Viscoelastic Constitutive Relation Research Articles

Forced vibration characteristics of viscoelastic variable stiffness laminated composite plates using time and frequency domain approaches

The linear forced vibration characteristics of viscoelastic variable stiffness laminated composite plates (VVSLC) are studied using a generalized Maxwell model and finite element method. The integral form of the viscoelastic constitutive relation is converted to the incremental form for finite element formulation based on the Reissner–Mindlin plate theory. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response directly in the time domain is obtained using shooting technique coupled with Newmark’s time integration method. The implementation of shooting technique for Boltzmann integral-based viscoelasticity for curvilinear fibre composite plates is done for the first time in this study. For the comparison purpose, the response/resonance frequency/modal loss factor is also obtained using equivalent complex modulus based viscoelastic correspondence principle. It is observed that the variation in fibre orientation and boundary conditions leads to significant variations in response, stress/moment resultant amplitude and the damping factor of the VVSLC plates. Further, the present time domain based approach is capable of predicting damping factor at all forcing frequencies whereas complex eigenvalue analysis can predict damping factor only at discrete resonance frequency. Based on the detailed studies, it is found that the curvilinear fibre composite plates depict a significant reduction in response/moment resultants compared to straight fibre composite plates.

Random flutter analysis of a novel binary airfoil with fractional order viscoelastic constitutive relationship

The fractional order model emerges naturally when the integer order model fails to meet practical requirements and finds extensive applications in engineering practice. However, there is still limited research on the random flutter of fractional order viscoelastic binary airfoils. This study aims to explore the random flutter of such airfoils under the influence of Gaussian white noise excitation by utilizing the pth moment Lyapunov exponent. Firstly, a two-degree-of-freedom fractional order stochastic dynamic equation is established for the viscoelastic binary airfoil through the introduction of fractional order damping. The singular perturbation method is then employed to derive the pth moment Lyapunov exponent of the system. An approximate analytical solution of the pth moment Lyapunov exponent is obtained through Fourier series expansion, which exhibits good agreement with numerical results obtained from Monte Carlo simulations. Subsequently, a comprehensive analysis is conducted to examine the impacts of fractional order, airflow velocity, noise intensity, and key structural parameters on the stochastic flutter of the fractional order viscoelastic binary airfoil. The findings indicate that compared with the typical binary airfoil model, the flutter speed of the fractional-order viscoelastic binary airfoil is significantly increased, the stability of the system is highly sensitive to the natural frequency of the system, and the fractional-order factor has a positive effect on the stochastic stability of the binary airfoil before the critical value. The above research results can guide the design optimization of the fractional-order viscoelastic airfoil.

Lifetime assessment of deep-sea manned cabin with PMMA pressure-resistant hull prior to micro-cracks

Fully-transparent cabin type, utilized in manned submersibles, is typically constructed using viscoelastic PMMA (polymethyl methacrylate) material. Accurate fatigue life assessment of manned cabin is of paramount importance to ensure structural safety. In this study, a calculation process for evaluating the fatigue life of PMMA manned cabins based on a continuum damage mechanics model combined with the viscoelastic constitutive relationship of the PMMA material in response to the absence of related specification is proposed. Fatigue tests of PMMA material are performed for determination of damage evolution model parameters and the calculation process is validated by the fatigue test data. Tailored subroutines are then developed by using ABAQUS software tool, respectively for structural analysis of a typical manned cabin with PMMA pressure-resistant hull, openings, hatches and penetration plates and simulating trapezoidal loading history acting on manned cabin. Subroutine coupling is accordingly employed to conduct parametric analysis on fatigue crack initiation life of the fully-transparent manned cabin with consideration of different outside pressure level and operation time and valuable results are obtained for identifying key areas of concern. This study intends to contribute to further cabins safety enhancement, as well as to design optimization and fatigue analysis of underwater structures involving PMMA materials.

