Constitutive equations convected 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... 

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 ... 

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. 

This is the constitutive equation or rheological equation of state for the elastic dumbbell suspensions. It is identical to the upper-convected Maxwell model, eq. 4.3.7. The molecular dynamics have led to a proper (frame-indifferent) time derivative and to a definition... 

Figure 3.11 gives plots of n /irjo versus A.jC that are predicted from two constitutive equations (1) the upper convected Maxwell model, and (2) the Oldroyd three-constant model. It is seen in Figure 3.11 that both models predict values of increasing very rapidly without bound as e increases, in contrast to the experimental results given in Figure 3.10. As a matter of fact, all the expressions summarized in Table 3.3 predict similar elongational behavior, which is considered to be physically unrealistic.

Giesekus (1982) summarized nicely a series of his papers on the formulation of a new class of constitutive equations. The origin of Eq. (3.23) comes from a modification of the upper convected Maxwell model as applied to a dilute polymer solution, namely... 

This is also the integral form of the differential constitutive equation called the upper convected Maxwell model , which is given in the next section. 

Just as there are various possible finite strain tensors, there are various time derivatives that can be used in place of the ordinary derivative of stress in Eq. 10.21 to satisfy the continuum mechanics requirements for a model to be able to describe large, rapid deformations in arbitrary coordinate systems. The derivative that yields a differential model equivalent to Lodge s Eq. 10.6 is the upper convected time derivative (defined in Eq. 11.19), and the resulting model is called the upper-convected Maxwell model. Other possibilities include the lower-convected derivative and the corotational derivative. Furthermore, a weighted-sum of two of these derivatives can be used to formulate a differential constitutive equation for polymeric liquids. In particular, the Gordon-Schowalter convected derivative [7] is defined in this manner. 

The major effect of polymer cooling is that it retards stress relaxation and, as mentioned previously in the chapter, some of the stress remains frozen-in even after the molding has completely solidified. This stress relaxation cannot be predicted using an inelastic constitutive equation (why ) the simplest equation that we can use for the purpose is the upper-convected Maxwell model. In the absence of flow, the use of this model yields... 

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

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]. 

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. 

---