Strain-rate Dependent Viscosity Research Articles

Alternative finite strain viscoelastic models: constant and strain rate-dependent viscosity

Alternative finite strain viscoelastic models: constant and strain rate-dependent viscosity

Partial Asymptotic Dimension Reduction for Steady State Non-Newtonian Flow with Strain Rate Dependent Viscosity in Thin Tube Structure

Partial Asymptotic Dimension Reduction for Steady State Non-Newtonian Flow with Strain Rate Dependent Viscosity in Thin Tube Structure

Mesh-free simulations of injection molding processes

In this paper, we introduce a mesh-free numerical framework using the finite pointset method for the modeling and simulation of injection molding processes. When compared to well-established mesh-based methods, which have been widely applied for these applications, our approach avoids the need for extensive preprocessing and enables accurate treatment of free surfaces and other associated phenomena. To accurately model the polymer injections, we consider a detailed material model, with temperature dependent viscosity and density, while also considering shear thinning behavior with a strain rate dependent viscosity. Our numerical investigations show that injection molding-specific problems such as the modeling of viscous flows and the fountain flow effect can be successfully implemented using our presented framework. For a thorough validation of our proposed model, we compare the simulated flow behavior with injection molding experiments, which are also performed in this work. The experimental setup considers the injection of a polymer melt into a spiral mold. The flow behavior is investigated experimentally at varying melt injection and wall temperature, with different threshold pressures. Our numerical simulations show a good comparison with these experimental results, both qualitatively and quantitatively. We also introduce a correction mechanism to ensure energy conservation, which has often been challenging in mesh-free approaches. This is the first time that the flow behavior in a mesh-free injection molding method has been experimentally validated and successfully applied to the simulation of an actual industrial vehicle component.

Steady state non-Newtonian flow with strain rate dependent viscosity in thin tube structure with no slip boundary condition

The steady state non-Newtonian flow, with strain rate dependent viscosity in a thin tube structure, with no slip boundary condition, is considered. Applying the Banach fixed point theorem we prove the existence and uniqueness of a solution. An asymptotic approximation is constructed and justified by an error estimate.

Steady state non-Newtonian flow with strain rate dependent viscosity in domains with cylindrical outlets to infinity

The paper deals with a stationary non-Newtonian flow of a viscous fluid in unbounded domains with cylindrical outlets to infinity. The viscosity is assumed to be smoothly dependent on the gradient of the velocity. Applying the generalized Banach fixed point theorem, we prove the existence, uniqueness and high order regularity of solutions stabilizing in the outlets to the prescribed quasi-Poiseuille flows. Varying the limit quasi-Poiseuille flows, we prove the stability of the solution.

Shear-induced porosity bands in a compacting porous medium with damage rheology

Shear-induced porosity bands have been observed experimentally and have been the subject of a number of theoretical and numerical analyses in which a number of rheological laws governing the partial melt system have been proposed. These bands have been suggested to be important in Earth’s interior in focussing melt to Earth’s mid-ocean ridges, in reducing the effective viscosity of the asthenosphere, and in affecting seismic and electrical properties. Recently, a linear analysis of the formation of melt bands has been presented in which the viscosity of the solid matrix depends on the grain size and a parameter characterizing the roughness of the grain-liquid interface For some parameter values, this “damage” rheology mimics the effect of very strongly strain-rate dependent viscosity which can produce low angle bands, similar to those seen in experiments. In the present paper, I show full nonlinear simulations of melt bands with damage rheology. In agreement with the linear analysis, low angle bands are possible when the grain size and grain roughness evolve rapidly compared with the deformation of the sample. The grain size field evolves to form bands where grain-size anticorrelates with porosity. The effective viscosity and electrical conductivity of bands are also investigated. For low angle bands, the effective viscosity relative to the mean viscosity decreases and the electrical conductivity anisotropy increases with strain, indicating significant strain and electrical conduction localization.

