Approaches to Nonlinear Viscoelastic Behavior

As mentioned earlier, there have been many attempts to develop mathematical models that would accurately represent the nonlinear stress-strain behavior of viscoelastic materials. This section will review a few of these but it is appropriate to note that those discussed are not all inclusive. For example, numerical approaches are most often the method of choice for all nonlinear problems involving viscoelastic materials but these are beyond the scope of this text. In addition, this chapter does not include circumstances of nonlinear behavior involving gross yielding such as the Luder s bands seen in polycarbonate in Fig. 3.7. An effort is made in Chapter 11 to discuss such cases in connection with viscoelastic-plasticity and/or viscoplasticity effects. The nonlinear models discussed here are restricted to a subset of small strain approaches, with an emphasis on the single integral approach developed by Schapery. 

Nonlinear Mechanical Models It is possible to represent nonlinear behavior by introducing nonlinear spring and damper elements into the derivation of differential stress-strain relations. For example, for the four-parameter fluid shown in Fig. 10.5, the spring moduli, damper viscosities and relaxation times are functions of stress, i.e., 

If only the spring moduli are nonlinear, a nonlinear generalized Kelvin model can be represented by. 

Nonlinear Creep Power Law It has been empirically observed that the creep of metals and other materials can be approximated using a creep power law of the form  

For steady state (or secondary) creep of both metals and polymers it is often assumed that n = 1.0. In this form Eq is a fitting parameter and is found by extrapolation of the linear (with time) secondary creep portion of the curve to zero time (Dillard, (1981)). Another form used by Findley (1976) is 

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Kinder, D.F. and Sternstein, S.S. (1976) A path-dependent variable approach to nonlinear viscoelastic behavior. Trans. Soc. Rheol, 20, 119-140. 

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