Incorporation of polymer diffusivity and migration into constitutive equations

Given a general one-particle constitutive equation for the stress tensor, we discuss how to incorporate the additional effects of polymer diffusivity and migration into that constitutive equation within the framework of continuum mechanics. For the example of an upper-convected Maxwell model representing the polymer contribution to the stress tensor of a dilute polymer solution, we describe i) how to modify the constitutive equation for the stress tensor to include diffusion and migration effects, ii) how to formulate a balance equation for the polymer mass density in order to describe the nonhomogeneous composition of the polymer solution resulting from migration, and iii) how to close the extended set of coupled equations by means of further constitutive equations for the migration velocity and the diffusion tensor. In order to guarantee the material objectivity for all equations, we formulate them in the body tensor formulation of continuum mechanics (and then translate them into Cartesian space). The proposed equations are compared to results of a recent kinetic-theory approach.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

Some tests of constitutive equations for entangled polymers based on planar-extension flow histories in a periodically constricted channel

Stress and velocity were determined locally by birefringence measurements and laser Doppler velocimetry for a mildly entangled polystyrene solution flowing at steady state in a rectangular channel with sinusoidally varying wall spacing. Having measured both the velocity and stress fields, we were able to test constitutive equations locally, i.e., without solving the equations of motion for the entire flow. Four were examined for the periodic planar extensions on the channel centerplane: the Newtonian model, the Lodge network model, the Doi-Edwards tube model, and the Wagner-Schaeffer modification of Doi-Edwards. High enough Weissenberg and Deborah numbers were reached to produce sizable departures from the Newtonian predictions. The Doi-Edwards model underpredicted the stress, as did Wagner-Schaeffer, although to a lesser extent. Predictions of the Lodge model were best of all, a surprising result in view of its inadequacy for simple shear deformations. The predictions of the Lodge model, without parameter adjustment, agreed remarkably well with the planar extension data over the accessible range for our apparatus: Deborah numbers up to 2.0, extensional Weissenberg numbers up to 6.5, and a maximum extension ratio of about 2.3.     

A new constitutive equation derived from network theory

A constitutive equation is derived from a Lodge—Yamamoto type of network theory for polymeric fluids. The network junctions are not assumed to move strictly as points of the continuum but allowed a certain “effective slip”. The rates of creation and destruction of junctions are assumed to depend on the instantaneous elastic energy of the network, or equivalently, the average extension of the network strand, in a simple manner. Agreement between model predictions and the I.U.P.A.C. data on L.D.P.E. is good.     

A Constitutive Model in Finite Viscoelasticity of Particle-reinforced Rubbers

Two series of tensile relaxation tests are performed on natural rubber filled with high abrasion furnace black. To fit observations, constitutive equations are derived for the nonlinear viscoelastic behavior of a particle-reinforced elastomer. A filled rubber is modeled as a composite medium, where inclusions with low concentrations of junctions are randomly distributed in the host matrix. The inclusions are treated as equivalent networks of macromolecules, where strands can separate from temporary junctions as they are thermally agitated. The bulk medium is thought of as a permanent network of chains. Unlike conventional concepts of transient networks, the concentration of strands in inclusions is assumed to be affected by mechanical factors: under active loading, inter-chain interactions weaken and some strands that were prevented from detachment from their junctions in a stress-free compound become free to separate from the junctions in a deformed medium. Unloading strengthens interactions between macromolecules, which results in an increase in the number of permanent strands. By using the laws of thermodynamics, stress–strain relations for a particle-reinforced rubber are developed. Adjustable parameters in the constitutive equations are found by fitting the experimental data. It is demonstrated that mechanical pre-loading and annealing of specimens at an elevated temperature noticeably affect concentrations of inclusions with various activation energies for rearrangement of strands.     

Finite viscoelasticity and viscoplasticity of semicrystalline polymers

Observations are reported on low-density polyethylene in uniaxial tensile and compressive tests with various strain rates and in tensile and compressive relaxation tests with various strains. A constitutive model is developed for the time-dependent response of a semicrystalline polymer at arbitrary three-dimensional deformations with finite strains. A polymer is treated as an equivalent network of chains bridged by junctions (entanglements between chains in the amorphous phase and physical cross-links at the lamellar surfaces). Its viscoelastic behavior is associated with separation of active strands from temporary junctions and merging of dangling strands with the inhomogeneous network. The viscoplastic response is attributed to sliding of junctions between chains with respect to their reference positions. Constitutive equations are derived by using the laws of thermodynamics. The stress–strain relations involve 6 material constants that are found by matching the observations.      

