The Schapery Single-Integral Nonlinear Model

The Schapery single integral approach (1964, 1969) is an outgrowth of the irreversible thermodynamic procedures developed by Biot, and others and is likely the most widely used technique to represent the nonlinear time-dependent behavior of polymers. The thermodynamic derivation of the fundamental equations needed to represent data is beyond the scope of this text but an excellent description of the original derivation is given by Hiel, et al. (1984). Schapery in 1997 also provides an updated mathematical approach that includes viscoplasticity effects. The purpose here is to introduce the method as a means of representing polymer data and provide a basic understanding of how to obtain the necessary material parameters from experiments. The development of equations here closely follows the description of Schapery (1969) and Lou and Schapery (1971). 

It is important to point out that the reason to develop a relatively simple and easy to use single integral method is not only to determine the necessary material parameters more easily, but to have a method that can be used with more ease and confidence in solving nonlinear boundary value problems to obtain stress, strain and displacement distributions for engineering design. This of necessity entails having a modified superposition approach as well as use of the time-shift principles discussed in Chapter 7. 

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