Skip to main content

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

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

References

  • Adams, G.R., Jager, A.J.: Petroscopic observations of rock fracturing ahead of stope faces in deep-level gold mine. J. South Afr. Inst. Min. Metall. 80(6), 204–209 (1980)

    Google Scholar 

  • Arnold, D.N., Boffi, D., Falk, R.S.: Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42(6), 2429–2451 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  • Arnold, D.N., Awanou, G., Qiu, W.: Mixed finite elements for elasticity on quadrilateral meshes. Adv. Comput. Math. 41(3), 553–572 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  • Becache, E., Joly, P., Tsogka, C.: A new family of mixed finite elements for the linear elastodynamic problem. SIAM J. Numer. Anal. 39(6), 2109–2132 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  • Blokhin, A.M., Dorovsky, V.N.: Mathematical Modelling in the Theory of Multivelocity Continuum. Nova Science Publishers, New York (1995)

    Google Scholar 

  • Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics. Springer, Berlin, Heidelberg (2013)

    MATH  Book  Google Scholar 

  • Chen, X.-G., Zhang, Q.-Y., Wang, Y., Liu, D.-J., Zhang, N.: Model test of anchoring effect on zonal disintegration in deep surrounding rock masses. Sci. World. J. (2013). doi:10.1155/2013/935148

    Google Scholar 

  • Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-holland Publishing Company, Amsterdam (1978)

    MATH  Google Scholar 

  • De Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. North-holland publishing company, Amsterdam (1962)

  • Dorovsky, V.N., Romensky, E.I., Sinev, A.V.: Spatially non-local model of inelastic deformations: applications for rock failure problem. Geophys. Prospect. 63(5), 1198–1212 (2015)

    Article  Google Scholar 

  • Godunov, S.K., Romenskii, E.I.: Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/Plenum Publishers, NY (2003)

    MATH  Book  Google Scholar 

  • Guzev, M.A.: Non-Euclidean Models of Elastoplastic Materials with Structure Defects. Lambert Academic Publishing, Saarbrucken (2010)

    Google Scholar 

  • Guzev, M.A., Paroshin, A.A.: Non-Euclidean model of the zonal disintegration of rocks around an underground working. J. Appl. Mech. Tech. Phys. 42(1), 131–139 (2001)

    MATH  Article  Google Scholar 

  • Lee, J.: Mixed methods with weak symmetry for time dependent problems of elasticity and viscoelasticity. PhD Thesis, University of Minnesota (2012)

  • Mura, T.: Micromechanics of Defects in Solids. Mechanics of Elastic and Inelastic Solids, 2nd edn. Kluwer Academic Publishers, Kluwer (1987)

    MATH  Book  Google Scholar 

  • Myasnikov, V.P., Gusev, M.A.: A geometrical model of the defect structure of an elastoplastic continuous medium. J. Appl. Mech. Tech. Phys. 40(2), 331–340 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  • Myasnikov, V.P., Gusev, M.A.: Thermomechanical model of elastic–plastic materials with defect structures. Theor. Appl. Fract. Mech. 33(3), 165–171 (2000)

    Article  Google Scholar 

  • Renard, Y., Pommier, J.: Getfem++. An open source generic C++ library for finite element methods. www.home.gna.org/getfem

  • Shemyakin, E.I., Fisenko, G.L., Kurlenya, M.V., Oparin, V.N., Reva, V.N., Glushikhin, F.P., Rozenbaum, M.A., Tropp, E.A., Kuznetsov, Y.S.: Zonal disintegration of rocks around under-ground workings, part I: data of in-situ observations. J. Min. Sci. 22(3), 157–168 (1986)

    Google Scholar 

  • Shemyakin, E.I., Fisenko, G.L., Kurlenya, M.V., Oparin, V.N., Reva, V.N., Glushikhin, F.P., Rozenbaum, M.A., Tropp, E.A., Kuznetsov, Y.S.: Zonal disintegration of rocks around underground workings, part II: rock fracture simulated in equivalent materials. J. Min. Sci. 22(4), 223–232 (1986)

    Google Scholar 

  • Shemyakin, E.I., Fisenko, G.L., Kurlenya, M.V., Oparin, V.N., Reva, V.N., Glushikhin, F.P., Rozenbaum, M.A., Tropp, E.A., Kuznetsov, Y.S.: Zonal disintegration of rocks around underground workings, part III: theoretical concepts. J. Min. Sci. 23(1), 1–6 (1987)

    Google Scholar 

  • Vtorushin, E.V., Dorovsky, V.N., Romensky, E.I.: Mixed finite element method applied to non-euclidean model of inelastic deformations. In: 49th US Rock Mechanics/Geomechanics Symposium 2015 American Rock Mechanics Association, vol 3, pp. 1977–1985 (2015)

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to E. V. Vtorushin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

  • Mixed finite elements
  • In-elasticity
  • Zonal disintegration phenomenon
  • Non-Euclidean continuum model

Mathematics Subject Classification

  • 74R20 Anelastic fracture and damage in the Mechanics of deformable solids