Normal Stress Coefficient Research Articles

Master curves for unidirectional flows of FENE-P fluids in rectilinear and curvilinear geometries

We demonstrate that velocity profiles for steady, unidirectional shear flows of the FENE-P (Finitely-Extensible Nonlinear Elastic, with Peterlin closure) fluid, undergoing canonical rectilinear (pressure-driven flow in a rectangular channel or a circular pipe) or curvilinear (in Taylor–Couette or Dean configurations) flows, obey universal master curves that are a function only of the ratio Wi/L , for a fixed solvent to solution viscosity parameter β. Here, Wi is the Weissenberg number defined as the product of the dumbbell relaxation time and an appropriate shear rate, while L is the ratio of the maximum extension of the polymer to its equilibrium root-mean-square end-to-end distance. The data collapse and the resulting master curves for the velocity profile is a generalization of the recent demonstration of master curves for polymer viscosity and first normal stress coefficient for a FENE-P fluid under steady simple shear flow (Yamani and McKinley, 2023). For pressure-driven channel and pipe flows, we derive simple analytical expressions for the velocity profiles, in the high shear-rate regime of Wi/L≫1, that readily elucidate the role of finite extensibility of the polymer on the velocity profiles. In the Wi/L≫1 regime, for all the flows considered, the limit of zero solvent (β=0) is shown to be singular, owing to the absence of a high-shear plateau in the total solution viscosity, resulting in very different velocity profiles for β=0 and β→0.

A novel statistical damage constitutive model of rock joints considering normal stress and joint roughness

The shear constitutive model of rock joints is of great significance to the stability analysis in rock engineering, and it is closely related to the normal stress ([Formula: see text]) and joint roughness coefficient ( JRC). However, the existing investigations seldom consider the influences of [Formula: see text] and JRC simultaneously. Therefore, a novel damage constitutive model considering the [Formula: see text] and JRC is developed in this work. In the presented model, it is assumed that the rock materials are composed of damaged and undamaged microunits, and the damage evolution law of the microunits conforms to the Weibull distribution in the shear process. Based on the proposed assumption, the constitutive relationship between shear stress and shear displacement is deduced. The evolutions of the mechanical parameters and damage variable versus [Formula: see text] and JRC are analyzed in detail. The proposed damage model that involves [Formula: see text] and JRC is verified by comparing theoretical values with the laboratory results. The results show that the damage constitutive model is in good agreement with the test results. Additionally, the influences of [Formula: see text] and JRC on the shear stress-displacement curves are studied. This work can provide a valuable theoretical method for analyzing the shear mechanical characteristics and damage evolution laws of rock joints.

A Theoretical and Experimental Study on Asperity Damage-Driven Strain Rate Dependency of Fractured Coal

AbstractAsperities within pre-existing fractures of coals can experience local damage during the fracture closure due to external loading. Previous research postulates that this local asperity damage can lead to strain rate-dependency without causing permanent deformation to the bulk of the coal specimens. This study aims to comprehensively investigate this behavior by developing a theoretical model that characterizes the strain rate-dependency driven by fracture asperity damage in coal. To achieve this objective, an initial series of micro-scale mechanical tests are conducted on joint specimens to establish a model for effective stress acting on asperities. Building upon this model, a theoretical foundation is further developed to describe the strain rate-dependent asperity damage evolution and resulting energy dissipation. These frameworks are subsequently incorporated into elasticity and damage mechanics to capture the strain rate-dependent stress–strain relationships. To validate the proposed model across multiple scales, additional triaxial tests on core-scale specimen and micro-scale mechanical tests on joint specimens are performed. The experimentally measured strain rate-dependency aligns well with the predictions of the proposed model, indicating a successful development of a robust model. The results of the model developed in this study reveal that the strain rate-dependency in fractured coals is governed by several factors, including asperity damage, mechanical properties of the coal specimens and effective stress acting on asperities of pre-existing fractures within the bulk of coal. Moreover, it is shown that the effective stress acting on asperities is significantly affected by both applied normal stress and joint roughness coefficient (JRC). The insights derived from this study demonstrate that the strain rate-dependency induced by micro-scale asperity damage of pre-existing fractures leads to observable strain rate-dependency in bulk specimens at core-scale and the proposed model can adequately capture this behavior.

