Normal Stress Coefficient Research Articles

Non-constant link tension coefficient in the tumbling-snake model subjected to simple shear.

The authors of the present study have recently presented evidence that the tumbling-snake model for polymeric systems has the necessary capacity to predict the appearance of pronounced undershoots in the time-dependent shear viscosity as well as an absence of equally pronounced undershoots in the transient two normal stress coefficients. The undershoots were found to appear due to the tumbling behavior of the director u when a rotational Brownian diffusion term is considered within the equation of motion of polymer segments, and a theoretical basis concerning the use of a link tension coefficient given through the nematic order parameter had been provided. The current work elaborates on the quantitative predictions of the tumbling-snake model to demonstrate its capacity to predict undershoots in the time-dependent shear viscosity. These predictions are shown to compare favorably with experimental rheological data for both polymer melts and solutions, help us to clarify the microscopic origin of the observed phenomena, and demonstrate in detail why a constant link tension coefficient has to be abandoned.

Molecular dynamics for linear polymer melts in bulk and confined systems under shear flow

In this work, we analyzed the individual chain dynamics for linear polymer melts under shear flow for bulk and confined systems using atomistic nonequilibrium molecular dynamics simulations of unentangled (C50H102) and slightly entangled (C178H358) polyethylene melts. While a certain similarity appears for the bulk and confined systems for the dynamic mechanisms of polymer chains in response to the imposed flow field, the interfacial chain dynamics near the boundary solid walls in the confined system are significantly different from the corresponding bulk chain dynamics. Detailed molecular-level analysis of the individual chain motions in a wide range of flow strengths are carried out to characterize the intrinsic molecular mechanisms of the bulk and interfacial chains in three flow regimes (weak, intermediate, and strong). These mechanisms essentially underlie various macroscopic structural and rheological properties of polymer systems, such as the mean-square chain end-to-end distance, probability distribution of the chain end-to-end distance, viscosity, and the first normal stress coefficient. Further analysis based on the mesoscopic Brightness method provides additional structural information about the polymer chains in association with their molecular mechanisms.

Communication: Appearance of undershoots in start-up shear: Experimental findings captured by tumbling-snake dynamics.

Our experimental data unambiguously show (i) a damping behavior (the appearance of an undershoot following the overshoot) in the transient shear viscosity of a concentrated polymeric solution, and (ii) the absence of a corresponding behavior in the transient normal stress coefficients. Both trends are shown to be quantitatively captured by the bead-link chain kinetic theory for concentrated polymer solutions and entangled polymer melts proposed by Curtiss and Bird, supplemented by a non-constant link tension coefficient that we relate to the nematic order parameter. The observed phenomena are attributed to the tumbling behavior of the links, triggered by rotational fluctuations, on top of reptation. Using model parameters deduced from stationary data, we calculate the transient behavior of the stress tensor for this "tumbling-snake" model after startup of shear flow efficiently via simple Brownian dynamics. The unaltered method is capable of handling arbitrary homogeneous flows and has the promising capacity to improve our understanding of the transient behavior of concentrated polymer solutions.

In search of physical meaning: defining transient parameters for nonlinear viscoelasticity

A complete set of model-independent viscoelastic functions for understanding responses to transient nonlinear rheological tests is presented, using large-amplitude oscillatory shear strain as a model nonlinear protocol. The derivation makes no assumptions about symmetries, and is therefore applicable to the responses to any input, allowing researchers to unambiguously define time-dependent moduli, viscosities, compliances, fluidities, and normal stress coefficients. A legend for interpreting the dynamic trajectories in modulus space is provided, along with explicit definitions of the rates at which the moduli change. These provide a quantitative mechanism to identify when, and by how much, a material response stiffens, softens, thickens, or thins while being deformed. In addition to providing analytical expressions for the moduli, the derivation requires the definition of a conceptually new term. This means there exist three, not two, time-dependent nonlinear viscoelastic functions by which any response can be fully described. The third function accounts for nonlinear properties such as yield stresses and the shifting of the strain equilibrium. This complete analysis scheme is unique in making a distinction between the strains in the lab and material frames. The quantitative sequence of physical process analysis, which is fully developed in this work, allows for comprehensive physical interpretations of responses to transient deformations of any kind to be made, including the steady alternance responses to large-amplitude oscillatory shear (LAOS), time-dependent oscillatory shear startup responses, and thixotropic and anti-thixotropic responses.

