Viscoelastic Plate Research Articles

Blow-up of Solutions of a Logarithmic Viscoelastic Plate Equation with Delay Term

In this paper, we consider a logarithmic viscoelastic plate equation with a delay term in a bounded domain. Under suitable conditions, we establish the blow-up of solutions in a finite time. Time delays often seem in thermal, economic phenomena, biological, chemical, physical, electrical engineering systems, mechanical applications and medicine.

Complex Dispersion Analysis of Topologically Protected Flexural Wave Transport in Periodic Viscoelastic Plates

Complex Dispersion Analysis of Topologically Protected Flexural Wave Transport in Periodic Viscoelastic Plates

Normal Impact of A Rigid Cone-shaped against A Viscoelastic Plate on Viscoelastic Foundation

This paper considers the problem of a low-velocity normal impact of a rigid cone-shaped upon a viscoelastic plate. The contact force is defined by the modified Hertz's contact law. Approximate solutions of the system of nonlinear integro-differential equations for the contact force and local indentation have been obtained.

Viscoelastic modeling and characterization of thin-ply composite laminates

Thin-ply composite laminates are promising materials for deployable spacecraft structures owing to their low mass and high folding curvature. However, the deployment force and the shape accuracy of the deployed configuration after a long duration of stowage is adversely affected by the inherent viscoelastic behavior of the polymer matrix. This paper presents a computational and experimental investigation into the effective viscoelastic response of thin-ply composite laminates made of woven carbon fabrics and an epoxy matrix intended for deployable spacecraft applications. A two-scale homogenization approach is utilized to model thin-ply composites as viscoelastic plates represented with relaxation ABD matrices. Micrographic analysis and relaxation tests are conducted to obtain the geometric and material characteristics that are incorporated into the model. The bending relaxation behavior is experimentally characterized using the recently developed column bending test methodology to compare against model predictions. The relaxation response is studied in two different orientations. The effects of ply phase arrangement, laminate thickness, fiber volume fraction, specimen width, and coordinate rotation are investigated.

Quasi-static and dynamic response of functionally graded viscoelastic plates

In this study, quasi static and dynamic response of functional graded material (FGM) viscoelastic plates under dynamic loads were investigated. The governing equations of composite plates were obtained with the help of Hamilton’s principle. Afterwards, time dependent partial differential equations were obtained by applying Navier solution method to these valid equations. These equations were transformed into Laplace space in order to solve time dependent partial differential equations. The transformation of the resulting calculations from Laplace domain into the time domain was conducted with the help of Modified Durbin algorithm. The obtained results were compared with various approaches It was shown that the results are in parallel with other approaches’ results, and in full agreement with the results of the finite element method (ANSYS). Then, quasi-static behavior was investigated for viscoelastic FGM because of there is not enough research on this subject. The sensitivity analysis demonstrated that the volume fractions and damping ratios have a significant effect on the quasi-static and dynamic response of viscoelastic FGM plates. The numerical sample results showed that the present method can solve the problem in a highly accurate, efficient, and easily by utilizing the Laplace transform without the use of natural frequencies or mode shapes.

Parametric vibration analysis of sector plates using perturbation technique

This paper aims to investigate the in-plane and out-of-plane free vibrations analysis of both solid and annular sector plates with time-dependent properties. The Fourier expansion and the perturbation technique are employed to solve the partial differential equations based on the first-order shear deformation theory. By defining a new parameter, the motion equations which have variable coefficients convert to equations with constant coefficients that lead to a closed-form solution for each order of the perturbation expansion. Due to the use of this displacement field, rapid convergence and high accuracy results can be easily obtained for both in-plane and out-of-plane natural frequencies. The influences of some key parameters including the inner to outer radius ratios, sector angle, thickness to radius ratios, and damping parameters on free vibration of the viscoelastic sector plates are illustrated and analyzed, which may serve as benchmark solutions in the future. Besides, the results are compared with the actual results from the literature, classic theory of plates, and finite element analysis. The present formulation yields many new and accurate results at a low computational cost for the study of mechanical and geometrical parameters.

Shifted Legendre polynomials algorithm used for the numerical analysis of viscoelastic plate with a fractional order model

An effective numerical algorithm is presented to analyze the fractional viscoelastic plate in the time domain for the first time in this paper. The viscoelastic behavior of the plate is described with fractional Kelvin–Voigt (FKV) constitutive model in three-dimensional space. A governing equation with three independent variables is established. Ternary unknown function in the governing equation is solved by deriving integer and fractional order differential operational matrices of the shifted Legendre polynomials. Error analysis and mathematical example are presented to verify the effectiveness and accuracy of proposed algorithm. Finally, numerical analysis of the plate under different loading conditions is carried out. Effects of the damping coefficient on vibration amplitude of the viscoelastic plate are studied. The results obtained are consistent with the current reference and actual situation. It shows that shifted Legendre polynomials algorithm is suitable for numerical analysis of fractional viscoelastic plates.