Measurement of 3D joint surface morphology and estimation for shear mechanical behavior: A theoretical model and experimental investigation

The majority of rock engineering instabilities are primarily due to shear slips and failures on the joint surfaces, and the measurement of joint surface morphology is crucial for shear strength evaluation. In this study, the high-precision 3D laser scanner was first used to characterize and digitally reconstruct the surface morphology of various joints. The evaluation method of joint surface roughness was modified according to the 2D profile data and shear direction. Then, the effects of normal stress and initial joint roughness coefficient on the shear mechanical behavior were investigated by experimental method. The damage evolution characteristics and failure mechanism of the joint surface are analyzed by acoustic emission monitoring and image binarization calculations. The research results indicate that as the normal stress and initial joint roughness coefficient increase, the peak shear stress and shear displacement, residual strength, and cumulative AE counts of rock joints display an increasing trend. Based on the evolution of mechanical parameters, a new mechanical model of rock joints to predict shear strength and deformation was developed, which was validated by comparing the theoretical prediction curves with test results. A good concordance was achieved between the shear test and theoretical results, indicating that the developed model can effectively predict the shear strength and deformation of joint surfaces under different normal stress and roughness. This research results are expected to provide a valuable guidance for early warning and evaluation of geological disasters.

Experimental and DEM analysis for the shear characteristics of interface between sand and irregular concrete under dynamic normal loading

To explore the dynamic shear property of sand–irregular concrete interface, a series of direct shear tests were conducted under dynamic normal loading. These tests were designed with various joint roughness coefficients (JRC), amplitudes and frequencies of dynamic normal loading. DEM models were established to analyze the microscopic mechanical behavior. The research results indicate that a larger amplitude and frequency of the dynamic normal loading would lead to a decrease in the interface shear stress and dilation. As the JRC increased, the interface shear strength first increased and then decreased, and the dilation increased. A rougher surface would reduce the phase shift between peak shear and normal stress. The phase shift between normal stress and friction coefficient was always half a normal loading cycle. A greater JRC would activate a more violent particle motion and a thicker shear band. The increase in effective roughness would lead to an increase in the deviation angle of the anisotropic direction. The evolution of force chains was observed in three-dimensional space. Weak force chains were screened out during the shearing process. As the shearing developed, the number of force chains decreased, the average length and strength of force chains continued to increase.

Global Well-Posedness and Vanishing Normal Stress Coefficients for the Hydrostatic Second-Grade Fluid Equations

Global Well-Posedness and Vanishing Normal Stress Coefficients for the Hydrostatic Second-Grade Fluid Equations

A practical guide to mitigate edge fracture instability in sheared polymer melts

The measurement of nonlinear shear response of viscoelastic materials is often hindered by edge fracture instabilities. The phenomenon was first addressed theoretically by Tanner and Keentok and ever since has attracted the interest of experimentalists and theorists alike. Despite progress, accounting for or mitigating edge fracture remains a challenge, in particular when dealing with strongly viscoelastic materials such as entangled polymer melts. Here, we present and compare different experimental attempts to delay edge fracture in a cone-and-plate (CP) geometry, including the use of an immiscible fluid bath around the sample (that reduces the stress and interfacial gradients in comparison with the sample/air interface), a cone-partitioned plate (CPP) fixture, and an outer collar attached to the sample's edge (in a CP or CPP fixture). Focusing on the torque signal, we find that the combination of CPP and collar provides the best results. This may indeed help measuring highly elastic materials over an extended range of shear rates and, importantly, contribute to reliably measuring the normal stress coefficients in a cone-partitioned plate tool. It is, therefore, hoped that this simple idea will be further pursued in the direction of improving our current rheometric capabilities.

