Skip to main content
SpringerLink
Log in

Influence of Concentrated Forces on an Interface Inclusion under the Conditions of Smooth Contact in the Inhomogeneous Transversely Isotropic Space

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We analyze the influence of concentrated forces on a disk-shaped inclusion operating the conditions of smooth contact in the plane of joint of two different transversely isotropic half spaces. The problem is reduced to the Riemann boundary-value problem with respect to a part of variables in the space of generalized functions. Its solution is constructed in the explicit form, which enables us to determine the dependence of translational and circular displacements of the inclusion and the jumps of stresses and displacements on the inclusion on concentrated forces and the relationship between the elastic constants of the half spaces. We also study the influence of presence of the concentrated forces (either in a single or in both half spaces) on the translational displacements and the jumps of normal stresses for different combinations of the materials of the half spaces and the shapes of the inclusion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K. S. Aleksandrov and T. V. Ryzhova, “Elastic properties of crystals. A survey,” Kristallografiya, 6, No. 2, 289–314 (1961).

    Google Scholar 

  2. V. V. Efimov, A. F. Krivoi, and G. Ya. Popov, “Problems on the stress concentration near a circular imperfection in a composite elastic medium,” Izv. Ros. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 42–58 (1998); English translation: Mech. Solids, 33, No. 2, 35–49 (1998).

  3. H. S. Kit and R. M. Andriichuk, “Problem of stationary heat conduction for a piecewise homogeneous space under the conditions of heat release in a disk-shaped domain,” Prykl. Probl. Mekh. Mat., Issue 10, 115–122 (2012).

  4. H. S. Kit and O. P. Sushko, “Problems of stationary heat conduction and thermoelasticity for a body with a heat permeable diskshaped inclusion (crack),” Mat. Met. Fiz.-Mekh. Polya, 52, No. 4, 150–159 (2009); English translation: J. Math. Sci., 174, No. 3, 309–321 (2011); https://doi.org/10.1007/s10958-011-0300-3.

  5. H. S. Kit and O. P. Sushko, “Axially symmetric problems of stationary heat conduction and thermoelasticity for a body with thermally active or thermally insulated disk inclusion (crack),” Mat. Met. Fiz.-Mekh. Polya, 53, No. 1, 58–70 (2010); English translation: J. Math. Sci., 176, No. 4, 561–577 (2011); https://doi.org/10.1007/s10958-011-0422-7.

  6. H. S. Kit and O. P. Sushko, “Distribution of stationary temperature and stresses in a body with heat permeable disk-shaped inclusion,” Met. Rozv. Prykl. Zadach Mekh. Deformiv. Tverd. Tila, Issue 10, 145–153 (2009).

  7. H. Kit and O. Sushko, “Stationary temperature field in a semiinfinite body with thermally active or thermally insulated disk-shaped inclusion,” Fiz.-Mat. Model. Inform. Tekhnol., Issue 13, 67–80 (2011).

  8. O. F. Kryvyi, “Mutual influence of an interface tunnel crack and an interface tunnel inclusion in a piecewise homogeneous anisotropic space,” Mat. Met. Fiz.-Mekh. Polya, 56, No. 4, 118–124 (2013); English translation: J. Math. Sci., 208, No. 4, 409–416 (2015); https://doi.org/10.1007/s10958-015-2455-9.

  9. O. F. Kryvyi, “Interface crack in the inhomogeneous transversely isotropic space,” Fiz.-Khim. Mekh. Mater., 47, No. 6, 15–22 (2011); English translation: Mater. Sci., 47, No. 6, 726–736 (2012); https://doi.org/10.1007/s11003-012-9450-9.

  10. O. F. Kryvyi, “Delaminated interface inclusion in a piecewise homogeneous transversely isotropic space,” Fiz.-Khim. Mekh. Mater., 50, No. 2, 77–84 (2014); English translation: Mater. Sci., 50, No. 2, 245–253 (2014); https://doi.org/10.1007/s11003-014-9714-7.

  11. O. F. Kryvyy, “Interface circular inclusion under mixed conditions of interaction with a piecewise homogeneous transversely isotropic space,” Mat. Met. Fiz.-Mekh. Polya, 54, No. 2, 89–102 (2011); English translation: J. Math. Sci., 184, No. 1, 101–119 (2012); https://doi.org/10.1007/s10958-012-0856-6.

