Constitutive models

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

Bingham plastics are fluids which remain rigid under the application of shear stresses less than a yield stress, Ty, but flow like a. simple Newtonian fluid once the applied shear exceeds this value. Different constitutive models representing this type of fluids were developed by Herschel and Bulkley (1926), Oldroyd (1947) and Casson (1959). 

Depending on the method of analysis, constitutive models of viscoelastic fluids can be formulated as differential or integral equations. 

Outline of a decoupled scheme for the differential constitutive models... 

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

Although the difference in final strength f, integrated through both the actual shock wave and the computational shock wave, will be mitigated by dynamic recovery (saturation) processes, this is still a substantial effect, and one that should not be left to chance. These are very important practical considerations in dealing with path-dependent, micromechanical constitutive models of all kinds. 

Nelson, I., Constitutive Models for Use in Numerical Computations, in Proceedings of the Dynamical Methods in Soil and Rock Mechanics, Vol. 2, Plastic and Long-Term Effects in Soils (edited by Gudehus, G.), A.A. Balkema, Rotterdam, 1978, pp. 45-99. 

Mars, W.V. et al.. Fatigue life analysis of an exhaust mount, in Constitutive Models for Rubber IV, Austrell, K., Ed., Swets Zeiflinger, The Netherlands, 2005, 23. 

Steinwegger, T., Flamm, M., and Weltin, U., A methodology for test time reduction in rubber part testing, in Constitutive Models for Rubber III, Busfield, J.J.C. and Muhr, A.H., Eds., A.A. Balkema, Lisse, The Netherlands, 2003, 27. 

The stress depends on the extent of reaction, p(tf), which progresses with time. However, it is not enough to enter the instantaneous value of p(t ). Needed is some integral over the crosslinking history. The solution of the mutation problem would require a constitutive model for the fading memory functional Gf Zflt, t p(t") which is not yet available. This restricts the applicability of dynamic mechanical experiments to slowly crosslinking systems. 

In the constitutional model of Ugi, rather than molecules, "ensemble of molecules (EM) are used in which the molecules can be either chemically different or identical. Like molecules, an EM has an empirical formula, which is the sum of the empirical formulae of the constiment molecules and describes the collection A of atoms within the EM under consideration. All the EM s which can be formed from A have the same empirical formula . Therefore, an EM(A) consists of one or more molecules which can be obtained from A using each atom which belongs to A only once. Moreover, a FIEM(A) or a family of isomeric EM, is the collection of all EM(A) and it is determined by the empirical formula . On the other hand, a chemical reaction, or a sequence of chemical reactions, is the conversion of an EM into an isomeric EM, and therefore a FIEM contains all EMs which are chemically interconvertible, as far as stoichiometry is concerned. In summary, a FIEM(A) contains, at least in principle, the whole chemistry of the collection A of atoms and since any collection of atoms may be chosen here, Ugi concludes that a theory of FIEM is, in fact, a theory of all chemistry. 

Viswanadhan, V. N., Ghose, A. K., Singh, U. C., and Wendoloski, J. J. (1999) Prediction of solvation free energies of small organic moleucles additive-constitutive models based on molecular fingerprints and atomic constants.. /. Chem. Inf. Comput. Sci. 39, 405-412. 

Bogue,D.C., Masuda,T., Einaga,Y., Onogi,S, A constitutive model for molecular weight and concentration effects in polymer blends. Polymer J. (Japan) 1,563-572 (1970). 

Constitutive Models for Rubber II Conference, Hanover, 10-12th Sept. 2001, Proceedings, Balkema Publishers. 

Besdo D, Ihlemann J, Kingston J G R, Muhr A H. Constitutive Models for Rubber III Conference, London, 15-17lh Sept. 2003, Proceedings, Balkema Publishers, p309. 

The field of transport phenomena is the basis of modeling in polymer processing. This chapter presents the derivation of the balance equations and combines them with constitutive models to allow modeling of polymer processes. The chapter also presents ways to simplify the complex equations in order to model basic systems such as flow in a tube or Hagen-Poiseulle flow, pressure flow between parallel plates, flow between two rotating concentric cylinders or Couette flow, and many more. These simple systems, or combinations of them, can be used to model actual systems in order to gain insight into the processes, and predict pressures, flow rates, rates of deformation, etc. 

This chapter gave an overview of how to simplify complex processes sufficiently to allow the use of analytical models for their analysis and optimization. These models are based on mass, momentum, energy and kinetic balance equations, with simplified constitutive models. At one point, as the complexity and the depth of these models increases by introducing more realistic geometries and conditions, the problems will no longer have an analytical solution, and in many cases become non-linear. This requires the use of numerical techniques which will be covered in the third part of this book, and for the student of polymer processing, perhaps in a more advanced course. 

As a constitutive model for the momentum balance, Stevenson chose a temperature dependent power law model represented by... 

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