Comparing finite viscoelastic constitutive relations and variational principles in modeling gastrointestinal soft tissue deformation

The mechanical attributes of soft tissues within the gastrointestinal (GI) tract are crucial for the effective operation of the GI system, and alterations in these properties may play a role in motility-related disorders. Various constitutive modeling approaches have been suggested to comprehend the response of soft tissues to diverse loading conditions. Among these, hyperelastic constitutive models based on finite elasticity have gained popularity. However, these models fall short in capturing rate- and time-dependent tissue properties. In contrast, finite viscoelastic models offer a solution to overcome these limitations. Nevertheless, the development of a suitable finite viscoelastic model, coupled with a variational formulation for efficient finite element (FE) implementation, remains an ongoing challenge. This study aims to address this gap by developing diverse finite viscoelastic constitutive relations and applying them to characterize soft tissue. Furthermore, the research explores the creation of compressible, nearly incompressible, and incompressible versions of viscoelastic constitutive relations, along with their variational formulation, to facilitate efficient FE implementation. The proposed model demonstrates remarkable accuracy in replicating experimental results, achieving an R2 value exceeding 0.99.

A time-domain integration method with equivalent complex damping model based on viscoelastic material constitutive relation

The complex damping model is only applied in frequency-domain. Based on the constitutive relation of viscoelastic materials, a vibration equation is equivalent to the constitutive relation of complex damping theory and the frequency response function satisfying the causality constraint. Compared with the existing equivalent complex damping model, the proposed equivalent damping model can consider more natural frequencies. The calculation formulas of Gauss precise integral method and improvement constant average acceleration method are derived. By the equivalent complex damping, the seismic time-history response of the typical example could carry on the numerical calculation. Compared with the Gauss precise integral calculation results of complex damping vibration equation, some conclusions can be analyzed. The proposed two methods could avoid divergent phenomenon in the time-domain numerical solution of complex damping vibration equation. The time-domain integral method of equivalent complex damping theory is stable and convergent. The Gauss precise integral method has strong equivalence and big computational complexity. The improvement constant average acceleration method has weak equivalence and small computational complexity.

Vibration Characteristics of a Functionally Graded Viscoelastic Fluid-Conveying Pipe with Initial Geometric Defects under Thermal–Magnetic Coupling Fields

In this study, the Kevin–Voigt viscoelastic constitutive relationship is used to investigate the vibration characteristics and stability of a functionally graded viscoelastic(FGV) fluid-conveying pipe with initial geometric defects under thermal–magnetic coupling fields. First, the nonlinear dimensionless differential equations of motion are derived by applying Timoshenko beam theory. Second, by solving the equilibrium position of the system, the nonlinear term in the differential equations of motion is approximated as the sum of the longitudinal displacement at the current time and longitudinal displacement relative to the position, and the equations are linearized. Third, these equations are discretized using the Galerkin method and are numerically solved under simply supported conditions. Finally, the effects of dimensionless temperature field parameters, dimensionless magnetic field parameters, thermal–magnetic coupling, initial geometric defect types, and the power-law exponent on the complex frequency of the pipe are examined. Results show that increasing the magnetic field intensity enhances the critical velocity of first-order mode instability, whereas a heightened temperature variation reduces the critical velocity of first-order diverge instability. Under thermal–magnetic fields, when the magnetic field intensity and temperature difference are simultaneously increased, their effects on the complex frequency can partially offset each other. Increasing the initial geometric defect amplitude increases the imaginary parts of the complex frequencies; however, for different types of initial geometric defect tubes, it exhibits the most distinct influence only on a certain order.

An adaptive coupling of PD-CCM model involving a viscoelastic constitutive relation for quasi-brittle material dynamic failure

A new viscoelastic bond-based peridynamic model (VBPD) is developed to describe the nonlinear deformation and dynamic failure of quasi-brittle materials under different loading rates. This model is established on the viscoelastic constitutive equation of classical continuum mechanics, which involves damage and strain rate effects. The model superbly generalizes the definition of strain rate in classical continuum mechanics phenomenally to the bond stretch rate in the bond-based peridynamics under one-dimensional strong impact loads. To describe the progressive damage of the peridynamic bond, The scalar value function related to bond stretch and bond stretch rate is introduced in the model to define the deformation of the peridynamic bond. Moreover, the model also considers the history of bond force under cyclic load. To improve the computational efficiency of simulation, the adaptive coupling of peridynamics and the classical continuum mechanics (PD-CCM) model established by the Morphing method is used to simulate the failure process of quasi-brittle materials. By combining the VBPD model and the PD-CCM model, the VBPD model is utilized to simulate the deformation and failure process of quasi-brittle materials under dynamic load with the least peridynamic subdomain. Finally, the validity of the proposed model is verified by three two-dimensional numerical examples.

