Equation viscoelastic constitutive

Model (material) parameters used in viscoelastic constitutive equations... 

The Maxwell class of viscoelastic constitutive equations are described by a simpler form of Equation (1.22) in which A = 0. For example, the upper-convected Maxwell model (UCM) is expressed as... 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

Equations of continuity and motion in a flow model are intrinsically connected and their solution should be described simultaneously. Solution of the energy and viscoelastic constitutive equations can be considered independently. 

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

The integrals in Equation (3.32) are found using a quadrature over the element domain The viscoelastic constitutive equations used in the described model are hyperbolic equations and to obtain numerically stable solutions the convection terms in Equation (3.32) are weighted using streamline upwinding as (inconsistent upwinding)... 

Predictions using the observed relaxation time spectrum at the gel point are consistent with further experimental observations. Such predictions require a constitutive equation, which now is available. Insertion of the CW spectrum, Eq. 1-5, into the equation for the stress, Eq. 3-1, results in the linear viscoelastic constitutive equation of critical gels, called the critical gel equation ... 

Molecular theories, utilizing physically reasonable but approximate molecular models, can be used to specify the stress tensor expressions in nonlinear viscoelastic constitutive equations for polymer melts. These theories, called kinetic theories of polymers, are, of course, much more complex than, say, the kinetic theory of gases. Nevertheless, like the latter, they simplify the complicated physical realities of the substances involved, and we use approximate cartoon representations of macromolecular dynamics to describe the real response of these substances. Because of the relative simplicity of the models, a number of response parameters have to be chosen by trial and error to represent the real response. Unfortunately, such parameters are material specific, and we are unable to predict or specify from them the specific values of the corresponding parameters of other... 

With viscoelastic models used by an increasing number of researchers, time and temperature dependence, as well as strain hardening and nonisotropic properties of the deformed parison can, in principle, be accounted for. Kouba and Vlachopoulos (97) used the K-BKZ viscoelastic constitutive equation to model both thermoforming and parison membrane stretching using two-dimensional plate elements in three-dimensional space. Debbaut et al. (98,99) performed nonisothermal simulations using the Giesekus constitutive equation. 

Kwon Y, Leonov AI (1995) Stability constraints in the formulation of viscoelastic constitutive equations. J Non-Newton Fluid Mech 58 25—46 Laius LA, Kuvshinskii EV (1963) Structure and mechanical properties of oriented amorphous linear polymers. Fizika Tverdogo Tela 5(11) 3113—3119 (in Russian)... 

A particularly useful expression for the complex dynamic moduli was introduced using operators and fractional derivatives for the viscoelastic constitutive equations ( ). For a rheologically simple system the complex shear modulus is... 

The viscoelastic constitutive equations are of hyperbolic type and it is well-known that numerical solutions require special care. The classical Galerkin method was very shown to be inadequate for such problems. Special techniques were developed based on the very general... 

Viscoelastic constitutive equations in the stream-tube method... 

Different viscoelastic constitutive equations were adopted for modelling the experimental data of both fluids ... 

Among the equations that govern a viscoelastic problem, only the constitutive equations differ formally from those corresponding to elastic relationships. In the context of an infinitesimal theory, we are interested in the formulation of adequate stress-strain relationships from some conveniently formalized experimental facts. These relationships are assumed to be linear, and field equations must be equally linear. The most convenient way to formulate the viscoelastic constitutive equations is to follow the lines of Coleman and Noll (1), who introduced the term memory by stating that the current value of the stress tensor depends upon the past history... 

Viscoelastic constitutive equations are used to model material properties. Viscoelastic theory combines the elements of elasticity and Newtonian fluids. The theory of viscoelasticity was developed to describe the behavior of materials which show intermediate behavior between solids and fluids. 

Finally, we cannot overlook the development of computational tools for the solution of problems in fluid mechanics and transport processes. Methods of increasing sophistication have been developed that now enable quantitative solutions of some of the most complicated and vexing problems at least over limited parameter regimes, including direct numerical simulation of turbulent flows so-called free-boundary problems that typically involve large interface or boundary deformations induced by flow and methods to solve flow problems for complex fluids, which are typically characterized by viscoelastic constitutive equations and complicated flow behavior. 

