Rheological Constitutive Equation Research Articles

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.

Local gradient theory of dielectrics incorporating polarization inertia and flexodynamic effect

A higher-grade theory of non-ferromagnetic thermo-elastic dielectrics which incorporates the local mass displacement, the heat flux gradient, polarization inertia, and flexodynamic effects is developed. The process of local mass displacement is associated with changes in material microstructure. Using the fundamental principles of continuum mechanics, electrodynamics, and non-equilibrium thermodynamics, the gradient-type constitutive equations are derived. Due to accounting for the polarization inertia, the rheological constitutive equation for the polarization vector is obtained. In the balance equation of linear momentum, an additional term with the second time derivative of the polarization vector appears in comparison with the classical theory. This term controls the influence of the dynamic flexoelectric effect on the mechanical motion of dielectric solids. The propagation of a plane harmonic wave is analyzed within the context of the developed theory. It is shown that the theory allows for capturing the experimentally observed phenomenon of high-frequency dispersion of a longitudinal elastic wave. The theory may be useful for modeling coupled processes in nanodielectrics and heterogeneous polarized systems.

RheologyNet: A physics-informed neural network solution to evaluate the thixotropic properties of cementitious materials

Thixotropic behaviors can be predicted by rheological partial differential equations (PDEs) of cementitious materials. The ability to solve the rheological PDEs of viscous fluids accurately and efficiently has become an emerging interest in research. However, due to the growing number of parameters in rheological constitutive equations and the non-ideal behavior of materials from experiments, solving the rheological PDEs becomes computationally costive and error-prone. We propose a physics-informed neural network (PINN)-based framework, RheologyNet, as a surrogate solution to predict the general thixotropic behavior of cementitious materials. The complex PDEs are embedded in the well-designed RheologyNet architecture to link macroscopic viscous flow behaviors and microstructural changes. Numerical experiments suggested that RheologyNet can accurately and efficiently predict the rheological properties of cementitious materials compared to the traditional Fully-connected Neural Network (FNN) and mechanistic Finite Element Analysis (FEA). Particularly, RheologyNet demonstrated great promise for simulating history-dependent thixotropic behaviors.

Effects of a polycarboxylate superplasticizer and temperature on the rheological properties of grout blended with seawater

A series of subsea tunnel projects with unprecedented scale is under vigorous construction. In the construction sector, grouting has been widely adopted as the primary method for disaster control, and superplasticizers are the additives commonly used in grouting engineering. However, during grouting, seawater intrusion and ambient temperature change lead to significant changes in the rheological properties of grouting materials and failure of the grouting scheme. Therefore, it is critical to study the rheological properties of seawater-mixed grout slurry containing a polycarboxylate superplasticizer (PCS) at different temperatures. In this study, the mechanism of a PCS and its effects on the rheological properties of biomass-activated grouts (BAGs) blended with seawater were revealed for the first time. Further, the rheological characteristics and constitutive equations of seawater-mixed grout slurry at 0–30 °C were determined. Results indicated that the PCS significantly reduced the shear stress and viscosity of seawater-mixed grout. The fluidity of grout slurry with 0.8 % PCS increased by 30.8 %. The power law model was applied to describe the rheological characteristics of seawater-mixed slurry with varying PCS contents. With increasing temperature, the rheological index increased, whereas the viscosity, shear stress, and consistency coefficient decreased. The constitutive equation of seawater-mixed BAGs containing 0.8 % PCS at different temperatures was also established for the first time. The findings of this study can provide a basis for grouting diffusion theory, numerical calculation, and the design of a grouting scheme for subsea tunnels at different temperatures.

