Viscoelastic constitutive behaviour

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

The viscoelastic nature of polymers generally determines rate and temperature dependence of their mechanical properties. At low strain levels, i.e. in a linear regime, this dependence is well described by intrinsic material properties defined within constitutive viscoelastic laws [1]. At high strains, in presence of failure processes, such as yielding or fracture, it is more difficult to establish a constitutive behaviour as well as to define material properties able to intrinsically characterise the failure process and its possible viscoelastic features. 

Relaxation in Nanocomposites. At concentrations above the percolation threshold polymer/nanoparticle interactions dominate the viscoelastic terminal behaviour of polymer nanocomposites. As has been reported for phenoxy based nanocomposites [8], the analysis of tan 5 relaxation at low frequencies constitutes a reliable rheological method to investigate the strength of phenoxy/nanoclay interactions. Moreover, since coordinates ((o)Max (tan 5)Max)) reflect the blocking effect of nanoparticles on polymer chains, the dependence of (o)Max with nanoparticles volume fraction can be used in the percolation equation X=Xq (volume fraction threshold [Pg.69]

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

Both of these models show contributions from the viscosity and the elasticity, and so both these models show viscoelastic behaviour. You can visualise a more complex combination of models possessing more complex constitutive equations and thus able to describe more complex rheological profiles. 

The set of relations (8.27) determines the fluxes as quasi-linear functions of forces. The coefficients in (8.27) are unknown functions of the thermodynamic variables and internal variables. We should pay special attention to the fourth relation in (8.27) which is a relaxation equation for variable The viscoelastic behaviour of the system is determined essentially by the relaxation processes. If the relaxation processes are absent (all the =0), equations (8.27) turn into constitutive equations for a viscous fluid. 

Instead of velocity gradients, displacement gradients can be used in relation (8.38). In this form, relations of the kind (8.38) are established on the basis of the phenomenological theory of so-called simple materials (Coleman and Nolle 1961). To put the theory into practice, function (8.38) should be, for example, represented by an expansion into a series of repeated integrals, so that, in the simplest case, one has the first-order constitutive relation (8.37). Let us note that the first person who used functional relations of form (8.38) for the description of the behaviour of viscoelastic materials was Boltzmann (see Ferry 1980). 

One can see that there are several forms for the representation of the constitutive relation of a viscoelastic liquid. Of course, we ought to say that all the types of constitutive relation we discussed in this section are equivalent. We can use any of them to describe the flow of viscoelastic liquids. However, the description of the flow of a liquid in terms of the internal variables allows one to use additional information, if it is available, about microstructure of the material, and, in fact, appears to be the simplest one for derivation and calculation. We believe that the form, which includes the internal variables, reflects a deeper penetration into the mechanisms of the viscoelastic behaviour of materials. From this point of view, all the representations of deformed material can be unified and classified. 

Notwithstanding the simplifying assumptions in the dynamics of macromolecules, the sets of constitutive relations derived in Section 9.2.1 for polymer systems, are rather cumbersome. Now, it is expedient to employ additional assumptions to obtain reasonable approximations to many-mode constitutive relations. It can be seen that the constitutive equations are valid for the small mode numbers a, in fact, the first few modes determines main contribution to viscoelasticity. The very form of dependence of the dynamical modulus in Fig. 17 in Chapter 6 suggests to try to use the first modes to describe low-frequency viscoelastic behaviour. So, one can reduce the number of modes to minimum, while two cases have to be considered separately. 

Constitutive equations of the Maxwell-Wiechert tjq)e have received a lot of attention as far as their ability to describe the linear viscoelastic behaviour of pol3maer melts is concerned. From a phenomenological point of view [1-4], these equations can be easily understood and derived using the multiple spring-dashpot mechanical analogy leading to the linear equation ... 

Nonlinear viscoelastic behaviour. To see the characteristic features of the constitutive equation (7.195), we approximate V (0 by... 

