Viscoelastic constitutive models

Viscoelasticity has already been introduced in Chapter 1, based on linear viscoelasticity. However, in polymer processing large deformations are imposed on the material, requiring the use of non-linear viscoelastic models. There are two types of general non-linear viscoelastic flow models the differential type and the integral type. 

Differential Viscoelastic Models. Differential models have traditionally been the choice for describing the viscoelastic behavior of polymers when simulating complex flow systems. Many differential viscoelastic models can be described by the general form 

The constants in eqn. (2.73) are defined in Table 2.7 for various viscoelastic models commonly used to simulate polymer flows. 

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

Shearing flows of the convected Jeffreys model. The convected Jeffreys model [6] or Oldroyd s B-fluid [54] is given by, 

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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. 

Solovyov SE, Virkler TL and Scott CE (1999) Rheology of acrylonitrile-butadiene-styrene polymer melts and viscoelastic constitutive models. J Rheol 43 977-90. 

Liu K, Ovaert TC (2011) Poro-viscoelastic constitutive modeling of unconfined creep of hydrogels using finite element analysis with integrated optimization method. J Mech Behav Biomed Mater 4 440-450... 

A linear viscoelastic constitutive model of dilute emulsion viscoelastic properties was proposed by Oldroyd [111, 112]. The model considered low deformation of monodispersed drops of one Newtonian liquid in another, with an interphase. Choi and Schowalter [113] extended their cell model to dilute emulsions with Newtonian matrix and viscoelastic drops under infinitesimally small oscillatory deformation. Oldroyd s model was modified by Palierne [126, 127] for dilute viscoelastic hquids emulsions with polydispersed spherical drops (thus, subject to small deformations) with constant interfacial tension coefficient, Vu, at concentrations below that where the drop-drop interactions start complicating the flow field, that is, < 0.1 ... 

Park S, Schapery R (1997) A viscoelastic constitutive model for particulate composites with growing damage. Int J Solids Struct 34 931-947... 

The main aspects of the nonlinear theory of elasticity are presented. As nonlinear elasticity, and, in particular, hyperelasticity, is such a useful tool in the description of the behavior of carbon black-filled rubber undergoing quasi-static loadings, the main methodologies for describing the behavior of materials subjected to large strains are introduced. Some of the results herein presented will be apphed to nonlinear viscoelastic constitutive models and discussed in subsequent review. 

Lion A, Kardelky C (2004) The Payne effect in finite viscoelasticity constitutive modelling based on Iractional derivatives and intrinsic time scales. Int J Plast 20 1313-1345... 

Bonet J (2001) Large strain viscoelastic constitutive models, hit J Solids Struct 38 2953-2968... 

In the section, a viscoelastic constitutive model for rPET polymer concrete is discussed. To model the mechanical response of polymers is difficult because of resin composition, stress level, temperature sensitivity and other factors. For a composite mixture of recycled polymers, the situation is more complicated than for virgin polymer concrete. Due to these factors, empirical formulae developed from the curve fitting of experimental data are most suitable for predicting the creep response of rPET polymer concrete. 

Equations (1.1) and (1.2) are not closed because of the presence of the extra stress tensor x. Therefore, one more equation is required, which is provided by a viscoelastic constitutive model [35, 49, 79). According to nonequilibritun thermodynamics, the most thermodynamically consistent way to describe the constitutive model is in terms of internal (structural) variables for which separate evolution equations are to be described [49]. The simplest case is when a single, second-order... 

The analytical determination of the gravity load distribution between the frame and the infill wall requires several considerations. First, a part of the gravity loads may be applied onto the RC columns before the construction of the infill walls because these walls could be constructed after the frame has been completed. Second, long-term effects such as concrete and masonry creep, concrete shrinkage, and brick masonry expansion with time due to water absorption can significantly affect the gravity load distribution. While refined finite element models with viscoelastic material properties can be employed for the determination of the gravity load distribution, the increased computational burden of such analyses may not necessarily produce results of increased accuracy due to lack of experimental data to allow the cahbration of multiaxial viscoelastic constitutive models. 

The finite element description of the nonlinear viscoelastic behavior of technical fabric was presented by Klosowski et al. [65]. The technical fabric called Panama used in this model was made of two polyester thread families woven perpendicularly to each other with the 2/2 weave. The long term uniaxial creep laboratory tests in directions were conducted at five different constant stress levels. The dense net model [66] together with the Schapery one-integral viscoelastic constitutive model [67] was assumed for the fabric behavior characterization and the least square method in the Levenberg-Marquardt variant was used for the parameters identification. 

Genieser, L. H., Brown, R. A., Armstrong, R. C. Comparison of mearused centerline stress and velocity fields with predictions of viscoelastic constitutive models. /. Rheol. (2003) 47, pp. 1331-1350... 

In this paper, a true 3D thermal flow solver incorporated with the Giesekus viscoelastic constitutive model is developed to simulate melt flow in the injection molding process. The mold filling of a typical plate part is analyzed to study the flow-induced residual stress. Finally, the comparison between experiment and simulation for an industrial part demonsfiates the capability of the present methodology. 

The goal of this work is to develop a model which accurately predicts variation in shape, temperature, stress, velocity, and ultimately structure for a polymer cast film between the slit die and the chill role. The mathematical model is similar to that used in the work of Sollogoub, et al. (Jrnl. non-Newtonian FI. Mech., 2006). Our approach varies from that effort in the treatment of the free surface and the choice of viscoelastic constitutive model. After describing the mathematical model and the solution procedure, comparisons of simulation results with previous results will be provided. 

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