Viscoelastic Constitutive Equation Research Articles

The effect of failure mechanics on the fatigue responses of lumbar intervertebral disc

Lumbar disc herniation is usually caused by the accumulation of long-term mechanical loads and sudden overload damage. Therefore, this study aims to illustrate how fatigue failure in lumbar spine segments is influenced by both cyclic loading magnitude and pre-existing damage. Eighty-six sheep intervertebral disc samples were divided into four groups to test the fatigue responses in healthy and damaged intervertebral discs. Both before and after fatigue loading, the specimens were performed on loading–unloading tests to analyze the viscoelasticity changes, while the specimens were performed on MRI examination to analyze the geometric and morphological changes. The Stress-Failure curve (SN curve) was examined, while the number of cycles to failure of damaged specimens was much smaller than that of healthy specimens at the same stress level during cyclic loading, and the relationship was approximately linear on a logarithmic scale. In addition, the healthy specimens will not accumulate fatigue failure if the compression force remains below 50% of the ultimate compressive tolerance (UTC). Before and after fatigue loading, the loading–unloading curves do not coincide and show obvious strain-rate-dependent viscoelastic characteristics, while the elastic modulus of the damaged specimen is significantly smaller. For magnetic resonance imaging, morphological changes included the changes of nucleus pulposus (NP) shape and area, while fatigue has a more significant effect on ruptured and herniated disc specimens. The dissipated energy of the intervertebral discs under cyclic loading was then calculated based on viscoelastic constitutive equations, which show that the load and preexisting damage both have significant effects on the dissipation rate.

Magneto-viscoelastic rod model for hard-magnetic soft rods under 3D large deformation: Theory and numerical implementation

The main purpose of this work is to develop a three-dimensional (3D) viscoelastic rod model for hard-magnetic soft (HMS) rods under large deformation which are widely used active structures in soft robotics. To do so, the Simo’s viscoelasticity theory has been rationally incorporated into the geometrically exact 3D curved rod model. The proposed model includes the deformation modes of axial tension, shear, bending, and torsion, which is applicable to the HMS rods with arbitrarily initial curved and twisted geometries under 3D large deformation. The viscoelastic constitutive equations of the HMS rod in the present formulation are formulated, which include the general relaxation functions. To obtain the expression for the magnetic load, the rotation-based magnetic free energy density is introduced, and the governing equations of the HMS rod with magnetic load and body force are presented. To obtain the numerical implementation, an implicit time integration algorithm that simply extends the generalized-α method for the rotation group, and the corresponding variational formulation and its linearization of the rod model are derived. To validate the model, five numerical examples, including 2D dynamic buckling, 3D static, and 3D dynamic problem are presented. The dynamic problems include the dynamic snap-through behavior of a bistable HMS arch and damped oscillation of a quarter arc cantilever under 3D deformation. The simulation results show good agreement with the results reported in the literature.

From micro to macro: Creep behaviour and closed-form expression of viscoelastic parameters for a rheological particle assembly

This study is devoted to analytically establish macro–micro quantitative relationships for creep characteristics of a rheological particle assembly, contributing to a more comprehensive insight into the interplay between different scales. To represent the various rheological behaviour in microscopic for a hexagonal close-packed (HCP) circular particle assembly, the contact responses are assumed as viscoelastic models. By extending the traditional homogenization method to the Laplace domain, the macroscopic one/two-dimensional creep equations of the rheological particle assembly, expressed with the microscopic contact parameters in a closed-form manner, are innovatively identified. It is found that the macroscopic creep behaviour is not always consistent with the microscopic one and in general exhibits more complex creep characteristics. The parametric analysis presents the quantitative influence of contact parameters on macroscopic creep behaviour. Extending the one-dimensional case to three dimensions (3D), the relaxation moduli as well as creep compliance, expressed in closed-form by contact parameters, are obtained in 3D viscoelastic constitutive equations. Distinct viscoelastic behaviours are observed for macroscopic shear and bulk responses. The proposed macro–micro quantitative relations of creep characteristics in this study benefit to deeply understand the macro–micro relations of viscoelastic properties, providing an alternative approach to calibrate microscopic parameters when involving the DEM simulations.

