Models of Viscoelastic Behavior

Numata and Kinjo (52) have shown rubber-modified isocyanurate-oxazolidone resins may be effectively modified with carboxyl-reactive nitrile liquids. The viscoelastic behavior of models using a polyglycidyl ether of phenol-formaldehyde novolac resin and di-phenylmethane-4,4 -diisocyanate is discussed. Such resins have suggested utility in thin films as electrical varnishes. 

We have relied heavily on the use of models in discussing the viscoelastic behavior of polymers in the transient and dynamic experiments of the last few sections. The models were mechanical, however, and while they provide a way for understanding the phenomena involved, they do not explicitly relate these phenomena to molecular characteristics. To establish this connection is the objective of this section. 

The dynamic viscoelasticity of particulate gels of silicone gel and lightly doped poly-p-phenylene (PPP) particles has been studied under ac excitation [55]. The influence of the dielectric constant of the PPP particles has been investigated in detail. It is well known that the dielectric constant varies with the frequency of the applied field, the content of doping, or the measured temperature. In Fig. 11 is displayed the relationship between an increase in shear modulus induced by ac excitation of 0.4kV/mm and the dielectric constant of PPP particles, which was varied by changing the frequency of the applied field. AG increases with s2 and then reaches a constant value. Although the composite gel of PPP particles has dc conductivity, the viscoelastic behavior of the gel in an electric field is qualitatively explained by the model in Sect. 4.2.1, in which the effect of dc conductivity is neglected. 

The combination of spring and dashpot in series is called the Maxwell model, and was in fact first investigated by the same Maxwell famous for his work on gases and molecular statistics. It is used to model the viscoelastic behavior of uncross-linked polymers. The spring is used to describe the recoverability of the chains that are elongated, and the dashpot the permanent deformation or creep (resulting from the uncross-linked chains irreversibly sliding by one another). 

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... 

Integral viscoelastic models. Integral models with a memory function have been widely used to describe the viscoelastic behavior of polymers and to interpret their rheological measurements [37, 41, 43], In general one can write the single integral model as... 

There are several models to describe the viscoelastic behavior of different materials. Maxwell model, Kelvin-Voigt model, Standard Linear Solid model and Generalized Maxwell models are the most frequently applied. 

Models of mechanical behavior of tissues have been difficult to develop primarily because of the time dependence of the viscoelasticity. Analysis of viscoelastic behavior of even simple polymers at strains greater than a few percent is not accurate. In addition, most tissues undergo strains larger than a few percent, which makes the analysis require an understanding of the elongation behavior. In this chapter we focus on using modeling techniques to analyze the physical basis for determination of the tensile behavior of ECMs found in connective tissue. 

Figure 8.1. Diagram showing Maxwell mechanical model of viscoelastic behavior of connective tissues. In this model an elastic element (spring) with a stiffness Em is in series with a viscous element (dashpot) with viscosity T m. This model is used to represent time dependent relaxation of stress in a specimen bold of fixed length.

In an earlier section, we have shown that the viscoelastic behavior of homogeneous block copolymers can be treated by the modified Rouse-Bueche-Zimm model. In addition, the Time-Temperature Superposition Principle has also been found to be valid for these systems. However, if the block copolymer shows microphase separation, these conclusions no longer apply. The basic tenet of the Time-Temperature Superposition Principle is valid only if all of the relaxation mechanisms are affected by temperature in the same manner. Materials obeying this Principle are said to be thermorheologically simple. In other words, relaxation times at one temperature are related to the corresponding relaxation times at a reference temperature by a constant ratio (the shift factor). For... 

On a global scale, the linear viscoelastic behavior of the polymer chains in the nanocomposites, as detected by conventional rheometry, is dramatically altered when the chains are tethered to the surface of the silicate or are in close proximity to the silicate layers as in intercalated nanocomposites. Some of these systems show close analogies to other intrinsically anisotropic materials such as block copolymers and smectic liquid crystalline polymers and provide model systems to understand the dynamics of polymer brushes. Finally, the polymer melt-brushes exhibit intriguing non-linear viscoelastic behavior, which shows strainhardening with a characteric critical strain amplitude that is only a function of the interlayer distance. These results provide complementary information to that obtained for solution brushes using the SFA, and are attributed to chain stretching associated with the space-filling requirements of a melt brush. 

Similar to Wu and Liao (75), Wu et al. (74) used a DMA (Model -242C, NETZSCH Co.) and a rheometer (HAAKE RS600, Thermo Electron Co.) to evaluate the viscoelastic behavior of the carboxylic-acid-functionalized MWCNTs reinforced PCL/PLA blend. Using DMA, it was observed that, with the increase of MWCNT loading, the Tg of the blend system shifted to higher temperatures. This agrees with the results obtained from the other studies discussed above and indicates the MWCNTs are compatible with the blend. The viscoelastic properties observed via rheometer were similar to those by Wu et al. (73), discussed above. 