The nonlinear viscoelastic response of carbon black-filled natural rubbers

Three series of tensile relaxation tests are performed on natural rubber filled with various amounts of carbon black. The elongation ratio varies in the range from λ=2.0 to 3.5. Constitutive equations are derived for the nonlinear viscoelastic behavior of filled elastomers. Applying a homogenization method, we model a particle-reinforced rubber as a transient network of macromolecules bridged by junctions (physical and chemical cross-links, entanglements and filler clusters). The network is assumed to be strongly heterogeneous at the meso-level: it consists of passive regions, where rearrangement of chains is prevented by surrounding macromolecules and filler particles, and active domains, where active chains separate from temporary nodes and dangling chains merge with the network as they are thermally agitated. The rate of rearrangement obeys the Eyring equation, where different active meso-domains are characterized by different activation energies. Stress–strain relations for a particle-reinforced elastomer are derived by using the laws of thermodynamics. Adjustable parameters in the constitutive equations are found by fitting experimental data. It is demonstrated that the filler content strongly affects the rearrangement process: the attempt rate for separation of strands from temporary nodes increases with elongation ratio at low fractions of carbon black (below the percolation threshold) and decreases with λ at high concentrations of filler.     

Electroosmotic flow of Eyring fluid in slit microchannel with slip boundary condition

In consideration of the electroosmotic flow in a slit microchannel, the con-stitutive relationship of the Eyring fluid model is utilized. Navier＇s slip condition is used as the boundary condition. The governing equations are solved analytically, yielding the velocity distribution. The approximate expressions of the velocity distribution are also given and discussed. Furthermore, the effects of the dimensionless parameters, the electrokinetic parameter, and the slip length on the flow are studied numerically, and appropriate conclusions are drawn.     

A dislocation density-based single crystal constitutive equation

Single crystal constitutive equations based on dislocation density (SCCE-D) were developed from Orowan’s strengthening equation and simple geometric relationships of the operating slip systems. The flow resistance on a slip plane was computed using the Burger’s vector, line direction, and density of the dislocations on all other slip planes, with no adjustable parameters. That is, the latent/self-hardening matrix was determined by the crystallography of the slip systems alone. The multiplication of dislocations on each slip system incorporated standard 3-parameter dislocation density evolution equations applied to each slip system independently; this is the only phenomenological aspect of the SCCE-D model. In contrast, the most widely used single crystal constitutive equations for texture analysis (SCCE-T) feature 4 or more adjustable parameters that are usually back-fit from a polycrystal flow curve. In order to compare the accuracy of the two approaches to reproduce single crystal behavior, tensile tests of single crystals oriented for single slip were simulated using crystal plasticity finite element modeling. Best-fit parameters (3 for SCCE-D, 4 for SCCE-T) were determined using either multiple or single slip stress–strain curves for copper and iron from the literature. Both approaches reproduced the data used for fitting accurately. Tensile tests of copper and iron single crystals oriented to favor the remaining combinations of slip systems were then simulated using each model (i.e. multiple slip cases for equations fit to single slip, and vice versa). In spite of fewer fit parameters, the SCCE-D predicted the flow stresses with a standard deviation of 14 MPa, less than one half that for the SCCE-T conventional equations: 31 MPa. Polycrystalline texture simulations were conducted to compare predictions of the two models. The predicted polycrystal flow curves differed considerably, but the differences in texture evolution were insensitive to the type of constitutive equations. The SCCE-D method provides an improved representation of single-crystal plastic response with fewer adjustable parameters, better accuracy, and better predictivity than the constitutive equations most widely used for texture analysis (SCCE-T).     

Inelastic constitutive equations for complex flows

A new class of inelastic constitutive equations is presented and discussed. In addition to the rate-of-strain tensor, the stress is assumed to depend also on the relative-rate-of-rotation tensor, a frame-indifferent quantity that brings information about the nature of the flow. The material functions predicted by these constitutive equations are given for simple shear and uniaxial extension. A special case of these equations takes the Newtonian form, except that the viscosity is a function of the invariants of both kinematic tensors on which the stress depends. This simple constitutive equation has potential applications in liquid flow process simulations, since it combines simplicity with the capability of responding independently to shear and extension, as real liquids seem to do. Finally, possible forms for the new viscosity function are discussed.     

Tractions, Balances, and Boundary Conditions for Nonsimple Materials with Application to Liquid Flow at Small-Length Scales

Using a nonstandard version of the principle of virtual power, we develop general balance equations and boundary conditions for second-grade materials. Our results apply to both solids and fluids as they are independent of constitutive equations. As an application of our results, we discuss flows of incompressible fluids at small-length scales. In addition to giving a generalization of the Navier–Stokes equations involving higher-order spatial derivatives, our theory provides conditions on free and fixed boundaries. The free boundary conditions involve the curvature of the free surface; among the conditions for a fixed boundary are generalized adherence and slip conditions, each of which involves a material length scale. We reconsider the classical problem of plane Poiseuille flow for generalized adherence and slip conditions.     