Well-posedness of the governing equations for a quasi-linear viscoelastic model with pressure-dependent moduli in which both stress and strain appear linearly

The response of a body described by a quasi-linear viscoelastic constitutive relation, whose material moduli depend on the mechanical pressure (that is one-third the trace of stress) is studied. The constitutive relation stems from a class of implicit relations between the histories of the stress and the relative deformation gradient. A-priori thresholding is enforced through the pressure that ensures that the displacement gradient remains small. The resulting mixed variational problem consists of an evolutionary equation with the Volterra convolution operator; this equation is studied for well-posedness within the theory of maximal monotone graphs. For isotropic extension or compression, a semi-analytic solution of the quasi-linear viscoelastic problem is constructed under stress control. The equations are studied numerically with respect to monotone loading both with and without thresholding. In the example, the thresholding procedure ensures that the solution does not blow-up in finite time.

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.

Dynamic fracture investigation of concrete by a rate-dependent explicit phase field model integrating viscoelasticity and micro-viscosity

To investigate dynamic fracture mechanisms of quasi-brittle materials, this work proposes a rate-dependent phase field model that integrates both macroscopic viscoelasticity and micro-viscosity to reflect the rate effects by free water and unhydrated inclusions. Based on the unified phase field theory, the model introduces a linear viscoelastic constitutive relation in effective stress space to consider the macro-viscosity of the bulk material. Additionally, the micro-force balance concept is utilized with the micro-viscosity to derive a parabolic phase field evolution law that accurately describes the dynamic micro-crack development. Explicit numerical solution schemes are established for the governing equations by developing VUEL and VUMAT subroutines in ABAQUS. This eliminates the convergence issue in implicit phase field modelling. Four typical benchmarks are investigated to validate the proposed model for macroscale and mesoscale heterogeneous problems. It is found that the proposed model can well capture the crack branching, delaying characteristic of micro-crack growth, and increase of macroscopic strength under higher strain rates. Using real meso‑structures from CT images, the complicated dynamic behaviour of concrete is investigated which yields deeper insight into stress wave propagation, crack evolution and load-carrying capacities.

Strain gradient viscoelasticity theory of polymer networks

A physically-based strain gradient viscoelasticity theory is proposed specially for polymer networks, accounting for the strain gradient effect and the history-dependent behavior, where the microstructure-dependence and history-dependence of stress and hyperstress are interpreted physically and quantitatively. The chain representation of the Helmholtz free energy density is transferred to the strain gradient continuum field representation through the geometric (spatial) connection between the chain stretch ratio and the continuum deformation measure. The necessity of using strain gradients to characterize asymmetric deformation of the microstructure of polymer networks serves as the origin of microstructure-dependent hyperstress, which is work-conjugated to the strain gradient field. The free energy per chain is assumed history-dependent to account for chain-environment interactions (e.g., friction between segments), which ultimately results in the history-dependent behavior of polymeric solids. A general constitutive relation is provided, which together with the assumed network structure, gives a concrete one. It is shown that independent of the assumed network structure, the stress, and the hyperstress share the same dimensionless relaxation function. A strain gradient viscoelastic constitutive relation based on the eight-chain network is applied to the analysis of polymer nanocomposites, where the complex moduli are rationally defined and calculated numerically. It is found that the strain gradient effect can be viewed as an amplifier that transforms the strain-induced stress into the effective stress.

Finite-difference modeling with topography using 3D viscoelastic parameter-modified free-surface condition

The free-surface boundary condition is a crucial aspect in the numerical modeling of (visco)elastic wave equations, especially when using a finite-difference (FD) method in the presence of surface topography. The parameter-modified method is a widely used approach to solve this problem. In this regard, the adaptive Poisson’s ratio parameter-modified method has proven to be effective in accurately simulating seismic surface waves within the FD discretization framework. Based on the equivalent medium theory, vacuum approximation, and mathematical limit, we develop a viscoelastic parameter-modified (VPM) method for the implementation of the free-surface boundary condition in the 3D viscoelastic wave equation. Our approach modifies the viscoelastic constitutive relation and density at the free surface and provides a formulation in terms of displacement and stress. We determine that our VPM method is more general than the viscoelastic stress-image method as it includes the latter as a limit case when the Poisson’s ratio equals zero. The presented free-surface method is represented implicitly within the FD grid, and we provide implementation details when simulating surface waves with topography in the standard staggered-grid FD scheme. We support our theory’s feasibility and accuracy through numerical examples.