  12. O. F. Kryvyy, “Singular integral relations and equations for a piecewise homogeneous transversely isotropic space with interphase defects,” Mat. Met. Fiz.-Mekh. Polya, 53, No. 1, 23–35 (2010); English translation: J. Math. Sci., 176, No. 4, 515–531 (2011); https://doi.org/10.1007/s10958-011-0419-2.

  13. O. F. Kryvyy, “Tunnel internal crack in a piecewise homogeneous anisotropic space,” Mat. Met. Fiz.-Mekh. Polya, 55, No. 4, 54–63 (2012); English translation: J. Math. Sci., 198, No. 1, 62–74 (2014); https://doi.org/10.1007/s10958-014-1773-7.

  14. O. F. Kryvyy, “Tunnel inclusions in a piecewise homogeneous anisotropic space,” Mat. Met. Fiz.-Mekh. Polya, 50, No. 2, 55–65 (2007).

  15. O. F. Kryvyi and Yu. O. Morozov, “Solution of the problem of heat conduction for the transversely isotropic piecewisehomogeneous space with two circular inclusions,” Mat. Met. Fiz.-Mekh. Polya, 60, No. 2, 130–141 (2017); English translation: J. Math. Sci., 243, No. 1, 162–182 (2019); https://doi.org/10.1007/s10958-019-04533-1.

  16. O. F. Kryvyi and Yu. O. Morozov, “Fundamental solutions for a piecewise-homogeneous transversely isotropic elastic space,” Mat. Met. Fiz.-Mekh. Polya, 63, No. 1, 122–132 (2020); English translation: J. Math. Sci., 270, No. 1, 143–156 (2019); https://doi.org/10.1007/s10958-023-06337-w.

  17. A. F. Krivoi, “Arbitrarily oriented defects in a composite anisotropic plane,” Visn. Odes’k. Derzh. Univ., Ser. Fiz.-Mat. Nauky, 6, Issue 3, 108–115 (2001).

  18. A. F. Krivoi, “Fundamental solution for a four-component composite anisotropic plane,” Visn. Odes’k. Derzh. Univ., Ser. Fiz.-Mat. Nauky, 8, Issue 2, 140–149 (2003).

  19. A. F. Krivoi and Yu. A. Morozov, “Solution of the problem of heat conduction for two coplanar cracks in a composite transversely isotropic space,” Visn. Donets’k. Nats. Univ., Ser. A. Pryrodn. Nauky, No. 1, 76–83 (2014).

  20. A. F. Krivoi and G. Ya. Popov, ”Interface tunnel cracks in a composite anisotropic space,” Prikl. Mat. Mekh., 72, No. 4, 689–700 (2008); English translation: J. Appl. Math. Mech., 72, No. 4, 499–507 (2008); https://doi.org/10.1016/j.jappmathmech.2008.08.001.

  21. A. F. Krivoi and G. Ya. Popov, ”Features of the stress field near tunnel inclusions in an inhomogeneous anisotropic space,” Prikl. Mekh., 44, No. 6, 36–45 (2008); English translation: Int. Appl. Mech., 44, No. 6, 626–634 (2008); https://doi.org/10.1007/s10778-008-0084-4.

  22. A. F. Krivoi, G. Ya. Popov, and M. V. Radiollo, “Certain problems of an arbitrarily oriented stringer in a composite anisotropic plane,” Prikl. Mat. Mekh., 50, No. 4, 622–632 (1986); English translation: J. Appl. Math. Mech., 50, No. 4, 475–483 (1986); https://doi.org/10.1016/0021-8928(86)90012-2.

  23. A. F. Krivoi and M. V. Radiollo, “Specific features of the stress field near inclusions in a composite anisotropic plane,” Izv. Akad. Nauk SSSR., Mekh. Tverd. Tela, No. 3, 84–92 (1984).

  24. R. M. Kushnir and Yu. B. Protsyuk, “Thermoelastic state of layered thermosensitive bodies of revolution for the quadratic dependence of the heat-conduction coefficients,” Fiz.-Khim. Mekh. Mater., 46, No. 1, 7–18 (2010); English translation: Mater Sci., 46, No. 1, 1–15 (2011); https://doi.org/10.1007/s11003-010-9258-4.

