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.     

A solution for the vertical variation of stress,rather than velocity,in a three-dimensional circulation model

A simple technique is presented that allows a numerical solution to be sought for the vertical variation of shear stress as a substitute for the vertical variation of velocity in a three-dimensional hydrodynamic model. In its most general form the direct stress solution (DSS) method depends only upon the validity of an eddy viscosity relation between the shear stress and the vertical gradient of velocity. The rationale for preferring a numerical solution for shear stress to one for velocity is that shear stress tends to vary more slowly over the vertical than velocity, particularly near boundaries. Consequently, a numerical solution can be obtained much more efficiently for shear stress than for velocity. When needed, the velocity profile can be recovered from the stress profile by solving a one-dimensional integral equation over the vertical. For most practical problems this equation can be solved in closed form. Comparisons are presented between the DSS technique, the standard velocity solution technique and analytical solutions for wind-driven circulation in an unstratified, closed, rectangular channel governed by the linear equations of motion. In no case was the computational effort required by the velocity solution competitive with the DSS when a physically realistic boundary layer was included. The DSS technique should be particularly beneficial in numerical models of relatively shallow water bodies in which the bottom and surface boundary layers occupy a significant portion of the water column.     

Selection of kernel functions used in a new constitutive equation for viscoelastic fluids

A selection of kernel functions is given to be used in a new integral constitutive equation proposed by Piau whereby the deviatoric stress is calculated from the integral of the history of the past intrinsic rate of rotation and rate of deformation tensors through a representation theorem. Piau has demonstrated the objectivity of a frame moving with a given particle whose axis are directed along the eigenvectors of the rate of deformation tensor. The use of such a framework provides a new approach in the attempt to reduce the computational difficulties associated with conventional constitutive equations written in co-deformational or co-rotational reference frames.The shear and primary normal-stress material functions and the extensional (elongational) stress growth function are defined for the proposed integral constitutive equation. These material functions are used to calculate the kernel functions using steady state, stress relaxation and stress growth data of Attané in simple shear flow for monodisperse polystyrene solutions. The shear and extensional stress growth data of Meissner for a polyethylene melt are also used to show the flexibility of the rheological model.The material functions are first written in terms of five monotonically decreasing functions of the time lag between the past and the present time. Then kernel functions are chosen such that when substituted in the new integral constitutive equation they yield the functions used to describe the data. A further condition imposed on the normalized kernel functions is that they be decreasing functions of time lag.     

The Dilute Rheology of Swimming Suspensions: A Simple Kinetic Model

A simple kinetic model is presented for the shear rheology of a dilute suspension of particles swimming at low Reynolds number. If interparticle hydrodynamic interactions are neglected, the configuration of the suspension is characterized by the particle orientation distribution, which satisfies a Fokker-Planck equation including the effects of the external shear flow, rotary diffusion, and particle tumbling. The orientation distribution then determines the leading-order term in the particle extra stress in the suspension, which can be evaluated based on the classic theory of Hinch and Leal (J Fluid Mech 52(4):683–712, 1972), and involves an additional contribution arising from the permanent force dipole exerted by the particles as they propel themselves through the fluid. Numerical solutions of the steady-state Fokker-Planck equation were obtained using a spectral method, and results are reported for the shear viscosity and normal stress difference coefficients in terms of flow strength, rotary diffusivity, and correlation time for tumbling. It is found that the rheology is characterized by much stronger normal stress differences than for passive suspensions, and that tail-actuated swimmers result in a strong decrease in the effective shear viscosity of the fluid.     

Computation of rheological properties of suspensions of rigid rods: stress growth after inception of steady shear flow

The orientation distribution and stress growth for a suspension of rigid rods (or dumbbells) in a Newtonian solvent are calculated for inception of steady shear flow. Galerkin's method, with spherical harmonics as trial functions, is used in the spatial coordinates to obtain a system of ordinary differential equations in time which is solved by the spectral method. The method is applicable over a wide range of dimensionless shear rates (Peclet numbers) and has been coded with standard system-solvent and eigensystem packages. For sufficiently large Peclet numbers, the results give the well known rigid-dumbbell prediction of an overshoot in the shear viscosity and normal stress differences. This overshoot is then followed by an undershoot. An explicit analytical approximation for the fluid stresses is presented which is reasonably accurate for Peclet numbers less than unity.     