Polymer Networks with Slip-links: 1. Constitutive Equations for an Uncross-linked Network

Constitutive equations are derived for the mechanical response of polymers at three-dimensional deformations with finite strains. A polymer is treated as an incompressible network of flexible chains with free ends whose motion at the micro-level is constrained by a random number of slip-links. The slip-links move affinely with macro-deformation, whereas chains can slide with respect to slip-links. When a free end of a chain slides through a slip-link, the slip-link disappears. Stress–strain relations are developed by using the laws of thermodynamics. They involve only one material constant with a transparent physical meaning.     

The viscoelastic and viscoplastic behavior of low-density polyethylene

Three series of tensile tests with constant cross-head speeds (ranging from 5 to 200 mm/min), tensile relaxation tests (at strains from 0.03 to 0.09) and tensile creep tests (at stresses from 2.0 to 6.0 MPa) are performed on low-density polyethylene at room temperature. Constitutive equations are derived for the time-dependent response of semicrystalline polymers at isothermal deformation with small strains. A polymer is treated as an equivalent heterogeneous network of chains bridged by temporary junctions (entanglements, physical cross-links and lamellar blocks). The network is thought of as an ensemble of meso-regions linked with each other. The viscoelastic behavior of a polymer is modelled as thermally-induced rearrangement of strands (separation of active strands from temporary junctions and merging of dangling strands with the network). The viscoplastic response reflects mutual displacement of meso-domains driven by macro-strains. Stress–strain relations for uniaxial deformation are developed by using the laws of thermodynamics. The governing equations involve five material constants that are found by fitting the observations. Fair agreement is demonstrated between the experimental data and the results of numerical simulation. It is shown that observations in conventional creep tests reflect not only the viscoelastic, but also the viscoplastic behavior of an ensemble of meso-regions.     

Vertical integration of three-phase flow equations for analysis of light hydrocarbon plume movement

A mathematical model is derived for areal flow of water and light hydrocarbon in the presence of gas at atmospheric pressure. Vertical integration of the governing three-dimensional, three-phase flow equations is performed under the assumption of local vertical equilibrium to reduce the dimensionality of the problem to two orthogonal horizontal directions. Independent variables in the coupled water and hydrocarbon areal flow equations are specified as the elevation of zero gauge hydrocarbon pressure (air-oil table) and the elevation of zero gauge water pressure (air-water table). Constitutive relations required in the areal flow model are vertically integrated fluid saturations and vertically integrated fluid conductivities as functions of air-oil and air-water table elevations. Closed-form expressions for the vertically integrated constitutive relations are derived based on a three-phase extension of the Brooks-Corey saturation-capillary pressure function. Closed-form Brooks-Corey relations are compared with numerically computed analogs based on the Van Genuchten retention function. Close agreement between the two constitutive models is observed except at low oil volumes when the Brooks-Corey model predicts lower oil volumes and transmissivities owing to the assumption of a distinct fluid entry pressure. Nonlinearity in the vertically integrated constitutive relations is much less severe than in the unintegrated relations. Reduction in dimensionality combined with diminished nonlinearity, makes the vertically integrated water and hydrocarbon model an efficient formulation for analyzing field-scale problems involving hydrocarbon spreading or recovery under conditions for which the vertical equilibrium assumption is expected to be a satisfactory approximation.     

Role of Crosslinking and Entanglements in the Mechanics of Silicone Networks

The goal of this study is to investigate the applicability of different constitutive models for silicone networks using comprehensive multiaxial experimental tests, including non-equibiaxial mechanical tests which introduce differential constraints on the networks in the two orthogonal directions, on samples prepared using various crosslinking densities. Uniaxial stress–strain experiments show that a decrease in crosslinker amounts used in the preparation of silicone networks lead to more compliant material response as compared to that obtained using higher amounts of crosslinker. Biaxial data were used to obtain fits to the neo- Hookean, Mooney-Rivlin, Arruda-Boyce and the Edward-Vilgis slip-link constitutive models. Our results show that the slip-link model, based on separation of the individual contributions of chemical crosslinks and physical entanglements, is better at describing the stress–strain response of highly crosslinked networks at low stretches as compared to other constitutive models. Modulus obtained using the slip-link model for highly crosslinked networks agrees with experimentally determined values obtained using uniaxial tension experiments. In contrast, moduli obtained using coefficients to the other constitutive models underpredict experimentally determined moduli by over 40 %. However, the slip-link model did not predict the experimentally observed stiffening response at higher stretches which was better captured using the Arruda-Boyce model.     

