A pragmatic approach for deriving constitutive equations endowed with pom–pom attributes

Abstract

The most important rheological and mathematical features of the pom–pom model are presently used to compare and improve other constitutive models such as the Giesekus and Phan-Thien–Tanner models. A pragmatic methodology is selected that allows derivation of simple constitutive equations, which are suited to possible software implementation. Alterations to the double convected pom–pom, Phan-Thien–Tanner and Giesekus models are proposed and assessed in rheometric flows by comparing model predictions to experimental data.

Appendices

Appendix

Stress equation for the MDCPP model and interpretation

Although this may bring the reader somewhat beyond the scope of the present paper, it can be instructive to derive the constitutive equation written in terms of extra-stress tensor for one of the suggested models. Presently, we select the MDCPP model as a candidate for this exercise. For this, let us define Σ as follows:

$$ \Sigma = \frac{{3G}} {{1 - \xi }}\Lambda ^{2} S $$

(6)

so that T can be rewritten as

$$ T = \Sigma - \frac{G} {{1 - \xi }}I. $$

(7)

In a few words, the methodology for obtaining the equation for Σ consists, at first, of using Eq. 6 for substituting S as a function of Σ and Λ in Eq. 18 in Table 3, and for substituting Λ as a function of tr(Σ). The subsequent part consists of a careful rewriting of the equation. Eventually, after appropriate simplifications, the constitutive equation for the MDCPP model can be written as a function of Σ as follows:

$$ \begin{array}{*{20}l} {{{\left[ {{\left( {{\text{1 - }}\frac{\xi } {{\text{2}}}} \right)}{\mathop \Sigma \limits^\nabla } + {\left( {\frac{\xi } {2}} \right)}{\mathop \Sigma \limits^\Delta }} \right]}} \hfill} \\ {{ + {\left[ {\frac{{2E}} {{\lambda _{s} }}{\left( {1 - {\sqrt {\frac{{3G}} {{{\left( {1 - \xi } \right)}tr{\left( \Sigma \right)}}}} }} \right)} + \frac{{3GE}} {{\lambda _{b} {\left( {1 - \xi } \right)}tr{\left( \Sigma \right)}}} - \frac{{2\xi }} {{tr{\left( \Sigma \right)}}}{\left( {\Sigma :D} \right)}} \right]}\Sigma = \frac{{EG}} {{\lambda _{b} {\left( {1 - \xi } \right)}}}I} \hfill} \\ \end{array} $$

(8)

where the quantity E is given by:

$$ E = \exp {\left( {\frac{2} {q}{\left( {{\sqrt {\frac{{1 - \xi }} {{3G}}tr{\left( \Sigma \right)} - 1} }} \right)}} \right)}. $$

(9)

From Eq. 8, one can attempt to connect the MDCPP model to the general network theory. Following the guidelines suggested, e.g. by Tanner and Nasseri (2003), we successively identify in Eq. 8 a transport of junctions, a rate of destruction of junctions and a rate of creation of junctions. The function controlling the rate of destruction of junctions is given by the coefficient of the second term on the left-hand side of Eq. 8. Since it is a rate, it has the dimension of a reciprocal time. Similarly, the function governing the rate of creation of junctions is given by the coefficient of I on the right-hand side of Eq. 8; here too, this function has the dimension of a reciprocal time.

A constitutive equation written in terms of Σ (or of extra-stress tensor T) can be of interest from the point of view of the general network theory. However, the corresponding form written as a function of orientation tensor S and stretching variable Λ allows an easier qualitative interpretation of the various mechanisms involved and undergone by the macromolecules of a polymer melt.