Nonlinear viscoelastic parameters

We may now identify some of the nonlinear viscoelastic parameters that (as mentioned at the start of this paper) are critical to adhesion. The most important property is the yield strength, Oy. After that, there is the property of elongational strengthening, i.e., the increase of force with rate of pulling on the end of a rod or fibre of the polymer and also the temperature coefficient of Oy. 

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... 

Molecular theories, utilizing physically reasonable but approximate molecular models, can be used to specify the stress tensor expressions in nonlinear viscoelastic constitutive equations for polymer melts. These theories, called kinetic theories of polymers, are, of course, much more complex than, say, the kinetic theory of gases. Nevertheless, like the latter, they simplify the complicated physical realities of the substances involved, and we use approximate cartoon representations of macromolecular dynamics to describe the real response of these substances. Because of the relative simplicity of the models, a number of response parameters have to be chosen by trial and error to represent the real response. Unfortunately, such parameters are material specific, and we are unable to predict or specify from them the specific values of the corresponding parameters of other... 

In the solid state deformation, the nonlinear viscoelastic effect is most clearly shown in the yield behavior. The activation volume tensor is a key parameter. In addition to the well known dependence of yield stress on temperature and strain rate, the functional relationships between yield, stress field, and physical aging are presented. 

It is necessary to state more precisely and to clarify the use of the term nonlinear dynamical behavior of filled rubbers. This property should not be confused with the fact that rubbers are highly non-linear elastic materials under static conditions as seen in the typical stress-strain curves. The use of linear viscoelastic parameters, G and G", to describe the behavior of dynamic amplitude dependent rubbers maybe considered paradoxical in itself, because storage and loss modulus are defined only in terms of linear behavior. 

One may use the linear viscoelastic data as a pure rheological characterization, and relate the viscoelastic parameters to some processing or final properties of the material inder study. Furthermore, linear viscoelasticity and nonlinear viscoelasticity are not different fields that would be disconnected in most cases, a linear viscoelastic function (relaxation fimction, memory function or distribution of relaxation times) is used as the kernel of non linear constitutive equations, either of the differential or integral form. That means that if we could define a general nonlinear constitutive equation that would work for all flexible chains, the knowledge of a single linear viscoelastic function would lead to all rheological properties. 

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. 

It is commonplace for rheologists to investigate the linear viscoelastic properties of materials and, indeed, this is certainly the case in the pharmaceutical and related sciences. Bird et al. (24) and Barnes et al. (2) have suggested several reasons for this, including the ability to derive speculative molecular structures of materials from their rheological response in the linear viscoelastic region and, additionally, the ability to relate the parameters derived from the linear viscoelastic response to quality control procedures and, in some instances, to clinical response (7,9,19,25-27). Furthermore, the mathematical principles associated with the linear viscoelastic response are less complex than those for nonlinear viscoelasticity, thus ensuring relatively simple interpretations of results. 

Duct flows of nonnewtonian fluids are described by the governing equations (Eq. 10.24-10.26), by the constitutive equation (Eq. 10.27) with the viscosity defined by one of the models in Table 10.1, or by a linear or nonlinear viscoelastic constitutive equation. To compare the available analytical and experimental results, it is necessary to nondimensionalize the governing equations and the constitutive equations. In the case of newtonian flows, a uniquely defined nondimensional parameter, the Reynolds number, is found. However, a comparable nondimensional parameter for nonnewtonian flow is not uniquely defined because of the different choice of the characteristic viscosity. 

The molecular approach which we will see eventually proved to be most successful in treating negative is based on the work of Doi [23]. Doi noted that the well established phenomenological theories for thermotropes (which he termed TLP for Ericksen, Leslie and Parodi [68]) which is successful in describing many dynamic phenomena in MLC nematics, is limited for polymeric liquid crystals in that it does not predict nonlinear viscoelasticity. Doi s approach determines the phenomenological coefficients from molecular parameters, so that the effects of, for example, molecular weight and concentration can be treated. He considers a single molecule (the test rod ) and notes that as concentration increases, constraints on its motion are imposed by collisions with other rods. This constraint can be modeled as a tube... 