Normal stress effects in the gravity driven flow of granular materials

In this paper, we study the fully developed gravity-driven flow of granular materials between two inclined plates. We assume that the granular materials can be represented by a modified form of the second grade fluid where the viscosity depends on the shear rate and volume fraction and the normal stress coefficients depend on the volume fraction. We also propose a new isotropic (spherical) part of the stress tensor which can be related to the compactness of the (rigid) particles. This new term ensures that the rigid solid particles cannot be compacted beyond a point, namely when the volume fraction has reached the critical/maximum packing value. The numerical results indicate that the newly proposed stress tensor has obvious and physically meaningful effects on both the velocity and the volume fraction fields.

Direct shear behavior of a plane joint under dynamic normal load (DNL) conditions

The effect of dynamic normal loading (DNL) on the direct shear behavior of a plane joint has been first time investigated using a dynamic direct shear box device. Multi-stage normal loading tests with superimposed normal cyclic excitation at different frequencies were conducted at constant horizontal shear velocity of 3000μm/min. Experimental results show, that peak shear force increases with increase of normal stress, and amplitude ratio between shear and normal stress decreases with the increase of normal stress. A significant phase shift between peak normal stress and peak shear stress is observed, which decreases with increasing normal load, while the relative time shift between peak normal stress and peak friction coefficient is nearly constant. Impact frequency has little effect on relative time shift. The dynamic normal stiffness is significantly higher than the static one. Finally, a new shear strength criterion is proposed for a joint under constant shear velocity and sinusoidal normal loading.

Stress in a dilute suspension of spheres in a dilute polymer solution subject to simple shear flow at finite Deborah numbers

A theoretical analysis shows that polymers interacting with particles in a shear flow experience enhanced streamwise stretch that grows in amplitude and spatial extent with increasing Deborah number. This results in shear thickening of the viscosity and first normal stress coefficient of a particle suspension in a Boger fluid.

Flow-Induced Orientation and Stretching of Entangled Polymers in the Framework of Nonequilibrium Thermodynamics

We provide a description of the Marrucci–Ianniruberto constitutive equation [Philos. Trans. R. Soc. London, A 2003, 361, 677−688] for the rheology of entangled polymer melts in the context of nonequilibrium thermodynamics and we properly extend it to account for a second normal stress difference by introducing a second order term in the relaxation tensor in terms of the conformation tensor. The modified model incorporates one additional parameter, the anisotropic mobility parameter α, which allows for nonvanishing predictions of the second normal stress coefficient. Application of the second law of thermodynamics and the requirement that the evolution equation must preserve the positive-definite nature of the conformation tensor between successive entanglement points along the chain for all times and all flow fields constrain the convective constraint release (CCR) parameter βccr to values strictly greater than one (βccr > 1) and the new parameter α to values in the interval 0 ≤ α ≤ 1 − βccr–1. The modifi...

Simulation of Individual Polymer Chains and Polymer Solutions with Smoothed Dissipative Particle Dynamics

In an earlier work (Litvinov et al., Phys.Rev.E 77, 066703 (2008)), a model for a polymer molecule in solution based on the smoothed dissipative particle dynamics method (SDPD) has been presented. In the present paper, we show that the model can be extended to three-dimensional situations and simulate effectively diluted and concentrated polymer solutions. For an isolated suspended polymer, calculated static and dynamic properties agree well with previous numerical studies and theoretical predictions based on the Zimm model. This implies that hydrodynamic interactions are fully developed and correctly reproduced under the current simulated conditions. Simulations of polymer solutions and melts are also performed using a reverse Poiseuille flow setup. The resulting steady rheological properties (viscosity, normal stress coefficients) are extracted from the simulations and the results are compared with the previous numerical studies, showing good results.

Effect of normal stresses on the results of thermoplastic mold filling simulation

The paper deals with the effect of the normal stresses on the predicted flow front during the filling stage of thermoplastic injection molding. The normal stresses are predicted using the non-linear Criminale-Ericksen-Filbey model (a variant of the second-order fluid rheological model with viscosity, first and second normal stress coefficients dependent upon magnitude of shear rate) incorporated into a comprehensive 3D simulation software for mold-filling analysis. The additional stress term allows the prediction of the so called ear-flow effect (melt racing on the edges of the cavity).