Hybridization of a Linear Viscoelastic Constitutive Equation and a Nonlinear Maxwell-Type Viscoelastoplastic Model, and Analysis of Poisson’s Ratio Evolution Scenarios under Creep

Hybridization of a Linear Viscoelastic Constitutive Equation and a Nonlinear Maxwell-Type Viscoelastoplastic Model, and Analysis of Poisson’s Ratio Evolution Scenarios under Creep

On the use of the Astarita flow field for viscoelastic fluids to develop a generalised Newtonian fluid model incorporating flow type (GNFFTy)

The two-dimensional, steady, homogeneous flow field proposed by Astarita (J. Rheol., vol. 35, 1991, pp. 687–689) is studied for a range of viscoelastic constitutive equations of differential form including the models due to Oldroyd (the upper and lower convected Maxwell; UCM/LCM), Phan-Thien and Tanner (simplified, linear form; sPTT) and Giesekus. As the flow is steady and homogeneous, the sPTT model results also give the FENE-P model solutions via a simple transformation of parameters. The flow field has the interesting feature that a scalar parameter may be used to vary the flow ‘type’ continuously from solid-body rotation to simple shearing to planar extension whilst the rate of deformation tensor $\boldsymbol{\mathsf{D}}$ remains constant (i.e. independent of flow type). The response of the models is probed in order to determine how a scalar ‘viscosity’ function may be rigorously constructed which includes flow-type dependence. We show that for most of these models – the Giesekus being the exception – the first and second invariants of the resulting extra stress tensor are linearly related, and for models based on the upper convected derivative, this link is simply via a material property, i.e. half the modulus. By defining a frame-invariant coordinate system with respect to the eigenvectors of $\boldsymbol{\mathsf{D}}$ , we associate a ‘viscosity’ for each of the flows to a deviatoric stress component and show how this quantity varies with the flow-type parameter. For elliptical motions, rate thinning is always observed and all models give essentially the UCM response. For strong flows, i.e. flow types containing at least some extension, thickening occurs and only a small element of extension is required to remove any shear thinning inherent in the model (e.g. as occurs in steady simple shearing for the sPTT/Giesekus models). Finally, a functional form of a viscosity equation which could incorporate flow type, but be otherwise inelastic, the so-called GNFFTy (generalised Newtonian fluid model incorporating flow type, pronounced ‘nifty’), is proposed. In the frame-invariant coordinate system proposed, this model is also capable of capturing normal-stress differences.

Effect of fluid elasticity and shear thinning on heat transfer from a heated sphere at moderate Reynolds numbers

The drag and heat transfer characteristics of an isothermal sphere in a FENE-P-type viscoelastic fluid have been studied in the steady axisymmetric flow regime. The governing equations, namely, mass, momentum, and energy equations, together with FENE-P type viscoelastic constitutive equations, have been solved using the finite volume method based opensource CFD code OpenFOAM over the following ranges of conditions: Reynolds number, 1 ≤ Re ≤ 100 ; Prandtl number, 1 ≤ Pr ≤ 50 ; Weissenberg number, 0 ​ ≤ Wi ≤ 10 and polymer extensibility parameter, 10 ≤ L 2 ≤ 500 for a fixed value of the polymer viscosity ratio of β = 0.5 . At low Reynolds numbers, fluid viscoelasticity causes an upward and downward shifting in the axial velocity profiles with reference to that in a Newtonian fluid along the upstream and downstream axis of the sphere, respectively. However, such a shift progressively diminishes with the increasing Reynolds number. Furthermore, a “velocity overshoot” and/or a “negative wake” is also observed in a viscoelastic polymer solution at low Reynolds numbers, and this is accentuated with the increasing Weissenberg number. The separation of boundary layers is seen to be suppressed with the increasing Weissenberg number. The drag coefficient decreases with the Reynolds number irrespective of the fluid type and the corresponding dependence on the Weissenberg number is found to be non-monotonic depending upon the values of Re and L 2. The average Nusselt number always increases with the increasing values of Re, Pr, and Wi and with the decreasing value of L 2. A limited number of simulations have also been carried out to show the effect of the temperature-dependent viscosity of the fluid on the flow dynamics and heat transfer characteristics. Finally, simple correlations for the drag coefficient and average Nusselt number are presented which not only capture the functional dependence but also can be used for the interpolation of the present results for the intermediate values of the governing parameters in a new application.