Construction of 2-2 Type Cement-Based Piezoelectric Composites’ Mechanic–Electric Relationship Based on Strain Rate Dependence

It has been found that the mechanic-electric response of cement-based piezoelectric composites under impact loading is nonlinear. Herein, we prepared a 2-2 cement-based piezoelectric composite material using cutting, pouring, and re-cutting. Then, we obtained the stress-strain and stress-electric displacement curves for this piezoelectric composite under impact loading using a modified split Hopkinson pressure bar (SHPB) experimental apparatus and an additional electrical output measurement system. Based on the micromechanics of the composite materials, we assumed that damage occurred only in the cement paste. The mechanical response relationship of the piezoelectric composite was calculated as the product of the viscoelastic constitutive relationship of the cement paste and a constant, where the constant was determined based on the reinforcement properties of the mechanical response of the piezoelectric composite. Using a modified nonlinear viscoelastic Zhu-Wang-Tang (ZWT) model, we characterized the stress-strain curves of the piezoelectric composite with different strain rates. The dynamic sensitivity and stress threshold of the linear response of the samples were calibrated and fitted. Thus, a mechanic-electric response equation was established for the 2-2 type cement-based piezoelectric composite considering the strain rate effects.

Numerical study of viscoelastic upstream instability

In this work, we report numerical results on the flow instability and bifurcation of a viscoelastic fluid in the upstream region of a cylinder in a confined narrow channel. Two-dimensional direct numerical simulations based on the FENE-P model (the finite-extensible nonlinear elastic model with the Peterlin closure) are conducted with numerical stabilization techniques. Our results show that the macroscopic viscoelastic constitutive relation can capture the viscoelastic upstream instability reported in previous experiments for low-Reynolds-number flows. The numerical simulations reveal that the non-dimensional recirculation length (LD) is affected by the cylinder blockage ratio (BR), the Weissenberg number (Wi), the viscosity ratio (β) and the maximum polymer extension (L). Close to the onset of upstream recirculation, LD with Wi satisfy Landau-type quartic potential under certain parameter space. The bifurcation may exhibit subcritical behaviour depending on the values of L2 and β. The parameters β and L2 have nonlinear influence on the upstream recirculation length. This work contributes to our theoretical understanding of this new instability mechanism in viscoelastic wake flows.

Multi-field formulations for solving plane problems involving viscoelastic constitutive relations

This article reports a multi-field numerical formulation for solving plane problems involving viscoelastic materials. Stress fields satisfying equilibrium equations are constructed using Airy’s potentials which are expressed as a linear combination of C2 basis functions. The strain field is derived from a continuous displacement field obtained from a linear combination of C0 basis functions. An appropriate linear combination of these stress and displacement basis functions is determined such that the resulting stress and strain fields satisfy the constitutive relation subjected to the satisfaction of the constraints arising from the boundary conditions. Since a viscoelastic constitutive relation involves stress, strain, and their rates, stress and displacement degrees of freedom or their rates can be considered as optimization variables for minimizing the error in satisfying the constitutive relation. Two Algorithms are proposed based on this choice of optimization variable. Accuracy and efficiency of the proposed algorithms are studied through five different boundary value problems involving four forms of the viscoelastic constitutive relations and for two loading histories. Using the developed rectangular element, viscoelastic beam bending problem is solved for the different constitutive relations studied.

Dynamic Stiffness Determination of the Rubber Bushing Fem Modelwith Viscoelastic Material Behaviour

Abstract Presented paper deals with determination of dynamic stiffness of rubber bushing. Viscoelasticity of rubber is introduced. Next rheological models are shown and determined. Theory of the viscoelastic constitutive relation are done with storage modulus and loss modulus. Rubber bushing FEM model was prepared for calculation of dynamic stiffness through dynamic harmonic analysis in MSC Marc. Results from simulations are compared to experimental measurement data.

GPS Determined Asymmetric Deformation Across Central Altyn Tagh Fault Reveals Rheological Structure of Northern Tibet

AbstractWe establish a continuous GPS transect crossing the central Altyn Tagh fault at 90°E with eight years of observations. GPS velocities along this profile and another one crossing the fault at 86°E suggest a fault slip rate of 12.4 ± 0.7 mm/yr, but with asymmetric straining of adjacent terrain. On the south side, ∼8.2 mm/yr of left‐lateral shear is absorbed across a region ∼210 km from the fault, but only ∼4.2 mm/yr is found on the north side. This estimate of slip rate is ∼30% larger than the consensus estimate of previous models. By treating the deforming regions as elastic plates with different thicknesses overlying a substrata that obeys a linear Maxwell viscoelastic constitutive relationship, we infer a viscosity of ∼5.1 × 1019 Pa s (between 3.5 and 9.1 × 1019 Pa s at 1‐σ) on the south side, beneath northern Tibetan Plateau. This low viscosity, compared to some estimates for the asthenosphere, concurs with the Tibetan Plateau being underlain by a relatively hot and weak lower crust and upper mantle. The effective elastic thickness on the south side is 16.5–20 km, which is significantly smaller than that of the Tarim Basin of >60 km.

