Rheological Law Research Articles

Simulation of fluid outflow from a channel with complex geometry

Topic of the paper is the simulation of fluid outflow from a channel with complex geometry. Subject of the study is the outflow of fluid from a channel with an arbitrary section, in particular, consisting of a parabolic inlet, cylindrical middle and hyperbolic outlet sections. The paper considers the integral form of determining the pressure and the average flow rate of the fluid in a channel with an arbitrary cross section. From the obtained expression for the pressure and from the equation of conservation of mass, given in integral form, it is necessary to determine the pressure and the average flow rate in successively located channels with a parabolic, cylindrical and hyperbolic section. Research methods are based on: Newton’s rheological law; the continuity equation and the Navier-Stokes equation, which are the basic equations of fluid motion; the method of mathematical modeling and the analytical method for their solution. The paper contains integral expressions for the hydrodynamic characteristics of a fluid flow in a channel with an arbitrary section. The pressure and average flow rate in the channel of successively located parabolic, cylindrical, hyperbolic segments are determined. Analytical expressions are obtained for the pressure and average flow rate of the liquid in a channel with an arbitrary cross section. As an example, the pressures and the average flow rate in the channel from the parabolic inlet, cylindrical middle and hyperbolic outlet sections are determined. In perspective, by substituting in them an arbitrary arithmetic expression of the channel cross-section, it is possible to determine the hydrodynamic characteristics of the flow with any geometry of the flow channel.

New Phenomenological Material Constitutive Models for the Description of the Ti6Al4V Titanium Alloy Behavior Under Static and Dynamic Loadings

The analysis and optimization of rapid or severe material forming process, where high gradients of plastic deformations, strain rates and temperatures are reached during the material flow, remains today a major challenge. Several light metallic alloys, as the titanium ones, are widely used in many industrial applications. Despite their wide spread adoption, particular phenomena are encountered during their machining: high plastic strains gradients, heavy strain rate localization, chip segmentation, accelerated tool wear, etc. Although the recent advances in the experimental devices, it is still difficult experimentally to investigate the instantaneous mesoscopic phenomena taking place during severe forming processes. Therefore, the use of numerical simulations, in addition to specific experimental measurements, presents an efficient alternative for a better understanding of machining processes. Given thought the significant sensitivity of the modelling to the reliable definition of the thermo-visco-plastic workpiece material behavior and referring to the physically based mesoscopic constitutive model proposed in previous works of Gavrus, this study focuses on formulating and identifying phenomenological rheological laws. Their ability to accurately reproduce the isotropic plastic behavior of the Ti6Al4V titanium alloy for both static and dynamic loadings states, as well as in a wide range of plastic strains, plastic strain rates and temperatures, is checked. A specific iterative non-linear regression method is used for the identification of all corresponding material parameters. Their adequacy is discussed and comparisons with experimental results of the literature are set up. A specific user material subroutine VUHARD© is implemented into the commercial code Abaqus®/Explicit. Numerical simulations of experimental compression tests are performed to valid the rheological models’ identification and the material flow prediction. A 2D Finite Element Modelling of the Ti6Al4V orthogonal cutting is performed and the accuracy of proposed constitutive models is examined.

On the Boundary Layer in the Case of Viscous Fluid Flowing Around Confuser with the Ladyzhenskaya Rheological Law

We consider the Prandtl type boundary layer equations for a fluid with the nonlinear Ladyzhenskaya rheological equation. In a nonstandard case of a fluid flow through a confuser (a tube of variable cross-section), we study the behavior of the boundary layer and derive an asymptotic estimate for the solution.

Effects of granular collisions on the rapid coarse-grained materials flow

A continuum model to predict the mobility of landslides is developed according to the formulation by Savage and Hutter as well as the main ‘microscopic aspects’ affecting the rapid sliding of granular masses. Conservation of mass and momentum equations, describing the geometry and velocity evolution during the motion, are re-written by considering the effects of collisions between solid grains (‘granular temperature’ and ‘dispersive pressures’) typically occurring within a basal thin ‘shear’ layer, causing a ‘fluidification’ effect and corresponding energy dissipations. To this purpose, a ‘frictional−collisional’ rheological law allowing to simulate the energy dissipation related to grain inelastic collisions and friction, simultaneously acting within the basal ‘shear layer’, is developed and applied. The interstitial pressures, and their possible increment along curved paths related to centrifugal force, and the mass variation due to erosion/deposition processes are also considered. The numerical simulation of a documented experimental test is finally carried out.