The fluid dynamics of a viscoelastic fluid dripping onto a substrate.

Extensional flows of complex fluids play an important role in many industrial applications, such as spraying and atomisation, as well as microfluidic-based drop deposition. The dripping-on-substrate (DoS) technique is a conceptually-simple, but dynamically-complex, probe of the extensional rheology of low-viscosity, non-Newtonian fluids. It incorporates the capillary-driven thinning of a liquid bridge, produced by a single drop as it is slowly dispensed from a syringe pump onto a solid partially-wetting substrate. By following the filament thinning and pinch-off process the extensional viscosity and relaxation time of the sample can be determined. Importantly, DoS allows experimentalists to measure the extensional properties of lower viscosity solutions than is possible with commercially available capillary break-up extensional rheometers. Understanding the fluid mechanics behind the operation of DoS will enable users to optimise and extend the performance of this protocol. To achieve this understanding, we employ a computational rheology approach, using adaptively-refined time-dependent axisymmetric numerical simulations with the open-source Eulerian code, Basilisk. The volume-of-fluid technique is used to capture the moving interface, and the log-conformation transformation enables a stable and accurate solution of the viscoelastic constitutive equation. Here, we focus on understanding the roles of surface tension, elasticity and finite chain extensibility in controlling the elasto-capillary (EC) regime, as well as the perturbative effects that gravity and substrate wettability play in setting the evolution of the self-similar thinning and pinch-off dynamics. To illustrate the interplay of these different forces, we construct a simple one-dimensional model that captures the initial rate of thinning when the dynamics are dominated by a balance between inertia and capillarity. This model also captures the structure of the transition region to the nonlinear EC regime in which the rapidly growing elastic stresses in the thread balance the capillary pressure as the filament thins towards breakup. Finally, we propose a fitting methodology based on the analytical solution for FENE-P fluids to improve the accuracy in determining the effective relaxation time of an unknown fluid.

Numerical simulation analysis of forming of continuous fiber reinforced polypropylene composites: Relaxation, creep, and hot stamping

In this study, the viscoelastic constitutive equation of continuous glass fiber reinforced polypropylene composites based on the generalized Maxwell model and SCFM model was constructed by combining genetic algorithm. Based on this, a multi-scale numerical model for describing material relaxation and creep is proposed, and the accuracy of different scale models was compared through experiments. The forming state of the workpiece under various stamping speeds and preheating temperatures was also examined using a hot stamping numerical model. The independent design of the hot stamping hemispherical mold was used to confirm the hot stamping model's accuracy. The study's findings demonstrate that the macroscopic model can accurately simulate the relaxation and creep of composite materials. The hot stamping simulation results are match up well with experimental results. Composite molding defects appeared at the waist of the part, and defects were accompanied by resin loss.

60 years of the Kaye–Bernstein, Kearsley, Zapas rheological constitutive law for polymers

The K–BKZ (Kaye–Bernstein, Kearsley, Zapas) rheological constitutive model is now 60 years old. The paper reviews the connections of the model and its variants with continuum mechanics and experimental evidence in polymer melt flow, presenting an up-to-date recap of research and major findings in the open literature. In the Introduction, an historical perspective is given on developments in the last 60 years of the K–BKZ model. Then, a section on mathematical modeling of polymer flows follows, including the governing equations of flow, the rheological constitutive equations (with emphasis on the viscoelastic integral constitutive equations of the K–BKZ type), dimensionless numbers controlling the flow, and relevant boundary conditions. The “Method of Solution” section reviews the major developments of numerical techniques for particle tracking and integral evaluation for the viscoelastic stresses. Finally, selected examples of successful application of the K–BKZ model in polymer flows are presented including considerations of wall slip and non-isothermal flows.