  25. F. Akbari, A. Khojasteh, and M. Rahimian “Three-dimensional interfacial Green’s function for exponentially graded transversely isotropic bi-materials,” Civ. Eng. Infrastruct. J., 49, No. 1, 71–96 (2016); https://doi.org/10.7508/ceij.2016.01.006.

    Article  Google Scholar 

  26. D. S. Boiko and Y. V. Tokovyy, “Determination of three-dimensional stresses in a semi-infinite elastic transversely isotropic composite,” Mech. Compos. Mater., 57, No. 4, 481–492 (2021); https://doi.org/10.1007/s11029-021-09971-0.

    Article  Google Scholar 

  27. P.-F. Hou, A. Y. T. Leung, and Y.-J. He, “Three-dimensional Green’s functions for transversely isotropic thermoelastic bimaterials,” Int. J. Solids Struct., 45, No. 24, 6100–6113 (2008); https://doi.org/10.1016/j.ijsolstr.2008.07.022.

    Article  Google Scholar 

  28. P.-F. Hou, Z.-S. Li, and Y. Zhang, “Three-dimensional quasi-static Green’s function for an infinite transversely isotropic pyroelectric material under a step point heat source,” Mech. Res. Comm., 62, 66–76 (2014); https://doi.org/10.1016/j.mechrescom.2014.08.008.

  29. P.-F. Hou, M. Zhao, J. Tong, and B. Fu, “Three-dimensional steady-state Green’s functions for fluid-saturated, transversely isotropic, poroelastic bimaterials,” J. Hydrology, 496, 217–224 (2013); https://doi.org/10.1016/j.jhydrol.2013.05.017.

  30. O. F. Kryvyi and Yu. O. Morozov, “Inhomogeneous transversely isotropic space under influence of concentrated power and temperature sources,” J. Phys.: Conf. Ser. Proc. of the 7th Internat. Conf. TPCM 2021 “Topical Problems of Continuum Mechanics” (October 4–8, 2021, Tsaghkadzor, Armenia), 2231, 012016 (2022); https://doi.org/10.1088/1742-6596/2231/1/012016.

  31. O. F. Kryvyi and Yu. O. Morozov, “The fundamental solution of the problem of thermoelasticity for a piecewise homogeneous transversely isotropic elastic space,” Doslid. Mat. Mekh., 25, No. 1(35), 16–30 (2020).

  32. O. Kryvyi and Yu. Morozov, “Interphase circular inclusion in a piecewise-homogeneous transversely isotropic space under the action of a heat flux,” in: E. Gdoutos (editor), Proc. of the First Internat. Conf. on Theoretical, Applied and Experimental Mechanics, Springer (2018), pp. 394–396; https://doi.org/10.1007/978-3-319-91989-8_94.

  33. O. Kryvyi and Yu. Morozov, “The influence of mixed conditions on the stress concentration in the neighborhood of interfacial inclusions in an inhomogeneous transversely isotropic space,” in: E. Gdoutos and M. Konsta-Gdoutos (editors), Proc. of the 3rd Internat. Conf. on Theoretical, Applied and Experimental Mechanics (Structural Integrity, 16), Springer (2020), pp. 204–209; https://doi.org/10.1007/978-3-030-47883-4_38.

  34. O. Kryvyi and Yu. Morozov, “The problem of stationary thermoelasticity for a piecewise homogeneous transversely isotropic space under the influence of a heat flux specified at infinity is considered,” J. Phys.: Conf. Ser. Proc. of the 6th Internat. Conf. “Topical Problems of Continuum Mechanics” (October 1–6, 2019, Dilijan, Armenia), 1474, 012025 (2019); https://doi.org/10.1088/1742-6596/1474/1/012025.

  35. O. Kryvyi and Yu. Morozov, “Thermally active interphase inclusion in a smooth contact conditions with transversely isotropic half spaces,” Frat. Integrita Strutt., 14, No. 52, 33–50 (2020); https://doi.org/10.3221/IGF-ESIS.52.04.

    Article  Google Scholar 

  36. O. Kryvyy, “The discontinuous solution for the piece-homogeneous transversal isotropic medium,” in: Modern Analysis and Applications, Ser. Operator Theory: Advances and Applications, Vol. 191, Birkhäuser, Basel (2009), pp. 395–406; https://doi.org/10.1007/978-3-7643-9921-4_25.

  37. R. Kumar and V. Gupta, “Green’s function for transversely isotropic thermoelastic diffusion biomaterials,” J. Therm. Stresses, 37, No. 10, 1201–1229 (2014); https://doi.org/10.1080/01495739.2014.936248.