Experimental data for polymer solutions have been reported by Osaki, Tamura, Kurata, and Kotaka (60), by Booij (12), and by Macdonald (50). Osaki et al. used polystyrene in toluene, polymethylmethacrylate in diethylphthalate, and poly-n-butylmethacrylate in diethylphthalate. Booij s data were for aluminum dilaurate in decalin and a rubbery ethylene-propylene copolymer in decalin. Macdonald s experiments were performed on several polystyrenes in several Aroclors and on polyisobutylene in Primol. Shortly after the original publication of the Japanese group, Macdonald and Bird (51) showed that a nonlinear viscoelastic constitutive equation was capable of describing quantitatively their data on both the non-Newtonian viscosity and the superposed-flow material functions. Other measurements and continuum model calculations have been described by Booij (12 a). 

Duct flows of nonnewtonian fluids are described by the governing equations (Eq. 10.24-10.26), by the constitutive equation (Eq. 10.27) with the viscosity defined by one of the models in Table 10.1, or by a linear or nonlinear viscoelastic constitutive equation. To compare the available analytical and experimental results, it is necessary to nondimensionalize the governing equations and the constitutive equations. In the case of newtonian flows, a uniquely defined nondimensional parameter, the Reynolds number, is found. However, a comparable nondimensional parameter for nonnewtonian flow is not uniquely defined because of the different choice of the characteristic viscosity. 

Work on designing profile extrusion dies is complicated by the effect known as die swell. In capillary flow, elastic effects cause the diameter of the extrudate to be greater than the capillary diameter. This effect depends on the length of the capillary as well as the processing conditions and must be taken into account when designing extrusion dies. To model such an effect requires a viscoelastic constitutive equation. There is a lack of appropriate models for which data is readily available and this has hindered the use of computer simulation in this field. Nevertheless, a great deal of literature exists on simulation. ... 

To help understand and quantitatively evaluate the secondary movement shown above, Debbaut et al. [75, 77] augmented this experimental work with a three-dimensional flow simulation that incorporated viscoelastic effects. The finite element method, using a 4-mode Giesekus model as the viscoelastic constitutive equation, was used for the simulation. The polymer used for the experiment and simulation was a low-density polyethylene. Figures 12.20 and 12.21 show the experimental observations and the numerical predictions of the deformations of the interface for the rectangular straight channel [78], and for the teardrop channel [75], respectively. 

R. M. Shay and J. M. Caruthers, A New Nonlinear Viscoelastic Constitutive Equation for Predicting Yield in Amorphous Solid Poljmiers , J. Rheol. 30, 871-827 (1986). 

Other Constitutive Modei Descriptions. The above work describes a relatively simple way to think of nonlinear viscoelasticity, viz, as a sort of time-dependent elasticity. In solid polymers, it is important to consider compressibility issues that do not exist for the viscoelastic fluids discussed earlier. In this penultimate section of the article, other approaches to nonlinear viscoelasticity are discussed, hopefully not abandoning all simplicity. The development of nonlinear viscoelastic constitutive equations is a very sophisticated field that we will not even attempt to survey completely. One reason is that the most general constitutive equations that are of the multiple integral forms are cumbersome to use in practical applications. Also, the experimental task required to obtain the material parameters for the general constitutive models is fairly daunting. In addition, computationally, these can be difficult to handle, or are very CPU-time intensive. In the next sections, a class of single-integral nonlinear constitutive laws that are referred to as reduced time or material clock-type models is disscused. Where there has been some evaluation of the models, these are examined as well. 

The uniaxial nonlinear viscoelastic constitutive equation of Schapery( > can be written for an isotropic material as... 

White, J.L. and Tanaka, H. (1981) Comparison of a plastic-viscoelastic constitutive equation with rheological measurements on a polystyrene melt reinforced with small particles, /. Non-Newtonian Fluid Mech., 8,1-10. 

We introduce the change of variables into Equations 11.6a-c and neglect nonlinear terms in the perturbation variables 0, 4, and n. (The nonlinearities here are quadratic, but they will not be quadratic for the energy equation or for a viscoelastic constitutive equation like the PTT model.) We thus obtain the following linear equations ... 

There have been attempts to explain the onset of melt fracture by using hnear stability theory, along the hnes discussed in the preceding chapter for the spinline. The procedure is the same in principle, but the computational problem to solve the relevant linear eigenvalue problem for channel flow with any viscoelastic constitutive equation is a very difficult one. Based on a number of successful solutions it appears that plane laminar flow of viscoelastic liquids at very low Reynolds numbers is stable to infinitesimal disturbances. 

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