Rheology of brain tissue and hydrogels: A novel hyperelastic and viscoelastic model for forensic applications

A new strain-energy function W, which possesses the strain energy expressible as a rational function of the principal invariants of the Cauchy tensor C, is proposed. It generates a hyperelastic constitutive equation with characteristics of brain tissues: a much stronger resistance to compression than to stretching and strongly nonlinear response in simple shear, including non-zero first and second normal differences. This model exponent α resembles the Ogden model in uniaxial stretching/compression and reveals plausible predictions for brain tissue with even values of α < 0 with sufficiently high magnitude (say, at α = −20). However, the dependence of the strain-energy function W on the principal invariants of C links it to hyperelastic hydrogel models (the Special and General Blatz–Ko models, neo-Hookean materials, incompressible Mooney–Rivlin and the Yeoh models). For α = −8, the present model reveals a compression/stretching behavior close to the tensorial Special Blatz–Ko model used for description of hydrogels. Furthermore, the present hyperelastic model is used as a kernel of the corresponding tensorial viscoelastic model with exponential fading memory. It belongs to the class of the integral Bernstein–Kearsley–Zapas (BKZ) models. In a number of important cases (the uniaxial stretching/compression, simple shear), it can be transformed into a differential viscoelastic model and predict viscoelastic liquid-like behavior under sustained deformations. The stress relaxation following an imposed strain reduces to the hyperelastic model with the elastic parameters exponentially fading in time. These tensorial hyperelastic and viscoelastic constitutive equations aim applications in modeling of blast-induced traumatic brain injuries and bullet penetration and spatter of brain tissue in forensic context.

Anticipating gelation and vitrification with medium amplitude parallel superposition (MAPS) rheology and artificial neural networks

Anticipating qualitative changes in the rheological response of complex fluids (e.g., a gelation or vitrification transition) is an important capability for processing operations that utilize such materials in real-world environments. One class of complex fluids that exhibits distinct rheological states are soft glassy materials such as colloidal gels and clay dispersions, which can be well characterized by the soft glassy rheology (SGR) model. We first solve the model equations for the time-dependent, weakly nonlinear response of the SGR model. With this analytical solution, we show that the weak nonlinearities measured via medium amplitude parallel superposition (MAPS) rheology can be used to anticipate the rheological aging transitions in the linear response of soft glassy materials. This is a rheological version of a technique called structural health monitoring used widely in civil and aerospace engineering. We design and train artificial neural networks (ANNs) that are capable of quickly inferring the parameters of the SGR model from the results of sequential MAPS experiments. The combination of these data-rich experiments and machine learning tools to provide a surrogate for computationally expensive viscoelastic constitutive equations allows for rapid experimental characterization of the rheological state of soft glassy materials. We apply this technique to an aging dispersion of Laponite® clay particles approaching the gel point and demonstrate that a trained ANN can provide real-time detection of transitions in the nonlinear response well in advance of incipient changes in the linear viscoelastic response of the system.

Numerical Simulation of Three-Dimensional Free Surface Flows Using the K-BKZ-PSM Integral Constitutive Equation.

This work introduces a novel numerical method designed to address three-dimensional unsteady free surface flows incorporating integral viscoelastic constitutive equations, specifically the K-BKZ-PSM (Kaye-Bernstein, Kearsley, Zapas-Papanastasiou, Scriven, Macosko) model. The new proposed methodology employs a second-order finite difference approach along with the deformation fields method to solve the integral constitutive equation and the marker particle method (known as marker-and-cell) to accurately capture the evolution of the fluid's free surface. The newly developed numerical method has proven its effectiveness in handling complex fluid flow scenarios, including confined flows and extrudate swell simulations of Boger fluids. Furthermore, a new semi-analytical solution for velocity and stress fields is derived, considering fully developed flows of a K-BKZ-PSM fluid in a pipe.

