The Maxwell model is one of the simplest viscoelastic constitutive equations. It exhibits a constant viscosity and a quadratic first normal-stress difference. In view of its rheological simplicity, it is recommended only when little information about the fluid is available, or when a qualitative prediction is sufficient. Despite its numerical simplicity, this model can be difficult from a computational point of view.

For the upper-convected Maxwell model, is computed from

(11–11)

The purely viscous component of the extra stress () is equal to zero in the Maxwell model.

The Oldroyd-B model is, like the Maxwell model, one of the simplest viscoelastic constitutive equations. It is slightly better than the Maxwell model, because it allows for the inclusion of the purely viscous component of the extra stress, which can lead to better behavior of the numerical scheme. Oldroyd-B is a good choice for fluids that exhibit a very high extensional viscosity.

For the Oldroyd-B model, is computed from Equation 11–11 and the purely viscous component is computed (optionally) from Equation 11–2. When a multi-mode viscoelastic model is used, the purely viscous component of the extra-stress tensor is defined through the first mode only; more precisely, the corresponding viscosity will be given by the product .

Most fluids are characterized by shear-thinning and non-quadratic first normal-stress difference. With the White-Metzner model, it is possible to reproduce such viscometric features.

The White-Metzner model computes from

(11–12)

and is computed (optionally) from Equation 11–2.

The relaxation time () and the viscosity () can be constant or represented by the power law or the Bird-Carreau law for shear-rate dependence.

The power-law representation of the total viscosity is

(11–13)

where is the consistency factor, is the power-law index, and is the natural time.

The Bird-Carreau representation of the viscosity is

(11–14)

where is the natural time (that is, inverse of the shear rate at which the fluid changes from Newtonian to power-law behavior) and is the power-law index.

and are then computed from Equation 11–3 and Equation 11–4.

The power-law representation of the relaxation time is

(11–15)

The Bird-Carreau representation of the relaxation time is

(11–16)

The Phan-Thien-Tanner (PTT) model is one of the most realistic differential viscoelastic models. It exhibits shear thinning and a non-quadratic first normal-stress difference at high shear rates.

The PTT model computes from

(11–17)

and is computed (optionally) from Equation 11–2. When a multi-mode viscoelastic model is used, the purely viscous component of the extra-stress tensor is defined through the first mode only; more precisely, the corresponding viscosity will be given by the product .

and are material parameters that control, respectively, the shear viscosity and elongational behavior. In particular, a nonzero value for leads to a bounded steady extensional viscosity.

Like the PTT model, the Giesekus model is one of the most realistic differential viscoelastic models. It exhibits shear thinning and a non-quadratic first normal-stress difference at high shear rates.

The Giesekus model computes from

(11–18)

and is computed (optionally) from Equation 11–2. When a multi-mode viscoelastic model is used, the purely viscous component of the extra-stress tensor is defined through the first mode only; more precisely, the corresponding viscosity will be given by the product .

is the unit tensor and is a material constant. A nonzero value for leads to a bounded steady extensional viscosity and a shear-rate dependence of the shear viscosity.

The FENE-P model is derived from molecular theories. In the simplest representation, the molecules are described as dumbbells, built of two spheres linked together by a nonlinear spring. Unlike in the Maxwell model, however, the springs are allowed only a finite extension, so that the energy of deformation of the dumbbell becomes infinite for a finite value of the spring elongation. This model predicts a realistic shear thinning of the fluid and a first normal-stress difference that is quadratic for low shear rates and has a lower slope for high shear rates.

The FENE-P model computes from a configuration tensor (state variable) on the basis of an algebraic equation, as follows:

(11–19)

where is computed from the following differential equation:

(11–20)

and is the ratio of the maximum length of the spring to its length at rest. In this expression, (or ) must be strictly larger than 1. Furthermore, as becomes infinite, the FENE-P model becomes equivalent to the Maxwell model. Finally, note that a slightly different expression exists for the FENE-P constitutive model, which differs only by the definition of .

is computed (optionally) from Equation 11–2.

See [4] for additional information about the FENE-P model.

