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Boundary Value Problems of Plane Elasticity Involving Orientations of Displacements and Tractions

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Abstract

This paper presents unconventional formulations of boundary problems of plane elasticity formulated in terms of orientations of tractions and displacements on a closed contour separating internal and external domains as the boundary conditions. These are combined with the conditions of continuity of tractions or displacements across the boundary. Therefore the magnitudes of neither tractions nor displacements are assumed on the contour. Four boundary value problems for both external and internal domains are investigated by analyzing the solvability of the corresponding singular integral equations. It is shown that all considered problems can have non-unique solutions expressed as linear combinations of particular solutions and, hence, contain free arbitrary parameters, the number of which is finite and determined by the contour orientations of tractions and/or displacements

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References

  1. Bonnet, M., Constantinescu, A.: Inverse problems in elasticity. Inverse Probl. 21, R1–R50 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Coblentz, D.D., Sandiford, M., Richardson, R.M., Zhou, S., Hillis, R.: The origins of the intraplate stress field in continental Australia. Earth Planet. Sci. Lett. 133, 299–309 (1995)

    Article  ADS  Google Scholar 

  3. Gakhov, F.D.: Boundary Value Problems. Dover, New York (1990)

    MATH  Google Scholar 

  4. Galybin, A.N., Mukhamediev, Sh.A.: Plane elastic boundary value problem posed on orientation of principal stresses. J. Mech. Phys. Solids 47(11), 2381–2409 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Galybin, A.N.: Plane boundary value problem posed by displacement and force orientations on a closed contour. J. Elast. 65, 169–184 (2002)

    Article  MathSciNet  Google Scholar 

  6. Galybin, A.N.: Stress fields in joined elastic regions: modelling based on discrete stress orientations. In: Lu, M. et al. (eds.) In-situ Rock Stress Measurement, Interpretation and Application, pp. 193–199. Taylor & Francis/Balkema, London/Rotterdam (2006)

    Google Scholar 

  7. Galybin, A.N.: An inverse problem of elastostatics in mechanics of composites. Compos. Sci. Technol. 68(5), 1188–1197 (2008)

    Article  Google Scholar 

  8. Heidbach, O., Tingay, M., Barth, A., Reinecker, J., Kurfess, D., Müller, B.: The 2008 release of the World Stress Map. Available online at www.world-stress-map.org (2008)

  9. Irša, J., Galybin, A.N.: Stress trajectories element method for stress determination from discrete data on principal directions. Eng. Anal. Bound. Elem. 34(5), 423–432 (2010)

    Article  Google Scholar 

  10. Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  11. Linkov, A.M.: Boundary Integral Equations in Elasticity Theory. Kluwer Academic, Dordrecht (2002)

    MATH  Google Scholar 

  12. Marin, L., Lesnic, D.: Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition. Comput. Methods Appl. Mech. Eng. 191(6), 3257–3270 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Marin, L., Elliott, L., Ingham, D.B., Lesnic, D.: Boundary element regularisation methods for solving the Cauchy problem in linear elasticity. Inverse Probl. Eng. Sci. 10(4), 335–357 (2002)

    Article  Google Scholar 

  14. Mukhamediev, Sh.A., Galybin, A.N., Brady, B.H.G.: Determination of stress fields in the elastic lithosphere by methods based on stress orientations. Int. J. Rock Mech. Min. Sci. 43(1), 66–88 (2006)

    Article  Google Scholar 

  15. Mukhamediev, Sh.A.: Galybin, A.N.: Where and how did the ruptures of December 26, 2004 and March 28, 2005 earthquakes near Sumatra originate? Dokl. Earth Sci. 406(1), 52–55 (2006)

    Article  ADS  Google Scholar 

  16. Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Groningen, Noordhoff (1953)

    MATH  Google Scholar 

  17. Schwab, A.A.: The inverse problem of elasticity theory: Application of the boundary integral equation for the holomorphic vector. Phys. Solid Earth 30(4), 342–348 (1994)

    MathSciNet  Google Scholar 

  18. Shvab, A.A.: Incorrectly posed static problems of elasticity. Mech. Solids 24(6), 98–106 (1989)

    Google Scholar 

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  1. Wessex Institute of Technology, Southampton, UK

    A. N. Galybin

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  1. A. N. Galybin
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Correspondence to A. N. Galybin.

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Galybin, A.N. Boundary Value Problems of Plane Elasticity Involving Orientations of Displacements and Tractions. J Elast 102, 15–30 (2011). https://doi.org/10.1007/s10659-010-9259-4

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  • DOI: https://doi.org/10.1007/s10659-010-9259-4

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