Rayleigh–Bénard convection of viscoelastic fluids in three-dimension finite cavity

The Rayleigh–Bénard convection (RBC) with viscoelastic fluids has captured wide attention in the past decades, caused by its ubiquitous uses in the heat transfer process. However, the numerical technique for simulating the viscoelastic fluid flows developed slowly until recent years because of the numerical instability raised by the hyperbolic characteristics of the viscoelastic constitutive equation. In the present work, a novel numerical scheme was developed to simulate the three-dimension (3D) RBC with viscoelastic fluids, where the quasi-linear treatment was applied to the hyperbolic terms of the momentum equation and the viscoelastic constitutive equation. The in-house solver was also optimized in the aspect of time cost. The results show that the critical value of convection onset in 3D is near to that in the two-dimension (2D); however, the flow pattern displays the wave characteristics in the horizontal direction. The viscoelastic kinetic energy budget of oscillating convection in 3D still conforms to the energy transport law of that in 2D [Zheng et al., Phys. Rev. Fluids 8, 023303 (2023)].

Study on Nonlinear Viscoelastic Constitutive Equations for Rock

Numerical simulation is one of the effective tools for the long-term stability assessment of underground structures. The reliability improvement of the numerical simulation requires precise constitutive equations to reproduce time-dependent behavior of rock such as loading-rate dependence, creep, and relaxation. These time-dependent behaviors of rock have nonlinearity between stress and strain rate and are influenced by surrounding environment. A number of constitutive equations have been proposed in previous studies; however, the equations based on the theory of linear viscoelasticity have difficulty reproducing accurately the time-dependent behavior of rock, and most of the equations based on the theory of viscoplasticity have been applied to the limited loading condition such as creep. In this article, nonlinear viscoelastic constitutive equations for rock proposed by the authors were introduced, and their exact solutions and application to experimental results were summarized. These equations can reproduce nonlinear time-dependent behavior and not only brittle but also ductile failures of rock and can be applied to the behavior under various loading conditions and experiment environment. In the equations, compliance or inelastic strain is assumed to be increased with time, but the functions and their combinations are of great variety. Hence, the differences of the solutions and applicability of the equations were detailed, and in addition future subjects were presented.

Damping optimization of viscoelastic thin structures, application and analysis

In this work, damping properties of bending viscoelastic thin structures are enhanced by topology optimization. Homogeneous linear viscoelastic plates are optimized and compared when modeled by either the Kirchhoff–Love or Reissner–Mindlin plate theories as well as by the bulk 3D viscoelastic constitutive equations. Mechanical equations are numerically solved by the finite element method and designs are represented by the level-set approach. High performance computing techniques allow to solve the transient viscoelastic problem for very thin 3D meshes, enabling a wider range of applications. The considered isotropic material is characterized by a generalized Maxwell model accounting for the viscoelasticity of both Young modulus and Poisson’s ratio. Numerical results show considerable design differences according to the chosen mechanical model, and highlights a counter-intuitive section shrinking phenomenon discussed at length. The final numerical example extends the problem to an actual shoe sole application, performing its damping optimization in an industrial context.

Numerical investigation of pressure drop within isothermal capillary rheometry for viscous and viscoelastic fluids

Abstract In capillary rheometry of a polymer melt, the total pressure drop consists of three primary components: shear viscosity, extensional viscosity, and normal stress differences. Traditionally, viscoelastic constitutive equations have been used to produce accurate predictions regarding the pressure drop. It is difficult to investigate the primary components in past experimental measurements and numerical computations. The useful improvement of the classic White-Metzner viscoelastic constitutive equation coupled with the weighted shear/extension viscosity is performed in isothermal capillary flow simulations. It is significant to analyze the extension-induced pressure drop for a short die and the shear-induced pressure drop for a long die, along with a small contribution of normal stress differences.

Non-isothermal simulation of a corner vortex within entry flow for a viscoelastic fluid

Abstract A modified White-Metzner viscoelastic constitutive equation is incorporated into the state-of-the-art Three Dimensional Computational Fluid Dynamics (3D-CFD) framework for performing isothermal and non-isothermal entry flow simulations of a polymer melt. As a result, the corner vortex becomes smaller with increased isothermal temperature, namely, the so-called temperature-induced vortex reduction. In addition, the vortex grows with raising wall temperatures, whereas the vortex reduction is found under high inlet temperatures. Through the visualized flow patterns, it is significant to investigate the dramatic variations of vortex size in relation to fluid temperature, weighted viscosity and extension rate, as well as Weissenberg number and Trouton ratio.

