Voigt Viscoelastic Model Research Articles

Nonlinear Vibration Analysis of Fractional Viscoelastic Nanobeam

PurposeConsidering the size-dependent influence ignored by classical continuum mechanics, a new non-classical Euler–Bernoulli beam model is proposed in this paper. The new fractional viscoelastic nanobeam model is set up using the fractional Kelvin–Voigt viscoelastic model and Hamilton’s principle. And the new model studies the total effects of nonlocal elasticity, modified couple stress, and surface energy.MethodsThe model represented the fractional integral-partial differential governing equation is solved by Galerkin’s and predictor–corrector methods. First, the effects of nonlocal elasticity, modified couple stress, surface energy, and their coupling impact on the nonlinear time response of free vibration of fractional viscoelastic nanobeam are analyzed. Then, in the frame of nonlocal couple-stress elasticity and surface energy theory, the effects of different parameters on the nonlinear time responses of free and forced vibration of fractional viscoelastic nanobeam are discussed.Results and ConclusionThe numerical results show that the fractional order must be considered in the modeling of viscoelastic nanobeam, and the system’s damping enhances by the increase of fractional order. Moreover, owing to the correlation between fractional order and excitation frequency, the nonlinear time responses of fractional order to free and forced vibration are different.

Free damping vibration of functionally graded porous viscoelastic nonlocal microbeam with thermal effect

Combining classic Kelvin–Voigt viscoelastic model and two-phase local/nonlocal integral models (TPNIM), two-phase local/nonlocal integral viscoelastic models (TPNIVM) are proposed to study the free thermo-damping vibration of functionally graded porous microbeam with four different porous distribution patterns. The differential governing equations, boundary conditions, and constitutive constraints are formulated. The axial stress due to environmental temperature variation is derived explicitly. Several nominal variables are introduced to simplify the mathematical formulae. Laplace transformation is applied to obtain the explicit expression for bending deflection and moment. On considering the boundary conditions and constitutive constraints, a complex coefficients nonlinear equation is obtained to determine the complex characteristic frequency. A two-step numerical method is proposed to solve the elastic vibration frequency and damping ratio. The influence of nonlocal parameters, porous distribution parameters, and environmental temperature variation on the elastic vibration frequency and damping ratio is investigated numerically for different boundary conditions. The numerical results show that, under different boundary conditions, the compressive thermal stresses lead to the decrease of nominal vibration frequencies, and the increase of critical viscos coefficient and the damping ratio. With the increase of and the decrease of , the damping ratio increases and decrease for strain-driven (εD) and stress-driven (σD) nonlocal models, respectively.

Relationship between nonlinear free vibration behavior and nonlinear forced vibration behavior of viscoelastic plates

In this paper, the relationship between nonlinear free vibration behavior and nonlinear forced vibration behavior of viscoelastic plates is studied based on Reddy higher order shear deformation theory and von Kármán geometric nonlinearity assumption. Kelvin–Voigt viscoelastic model is adopted to describe the viscoelasticity of the plate. According to the geometric equations and constitutive equations, the elastic potential energy, kinetic energy and virtual work done by the external force and viscous damping force are obtained. Furthermore, the nonlinear governing differential equations of the system are achieved by using Hamilton’s principle. On the basis of the harmonic balance method and Runge–Kutta method, the nonlinear free vibration responses and nonlinear forced vibration responses of the system are presented, respectively. The major innovation of the manuscript is that the nonlinear frequency ratio derived from the nonlinear free vibration of the system, is in good agreement with the resonant frequency ratio obtained from the nonlinear forced vibration of the system. Also, the viscous damping coefficient has a little effect on the nonlinear frequency ratio.

Research on the Attenuation Characteristics of High-Frequency Elastic Waves in Rock-Like Material.

In order to study the frequency-dependent attenuation characteristics of high-frequency elastic waves in rock-like materials, we conducted high-frequency elastic wave attenuation experiments on marble, granite, and red sandstone rods, and investigated the frequency dependence of the attenuation coefficient of high-frequency elastic waves and the frequency dependence of the attenuation of specific frequency components in elastic waves. The results show that, for the whole waveform packet of the elastic wave signal, the attenuation coefficient and the elastic wave frequency have an approximate power relationship, with the exponents of this power function being 0.408, 0.420, and 0.384 for marble, granite, and red sandstone, respectively, which are close to 1/2 the exponent value obtained theoretically by the Kelvin–Voigt viscoelastic model. However, when the specific frequency components are tracked during the elastic wave propagation, the exponents of the power relationship between the attenuation coefficient and frequency are 0.982, 1.523, and 0.860 for marble, granite, and red sandstone, respectively, which indicate that the relationship between the attenuation coefficient and frequency is rock-type dependent. Through the analysis of rock microstructure, we demonstrate that this rock-type-dependent relationship is mainly caused by the scattering attenuation component due to the small wavelength of the high-frequency elastic wave. Therefore, the scattering attenuation component may need to be considered when the Kelvin–Voigt model is used to describe high-frequency elastic wave attenuation in rock-like materials. The results of this research are of good help for further understanding the attenuation characteristics of high-frequency elastic waves in rock-like materials.

