Constitutive equations Maxwell model

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... 

A frequently used example of Oldroyd-type constitutive equations is the Oldroyd-B model. The Oldroyd-B model can be thought of as a description of the constitutive behaviour of a fluid made by the dissolution of a (UCM) fluid in a Newtonian solvent . Here, the parameter A, called the retardation time is de.fined as A = A (r s/(ri + s), where 7]s is the viscosity of the solvent. Hence the extra stress tensor in the Oldroyd-B model is made up of Maxwell and solvent contributions. The Oldroyd-B constitutive equation is written as... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

In order to describe the material properties as a function of frequency for a body that behaves as a Maxwell model we need to use the constitutive equation. This is given in Equation (4.8), which describes the relationship between the stress and the strain. It is most convenient to express the applied sinusoidal wave in the exponential form of complex number notation ... 

Using a Maxwell model as a constitutive equation for a viscoelastic fluid, one can show that the instantaneous shear stress at the wall is smaller in the viscoelastic fluid than in the corresponding Newtonian fluid. 

Together with Eq. 3.3-17, Eq. 3.3-16 is the White-Metzner constitutive equation, which has been used frequently as a nonlinear viscoelastic model. Of course, for small deformations, X(i) = dx/dt, and the single Maxwell fluid equation (Eq. 3.3-9) is obtained. 

The reality, however, is not as simple as that. There are several possibilities to describe viscosity, 77, and first normal stress difference coefficient, P1. The first one originates from Lodge s rheological constitutive equation (Lodge 1964) for polymer melts and the second one from substitution of a sum of N Maxwell elements, the so-called Maxwell-Wiechert model (see Chap. 13), in this equation (see General references Te Nijenhuis, 2005). 

In Fig. 15.27, the transient extensional viscosity of a low-density polyethylene, measured at 150 °C for various extensional rates of strain, is plotted against time (Munstedt and Laun, 1979). Qualitatively this figure resembles the results of the Lodge model for a Maxwell model in Fig. 15.26. For small extensional rates of strain (qe < 0.001 s ) 77+(f) is almost three times rj+ t). For qe > 0. 01 s 1 r/+ (f) increases fast, but not to infinite values, as is the case in the Lodge model. The drawn line was estimated by substitution of a spectrum of relaxation times of the polymer (calculated from the dynamic shear moduli, G and G") in Lodge s constitutive equation. The resulting viscosities are shown in Fig. 15.28 after a constant value at small extensional rates of strain the viscosity increases to a maximum value, followed by a decrease to values below the zero extension viscosity. 

In the DGM, the solid phase is modeled as giant dust molecules held motionless in space with which the diffusing gas molecules collide. The constitutive equations governing the diffusion molar flux intensities Nf for both MTPM and DGM are the generalized Maxwell-Stefan equations... 

The following summary is from Jou and Casas-Vazquez (2001). In the extended nonequilibrium thermodynamics for a binary liquid mixture, the viscous pressure tensor Pv and the diffusion flux J are considered as additional independent variables. The viscous pressure tensor, Pv, by the simplest Maxwell model, is defined by the following constitutive equation ... 

The rheological constitutive equation of the Rouse model is that of an upper-convected Maxwell model, with the consequence that steady-state elongational flow only exists for strain rates lower than l/(2A,i). The steady-state elongational wscosity depends then on strain rate ... 

In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical treatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11]. 

It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan-Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory munerical results were also obtained with multi-mode integral constitutive equations using a spectnun of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [19]. 

At present two models are available for description of pore-transport of multicomponent gas mixtures the Mean Transport-Pore Model (MTPM)[4,5] and the Dusty Gas Model (DGM)[6,7]. Both models permit combination of multicomponent transport steps with other rate processes, which proceed simultaneously (catalytic reaction, gas-solid reaction, adsorption, etc). These models are based on the modified Maxwell-Stefan constitutive equation for multicomponent diffusion in pores. One of the experimentally performed transport processes, which can be used for evaluation of transport parameters, is diffusion of simple gases through porous particles packed in a chromatographic column. 

