viscoelastic corrugated circular plates with shallow sinusoidal corrugations

Wang

et al. 2011

Arch Appl Mech

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An integrated mathematic model and an efficient algorithm on the dynamical behavior of homogeneous viscoelastic corrugated circular plates with shallow sinusoidal corrugations are suggested. Based on the nonlinear bending theory of thin shallow shells, a set of integro-partial differential equations governing the motion of the plates is established from extended Hamilton's principle. The material behavior is given in terms of the Boltzmann superposition principle. The variational method is applied following an assumed spatial mode to simplify the governing equations to a nonlinear integro-differential variation of the Duffing equation in the temporal domain, which is further reduced to an autonomic system with four coupled first-order ordinary differential equation by introducing an auxiliary variable. These measurements make the numerical simulation performs easily. The classical tools of nonlinear dynamics, such as Poincaré map, phase portrait, Lyapunov exponent, and bifurcation diagrams, are illustrated. The influences of geometric and physical parameters of the plate on its dynamic characteristics are examined. The present mathematic model can easily be used to the similar problems related to other dynamical system for viscoelastic thin plates and shallow shells.

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Stability of a viscoelastic plate under dynamic loading

Cited by 14 publications

References 3 publications

Dynamic Stability of Viscoelastic Plates under Increasing Compressing Loads

Dynamic Stability of Viscoelastic Plates under Increasing Compressing Loads

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

A dynamical model for nonlinear viscoelastic corrugated circular plates with shallow sinusoidal corrugations

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