Viscoelastic Behavior and Constitutive Relation of Heavy Crude Oils.

Heavy crude oil exhibits very complex viscoelastic behaviors due to its complex composition of resins, asphaltenes, saturates, and aromatics. It has a great influence on oil production and transportation. In this work, the viscoelastic behaviors of three different heavy crude oils were measured using a rotational rheometer. In conclusion, all of these heavy crude oils display linear viscoelastic behaviors in the experimental range. The loss modulus (E″) of the three crude oils decreased as the experimental temperature increased, and the variation trends of the three crude oils were basically the same. However, the experimental temperature has almost no effect on the storage modulus (E′), which always retained a constant value of 0.4 Pa. Furthermore, the storage modulus (E′) and loss modulus (E″) increase as the angular frequency increases. To describe the physical deformation characteristics of viscoelastic materials, the generalized Maxwell model and the fractional derivative Maxwell model are used to establish the constitutive relation of heavy crude oil. In conclusion, the generalized Maxwell model and the fractional derivative Maxwell model can predict the experimental results very well. All of the square of the correlation coefficient (R2) values are greater than 0.95. However, the number of fitting parameters for the fractional derivative Maxwell model is less than that for the fourth-order generalized Maxwell model which can save the calculating time. Therefore, the fractional derivative Maxwell model is suggested to describe the viscoelastic behavior of heavy crude oil in industrial applications.

Free vibration characteristics of viscoelastic nano-disks based on modified couple stress theory

This paper investigates the vibrational behavior of a viscoelastic and size-dependent nano-disk based on the modified couple stress theory (MCST). The material characteristics in nano-scale are modeled according to Zener viscoelastic constitutive relation. In addition, displacement components are defined based on classical plate theory. Leaderman integral is also used to determine the viscous parts of the stress tensor. Hamilton’s principle is utilized to derive the governing equations of motion for specifying the strain, kinetic energy, and viscous work. The obtained equations are discretized with the help of the Galerkin method and decoupled through the diagonalization procedure. Laplace transformation is employed to solve the resulting equations in differential–integral form. The damping ratio, the imaginary part and real part of the Eigen frequency of the considered nano-disk are calculated to investigate the effects of influential parameters on the nano-disk vibrational behavior. These parameters include nonlocal parameter boundary conditions, geometric constant, power constant, and element relaxation coefficient. Results obtained on different mode shapes indicate that increasing the dimensionless element relaxation coefficient is followed by a decrease in the imaginary part of the Eigen frequency regarding the energy dissipation as well as a decrease in the real part of the Eigen frequency. Furthermore, increasing the h/l ratio is accompanied by variations in the imaginary part, real part, and damping ratio. According to the results, the effect of damping on vibrational behavior of the nano disk is more distinguished for smaller values of h/l.

Continuum mechanics modeling of complex fluid systems following Oldroyd's seminal 1950 work

A review of some key developments in the continuum mechanics – based macroscopic modelling of complex fluid flows, following the pioneering work of Oldroyd in the 1950s will be presented. We start by reviewing the major developments achieved by Oldroyd, namely how his work established rules for consistency in developing continuum stress constitutive models extending the material into objective time derivatives for tensor quantities, with first application the stress tensor in viscoelastic constitutive relationships. We then show how the impact of his work to viscoelastic fluid flows has been amplified from the kinetic theory work of Bird and others who established a firm macromolecular structural connection. The use of objective time derivatives was then extended to include second order conformation tensors and through them the connection was made to internal deformation energy by Marrucci. We then further demonstrate the impact of this work by analysing the models through the microstructural descriptions offered within the non-equilibrium thermodynamics formalism. We then conclude with a brief mentioning of direct applications of the original Oldroyd-B model (through stability analyses, etc.) towards bettering our understanding of the rheological behaviour of complex fluid systems.