Voigt Viscoelastic Model Research Articles

On Regularity of Weak Solutions to a Generalized Voigt Model of Viscoelasticity

The existence and uniqueness of a strong solution to the initial-boundary value problem for a system of fluid dynamics equations that is a fractional analogue of the Voigt viscoelasticity model in the plane case are established. The rheological equation of this model involves fractional derivatives.

A fractional viscoelastic model for vibrational analysis of thin plate excited by supports movement

In this paper, a fractional viscoelastic model is presented for vibration analysis of a fully simply supported plate excited by the supports movement. To govern the steady-state condition for the plate vibration, the lower limit of the integral in Riemann-Liouville fractional derivative is assumed in minus infinity. The Voigt viscoelastic model is employed for describing the damping effects. Considering the Kirchhoff hypothesis the governing equation of the viscoelastic plate is derived. An analytical solution for the dynamic response of the fractional viscoelastic plate under the support excitation is proposed. The effect of fractional derivative order and damping coefficient on the amplitude-frequency characteristic of the plate is investigated. Also, the obtained natural frequencies for elastic plate (by setting the fractional derivative order to zero) and classical viscoelastic plate (by setting the fractional derivative order to one) are compared with existing results in the literature.

A New Theoretical Model to Study the Closing Bounce Characteristics of the Electromagnetic Relay Under Capacitive Loads

Contact bounce is a common phenomenon in the closing process of the electromagnetic relay under capacitive loads. The investigations on the contact characteristics are mainly conducted by experimental methods. However, the experimental method is extremely complicated and time-consuming in structural design, and it cannot clearly reveal the mechanism of contact bounce. Therefore, a new theoretical model is proposed to investigate the dynamic characteristics of the electromagnetic relay under capacitive loads. The model allows for simulating dynamic process and collecting bounce parameters in a simple and effective way compared with conventional simulation and experimental methods. The physical mechanism of contact bounce for the electromagnetic relay is presented. The nonlinear contact force between the fixed and moving contacts is evaluated according to the Kelvin–Voigt viscoelastic contact model. The equations of motion considering the coupling of electromagnetic and mechanical fields are established. The theoretical results obtained in this study are validated by comparison with experimental data. The influences of the structural parameters, coil current, surge current, and contact resistance on the contact bounce are analyzed. The numerical results well demonstrate that the contact bounce can be effectively weakened by adjusting the parameters of contact spring and the mass of the moving contact.

Investigating vibrations of viscoelastic fluid-conveying carbon nanotubes resting on viscoelastic foundation using a nonlocal fractional Timoshenko beam model

On the basis of fractional viscoelasticity, the size-dependent free-vibration response of viscoelastic carbon nanotubes conveying fluid and resting on viscoelastic foundation is studied in this article. To this end, a nonlocal Timoshenko beam model is developed in the context of fractional calculus. Hamilton’s principle is applied in order to obtain the fractional governing equations including nanoscale effects. The Kelvin–Voigt viscoelastic model is also used for the constitutive equations. The free-vibration problem is solved using two methods. In the first method, which is limited to the simply supported boundary conditions, the Galerkin technique is employed for discretizing the spatial variables and reducing the governing equations to a set of ordinary differential equations on the time domain. Then, the Duffing-type time-dependent equations including fractional derivatives are solved via fractional integrator transfer functions. In the second method, which can be utilized for carbon nanotubes with different types of boundary conditions, the generalized differential quadrature technique is used for discretizing the governing equations on spatial grids, whereas the finite difference technique is used on the time domain. In the results, the influences of nonlocality, geometrical parameters, fractional derivative orders, viscoelastic foundation, and fluid flow velocity on the time responses of carbon nanotubes are analyzed.

Asymmetric Large Deformation Superharmonic and Subharmonic Resonances of Spiral Stiffened Imperfect FG Cylindrical Shells Resting on Generalized Nonlinear Viscoelastic Foundations

This paper is devoted to superharmonic and subharmonic behavior investigation of imperfect functionally graded (FG) cylindrical shells with external FG spiral stiffeners under large amplitude excitations. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler–Pasternak foundation augmented by a Kelvin–Voigt viscoelastic model with a nonlinear cubic stiffness, to account for the vibration hardening/softening phenomena and damping considerations. The von Kármán strain-displacement kinematic nonlinearity is employed in the constitutive laws of the shell and stiffeners. The external spiral stiffeners of the cylindrical shell are modeled according to the smeared stiffener technique. The coupled governing equations are solved by using Galerkin’s method in conjunction with the stress function concept. The multiple scales method is utilized to detect the subharmonic and superharmonic resonances and the frequency–amplitude relations of the 1/3 and 1/2 subharmonic and 3/1 and 2/1 superharmonic resonances of the system. Finally, the influences of the stiffeners helical angles, foundation type, coefficient of the nonlinear elastic foundation, material distribution, and excitation amplitude on the system resonances are investigated comprehensively.

