Hopkins-Hamming Method

Appendix 4.A — The Hopkins—Hamming Method for the Conversion of G(t) into J(t)... 

In this equation a negative transition term is omitted, which vanishes at large times. The method of Hopkins and Hamming is justified by the fact that, for a description of steady flow, the upper integration limit t in eq. (2.3) can always be chosen so large that the intervals of r, where G (t) differs from zero, and where the transient term of / (t— x) differs from zero, can be separated. This is the reason, why influence of the transition term of J (t— x) disappears in the final result. 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

The convolution integral Eq. (4.73) may be solved nmnerically by the method of Hopkins and Hamming. The method is discussed in Appendix 4.A. Shown in Fig. 4.13 is the comparison of the G i) and curves of a nearly monodisperse polystyrene melt with = 4.69 x 10 obtained at 114.5°C (detailed in Chapter 14). The G t) line shape has more features — meaning different processes are better separated or resolved — than the curve, which are smeared by the con-... 

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