Equations of Symmetric MHD-Boundary Layer of Viscous Fluid with Ladyzhenskaya Rheology Law

One considers flow past a body in an electrically conductive viscous fluid in magnetic field, the fluid being subject to a nonlinear rheological law. Solutions of the corresponding system of magnetohydrodynamic boundary layer are examined in a neighborhood of a frontal critical point.

Rheological Law and Mechanism for Superplastic Deformation of Ti-6Al-4V.

The behaviors of and mechanisms acting in Ti–6Al–4V alloy during low-temperature superplastic deformation were systematically studied by using a Gleeble-3800 thermocompression simulation machine. Focusing on the mechanical behaviors and microstructure evolution laws during low-temperature superplastic compression tests, we clarified the changing laws of the strain rate sensitivity index, activation energy of deformation, and grain index at varying strain rates and temperatures. Hot working images based on the dynamic material model and the deformation mechanism maps involving dislocation quantity were plotted on the basis of PRASAD instability criteria. The low-temperature superplastic compression-forming technique zone and the rheological instability zone of Ti–6Al–4V were analyzed by using hot processing theories. The dislocation evolution laws and deformation mechanisms of the grain size with Burgers vector compensation and the rheological stress with modulus compensation during the low-temperature superplastic compression of Ti–6Al–4V were predicted by using deformation mechanism maps.

Equations of Symmetric Boundary Layer for the Ladyzhenskaya Model of a Viscous Medium in the Crocco Variables

We consider the system of boundary layer equations governing a viscous medium subject to the nonlinear rheological law in the sense of Ladyzhenskaya. Owing to the use of the Crocco transformation for reducing the system to a single quasilinear equation, it becomes possible to study both stationary and nonstationary boundary layers. We obtain asymptotic estimates for the solution.

On Hiemenz flow of Maxwell incompressible viscoelastic medium

Steady axisymmetric flow of a viscoelastic incompressible fluid near the critical point of the plane surface is considered. Model of Maxwellian fluid with upper convected derivative in the rheological constitutive law is used. Velocity and extra stresses field are discussed for various Weissenberg numbers. The subclass of solutions in a stationary case is completely described. The asymptotic solutions for small Weissenberg numbers are compared to the solutions of the complete nonlinear problem. It is found that the fluid elasticity decreases the boundary layer thickness.

Research on Rheological Properties of Rockfill Material under Stress Unloading

A number of super high earth-rock dams will be built in China in the near future. Deformation control is the key issue in the safety construction and operation of these major water conservancy projects.In this paper, the effect of stress loading process on the subsequent rheological development is studied, and a new accumulation rule of rockfill rheology is proposed, that is, the current rheological variable of rockfill unit at a certain time is equal to the hardening rheological variable that can cause additional hardening of rockfill at that time.Then, the influence of stress unloading process on the subsequent rheological development law is further studied. It is found that after stress unloading, there are positive rheological zone, no rheological zone and reverse rheological zone in the rheological behavior of rockfill material. The rheological accumulation rule proposed in this paper is also applicable to the case of stress unloading.

On steady two-dimensional analytical solutions of the viscoelastic Maxwell equations

Stationary two-dimensional flow near a free critical point of an incompressible viscoelastic Maxwell medium with upper, lower, and corotational convective derivatives in the rheological constitutive law is considered. Analysis of the analytical unstationary solution found earlier (S. V. Meleshko, N. P. Moshkin, and V. V. Pukhnachev, On exact analytical solutions of equations of Maxwell incompressible viscoelastic medium. Int. J. Non-Lin. Mech., 105:152–157, 2018) provides a new class of stationary solutions. The solutions found comprise both already known as well as substantially new solutions. Nonsingular solutions of the stress tensor at the critical point and bounded at infinity are constructed. Exact analytical formulae for the stress tensor with the Weissenberg number Wi=1/2 are obtained.

