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A simplified model of interaction between tectonic plates


In our approach, the tectonic plates are considered bounded surfaces. If in one place, the plate undergoes irreversible deformations, these appear because of external factors. According to the mechanics of continuous medium, at any point the stress tensor has six independent components. However, lack of third dimension allows reducing the number of perturbation parameters to five. Close to another neighboring plate, there are two degrees of freedom: the distance of draw near δ and the relative rotational angle γrel. For the final potential energy, equal to the initial potential energy plus the work done by the exterior forces, we obtain a real function, which depends on two (kinematic) variables and five parameters. When the final state is irreversible, that is to say, when it corresponds to a new distribution of equilibrium points, then, for the potential function (final potential energy) the catastrophe theory provides four and only four kinds of catastrophes: D±6 and E±6. The first two are useless due to the inconsistencies to be interpreted in our context. However, E6 and E-6 fit completely in a scheme of collisions based on elementary mechanical concepts such as: linear springs, absolutely rigid levers and forces. The formal behavior for these cases is x3±y4+t1y +t2y2+x(t3+t4y+t5y2) respectively, where x and y are practically other notations for the variables δ and γrel. The parameters t1, t2, t3, t4 and t5 are interpreted by the breakdown of the catastrophes E±6 into the “sub” catastrophes A2 plus A±3 respectively. In this way, the first and the last terms are a catastrophe caused by the mutual approach of the plates while the sum of the all other three terms describes the irreversible processes due to the tangential rubbing of the plates. The results are given as shown in the following figure. (Go to Search and Discovery to view figures). The mechanical equivalences are constructed matching the germs' expressions derived from the formulae of the catastrophes E±6 with the interaction potential energy plus the work of an external force. This simplified model, among other things, allows some useful applications of statistical methods and the probability theory (see the related abstract “Description of geological processes starting from Markov chains” sent as well for the AAPG Europe Region Conference – 2015 Lisbon).