Laun's rule for predicting the first normal stress coefficient in complex fluids: A comprehensive investigation using fractional calculus

Laun's rule [H. M. Laun, “Prediction of elastic strains of polymer melts in shear and elongation,” J. Rheol. 30, 459–501 (1986).] is commonly used for evaluating the rate-dependent first normal stress coefficient from the frequency dependence of the complex modulus. We investigate the mathematical conditions underlying the validity of Laun's relationship by employing the time-strain–separable Wagner constitutive formulation to develop an integral expression for the first normal stress coefficient of a complex fluid in steady shear flow. We utilize the fractional Maxwell liquid model to describe the linear relaxation dynamics compactly and accurately and incorporate material nonlinearities using a generalized damping function of Soskey–Winter form. We evaluate this integral representation of the first normal stress coefficient numerically and compare the predictions with Laun's empirical expression. For materials with a broad relaxation spectrum and sufficiently strong strain softening, Laun's relationship enables measurements of linear viscoelastic data to predict the general functional form of the first normal stress coefficient but often with a noticeable quantitative offset. Its predictive power can be enhanced by augmenting the original expression with an adjustable power-law index that is based on the linear viscoelastic characteristics of the specific material being considered. We develop an analytical expression enabling the calculation of the optimal power-law index from the frequency dependence of the viscoelastic spectrum and the strain-softening characteristics of the material. To illustrate this new framework, we analyze published data for an entangled polymer melt and for a semiflexible polymer solution; in both cases our new approach shows significantly improved prediction of the experimentally measured first normal stress coefficient.

Cyclic shear behavior of en-echelon joints under constant normal stiffness conditions

To reveal the mechanism of shear failure of en-echelon joints under cyclic loading, such as during earthquakes, we conducted a series of cyclic shear tests of en-echelon joints under constant normal stiffness (CNS) conditions. We analyzed the evolution of shear stress, normal stress, stress path, dilatancy characteristics, and friction coefficient and revealed the failure mechanisms of en-echelon joints at different angles. The results show that the cyclic shear behavior of the en-echelon joints is closely related to the joint angle, with the shear strength at a positive angle exceeding that at a negative angle during shear cycles. As the number of cycles increases, the shear strength decreases rapidly, and the difference between the varying angles gradually decreases. Dilation occurs in the early shear cycles (1 and 2), while contraction is the main feature in later cycles (3−10). The friction coefficient decreases with the number of cycles and exhibits a more significant sensitivity to joint angles than shear cycles. The joint angle determines the asperities on the rupture surfaces and the block size, and thus determines the subsequent shear failure mode (block crushing and asperity degradation). At positive angles, block size is more greater and asperities on the rupture surface are smaller than at nonpositive angles. Therefore, the cyclic shear behavior is controlled by block crushing at positive angles and asperity degradation at negative angles.

Designing a swimming rheometer to measure the linear and non-linear properties of a viscoelastic fluid

At low Reynolds numbers, “swirlers” – swimmers with an axisymmetric “head” and “tail” counterrotating about the axis of symmetry – generate no net propulsion in a Newtonian fluid as a consequence of the “scallop theorem”. Viscoelasticity in the suspending fluid breaks the time-reversibility and allows swirlers to propel themselves, with the swim speed being a function of swimmer geometry, fluid elasticity, and swimming gait. Using analytical theory and numerical simulations, we study the unsteady motion of a freely-suspended self-propelled swirler though viscoelastic fluids described by the Giesekus model, allowing for general axisymmetric geometry and time-dependent tail rotation rate. We show the steady swim speed can be calculated for general arbitrary axisymmetric geometries at low Deborah number via the reciprocal theorem and the solution of two Newtonian flow problems. In this “weak flow” limit, we analytically determine the swim speed and its dependence on the parameters of the Giesekus fluid which in turn are related to the primary and secondary normal stress coefficients Ψ1 and Ψ2. Furthermore, at low De, we derive the unsteady swim speed as a function of a specified unsteady tail rotation rate and the material properties of the suspending fluid. We show that for a particular tail rotation rate, the unsteady swim speed can be analyzed to recover the spectrum of fluid relaxation times, analogous to small-amplitude oscillatory shear measurements on a benchtop rheometer. This study expands upon the design space for a “swimming rheometer” by increasing its functionality to make and interpret rheological measurements.

