Modeling the Viscoelastic Behavior

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). 

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... 

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]. 

FIGURE 40.32 Modeling the viscoelastic behavior of wood with the number of Kelvin elements in series. 

Hookean spring n. A concept visualized as a coil spring whose extension is proportional to the applied load, useful by analogy in modeling the viscoelastic behavior of polymers. Serway RA, Faugh JS, Bennett CV (2005) College physics. Thomas, New York. 

As mentioned above, interfacial films exhibit non-Newtonian flow, which can be treated in the same manner as for dispersions and polymer solutions. The steady-state flow can be described using Bingham plastic models. The viscoelastic behavior can be treated using stress relaxation or strain relaxation (creep) models as well as dynamic (oscillatory) models. The Bingham-fluid model of interfacial rheological behavior (27) assumes the presence of a surface yield stress, cy, i.e.. 

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. 

In order to understand the effects of filler loading and filler-filler interaction strength on the viscoelastic behavior, Chabert et al. [25] proposed two micromechanical models (a self-consistent scheme and a discrete model) to account for the short-range interactions between fillers, which led to a good agreement with the experimental results. The effect of the filler-filler interactions on the viscoelasticity... 

These relationships show that to model the mechanical behavior of a given viscoelastic material we need to know the function(s) describing the time variation E(t), G(t), D(t), J(t) of the moduli or compliances under consideration. 

The use of Cole-Cole plots is not very developed in practice, despite the fact that they open the way for the modeling of the viscoelastic behavior in dynamic as well as in static loading cases (through Laplace transform). By contrast, these plots could be interesting from the fundamental point of view if certain parameters would reveal a clear dependence with the crosslink density. The effects of crosslinking are difficult to detect on the usual viscoelastic properties, except for the variation of the rubbery modulus E0. 

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. 

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... 

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. 

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. 

The mechanical response of viscoelastic materials to mechanical excitation has traditionally been modeled in terms of elastic and viscous components such as springs and dashpots (1-3). The corresponding theory is analogous to the electric circuit theory, which is extensively described in engineering textbooks. In many respects the use of mechanical models plays a didactic role in interpreting the viscoelasticity of materials in the simplest cases. However, it must be emphasized that the representation of the viscoelastic behavior in terms of springs and dashpots does not imply that these elements reflect the molecular mechanisms causing the actual relaxation... 

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). 

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. 

In trying to model the response behavior of a viscoelastic material, the next step is to identify what kind of combination of elastic and viscous behavior is realistic. One approach would be to postulate a linear combination of the two. If we represent the shear modulus by G, the two shear terms could be written as... 

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