Reflections on inflections

In plastics processing, the single most important rheological property is the steady shear viscosity curve: the logarithm of the steady shear viscosity versus the logarithm of the shear rate. This curve governs the volumetric flowrate through any straight channel flow, and thus governs the production rate of extruded plastics. If the shear rate is made dimensionless with a characteristic time for the fluid (called the Weissenberg number, Wi), then we can readily identify the end of the Newtonian plateau of a viscosity curve with the value Wi≈1. Of far greater importance, however, is the slope at the point where the viscosity curve inflects, (n-1), where n is called the shear power-law index. This paper explores the physics of this point and related inflections, in the first and second normal stress coefficients. We also discuss the first and second inflection pairing times, λ′B and λ″B. First, we examine the generalized Newtonian fluid (Carreau model). Then, we analyze the more versatile model, the corotational Oldroyd 8-constant model, which reduces to many simpler models, for instance, the corotational Maxwell and Jeffreys models. We also include worked examples to illustrate the procedure for calculating inflection points and power-law coefficients for all three viscometric functions, \(\eta \left( {\dot \gamma } \right)\), \({\Psi _1}\left( {\dot \gamma } \right)\) and \({\Psi _2}\left( {\dot \gamma } \right)\).

Frictional strength and heat flow of southern San Andreas Fault

Frictional strength and heat flow of faults are two related subjects in geophysics and seismology. To date, the investigation on regional frictional strength and heat flow still stays at the stage of qualitative estimation. This paper is concentrated on the regional frictional strength and heat flow of the southern San Andreas Fault (SAF). Based on the in situ borehole measured stress data, using the method of 3D dynamic faulting analysis, we quantitatively determine the regional normal stress, shear stress, and friction coefficient at various seismogenic depths. These new data indicate that the southern SAF is a weak fault within the depth of 15 km. As depth increases, all the regional normal and shear stresses and friction coefficient increase. The former two increase faster than the latter. Regional shear stress increment per kilometer equals 5.75 ± 0.05 MPa/km for depth ≤15 km; regional normal stress increment per kilometer is equal to 25.3 ± 0.1 MPa/km for depth ≤15 km. As depth increases, regional friction coefficient increment per kilometer decreases rapidly from 0.08 to 0.01/km at depths less than ~3 km. As depth increases from ~3 to ~5 km, it is 0.01/km and then from ~5 to 15 km, and it is 0.002/km. Previously, frictional strength could be qualitatively determined by heat flow measurements. It is difficult to obtain the quantitative heat flow data for the SAF because the measured heat flow data exhibit large scatter. However, our quantitative results of frictional strength can be employed to investigate the heat flow in the southern SAF. We use a physical quantity P f to describe heat flow. It represents the dissipative friction heat power per unit area generated by the relative motion of two tectonic plates accommodated by off-fault deformation. P f is called “fault friction heat.” On the basis of our determined frictional strength data, utilizing the method of 3D dynamic faulting analysis, we quantitatively determine the regional long-term fault friction heat at various seismogenic depths in the southern SAF. The new data show that as depth increases, regional friction stress increases within the depth of 15 km; its increment per kilometer equals 5.75 ± 0.05 MPa/km. As depth increases, regional long-term fault friction heat increases; its increment per kilometer is equal to 3.68 ± 0.03 mW/m2/km. The values of regional long-term fault friction heat provided by this study are always lower than those from heat flow measurements. The difference between them and the scatter existing in the measured heat flow data are mainly caused by the following processes: (i) heat convection, (ii) heat advection, (iii) stress accumulation, (iv) seismic bursts between short-term lull periods in a long-term period, and (v) influence of seismicity in short-term periods upon long-term slip rate and heat flow. Fault friction heat is a fundamental parameter in research on heat flow.

Explosive instabilities for a generalized second grade fluid

We consider a general model for a non-Newtonian fluid of second grade which is capable of describing shear thinning and shear thickening effects. The model incorporates parameters m and n which are varied to account for which effect, shear thinning or thickening, is desired in the tangential and normal shear rate directions, although here we restrict attention to the shear thickening case. By constructing a generalized energy-like equation and manipulating this we show that the solution may exhibit finite-time blow up if the initial data are regular enough for the solution to exist for a sufficiently long time. The nature of the blow up, or nonexistence, behavior depends critically on the normal stress coefficients α1 and α2, and on the parameters m and n.