Hydroelastic analysis of floating long viscoelastic plate in shallow water

The present study investigates the interaction of gravity waves heading on a large viscoelastic floating structure of negligible draft in shallow water region. The flexible property of the plate is modeled using well known Euler-Bernoulli beam equation, whereas the viscosity is modeled using complex flexural rigidity parameter. The problem is solved using the eigenfunction expansion procedure. Due to the presence of complex flexural rigidity, the dispersion relation in the region covered by the plate is complex in nature, and therefore the roots of the dispersion relation are complex types. The location of the roots has been found using Matrix analysis. Finally, various results of physical interests are studied for a wide range of structural and wave parameters.

Numerical modeling for viscoelastic sandwich smart structures bonded with piezoelectric materials

Regarding to viscoelastic laminated piezo smart plates and shells, electro-mechanically coupled finite element (FE) formations are derived based on zig-zag hypothesis for vibration and damping analyses. Three lamination configurations are considered in the model, where both elastic layers and viscoelastic layers are assumed to be Reissner–Mindlin plate. The quadratic element is adopted for FE modeling, each node has seven mechanical degrees of freedom (DOFs), which considers the complex shear modulus of viscoelastic materials. The analytical model is first validated by FE simulation of viscoelastic sandwich plates. Later, the effects of various parameters on the damping performance of piezo-viscoelastic laminated cylindrical structures are carried out, e.g. boundary conditions, reinforcement orientation angle, thickness of the viscoelastic core and structural curvature. The present FE modeling and analyses provide a powerful tool to accurately investigate the behaviors of viscoelastic laminated structures.

Analysis and Comparison of Nonlinear Dynamic Behavior of Viscoelastic and Elastic Rectangular Plates under Supersonic Aerodynamic Flow

Analysis and Comparison of Nonlinear Dynamic Behavior of Viscoelastic and Elastic Rectangular Plates under Supersonic Aerodynamic Flow

Global existence of solutions for a coupled viscoelastic plate equation with degenerate damping terms

We consider a coupled viscoelastic plate nonlinear equations with degenerate damping terms on a bounded domain in Rn. We establish the global existence results under different conditions.

BLOW-UP RESULTS FOR A VISCOELASTIC PLATE EQUATION WITH DISTRIBUTED DELAY

In this paper, we consider a nonlinear viscoelastic plate equation with distributed delay. Under suitable conditions, we obtain the blow-up of solutions with distributed delay and source terms.

Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness

The research object of this work is an orthotropic viscoelastic plate with an arbitrarily varying thickness. The plate was subjected to dynamic periodic load. Within the Kirchhoff–Love hypothesis framework, a mathematical model was built in a geometrically nonlinear formulation, taking into account the tangential forces of inertia. The Bubnov–Galerkin method, based on a polynomial approximation of the deflection and displacement, was used. The problem was reduced to solving systems of nonlinear integrodifferential equations. The solution of the system was obtained for an arbitrarily varying thickness of the plate. With a weakly singular Koltunov–Rzhanitsyn kernel with variable coefficients, the resulting system was solved by a numerical method based on quadrature formulas. The computational algorithm was developed and implemented in the Delphi algorithmic language. The plate’s dynamic stability was investigated depending on the plate’s geometric parameters and viscoelastic and inhomogeneous material properties. It was found that the results of the viscoelastic problem obtained using the exponential relaxation kernel almost coincide with the results of the elastic problem. Using the Koltunov–Rzhanitsyn kernel, the differences between elastic and viscoelastic problems are significant and amount to more than 40%. The proposed method can be used for various viscoelastic thin-walled structures such as plates, panels, and shells of variable thickness.

Three-dimensional static analysis of a viscoelastic rectangular functionally graded material plate embedded between piezoelectric sensor and actuator layers

The three-dimensional bending behavior of a viscoelastic functionally graded material (FGM) layer embedded between piezoelectric layers and subjected to an electric field as well as a uniform transverse pressure is studied. An analytical solution is computed for a simply supported viscoelastic smart FGM plate using the state space technique along the thickness direction and a Fourier expansion along the in-plane coordinates. The governing differential equations in the time domain are transformed to the Laplace domain, solved in the Laplace domain, and transformed back to the time domain using the inverse Laplace transform. In the present study, the relaxation modulus of the FGM layer is assumed in the form of a Prony series, and it varies according to the power law in the thickness direction. The validity of the proposed approach is assessed by comparison of the numerical results of the approach with those of published works in the literature. The effects of geometric dimensions, electromechanical loads, and relaxation time constant on the behavior of the smart plate are investigated.

Vibrations of a viscoelastic isotropic plate under periodic load without considering the tangential forces of inertia

A mathematical model of the problem of viscoelastic isotropic plate vibrations based on the Kirchhoff-Love hypothesis in a geometrically nonlinear formulation was presented. The mathematical model was built without considering the tangential forces of inertia. To describe the viscoelastic properties of the plate material, a weakly singular Koltunov-Rzhanitsyn kernel with three different rheological parameters was chosen. To solve the problem of parametric vibrations of a viscoelastic plate with a weakly singular relaxation kernel, a numerical method based on the use of quadrature formulas was applied. A discrete model of this problem was first constructed using the Bubnov-Galerkin method; i.e., a system of integro-differential equations with variable coefficients was obtained, and then, using a numerical method based on the elimination of a singularity of the kernel, the problem of parametric vibrations of viscoelastic rectangular plates was solved. The influence of the viscoelastic properties of the material and the variability of the plate thickness on the oscillatory process was shown.