Line contact lubricated by a fluid described by non-Newtonian Giesekus model

Abstract Modelling hydrodynamic lubrication helps engineers and material scientists designing lubricants that minimize viscous energy losses without compromising the durability of engine components. This type of modelling requires selecting lubricant rheological constitutive equations that provide realistic information on fluid behaviour within tribological contact zones. Due to complexity of differential constitutive equations the modelling effort was limited to generalized Newtonian or Oldroyd-B fluids. In this paper, we present more realistic description of hydrodynamic lubrication using the Giesekus fluid capturing both viscous shear thinning and the development of elastic stresses at high Weissenberg numbers. The problem analysis is done using a modified regular perturbation method with respect to the small mobility parameter $\alpha $. The perturbation analysis modification is caused by the fact that the problem contains several a priori unknown boundaries. The actual perturbation solutions for functions of pressure and gap, as well as the coordinate of exit from the contact and the exit film thickness, are obtained as linear functions of $\alpha $. The effects of the model parameters such as the mobility factor, relaxation time and the polymer contribution to viscosity on pressure, shear and extensional viscosities within the contact zone are analysed. It is shown that the presence of polymers helps to increase the gap and reduce the viscous losses associated with hydrodynamic friction. This result is consistent with empirical observations of fuel economy enhancement for polymer-containing lubricants.

A universal rheological constitutive equation of magnetorheological fluids with a wide shear rate range

As an intelligent material, the magnetorheological fluid (MRF) has increased use in various fields, e.g., the semi-active suspension of vehicles. However, the efficient utilization and optimization of MRFs depend on a reliable constitutive relationship over a wide shear rate range, which existing models cannot fully and satisfactorily explain. This paper proposes a new non-Newtonian fluid constitutive equation that combines exponential and linear terms to describe the general characteristics of MRFs to tackle this problem. An effective parameter identification approach is realized by adopting model splitting to reduce the dimension of the optimized parameter space combined with a reasonable initial guess. A well-designed experiment for a commercial MRF is conducted to verify the proposed constitutive equation, indicating an improvement in accuracy of approximately 43 % compared with the Herschel-Bulkley model and its capability to reveal the rheological behaviors of MRFs in the wide shear rate range of 0.3–30000 s−1. The high compatibility between the proposed model and the Bingham model is supported by the high correlation coefficients of the dynamic yield stress and viscosity coefficient identified by this model and the existing data processing method based on the Bingham model, 0.9949 and 0.9160, respectively. Owing to its decoupled parameter structure, the extension to a multivariable framework has been implemented to consider the magnetorheological effect with good agreement. Finally, the application of magnetorheological device design has demonstrated the versatility and capacity of the proposed model for developing and utilizing a new generation of MRFs.

Application and Optimization of the Rheological Model for a Hydrophobically Associating Dendrimer Polymer.

Polymer flooding is one of the most important enhancing oil recovery (EOR) technologies in the world. With the optimization of polymer synthesis, the performance of polymer solutions has been greatly improved, which can adapt to more complex oil and gas reservoirs. However, with the continuous improvement of the properties of polymer solutions, the elastic property of polymer solutions is significantly improved, and the rheological law has also changed. This series of changes affects the application of polymer flooding reservoir numerical simulation technology. Therefore, constructing an accurate description model and precise limitation conditions is particularly important. The rheological curve with a wide shear range (0.1~10,000 s−1) and the viscoelasticity of the two polymers (partially hydrolysed polyacrylamide (HPAM) and dendritic hydrophobic association polymer (DHAP)) were analyzed and tested by a rotating rheometer. The results showed that under the experimental conditions, the rheological curve of both polymers can be described by the Carreau rheological model. Meanwhile, the structural viscosity of the hydrophobically associating polymer solution (DHAP) greatly improved the elasticity of the solution and led to the change of elastic modulus. Considering the influence of elastic characteristics on the rheological curve, the relaxation time spectrum derived from small vibration experimental data was used to limit the characteristic relaxation time, that is, the value range of λ. It was observed that the experimental data were highly matched with the nonlinear regression fitting curve of the Carreau rheological model. Therefore, the relationship between different test parameters should be fully considered while studying the rheological constitutive equation of viscoelastic fluid, so as to optimize and improve the equation of it.