점탄성 고분자 액체의 정상유동함수와 과도적 유동함수의 상관관계 연구: Gleissle 밀러 관계식들의 실험적 검증 및 이론적 고찰

The objective of this study is to systematically investigate the relationships between steady flow functions and transient flow functions for viscoelastic polymer liq- uids. Using a strain-controlled rheometer (Advanced Rheometric Expansion System (ARES)), the steady shear flow properties and the transient shear flow properties of concen- trated poly(ethylene oxide) (PEO) solutions have been measured over a wide range of shear rates and times. The validity of the three forms of the Gleissle mirror relations was examined by comparing them with the experimentally obtained results. In addition, the effect of nonlinearity on the applicability of these Gleissle mirror relations was discussed from a theoretical view-point by introducing the concept of a nonlinear strain measure. The main findings obtained from this study can be summarized as follows: (1) A nonlinear strain measure is decreased with an increase in strain magnitude, after reaching the maxi- mum value at small strain range. This behavior is quite different from the theoretical pre- diction to satisfy the conditions of the Gleissle mirror relations. (2) The first mirror relation describing the equivalence between steady shear flow viscosity and shear stress growth coefficient is valid over a wide range of shear rates and is hardly affected by the nonlinear- ity of polymer solutions. (3) The second mirror relation expressing the equivalence between first normal stress coefficient and first normal stress growth coefficient is also applicable over a wide range of shear rates. This relation is, however, significantly influ- enced by the degree of nonlinearity (i.e., shape of a nonlinear strain measure) of polymer solutions. (4) The third mirror relation can be regarded as a very useful empirical model to predict the first normal stress coefficient from steady shear flow viscosity data, provided that an appropriate value of a shift factor is given.

Modified Single-Mode Leonov Rheological Equations for Polymer Melts and Solutions

The shear viscosity and the normal stress coefficients are important parameters in the flow of polymer melts and polymer solutions. Based on the Leonov model, modified single-mode rheological equations are presented by introducing relaxation time and temperature functions, and the shear viscosity and the normal stress coefficients are predicted. Without a complex statistical simulation, the experimental data of a low-density polyethylene melt, a poly(ethylene oxide) solution and a mixed decalin/polybutene oil solution were compared to verify the modified equations in very wide range of deformation rates. Furthermore, based on the equations, the relationship between the stress overshoot and the temperature is discussed. In addition, the predicted shear thinning behavior for the modified equations is also compared with other single-mode models.

Density-depth model of the continental wedge at the maximum slip segment of the Maule Mw8.8 megathrust earthquake

Complexities in the rupture process during a megathrust earthquake can be attributed to the combined effect of inhomogeneous distribution of stress accumulated during the interseismic period and inhomogeneous rheology of the seismogenic contact. We modeled the free-air gravity field of the southern Central Chile convergent margin along five 2-D profiles that cross the patch of highest slip during the Chilean 2010 megathrust earthquake in order to analyze variability in the density and shape of the continental wedge and its relationship with seismotectonics. We also analyzed the bathymetry to derive the long-term interplate friction coefficient. The results show that the high slip patch during the Maule earthquake corresponds to a segment of the margin characterized by (1) low densities in the continental wedge, (2) low vertical loading over the inter-plate contact, (3) a well-developed shelf basin and, (4) low taper angles consistent with a low effective basal friction coefficient. We interpret the correlation between these parameters in terms of the total potential energy change during the earthquake and conclude that if the normal stress or frictional coefficient are low, then a large slip does not necessarily imply a large amount of coseismic work. Heterogeneities in density of the continental basement can therefore be related to complexities in the pattern of coseismic slip and in the aftershock distribution. Locally, a subducted seamount or seaward spur of high-density continental crust may be present near the high slip patch.

Shear rheology and structural properties of chemically identical dendrimer-linear polymer blends through molecular dynamics simulations.