Equation (10) cannot be applied until A, the equivalent relaxation time for the fluid, is known. However, A is defined by the linear Maxwell model, and actual polymer solutions exhibit marked nonlinear viscoelastic properties [5,6,7]. For both fresh and shear degraded solutions of Separan AP 30 polyacrylamide, which exhibit pronounced drag reduction in turbulent flow, Chang and Darby [8] have measured the nonlinear viscosity and first normal stress functions, and Tsai and Darby [6] have reported transient elastic properties of similar solutions, A nonlinear hereditary integral function containing six parameters has been proposed to represent the measured properties [8], The apparent viscosity function predicted by this model is ... 

All previously cited biocompatible features related to fibers and fiber interlacing could be then more precisely controlled. Stmctural parameters could be specifically toned, such as fiber orientation for cell guidance, or even tuned for their related properties, such as fiber orientation for mechanical behavior improvement. The ultimate goal to reproduce anisotropy, inhomogeneity, nonlinearity, viscoelasticity of native tissue while providing an ideal stmctural scaffold for cell culturing is within reach. 

One important issue in dealing with the nonlinear viscoelastic response of materials is the amount of data needed to determine the material parameters... 

Other Constitutive Modei Descriptions. The above work describes a relatively simple way to think of nonlinear viscoelasticity, viz, as a sort of time-dependent elasticity. In solid polymers, it is important to consider compressibility issues that do not exist for the viscoelastic fluids discussed earlier. In this penultimate section of the article, other approaches to nonlinear viscoelasticity are discussed, hopefully not abandoning all simplicity. The development of nonlinear viscoelastic constitutive equations is a very sophisticated field that we will not even attempt to survey completely. One reason is that the most general constitutive equations that are of the multiple integral forms are cumbersome to use in practical applications. Also, the experimental task required to obtain the material parameters for the general constitutive models is fairly daunting. In addition, computationally, these can be difficult to handle, or are very CPU-time intensive. In the next sections, a class of single-integral nonlinear constitutive laws that are referred to as reduced time or material clock-type models is disscused. Where there has been some evaluation of the models, these are examined as well. 

The Schapery Model. One of the earliest models of the nonlinear viscoelastic response of pol5nners to use the concept of a reduced time is due to Schapery (147-149). The model is based on thermodynamic considerations and has a form similar to the Boltzmann superposition principal described previously. The model time dependences, except for the shift factors, are the same as those obtained in the linear response regime. Hence, the model is relatively easy to implement and to determine the relevant material parameters. It results in a generalization of the generalized superposition principal developed by Leaderman (150). 

The strain-clock term (eq. 120) is a function of the entire deformation history. McKenna and Zapas used the strain-clock formalism to describe the torsional response of a PMMA material in two-step strain histories (112). The difficulty arises because the determination of material parameters requires at least the data for both the first and second step responses. Furthermore, McKenna and Zapas also assumed that the clock-form for the torque response and for the normal force response was the same. Their results were consistent with this assumption, as shown in Figure 62. However, that work also indicates that considerable experimental data are required to use the model—a constant issue in nonlinear viscoelasticity. One other interesting thing that came from the work of McKenna and Zapas (112-114) was the verification of equation 60 for the normal force. This is seen in Figure 63, where the normal force response after the half-step history is plotted against the duration of the first step for two different isochrones. As seen, for times beyond 1677 s and for both short and long isochrones, the response is both independent of the duration of the first step and the same as if the material had... 

Plasticity and Viscoplasticity and Other Models. As discussed above, the alternative representation of the nonlinear viscoelastic response of polymers is that of plasticity and viscoplasticity. In some respects, these models could be recast as viscoelastic models and they would be equivalent to some of the models discussed above. However, the perspective that glassy polymers are really fluids and do follow time-temperature superposition is lost with these models. Hence, the physical interpretation of material parameters, in this author s opinion, becomes very qnestionable. Therefore, only references to the major papers on polymer plasticity and viscoplasticity are given (174-177). 

The viscoelasticity can be categorized as either linear or nonlinear, but only the linear viscoelasticity can be described theoretically with uncomplicated mathematics. The fundamental viscoelastic parameters of a linear viscoelastic system do not depend on the magnitude of the stress or strain. Therefore, the linear viscoelastic regime is always used for studying the mechanical properties of viscoelastic blended materials. One of the accepted techniques for investigating the viscoelastic behaviours of natural rubber blended materials is the... 

Analysis of crack growth in viscoelastic media is mainly limited to linear isotropic, homogeneous materials. Schapery(34) proposed the use of parameters similar to the J integral for quasi-static crack growth in a class of nonlinear viscoelastic materials subject to finite strains. 