Dynamic Analysis of Layered Functionally Graded Viscoelastic Deep Beams with Different Boundary Conditions Due to a Pulse Load

This paper studies the dynamic viscoelastic response of functionally graded (FG) thick 2D cantilever and simply supported beams under dynamic pulse load, for the first time. A point load applied at a specific spatial point is described as a time-pulse sinusoidal load. Two-dimensional plane-stress constitutive equation is exploited to describe the local stress–strain relation through the beam. The gradation of material is depicted by generalized power law function through the layer thickness across beam thickness. The Kelvin–Voigt viscoelastic model is proposed to describe material damping of structure. Lagrange’s equation is employed to derive governing motion equation. A finite element method (FEM) is exploited to discretize the spatial domain of 2D beam structure by using 12-node 2D plane element. Numerical Newmark implicit time integration method is proposed to solve the equation of motion incrementally and get the response of beam structure. Two types of boundary conditions are considered in the numerical examples. In numerical results, effects of stacking sequence, geometry parameters and material gradation index and viscoelasticity coefficients on the displacement-time response of layered functionally graded viscoelastic deep beams for different boundary conditions.

Dynamic analysis of viscoelastic functionally graded porous thick beams under pulse load

Due to the significant effect of porosity on the mechanical response of functionally graded (FG) structures, this paper presents a comprehensive model to investigate the vibration response of FG porous thick beam under the dynamic sine pulse load including the damping effect by using adopted finite element model, for the first time. The multilayer thick beam is modeled as two-dimensional plane stress problem. The distribution of material gradation through the graded layer is described by the power law function, and the porosity is depicted by three different distributions (i.e., symmetric-distribution, X-distribution and O-distribution). The damping effect is included in the model by using the Kelvin–Voigt viscoelastic constitutive model. Constitutive equations, gradation and porosity functions are described in detail. Forced motion equations are derived by using Lagrange energy principles. Twelve-node 2D plane element with 3 × 3 integration points is proposed to discretize the beam and get the element matrices and force vectors. The numerical time integration method of Newmark is proposed to solve the system numerical and get the displacement response of the structure. Effects of layer stacking sequence, material gradation index and porosity parameter on the dynamic’s response of thick FG porous damped beam are presented. The presented mathematical model is useful in analysis and design of nuclear, marine, vehicle and aerospace structures those manufactured from functionally graded materials.

On the Dynamics of a Visco–Piezo–Flexoelectric Nanobeam

The fundamental motivation of this research is to investigate the effect of flexoelectricity on a piezoelectric nanobeam for the first time involving internal viscoelasticity. To date, the effect of flexoelectricity on the mechanical behavior of nanobeams has been investigated extensively under various physical and environmental conditions. However, this effect as an internal property of materials has not been studied when the nanobeams include an internal damping feature. To this end, a closed-circuit condition is considered taking converse piezo–flexoelectric behavior. The kinematic displacement of the classical beam using Lagrangian strains, also applying Hamilton’s principle, creates the needed frequency equation. The natural frequencies are measured in nanoscale by the available nonlocal strain gradient elasticity model. The linear Kelvin–Voigt viscoelastic model here defines the inner viscoelastic coupling. An analytical solution technique determines the values of the numerical frequencies. The best findings show that the viscoelastic coupling can directly affect the flexoelectricity property of the material.

Dynamic Response of Fractionally Damped Viscoelastic Plates Subjected to a Moving Point Load

AbstractThe dynamic response of fractionally damped viscoelastic plates subjected to a moving point load is investigated. In order to capture the viscoelastic dynamic behavior more accurately, the material is modeled using the fractionally damped Kelvin–Voigt model (rather than the integer-type viscoelastic model). The Riemann–Liouville fractional derivative of order 0 < α ≤ 1 is applied. Galerkin's method and Newton–Raphson technique are used to evaluate the natural frequencies and corresponding damping coefficients. The structure is subject to a moving point load, traveling at different speeds. The modal summation technique is applied to generate the dynamic response of the plate. The influence of the order of the fractional derivative on the free and transient vibrations is studied for different velocities of the moving load. The results are compared with those using the classical integer-type Kelvin–Voigt viscoelastic model. The results show that an increase in the order of the fractional derivative causes a significant decrease in the maximum dynamic amplification factor, especially in the “dynamic zone” of the normalized sweep time. The dynamic behavior of the plate is verified with ansys.