We present nonequilibrium molecular dynamics (NEMD) simulation results for the miscibility, structural properties, and melt rheological behavior of polymeric blends under shear flow. The polymeric blends consist of chemically identical linear polymer chains (187 monomers per chain) and dendrimer polymers of generations g = 1-4. The number fraction x of the dendrimer species is varied (4%, 8%, and 12%) in the blend melt. The miscibility of blend species is measured, using the pair distribution functions gDL, gLL, and gDD. All the studied systems form miscible blend melts under the conditions investigated. We also study the effect of shear rate γ̇ and dendrimer generation on inter-penetration between blend species for different blend systems. The results reveal that shear flow increases the interpenetration of linear chains toward the core of the dendrimers. We also calculate the shear-rate dependent radius of gyration and ratios of the eigenvalues of the gyration tensor to study the shear-induced deformation of the molecules in the blend. Melt rheological properties including the shear viscosity and first and second normal stress coefficients obtained from NEMD simulations at constant pressure are found to fall into the range between those of pure dendrimer and pure linear polymer melts.

Melt rheological properties of PBT/SEBS and reactively compatibilized PBT/SEBS/SEBS‐g‐MA polymer blends

ABSTRACTMelt rheological properties of PBT/SEBS and PBT/SEBS/SEBS‐g‐MA blends at SEBS volume fraction (Φd) = 0.00–0.38 were studied at 240°C, 250°C and 260°C using a capillary rheometer. The compatibilizer SEBS‐g‐MA addition resulted in significant reduction in the dynamic interfacial tension which in turn led to increased phase adhesion. The power law exponent n decreased with increasing Φd and increasing temperature for both the compatiblized and uncompatiblized blends. The consistency index of PBT/SEBS increased with increasing Φd but were smaller than those of PBT/SEBS/SEBS‐g‐MA blends. Melt elasticity such as die swell and first normal stress difference increased with Φd. Variations of first normal stress coefficient function (ψ1), recoverable shear strain (γR), relaxation time (λ), and shear compliance (Jc) values versus shear rate were analyzed. © 2014 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2015, 132, 41402.

A fractional K-BKZ constitutive formulation for describing the nonlinear rheology of multiscale complex fluids

The relaxation processes of a wide variety of soft materials frequently contain one or more broad regions of power-law like or stretched exponential relaxation in time and frequency. Fractional constitutive equations have been shown to be excellent models for capturing the linear viscoelastic behavior of such materials, and their relaxation modulus can be quantitatively described very generally in terms of a Mittag–Leffler function. However, these fractional constitutive models cannot describe the nonlinear behavior of such power-law materials. We use the example of Xanthan gum to show how predictions of nonlinear viscometric properties, such as shear-thinning in the viscosity and in the first normal stress coefficient, can be quantitatively described in terms a nonlinear fractional constitutive model. We adopt an integral K-BKZ framework and suitably modify it for power-law materials exhibiting Mittag–Leffler type relaxation dynamics at small strains. Only one additional parameter is needed to predict nonlinear rheology, which is introduced through an experimentally measured damping function. Empirical rules such as the Cox–Merz rule and Gleissle mirror relations are frequently used to estimate the nonlinear response of complex fluids from linear rheological data. We use the fractional model framework to assess the performance of such heuristic rules and quantify the systematic offsets, or shift factors, that can be observed between experimental data and the predicted nonlinear response. We also demonstrate how an appropriate choice of fractional constitutive model and damping function results in a nonlinear viscoelastic constitutive model that predicts a flow curve identical to the elastic Herschel-Bulkley model. This new constitutive equation satisfies the Rutgers-Delaware rule, which is appropriate for yielding materials. This K-BKZ framework can be used to generate canonical three-element mechanical models that provide nonlinear viscoelastic generalizations of other empirical inelastic models such as the Cross model. In addition to describing nonlinear viscometric responses, we are also able to provide accurate expressions for the linear viscoelastic behavior of complex materials that exhibit strongly shear-thinning Cross-type or Carreau-type flow curves. The findings in this work provide a coherent and quantitative way of translating between the linear and nonlinear rheology of multiscale materials, using a constitutive modeling approach that involves only a few material parameters.

Determination of Wheel-Rail Contact Characteristics by Creating a Special Program for Calculation

Abstract The authors in this paper describe the steps of creating a special program in GUI tool in Matlab. The program is designed to calculate the main properties of wheel-rail contact zone, such as: contact ellipse dimensions, normal stress and friction coefficients. All the relevant equations, which were introduced by different researchers, are firstly presented and modified to be applicable to the programming environment, and then the program was built. In the end, the program working quality is discussed and some expected future developments on this program are suggested. The proposed program can make the comparison between theoretical and experimental results, when they are available, easier and faster.