A Novel Nonlinear Dynamic Model for Predicting Contact Bounce of Electromagnetic Relays With Flexible Spring Components

Contact bounce is a common phenomenon in the closing process of electromagnetic relays, which seriously affects the electrical contact reliability of the system. The whole closing process of flexible spring components consists of the free movement stage and the contact bounce stage, which increases the complexity of analyzing the electrical contact behaviors of electromagnetic relays. In this study, a novel nonlinear dynamic model of the flexible spring components in electromagnetic relays is established to analyze the physical mechanism of the electrical contact bounce. Considering the electromagnetic effect, flexible deformation, contact collision, and welding force, the proposed mathematical model can capture the dynamic closing process accurately and collect bounce parameters efficiently. The Kelvin–Voigt viscoelastic contact model is employed to evaluate the nonlinear contact force between the static and movable contacts. Based on the Galerkin method, the nonlinear equations of motion for the movable spring within electromagnetic relays are formulated. A series of experimental work are designed to measure the typical waveforms of displacements of the movable contact and currents in the coil during the closing process of the movable spring. Furthermore, the theoretical results are validated by comparing with the experimental data. The dependences of some key factors, including the structural, material, and electrical contact parameters on the contact bounce characteristics, are investigated explicitly. The theoretical results attest that the proposed model can accurately predict the dynamic behaviors of the movable spring, and the contact bounce can be effectively suppressed by adjusting the parameters of the flexible spring component.

Viscoelastic rheological models for guided acoustic waves: Single mode at high frequency and second harmonic generation of Lamb and SH modes

The previous published work by Othmani et al. invites a further useful discussion on how the viscoelastic rheological model affects the second harmonic and single modes of guided waves. However, this paper has two main objectives. The first objective is to establish a simple relationship between the Rayleigh waves velocity (VR), fundamental Lamb modes (A0 and S0), and the shear wave velocity (VS) in unidirectional viscoelastic carbon–epoxy at high frequency. The second objective is to study the second harmonic Lamb modes. Numerical codes based on Legendre polynomial and Newton–Raphson methods are developed and implemented to predict dispersion and attenuation curves. The second harmonic generation was investigated with the hysteretic and Kelvin–Voigt viscoelastic models. Results showed that rheological models have a much larger effect on the attenuation curves than the ones on the dispersion trajectories. Additionally, the optimal angles of Lamb and SH modes in unidirectional carbon–epoxy under Snell‐Descartes law was investigated. The present numerical results may render as a useful message for the rheological community.

Nonlinear dynamic responses of porous FG sandwich cylindrical shells with a viscoelastic core resting on a nonlinear viscoelastic foundation

The nonlinear vibration, including primary resonance (PR) of porous FG sandwich (PFGS) cylindrical shells with a viscoelastic core is investigated in this research. The structure considered is on a nonlinear viscoelastic foundation composing of two Pasternak and Winkler foundations augmented with a nonlinear cubic stiffness by Kelvin–Voigt viscoelastic model. Two types of PFGS cylindrical shells are considered and viscoelastic core is made of Kelvin–Voigt type material. Utilizing Donnell’s shell theory, von-Kármán equation and the Galerkin’s method, a discretized governing equation is derived and verified. The multiple scales method is applied to investigate the nonlinear PR of the shells.