Multi-scale modelling of dilute viscoelastic liquids: Atomistic to mesoscale mapping of polymer solutions

A framework predicting the rheological (storage and loss moduli, first normal stress coefficient, and relaxation time) and transport (viscosity, diffusivity) properties of non-Newtonian dilute polymer solutions at mesoscales (e.g. ∼ns to μs) from the atomistic-scale molecular behaviour is presented. More specifically, the rheological behaviour differences of OCP and PMA polymer solutions in PAO-2 oil are simulated using both atomistic molecular dynamics (MD) and many-body dissipative particle dynamics (mDPD) within a temperature range of 313–373 K. The simulation methodology described is able to distinguish itself from the standard DPD model by accurately reproducing the shear-thinning with high sensitivity, as seen in the atomistic MD simulations at high shear rates (e.g. 108−1013 s−1). It is shown that the model is well-suited to compute properties such as first normal stress differences and relaxation times that are difficult to estimate at atomistic scales due to the low signal-to-noise ratio. Moreover, the Schmidt numbers (>103) are predicted with high accuracy when compared with the values from atomistic-scale simulations. The proposed model is able to predict relaxation times of dilute polymer mixtures that are difficult to be obtained using state-of-the-art rheometers. Finally, it is found that the terminal relaxation time of PMA polymer chain does not vary monotonically as a function of temperature, unlike in the case of OCP; this is significant for describing viscoelastic behaviour at macroscales where satisfactory constitutive equations are not available.

General rigid bead-rod theory for steady-shear flow

General rigid bead-rod theory yields uniquely the relation between macromolecular architecture and complex viscosity. For this, it relies on the analytical solution of the general diffusion equation for small-amplitude oscillatory shear flow of Bird et al. [Dynamics of Polymeric Liquids, 2nd ed. (Wiley, New York, 1987), Vols. 1–2]. Unfortunately, this general diffusion equation has yet to be solved for any other flow field. In this paper, we do so for steady-shear material functions, namely, viscosity and first normal stress coefficient. We, thus, explain the non-Newtonian behaviors of macromolecular suspensions of any axisymmetric design in steady-shear flow.

Confined steady simple shear flow of polymeric liquids

In a confined simple shear flow, the macromolecules of a polymeric liquid reorient near the walls so that the measured viscosity decreases. For instance, in a small-amplitude oscillatory shear flow, the real part of the complex viscosity decreases with confinement, and macromolecular orientation explains this. These effects in oscillation have been explained analytically for a rigid dumbbell suspension and, for a confined small-amplitude oscillatory shear flow, the summation coefficients have been determined. By contrast, for the confined steady shear flow, the summation coefficients are undetermined. In this paper, we determine these coefficients and use them to evaluate the steady shear (i) viscosity and (ii) normal stress coefficients for a rigid dumbbell suspension. We find that the zero-shear viscosity and the zero-shear first normal stress coefficients decrease with confinement. We further find that the dimensionless (i) steady shear viscosity curve increases with confinement and (ii) first normal stress coefficient first decreases with light confinement and then increases with greater confinement. We confirm our theory, at low confinement, by comparing with our new measurements of the confined zero-shear viscosity of a polystyrene solution.

Effect of particle regularity on the shear response of geogrid-granular material interface under cyclic normal loading

The investigation of shear behaviour on geogrid-granular material interface under cyclic normal loading (CNL) is significant in seismic design of reinforcement structure. The cyclic shear mechanism about the effect of particle overall regularity on reinforced soil interface was analysed using dynamic direct shear apparatus. Five mixtures of round spherical media and crushed limestone with different percentages were selected to obtain various particle overall regularity. By comparing the shear behaviour under various overall regularity, the interface shear stress and vertical displacement under the CNL condition were smaller than those under the static loading condition. The decrement in interface shear stress of lower overall regularity was smaller than that of higher overall regularity. The interface peak shear stress, final vertical displacement decreased with the increase of cyclic normal loading amplitude and overall regularity. The effect of initial normal loading on ratio of strength reduction and energy consumption was reverse to that of the overall regularity. Based on the constant relative phase shift of the peak normal stress and friction coefficient for various overall regularity, the prediction formula was proposed. The test results improve the understanding on how normal loading perturbations and particle overall regularity affect the shear behaviour of reinforced soil interface.

