Nonlinear Viscoelastic Behavior

It has already been emphasized in Section A1 above that marked departures from linear viscoelasticity appear in crystalline polymers at small strains. For tensile stress relaxation of single crystal mats of polyethylene, the ratio of stress to strain decreases more rapidly with time at higher extensions, in the range from e = 0.0003 to 0 003 the degree of nonlinearity increases markedly with decreasing temperature in the range from 40 to 10 C. For polyethylene crystallized in bulk, the temperature dependence of the nonlinearity is in the same direction, but for polyethylene terephthalate it is the opposite. Extensive studies of tensile creep of polypropylene have been made by Turner.  

An example of nonlinear stress relaxation is shown in Fig. 16-17, where the ratio of time-dependent tensile stress to tensile strain is plotted logarithmically against time for different strains for cellulose monofilaments. (In this case the structure is no doubt preoriented.) The differences can be interpreted as due to a decrease in relaxation times with increasing stress, and the curves can be combined approximately into a composite curve by plotting with reduced variables, with a shift factor Os which decreases very rapidly with increasing strain. It is doubtful, how-ever, 2 whether the latter can be entirely related to fractional free volume in crystalline polymers as it is for amorphous polymers (Section Cl of Chapter 15). 

Nonlinearity in creep is associated with severe deviations from the Boltzmann superposition principle in creep recovery. An example of extreme effects in a crystalline polymer is shown in Fig. 16-18 for recovery of polyethylene following partial stress relaxation at constant strain for various times and strain magnitudes. It is clear that recovery is much slower at large strains but is somewhat faster for shorter durations of the initial straining. In general, strains less than 0.01% appear to be required for conformity to the Boltzmann superposition principle in this system.  

If the viscoelastic behavior is nonlinear, stress-strain curves at constant rate of loading or deformation will be so a fortiori, since they can depart from linearity even without this complication. Calculations by Van Holde show that the nonlinearity of tensile creep in nitrocellulose implies a stress-strain curve at constant rate of loading with a sharp change in slope at strains of about 5% which resembles the apparent yield points observed in such experiments on many textile fi-bers. 5 6 

Another consequence of nonlinearity is that the relaxation modulus can no longer be calculated by differentiation of a stess-strain curve at constant rate of strain the tensile analog of equation 59 of Chapter 3 is not applicable. Examples of the 

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FT rheometry is a powerful technique to document the nonlinear viscoelastic behavior of pure polymers as observed when performing large amplitude oscillatory strain (LAOS) experiments. When implemented on appropriate instmments, this test technique can readUy be applied on complex polymer systems, for instance, filled mbber compounds, in order to yield significant and reliable information. Any simple polymer can exhibit nonlinear viscoelastic properties when submitted to sufficiently large strain in such a case the observed behavior is so-called extrinsic... 

Figure 30.14 shows an interesting aspect of RPA-FT experiments, i.e., the capability to quantify the strain sensitivity of materials through parameter B of ht Equation 30.3. As can be seen, curatives addition strongly modifies this aspect of nonlinear viscoelastic behavior, with furthermore a substantial change in strain history effect. Before curatives addition, mn 2 data show very lower-strain... 

Binary fluorides, methods of preparing noble-gas, 77 335-336 Binary heterogeneous polymer blends compliance of, 20 347-348 moduli of, 20 346-347 nonlinear viscoelastic behavior of, 20 348 yield and/or tensile strength of, 20 348-349... 

Accordingly, given the necessity from equilibrium coil dimensions that bt> 1, the shear rate and frequency departures predicted by FENE dumbbells are displaced from each other. Moreover, the displacement increases with chain length. This is a clearly inconsistent with experimental behavior at all levels of concentration, including infinite dilution. Thus, finite extensibility must fail as a general model for the onset of nonlinear viscoelastic behavior in flexible polymer systems. It could, of course, become important in some situations, such as in elongational and shear flows at very high rates of deformation. 

T. A. Tervoort, E. T. J. Klompen, and L. E. Govaert, A Multi-mode Approach to Finite, Three-dimensional Nonlinear Viscoelastic Behavior of Polymer Glasses, J. Rheol., 40, 779 (1996). 

At small stresses and strains, glassy PC exhibits linear viscoelastic behavior. The limit of applicability of the theory of linear viscoelasticity has been investigated by Yannas et al. over the temperature range 23 °C-130 °C. The critical strain at which, within the precision of their measurement, deviations from the linear theory occur has been found to diminish from about 1.2% at 23 °C to about 0.7 % at 130 °C. According to Jansson and Yannas the transition from linear to nonlinear viscoelastic behavior is marked by the onset of significant rotation around backbone bonds. 