Isomorph invariance of Couette shear flows simulated by the SLLOD equations of motion

Non-equilibrium molecular dynamics simulations were performed to study the thermodynamic, structural, and dynamical properties of the single-component Lennard-Jones and the Kob-Andersen binary Lennard-Jones liquids. Both systems are known to have strong correlations between equilibrium thermal fluctuations of virial and potential energy. Such systems have good isomorphs (curves in the thermodynamic phase diagram along which structural, dynamical, and some thermodynamic quantities are invariant when expressed in reduced units). The SLLOD equations of motion were used to simulate Couette shear flows of the two systems. We show analytically that these equations are isomorph invariant provided the reduced strain rate is fixed along the isomorph. Since isomorph invariance is generally only approximate, a range of strain rates were simulated to test for the predicted invariance, covering both the linear and nonlinear regimes. For both systems, when represented in reduced units the radial distribution function and the intermediate scattering function are identical for state points that are isomorphic. The strain-rate dependent viscosity, which exhibits shear thinning, is also invariant along an isomorph. Our results extend the isomorph concept to the non-equilibrium situation of a shear flow, for which the phase diagram is three dimensional because the strain rate defines a third dimension.

The role of magma chamber-fault interaction in caldera forming eruptions

This paper examines the role of the position and orientation of a regional fault in the roof of a magma chamber on stress distribution, mechanical failure, and dyking using 2D finite element numerical simulations. The study pertains to the magma chamber behavior in the relatively short time intervals of several hundreds to thousand of years. The magma chamber is represented as an elliptical inclusion (eccentricity, a/b = 0.12) at a relative depth, H/a, of 0.9. The fault has a 45° dip and is represented by a frictionless fracture. The temperature field in the host rock is calculated assuming a quasisteady-state thermal regime that develops through periodic episodes of magma supply. The rheology of the surrounding rocks is treated using viscoelasticity with temperature activated strain-rate dependent viscosity. Strain weakening of the rocks in the ductile zone is described within the frame of the Dynamic Power Law model . The magma pressure is coupled with the deformation of the rock mass hosting the chamber, including the fault. The variation of magma pressure in response to magma supply and chamber deformation is calculated in the elastic and viscoelastic regimes. The latter corresponds to slow filling, while the former represents a filling time much less than the viscous relaxation time scale. The resulting “equation of state” for the magma chamber couples the magma pressure with the chamber volume in the elastic regime, and with the filling rate for the viscoelastic regime. Analysis of stresses is used to predict dyke propagation conditions, and the mechanical failure of the chamber roof for different fault positions and magma overpressures. Results show that an outward dipping fault located on the periphery of the chamber roof hinders the propagation of dykes to the surface, causing magma to accumulate under the footwall of the fault. At high to moderate overpressures (30–40 MPa), the fault causes localized shear failure and chamber roof collapse that might lead to the first stage of a caldera-forming eruption.

Toward reliable calculations of heat and plastic flow during friction stir welding of Ti-6Al-4V alloy

Abstract Heat transfer and visco-plastic flow during friction stir welding of Ti-6Al-4V alloy have been modeled in three dimensions by numerically solving the equations of conservation of mass, momentum and energy using temperature dependent thermo-physical properties and temperature and strain-rate dependent viscosity values. The computed results showed that five important model parameters, i. e., the spatially variable friction coefficient, the spatially variable slip between the tool and the workpiece, the extent of viscous dissipation, the mechanical efficiency and the spatially variable heat transfer rate from the bottom surface of the workpiece significantly affected both the temperature fields and the computed torque on the tool. An important problem in the modeling of friction stir welding is that the values of these five parameters cannot be specified from fundamental principles and, and as a result, computed results are not always accurate. Here we show that by combining the heat transfer and plastic flow model with a genetic algorithm based optimization scheme, the values of the five uncertain parameters can be determined from a limited volume of experimental data so that the model predictions of peak temperatures and cooling rates match well with the experimental results. The computed results show that for the welding conditions reported in this paper, close to sticking condition prevailed at the tool – workpiece interface for all the experiments. The extent of viscous dissipation converted to heat was fairly low indicating lack of intimate atomic mixing in the stir zone. Computed three dimensional pressure distributions and streamlines were consistent with defect-free reliable welds for all conditions of welding studied.

