The problem of construction of the fundamental solutions for a piecewise-homogeneous transversely isotropic space is reduced to a matrix Riemann problem in the space of slowly increasing distributions. We propose a method for the solution of this problem. As a result, in the explicit form, we obtain expressions for the components of the vector of fundamental solution and simple representations for the components of the stress tensor and the vector of displacements in the plane of joint of transversely isotropic elastic half spaces subjected to the action of concentrated normal and tangential forces. We study the fields of stresses and displacements in the plane of joint of the half spaces. In particular, for some combinations of materials, we present the numerical values of the coefficients of influence of concentrated forces on the stresses and displacements. We also establish conditions under which the normal displacements are absent in the plane of joint of transversely isotropic elastic half spaces.
Similar content being viewed by others
References
K. S. Aleksandrov and T. V. Ryzhova, “Elastic properties of crystals. A review,” Kristallografiya, 6, Issue 2, 289–314 (1961).
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, No. 2, 42–58 (1998); English translation: Mech. Solids, 33, No. 2, 35–49 (1998).
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 circular domain,” Prykl. Probl. Mekh. Mat., Issue 10, 115–122 (2012).
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); 10.1007/s10958-011-0300-3.
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); 10.1007/s10958-011-0422-7.
H. S. Kit and O. P. Sushko, “Distribution of stationary temperature and stresses in a body with heat-permeable disk-shaped inclusion,” Met. Rozv’yaz. Prykl. Zadach Mekh. Deformivn. Tverd. Tila, Issue 10, 145–153 (2009).
H. Kit and O. Sushko, “Stationary temperature field in a semiinfinite body with thermally active and heat-insulated disk-shaped inclusion,” Fiz.-Mat. Model. Inform. Tekhnol., Issue 13, 67–80 (2011).
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); 10.1007/s10958-015-2455-9.
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); 10.1007/s11003-012-9450-9.
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); 10.1007/s11003-014-9714-7.
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); 10.1007/s10958-012-0856-6.
O. F. Kryvyy, “Singular integral relations and equations for a piecewise homogeneous transversely isotropic space with interface defects,” Mat. Met. Fiz.-Mekh. Polya, 53, No. 1, 23–35 (2010); English translation: J. Math. Sci., 176, No. 4, 515–531 (2011); 10.1007/s10958-011-0419-2.
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); 10.1007/s10958-014-1773-7.
O. F. Kryvyi, “ Tunnel inclusions in a piecewise homogeneous anisotropic space,” Mat. Met. Fiz.-Mekh. Polya, 50, No. 2, 55–65 (2007).
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); 10.1007/s10958-019-04533-1.
. A. F. Krivoi, “Arbitrarily oriented defects in a composite anisotropic plane,” Visn. Odesk. Derzh. Univ., Ser. Fiz.-Mat. Nauky, 6, Issue 3, 108–115 (2001).
. A. F. Krivoi, “Fundamental solution for a four-component anisotropic plane,” Visn. Odesk. Derzh. Univ. Ser., Fiz.-Mat. Nauky, 8, Issue 2, 140–149 (2003).
A. F. Krivoi and G. Ya. Popov, “Solution of the problem of heat conduction for two coplanar cracks in a composite transversely isotropic space,” Visn. Donetsk. Nats. Univ., Ser. A, Pryrodn. Nauky, No. 1, 76–83 (2014).
A. F. Krivoi and G. Ya. Popov, “Interface tunnel cracks in a composite anisotropic space,” Prykl. Mat. Mekh., 72, No. 4, 689–700 (2008); English translation: J. Appl. Math. Mech., 72, No. 4, 499–507 (2008); 10.1016/j.jappmathmech.2008.08.001.
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); 10.1007/s10778-008-0084-4.
A. F. Krivoi, G. Ya. Popov, and M. V. Radiollo, “Certain problems of an arbitrarily oriented stringer in a composite anisotropic plane,” Prykl. Mat. Mekh., 50, No. 4, 622–632 (1986); English translation: J. Appl. Math. Mech., 50, No. 4, 475–483 (1986); 10.1016/0021-8928(86)90012-2.
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).
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); 10.1007/s11003-010-9258-4.
P.-F. Hou, A. T. Y. Leung, and Y.-J. He, “Three-dimensional Green’s functions for transversely isotropic thermoelastic biomaterials,” Int. J. Solids Struct., 45, No. 24, 6100–6113 (2008); https://doi.org/10.1016/j.ijsolstr.2008.07.022.
O. F. Kryvyi and Yu. Morozov, “Thermally active interface inclusion in a smooth contact conditions with transversely isotropic half-spaces,” Frat. Integr. Strutt., 14, No. 52, 33–50 (2020); DOI: https://doi.org/10.3221/IGF-ESIS.52.04.
O. F. 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., 1474, 012025, Proc. of the 6th Internat. Conf. “Topical Problems of Continuum Mechanics” (October 1–6, 2019, Dilijan, Armenia); DOI:10.1088/1742-6596/1474/1/012025.
O. F. 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 ICTAEM-2020. Structural Integrity, Vol. 16, (2020), pp. 204–209; 10.1007/978-3-030-47883-4_38.
O. F. Kryvyi and Yu. O. Morozov, “Interface circular inclusion in a piecewise-homogeneous transversely isotropic space under the action of a heat flux,” in: E. Gdoutos (editor), Proc. of the 1st Internat. Conf. on Theoretical, Applied and Experimental Mechanics ICTAEM-2018 (June 17–20, 2018, Cyprus, Greece), Springer (2018), pp. 394–396; 10.1007/978-3-319-91989-8_94.
O. Kryvyy, “The discontinuous solution for the piece-homogeneous transversal isotropic medium,” Oper. Theory: Adv. Appl., 191, 395–406 (2009); 10.1007/978-3-7643-9921-4_25.
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.
R. Kushnir and B. Protsiuk, “A method of the Green’s functions for quasistatic thermoelasticity problems in layered thermosensitive bodies under complex heat exchange,” Oper. Theory: Adv. Appl., 191, 143–154 (2009); https://doi.org/10.1007/978-3-7643-9921-4_9.
X.-F. Li and T.-Y. Fan, “The asymptotic stress field for a ring circular inclusion at the interface of two bonded dissimilar elastic half-space materials,” Int. J. Solids Struct., 38, Nos. 44-45, 8019–8035 (2001); https://doi.org/10.1016/S0020-7683(01)00010-5.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 63, No. 1, pp. 122–132, January–March, 2020.
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.
About this article
Cite this article
Kryvyi, О.F., Morozov, Y.О. Fundamental Solutions for a Piecewise-Homogeneous Transversely Isotropic Elastic Space. J Math Sci 270, 143–156 (2023). https://doi.org/10.1007/s10958-023-06337-w
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06337-w