Takayanagi, M., Harima, H., and Iwata, Y. (1963). Viscoelastic behavior of polymer blends and its eomparison with model experiments. Mem. Faculty Eng. Kyushu Univ. 23, 1-13. 

The damping of the composite structure will be affected by the thicknesses of the various layers, stiffnesses of the base and top plates, and the viscoelastic properties of the constrained layer (12). In the present instance (13), it was desired to develop a a broad-band material to damp a model composite structure consisting of a 2.54 cm. base plate (H ), 0.079 cm. polymer layer (H ), and 0.159 cm. cover (Ho)fi oase and cover were composed oi brass with a modulus of 10 Pa. In this instance, the only variable was the viscoelastic behavior of the polymer layer. A temperature range from 0 to 20 degrees Centigrade and a frequency range from 100 Hz to 10 kHz were desired. 

A simple and flexible modeling scheme to quantitatively characterize complete viscoelastic properties of pol3nmers is readily useable with FTMA data. It contains the essential features of the frequency-temperature equivalence of the viscoelastic behaviors of pol3nmers and is easily adapted to modern computer systems. 

As is well known, springs and dashpots represent, respectively, ideal elastic and viscous responses to step stress perturbations. In a similar way, a combination of the two can be used to describe the viscoelastic behavior of materials. The Maxwell model, a spring in series with a dashpot, is the more immediate idealization of this behavior (Fig. 10.1). 

We can generalize the analogy by considering the viscoelastic materials as a continuum where the theory of transmission lines can be applied. In this way, a continuous distribution of passive elements such as springs and dash-pots can be used to model the viscoelastic behavior of materials. Thus the relevant equations for a mechanical transmission line can be written following the same patterns as those in electrical transmission lines. By representing the impedance and admittance per unit of length by g and j respectively, one has... 

The viscoelastic properties of the crystalline zones are significantly different from those of the amorphous phase, and consequently semicrystalline polymers may be considered to be made up of two phases each with its own viscoelastic properties. The best known model to study the viscoelastic behavior of polymers was developed for copolymers as ABS (acrylonitrile-butadiene-styrene triblock copolymer). In this system, spheres of rubber are immersed in a glassy matrix. Two cases can be considered. If the stress is uniform in a polyphase, the contribution of the phases to the complex tensile compliance should be additive. However, if the strain is uniform, then the contribution of the polyphases to the complex modulus is additive. The... 

David and Augsburger (63) studied the decay of compressional forces for a variety of excipients, compressed with flat-faced punches on a Stokes rotary press. They found that initial compressive force could be subject to a fairly rapid decay and that this rate was dependent on the deformation behavior of the excipient for the materials studied, they found that maximum loss in compression force was for compressible starch and MCC, which was followed by compressible sugar and DCP. This was attributed to differences in the extent of plastic flow. The decay curves were analyzed using the Maxwell model of viscoelastic behavior. Maxwell model implies first order decay of compression force. 

Several researchers reported viscoelastic behavior of yeast suspensions. Labuza et al. [9] reported shear-thinning behavior of baker s yeast (S. cerevisiae) in the range of 1 to 100 reciprocal seconds at yeast concentrations above 10.5% (w/w). The power law model was successfully applied. More recently, Mancini and Moresi [10] also measured the rheological properties of baker s yeast using different rheometers in the concentration range of 25 to 200 g dm. While the Haake rotational viscometer confirmed Labuza s results on the pseudoplastic character of yeast suspension, the dynamic stress rheometer revealed definitive Newtonian behavior. This discrepancy was attributed to the lower sensitivity of Haake viscometer in the range of viscosity tested (1.5 to 12 mPa s). Speers et al. [11] used a controlled shear-rate rheometer with a cone-and-plate system to measure viscosity of... 

The linear viscoelastic behavior of the pure polymer and blends has already been described quantitatively by using models of molecular dynamics based on the reptation concept [12]. To describe the rheological behavior of the copolymers in this study, we have selected and extended the analytical approach of Be-nallal et al. [13], who describe the relaxation function G(t) of Hnear homopolymer melts as the sum of four independent relaxation processes [Eq. (1)]. Each term describes the relaxation domains extending from the lowest frequencies (Gc(t)) to the highest frequencies (Ghf( )), and is well defined for homopolymers in Ref [13]. 

This model indicates that the modulus of the polymer is the result of the individual moduli of each element, E, and the stress depends on the relaxation times, X, of each element. This equation is a better approximation to the behavior of polymers. To model the viscoelastic behavior of polymers, other models have been proposed, such as the Kelvin model [12]. 

The four-parameter model provides a crude quahtative representation of the phenomena generally observed with viscoelastie materials instantaneous elastie strain, retarded elastic strain, viscous flow, instantaneous elastie reeovery, retarded elastie reeovery, and plastic deformation (permanent set). Also, the model parameters ean be assoeiated with various molecular mechanisms responsible for the viscoelastic behavior of linear amorphous polymers under creep conditions. The analogies to the moleeular mechanism can be made as follows. 

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