Prediction viscoelastic response

The response of this model to creep, relaxation and recovery situations is the sum of the effects described for the previous two models and is illustrated in Fig. 2.39. It can be seen that although the exponential responses predicted in these models are not a true representation of the complex viscoelastic response of polymeric materials, the overall picture is, for many purposes, an acceptable approximation to the actual behaviour. As more and more elements are added to the model then the simulation becomes better but the mathematics become complex. 

The reptation model for polymer diffusion would predict that the thickness of the gel phase reflects the dynamics of disentanglement. The important factors here are chain length, solvent quality and temperature since they affect the dimensions of the polymer coils in the gel phase. The precursor phase, on the other hand, depends upon solvency and temperature only through the osmotic force it can generate in the system and the viscoelastic response of the system in the region of the front. These factors should be independent of the PMMA molecular weight. 

We have developed the idea that we can describe linear viscoelastic materials by a sum of Maxwell models. These models are the most appropriate for describing the response of a body to an applied strain. The same ideas apply to a sum of Kelvin models, which are more appropriately applied to stress controlled experiments. A combination of these models enables us to predict the results of different experiments. If we were able to predict the form of the model from the chemical constituents of the system we could predict all the viscoelastic responses in shear. We know that when a strain is applied to a viscoelastic material the molecules and particles that form the system gradual diffuse to relax the applied strain. For example, consider a solution of polymer... 

It appears that the formal theories are not sufficiently sensitive to structure to be of much help in dealing with linear viscoelastic response Williams analysis is the most complete theory available, and yet even here a dimensional analysis is required to find a form for the pair correlation function. Moreover, molecular weight dependence in the resulting viscosity expression [Eq. (6.11)] is much too weak to represent behavior even at moderate concentrations. Williams suggests that the combination of variables in Eq. (6.11) may furnish theoretical support correlations of the form tj0 = f c rjj) at moderate concentrations (cf. Section 5). However the weakness of the predicted dependence compared to experiment and the somewhat arbitrary nature of the dimensional analysis makes the suggestion rather questionable. 

Figure 4 depicts the imaginary part of the frequency-dependent viscosity which clearly demonstrates the bimodality of the viscoelastic response. In the same figure the prediction from the Maxwell s relation have also been plotted. In the latter the relaxation time xs is calculated by the well-known... 

But it is difficult to apply zero shear predictions to measurements that have been performed at low, but nonzero, shear rates. Neither s nor u can be decisively determined at a fixed frequency near the gel point the only way to truly obtain s and u exponents from experiments is through the use of a theory capable of predicting the entire frequency dependence of the viscoelastic response. 

The master curves and shift factors of transient and dynamic linear viscoelastic responses are calculated for linear, semi-crystalline, and cross-linked polymers. The transition from a WLF dependence to an Arrhenius temperature dependence of the shift factor in the vicinity of Tg is predicted and is related to the temperature dependence of physical aging rate. 

With polymers, complications may potentially arise due to the material viscoelastic response. For glassy amorphous polymers tested far below their glass transition temperature, such viscoelastic effects were not found, however, to induce a significant departure from this theoretical prediction of the boundary between partial slip and gross slip conditions [56]. 

Analysis of these effects is difficult and time consuming. Much recent work has utilized two-dimensional, finite-difference computer codes which require as input extensive material properties, e.g., yield and failure criteria, and constitutive laws. These codes solve the equations of motion for boundary conditions corresponding to given impact geometry and velocities. They have been widely and successfully used to predict the response of metals to high rate impact (2), but extension of this technique to polymeric materials has not been totally successful, partly because of the necessity to incorporate rate effects into the material properties. In this work we examined the strain rate and temperature sensitivity of the yield and fracture behavior of a series of rubber-modified acrylic materials. These materials have commercial and military importance for impact protection since as much as a twofold improvement in high rate impact resistance can be achieved with the proper rubber content. The objective of the study was to develop rate-sensitive yield and failure criteria in a form which could be incorporated into the computer codes. Other material properties (such as the influence of a hydrostatic pressure component on yield and failure and the relaxation spectra necessary to define viscoelastic wave propagation) are necssary before the material description is complete, but these areas will be left for later papers. 

Doi and Edwards (1978a, 1979, 1986) developed a constitutive equation for entangled polymeric fluids that combines the linear viscoelastic response predicted by de Gennes... 

Having discussed the viscoelastic responses of simple mechanical models, we may now consider molecular theories. In this treatment it will be shown that the results of molecular theories can, in fact, be couched in terms of the mechanical models already presented. The molecular theories predict the distribution of relaxation times and partial moduli associated with each relaxation time (r/s and Els for all z s), which we treated as unknowns or parameters in the previous discussion. Thus, although molecular theories are not based on mechanical models, the results of these treatments may be presented in terms of the parameters of these models. Since, as we have already shown, it is possible to develop expressions giving the viscoelastic responses of the models to various types of deformations, the predictions of the molecular theories are obtainable through the known responses of these models. 