Extended irreversible thermodynamics,generalized hydrodynamics,kinetic theory,and rheology

The gist of extended irreversible thermodynamics and generalized hydrodynamics is presented within the context of rheology of complex molecules (e.g., polymers) in this paper. Then, the constitutive equation for stress developed for polyatomic fluids in a previous paper is applied to rheology of polymeric fluids. This constitutive equation is fully consistent with the thermodynamic laws. It is shown that the collision bracket integrals appearing in the constitutive equation can be recast in terms of friction tensors of beads and equilibrium force-force correlation functions if the momentum relaxation is much faster than the configuration relaxation and there exist such relaxation times. The force-force correlation functions reduce to those related to the mean square radius of gyration of the polymer if the Hookean model is taken for forces. By treating the recast collision bracket integrals in the constitutive equation as empirical parameters, we analyze some experimental data on shear rate and elongation rate dependence of polymeric melts and obtain excellent agreement with experiment. We show that the empirical parameters can be related to the zero shear rate viscosity and the ratio of the secondary to the primary normal stress coefficient. Therefore, for the plane Couette flow geometry considered in the paper, the constitutive equation is completely specified by the limiting material functions at zero shear rate and relaxation times.Work supported in part by the Natural Sciences and Engineering Research Council of Canada and Fonds FCAR, Quebec. This paper was presented at the Symposium on Recent Developments in Structured Continua II held at Magog, Quebec, Canada, May 23–25, 1990.     

Pulsatory Turbulent Compressible Fluid Flow and Propagation of Pressure Waves in a Channel

The properties of the damping coefficient and phase velocity of propagation of small-amplitude pressure waves as functions of the oscillation frequency are investigated for the turbulent flow of a weakly compressible fluid in a circular pipe. The wall friction is found by solving numerically the equation of motion and the relaxation equations for the turbulent shear stress and viscosity which provide the basis for a turbulent transfer model developed for unsteady conditions. The properties are explained in terms of an analysis of the calculated data on turbulent transfer. The results obtained are compared with experiments.     

Finite viscoplasticity of non-affine networks: stress overshoot under shear

A constitutive model is derived for the viscoplastic behavior of polymers at finite strains. A polymer is treated as an equivalent network of chains bridged by permanent junctions. The elastic response of the network is attributed to elongation of strands, whereas its plastic behavior is associated with sliding of junctions with respect to their reference positions. A new kinetic equation is proposed that expresses the rate of sliding of junctions in terms of the Cauchy stress tensor. Constitutive equations for an equivalent non-affine network are developed by using the laws of thermodynamics, where internal dissipation of energy reflects two processes at the micro-level: sliding of chains along entanglements and friction of strands between junctions. A similarity is revealed between these relations and the pom-pom model. The governing equations are applied to study stress overshoot at simple shear of an incompressible medium. Adjustable parameters in the stress-strain relations are found by fitting experimental data on polycarbonate melt reinforced with short glass fibers and polystyrene solution. Fair agreement is demonstrated between the observations and the results of numerical simulation.Received: 6 January 2003, Accepted: 28 April 2003     

Deformations of an elastic,internally constrained material. Part 1: Homogeneous deformations

The nonlinear elastic response of a class of materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite deformation of a variety of metals is investigated. The purely kinematical consequences of the Bell constraint are discussed, and restrictions on the full range of compatible deformations are presented in geometrical terms. Then various forms of the constitutive equation relating the stress and stretch tensors for an isotropic elastic Bell material are presented. Inequalities on the mechanical response functions are introduced. The importance of these in applications is demonstrated in several examples throughout the paper.This paper focuses on homogeneous deformations. In a simple illustration of the theory, a generalized form of Bell's empirical rule for uniaxial loading is derived, and some peculiarities in the response under all-around compressive loading are discussed. General formulae for universal relations possible in an isotropic elastic, Bell constrained material are presented. A simple method for the determination of the left stretch tensor for essentially plane problems is illustrated in the solution of the problem of pure shear of a materially uniform rectangular block. A general formula which includes the empirical rule found in pure shear experiments by Bell is derived as a special case. The whole apparatus is then applied in the solution of the general problem of a homogeneous simple shear superimposed on a uniform triaxial stretch; and the great variety of results possible in an isotropic, elastic Bell material is illustrated. The problem of the finite torsion and extension of a thin-walled cylindrical tube is investigated. The results are shown to be consistent with Bell's data for which the rigid body rotation is found to be quite small compared with the gross deformation of the tube. Several universal formulas relating various kinds of stress components to the deformation independently of the material response functions are derived, including a universal rule relating the axial force to the torque.Constitutive equations for hyperelastic Bell materials are derived. The empirical work function studied by Bell is introduced; and a new constitutive equation is derived, which we name Bell's law. On the basis of this law, we then derive exactly Bell's parabolic laws for uniaxial loading and for pure shear. Also, form Bell's law, a simple constitutive equation relating Bell's deviatoric stress tensor to his finite deviatoric strain tensor is obtained. We thereby derive Bell's invariant parabolic law relating the deviatoric stress intensity to the corresponding strain intensity; and, finally, Bell's fundamental law for the work function expressed in these terms is recovered. This rule is the foundation for all of Bell's own theoretical study of the isotropic materials cataloged in his finite strain experiments on metals, all consistent with the internal material constraint studied here.     

