Linear Nonlinear viscoelastic behavior

This chapter focuses on the non-linear viscoelastic behavior of rubber composites and nanocomposites. Here, we have discussed about the effect of individual fillers (mineral fillers, nanotubes, carbon nanofillers, fibrous nanofiUers, biofillers, special structured fillers viz. nanorods, nanowires, nanoflowers etc.) on the linear/ nonlinear viscoelastic behavior of rubber composites. Moreover, as this chapter is more concerned on the non-linear viscoelastic behavior, we have also discussed the effect of hybrid fillers on the nonUnear viscoelastic behavior of rubber composites in more detail. 

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. 

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. 

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). 

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. 

The stresses in an adhesive joint depend, once a constitutive model is chosen, on the geometry, boundary conditions, the assumed mechanical properties of the regions involved, and the type and distribution of loads acting on the joint. In practice, most adhesives exhibit, depending on the stress levels, nonlinear-viscoelastic behavior, and the adhetends exhibit elastoplastic behavior. Most theoretical studies conducted to date on the stress analysis of adhesively bonded joints have made simplifying assumptions of linear and elastic and/or viscoelastic behavior in the interest of tracking solutions. 

The present discussion has a twofold objective First, to review the literature in the stress analysis of adhesive joints using the finite-element method. Second, to present a finite-element computational procedure for adhesive joints experiencing two-dimensional deformation and stress fields. The adherends are linear elastic and can undergo large deformations, and the adhesive experiences linear strains but nonlinear viscoelastic behavior. Following these general comments, a review of the literature is presented. The technical discussion given in the subsequent sections comes essentially from the authors research(i 2> conducted for the Oifice of Naval Research. 

This section considers the behavior of polymeric liquids in steady, simple shear flows - the shear-rate dependence of viscosity and the development of differences in normal stress. Also considered in this section is an elastic-recoil phenomenon, called die swell, that is important in melt processing. These properties belong to the realm of nonlinear viscoelastic behavior. In contrast to linear viscoelasticity, neither strain nor strain rate is always small, Boltzmann superposition no longer applies, and, as illustrated in Fig. 3.16, the chains are displaced significantly from their equilibrium conformations. The large-scale organization of the chains (i.e. the physical structure of the liquid, so to speak) is altered by the flow. The effects of finite strain appear, much as they do when a polymer network is deformed appreciably. 

Zhu ZY, Thompscm T, Wang SQ, von Meerwall ED, Halasa A (2005) Investigating linear and nonlinear viscoelastic behavior using model sUica-particle-filled polybutadiene. Macrranole-cules 38 8816-8824... 

Abstract The nonlinear viscoelastic behavior of cured rubber is quite different from that of uncured compound, since the presence of crosslink networks. The factors for the influence of the crosslink networks on the nonlinear viscoelastic behaviors of cured rubbers are very complex and obscure. One of the reasons is that the crosslink networks may be consisted of several different types of networks. However, there are few literatures reporting the nonlinear viscoelastic behaviors of cured mbbers with mutle-networks. We reviewed the literatures dedicated to the topic of the non-linear viscoelasticity of simplest mutle-networks—double-network and summarized the useful information as much as possible in the present paper. Song s transient double-network model, double-network formed by twice curing and the specific crosslink network formed in metal salts of unsaturated carboxylic acids reinforced rubbers are introduced in detail. 

The Quasi-Linear Viscoelastic (QLV) model has proven to be a successful phenomenological model for describing the nonlinear viscoelastic behavior of solids [186-188]. 

Chapter 4 investigates the rheological and the dynamic mechanical properties of rubber nanocomposites filled with spherical nanoparticles, like POSS, titanium dioxide, and nanosilica. Here also the crucial parameter of interfacial interaction in nanocomposite systems under dynamic-mechanical conditions is discussed. After discussing about filled mono-matrix medium in the first three chapters, the next chapter gives information about the nonlinear viscoelastic behavior of rubber-rubber blend composites and nanocomposites with fillers of different particle size. Here in Chap. 5 we can observe a wide discussion about the influence of filler geometry, distribution, size, and filler loading on the dynamic viscoelastic behavior. These specific surface area and the surface structural features of the fillers influence the Payne effect as well. The authors explain the addition of spherical or near-spherical filler particles always increase the level of both the linear and the nonlinear viscoelastic properties whereas the addition of high-aspect-ratio, fiberlike fillers increase the elasticity as well as the viscosity. 

This is known as nonlinear viscoelastic behavior, and it is die subject of the next chtqiter. Here we consider only relatively small strains, such that the relaxation modulus is independent of strain Note that we can also define linear viscoelasticity for a particular... 

In Chapter 4, it was noted that linear viscoelastic behavior is observed only in deformations that are very small or very slow. The response of a polymer to large, rapid deformations is nonlinear, which means that the stress depends on the magnitude, the rate and the kinematics of the deformation. Thus, the Boltzmann superposition principle is no longer valid, and nonlinear viscoelastic behavior cannot be predicted from linear properties. There exists no general model, i.e., no universal constitutive equation or rheological equation of state that describes all nonlinear behavior. The constitutive equations that have been developed are of two basic types empirical continuum models, and those based on a molecular theory. We will briefly describe several examples of each type in this chapter, but since our primary objective is to relate rheological behavior to molecular structure, we will be most interested in models based on molecular phenomena. The most successful molecular models to date are those based on the concept of a molecule in a tube, which was introduced in Chapter 6. We therefore begin this chapter with a brief exposition of how nonlinear phenomena are represented in tube models. A much more complete discussion of these models will be provided in Chapter 11. 

Theoretical efforts have been made to develop/refine the tube model that can consistently desaibe the linear and nonlinear viscoelastic behavior of entangled polymers. The refined tube model(s) and the remaining problem(s) are briefly explained below. 

Viscoelastic materials can follow at least three different behaviors, i.e. linear vis-coleasticity, nonlinear viscoelasticity and anelastic behaviour. 

Measurement of linear viscolelastic properties is a useful way of gaining information about a food s micro structure and how this influences the food s rheological character (Narine and Marangoni, 1999 Gunsekaran and Ak, 2002). However, many processing operations, and mastication, involve large, rapid deformations during which viscoelastic behavior is nonlinear. 

Although attempts to measure and interpret nonlinear behavior are potentially useful, there are few reports in the literature on the measurement of the nonlinear viscoelastic properties of foods. This has been due to a lack of both suitable instrumentation and suitably developed theory nonlinear behavior, the predominant form of which is the exhibition of normal stresses, and a dependence of viscosity on shear rate, is much more complex than linear behavior (Gunasekaran and Ak, 2002). 

---