Simulation of energy transfers in waves generated by granular slides

This paper presents a multi-fluid Navier-Stokes modeling of the waves generated by two granular slides (subaerial and submarine) which were previously studied experimentally and a pure synthetic submarine case used for results interpretation. In the numerical model, air and water are considered Newtonian fluids. The slide is modeled as a Newtonian fluid whose viscosity is adjusted to fit the experimental results. Once the viscosity is adjusted, the first and the second waves are shown to be accurately reproduced by the model even though the computed slide is slower. For the subaerial case, the viscosity value found is shown to be consistent with the granular μ(I) rheological law. The second part of this work focuses on the energy transfers between a slide and its generated waves. Energy balance is computed in each phase. The wave energy is evaluated in the wave propagation zone. Energy dissipation, kinematic and potential energies are taken into account in the computation of energy transfer ratio allowing for a better understanding of the phenomena. In light of these results, the wave train generation process is discussed as well as the importance of the slide dynamics in the wave generation stage. The amount of energy transferred to wave is not constant with time and the transfer rate depends strongly on the definition of this rate as well as the case considered. For instance, in the subaerial case simulated, the energy transferred to surface waves is 30% of the energy transferred to water at the time the transfer stops, but this conversion rate is only equal to 4% of the overall available potential slide energy at the end of the process. For the two submarine cases simulated, the corresponding values, equal in both cases, are 2% and 1%, respectively. The simulation results also show that the slide energy is transferred to the water in a short period of time at the beginning whatever the case considered. This observation may be related to the initial nil slide velocity (subaerial case) and the relatively large slope values considered (both cases). Nevertheless, the results illustrate the importance of accurate simulation of the slide dynamics within the wave generation process.

Quantitative impact of structural inheritance on present-day deformation and seismicity concentration in intraplate deformation zones

Structural inheritance (i.e. paleo-tectonic) areas, acting as weakened domains, appear to be a key element localizing the seismicity in intraplate deformation zones. However, the impact of structural inheritance on the observed present-day seismicity and strain rate concentration remains to be quantified. In this study, we quantify through 2D numerical modeling the localization and amplification factor of upper crustal strain rates induced by structural inheritance. Our 2D models are constrained by intraplate velocity boundary conditions and include rheology laws that accounts for inherited strain weakening in both the brittle and ductile layers of the lithosphere. The role of structural inheritance is investigated for different localization of the weakened domain in the lithosphere. For an average intraplate geotherm (Moho temperature ca. 500°C), brittle weakening (i.e. inherited faults) alone induces a limited amplification factor of upper crustal strain rates of ca. 4. Ductile weakening can increase the amplification factor to ca. 7 when localized in the lower crust, but has no effect when localized in the lithospheric mantle. Overall, the amplification factors of upper crustal strain rates vary between 1 and 27 depending on the location of the weakened area in the lithosphere and on the different possible net driving forces, crustal strengths, amounts of weakening, and geotherms. These model amplification factors are in reasonable agreement with those derived from GPS and seismicity data over large spatial scale (several hundreds of kilometers) in North America.

Strain localization by shear heating and the development of lithospheric shear zones

Analogue and numerical models show that strong or weak domains in a deforming ductile material cause stress concentrations that may promote strain localization. Such domains commonly occur in the lithosphere through variations in composition or mineral fabric. Here we use a 2D plane-stress, non-Newtonian, viscous model to explore how strain localization develops from an initial isolated weak inclusion. We use a temperature-dependent rheological law for which the material weakens as a result of work done by shear converted to heat. The progress of strain localization follows a power-law growth that is strongly non-linear and may be regarded as an instability. Although this localization mechanism is ultimately limited by thermal diffusion, this parametrization permits a robust criterion for the conditions in which localized shear zones can form within the lithosphere. Shear zones in the lower crust are typically depicted as the downward continuation of faults. We argue that the depth-extent of narrow shear zones within the lithosphere is limited by the stability criteria that we infer from 2D numerical experiments. When applied to the rheological laws for common lithospheric minerals, the combination of temperature and stress-dependence provides a direct means of predicting the depth below which the localization instability does not occur. For an olivine based rheology, the maximum depth at which rapid localization is expected is in the range of ~20 to 60 km, depending on heat flow, strain-rate and water fugacity. We apply our calculations to two major continental strike-slip zones, the San Andreas Fault and North Anatolian Fault, and compare our predicted maximum localization depths with published seismological images. Strain localization in the lower crust requires a dry rheology comparable to plagioclase. Observations that imply localized strain in the uppermost mantle beneath these fault zones are consistent with the localization criteria and the rheological properties of dry olivine.

The Phan-Thien and Tanner Model Applied to the Lubrication of Knee Prostheses.