New constitutive equations for the rapid flow of granular materials—II

This paper compliments a previous paper, (J. Non-Newtonian Fluid Mech.,9(1981) 147), which discussed new constitutive equations for rapid collisionally maintained flow of granular materials as a non-Newtonian microfluid in which the gradient of microrotation played an important role as a kinematic variable. In this article we discuss another variation of such constitutive equations; one for which we do not consider the effects of gradients of the mean microrotation of grains. Derived are expressions for the dispersive normal and shear stresses for a plane shear rapid flow of a granular material in the presence of intergranular slip and friction. These expressions reduce to classical results if friction is set equal to zero. A graph of the variation of stresses versus the friction factor is also presented which reveals a kind of choking phenomenon at larger values of μ_k.     

Slip flow due to a stretching cylinder

The slip flow due to a stretching cylinder is studied. A similarity transform reduces the Navier-Stokes equations to a set of non-linear ordinary differential equations. Asymptotic solutions for large Reynolds number and small slip show the problem can be related to the existing two-dimensional stretching cases. Due to algebraic decay, the equations are further transformed through a compressed variable, and then integrated numerically. It is found that slip greatly reduces the magnitudes of the velocities and the shear stress.     

Permanent Waves in Slow Free-Surface Flow of a Herschel–Bulkley fluid

Unsteady flow of a viscoplastic fluid on an inclined plane is examined. The fluid is described by the three-parameter Herschel–Bulkley constitutive equation. The set of equations governing the flow is presented, recovering earlier results for a Bingham fluid and steady uniform motion. A permanent wave solution is then derived, and the relation between wave speed and flow depth is discussed. It is shown that more types of gravity currents are possible than in a Newtonian fluid; these include some cases of flows propagating up a slope. The speed of permanent waves is derived and the possible surface profiles are illustrated as functions of the flow behavior index.     

The “decaying springs” model for irreversibility in polymer melts

In this work, entanglements in a polymer melt are modeled as a system of parallel springs which form and decay spontaneously. The springs are assumed to be nonlinear, and a certain fraction of them is torn apart by a certain strain.Based on these assumptions, a model of behavior in simple shear is developed. This model is shown to predict a behavior comprising that of a Wagner fluid, and is generalized to a tensorial model of single integral type. The integrand depends on a product of a material function, modeling reversible behavior, and a material functional which takes irreversible processes into account.Irreversibility of network disentanglement, which may occur when deformation changes or reverses direction, can be modeled in this way. It is shown that the two well-known Wagner constitutive equations with and without irreversibility assumptions are special cases of the model developed. In case of a deformation which does not change directions, the new material function and the material functional are multiplied to yield Wagner's damping function.When the rate of spring formation is a function of temperature, the developed model is shown to predict thermorheologically simple behavior. A constitutive equation for non-isothermal flow of polymers is developed with this assumption.     

基于格子Boltzmann两相模型的粘弹流体挤出胀大分析

挤出胀大的数值模拟是非牛顿流体研究中具有挑战性的问题．本文运用格子Boltzmann方法(LBM)分析Oldroyd-B和多阶松弛谱PTT粘弹流体的挤出胀大现象,采用颜色模型模拟出口处粘弹流体和空气的两相流动,通过重新标色获得两种流体的界面,并最终获得胀大的形状．Navier-Stokes方程和本构方程的求解采用双分布函数模型．将胀大的结果与解析解、实验解和单相自由面LBM结果进行了比较,发现格子Boltzmann两相模型结果与解析解和实验结果相吻合,相比于单相模型,收敛速度更快,解的稳定性更高．研究了流道尺寸对胀大率的影响,并对挤出胀大的内在机理进行了分析．     

A three-dimensional non-linear constitutive law for magnetorheological fluids, with applications

In this paper we first summarize the magnetic and mechanical balance equations for magnetorheological fluids undergoing steady motion in the presence of a magnetic field. A general three-dimensional non-linear constitutive law for such a fluid is given for the case in which the magnetic induction vector is used as the independent magnetic variable. The equations are needed for the analysis of boundary-value problems involving fluids with dispersed micron-sized ferrous particles subjected to a time-independent magnetic field. For illustration, the equations are applied, in the case of an incompressible fluid, to the solution of some basic problems. We consider unidirectional flow in a region confined by two infinite parallel plates with a magnetic field applied perpendicular to the plates. Next, we examine two problems involving a circular cylindrical geometry with the fluid occupying the region between two concentric cylinders: axial flow subjected to an axial magnetic field and circumferential flow with a circumferential field. After making some simplifying assumptions on the constitutive law and choosing material parameters, numerical solutions for the velocity profiles are illustrated.