A Unifying Perspective on the Quasi-steady Analysis of Magnetorheological Dampers

A magnetorheological (MR) fluid, modeled as a Bingham plastic (BP) material, is characterized by a field dependent yield stress, and a (nearly constant) postyield plastic viscosity. Based on viscometric measurements, such a BP model is an idealization to physical MR behavior, albeit a useful one. A better approximation involves modifying both the preyield and postyield constitutive behavior as follows: (1) assume a high viscosity preyield behavior when the shear stress is less than the transition stress, and (2) assume a power law fluid (i.e., strain rate dependent viscosity) when the shear stress is greater than the transition stress. Assuming a power law fluid in postyield allows the model to account for shear thinning behavior exhibited by MR fluids at higher strain rates. Such an idealization for MR fluid constitutive behavior is called an viscous-power law model, or a Herschel—Bulkley (HB) model with preyield viscosity. This study develops a quasi-steady analysis for such a constitutive MR fluid behavior applied to an MR flow mode damper. Closed form solutions are developed for the fluid velocity, as well as key performance metrics, such as damping capacity and dynamic range (ratio of field-on to field-off force). For the given fluid properties and flow mode damper geometry, the fluid velocity profile and gradient, and the relationship of the damper force and piston velocity are analyzed. In addition, specializations to existing models, such as the HB, biviscous, and BP models, are shown to be easily captured by this model when physical constraints (idealizations) are placed on the rheological behavior of the MR fluid.

On the effect of temperature and strain-rate dependent viscosity on global mantle flow, net rotation, and plate-driving forces

SUMMARY Global circulation models are analysed using a temperature and strain-rate dependent rheology in order to refine previous estimates of the nature of mantle flow and plate driving forces. Based on temperature inferred from a tectonic model and seismic tomography, the suboceanic viscosity is lower than underneath continents by ∼ one order of magnitude. If net-rotations of the lithosphere with respect to a stable lower mantle reference frame are accounted for, the patterns of flow in the upper mantle are similar between models with layered and those with laterally varying viscosity. The excited net rotations scale with the viscosity contrast of continental roots to the ambient mantle; this contrast is dynamically limited by the power-law rheology. Surface net rotations match the orientation of hotspot reference-frame Euler poles well, and amplitudes are of the right order of magnitude. I compare prescribed surface velocity models with free-slip computations with imposed weak zones at the plate boundaries; velocity fields are generally consistent. Models based on laboratory creep laws for dry olivine are shown to be compatible with average radial viscosity profiles, plate velocities in terms of orientation and amplitudes, plateness of surface velocities, toroidal:poloidal partitioning, and fabric anisotropy formation under dislocation creep in the upper mantle. Including temperature-dependent variations increases the relative speeds of oceanic versus continental lithosphere, makes surface velocities more plate-like, and improves the general fit to observed plate motions. These findings imply that plate-driving force studies which are based on simpler mantle rheologies may need to be revisited.