An Inverse Analysis Method Applied to Optimization of Specimen’s Shape for Performing Hot Rapid Crushing Tests from Homogeneous Initial Temperature Field

Specific experimental tests with loadings conditions close to those of industrial fast forming processes as rapid forging, rapid stamping or high speed machining, characterized by large plastic strains, localized deformations and important gradients of strain rates, strain and temperature, requires to analyses material flow behavior at different initial temperatures. One of the more important conditions to obtain intrinsic rheological constitutive equations is to have a quasi-homogenous initial temperature distribution and especially to keep constant the material microstructure during the specimens heating. The rapid induction heating seems to be one of the most reliable processes. This scientific study proposes an inverse analysis technique based on numerical finite element modelling to define on the thermal point of view, optimal specimen shapes for performing hot rapid crushing tests from homogenous initial temperature field.

Elasticity and relaxation properties of oral fluid

The research is aimed at crating a method for rheological testing of viscoelastic fluids, the droplets of which, when stretched, form thinning filaments, i.e. exhibit the property of spinning. The typical example of such fluid is oral fluid (or mixed saliva). Among other things, the rheological features of the oral fluid actively affect the mechanisms of transmission of infection by airborne droplets. In this work, the oral fluid was studied both from the point of view of a participant in the transmission of infection and as a model for studying the rheology of viscoelastic fluids. The method is based on video recording of the stretching of a drop of the test liquid between the legs of the tweezers and the subsequent spontaneous thinning of the formed capillary filament of the liquid. The process is controlled by the competition of forces of inertia, elasticity, capillarity. By analyzing the video recording, it is possible to trace the contribution of each factor and, within the framework of the Oldroyd/Maxwell rheological constitutive equation (rheological model), determine the numerical values of all model constants. The obtained rheological characteristics of the oral fluid make it possible to theoretically model the processes of the formation of drops of oral fluid during sneezing, coughing, talking, as well as the processes of collision of drops with protective masks, filters and other obstacles. In general, the proposed method of rheological testing is applicable for studying a wide class of viscoelastic fluids, including biological ones. Among the latter are bronchial sputum, synovial fluid, reproductive fluids, and others. The method is distinguished by the simplicity of the experiment and the use of elementary equipment.

Flows of Linear Polymer Solutions and Other Suspensions of Rod-like Particles: Anisotropic Micropolar-Fluid Theory Approach.

We formulate equations governing flows of suspensions of rod-like particles. Such suspensions include linear polymer solutions, FD-virus, and worm-like micelles. To take into account the particles that form and their rotation, we treat the suspension as a Cosserat continuum and apply the theory of micropolar fluids. Anisotropy of suspensions is determined through the inclusion of the microinertia tensor in the rheological constitutive equations. We check that the model is consistent with the basic principles of thermodynamics. In addition to anisotropy, the theory also captures gradient banding instability, coexistence of isotropic and nematic phases, sustained temporal oscillations of macroscopic viscosity, shear thinning and hysteresis. For the flow between two planes, we also establish that the total flow rate depends not only on the pressure gradient, but on the history of its variation as well.

On the Influence of Viscoelastic Modeling in Fluid Flow Simulations of Gum Acrylonitrile Butadiene Rubber.

Computational fluid dynamics (CFD) simulation is an important tool as it enables engineers to study different design options without a time-consuming experimental workload. However, the prediction accuracy of any CFD simulation depends upon the set boundary conditions and upon the applied rheological constitutive equation. In the present study the viscoelastic nature of an unfilled gum acrylonitrile butadiene rubber (NBR) is considered by applying the integral and time-dependent Kaye–Bernstein–Kearsley–Zapas (K-BKZ) rheological model. First, exhaustive testing is carried out in the linear viscoelastic (LVE) and non-LVE deformation range including small amplitude oscillatory shear (SAOS) as well as high pressure capillary rheometer (HPCR) tests. Next, three abrupt capillary dies and one tapered orifice die are modeled in Ansys POLYFLOW. The pressure prediction accuracy of the K-BKZ/Wagner model was found to be excellent and insensitive to the applied normal force in SAOS testing as well as to the relation of first and second normal stress differences, provided that damping parameters are fitted to steady-state rheological data. Moreover, the crucial importance of viscoelastic modeling is proven for rubber materials, as two generalized Newtonian fluid (GNF) flow models severely underestimate measured pressure data, especially in contraction flow-dominated geometries.