Swelling and thermal behavior of a cross-linked polymer networks poly(2-phenoxyethyl acrylate): exploitation by the Voigt viscoelastic model

The response-ability to the external and internal environmental condition qualifies the smart polymer material to be successfully applied in artificial muscle, drug delivery and water treatment. A three-dimensional polymer network based on the phenyl monomer, 2-phenoxy ethyl acrylate (PEA), was polymerized under ultraviolet (UV) radiation using the Darocur 1173 as an initiator of the polymerization reaction, and the 1,6-hexane-diol diacrylate (HDDA) as a chemical cross-linker. The analysis of the thermophysical properties of this elaborated polymeric material represents the main goal of the present work. The infrared spectroscopy (FTIR) and the dynamic scaling calorimetry (DSC) were used to examine, respectively, the vibration of the carbon double bond (C=C) and the glass transition temperature Tg. It is found that parameters, the rate of reticulation and monomer composition have a remarkable effect on the variation of Tg. The swelling behavior of the cross-linked poly(PEA/HDDA) was investigated; the effects of solvent nature, the degree of cross-linking, temperature and chemical structure have a significant influence on material swelling properties. The theoretical investigation based, especially on the Voigt model, Flory–Rehner theory and solubility approximation, permits to give much information about several parameters such as the swelling time constant, swelling equilibrium constant, solubility prediction, entropy and enthalpy. It ended interestingly that the Voigt model shows a good agreement with swelling experimental results.

A new approach for smart control of size-dependent nonlinear free vibration of viscoelastic orthotropic piezoelectric doubly-curved nanoshells

• The active vibration control of a viscoelastic orthotropic piezoelectric doubly-curved nanoshell is studied. • A velocity feedback control law is introduced to obtain an efficient active damping for the present system. • The nonlinear vibration response of the system with surface effects is reduced due to the effect of active damping. • There exists a critical value of width to thickness ratio above which the natural frequency will monotonously increase. The focus of this paper is on the active control of nonlinear free vibration of viscoelastic orthotropic piezoelectric doubly-curved smart nanoshells with surface effects. To achieve an efficient active damping in the vibration control, a velocity feedback control law is introduced to carry out the present study. Within the framework of surface piezoelectricity theory and Kelvin–Voigt viscoelastic model, the nonlinear equations of motion, including the surface effect and internal damping effect, are derived from the generalized Hamilton's principle. By using the harmonic balance method to solve the equations of motion, the nonlinear vibration responses of the system with and without active vibration control are presented. The effects of the surface energy, viscous damping coefficient, initial displacement, aspect ratio and control gain on the nonlinear vibration responses of the system are studied. Numerical results show that these parameters, especially the control gain, play a significant role in the active vibration control of a viscoelastic orthotropic piezoelectric doubly-curved nanoshell.

A priori error estimates of fully discrete finite element Galerkin method for Kelvin–Voigt viscoelastic fluid flow model

In this article, a finite element Galerkin method is applied to the Kelvin–Voigt viscoelastic fluid model, when its forcing function is in L∞(L2). Some new a priori bounds for the velocity as well as for the pressure are derived which are independent of inverse powers of the retardation time κ. The second order error estimate in L∞(L2)-norm, the first order error estimate in L∞(H01)-norm for the velocity and the first order error estimate in L∞(L2)-norm for the pressure for the semidiscrete method are derived which hold uniformly with respect to κ as κ→0 with the initial condition only in H2∩H01. Further under the smallness assumption on the data, these error estimates are shown to be uniform in time as t↦∞. For the complete discretization of the semidiscrete system, a first-order accurate backward Euler method is applied and fully discrete error estimates are established. Finally, numerical experiments are conducted to verify the theoretical results. The results derived in this article are sharper than those derived earlier for finite element analysis of the Kelvin–Voigt fluid model in the sense that the error estimates in this article hold true uniformly even as κ→0.