Nonlinear post‐buckling analysis of viscoelastic nano‐scaled beams by nonlocal integral finite element method

AbstractThe viscoelastic buckling and nonlinear post‐buckling behavior of nano‐scaled beams are analyzed using the nonlocal integral elasticity theory. Eringen's nonlocal theory is one of the well‐known and popular size‐dependent theories, which has been used by several researchers to study the mechanical behavior of, mostly, the elastic nanostructures. A finite element method is developed using Hamilton's principle based on the two‐phase nonlocal integral theory and taking into account the buckling related terms and viscoelastic effects. The corresponding formulations are derived by implementing the Euler–Bernoulli beam theory and Kelvin–Voigt viscoelastic model. Furthermore, for analyzing the post‐buckling and viscoelastic buckling behavior, the nonlinear strains, and initial displacement (imperfection) have been considered. By employing the variational relations, the governing equations are obtained and then solved numerically using the finite difference and Newton–Raphson methods. Because of the finite element nature of the current method, various boundary conditions can be properly implemented. The results of the current study are compared with those available in the literature and the effects of nonlocal parameter, viscoelastic parameter, axial compressive load and boundary conditions on the viscoelastic buckling, and postbuckling behavior have been investigated.

Time-domain asymptotic homogenization for linear-viscoelastic composites: mathematical formulae and finite element implementation

Asymptotic homogenization (AH) is a rigorous method used for modeling multiscale mechanical behavior in periodic composites. Unlike conventional AH methods, which predict the effective viscoelasticity using the elastic–viscoelastic correspondence principle, an AH method was proposed to directly calculate the effective material properties in the time domain, thereby enabling the complex transform to be omitted between the time and Laplace domains. This time-domain AH method is reformulated based on the integral form rather than the differential form of the Kelvin–Voigt viscoelastic model. Thus, multiple time-domain-based characteristic elasticity and viscosity displacement tensors are replaced by one integral form characteristic displacement tensor in the polymer matrix composite. Using mathematical formulae, a numerical implementation method was then developed by establishing location- and time-related stress loads. An in-house Python subroutine was also programmed to compute the effective viscoelastic properties from the finite element implementation results. Unidirectional and woven fabric composite homogenization results were comparable to those obtained using the conventional representative volume element method, indicating high effectiveness and accuracy. The results suggest that the proposed time-domain AH method is a powerful tool for modeling the two-scale mechanical behavior of viscoelastic composites and is compatible with the existing commercial software.

Nonlinear Models of Thermo-Viscoelastic Materials.

The paper develops a general scheme for viscoelastic materials, where the constitutive properties are described by means of measures of strain, stress, heat flux, and their time derivatives. The constitutive functions are required to be consistent with the second law of thermodynamics. Indeed, a new view is associated with the second law: the non-negative expression of the entropy production is set equal to a further constitutive function. The introduction of the entropy production as a constitutive function allows for a much wider range of models. Within this range, a scheme to obtain nonlinear models of thermo-viscoelastic materials subject to large deformations is established. Notably, the Kelvin–Voigt, Maxwell, Burgers, and Oldroyd-B viscoelastic models, along with the Maxwell–Cattaneo heat conduction, are obtained as special cases. The scheme allows also for modelling the visco-plastic materials, such as the Prandtl–Reuss work-hardening function and the Bingham–Norton fluid.

Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Beams.

Fractional models and their parameters are sensitive to intrinsic microstructural changes in anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we study the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelasticity is described by a distributed-order fractional model. We employ Hamilton's principle to obtain the equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin-Voigt viscoelastic model of order α. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, where the linear counterpart is numerically integrated through a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, yielding a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of α-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law decay rates, amplitude super-sensitivity at free vibration, and bifurcation in steady-state amplitude at primary resonance.

Analysis of a quarter car suspension based on a Kelvin–Voigt viscoelastic model with fractional-order derivative

The dynamical analysis of a quarter car suspension based on the Kelvin–Voigt​ viscoelastic model with fractional order derivative is analytically and numerically studied in this paper. Depending on the non-dimensional quadratic stiffness parameter, the potential configuration associated with the system under study is possible to obtain monostable and, asymmetric bistable configurations. Thanks to the multiple time scale method, the effects of viscoelastic parameters on the vibration amplitude of the quarter car suspension model are analytically investigated and the numerical simulations confirmed the analytical results. Moreover, the vibration amplitude of the quarter car suspension model can display periodic and dissipative chaotic behaviors by varying the non-dimensional quadratic stiffness, the fractional-order, and the road profile parameters. One scroll and double scroll chaotic attractors are generated by varying the non-dimensional quadratic stiffness parameter. Finally, the threshold conditions for the appearance of homoclinic bifurcation (left well and right well) in the case of asymmetric bistable potential are investigated using Melnikov’s criterion. Melnikov’s predictions are validated numerically by using the basin of attraction of initial conditions. It is found that the decrease of fractional order contributes to increasing the regular motion domain up to a critical value of the fractional-order and after it, decreases rather.