Systematic derivation of hydrodynamic equations for viscoelastic phase separation

We present a detailed derivation of a simple hydrodynamic two-fluid model, which aims at the description of the phase separation of non-entangled polymer solutions, where viscoelastic effects play a role. It is directly based upon the coarse-graining of a well-defined molecular model, such that all degrees of freedom have a clear and unambiguous molecular interpretation. The considerations are based upon a free-energy functional, and the dynamics is split into a conservative and a dissipative part, where the latter satisfies the Onsager relations and the second law of thermodynamics. The model is therefore fully consistent with both equilibrium and non-equilibrium thermodynamics. The derivation proceeds in two steps: firstly, we derive an extended model comprising two scalar and four vector fields, such that inertial dynamics of the macromolecules and of the relative motion of the two fluids is taken into account. In the second step, we eliminate these inertial contributions and, as a replacement, introduce phenomenological dissipative terms, which can be modeled easily by taking into account the principles of non-equilibrium thermodynamics. The final simplified model comprises the momentum conservation equation, which includes both interfacial and elastic stresses, a convection–diffusion equation where interfacial and elastic contributions occur as well, and a suitably convected relaxation equation for the end-to-end vector field. In contrast to the traditional two-scale description that is used to derive rheological equations of motion, we here treat the hydrodynamic and the macromolecular degrees of freedom on the same basis. Nevertheless, the resulting model is fairly similar, though not fully identical, to models that have been discussed previously. Notably, we find a rheological constitutive equation that differs from the standard Oldroyd-B model. Within the framework of kinetic theory, this difference may be traced back to a different underlying statistical-mechanical ensemble that is used for averaging the stress. To what extent the model is able to reproduce the full phenomenology of viscoelastic phase separation is presently an open question, which shall be investigated in the future.

Effects of moisture content and tillage methods on creep properties of paddy soil.

The rheological properties parameters of paddy soil affect the interaction between the tillage tools and soil, thus influencing the operation quality and power consumption. In order to study the effects of tillage methods and moisture content on the rheological properties parameters of paddy soil in the middle and lower reaches of the Yangtze River, uniaxial compression creep tests of paddy soils with four moisture contents under no tillage (moisture contents: 26.71%, 24.52%, 23.26%, 21.28%) and plough tillage (moisture contents: 26.77%, 25.55%, 23.40%, 20.56%) were carried out using a TMS-PRO texture analyzer. The creep properties curves obtained from the tests, and the rheological constitutive equation of paddy soil under compression was established by Burgers viscoelastic model. Respectively, the quantitative change rules of creep properties of paddy soil with different moisture contents under different tillage methods and the correlation between these parameters were explored. The results showed that the moisture content under the three-year plough tillage and no tillage methods had significant influence on the rheological properties parameters of paddy soil (P < 0.05). The instantaneous elastic modulus, delay elastic modulus, and viscosity coefficient of the two paddy soils (no tillage and plough tillage soils) decreased with the increase of moisture content. However, the variation rules of relaxation time and delay viscosity coefficient with moisture content differed between these two paddy soils. Specifically, the strain rate of the two paddy soils decreased as moisture content decreased, where the total strain combines elastic strain, viscous strain, and viscoelastic strain. The initial strain rate and steady strain rate of the plough tillage paddy soils were lower than that of the no tillage paddy soils. The established creep model equation could be used to obtain viscoelastic rheological parameters of paddy soil in a wide range. The fitting equations between rheological parameters and moisture content were introduced into Burgers model, and the coupling equations between creep deformation and moisture content and time were derived, which could be used to predict the creep properties and deformation behavior of paddy soil in a certain range of no tillage and ploughed field. Overall, this study has a certain theoretical significance for the development and improvement of paddy soil rheology theory, and can also provide theoretical basis and technical support for the research of agricultural machinery design optimization, field water, soil conservation, soil tillage and compaction related simulation analysis in the middle and lower reaches of the Yangtze River.