Nonlinear dynamics of a viscoelastic traveling beam with time-dependent axial velocity and variable axial tension

In this paper, stability, various bifurcations and chaotic behavior of a traveling Euler beam are studied. Kelvin–Voigt viscoelastic model is adopted for the beam material. The beam has time dependency in velocity as well as tension in axial direction which is considered for the first time for traveling viscoelastic beam. The multi-frequency parametric resonance is assumed to be comprised of simultaneous principal parametric resonance of first mode due to the time dependency of speed, while the principal parametric resonance of second mode is due to the variable tension in the axial direction. The two-frequency parametric resonance is considered along with 3:1 internal resonance. Such multi-resonant case has not been considered so far in any available literature. The higher-order integro-partial differential equation of motion is attacked with direct method of multiple scales. Solvability condition is incorporated to get two complex variable modulation equations for amplitude and phase. From these two equations, a set of normalized reduced equations are derived through Cartesian transformation which are subsequently utilized to get numerical solutions for the nonlinear system. Continuation algorithm is used to depict the effect of system parameters such as fluctuating tension component, fluctuating velocity component, internal frequency detuning parameter, different damping on the frequency response curves, their stability and bifurcation analysis. This research work reveals interesting results which are not found so far in the existent literature.

Characterizing the viscoelastic properties of a tissue mimicking phantom for ultrasound elasticity imaging studies

With the increased use of ultrasound (US) imaging in biomedical fields, the ability to provide phantoms that are capable of mimicking desired properties of soft tissues is important. Polyvinyl alcohol (PVA) is a polymer of great interest because of its relatively simple chemical structure, ease of processing, and satisfactory acoustic and durability properties. PVA may be useful as a phantom material in the study of ultrasound elasticity imaging (UEI) for liver fibrosis. A PVA-based US phantom was constructed through a freeze–thaw process, and measurements of mechanical properties of PVA were carried out with a rotational rheometer. The storage modulus and the loss modulus of PVA were obtained, and the elasticity and viscosity of PVA phantom were estimated by using Voigt viscoelastic model. The obtained viscoelastic results were compared with those of liver at fibrosis stage. Our study suggests that it is possible to construct a PVA based phantom that mimics fibrous liver for the purpose of UEI imaging studies.

Mathematical modelling of microtubule-tau protein transients: Insights into the superior mechanical behavior of axon

• The theory of shear lag model is extended. • Viscoelastic shear lag model is developed to illustrate the mechanical behavior of axon under transient tensile loading. • The paper explains how the vulnerable axons protect themselves from overall damage over the course of evolution. • The paper explains how axons achieve an outstanding mechanical balance of high stiffness and toughness. Axons with staggered microtubules cross-linked by tau protein possess a remarkable mechanical balance of high specific stiffness and toughness. Owing to their viscoelastic nature, axons exhibit stress rate-dependent mechanical behavior, which is relevant to their selective vulnerability to damage in traumatic brain injury. A Kelvin–Voigt viscoelastic shear lag model is developed to elucidate the mechanical responses of axons under transient tensile force. Analytical closed-form expressions are derived to characterize the relative sliding, stress transfer and failure mechanism between microtubule and tau protein while the axon is stretched transiently. The results from the theoretical solutions elucidate how the MT-tau interface length and stress rate affect the mechanical responses of axon. It is found that axonal failure mechanism may be different under different loading conditions. Long microtubules are more vulnerable to rupture at high stress rate, yet short microtubules are likely to detach from microtubule bundles under large deformations. In the view of multi-level failure of axon, it is illustrated how the vulnerable axons protect themselves from overall damage, and how the axon can simultaneously achieve an outstanding mechanical balance of high specific stiffness and toughness.

Semi-analytical investigation on dynamic response of viscoelastic single-walled carbon nanotube in nanoparticle delivery

Dynamic response of simply supported single-walled carbon nanotubes (SWCNTs) with moving nanoparticle is studied based on nonlocal continuum mechanics formulations. The new differential equation of motion is proposed by considering the effect of inertia and Coriolis forces due to nanoparticle movement. The viscoelastic behavior of the SWCNTs is established by Kelvin–Voigt viscoelastic model. The governing differential equation obtained from Newton’s second law is discretized using the Galerkin method, and then, the effect of various parameters on dynamic response of the system is evaluated. Results show that the proposed moving mass model considering interaction and friction force effects between nanoparticle and SWCNTs reveals significant influence on dynamic response of the system. The dynamic amplitude obtained from moving mass model is almost 30% higher than the corresponding value obtained from moving load model. Moreover, the viscoelastic behavior eliminates the oscillations caused by natural vibrations, from the transient response of the SWCNTs. In addition, the friction between nanoparticle and SWCNTs has significant effect on dynamic response of the system, especially the steady-state response.