The non linear viscoelasticity of various particles filled rubber is addressed in range of studies. It is found that the carbon black filled-elastomer exhibit quasi-static and dynamic response of nonlinearity. Hartmann reported a state of stress which is the superposition of a time independent, long-term, response (hyperelastic) and a time dependent, short-term, response in carbon black filled-rubber when loaded with time-dependent external forces. The short term stresses were larger than the long term hyperelastic ones. The authors had done a comparative study for the non linear viscoelastic models undergoing relaxation, creep and hysteresis tests [20-22]. For reproducible and accurate viscoelastic parameters an experimental procedure is developed using an ad hoc nonlinear optimization algorithm. 

Abstract This chapter describes the influence of three-dimensional nanofillers used in elastomers on the nonlinear viscoelastic properties. In particular, this part focuses and investigates the most important three-dimensional nanoparticles, which are used to produce rubber nanocomposites. The rheological and the dynamic mechanical properties of elastomeric polymers, reinforced with spherical nanoparticles, like POSS, titanium dioxide and nanosdica, were described. These (3D) nanofillers in are used polymeric matrices, to create new, improved rubber nanocomposites, and these affect many of the system s parameters (mechanical, chemical, physical) in comparison with conventional composites. The distribution of the nanosized fillers and interaction between nanofUler-nanofiUer and nanofiller-matrix, in nanocomposite systems, is crucial for understanding their behavior under dynamic-mechanical conditions. 

Many of the characteristic parameters, in obtained nanocomposites, were increased in their values by the addition of nanofiller. The viscosity and the storage modulus of PU/MWCNT and PU/Nanoclay systems increased in comparison to the unmodified polymer. In turn, for PU/POSS nanocomposites a strong influence of this type of nanofiller was observed, on glass transition temperature and PU toughness. Only for PU nanocomposites containing MWCNT and nanoclay, nonlinear viscoelastic behavior was observed. Generally, small filler particles maximize the interfacial area and provide great reinforcement. However, for the POSS nanofiller, the authors didn t observe such improvement, because this nanoparticles, in a polyurea matrix, doesn t behave like a conventional nanofiller, but rather like a chemically reactive additive [49],... 

The nonlinear viscoelastic behavior of elastomers is usually related to their inner structure interaction, namely, the interaction between the matrix molecules, the interaction between the matrix molecules and fillers, and the interaction between the fillers. Of course, the effect of characteristics of the inner structure on the nonlinear viscoelastic behavior of elastomers cannot be ignored, since it is also related to portion of the energy dissipated during dynamic deformation. For instance, the filler parameters are important which influence the dynamic properties of rubber compounds, dynamic hysteresis in particular, as well as their temperature... 

The nonlinear viscoelastic behavior of the composites of natural rubber filled with surface-modified nanosilica was studied with reference to silica loading [191]. The effect of temperature on the nonlinear viscoelastic behavior has been investigated. It was observed that Payne effect becomes more pronounced at higher silica loading. The filler characteristics such as particle size, specific surface area, and the surface structural features were found to be the key parameters influencing the Payne effect. A nonlinear decrease in storage modulus with increasing strain was observed for unfilled compounds also. The results reveal that the mechanism includes the breakdown of different networks namely the filler — filler network, the... 

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. 

Consider a nonlinear viscoelastic material which is well modeled by the Schapery approach. Would it be possible to determine all seven (7) material parameters only using creep tests That is, not using recovery (unloading) data or a multiple steps in stress Give a detailed explanation for your answer. 

In oscillatory measurements one carries out two sets of experiments (i) Strain sweep measurements. In this case, the oscillation is fixed (say at 1 Hz) and the viscoelastic parameters are measured as a function of strain amplitude. This allows one to obtain the linear viscoelastic region. In this region all moduli are independent of the appUed strain amplitude and become only a function of time or frequency. This is illustrated in Fig. 3.50, which shows a schematic representation of the variation of G, G and G" with strain amplitude (at a fixed frequency). It can be seen from Fig. 3.49 that G, G and G" remain virtually constant up to a critical strain value, y . This region is the linear viscoelastic region. Above y, G and G start to fall, whereas G" starts to increase. This is the nonlinear region. The value of y may be identified with the minimum strain above which the "structure of the suspension starts to break down (for example breakdown of floes into smaller units and/or breakdown of a structuring agent). 

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