Burgers viscoelastic model-based variable stiffness design of compliant clamping mechanism for leafy greens harvesting

To prevent mechanical damage during the feeding and conveying process for leafy greens when roots harvesting, it is necessary to redesign the clamping mechanism in consideration of the rheological properties of leafy greens under the mechanical compression. This study takes spinach, a typical representative of leafy greens, as the research object. Firstly, the rheological constitutive equation for leafy greens under clamping is derived using the Burgers viscoelastic model, and the equation coefficient is identified using a spinach creep test. The damage constraints for the feeding process for different feeding widths are constructed with deformation energy based on the rheological constitutive equation. Secondly, a compliant clamping mechanism with variable stiffness properties is proposed, comprising a clamping floating mechanism and a compact cam pendulum mechanism. This mechanism maintains a preconfigured nonlinear force-displacement curve by utilising off-the-shelf torsional spring and cam profile. Furthermore, the mathematical models of interaction between this mechanism and leafy greens are constructed, with which the low-damage and stable clamping as the optimisation objectives are analysed, and the optimal parameters are achieved with particle swarm optimisation algorithm. Finally, an interactive simulation model of the clamping belt with floating roller supported by the compliant clamping mechanism and the Burgers viscoelastic model is built to simulate the feeding process. The dynamic characteristics of the clamping force are analysed for three feeding widths represented by parallel width of spinach. The experimental results show that relative error of the maximum clamping forces between simulation analysis and theoretical computation is less than 10%, and the relative error between test and theoretical computation is less than 13%, indicating the validity of the simulation model and feasibility of the designed compliant clamping.

PIV-based estimation of viscosity and pressure fields for a steady pseudoplastic flow

Targeting a pseudoplastic fluid, we propose a method to simultaneously estimate viscosity and pressure fields from the velocity field data of particle image velocimetry (PIV). As a two-dimensional case study, we chose a steady two-dimensional wake structure behind a circular cylinder where local viscosity and pressure are dynamically coupled. The method involves PIV, momentum conservation equation of non-Newtonian fluids and rheological constitutive equations. The CMC (Carboxy Methyl Cellulose) aqueous solution of weight concentration of 0.1% is used as a test case of pseudoplastic fluid. Viscosity distribution is obtained through power law model and Carreau-Yasuda model as the constitutive equation. Pressure distribution is then calculated by substituting the viscosity into the momentum conservation equation. Applied results show pressure-lowering at the vortex cores and viscosity-lowering at their perimeters, stabilizing vortex attachment to the cylinder in the pseudoplastic fluid. We also analyze error propagation characteristics to conclude the feasibility of the present method and highlight the difference in error propagation characteristics during pressure estimation between Newtonian and pseudoplastic fluid flows.

Toppling mechanical characterization based on rheological test of the cantilever beam

This study aims to explain the time-sensitive deformation of rock mass toppling and analyze its mechanical characteristics. A mechanical model of the toppling rock beam is abstracted by performing a force analysis of the countertendency layered slope, and the bending rheological test of the cantilever beam is carried out. The flexural rheological model and constitutive equation of the rock cantilever are identified from the experimental results. The strain (ε = 0) at the ultimate depth of development of the toppling deformation and the strain acceleration (a ≥ a2) during rock beam dumping bending are selected to obtain the development depth LF and bending depth LZ of the toppling deformation, and LF ≥ LZ,LF > 0,LZ > 0 is used to obtain the development conditions of the toppling deformation. The time-sensitive characteristics of toppling differ from those of other types of slope deformation. This study could serve as a reference for elucidating the mechanical mechanism and evaluating the stability of the toppling deformation.