It can be shown using Eq. (1-20) that the upper-convected Maxwell equation is equivalent to the Lodge integral equation, Eq. (3-24), with a single relaxation time. This is shown for the case of start-up of uniaxial extension in Worked Example 3.2. Thus, the simplest temporary network model with one relaxation time leads to the same constitutive equation for the polymer contribution to the stress as does the elastic dumbbell model. 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

Note that the simple Hooke s law behavior of the stress in a solid is analogous to Newton s law for the stress of a fluid. For a simple Newtonian fluid, the shear stress is proportional to the rate of strain, y (shear rate), whereas in a Hookian solid, it is proportional to the strain, y, itself. For a fluid that shares both viscous and elastic behavior, the equation for the shear stress must incorporate both of these laws— Newton s and Hooke s. A possible constitutive relationship between the stress in a fluid and the strain is described by the Maxwell model (Eq. 6.3), which assumes that a purely viscous damper described by Eq. 6.1 and a pure spring described by Eq. 6.2 are connected in series (i.e., the two y from Eqs. 6.1 and 6.2 are additive). 

In fact, Equation 5.281 describes an interface as a two-dimensional Newtonian fluid. On the other hand, a number of non-Newtonian interfacial rheological models have been described in the literature. Tambe and Sharma modeled the hydrodynamics of thin liquid films bounded by viscoelastic interfaces, which obey a generalized Maxwell model for the interfacial stress tensor. These authors also presented a constitutive equation to describe the rheological properties of fluid interfaces containing colloidal particles. A new constitutive equation for the total stress was proposed by Horozov et al. ° and Danov et al. who applied a local approach to the interfacial dilatation of adsorption layers. 

The equation for solvent transport consists of a diffusional term and a term due to osmotic pressure. The osmotic pressure term arises by using linear irreversible diermodynamics arguments (20). The osmotic pressure is relat to the viscoelastic properties of the polymer through a constitutive equation. In our analysis, the Maxwell element has been used as the constitutive model. Thus, the governing equations for solvent transport in the concentrated regime are... 

Consider the mechanical behavior of a model in which a spring and dashpot are placed in series, which is known as the Maxwell model (Fig. 5.13). Recognizing that the overall strain rate is the sum of the contributions from the spring and the dashpot, the constitutive equation can be written... 

Maxwell s model was initially proposed to characterize the permittivity of a dielectric in 1873 [114]. However, because the constitutive equations governing electrical potential and flux through a membrane material are similar, the theory has been extended to the hybrid membrane desralption. The mathranatical expression is given by... 

Apelian, M. R., e. a. (1988). hnpad of the constitutive equation and singularity on the calculation of stick-slip flow The modified upper-convected maxwell model, /. Non-Newtonian Fluid Mech. 27 299-321. 

The onset of ductile failure in sohds is determined by the Considire construction, in which a maximum in the stress-strain curve causes an instability that manifests itself as a neck. This concept is unhkely to be apphcable to the onset of necking in polymer melts. All constitutive equations, including the Maxwell model. Equation 9.16, predict a maximum in the stress-strain curve for stretching at a constant stretch rate, and this maximum normally occurs prior to the attainment of steady state. Hence, hteral interpretation of the construction as a sufficient condition for failure would imply that uniform uniaxial extensional experiments could never be carried out past the force maximum, which often corresponds to a relatively low strain such an interpretation is clearly contrary to substantial experimental experience in extensional rheometry, and several experimental studies focusing specifically on the Considere construction have shown that it does not predict the experimental onset of necking in melts. 

The combination of Equations 4.30-4.32 and the elimination of the subscripts for the Maxwell and Kelvin models give the third-order linear differential constitutive equation ... 

An equation between stress and strain can be obtained for any mechanical model by using equilibrium and kinematic equations for the system and constitutive equations for the elements. For a Maxwell fluid, equilibrium... 

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