To study the kinetics of temperature stresses in polymers, to analyse the influence of various factors on the flow of the examined processes, and to model the relaxation behaviour in polymers, a nonlinear constitutive differential equation is used in the paper. This equation was proposed by G.I. Gurevich [1], who called it the nonlinear generalized Maxwell equation out of respect for J. Maxwell s ideas [2] that served as a partial basis for deducing the equation. Total deformation is regarded as the sum of elastic, viscoelastic and temperature deformations ... 

The viscoelastic and viscoplastic nature of polymeric materials that constitutes the composite matrix makes their mechanical behaviour time-or rate-dependent. Polymers are also temperature- and moisture-dependent displaying, in certain cases, large stiffness variations. Physical ageing is also an important issue that is often ignored for simplification purposes. A recent overview concerning these important matters is given elsewhere [9]. 

Hooke s law describes the behaviour of a linear elastic solid and Newton s law that of a linear viscous liquid. A simple constitutive relation for the behaviour of a linear viscoelastic solid is obtained by combining these two laws ... 

For polyacrylamide there are two rheological effects which can be explained in terms of its random coil structure. Firstly, it was discussed above that polyacrylamide is much more sensitive than xanthan to solution salinity and hardness. This is explained by the fact that the salinity causes the molecular chain to collapse, which results in a much smaller molecule and hence in a lower viscosity solution. The second effect which can be explained in terms of the polyacrylamide random coil structure is the viscoelastic behaviour of this polymer. This is shown both in the dynamic oscillatory measurements and in the flow through the stepped capillaries (Chauveteau, 1981). When simple models of random chains are constructed, such as the Rouse model (Rouse, 1953 Bird et al, 1987), the internal structure of these bead and spring models gives rise to a spectrum of relaxation times, Analysis of this situation shows that these relaxation times define response times for the molecule, as indicated in the simple Maxwell model for a viscoelastic fluid discussed above. Thus, because of the internal structure of a flexible coil molecule, one would expect to observe some viscoelastic behaviour. This phenomenon is discussed in much more detail by Bird et al (1987b), in which a range of possible molecular models are discussed and the significance of these to the constitutive relationship between stress and deformation rate and deformation history is elaborated. 

Kontou and Spathis [44] carried out an investigation into the relationship between long-term viscoelasticity and viscoplastic responses of two types of ethylene-vinyl acetate metallocene-catalysed linear low-density polyethylene using DSC, DMTA and tensile testing. A relaxation modulus function with respect to time was obtained from values of relaxation spectra and treated as a material property. This relaxation modulus function was used to describe the corresponding tensile data and a constitutive analysis, which accounts for the viscoelastic path at small strains and the viscoplastic path at high strains, was employed to predict the tensile behaviour of the ethylene polymers (see also [45 9]). 

The traditional discussion of mechanical (spring and dashpot) models and the related topic of differential forms of the constitutive equations will not be included here, but are treated extensively in several older references, Gross (1953), Ferry (1970), Bland (1960) for example. See also Nowacki (1965), Flugge (1967) and Lockett (1972). A consistent development of the theory is possible without these concepts. However, they do provide insights into the nature of viscoelastic behaviour and physically motivate exponential decay models. 

Much research in the last few decades focused on the simulation of LCPs for various processes. Suitable rheological constitutive equations are required for this simulation. Leslie-Ericksen (LE) theory describes the flow behaviour and molecular orientation of many LCPs. LE model is limited to low shear rates and weak molecular distortions. But at high shear rate, the rate of molecular distortions is too fast. Doi and Edwards developed their model to describe the complex dynamics of macromolecules at high shear rate (Doi and Edwards 1978). Doi theory is applicable for lyotropic LCPs of small and moderate concentrations. Due to the complex nature of Doi theory, it is also challenging for simulation. Leonov s continuum theory of weak viscoelastic nematodynamics, developed on the basis of thermodynamics and constitutive relations, consider the nematic viscoelasticity, deformation of molecules as well as evolution of director. 

A simpler and effective approach was given by Schapery [149]. The theory has proven to describe reasonably well the non-linear viscoelastic behaviour for several polymers [150]. The constitutive equation in terms of strain is restricted to small strains by the rmderlying thermodynamic theory. For the unidirectional case is given as... 

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