Long-term viscoelastic response

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

Finally, it is worth mentioning another approach used to describe nonlinear viscoelastic solids nonlinear differential viscoelasticity [49, 178, 179]. This theory has been successfully applied to model finite amplitude waves propagation [180-182]. It is the generalization to the three-dimensional nonlinear case of the rheological element composed by a dashpot in series with a spring. Thus in the simplest case, the stress depends upon the current values of strain and strain rate rally. In this sense, it can account for the nonlinear short-term response and the creep behavior, but it fails to reproduce the long-term material response (e.g., relaxation tests). The so-called Mooney-Rivlin viscoelastic material [183] and the incompressible version of the model proposed by Landau and Lifshitz [184] belraig to this class. 

Kontou and Spathis [44] carried out an investigation into the relationship between long-term viscoelasticity and viscoplastic responses of two types of ethylene-vinyl acetate metallocene-catalysed linear low-density polyethylene using DSC, DMTA and tensile testing. A relaxation modulus function with respect to time was obtained from values of relaxation spectra and treated as a material property. This relaxation modulus function was used to describe the corresponding tensile data and a constitutive analysis, which accounts for the viscoelastic path at small strains and the viscoplastic path at high strains, was employed to predict the tensile behaviour of the ethylene polymers (see also [45 9]). 

It is now well established that the thermomechanical response of glassy polymers and their composites is viscoelastic at temperatures near to, and above the glass transition temperature. Therefore, an accurate long term durability model at elevated temperatures for resins and reinforced plastics (FRP) must necessarily include viscoelastic behavior. This is especially true... 

A comprehensive analytical model for predicting long term durability of resins and of fibre reinforced plastics (FRP) taking into account viscoelastic/viscoplastic creep, hygrothermal effects and the effects of physical and chemical aging on polymer response has been presented. An analytical tool consisting of a specialized test-bed finite element code, NOVA-3D, was used for the solution of complex stress analysis problems, including interactions between non-linear material constitutive behavior and environmental effects. 

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]. 

In mathematical models of healthy human joints, cartilage is often represented as a single-phase, elastic material with homogeneous and isotropic properties. This approximation is valid, provided that only the short-term response of the tissue is of interest when cartilage is loaded for 1 to 5 seconds, its response is more or less elastic (Hayes and Bodine, 1978 Hori and Mockros, 1976 Mak, 1986). In the long term, however, say more than 1 minute, the response of the tissue is dominated by the nonlinear, viscoelastic properties of creep and stress relaxation (Hayes and Mockros, 1971 Mow et al., 1984). 

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... 

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. 

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. 

Polymer properties exhibit time-dependent behavior, which is dependent on the test conditions and polymer type. Figure 1.7 shows a typical viscoelastic response of a polymer to changes in testing rate or temperature. Increases in testing rate or decreases in temperature cause the material to appear more rigid, while an increase in temperature or decrease in rate will cause the material to appear softer. This time-dependent behavior can also result in long-term effects such as stress relaxation or creep. These two time-dependent behaviors are shown in Fig. 1.8. Under a fixed displacement, the stress on the material will decrease over time, and this is called stress relaxation. This behavior can be modeled nsing a... 

Ductile deformation requires an adequate flexibility of polymer chain segments in order to ensure plastic flow on the molecular level. It has been long known that macromoleculai- chain mobility is a crucial factor decisive for either brittle or ductile behavior of a polymer [93-95]. An increase in the yield stress of a polymer with a decrease of the temperature is caused by the decrease of macromoleculai chain mobility, and vice versa the yield stress can serve as a qualitative measure of macromolecular chain mobility. It was shown that the temperature and strain rate dependencies of the yield stress are described in terms of relaxation processes, similarly as in linear viscoelasticity. Also, the kinetic elements taking pai-t in yielding and in viscoelastic response of a polymer are similar segments of chains, part of crystallites, fragments of amorphous phase. However, in crystalline polymei-s above their glass transition temperature the yield stress is determined by the yield stress required for crystal deformation... 

Another interesting aspect of linear viscoelasticity is that it can be extended to enable predictions at different temperatures. Further, the long-term, time-dependent response at a given temperature can be predicted based on material characterization at several temperatures. The approach to developing these predictions is based on a time-temperature superposition principle [20], which has been shown to work well in a restricted temperature range as long as the requirement of infinitesimal strains is satisfied. 

The creep response according to the Burgers model in Fig. 34.4 covers all elementary aspects of time-dependant viscoelastic behavior including instantaneous elastic strain, secondary steady state creep in the long-term area, and a delayed elastic strain transition behavior that can be, for example, fitted to experimental data according to the choice of the t/i, Ei, ijz, and Ez parameters. 

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