In many applications, plastic parts carry reasonably constant mechanical loads over periods up to few years. The polymer will creep during the lifetime of the part. At moderate load levels, long-term prediction of creep from short-term tests is possible, because the viscoelastic response of polymers (creep, stress relaxation) measured at different temperatures superimpose when shifted along the time axis [24]. 

The prediction of the long-term viscoelastic response of a polymer part subjected to any given temperature history is possible through the integration of... 

Transient Response Creep. The creep behavior of the polsrmeric fluid in the nonlinear viscoelastic regime has some different features from what were found with the linear response regime. First, there are no ready means of relating the creep compliance to the relaxation modulus as was done in the linear viscoelastic case. In fact, the relationship between the relaxation properties and the creep properties depends entirely on the exact constitutive relationship chosen for the response of the material, and numerical inversion of the specific constitutive law is ordinarily necessary to predict creep response from the relaxation... 

At this point the reptation theory makes some strong predictions about the viscoelastic response in the linear regime, viz, the viscosity varies as and the ratio of Js°/Gn = 6/5 = 1.2. Note that the molecular weight dependence of the viscosity has already been discussed above, and recall that, experimentally, the viscosity varies as N -. In addition, the ratio is observed experimentally... 

G. C. Papanicolaou, S. P. Zaoutsos, and A. H. Cardon, Prediction of the Nonlinear Viscoelastic Response of Unidirectional Fiber Composites Compos. Sci. Technol. 59, 1311-1319 (1999). 

A further set of tests was conducted in order to evaluate the accuracy of the finite-element code for the case where creep is followed by creep recovery. A qualitative depiction of the loading and the resulting creep strain is given in Figure 11. Rochefort and Brinson O) presented experimental data and analytical predictions on the creep and creep recovery characteristics of FM-73 adhesive at constant temperature. The Schapery parameters necessary to characterize the viscoelastic response of FM-73 at a fixed temperature of 30 °C are obtained by applying a least-squares curve fit to the data presented in Reference 50. The resulting analytical expressions for the creep compliance function D(i ), the shift function and the nonlinear parameters go> 82 presented in Table 4. 

The theory has only a single adjustable parameter, which corresponds to the Rouse time (the characteristic relaxation time for an unconfined chain) of the polymer, and it does a quite reasonable job of predicting the hnear viscoelastic response and the transient and steady-state shear and normal stresses in simple shear, ft is not as good as more complex tube-based models hke the pom-pom model, and it cannot be used for nonviscometric flows because of the absence of a continuum representation, but it contains structural details and is very useful for providing insight into the mechanics of slip. 

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation (http //en.wikipedia.org/wiki/Visco-elasticity). Linear viscoelastic behavior is exhibited by a material when it is subjected to a very small or very slow deformation. So when a viscoelastic material is subjected to a deformation that is neither very small nor very slow, its behavior in no longer linear, and there is no universal rheological constitutive equation that can predict the response of the material to such a deformation [18]. Nonhnear viscoelastic behavior is more important than linear properties of mbber/polymer nanocomposites as the industrial processing of viscoelastic materials (mbbers/polymers) always involves large and rapid deformations in which the behavior is nonlinear. 

The effects of a number of environmental factors on viscoelastic material properties can be represented by a time shift and thus a shift factor. In Chapter 10, a time shift associated with stress nonlinearities, or a time-stress-superposition-principle (TSSP), is discussed in detail both from an analytical and an experimental point of view. A time scale shift associated with moisture (or a time-moisture-superposition-principle) is also discussed briefly in Chapter 10. Further, a time scale shift associated with several environmental variables simultaneously leading to a time scale shift surface is briefly mentioned. Other examples of possible time scale shifts associated with physical and chemical aging are discussed in a later section in this chapter. These cases where the shift factor relationships are known enables the constitutive law to be written similar to Eq. 7.53 with effective times defined as in Eq. 7.54 but with new shift factor functions. This approach is quite powerful and enables long-term predictions of viscoelastic response in changing environments. 

Recently, Matadi Boumbimba et al. [12] proposed a temperature- and frequency-dependent version of the rule of mixtures to describe the viscoelastic response, in terms of storage modulus, of PMMA/Cloisite 20A and SOB. In the present work, to predict the effective viscoelastic response of polymer-based nanocomposites, the elastic-viscoelastic correspondence principle [11] is applied to our micromechanical model. The two implicit equations (5) become ... 

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