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Application of mixed finite elements to spatially non-local model of inelastic deformations

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Abstract

Rock behaviour frequently does not fit the classical theory of continuum mechanics because of rock aggregated granular structure. Particularly, rock fracturing may be accompanied by zonal disintegration formation. The key to building the non-classic model of rock fracturing is the granulated structure. Deformations of solid bodies with microscopic flaws can be described within the scope of non-Euclidean geometry, and non-trivial deformation incompatibility can be referred to as a fracture parameter. The non-Euclidean continuum model used in this paper enables the prediction of the zones initializing and developing as a periodic structure. The non-Euclidean description of phenomenon initiates an appearance of two new material constants. The coupled model must comprise the fourth-order parabolic equation on disintegration thermodynamic parameter be solved with the classical hyperbolic system of equations for the dynamics of continuous media. In this paper, the mixed finite element method is applied to approximate the equations and to model the zonal disintegration phenomenon numerically. The 2D model problem of disintegration zone formation was solved numerically. The zone magnitude and site that can be described by the term ‘disintegration scale’ are determined by values of new constants. Therefore, the numerical model based on the new non-Euclidean continuum model is capable of predicting formation of a disintegration field periodic structure. The second spatial direction of disintegration parameter field propagation is ascertained that allows the model to be applied to various problems of fracture mechanics of rocks.

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Acknowledgments

The author would like to thanks: Baker Hughes researcher Dr Vitaliy Dorovsky and Sobolev IGM SB RAS researcher Dr Evgeny Romensky for their scientific advice on the model, the manuscript review and critical comments; Universite de Lyon, CNRS professor Dr Yves Renard for his scientific and technical support of GetFEM\(++\) framework.

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  1. Baker Hughes, Kytatelatze av. 4A, Novosibirsk, 630090, Russia

    E. V. Vtorushin

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  1. E. V. Vtorushin
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Vtorushin, E.V. Application of mixed finite elements to spatially non-local model of inelastic deformations. Int J Geomath 7, 183–201 (2016). https://doi.org/10.1007/s13137-016-0083-2

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  • DOI: https://doi.org/10.1007/s13137-016-0083-2

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