Modelling and Analysis of the Dispersal of a Polymeric Pollutant Injected into a Channel Flow of a Newtonian Liquid

The transient dynamics of nonlinear dispersion of a polymeric pollutant ejected by an external source into a laminar flow of a Newtonian liquid flowing through a rectangular channel is investigated. The Boussinesq approximation is assumed for the density variation with pollutant concentration. The governing equations of mass and momentum conservation are coupled to the pollutant concentration equation as well as to the viscoelastic constitutive model for the polymer stresses. The Oldroyd-B viscoelastic constitutive model is employed to model the deformation and characteristics of the polymer stresses. The coupled system of nonlinear partial differential equations is solved numerically using robust and efficient semi-implicit finite difference methods (FDM). Solutions are presented in graphical form for various parameter values. The model can be a useful tool in understanding the dynamics of domestic and industrial pollution situations that may arise from improper discharge of long-chain hydrocarbon products into, say, water drainage systems. The novelty of this investigation is in the modelling of the long-chain hydrocarbon-product pollutants via appropriate viscoelastic (polymeric) constitutive equations. In general, it is observed that parameters which increase (decrease) the flow velocity correspondingly increase (respectively decrease) the wall shear stress. Similarly, it is observed that parameters which increase (decrease) the polymer concentration correspondingly increase (respectively decrease) the mass transfer rates. The wall shear stress and mass transfer are measurable quantities. In this respect, our work offers such measurements as predictive tools to detect the scale of contamination.

Effect of cavity aspect ratio on mixed convective heat transfer phenomenon inside a lid-driven cavity due to elastic turbulence

This study performs extensive numerical simulations to investigate how the aspect ratio (AR) of a lid-driven cavity influences the onset of elastic instability and elastic turbulence and the subsequent mixed convective heat transfer rate inside it. To this end, we utilize the finite volume method based open source code OpenFOAM along with Rheotool to solve the mass, momentum, energy, and viscoelastic constitutive equations. We find that the dependency of the cavity AR on the heat transfer rate is highly complicated depending upon the values of the Richardson (Ri) and Prandtl numbers (Pr). At low values of Ri, the heat transfer rate continuously decreases with AR irrespective of the value of the Prandtl number and the fluid type, i.e., Newtonian or viscoelastic. The same trend is also observed at high values of Ri and low values of Pr. At these combinations of Ri and Pr, the heat transfer rate is always higher in viscoelastic fluids than in Newtonian fluids due to the presence of elastic turbulence in the former fluids. However, a different trend is observed at high values of both Ri and Pr. At this combination of Ri and Pr, the heat transfer rate increases with AR in Newtonian fluids, whereas it decreases in viscoelastic fluids. Therefore, at high values of AR, Ri, and Pr, the heat transfer rate is higher in Newtonian fluids than that in viscoelastic fluids despite the presence of elastic turbulence in the latter fluids. This is in contrast to the assumption that the elastic turbulence phenomenon always increases the rate of transport processes. A possible explanation for this behavior is provided in this study. Along with the heat transfer aspects, we also provide a detailed discussion on how the cavity aspect ratio influences the corresponding flow dynamics inside the cavity. In particular, we find that the onset of the elastic instability (and the subsequent elastic turbulence) phenomenon is delayed to higher values of the Weissenberg number as the cavity aspect ratio increases. This is in line with prior experimental studies reported in the literature.

Numerical Simulation on Structural Integrity of Grain with Large Increased Area Ratio for Solid Rocket Motor

To evaluate the structural integrity of a type of solid rocket motor grain with a largely increased area ratio using free-loading modified double-base grain, the Schapery integral nonlinear viscoelastic constitutive equation with damage is derived, and the numerical simulation model of solid rocket motor grain is constructed. According to the mission profile of the grain during the equipment period, the numerical simulation research on the mechanical response characteristics of the grain under three typical working conditions, namely, platform vibration, underwater explosion impact, and ignition pressurization during the launch is carried out. The simulation results show that under the three typical working conditions, the grain will not produce structural damage and defects, and the mechanical response of the grain under the platform vibration and explosion impact load is far less than that under the launch ignition pressurization. The ignition pressurization should be considered in the grain structural integrity design of this type or similar structure solid rocket motor.