# Partially closure of rectilinear crack emanating from contour of circular hole in stringer plate

## Abstract

The technical holes existing in plates create an increased concentration of stress in the plate. In present article, a thin plate with a circular hole from which a rectilinear crack emanates is studied. The plate is reinforced by stringers. The model of crack with interfacial bonds in end zone is used. The plate and reinforcing ribs are made of different elastic and isotropic materials. It is assumed that the stringers are not bending and their thickness does not change during deformation. The plate is assumed to be unbounded and subjected to stretching at infinity. The case of partial crack closure is considered. The action of the stringers is replaced by unknown equivalent concentrated forces applied at the points of connection of the ribs and the plate. To solve the problem under consideration, the method of solution of the elastic problem and the method of construction in explicit form of the Kolosov - Muskhelishvili potentials corresponding to unknown normal displacements along a rectilinear crack are combined. To determine the parameters that characterize the crack closure, a singular integral equation is obtained and converted to a finite nonlinear algebraic system. To determine the unknown equivalent concentrated forces, Hooke's law is used. Solution of the algebraic system was obtained using the method of successive approximations. Directly from the solution of the obtained algebraic systems the cohesive forces in the bonds, contact stresses and size of the crack contact zone were found. Using the obtained relations it is possible to solve the inverse problem, i.e. to determine the characteristics and stress state of the stringer-reinforced thin plate with a circular hole at which the predetermined contact area of the faces of the rectilinear crack emanating from the hole is reached.

## Keywords

Institute of Mathematics and Mechanics of Azerbaijan NAS

Author for correspondence.

Cand. Sci. (Eng.), Leading Researcher Associate of the Creep Theory Department, Institute of Mathematics and Mechanics of the NAS of Azerbaijan. Scientific interests: theory of elasticity, fracture mechanics of plates