Application of quartz crystal microbalance with dissipation (QCM-D) to study low-temperature adsorption and fouling of milk fractions on stainless steel

In food manufacturing, fouling of surfaces has deleterious effects on the efficiency of operations. The intensity of fouling varies with product formulation and processing temperature. Quartz crystal microbalance with dissipation (QCM-D) is a small-scale tool with the potential to predict fouling on a larger scale. The adsorption of milk fractions on stainless steel was investigated at 25, 50, and 65 °C using QCM-D. Mass density was quantified using the Voigt viscoelastic model and a modified Hill model. Permeate (pH 6.6) was unstable at 65 °C, near the precipitation temperature of calcium phosphate. Acid whey (pH 4.6) demonstrated significant constant-rate adsorption at long processing times. It is anticipated that linear adsorption rates at extended times can be used to predict fouling propensity at a commercial scale. These results provide the foundation for the use of QCM-D to study milk adsorption and fouling at higher temperatures (>65 °C).

Combination resonance analysis of imperfect functionally graded conical shell resting on nonlinear viscoelastic foundation in thermal environment under multi-excitation

In this work, nonlinear forced vibrations of truncated conical shells are presented using a semi-analytical method. The material properties are varied along the thickness direction as a power law distribution. The functionally graded truncated conical shells are exposed to external harmonic load and placed in the thermal environment and have an initial imperfection. Furthermore, the functionally graded truncated conical shells rests on generalized nonlinear viscoelastic foundations which consisted of a Winkler and Pasternak foundation parameters augmented by a Kelvin–Voigt viscoelastic model and a nonlinear cubic stiffness. The fundamental equations are extracted using first-order shear deformation theory in conjunction with nonlinear von Kármán relationships. The partial differential equations of truncated conical shells are reduced through Galerkin’s method, and the result is extracted using the multiple scales method. To analyze the resonance analyses, a two-term external excitation is considered. In this regard, various secondary resonances are investigated, and finally, the analyses about combination resonances are represented. To investigate the presented approach, a comparison study is performed with those addressed by other researchers. To analyze the nonlinear combination resonance behavior of truncated conical shells, the effect of geometrical characteristics, material properties, power law index, thermal effects, external load amplitude, and initial imperfection are examined. Finally, the steady-state responses of the nonlinear system are analyzed. As one of the most interesting results, the softening behavior of truncated conical shells with inverse quadratic distribution is the most, and for the quadratic distribution is the least.

Damped forced vibration analysis of layered functionally graded thick beams with porosity

The following article presents the damped forced vibration of layered functionally graded thick beams including material porosities. In case of very thick beams, beam theories fail to satisfy boundary conditions and to predict the mechanical response accurately. So, the two-dimensional (2D) plane continuum model is exploited to model a thick functionally graded layered beam. The beam is composed from three- layers with functionally graded porous materials. The porosity is described by three different distribution models through the layer thickness. Applied forces to the functionally graded beam are assumed to be sinusoidal harmonic point load in time domain. The Kelvin—Voigt viscoelastic constitutive model is used to simulate damping effect. The governing equations are obtained by using Lagrange's equations. In frame of finite element analysis, twelve .node 2D plane element is exploited to discretize the space domain of thick beam. In the solution of the dynamic problem, the Newmark average acceleration method is used. Numerical studies illustrate effects of porosity distribution, stacking sequence, and graduation constant on the dynamic responses of layered functionally graded porous thick beams. The results show that the porosity function, stacking sequences and the damping ratio have a vital role in dynamic response of functionally graded beams. The proposed model can be used in nuclear, marine, and aerospace technologies.

A quadrature-free Legendre polynomial approach for the fast modelling guided circumferential wave in anisotropic fractional order viscoelastic hollow cylinders

Compared to the traditional integer order viscoelastic model, a fractional order derivative viscoelastic model is shown to be advantageous. The characteristics of guided circumferential waves in an anisotropic fractional order Kelvin–Voigt viscoelastic hollow cylinder are investigated by a quadrature-free Legendre polynomial approach combining the Weyl definition of fractional order derivatives. The presented approach can obtain dispersion solutions in a stable manner from an eigenvalue/eigenvector problem for the calculation of wavenumbers and displacement profiles of viscoelastic guided wave, which avoids a lot of numerical integration calculation in a traditional polynomial method and greatly improves the computational efficiency. Comparisons with the related studies are conducted to validate the correctness of the presented approach. The full three dimensional spectrum of an anisotropic fractional Kelvin–Voigt hollow cylinder is plotted. The influence of fractional order and material parameters on the phase velocity dispersion and attenuation curves of guided circumferential wave is discussed in detail. Moreover, the difference of the phase velocity dispersion and attenuation characteristics between the Kelvin–Voigt and hysteretic viscoelastic models is also illustrated. The presented approach along with the observed wave features should be particularly useful in non-destructive evaluations using waves in viscoelastic waveguides.