LAMINAR IMPINGING JET HEAT TRANSFER WITH A PURELY VISCOUS INELASTIC FLUID

Laminar impinging flow heat transfer is considered with a purely viscous inelastic fluid. The rheology of the fluid is modeled using a strain rate dependent viscosity coupled with asymptotic Newtonian behavior in the zero shear limit. The velocity and temperature fields are computed numerically for a confined laminar axisymmetric impinging flow. Important features of the non-Newtonian developing flow field are described and contrasted with the Newtonian situation. It is demonstrated that very small departures from Newtonian rheology lead to qualitative changes in the Nusselt number distribution along the impinging surface. In particular, a mildly shear thinning fluid displays a pronounced off-stagnation point heat transfer maxima, a feature that is not observed with a Newtonian fluid. Hence, Newtonian fluid approximations cannot adequately describe experimental heat transfer measurements in such situations even though they may be deemed acceptable in terms of describing the velocity field in the incoming nozzle. Numerical results are presented to analyze the effect of the dimensionless nozzle-to-plate distance, the rheological parameters, and the Reynolds and Prandtl numbers on the magnitude of the off-stagnation point peak heat transfer rate. The influence of the rheology of the fluid is particularly significant at low nozzle-to-plate distances since the mean strain rate in the flow field increases as the nozzle-to-plate distance is reduced. The numerical heat transfer results are interpreted in the context of the developing flow field.

Non-Newtonian viscous oscillating free surface jets, and a new strain-rate dependent viscosity form for flows experiencing low strain rates

A model for oscillating free surface jet flow of a fluid from an elliptical orifice, together with experimental measurements, can be exploited to characterize the elongational viscosity of non-Newtonian inelastic fluids. The oscillating jet flow is predominantly elongational, with a small strain that oscillates rapidly between large and zero strain rates. We find that to reproduce the experimentally observed steady oscillating jet flow in model simulations, the assumed form of the non-Newtonian viscosity as a function of strain rate must have zero gradient, i.e., be Newtonian, at zero strain rate (a behavior exhibited, in general, by real inelastic fluids). We demonstrate that the Cross, Carreau, Prandtl-Eyring, and Powell-Eyring forms, although they have finite viscosity at zero strain rate, have either nonzero or even unbounded gradient at zero, and hence are unable to model oscillating jet behavior. We propose a new non-Newtonian viscous form which has all of the desirable features of existing forms (high and low strain rate plateaus, with adjustable location and steepness of the transition) and the additional feature of Newtonian behavior at low strain rates.

The Rheology of Liquid Pentane Isomers by Non-Equilibrium Molecular Dynamics Simulations

In this paper, non-equilibrium molecular dynamics (NEMD) simulations of planar Couette flow are reported for an expanded collapsed atom model for liquid pentane isomers at 273.15 K. The strain rate dependent viscosity for liquid pentane isomers exhibits shear-thinning and a linear dependence on γ1/2. Newtonian viscosities for liquid pentane isomers obtained by a linear extrapolation to zero strain rate are: 0.256cP for normal pentane, 0.219cP for isopentane, and 0.168cP for neopentane. The strain rate dependent pressure difference and normal stress difference vary nearly linearly with the γ3/2 law and the γ law, respectively, for all three liquid pentane isomers. The overall trend of the square of radius of gyration and end-to-end distance for normal pentane is a linear increase with strain rate. For isopentane, the trend hardly changes for the range of shear rate in this study. The alignment angle decreases with increasing strain rate and the alignment angle of the straight chain alkane is less than that of the branched chain alkane. The average percentage of C‒C‒C‒C trans for normal pentane as a function of strain rate is in excellent correlation with the square of the radius of gyration and the average end-to-end distance. Applying the strain rate in the x-direction, the alignment angle is forced to decrease and the percentage of C‒C‒C‒C trans increases with increasing strain rate.

The Rheology of 6-Propyl Duodecane and 5-Dibutyl Nonane by Non-Equilibrium Molecular Dynamics Simulations