  38. R. Kushnir, “Thermal stresses – advanced theory and applications,” J. Therm. Stresses, 33, No. 1, 76–78 (2010); https://doi.org/10.1080/01495730903538421.

    Article  Google Scholar 

  39. R. Kushnir and B. Protsiuk, “A method of the Green’s functions for quasistatic thermoelasticity problems in layered thermosensitive bodies under complex heat exchange,” in: Modern Analysis and Applications, Ser. Operator Theory: Advances and Applications, Vol. 191, Birkhäuser, Basel (2009), pp. 143–154; https://doi.org/10.1007/978-3-7643-9921-4_9.

  40. X.-F. Li and T.-Y. Fan, “The asymptotic stress field for a rigid circular inclusion at the interface of two bonded dissimilar elastic half space materials,” Int. J. Solids Struct., 38, No. 44-45, 8019–8035 (2001); https://doi.org/10.1016/S0020-7683(01)00010-5.

    Article  Google Scholar 

  41. V. Mantič, L. Távara, J. E. Ortiz, and F. París, “Recent developments in the evaluation of the 3D fundamental solution and its derivatives for transversely isotropic elastic materials,” Electron. J. Bound. Elem., 10, No. 1, 1–41 (2012); https://doi.org/10.14713/ejbe.v10i1.1116.

  42. E. Pan and W. Chen, “Green’s functions in a transversely isotropic magnetoelectroelastic bimaterial space,” in: A. Pan and W. Chen, Static Green’s Functions in Anisotropic Media, Chapter 7, Cambridge Univ. Press (2015), pp. 220–259; https://doi.org/10.1017/CBO9781139541015.008.

  43. K. Sahebkar and M. Eskandari-Ghadi, “Displacement ring load Green’s functions for saturated porous transversely isotropic trimaterial full-space,” Int. J. Numer. Anal. Meth. Geomech., 41, No. 3, 359–381 (2017); https://doi.org/10.1002/nag.2560.

    Article  Google Scholar 

  44. Yu. Tokovyy, “Direct integration of three-dimensional thermoelasticity equations for a transversely isotropic layer,” J. Therm. Stresses, 42, No. 1, 49–64 (2019); https://doi.org/10.1080/01495739.2018.1526150.

  45. Yu. V. Tokovyy and C. C. Ma, “Three-dimensional elastic analysis of transversely isotropic composites,” J. Mech., 33, No. 6, 821–830 (2017); https://doi.org/10.1017/jmech.2017.91.

    Article  Google Scholar 

  46. Z. Q. Yue, “Elastic fields in two joined transversely isotropic solids due to concentrated forces,” Int. J. Eng. Sci., 33, No. 3, 351–369 (1995); https://doi.org/10.1016/0020-7225(94)00063-P.

    Article  MathSciNet  Google Scholar 

  47. Y. Zafari, M. Shahmohamadi, A. Khojasteh, and M. Rahimian, “Asymmetric Green’s functions for a functionally graded transversely isotropic tri-material,” Appl. Math. Model., 72, 176–201 (2019); https://doi.org/10.1016/j.apm.2019.02.038.

  48. Y.-F Zhao., M.-H. Zhao, E. Pan, and C.-Y. Fan, “Green’s functions and extended displacement discontinuity method for interfacial cracks in three-dimensional transversely isotropic magneto-electro-elastic bi-materials,” Int. J. Solids Struct., 52, 56–71 (2015); https://doi.org/10.1016/j.ijsolstr.2014.09.018.

Download references

Author information

Authors and Affiliations

  1. “Odesa Maritime Academy” National University, Odesa, Ukraine

    O. F. Kryvyi

  2. Odesa National Polytechnic University, Odesa, Ukraine

    Yu.O. Morozov

Authors
  1. O. F. Kryvyi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu.O. Morozov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. F. Kryvyi.

Additional information

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 64, No. 4, pp. 68–81, October–December, 2021.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kryvyi, O.F., Morozov, Y. Influence of Concentrated Forces on an Interface Inclusion under the Conditions of Smooth Contact in the Inhomogeneous Transversely Isotropic Space. J Math Sci 279, 197–212 (2024). https://doi.org/10.1007/s10958-024-07005-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-024-07005-3

Keywords

Search

Navigation