Study on rheological behavior and microstructural evolution of Al-6Mg-0.4Mn-0.15Sc-0.1Zr alloy by isothermal compression

The effects of trace amounts of Sc/Zr on rheological properties and microstructural evolution of the Al-6Mg-0.4Mn-0.15Sc-0.1Zr alloy under different deformation conditions were studied by isothermal compression tests at deformation temperatures of 280 °C ∼ 460 °C and strain rates of 0.001 ∼ 10 s−1. A constitutive model based on the hyperbolic sine function was established. The dependence of the flow stress on strain, strain rate, and deformation temperature was described. Using experimental data and dynamic material model, machining diagrams of the alloy with strains of 0.3 and 0.5 were obtained, and the hot workability of the alloy was verified. The results showed that the deformation temperature and strain rate had significant effects on the deformation behavior of the alloy. The flow stress of the alloy increased with the increase in strain rate and decreased with the increase in deformation temperature. Under high-temperature and low-strain-rate conditions, the alloy tended to undergo dynamic recrystallization, and the volume fraction and grain size increased with the increase in deformation temperature and the decrease in strain rate. Based on the analysis of the microstructural evolution, the construction of the machining diagram, and the solution of the rheological constitutive equation, suitable values of the deformation temperature were determined to be 380 °C ~ 460 °C. Moreover, suitable values of the strain rate were determined to be 0.001 and 0.3 s−1. The deformation activation energy of 161.28 kJ mol−1 was obtained.

Constitutive modeling of polymers accounting for their hyperelasticity, plasticity, creep and viscoelastic relaxation

The hyperelastic Yeoh model has been generalized to account for creep, plasticity and viscoelasticity of polymers. The general tensorial model developed is applied to several rheometric situations: the tensile test used to measure the stress-strain curve in tension, as well as the creep and recovery tests. The resulting equations are compared to the experimental results acquired in the present work for several monolithic synthetic fibers used as specimens. The comparison revealed that the proposed phenomenological rheological constitutive equation is capable of reproducing the experimental data with a uniformly valid set of physical parameters. Moreover, it was possible to accurately predict the residual plastic deformation of the fibers.

The Impulsive Motion of Flat Plate in Generalized Second Grade Fluid with Anomalous Diffusion

The flow adjacent to a wall rapidly set in motion for a generalized second-grade fluid with anomalous diffusion is examined. For the elucidation of such a fluid, the fractional-order derivative approach in the constitutive relationship model is presented because models based on ordinary differential equations have a relatively limited class of solutions, which does not provide compatible description of the complex systems in general. The current model of second-order fluid involving fractional calculus is based on the formal replacement of the first-order derivative in ordinary rheological constitutive equation by fractional derivative of a non-integer order. In addition, the time-fractional equation considered in this article describes the anomalous sub-diffusion. In this article, the velocity and stress field of generalized second-grade fluid with fractional anomalous diffusion are studied by fractional partial differential equations. Analytic solutions are given in closed form, from these differential equations in terms of the generalized G-functions or Fox's H-function with the discrete Laplace transform technique. Thus, many previous and classical results, namely, the solution of fractional diffusion equation obtained by Wyss, the classical Rayleigh’s time-space regularity solution, the relationship between velocity field and stress field obtained by Bagley and Torvik, are represented by particular cases of our proposed derivation.

Simulation-Based Approach for Probing Rheology-Processing-Structure Relationships in Foam Blow Molding

AbstractThe broad objective of this work was to demonstrate a modelling and simulation framework for foam blow molding using commercially available simulation software. The simulation framework would have to account for the initial morphology of the foam, the relationship between the morphology and the rheological and deformation characteristics of the foam at high temperatures and high strains that are typically encountered during blow molding, and correlate the strains developed during blow molding to the morphological aspects in the resulting blow molded part. These aspects are addressed in this paper using simulations of uniaxial tensile deformation of a virtual representative volume element of a foam microstructure (rendered in DIGIMAT-FE) to derive the nonlinear tensile response of the foam at high temperatures (using ABAQUS). The resulting simulated stress-strain curve is employed to parameterize a nonlinear rheological constitutive equation. These parameters are then employed for the homogenized representation of the foam in the blow molding simulation carried out in B-SIM, a commercially available simulation software for blow molding. The regions where the simulated parison has undergone primarily uniaxial elongation are then mapped back to the expected local foam morphology using the transfer functions derived from the RVE simulations. These steps result in a preliminary and simple demonstration of the simulation framework, and offer a template that can be detailed further with experimental rheological information on actual foamed parisons, and more detailed post-processing algorithms to correlate multiaxial elongations with microstructure.