Integral equation of elastic medium containing a deformable thread-like inclusion

  • Ia. M. Pasternak Lutsk National Technical University
  • H. T. Sulym Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine
Keywords: thread-like inclusion, integral equation, regularization


UDC 517.968.23: 539.3

In this study, we propose a method of mathematical modeling of deformable thread-like (wire) inclusions based on replacing their influence on the elastic medium with the tension/compression forces distributed along their axis. We derive a regularized integral equation of the problem, which is external with respect to the inhomogeneity, and also a mathematical model of inhomogeneity, which, taking into account the contact conditions, makes it possible to determine the desired distribution of forces along the axis of inhomogeneity. An approach to solving these equations is proposed. For the case of a rectilinear elastic thread-like inclusion, an approximate solution of the problem is obtained in a close analytical form.


A. N. Komarenko, Y. A. Lukovskyi, S. F. Feshchenko, K zadache o sobstvennykh znachenyiakh s parametrom v kraevykh uslovyiakh, Ukr. mat. zhurn., 17, № 6, 22 – 30 (1965).

H. T. Sulym, Osnovy matematychnoi teorii termopruzhnoi rivnovahy deformivnykh tverdykh til z tonkymy vkliuchenniamy, Doslid.-vyd. tsentr NTSh, Lviv (2007).

A. F. Ulitko, N. E. Kachalovska, Osoblyvosti napruzhenoho stanu pruzhnoho seredovyshcha, shcho mistyt zhorstke „holkopodibne” vkliuchennia, Dop. AN URSR. Ser. A, № 5, 44 – 48 (1987).

V. T. Grinchenko, A. F. Ulitko, On local singularities in mathematical models of physical fields, J. Math. Sci., 97, № 1, 3777–3795 (1999), DOI:

K. M. Liew, Z. Pan, L.-W. Zhang, The recent progress of functionally graded CNT reinforced composites and structures, Sci. China. Phys., Mech., Astron., 63, Article 234601 (2020); DOI:

Ia. Pasternak, N. Ilchuk, H. Sulym, O. Andriichuk, Boundary integral equations for anisotropic elasticity of solids containing rigid thread-like inclusions, Mech. Res. Commun., 100, Article 103402, 1 – 6 (2019). DOI:

R. A. Sauer, G. Wang, S. Li, The composite eshelby tensors and their applications to homogenization, Acta Mech., 197, 63 – 96 (2008); DOI:

H. Sulym, Ia. Pasternak, R. Pasternak, Boundary element analysis of multifield materials, Sci. Thesis № 274. Library of Mechanics, Printing House Białystok Univ. Technology, Bialystok (2015).

Ia. M. Pasternak, H. Sulym, Thermoelasticity of solids containing thread-like inhomogeneities. I. Nondeformable thread-like inclusions, Int. J. Solids and Structures, 232, 111176, 1 – 12 (2021); DOI:

How to Cite
Pasternak , I. M., and H. T. Sulym. “Integral Equation of Elastic Medium Containing a Deformable Thread-Like Inclusion”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 10, Oct. 2021, pp. 1391-03, doi:10.37863/umzh.v73i10.6785.
Research articles