As remarked earlier, the nonlinear viscoelastic behavior of entangled wormy micellar solutions is similar to that of entangled flexible polymer molecules. Cates and coworkers (Cates 1990 Spenley et al. 1993, 1996) derived a full constitutive equation for entangled wormy micellar solutions, based on suitably modified reptation ideas. The stress tensor obtained from this theory is (Spenley et al. 1993)... 

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

Linear viscoelasticity is valid only imder conditions where structural changes in the material do not induce strain-dependent modulus. This condition is fulfilled by amorphous polymers. On the other hand, the structural changes associated with the orientation of crystalline polymers and elastomers produce anisotropic mechanical properties. Such polymers, therefore, exhibit nonlinear viscoelastic behavior. 

This equation represents nonlinear viscoelastic behavior. For simplicity of analysis it is often reduced to the form... 

The arterial circulation is a multiply branched network of compliant tubes. The geometry of the network is complex, and the vessels exhibit nonlinear viscoelastic behavior. Flow is pulsatile, and the blood flowing through the network is a suspension of red blood cells and other particles in plasma which exhibits complex non-Newtonian properties. Whereas the development of an exact biomechanical description of arterial hemodynamics is a formidable task, surprisingly useful results can be obtained with greatly simplified models. 

Plazek, 1986 Plazek and Choy, 1989 Plazek and Frund, 1990 Plazek and Chay, 1991) as well as some polybutadienes and fluorinated elastomers (Plazek et al, 1983 Plazek et al., 1988 Plazek and Rosner, 1998). Some nonlinear viscoelastic behavior is discussed. 

Measurements of linear and nonlinear viscoelastic behavior of elastomers have a long history. Instead of reviewing the works done in the past by various workers... 

Because of the complications caused by the stress-induced orientation of clay platelets resulting in different rheological responses, the studies of CPNC flow focus on smaU-amplitude oscillatory shear flow (SAGS). As the discussion on the steady-state flow indicates, there is a great diversity of structures within the CPNC family. Whereas some nanocomposites form strong three-dimensional structures, others do not thus while nonlinear viscoelastic behavior is observed for most CPNCs, some systems can be smdied within the linear regime. 

Zhu, Z., Thompson, T, Wang, S. -Q., von Meerwall, E. D., and Halasa, A., Investigating linear and nonlinear viscoelastic behavior using model silica-particle-fllled polybutadiene. Macromolecules, 38, 8816-8824 (2005). 

For a deeper understanding of the strongly nonlinear viscoelastic behavior of filler reinforced elastomers it is... 

As the stress-strain linearity limit of most thermoplastics and their blends is very low, nonlinear viscoelastic behavior of heterogeneous blends needs to be considered in most cases. The nonlinearity is at least partly ascribed to the fact that the strain-induced expansion of materials with Poisson s ratio smaller than 0.5 markedly enhances the fractional free volume (240). Consequently, the retardation times are perpetually shortened in the course of a tensile creep in proportion to the achieved strain. Thus, knowledge of creep behavior over appropriate intervals of time and stress is of great practical importance. The handling and storage of the compliance curves D (t,a) in a graphical form is impractical, so numerous empirical functions have been proposed (241), eg. 

Nonlinear viscoelastic behavior is found in many molten plastics. Theoretical (67) as well as practical approaches address this issue, including a sliding plate normal-thrust rheometer (68,69). 

Constitutive Description of Polymer Melt Behavior K-BKZ and DE Descriptions. Although there are many nonlinear constitutive models that have been proposed, the focus here is on the K-BKZ model because it is relatively simple in structure, can be related conceptually to finite elasticity descriptions of elastic behavior, and because, in the mind of the current author and others (82), the model captures the major features of nonlinear viscoelastic behavior of polymeric fluids. In addition, the reptation model as proposed by Doi and Edwards provides a molecular basis for understanding the K-BKZ model. The following sections first describe the K-BKZ model, followed by a description of the DE model. 

G. B. McKenna and L. J. Zapas, Nonlinear Viscoelastic Behavior of Poly(methyl methacrylate) in Torsion cZ Rheol. 23, 151-166 (1979). 

H. Lu and W. G. Knauss, The Role of Dilatation in the Nonlinearly Viscoelastic Behavior of Pmma under Multiaxial Stress States Mec/j. Time-Dependent Mails. 2,307—334... 

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