Nonlinear vibration of fractional Kelvin–Voigt viscoelastic beam on nonlinear elastic foundation

The main objective of the present study is to analyze the nonlinear vibration behavior of fractional Kelvin–Voigt viscoelastic beams on a nonlinear elastic foundation under harmonic excitation. Attention is concentrated on the primary, superharmonic, and subharmonic resonances with the fractional Kelvin–Voigt constitutive model. To describe some viscoelastic material behaviors such as biomaterials, intelligent and polymeric materials, the fractional Kelvin–Voigt constitutive equation is used. Governing equations are derived using Hamilton’s principle based on the Euler–Bernoulli beam theory, with nonlinear elastic foundation and Von Karman’s nonlinearities due to stretching. These nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by the Galerkin projection technique. The method of multiple scales is utilized to obtain the response of beams under hard and soft excitations. Results are verified by the available literature. A parametric analysis is conducted to determine the influence of the fractional Kelvin–Voigt viscoelastic model on primary and secondary resonances. The numerical results illustrate that effect of the fractional Kelvin–Voigt model on the frequency-response and amplitude-response is remarkable. Therefore, the obtained results provide a useful benchmark for further nonlinear analysis of fractional Kelvin–Voigt viscoelastic beams with elastic foundations.

Hygrothermal forced vibration of a viscoelastic laminated plate with magnetostrictive actuators resting on viscoelastic foundations

The article presents a vibration analysis of a laminated composite sandwich plate containing viscoelastic layers in the core and faces of the plate. The sandwich plate includes magnetostrictive actuating layers, rests on a three-parameter viscoelastic medium, and undergoes to hygrothermal environmental conditions and in-plane forces in $$x$$ and $$y$$ directions. The viscoelastic layers are considered as Kelvin–Voigt viscoelastic model. The partial differential equations system is deduced according to the dynamic version of the principle of virtual displacements and is based on a theory that accounts for the exponential distribution of transverse shear deformation. The analytical Navier’s solution type is obtained to investigate the parametric effect of the location of smart layers, lamination schemes, modes, the feedback gain control value, viscoelastic structural damping, thickness ratio, the aspect ratio, viscoelastic layer thickness-to-magnetostrictive layer thickness ratio, foundation, in-plane forces, temperature and moisture on vibration characteristics of the sandwich plate. The outcomes show that the deformation and vibration in the structure can be controlled by changing the smart layer position and value of the feedback gain control parameter as well as the studied system stability can be affected by the value of viscoelastic structural damping and foundation constants. Besides, the absorption of moisture in matrix material increases the deflection of structure and increases the damping time. The results of this study can contribute to the development of the design of smart structural applications, especially those that require foundations to support them and deal with their dynamic response as well as provide a reference framework for studies of the impacts of humid environments on such structures in the presence of in-plane forces, especially, in the applications subjected to various forces and environmental conditions. Many of these systems require foundations to support the structures and deal with their dynamic response such as civil and mechanical structural applications and the aerospace and aeronautical vehicles.

Vibration Analysis of Viscoelastic FGM Nanoscale Plate Resting on Viscoelastic Medium Using Higher-order Theory

The present article aims essentially to present an analytical and numerical method which makes it possible to study the damped vibrations of viscoelastic FGM nanoplates resting on viscoelastic foundations. A new model for the higher-order shear deformation plate theory is coupled with the internal Kelvin - Voigt viscoelastic model and the three-parameter viscoelastic foundation model for the purpose of reducing and minimizing the vibration response of the system. It is widely admitted that the mechanical properties of these new functionally gradient materials (FGMs) vary according to the thickness of the plate and depend on its volume fraction. The use of FGM plates seems to be an ideal solution for the study of free vibrations because of their multifunctionality that is fully integrated with the nonlocal Eringen effect. The dynamic response of such a complex system has been investigated by varying the aspect ratio of the plate, the mechanical characteristics of the material used, the internal and external damping and the foundation rigidity. The results obtained, with and without the nonlocal effect, were compared with those of different models of higher-order theories and under various boundary conditions; they were found to be in good agreement with those reported in the literature.

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.