A two variable refined plate theory for orthotropic plate analysis

A new theory, which involves only two unknown functions and yet takes into account shear deformations, is presented for orthotropic plate analysis. Unlike any other theory, the theory presented gives rise to only two governing equations, which are completely uncoupled for static analysis, and are only inertially coupled (i.e., no elastic coupling at all) for dynamic analysis. Number of unknown functions involved is only two, as against three in case of simple shear deformation theories of Mindlin and Reissner. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Well studied examples, available in literature, are solved to validate the theory. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories having more number of unknown functions.     

Rheological properties for inelastic Maxwell mixtures under shear flow

The Boltzmann equation for inelastic Maxwell models is considered to determine the rheological properties in a granular binary mixture in the simple shear flow state. The transport coefficients (shear viscosity and viscometric functions) are exactly evaluated in terms of the coefficients of restitution, the (reduced) shear rate and the parameters of the mixture (particle masses, diameters and concentration). The results show that in general, for a given value of the coefficients of restitution, the above transport properties decrease with increasing shear rate.     

Constitutive equations in finite viscoplasticity of semicrystalline polymers

Three series of uniaxial tensile tests with constant strain rates are performed at room temperature on isotactic polypropylene and two commercial grades of low-density polyethylene with different molecular weights. Constitutive equations are derived for the viscoplastic behavior of semicrystalline polymers at finite strains. A polymer is treated as an equivalent network of strands bridged by permanent junctions. Two types of junctions are introduced: affine whose micro-deformation coincides with the macro-deformation of a polymer, and non-affine that slide with respect to their reference positions. The elastic response of the network is attributed to elongation of strands, whereas its viscoplastic behavior is associated with sliding of junctions. The rate of sliding is proportional to the average stress in strands linked to non-affine junctions. Stress–strain relations in finite viscoplasticity of semicrystalline polymers are developed by using the laws of thermodynamics. The constitutive equations are applied to the analysis of uniaxial tension, uniaxial compression and simple shear of an incompressible medium. These relations involve three adjustable parameters that are found by fitting the experimental data. Fair agreement is demonstrated between the observations and the results of numerical simulation. It is revealed that the viscoplastic response of low-density polyethylene in simple shear is strongly affected by its molecular weight.     

EIGEN THEORY OF VISCOELASTIC MECHANICS FOR ANISOTROPIC SOLIDS

Anisotropic viscoelastic mechanics is studied under anisotropic subspace. It is proved that there also exist the eigen properties for viscoelastic medium. The modal Maxwell's equation, modal dynamical equation (or modal equilibrium equation) and modal compatibility equation are obtained. Based on them, a new theory of anisotropic viscoelastic mechanics is presented. The advantages of the theory are as follows: 1) the equations are all scalar, and independent of each other. The number of equations is equal to that of anisotropic subspaces, 2) no matter how complicated the anisotropy of solids may be, the form of the definite equation and the boundary condition are in common and explicit, 3) there is no distinction between the force method and the displacement method for statics, that is, the equilibrium equation and the compatibility equation are indistinguishable under the mechanical space, 4) each model equation has a definite physical meaning, for example, the modal equations of order one and order two express the volume change and shear deformation respectively for isotropic solids, 5) there also exist the potential functions which are similar to the stress functions of elastic mechanics for viscoelastic mechanics, but they are not man-made, 6) the final solution of stress or strain is given in the form of modal superimposition, which is suitable to the proximate calculation in engineering.     

The relationship between simple shear and pure shear for a simple fluid without coupling effects

Similarities between simple shear and pure shear or planar extension are exploited to derive equations relating stress in pure shear at constant extension rate to the stress in simple shear at constant shear rate. For the class of materials considered it follows that there are only two independent material functions required to describe simple shear. The relationships derived may also be used to estimate the ratio of first to second normal-stress differences in simple shear using experimental results from pure shear experiments.     