Exact analysis of imperfect angle-ply laminated panels with surface-bonded piezoelectric layers

The electromechanical behavior of an angle-ply laminated plate in cylindrical bending with surface-bonded piezoelectric layers is investigated. The interlaminar bonding of this smart laminate is described by a Kelvin–Voigt viscoelastic model. Besides, a matrix reduction method is employed to construct the transfer relations between the piezoelectric layer and the elastic strip. Based on the state-space approach, a state differential equation of the interfacial sliding displacements with respect to time variable is derived, from which the exact solutions are obtained in the time domain. Comparison with the existing techniques validates the high efficiency and excellent accuracy of the present analysis. Furthermore, the numerical results indicate that the intelligent laminate may lose partly or completely the load-bearing capacity and the function of sensor/actuators with time elapsing because of the viscoelastic interfaces.

On the dynamics of imperfect shear deformable microplates

This paper investigates the nonlinear forced dynamical behaviour of a geometrically imperfect viscoelastic shear-deformable microplate. The third-order shear deformation plate theory and the Kelvin–Voigt viscoelastic model are utilised in the framework of the modified version of the couple-stress theory to develop a model for the microsystem. The developed model is in the form of partial differential equations (PDEs) and accounts for geometric nonlinearities, damping nonlinearities, micro-scale size effects, and initial imperfection. Five coupled PDEs are derived for the five independent displacements and rotations. These PDEs are truncated to a set of nonlinearly coupled ordinary differential equations via application of a two-dimensional modal decomposition based on the Galerkin technique. The final set of equations consists of quadratic and cubic nonlinear terms for both damping and stiffness. An efficient numerical algorithm based on a continuation scheme is utilised to analyse the nonlinear forced vibration characteristics of such complicated system. The effects imperfection amplitude, damping nonlinearities, and micro-scale size on forced resonant vibration response are highlighted.

Nonlocal Thermal and Mechanical Buckling of Nonlinear Orthotropic Viscoelastic Nanoplates Embedded in a Visco-Pasternak Medium

This paper discusses the thermal and mechanical buckling of simply supported and clamped orthotropic viscoelastic graphene sheets (nanoplates) embedded in a visco-Pasternak elastic medium. For this purpose, the nonlocal continuum mechanical model is employed with two-variable plate theory. The material of the present nanoplate is assumed to be orthotropic and viscoelastic. The modified nonlinear Kelvin–Voigt viscoelastic model is utilized to formulate the constitutive relations depending on the viscoelastic structural damping coefficient. Moreover, the visco-Pasternak elastic medium is composed of both viscoelastic and shear layers. The viscoelastic layer includes a set of dashpots and elastic springs connected in parallel. In accordance with the two-variable theory, two governing equations are derived via Hamilton’s principle. These equations are analytically solved for various boundary conditions to obtain the explicit solution for critical buckling temperature and buckling load. The present buckling load and buckling temperature both are compared well with the published ones in the literature. In addition, various numerical studies are thoroughly carried out, concentrating on the influences of the plate geometric, nonlocal parameter, structural damping coefficient, elastic foundation parameters, foundation damping parameter and boundary conditions on the critical buckling load and temperature of the nanoplates. The results show that the involvement of the viscidity of the nanoplate and viscoelastic foundation enhances the strength of the nanoplates and therefore increases the resistance of them to external loads.

Nonlinear primary resonances of lattice sandwich beams with pyramidal truss core and viscoelastic surfaces

In this paper, the nonlinear forced vibration properties of the lattice sandwich beam with pyramidal truss core and viscoelastic surfaces subjected to harmonic external excitation are investigated. The nonlinear dynamic model of the lattice sandwich beam is formulated based on the Kelvin–Voigt viscoelastic model and discretized into nonlinear ordinary equations with multiple degrees of freedom by using the assumed mode method. The nonlinear amplitude–frequency curves of the steady-state responses of the beam are obtained by an iterative algorithm. The effects of the external excitation amplitude, the inclination angle of the truss core, and the viscoelastic coefficient of the surface material on the nonlinear forced vibration behaviors of the beam are analyzed. From the research results, it can be seen that the lattice sandwich beam shows very rich and novel nonlinear dynamic behavior. The viscoelastic damping of the surfaces can decrease the resonance amplitudes of most modes of the beam. An acceptable optimal inclination angle considering the equivalent mass density of the pyramidal truss core and the resonance amplitude of the beam is obtained. The evolution rules of the lattice sandwich beam with the material and structural parameters obtained from the preset study are helpful for the engineering applications of this kind of lightweight sandwich structures.