The Equations Left Behind

The latest Rheology Bulletin (link not available yet) announced the death of Professor Dr. Joachim Meissner this year.  During graduate school, I often spoke Meissner's name, because of his work in constitutive equations.  In 1972, Arthur Lodge and Meissner used phenomenological arguments to show that the first normal stress difference and shear stress are proportional to each other when a single strain is applied at time zero [1].

This relationship is supported by many experiments and constitutive equations.  During graduate school, I used the above equation to infer the sample thickness in my optical rheometer [2].

In addition to single-step strains, I and others did work on reversing double-step strains, where a second strain is in the opposite direction to the first.  Osaki and co-workers [3] proposed a relationship similar to the equation shown above:

When Venerus and Kahvand published their study of mechanical double-step strain rheology [4], Lodge somehow heard of the work and proved and generalized the second equation for multiple reversing step strains where the absolute value of the strain is constant.  Following this chain of events, I started referring to the second equation as the Osaki-Lodge relationship.  (I always thought this chain of events showed scientific collaboration at its best.)

[1] Lodge, A.S.; Meissner, J. Rheol. Acta 1972, 11, 351-352.

[2] Brown, E.F. et al., Rheol. Acta 1995, 34, 221-234.

[3] Osaki, K. et al. J. Polym. Sci., Part B: Polym Phys. 1981, 19, 517-527.

[4] Venerus, D.C.; Kahvand, H. J. Rheol. 1994, 38, 1297-1315 (with an appendix by Lodge).