Leonov model

Fig. 12.37 First normal stress difference vs. shear stress, as predicted by the Leonov model. [Reprinted by permission from H. Mavridis and R. N. Shroff, Multilayer Extrusion Experiments and Computer Simulation, Polym. Eng. Sci., 34, 559 (1994).]...

Newtonian model, isothermal Power-law model, nonisothermal Newtonian, nonisothermal, gravity effects included Maxwell and Leonov models, nonisothermal... 

Tervoort and co-workers Model. While the above models make reasonable predictions of the stress-strain behavior in monotonic loading conditions, a main drawback to them is that they use only a single stress-dependent characteristic (relaxation) time. As a consequence, the predicted behavior tends to show a sharp transition between elastic (solid-Hke) and plastic (fluid-like) behavior. However, it is found in practice that all poljnners exhibit behavior consistent with a spectrum of relaxation times and this is clearly going to affect the stress-strain response at constant strain rate. In an effort to address this inconsistency Tervoort and co-workers (40) have developed a modified compressible Leonov model. ... 

R. K. Upadhyay and A. I. Isayev, Elongational flow behavior of polymeric fluids according to the Leonov model, Rheol. Acta 22, 557-568 (1983). 

There have been numerous studies on the film-blowing process. Since the initial thin-shell approximation proposed by Pearson and Petrie [125, 126] with the Newtonian model assumed for deformation, various rheological models have been incorporated in simulations, such as the power-law model [127,128], a crystallization model [129], the Maxwell model [130-133], the Leonov model [133], a viscoplasti-c-elastic model [134], the K-BKZ/PSM model [135-137], and a nonisothermal viscosity model [138]. A complete set of experimental data was reported by Gupta [139] for the Styron 666 polystyrene and by Tas [140] for three different grades of LDPE. 

Differential models obtained by replacing the ordinary time derivative in Eq. 10.21 by one that can describe large, rapid deformations are able to describe some nonlinear viscoelastic phenomena, but only qualitatively. To improve on such models, it is necessary to introduce additional nonlinearity into the equation. In the popular Phan-Thien/Tanner model, the Gordon-Schowalter convected derivative is used, and nonlinearity is introduced by multiplying the stress term by a function of the trace of the stress tensor. The Giesekus and Leonov models are other examples of nonlinear differential models. All of the models mentioned above are described in the monograph by Larson [7j. 

The model is based on an earlier one by the same authors, the compressible Leonov model (59). In this, the behavior is modeled with a single Maxwell... 

A purely viscous non-Newtonian approach was followed by Han and Park (1975b). They used the power-law model and the energy equation, assuming that the effects of crystallization were insignificant. The agreement of this model with experimental data in terms of the bubble radius and thickness as a function of the axial distance for LDPE and HDPE was reported to be reasonable. In terms of viscoelastic models, Luo and Tanner (1985) considered the Leonov model, and Cain and Denn (1988) considered the upper convected Maxwell and Marrucci models in nonisothermal cases of film blowing. In some of the cases analyzed, multiple steady-state solutions were present (see also Problem 9C.2). 

The fitting/predictive capabilities of three models (extended Pom-Pom, PTT-XPP and modified Leonov model) are tested for both, steady as well as transient shear and uniaxial extensional flows of mLLDPE and HDPE. The applicability of these constitutive equations has been investigated from the coextiusion flow modeling point of view. Finally, the FEM and modified Leonov model has been employed for the stress analysis in the coextrusion flow domain and predicted stress fields have been compared with the stress measurements fl om the flow coextrusion visualization cell. 

Leonov model This constitutive equation is based on irreversible thermodynamic arguments resulting from the theory of rubber elasticity [5]. Mathematically it relates the stress to the elastic strain stored in the polymer melt as ... 

Leonov, A. V., Stygar, O. V. (2001). Mathematical modeling of organogenic material biotransformation processes for studying the conditions of water eutrophication in the Caspian Sea surface layer. Water Resources, 28(5), 532-555... 

Acknowledgements The authors wish to thank Prof. A. I. Leonov for his help in the establishing of the theoretical part of this work, Dr. A. N. Prokunin for his valuable assistance in studying the rheological properties of plastisols and helpful discussions of mathematical modelling, and Dr. V. A. Teleshov who has provided the possibility of experimental checking of theoretical predictions. 

Leonov AI (1994) On a self-consistent molecular modelling of linear relaxation phenomena in polymer melts and concentrated solutions. J Rheol 38( 1) 1—11 Liu B, Diinweg B (2003) Translational diffusion of polymer chains with excluded volume and hydrodynamic interactions by Brownian dynamics simulation. J Chem Phys 118(17) 8061-8072... 

Ayzatullin TA, Leonov AV, Shaporenko SN (2003) Mathematical modelling of the formation and evolution of the Black Sea anoxic zone. In Aktualnie prob-lemy okeanologii (Current problems in oceanography). Nauka, Moscow, p 431 (in Russian)... 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

The results set out in 3. show that a fiiction law must make allowance for the remarks made in 3.2 in order to represent fiiction with macroscopic shp in the case of polymer melts. An initial approach was made by Chernyak and Leonov in 1986 [18], and then Leonov in 1990 [20]. They proposed relations for modelling the bellshaped curve with its maximum and minimi minima. It appeared worthwhile to adapt these relations to take into accoimt the existence of a positive stress at rest and the decrease in stress at the wall when the slip velocity increases, for low shp regimes. With given temperature and pressure, these relations are written as follows ... 

Leonov A. I., Rheol. Acta, "A linear model of the stick slip phenomena in... 

Leonov [1994] introduced kinetics of interactions into his rheological equation of state. The new relation can describe systems with a dynamic yield stress, without resorting to a priori introducing the yield stress as a model parameter (as it has been done in earlier models). 

Time dependency also enters into the consideration of the rheological response of any viscoelastic system. In the steady-state testing of such materials as molten polymers, the selected time scale should be sufficiently long for the system to reach equilibrium. Frequently, the required period, t > 10" sec, is comparable to that in thixotropic experiments. More direct distinctions between these two types of flow are the usual lack of elastic effects and larger strain values at equilibrium observed for the thixotropic materials (see Table 7.4). There is a correlation between these two phenomena, and theories of viscoelasticity based on thixotropic models have been formulated by Leonov [1972, 1994]. Inherent to the concept of thixotropy is the yield stress. 

The formulations of (47) and (55) have been criticized by Leonov [L9, Lll], among others, as not being tested for consistency with the second law of thermodynamics. For Newtonian fluids, such testing requires a positive shear viscosity. For a linear viscoelastic material, one may show that the relaxation modulus function must be always positive to satisfy the second law. The requirements for Eqs. (47) and (55) are not so clear. Leonov has sought to develop nonlinear viscoelastic rheological models based on thermodynamic arguments. 

Simhambhatla M, Leonov AI (1995) On the rheological modeling of filled polymers with particle-matrix interactions. Rheol Acta 34 329-338... 

Simhambhatia, M. and Leonov, A.l. (1995) On the rheological modeling of viscoelastic polymer liquids with stable constitutive equations. Rheol. Acta, 34, 259-273. 

---