9 B. Vahabzadeh St., Baku, AZ1141, Azerbaijan

## References

1. Mirsalimov V.M. (1977). Issledovanie predel’nogo polya napryazhenij vozle treshhin, iskhodyashhih iz konturov otverstij perforirovannoj plastiny [Study of maximum stress field alongside cracks emerging from contours of openings in a perforated plate]. Journal of Applied Mechanics and Technical Physics, (2), 147–154. (In Russ.)
2. Mirsalimov V.M. (1979). Uprugoplasticheskoe ravnovesie plastiny, oslablennoj dvojakoperiodicheskoj sistemoj kruglyh otverstij i treshhinami, vyhodjashhimi na kontury otverstij [Elastic-plastic equilibrium of the plate with the double-periodical system of the round orifices and cracks running to the orifices contour]. Izvestiya AN AzSSR. Seriya Fiz.-tekh. i mat. nauk, (2), 118–125. (In Russ.)
3. Mirsalimov V.M. (1980). Hrupkoe razrushenie plastiny, oslablennoj perio-dicheskoj sistemoj kruglyh otverstij s vyhodyashhimi na ih kontury treshhinami [Brittle fracture of a plate weakened by a periodic system of circular holes with cracks emanating from their contours]. International Applied Mechanics, 16(11), 992–997. (In Russ.)
4. Mir-Salim-zadeh M.V. (2003). Fracture of an elastic rib reinforced plate weakened by a circular cracked hole. International Journal of Fracture, 122, L113–L117.
5. Yan X. (2006). Cracks emanating from circular hole or square hole in rectangular plate in tension. Engineering Fracture Mechanics, 73(12), 1743–1754.
6. Abdelmoula R., Semani K., Li J. (2007). Analysis of cracks originating at the boundary of a circular hole in an infinite plate by using a new conformal mapping approach. Applied Mathematics and Computation, 188(2), 1891–1896.
7. Mirsalimov V.M., Shahbandaev E.G. (2008). Predel'noe ravnovesie teplovydeljajushhej sredy s periodicheskoj sistemoj otverstij i prjamolinejnyh treshhin [Limit equilibrium of heat-generating medium with a periodic system of holes and rectilinear cracks]. Vestnik I. Yakovlev Chuvach State Pedagogical University. Series: Mechanics of a limit state, (1), 98–107. (In Russ).
8. Mir-Salim-zade M.V. (2008). Predel’noe ravnovesie plastiny s regulyarnoj sistemoj stringerov i ishodyashhimi iz krugovogo otverstiya treshhinami [Ultimate state of a plate with a regular system of stringers and cracks issuing from a circular hole]. Journal of Machinery Manufacture and Reliability, 37, 44–51. (In Russ.)
9. Shahbandaev E.G. (2008). On partial closing of cracks in heat-releasing medium weakened by a periodic system of circular holes. Proceedings of IMM of NAS of Azerbaijan, XXIX(XXXVII), 215–224.
10. Chen Y.Z., Lin X.Y., Wang Z.X. (2009). A semianalytic solution for multiple curved cracks emanating from circular hole using singular integral equation. Applied Mathematics and Computation, 213, 389–404.
11. Guo J.-H., Lu Z.-X., Feng X. (2010). The fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials. Acta Mechanica, 215(1–4), 119–134.
12. Tong D.H., Wu X.R. (2013). Determination of crack surface displacements for cracks emanating from a circular hole using weight function method. Fatigue & Fracture of Engineering Materials & Structures, 36, 340–348.
13. Hasanov F.F. (2013). Modelirovanie zarozhdenija treshhin sdviga v tele, oslablennom periodicheskoj sistemoj kruglyh otverstij [Modeling of shear crack nucleation in a body, weakening by periodic system of circular holes]. Journal of mechanical engineering, 16(3), 29–37. (In Russ.)
14. Iskenderov R.A. (2013). Zarozhdenie treshhiny pri poperechnom izgibe izotropnoj plastiny, oslablennoj periodicheskoj sistemoj krugovyh otverstij [The crack nucleation in the isotropic plate, weakened by a periodical system of circular holes under transverse bending]. Structural Mechanics of Engineering Constructions and Buildings, (3), 18–28. (In Russ.)
15. Mirsalimov V.M., Akhmedova M.V. (2013). Uprugoplasticheskoe razrushenie tonkoj plastiny, oslablennoj periodicheskoj sistemoj krivolinejnyh otverstij [Elastoplastic fracture of a thin plate, weakened by periodic system of the curvilinear holes]. I. Yakovlev Chuvach State Pedagogical University Bulletin. Series: Mechanics of a limit state, (1), 133–144. (In Russ.)
16. Liu T. J.-C. (2014). Joule heating behaviors around through crack emanating from circular hole under electric load. Engineering Fracture Mechanics, 123, 2–20.
17. Yang J., Li X. (2016). Analytic solutions of problem about a circular hole with a straight crack in onedimensional hexagonal quasicrystals with piezoelectric effects. Theoretical and Applied Fracture Mechanics, 82, 17–24.
18. Mirsalimov V.M. (2017). Cracks with interfacial bonds in perforated heat-releasing nuclear fuel. International Journal of Damage Mechanics. https://doi.org/10.1177/ 1056789517713072.
19. Mirsalimov V.M. (1986). Nekotorye zadachi konstrukcionnogo tormozheniya treshhin [Some problems of structural arrest of cracks]. Materials Science, 22, 84–88. (In Russ.)
20. Savruk M.P., Kravets V.S. (1995). Reinforcement of a thin cracked plate by a system of parallel stringers. Materials Science, 30, 95–104.
21. Mir-Salim-zada M.V. (2010). Modelirovanie chastichnogo zakrytiya treshhin v perforirovannoj izotropnoj srede, usilennoj regulyarnoj sistemoj stringerov [Modeling of partial closure of cracks in a perforated isotropic medium reinforced by a regular system of stringers]. Journal of Applied Mechanics and Technical Physics, 51, 148–159. (In Russ.)
22. Muskhelishvili N.I. (1977). Nekotorye osnovnye zadachi matematicheskoj teorii uprugosti [Some basic problems of mathematical theory of elasticity]. Moscow, Nauka Publ., 707.
23. Panasyuk V.V., Savruk M.P., Datsyshin A.P. (1976). Raspredelenie naprjazhenij okolo treshhin v plastinah i obolochkah [Distribution of stresses near cracks in plates and shells]. Kiev, Naukova Dumka Publ., 443. (In Russ.)
24. Mirsalimov V.M. (1987). Neodnomernye uprugoplasticheskie zadachi [Non-one-dimensional elastoplastic problems]. Moscow, Nauka Publ., 256. (In Russ.)