Odd viscosity in chiral passive suspensions

Prior studies have revealed that nonzero odd viscosity is an essential property for chiral active fluids. Here we report that such an odd viscosity also exists in suspensions of non-active or non-externally-driven but chirally-shaped particles. Computational simulations are carried out for monolayers of dense ratchets in simple shear and planar extensional flows. The contact between two ratchets can be either frictionless or infinitely-frictional, depending on their teeth and sliding directions at the contact point. Our results show that the ratchet suspension has the intermediate shear/extensional viscosity as compared with the suspensions of smooth and gear-like particles. Meanwhile, the ratchet suspensions show nonzero even and odd components of the first normal stress coefficient regarding the flow rate, which indicates the mixed feature of conventional complex fluids and chiral viscous fluids.

A freely suspended robotic swimmer propelled by viscoelastic normal stresses

We present a self-propelled axisymmetric low-Reynolds-number swimmer (force- and torque-free) that, while unable to swim in a Newtonian fluid, propels itself in a non-Newtonian fluid as a result of fluid elasticity. The propulsion gait is pure ‘swirling’ and is demonstrated using a robotic swimmer, comprised of a ‘head’ sphere and a ‘tail’ sphere, whose swimming speed is shown to demonstrate good agreement with a microhydrodynamic asymptotic theory and numerical simulations. Schlieren imaging demonstrates that propulsion of the swimmer is driven by a strong viscoelastic jet at the tail, which develops due to the fore–aft asymmetry of the swimmer. Optimized cylindrical and conic tail geometries are shown to double the propulsive signal, relative to the optimal spherical tail. We show that we can use observations of this robot to infer rheological properties of the surrounding fluid. We measure the primary normal stress coefficient $\varPsi _1$ at shear rates ${<}1\,{\rm Hz}$ , and show reasonable agreement with extrapolated bench-top measurements ( $0.8\unicode{x2013}1.2\,{\rm Pa}\,{\rm s}^2$ difference). We also derive how the head rotation may be used to measure the second normal stress coefficient, $\varPsi _2$ . The study demonstrates the exciting potential for a ‘swimming rheometer’, examining the intersection between swimming and sensing in complex fluids.

Hydrodynamic interaction and complex viscosity of multi-bead rods

One good way to explain the elasticity of a polymeric liquid is to just consider the orientation distribution of the macromolecules. When exploring how macromolecular architecture affects the elasticity of a polymeric liquid, we find the general rigid bead–rod theory to be both versatile and accurate. This theory sculpts macromolecules using beads and rods. Whereas beads represent points of Stokes flow resistances, the rods represent rigid separations. In this way, how the shape of the macromolecule affects its rheological behavior in suspension is determined. Until recently, general rigid bead–rod theory has neglected interferences of the Stokes flow velocity profiles between nearby beads. We call these hydrodynamic interactions, and we here employ our new method for exploring how these interactions affect the complex viscosity of suspensions of multi-bead rods. These multi-bead rods are also called shish-kebabs. We use the center-to-center distance between adjacent beads as the characteristic length. We proceed analytically, beginning with a geometric expression for the shish-kebab bead positions. Our analytical solution for the complex viscosity presents as one for N=3,5,6,7,8,…, one for N=4, and another for the rigid dumbbell, N=2. We find that for shish-kebabs, hydrodynamic interactions (i) increase zero-shear viscosity, (ii) increase zero-shear first normal stress coefficient, (iii) decrease the real part of the dimensionless complex viscosity, and (iv) increase minus the dimensionless imaginary part. We find that the combination of (iii) and (iv) explains crossovers of the parts of the complex viscosity. We further find that for a monodisperse polystyrene solution, the general rigid bead–rod theory with hydrodynamic interaction, for both parts of the complex viscosity, provides stunning improvement over without.

