Consideration of the width of crack faces contact zone under bending of a piecewise-homogeneous isotropic plate when assessing the stress-strain state in its tip

Keywords

stress-strain state, crack, piecewise-homogeneous plate, contact domain, complex potentials, contact force.

Cite as

Yatsyk I. M., Zvizlo I. S., and Slobodian M. S. Consideration of the width of crack faces contact zone under bending of a piecewise-homogeneous isotropic plate when assessing the stress-strain state in its tip. Physicochemical Mechanics of Materials. 2026. 62(1), 112-119.

https://doi.org/10.15407/pcmm2026.01.112

Abstract

The stress-strain state of an infinite, piecewise-homogeneous isotropic plate containing a crack, oriented perpendicular to the material interface, is analyzed. The crack surfaces are not subjected to external loads. The plate experiences uniformly distributed bending mo­ments. It is assumed that the crack faces are in smooth contact along their entire length within a two-dimensional region of constant width near the upper edge of the plate. This contact condition allows the problem to be expanded into two interconnected subproblems: a plane problem of elasticity and a plate bending problem based on the Kirchhoff–Love theory. By applying complex variable methods and complex potentials, a system of singular integral equations is derived, which is then solved numerically using the mechanical quadrature me­thod. The numerical analysis includes evaluation of the contact forces between the crack faces and graphical representation of their dependence on various problem parameters.

References

1. M.P. Savruk, Two-dimensional Elasticity Problems for Bodies with Cracks [in Russian], Naukova Dumka, Kyiv (1981).
2. I.A. Prusov, Conjugation Method in Plate Theory [in Russian], Belorus Universytet Publ. House, Minsk (1974).
3. L.T. Berezhnytskyi, M.V. Delyavskui, and V.V. Panasyuk, Bending of Thin Plates with Crack-type Defects [in Russian], Naukova Dumka, Kyiv (1979).
4. R.S. Alwar and K.N. Ramachandran Nambissan, “Influence of crack closure on the stress intensity factor for plates subjected to bending – A 3-D finite element analysis,” Eng. Fract. Mech. 17, Is. 4, 323-333 (1983). https://doi.org/10.1016/0013-7944(83)90083-8
5. D.P. Jones and J.L. Swedlow, “The influence of crack closure and elasto-plastic flow on the bending of a cracked plate,” Int. J. Fract., 11, Is. 6, 897-914 (1975). https://doi.org/10.1007/BF00033836
6. Jr.F.S. Heming, “Sixth order analysis of crack closure in bending of an elastic plate,” Int. J. Fract., 16, Is. 4, 289-304 (1980). https://doi.org/10.1007/BF00018233
7. M.L. Williams, “The bending stress distribution at the base of a stationary crack,” Trans ASME. J. Appl. Mech., 28, 78-82 (1961). https://doi.org/10.1115/1.3640470
8. V.K. Opanasovych, “Bending of a plate with a through rectilinear crack taking into account the width of the contact area of its surfaces,” [in Ukrainian], Naukovi Notatky Lytskogo Tekcnichnogo Universytetu [in Ukrainian], 20, Is. 2, 123-127 (2007).
9. V.K. Opanasovych and I. Yatsyk, “Bending of a Reissner plate with a through crack taking into account the width of the contact area of its edges,” Visnyk Lvivskogo Universytetu [in Ukrainian], 69, 125-135 (2008).
10. I.P. Shatskii, “Bending of a plate weakened by a cut with contacting edges,” [in Ukrainian], Dopovidi Akademii Nauk Ukrainy, Ser. Fizyko-Maematychni ta Tekhnichni Nauky, Is. 7, Is. 7, 49-51 (1988).
11. I.P. Shats’kii and V.V. Perepichka, “Bending of a semiinfinite plate weakened by a crack with contacting edges,” Soviet Materials Science, 28, Is. 2, 154-157 (1992). https://doi.org/10.1007/BF00727310
12. I. Shatskii, V. Perepichka, T. Dalyak, and A. Shcherbii, “Problems of the theory of plates and shells with interconnected boundary conditions on sections,” Mathematical Problems of Mechanics of Inhomogeneous Structures [in Ukrainian], 2, 51-54 (2000).
13. J.P. Dempsey, I.I. Shekhtman, and L.I. Slepyan, “Closure of a through crack in a plate under bending,” Int. J. Solids Struct., 35, Iss. 31-32, 4077-4089 (1998). https://doi.org/10.1016/S0020-7683(97)00302-8
14. M.J. Young and C.T. Sun, “Influence of crack closure on the stress intensity factor in bending plates – A classical plate solution,” Int. J. Fract. 55, 81-93 (1992). https://doi.org/10.1007/BF00018034
15. M. Slobodian, I. Zvizlo, O. Bilash, M. Sorokatyi, O. Petruchenko, and L. Markevych, “Bending of a piecewise homogeneous plate with a circular interfacial materials separation zone and radial crack considering the strip contact of its edges,” Vibroengineering Proc., 55, 54-59 (2024). https://doi.org/10.21595/vp.2024.24515
16. M.S. Slobodian and M.I. Shainoga, “Stress-strain state of bent plate with system of cracks with faces contacting within constant-height zone,” Int. Appl. Mech., 60, Is. 2, 188-202 (2024). https://doi.org/10.1007/s10778-024-01273-x
17. N.I. Muskhelishvili, Some basic Problems of the Mathematical Theory of Elasticity, Berlin: Springer (1977). https://doi.org/10.1007/978-94-017-3034-1
18. V.V. Panasyuk, M.P. Savruk, and A.P. Datsyshyn, Distribution of Stresses Around Cracks in Plates and Shells [in Russian], Naukova Dumka, Kyiv (1976).
19. V.K. Opanasovych, and I.S. Zvizlo, “Bending of a piecewise homogeneous isotropic plate with a crack perpendicular to the material interface, taking into account the contact of the edges,” [in Ukrainian], Systemni Technologii, 4, Is. 21, 124-129 (2002).