Hertz's task – the task of depending on the deformation in contact of two elastic bodies from the force with which they are pressed against each other – is one of the classic tasks of the theory of elasticity. In Volume 7 "The Theory of Elasticity" of the Landau and Lifshitz Course, several key paragraphs of the first chapter are given to this task. Interestingly, the approach developed by Landau, unlike the original solution of Hertz (1882), allows us to turn to a more private, but significantly more complex task of analytical calculation of viscous dissipation in the collision of viscous-elastic bodies.
Previously, this task had already been considered in a number of works, but due to the fallacy of approaches, the results of those works were physically implausible. From the main and common to all:
first, the disappearance of dissipation for materials, like rubber, which are close to 1/2 of the Poisson coefficient (which corresponds to the weak compression of the material compared to its malleability of shifting deformations),
secondly, the violation for viscous tensions of Newton's third law in the case of contact of bodies from different material.
In collaboration with Professor Nikolai V. A diamond approach similar to Landau's approach to Hertz's task has been developed for the task of contacting viscous-elastic bodies. As part of this approach, the problem of dynamic contact of viscous-elastic bodies of an arbitrary convex shape was achieved. The approach is mathematically and physically rigorous. The results do not reveal any "non-physical" features. The works are published in The European Physical Journal E and Europhysics Letters (pdf files of works are available below). These works are undoubtedly more private than Hertz's 1882 work, but one can afford some courage to argue that, like Hertz's work, they can be regarded as classic works of theoretical physics.
At present, work in this area is continuing and new interesting results are being prepared for publication.
- D.S. Goldobin, E.A. Susloparov, A.V. Pimenova, and N.V. Brilliantov, Collision of viscoelastic bodies: Rigorous derivation of dissipative force, Eur. Phys. J. E 38, 55 (2015). Pdf
- N.V. Brilliantov, A.V. Pimenova, and D.S. Goldobin, A dissipative force between colliding viselasticco bodies: Rigorous approach, Europhys. Lett. 109, 14005 (2015). Pdf
Prof. Nikolai V. Brilliantov, University of Leicester
About some works of N.V. Diamond in the Russian media:
http://tvkultura.ru/article/show/article_id/141022/http://lenta.ru/news/2009/04/29/enceladus/http://lenta.ru/news/2008/02/06/enceladus/