## 3D models of pebble abrasion and the change of equilibrium class due to abrasion – running

One of the central questions of sendimentolgy is the geometry of pebbles and its origin. The problem can be investigated by the tools of applied mechanics, especially by models describing statics of rigid bodies and the geometry of convex bodies. These tools lead to new methods applicable in geology. In our work we study how to make an informative classification system based on mechanical properties (equilibrium points). Furthermore, we investigate the abrasional process and the connection between observed morphology, the parameters of the abrasion and the geological locality.

Abrasion of one pebble is modeled by the partial differential equation by Bloore [1]. In the literature there are several papers investigating each coefficient (mean curvature, Gaussian curvature, uniform term) of the equation separately, but just a very little is known about the solutions of the general form. Numerical simulation of this equation needs a high computational capacity, either we use the level set method, which is the general tool for simulating geometric PDEs, or we choose some other approach for the discretization of the PDE.

In our earlier work [2] we pointed out that a stochastic method, called chopping algorithm might be used to model the physical process. This method does not require a direct spatial discretization. In [2], based on plenty of numerical simulations we showed, that the bifurcation diagram of the two dimensional abrasion computed by the direct method and by the stochastic method is similar. Due to the high computational need a finer comparison or the three dimensional case both require the super computing environment.

This project aims to model abrasion of one pebble in three dimension by a direct discretization of the Bloore equation using the level-set and/or the finite element method and by the chopping algorithm. Our final goal is to compare the shape evolution from the direct method and the stochastic algorithm. To reach our goal we need a large number of numerical simulations. We also propose to compare our numerical outcomes to experimental results. Earlier [3,4] we derived a classification system of pebbles based on the number of their equilibrium points. By our simulations we also would like to investigate the change of the equilibrium class due to abrasion.

[1] F. J. Bloore, The Shape of Pebbles, Mathematical Geology 9, (1977) 113-122.

[2] G. Domokos, A.A. Sipos, P. Várkonyi, Continuous and discrete models for abrasion processes,

Periodica Polytechnica Architecture 40, (2009) 3-8.

[3] P.L. Várkonyi, G. Domokos, Static equilibria of rigid bodies: dice, pebbles and the Poincaré-Hopf Theorem J. Nonlinear Sci. 16, (2006) 255-281.

[4] T. Szabó, G. Domokos: A new classification system for pebble (and crystal) shapes based on static equilibrium points. Central European Geology 53, (2010) 1-19.

- Project owner:
- Dr. Sipos András Árpád (Szilárdságtani és Tartószerkezeti Tanszék)
- Members:
- Szilárdságtani és Tartószerkezeti Tanszék (ÉPK-SZT)