Study on the generalized prandtl-reuss constitutive equation and the corotational rates of stress tensor

I.IntroductionTilepl'ogl'ess11as.toifcertainextent,beenmadeintheelastic-plasticconstitutivetheoryatII[litedefbrlllations.Coil'paredwitllotherconstitutiverelations,thegeneralizedPrandtlReuss(P-R)equatiollsareextensivelystudiedandwidelyapplied.IndevelopingthegeneralizedP-Requation.itisusuallyassumedthatthedeformationrate(thesymmetricpartorvelocitygradiellt)isdecolllposedintotheelasticpartandplasticpart.TheplasticLIcf\'l.llliltlollrittcobeystilenormalfi(,xvrilleasillthecaseofinfinitcsilllnld…     

Experimental investigations into an isothermal spinning threadline: Extensional rheology of a Separan AP 30 solution in glycerol and water

Summary Experimental observations on a steady isothermally extending filament of a water/glycerol solution of Separan AP 30 are presented. Photographic records were analysed to give filament diameter (and hence filament speed) as a function of distance below the extrusion die (a glass capillary). Measurements of inline tension were also made. When effects of weight, surface tension and air drag were accounted for, the extensional stress at every point along the filament could be calculated. Results for stress versus extension rate are presented for various flow situations.Independent rheogoniometric measurements of simple shear viscosity, first and second normal stress differences, and of a crude relaxation time were also made at comparable rates of deformation.Comparison shows that apparent extensional viscosities are several orders of magnitude larger than corresponding simple shear viscosities. After discussion, no conclusion can be drawn about what constitutive equation is most suitable to describe the results.An analysis to predict air drag is given.With 18 figures     

Numerical simulation of 3-D turbulent flows over dredged trenches

A 3-D free surface flow in open channels based on the Reynolds equations with thek-ε turbulence closure model is presented in this paper. Insted of the “rigid lid” approximation, the solution of the free surface equation is implemented in the velocity—pressure iterative procedure on the basis of the conventional SIMPLE method. This model was used to compute the flow in rectangular channels with trenches dredged across the bottom. The velocity, eddy viscosity coefficient, turbulent shear stress, turbulent kinetic energy and elevation of the free surface can be obtained. The computed results are in good agreement with previous experimental data.     

A rate type method for large deformation problems of nonlinear elasticity

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Constitutive relationships for polymeric materials with power-law distributions of relaxation times

The linear relaxation modulus of polydisperse polymer melts and solutions can often be approximated by a power law,ct ^–m over some range of time,t. If, in addition, the nonlinear rheology is given by a separable integral equation, with a strain-dependent factor typical of those observed experimentally, then some commonly observed empirical rules and equations can be readily derived as approximations, namely the Cox-Merz relationship between complex viscosity and steady-state shear viscosity, Bersted's predictions of steady shear stress and first normal-stress difference from a truncated spectrum of linear relaxation times, and the observation of Koyama and coworkers that the ratio of the nonlinear to the linear time-dependent elongational viscosity is independent of strain rate, over a range of strain rates outside the linear regime.     

Non-oscillation Criteria for Hypoelastic Models under Simple Shear Deformation

Ten objective rates, spinning or non-spinning, are critically examined from the viewpoint of Sturm's theorems in ordinary differential equations. Upon developing implication relations of oscillatory, non-oscillatory, and disconjugate behavior, we establish oscillation and non-oscillation criteria which pick out the objective stress rates that lead to oscillatory and non-oscillatory responses in simple shear deformation, respectively. Among the hypoelastic equations associated with the spinning objective rates examined, the Jaumann equation is an oscillatory minorant, the homogeneous Xiao–Bruhns–Meyers equation is a non-oscillatory majorant, and the homogeneous Green–Naghdi equation is a disconjugate majorant. If (Sturm comparable) non-spinning objective rates are also taken into consideration, then the Durban–Baruch equation becomes an oscillatory minorant, but the other two equations remain to play the same roles. The Jaumann equation is a Sturm majorant for all the other nine homogeneous hypoelastic equations, and the homogeneous Szabó–Balla-2 equation is a Sturm minorant for all the other nine homogeneous hypoelastic equations. Most of the solutions of the zeroth-grade hypoelastic equations at simple shear have already been published, except for those of Szabó and Balla, to which the closed-form solutions are derived here. Moreover, all solutions are extended to include the effect of initial stresses. This revised version was published online in August 2006 with corrections to the Cover Date.