A viscoelastic constitutive equation, referred to as the POM-POM model, was introduced by McLeish and Larson [25]. The pom-pom molecule consists of a backbone to which q arms are connected at both extremities. In a flow, the backbone orients in a Doi-Edwards reptation tube consisting of the neighboring molecules, while the arms may retract into that tube when high deformation is applied. The concept of the pom-pom macromolecule makes the model suitable for describing the behavior of branched polymers. The approximate differential form of the model is based on equations of macromolecular orientation, and macromolecular stretching in relation to changes in orientation [25].

The model has undergone subsequent changes and improvements to make it suitable for software implementation and to introduce a nonzero second normal-stress difference. The formulation implemented in Ansys Polydata, referred to as DCPP, is developed by Clemeur, Rutgers, and Debbaut [7].

The DCPP model computes from an orientation tensor, and a stretching scalar (states variables), on the basis of the following algebraic equation:

(11–21)

where is the shear modulus and is a nonlinear material parameter (the nonlinear material parameter will be introduced later on). The state variables S and are computed from the following differential equations:

(11–22)

(11–23)

In these equations, and are the relaxation times associated to the orientation and stretching mechanisms respectively. The ratio, should be within the range 2 to 10.

In the stretching (Equation 11–23), characterizes the number of dangling arms (or priority) at the extremities of the pom-pom molecule or segment. It is an indication of the maximum stretching that the molecule can undergo, and therefore of a possible strain hardening behavior. can be obtained from the elongational behavior. is a nonlinear parameter that enables the introduction of a nonvanishing second normal stress difference in the DCPP model.

A multi-mode DCPP model can also be defined where the total viscoelastic extra-stress tensor given by Equation 11–21 will be split into individual contributions. Each contribution will involve an orientation tensor governed by Equation 11–22 and a stretching variable governed by Equation 11–23. A few guidelines are required for the determination of the several linear and nonlinear parameters.

Consider a multi-mode DCPP model characterized by N modes sorted with increasing values of relaxation times (increasing seniority). The linear parameters, and in Equation 11–21 and Equation 11–22 characterize the linear viscoelastic behavior of the model and can be determined with the usual procedure. Then the relaxation times () for stretching should be determined. Following Inkson et al. [19], should be equal to times the average number of entanglements of backbone section. Though this value is usually unknown, it should be within the range of 2 to 10. For a completely unentangled polymer segment you may accept the physical limit of = . should also satisfy the constraint , since sets the fundamental diffusion time for the branch point controlling the relaxation of polymer segment (i).

The parameter indicating the number of dangling arms (or priority) at the extremities of a pom-pom segment , also indicates the maximum stretching that can be undergone by that segment, and therefore its possible strain hardening behavior. For a multi-mode DCPP model, both seniority and priority are assumed to increase together towards the inner segments; hence should also increase with . The parameter can be obtained from the elongational behavior.

is a fifth set of nonlinear parameters. They control the ratio of second to first normal stress differences. The value of parameter should range between 0 and 1. For moderate values, corresponds to twice the ratio of the second to the first normal stress difference, and may decrease with increasing seniority.

As opposed to the models whose constitutive equations are directly expressed in terms of an extra-stress tensor, the planar flow of a pom-pom fluid may exhibit a stress component in the transverse direction. For such planar and axisymmetric flows, four components will be evaluated for the orientation tensor .

is computed (optionally) from Equation 11–2.

Elastomers are usually filled with carbon black and/or silicate. From the point of view of morphology, macromolecules at rest are trapped by particles of carbon black, via electrostatic forces of the van der Waals type. Under a deformation field, electrostatic bonds can break and macromolecules become free, while a reverse mechanism may develop when the deformation ceases. Thus the macromolecular system may consist of trapped and free macromolecules, with a reversible transition from one state to the other.

Leonov and Simhambhatla have developed a rheological model for the simultaneous prediction of the behavior of trapped and free macromolecular chains. The model is for filled elastomers, and involves two tensor quantities and a scalar one. One of the tensor quantities focuses on the behavior of the free macromolecular chains of the elastomer, while the other focuses on the trapped macromolecular chains.

The model exhibits a yielding behavior. It is intrinsically nonlinear, since the nonlinear response develops and is observable at early deformations. By comparison, models of the Oldroyd family and the POM-POM [DCPP] model are extrinsically nonlinear, since the nonlinear regime develops with kinematics.