Dynamic Vibration Extinguished on a Viscously Elastic Base

The aim of the work is to develop algorithms and a set of programs for studying the dynamic characteristics of viscoelastic thin plates on a deformable base on which it is installed with several dynamic dampers. The theory of thin plates is used to obtain the equation of motion for the plate. The relationship between the efforts and the stirred plate obeys in the hereditary Boltzmann Voltaire integral. With this, a system of integro-differential equations is obtained which is solved by the method of complex amplitudes. As a result, a transcendental algebraic equation was obtained to determine the resonance frequencies, which is solved numerically by the Muller method. To determine the displacement of the point of the plate with periodic oscillations of the base of the plate, a linear inhomogeneous algebraic equation was obtained, which is solved by the Gauss method. The amplitude - frequency response of the midpoint of the plate is constructed with and without regard to the viscosity of the deformed element. The dependence of the stiffness of a deformed element on the frequency of external action is obtained to ensure optimal damping of vibrational vibrations of the plate.

Modeling lamb wave propagation in visco-elastic composite plates using a fifth-order plate theory

A new extension of the shear deformation theory to fifth order in order to calculate the spectrum of Lamb waves in orthotropic media over a wide frequency range is developed and analyzed. The aspiration of the proposed method is to create an alternative framework to exhaustive 3D elasticity based solutions by increasing computational efficiency without losing accuracy, nor robustness. A new computational framework is introduced which allows to estimate the dispersion curves for the first nine symmetric and nine anti-symmetric Lamb modes. Analytically calculated dispersion curves using 5-SDT for different propagation directions and polar plots for selected frequency of different materials are compared with the results from both the semi analytical finite element method, and lower order shear deformation theories. Careful analysis for individual laminae and for symmetric composite laminates exhibits a good agreement between the new higher order plate theory and the semi analytical finite element method over an extensive frequency range. In addition, attenuation plots show that the proposed method can also be used for visco-elastic materials (or highly damped materials). The advantage of the new higher order plate theory and its numerical implementation is that it is much more computationally efficient compared to comprehensive methods as Lamb wave polar plots of composite plates as function of incidence angle, polar angle and frequency can be calculated in less than a second on a standard laptop. Consequently, the use of this framework in inversion routines opens up the possibility of quasi real-time Structural Health Monitoring for visco-elastic composites covering a sufficiently wide frequency range.

Frequency control of cross-ply magnetostrictive viscoelastic plates resting on Kerr-type elastic medium

Frequency control of cross-ply magnetostrictive viscoelastic plates resting on Kerr-type elastic medium

Blunt body impact onto viscoelastic floating ice plate with a soft layer on its upper surface

Two-dimensional vertical impact of a rigid blunt body onto a floating ice plate is studied. The problem is coupled and unsteady. The liquid is inviscid, incompressible, and of infinite depth. The ice floe is modeled as a thin viscoelastic plate of constant thickness. The plate edges are free of bending stresses and shear forces. The upper surface of the plate is covered with a viscoelastic layer of constant small thickness and negligible inertia. The reaction force of this soft layer is predicted by a nonlinear and one-dimensional Winkler-Kelvin-Voigt model, which does not permit a contact between the rigid body and the ice plate. The soft layer may describe either the presence of snow on the ice or a layer of crushed ice in the place of impact, or can be considered as a way of regularization of problems with concentrated loads. The rigid body touches the upper surface of the soft layer and then suddenly starts to move downward with constant velocity. It is shown that the strains in the ice plate caused by the impact are weakly dependent on the characteristics of the soft layer. The magnitudes and distributions of the strains are studied depending on the length of the ice plate, retardation time of the ice model, thickness of the plate, shape of the rigid body, place of impact, and the impact speed. The value of the retardation time in the soft layer model is discussed with relation to the ice crushing by impact.

Beam-steering ultrasonic guided waves in a bone-mimicking plate by time-delaying the excitation of the elements in a multi-element array: a numerical study

We present a numerical simulation of the beam-steering of ultrasonic guided waves in an isotropic and viscoelastic solid plate, which mimics bovine cortex. The excitation was modeled by a group of five finite-size emitters, each exercised a normal force to the bone plate. Beam steering was achieved by delaying the emitters’ firing. The simulation technique was implemented by a semi-analytical finite element scheme to compute the wave fields. At small steering angles, the simulated time-offset signals show mainly two groups of arrivals. The first group is the fast-traveling and high-frequency bulk waves and the second one is slow-traveling and low-frequency guided waves. The fast-traveling waves gradually diminish with increasing steering angles, in agreement with the excitation function of the source influence theory. The frequency-phase velocity dispersion maps also illustrate the phenomenon. The study has demonstrated that the lowest order Lamb asymmetrical mode, A 0, which is useful for bone characterization, can best be excited when the cortical bone thickness is thin, the beam angle is large, and the excited frequency is low.