Constitutive equations for polymer melts and solutions

Journal of Polymer Science: Polymer Physics Edition, 1980

1980

International Journal of Plasticity, 2008

Classical models based on the thermodynamics of irreversible process with internal variables dedicated to the inelastic analysis of metallic structures are modified and then used for modeling the mechanical behavior of polymers. The major difference comes from the expression of the yield criterion. Indeed, a generalized yield criterion, based on the parabolic Drucker and Prager criterion, is proposed including the first invariant of the stress tensor as well as the second invariant and the third invariant of the deviatoric part of the stress tensor. Close agreement between experimental data and yielding predictions is obtained for various polymers loaded under different states of stress. It has been established that the temperature T, the strain rate _ s, the critical molecular mass M c and the degree of crystallinity X c do not affect the parameter m of the proposed yield function. Furthermore, viscoplastic constitutive equations are developed in the framework of the general principles of thermodynamics with internal variables for generalized materials considering only the kinematic hardening rule. Experimental data obtained under different loading conditions are well reproduced by the proposed model. An accurate identification of the model parameters and the introduction of the isotropic hardening variable into the yield function and the drag stress will improve the predictions of the overall mechanical behavior of polymers especially the unloading path.

ISRN Polymer Science, 2013

The K-BKZ constitutive model is now 50 years old. The paper reviews the connections of the model and its variants with continuum mechanics and experiment, presenting an up-to-date recap of research and major findings in the open literature. In the Introduction a historical perspective is given on developments in the last 50 years of the K-BKZ model. Then a section follows on mathematical modeling of polymer flows, including governing equations of flow, rheological constitutive equations (with emphasis on viscoelastic integral constitutive equations of the K-BKZ type), dimensionless numbers, and boundary conditions. The Method of Solution section reviews the major developments of techniques necessary for particle tracking and calculation of the integrals for the viscoelastic stresses in flow problems. Finally, selected examples are given of successful application of the K-BKZ model in polymer flows relevant to rheology.

Journal of Non-Newtonian Fluid Mechanics, 1990

The Hamiltonian formulation of equations in continuum mechanics through Poisson brackets was used in Ref. 1 to develop a constitutive equation for the stress and the order parameter tensor for a polymeric liquid crystal. These equations were shown to reduce to the homogeneous Doi equations as well as to the Leslie-Ericksen-Parodi (LEP) constitutive equations under small deformations [l]. In this paper, these equations are fitted against the non-homogeneous Doi equations through the simulation of the spinodal decomposition of the isotropic state when it is suddenly brought into a parameter region in which it is thermodynamically unstable. Linear stability analysis reveals the wavelength of the most unstable fluctuation as well as its initial growth rate. Results predicted from this theory compare well with the predictions of Doi for the spinodal decomposition using an extended molecular rigid-rod theory in terms of the distribution function.. This completes the development of a generalized constitutive equation for polymeric liquid crystals initiated in Part 1.

Refinements of classical theories for entangled or crosslinked polymeric systems have lead to incommensurable models for rubber networks and polymer melts, contrary to experimental evidence, which suggests a great deal of similarity. Uniaxial elongation and compression data of linear and branched polymer melts as well as of crosslinked rubbers were analyzed with respect to their nonlinear strain measure. This was found to be the result of two contributions: (1) affine orientation of network strands, and (2) isotropic strand extension. Network strand extension is caused by an increasing restriction of lateral movement of polymer chains due to deformation, and is modelled by a molecular stress function f, which in the tube concept of Doi and Edwards is the inverse of the relative tube diame- ter. Up to moderate strains, f 2 is found to be linear in the average stretch for melts as well as for rubbers (the Linear Molecular Stress Function Theory), which corresponds to a constant tube v...

Journal of Applied Mechanics and Technical Physics, 1996

In order to model polymer fluid flows within the framework of continuum mechanics, it is necessary to write a theological state equation that establishes a relationship between the stress tensor for a polymer system and the velocity-gradient tensor. This can be done either by a phenomenological approach [1], generalizing the available experimental data, or by using some model concepts of the structure of polymer materials [2][3][4][5][6][7][8][9][10][11][12][13]. However, both approaches will probably not provide us .with a simple enough rheological constitutive relation suitable for a description of various flows of linear polymer solutions and melts. Therefore, the problem of construction of a succession of rheologieal constitutive relations taking new and more subtle effects into account at each step is of great importance. The success of such a procedure is determined by the selection of an initial approximation and by the rules of transition to subsequent approximations.

Journal of Non-Newtonian Fluid Mechanics, 2004

In our previous publication, we presented a molecular model to describe the dynamics of the interfacial layer between a flowing polymer melt and a die wall. We showed that the ensemble-averaged behavior of polymer molecules adsorbed on the wall could be successfully described in terms of the so-called bond vector probability distribution function (BVPDF). The BVPDF couples the chain orientation and chain stretch on the level of single segment, and thus is an extension of the orientation distribution function of Doi and Edwards introduced for inextensible chains. In this paper, the developed formalism is extended to molecules in the polymer bulk. We show how the well-known Doi and Edwards theory (DE) for inextensible chains based on the orientation distribution function can be naturally extended to include chain stretch and (convective) constraint release (CCR). The final constitutive equation accounts for such mechanisms on polymer chains as reptation, retraction, convection, contour length fluctuations, and (convective) constraint release. It is valid for both linear and non-linear flow regimes. The proposed theory is quantitative, and contains the same input parameters as the original DE model. As an application of the full theory, a simple equation of motion for the stress tensor is derived. Despite the simplicity, its predictions are found to be in good agreement with available experimental data over a wide range of flow regimes and histories. (M.A. Tchesnokov). to be in an excellent agreement with experimental data. The DE model, however, failed to predict other non-linear shear properties, such as the steady-state viscosity or the relaxation of stress after cessation of steady shearing.

Macromolecular Theory and Simulations, 2004

Il ritratto di Giulia Varano bambina, 2021

il ritrAtto di GiuliA dA VArAno bAmbinA di dosso dossi ritrovato e restituito a Camerino 1