Understanding viscoelastic flow instabilities: Oldroyd-B and beyond

The Oldroyd-B model has been used extensively to predict a host of instabilities in shearing flows of viscoelastic fluids, often realized experimentally using polymer solutions. The present review, written on the occasion of the birth centenary of James Oldroyd, provides an overview of instabilities found across major classes of shearing flows. These comprise (i) the canonical rectilinear shearing flows including plane Couette, plane and pipe Poiseuille flows; (ii) viscometric shearing flows with curved streamlines such as those in the Taylor-Couette, cone-and-plate and parallel-plate geometries; (iii) non-viscometric shearing flows with an underlying extensional flow topology such as the flow in a cross-slot device; and (iv) multilayer shearing flows. While the underlying focus in all these cases is on results obtained using the Oldroyd-B model, we also discuss their relation to the actual instability, and as to how the shortcomings of the Oldroyd-B model may be overcome by the use of more realistic constitutive models. All the three commonly used tools of stability analysis, viz., modal linear stability, nonmodal stability, and weakly nonlinear stability analyses are discussed, with supporting evidence from experiments and numerical simulations as appropriate. Despite only accounting for a shear-rate-independent viscosity and first normal stress coefficient, the Oldroyd-B model is able to qualitatively predict the majority of instabilities in the aforementioned shearing flows. The review also highlights, where appropriate, open questions in the area of viscoelastic stability.

Mesoscopic theoretical modeling and experimental study of rheological behavior of water-based drilling fluid containing associative synthetic polymer, bentonite, and limestone

Employing an effective rheological model for the flow of drilling fluid that can accurately predict changing conditions is of significant importance in drilling fluid optimization. Traditional generalized Newtonian models cannot predict the time change condition, viscoelastic behavior, role of each component, or microstructural behaviors within the fluid. Consequently, the present research aims to develop constitutive equations in the framework of generalized bracket formalisms and the extra tensor concept that connect the microscopic and macroscopic properties and can overcome the aforementioned problems of traditional rheological models. The developed model is applicable for drilling fluid as a suspension system containing associative synthetic polymer, bentonite, and limestone suspended in water; where simple structures of flexible dumbbells, disks, and hard spheres, respectively, are representative of each component in the system. Five samples of drilling fluid with different additive concentrations were prepared and rheological testing was performed on these samples. To obtain the fixed parameters of the model, scanning electron microscope and particle size analyses were conducted on the dried powder and dispersed particles of bentonite and limestone to characterize the shape and size of the particles. The adjustable parameters of the model were then obtained by fitting it using the gathered experimental rheological data. The outcomes of the study revealed that the novel developed model can accurately predict rheological material functions, including shear viscosity and the first normal stress coefficient under transient and steady-state conditions. Furthermore, the presented model is capable of distinguishing the contribution of each component in the drilling fluid rheology.

Rouse model with fluctuating internal friction

A coarse-grained bead-spring-dashpot chain model with the dashpots representing the presence of internal friction is solved exactly numerically for the case of chains with more than two beads. Using a decoupling procedure to remove the explicit coupling of a bead’s velocity with that of its nearest neighbors, the governing set of stochastic differential equations are solved with Brownian dynamics simulations to obtain material functions in oscillatory and steady simple shear flow. Simulation results for the real and imaginary components of the complex viscosity have been compared with the results of previously derived semi-analytical approximations, and the difference in the predictions is seen to diminish with an increase in the number of beads in the chain. The inclusion of internal friction results in a nonmonotonous variation of the viscosity with shear rate, with the occurrence of continuous shear-thickening following an initial shear-thinning regime. The onset of shear-thickening in the first normal stress coefficient is pushed to lower shear rates with an increase in the internal friction parameter.