In recent papers, we reported non-equilibrium molecular dynamics (NEMD) simulations of planar Couette flow for liquid n- and i-butane, and liquid n-decane and 4-propyl heptane, using two collapsed atom models and an atomistically detailed model. It was found that the collapsed atom models predict the viscosities of the n-butane and n-decane quite well, and that the atomistically detailed model does not yield quantitative agreement with the viscosity of the n-alkanes or the branched alkanes, but it does have the one positive feature that the calculated viscosities of the branched alkanes are higher than these of the n-alkanes. In the present paper, we report results of NEMD simulations of planar Couette flow for liquid 6-propyl duodecane and 5-dibutyl nonane at 296 K and 0.782 g/cc, using an expanded collapsed atom model for simplicity. The strain rate dependent viscosity shows three different regions—two shear thinning ones and a Newtonian one. The slopes of the log-log plot for the branched-chain alkanes at high strain rate are different from those at intermediate strain rate, which is characterized as a rheological behavior of branched-chain alkanes. The Newtonian viscosity of the branched-chain alkanes can be extrapolated from the plateau value of the strain rate dependent viscosity at low strain rate as for straight-chain alkanes [J. Chem. Phys., 105, 1214 (1996)]. The results indicate that more-branched alkanes have a larger viscosity than less-branched C17 alkanes.

Response of constrained and unconstrained bubbles to lithotripter shock wave pulses.

The Gilmore formulation for spherical bubble dynamics [F. R. Gilmore, The Growth or Collapse of a Spherical Bubble in a Viscous Compressible Liquid (California Institute of Technology, Pasadena, CA, 1952), Rep. No. 26-4] is used to investigate the response of air bubbles to a variety of lithotripter shock waveforms. A modification of the Gilmore model is proposed to account for partial constraint of the bubble expansion that can be caused by bubble coatings (such as in echo contrast agents) and by tissues or vessels surrounding bubbles in vivo. In the modified formulation, a viscoelastic membrane is assumed to exist at the bubble interface to include the possible effects of the nonlinear elasticity and strain rate dependent viscosity on the bubble response. The stress induced in the membrane is assumed to be an exponential function of the bubble radius, which tends to restrict the bubble expansion. The viscosity is assumed to increase with the strain rate. In the absence of the membrane, the maximum bubble wall pressure induced by a negative (tensile) pulse is much larger than that induced by a positive (compressive) pulse of the same pressure waveform and amplitude. This difference increases with decreasing initial bubble radius. The addition of the viscoelastic membrane significantly decreases the predicted maximum bubble pressure and the difference in response between the positive and negative pulses. The effect of the time delay between double pulses (positive followed by negative or negative followed by positive) is also investigated for unconstrained bubbles.

Isostatic recovery and the strain rate dependent viscosity of the Earth's mantle

This paper is concerned with the interpretation of isostatic recovery data in terms of the flow properties of the earth's mantle. A hydrodynamic analysis is first presented that allows straightforward calculation of the relaxation time for isostatic recovery within a mantle in which the viscosity varies continuously with depth. However, it transpires that no curve of this type (i.e., choice of a reference viscosity and a rate of change of viscosity with depth) can of itself adequately explain the available observational data from the Fennoscandian and Laurentide ice sheets and the pluvial Lake Bonneville. Proceeding onward it is then demonstrated that the strain rates within such flows are in fact greater than the critical strain rate envisaged by Weertman (1970) in his theoretical rheological model of the mantle. Below this critical value, diffusion creep is the dominant flow process, and the flow can be modeled by a Newtonian viscosity. But above this value, dislocation glide takes over, and the viscosity exhibits a decrease with increasing strain rate. This feature is then incorporated into the theoretical model, and the isostatic recovery data are interpreted in such a way as to provide experimental values of the strain rate dependent viscosity that can be compared with the values in Weertman's rheological model. It is demonstrated that the data become most self-consistent and exhibit the most satisfactory agreement with Weertman's model when the increase of mantle viscosity with depth is given roughly by exp (5 × 10^(−4)z), where z is the depth in kilometers. Thus in addition, the analysis would appear to provide some verification of Weertman's model of the mantle flow properties. It is further demonstrated that the much larger increase of viscosity with depth predicted by McConnell (1968) and others from previous analyses of isostatic recovery data is an artifice induced by the nature of such flows in which the strain rate decreases with depth; this led to an apparent increase of viscosity that is much larger than the actual variation.