This work aims to provide a contribution to determine a proper model for the study of fluid film lubrication for the reduction of knee prostheses failure due to polyethylene wear. The Phan-Thien and Tanner (PTT) rheological law and the elastic deformation of the articular surfaces were considered in this modeling. The governing equations were solved numerically for different geometries and different Weissenberg numbers. The lubrication approximation applied to the PTT rheological law leads to an expression for the apparent viscosity similar to the Cross model. The results attest the importance of considering the non-Newtonian behavior of the synovial fluid, the elastic deformation, and the geometrical features of the prostheses to obtain quantitative information.

Creep mechanisms in the lithospheric mantle inferred from deformation of iron-free forsterite aggregates at 900–1200 °C

To further constrain the plasticity of rocks in the uppermost lithospheric mantle, deformation experiments were carried out on forsterite aggregates using a gas-medium apparatus (Paterson press) at 300 MPa, 900–1200 °C and nearly constant strain rates of ~10−5 s−1. The starting material was a synthetic iron-free forsterite aggregate with an average grain size of ~2.8 μm and ~2–3% of iron-free enstatite. Eight deformation experiments were performed as well as an additional static annealing to characterize grain growth. The maximum stresses obtained range from ~480 to 1870 MPa. Below 1000 °C, where stress significantly exceeds confining pressure, and based on microstructural observations, grain boundary mediated creep is observed, with evidences of sliding and cavitation (gaping) at grain boundaries. At 1050–1200 °C, where pseudo-steady state could be achieved, the microstructures are very different and show evidences of dislocation activity, resulting from the activation of several dislocation slip systems with increasing temperature.When compared to rheology laws previously obtained from similar experiments, the temperature dependence of iron-free olivine creep is similar to the one of its iron-bearing counterpart at high temperature (∼1200 °C); at temperatures ≤ 1000 °C, however, the strength of iron-free olivine is higher than for iron-bearing olivine. The deformation-induced textures obtained show that grain boundary sliding (GBS) lead to cavitation, which was activated in response to large differential stresses, i.e., beyond the Goetze criterion. Given these high-stress conditions, our results cannot be directly applied to deformation of the Earth's mantle at large scale. Nevertheless, they highlight the key role played by grain boundaries in producing strain at lithospheric temperatures, when crystal-plastic mechanisms remain inefficient.

Numerical simulation of creep fracture with internal time-dependent hyperelastic-Kelvin cohesive bonds

The macro brittle creep is closely related to the fracturing process on the micro scale. The virtual internal bond (VIB) is a microstructure-based continuum modeling method. It considers a solid as a bond network on the micro scale. The macro constitutive relation is directly derived from the micro bond potential, which intrinsically contains the micro fracture mechanisms. In this study, the VIB is firstly extended for modeling viscoelasticity, and then it is applied to creep fracture simulation. To reflect the time-dependence property of creep, a viscous bond is introduced to join a time-dependent hyperelastic bond in parallel to constitute a hybrid hyperelastic-Kelvin bond. Based on the hybrid bond potential, the macro viscohyperelastic constitutive relation is derived. The corresponding relationship between the micro bond parameters and the macro material constants is theoretically calibrated. Through this model, the typical three-stage feature of the brittle creep is well reproduced and the creep fracture is effectively simulated. The simulation results suggest that the fast and unstable fracture growth leads to the tertiary stage of creep. The viscosity mainly affects the deformation rate in the primary and tertiary stage of creep. The constitutive relation of the VIB stems from the 1D micro bond; thus, the rheology model derived from the cluster of rheological elements (e.g., spring, dashpot) is easily incorporated into the VIB framework, avoiding the 1D-to-3D generalization of the rheology law. For both the micro viscosity and fracture mechanisms that are incorporated into the macro constitutive relation, the present VIB can simulate complex creep fractures without a separate fracture criterion. It has great potential to simulate fracture propagation in a more extensive viscohyperelastic material.

Rheological information determined from cleavage refraction in naturally deformed interlayered quartzites and phyllites

Abstract Cleavage refraction angles are used to estimate effective viscosity contrasts between interlayered quartzites and phyllites within the Baraboo Syncline, Wisconsin, USA. Both types of layers contain two major phases, quartz and pyrophyllite, with minor amounts of hematite. Quartz (with minor hematite) behaves as the strong phase and pyrophyllite acts as the weak phase. Cleavage refraction directly relates to mineralogy with a linear relationship between bedding/cleavage angle and strong-phase concentration. Mineralogy exerts first-order control over effective viscosity contrasts, which are generally small, in most cases <10. Effective viscosity contrasts are consistent across the fold, so are likely not to be highly strain dependent and indicate approximate linear viscous rheology. Microstructures suggest deformation was dominated by dislocation creep in layers with high quartz concentrations and diffusive mass transfer in layers with lower quartz concentrations, and that the transition of the deformation mechanism is gradual. Thus, the rheological flow laws at the small scale may not reflect the bulk flow law at the large scale over the span of the deformation. Effective viscosity contrasts allow an evaluation of samples compared to theoretical two-phase mixtures. The analysed samples most closely resemble the Reuss bound of two-phase mixtures, regardless of the mineralogy.

