The Schapery Equation

Using irreversible thermodynamic (or energy) descriptions of the state of a viscoelastic material subjected to external loads, R. A. Schapery (1964, 1966) developed the following single-integral representation for strains due to a variable stress input. 

The Schapery method given by Eqs. 10.25 and 10.26 is a mathematical definition of a time-stress-superposition-principle or TSSP that is analogous to the TTSP. Later it will be shown how to obtain stress dependent compliance and modulus master curves from experimental data using TSSP much in the same manner as temperature dependent master curves were determined from experimental data using the TTSP. 

An analogous equation for stress under a variable input of strain was also developed by Schapery and is given by, 

It should be noted that the Boltzman superposition integral for linear viscoelasticity is recovered in Eq. 10.25 if the nonlinear parameters are each identically equal to one, i.e.. 

Further, if all parameters except are unity Knauss s free volume model (Knauss and Emri, 1981) is recovered in which. 

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Polymeric materials have relatively large thermal expansion. However, by incorporating fillers of low a in typical plastics, it is possible to produce a composite having a value of a only one-fifth of the unfilled plastics. Recently the thermal expansivity of a number of in situ composites of polymer liquid crystals and engineering plastics has been studied [14,16, 98, 99]. Choy et al [99] have attempted to correlate the thermal expansivity of a blend with those of its constituents using the Schapery equation for continuous fiber reinforced composites [100] as the PLC fibrils in blends studied are essentially continuous at the draw ratio of 2 = 15. Other authors [14,99] observed that the Takayanagi model [101] explains the thermal expansion. 

The Schapery Equation for a Two Step Stress Input Determination of the material parameters necessary for the application of the Schapery Equation are best done by using creep-recovery data and will be demonstrated in a later section. Toward that end, we develop the specific form for the Schapery equation with a simple two-step load. In this section, we assume a general two step stress distribution such that. 

Using the nonlinear creep and creep recovery data given below, find the seven material parameters (Dg, Dj, n, go, gi, g2 aj needed for representation by the Schapery equations. 

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