Extended information on the Leonov model can be found in the PhD thesis [31] by Murthy V. Simhambhatla (supervised by Arkadii I. Leonov) at the University of Akron, as well as in [30]. Hereafter, we briefly recall the Leonov model, as well as its important features.

In a single-mode approach, the total stress tensor can be decomposed as the sum of free and trapped contributions, as follows:

(11–24)

In the previous equation, subscripts and refer to the free and trapped parts, respectively. Each of these contributions obeys its own equation. In particular, they invoke their own deformation field described by means of Finger tensors. An elastic Finger tensor is defined for the free chains, which obeys the following equation:

(11–25)

where is the relaxation time and is the unit tensor. and are the first invariant of and , respectively, and are defined in the following manner:

(11–26)

(11–27)

Equation 11–25 also involves a material function , which describes the deformation history dependence. Although Leonov [23] has suggested various expressions, the following expression has been deemed to be endowed with enough regularity, for the time being:

(11–28)

The parameter must be 0. This parameter slightly increases the amount of shear thinning.

Similarly, an elastic Finger tensor is defined for the trapped chains, which obeys the following equation:

(11–29)

where and are the first invariant of and , respectively, and are defined in the following manner:

(11–30)

(11–31)

In the equation for the trapped chains, the variable quantifies the degree of structural damage (debonding factor), and is the fraction of the initially trapped chains that are debonded from the filler particles during flow. The function is a scaling factor for the relaxation time , and is referred to as the "mobility function". A phenomenological kinetic equation is suggested for :

(11–32)

In Equation 11–32, is the local shear rate (or equivalently, the generalized shear rate) defined as the second invariant of the rate of deformation tensor , while is the generalized shear at which yielding occurs (yielding strain). Also, is a dimensionless time factor, which may delay or accelerate debonding. For the mobility function appearing in the constitutive Equation 11–29 for , the following form is suggested, based on [8]:

(11–33)

where is a power index in the mobility function.

Let us momentarily discard the parameter . The above selection for the mobility function endows the rheological properties with a yielding behavior. When is large (or unbounded), the algebraic term dominates the constitutive equation for , and the solution is expected to be . When is vanishing, becomes governed purely by a transport equation. This may lead to trouble when solving a complex steady flow. Indeed, boundary conditions may become inappropriate when secondary motions (vortices) are encountered, and the system matrix may become singular. Finally, when is small and nonvanishing, the situation is similar to the one involving a very long relaxation time . Based on this, parameter can be understood as the value of the mobility function under no debonding. The original equation is recovered from [8] when . Actually, for nonvanishing , the suggested Equation 11–33 departs from the reference one only by a small amplitude, and at low values of .

Finally, in order to relate the Finger tensors to the corresponding stress tensor, potential functions are required. For and , the following expressions are suggested:

(11–34)

(11–35)

where is the shear modulus, while and are a coefficient and a power index in the potential function, respectively. Also, and .

It is interesting to note that has no effect on the shear viscosity, while it contributes to a decrease of the elongational viscosity. On the other hand, the parameter increases both shear and elongational viscosities. From there, stress contributions from free and trapped chains in Equation 11–24 are given by the following equations, respectively:

(11–36)

(11–37)

where parameter is the initial ratio of free to trapped chains in the system. A vanishing value of indicates that all chains are trapped at rest, while a large value of indicates a system that essentially consists of free chains. These stress components are subsequently incorporated into the momentum equation, and are subjected to appropriate initial and boundary conditions for , and . A comparison of the formulations developed by Leonov (see [31] and [30]) with the extra-stress contributions and that result from Equation 11–36 and Equation 11–37 reveals that a change has been introduced in the latter, which is motivated by the need of obtaining zero stresses at rest. As a matter of fact, the change does not affect the predictions of velocity, swelling, and total stress tensor. It affects the spherical part of the contributions and , which can be compensated via the pressure.

is computed from Equation 11–2. However, it is important to mention that the additional viscosity factor should not vanish when starting a calculation with the Leonov constitutive model. Optionally, an evolution function can be subsequently applied to this parameter.