A depth average SPH model including μ(I) rheology and crushing for rock avalanches

SummaryClassical depth‐integrated smoothed particle hydrodynamics (SPH) models for avalanches are extended in the present work to include a μ(I)− rheological model enriched with a fragmentation law. With this improvement, the basal friction becomes grain distribution dependent. Rock avalanches, where grain distribution tends to change with time while propagating, are the appropriate type of landslide to apply the new numerical proposal. The μ(I)− rheological models considered in the present work are those of Hatano and Gray, combined with two different fragmentation laws, a hyperbolic and a fractal‐based law. As an application, Frank avalanche, which took place in Canada in 1903, is analyzed under the scope of the present approach, focusing in the influence of the rheological and fragmentation laws in the evolution of the avalanche.

Investigation of surge-derived pyroclastic flow formation by numerical modelling of the 25 June 1997 dome collapse at Soufrière Hills Volcano, Montserrat

Deposits from ash-cloud surges associated with dome collapse can, under certain conditions, be remobilised to form surge-derived pyroclastic flows (SDPFs). Using numerical modelling, we reproduce the emplacement of these flows and investigate the conditions that favour their genesis. We use the new version of the numerical model VolcFlow, which simulates the two components of a pyroclastic flow: the basal avalanche and the overriding ash-cloud surge. The basal avalanche (primary block-and-ash flows and SDPFs) are simulated using three previously published rheological laws: plastic, frictional and frictional velocity-weakening rheologies. Applied to the 25 June 1997 dome collapse at Soufriere Hills Volcano, the models reproduce to different degrees the deposit footprints formed by the block-and-ash flows, the ash-cloud surges and the SDPFs. In the plastic model, SDPFs occur if the ash-cloud surge deposit exceeds a threshold thickness that allows it to remobilise and flow. In the frictional models, SDPFs occur only if ash-cloud surge deposition takes place on a slope exceeding the friction angle of the ash. Results also highlight that SDPFs appeared so clearly in 1997 at Montserrat due to a combination of topographic factors: (i) a bend in the Mosquito Ghaut drainage that allowed the ash-cloud surges to detach, (ii) a depositional area on the watershed between the eastern and western drainage channels and (iii) a network of tributaries that drained all the remobilised mass into Dyer’s River to form a single, large SDPF. Our model could be a promising tool for the future forecasting of hazards posed by surge-derived pyroclastic flows.

Определение дисперсной фазы буровых растворов на полимерной основе

The purpose of the article is determination of the volume of the polymer solution dispersed phase, estimation of their subordination degree to the power law of fluid flow, as well as determination of the rheological model of mud flow in order to calculate the pressure losses under fluid movement in the elements of the circulation system. Water-soluble polymers are widely used in the composition of drilling fluids and are also used independently as flushing fluids for drilling wells. Despite the wide application range of polymers, when drilling little attention is paid to the volume of the dispersed phase resulting from the dissolution of polymer in water. Particular interest is caused by so-called dilute solutions (term used in the theory of polymer solutions). They are little mentioned in drilling but their number of macromolecular coils of the polymer reaches such a value that they do not interact with each other when the fluid flows. Using a viscometric method, the molecular mass and the characteristic viscosity of each solution used for the dispersion phase volume determination are estimated. The parameters of the rheological power law are determined through the application of rotametric viscometry. As a result, it has been found out that the solutions of the considered polymers prepared for the study are referred to diluted. On example of one of them the possibility of determining the relative volume of macromolecular coils of polymer with the immobilized solvent (water) is shown. It is demonstrated that a change in the specified volume causes a change in the parameters of the rheological power law of researched solution flow. The proposed methods for determining the relative volume of the dispersed phase of drilling muds based on dilute polymer solutions allows to adjust the rheological properties of flushing fluids and set the parameters of the rheological power law describing the flow of pseudoplastic fluids.