Stress distribution in a half-plane with a circular hole under tension at an angle to a straight edge
DOI:
https://doi.org/10.31734/agroengineering2024.28.117Keywords:
concentration of stresses near holes, biharmonic functions of stresses, bipolar coordinatesAbstract
In modern industries such as project construction, aircraft manufacturing, shipbuilding, and machine engineering, thin elastic plates are commonly used. Depending on various factors, these plates feature different types of holes, which can lead to strain concentration when loaded near these holes. This concentration of strain can have an unfavorable impact on the durability of the component. The distribution of strains around the contours of the holes is uneven, leading to small sections that experience high strain levels. These critical sections are often where brittle cracks or plastic deformations develop, potentially resulting in structural failure.
Therefore, studying the distribution of strains near curved holes is essential from both theoretical and engineering perspectives. This work presents a solution to the problem of stress concentration in a half-plane with a circular hole, subjected to stretching at an arbitrary angle to a straight edge. The problem is approached by utilizing the main stress function, which corresponds to the stress state in the half-plane without the hole. A second biharmonic function is added to this function, accounting for the additional stress state introduced by the hole's presence. The task involves finding a biharmonic stress function that satisfies the boundary conditions along the contour of the hole, on the straight edge, and at infinity. The novelty of this research lies in its provision of insights regarding the impact of the edge orientation of the half-plane concerning the tensile load field on stress concentration. The solution is
presented in bipolar coordinates, and formulas for the stresses along the contour of the hole and on the straight edge are derived. The stress values obtained for various key points along the hole’s contour and the straight edge for specific cases provide a basis for establishing the coefficient of strain concentration near the holes. This information can be beneficial in engineering practices during the design and development of components in the aircraft, shipbuilding, and machine manufacturing industries.
References
Beihul, O. O., & Lepetova, H. L. (2014). Metody teorii pruzhnosti dlia doslidzhennia ta rozrahunkiv metalurhiinoho obladnannda: Navch. posib. Dniprodzerzhynsk: Dniprodzerzh. derg. techicn. un-t.
Dovbnia, K., & Vrublevskyi, V. (2018). Doslidzhennia napriazhonnoho stanu v ortotropnii plastyni z dvoma kruhovymy otvoramy ta trishchynoiu. Suchasni problemy mekhaniky i matematyky: Materialy Mizhnar. nauk. konf. (m. Lviv, 22-25 trav. 2018 r.). (T. 2, s. 33-35). Lviv: IPPMM.
Gart E., Semencha, O. (2023). Skinchennoelementnyi analiz napruzheno-deformovanogo stanu tonkykh plastyn, tsylindrichnykh i konichnykh obolonok z otvoramy i strichkovymy vkluchenniamy. Suchasni problemy mechaniky i matematyky - 2023: Materialy mizhnar. nauk. konf. (m. Lviv, 23-25 trav. 2023 r., s. 173–174). Lviv: IPPMM.
Kaloerov, S. A. (1998). Reshenije osnovnych zadach teorii uprugosti dlija poluploskosti c otverstijami I trechchinami. Teoreticheskaja i prikladnaja mechanica, 28, 157–171.
Kaloerov, S. A. (2004). Priblizhonnyi metod issledovanija naprizhonnogo sosyoyaniya izotropnoy poluploskosti i polosy s otverstijami i trechinami. Teoreticheskaia i prikladnaia mekhanica, 39, 83–93.
Kaloerov, S. A., & Avdiushina, E. V. (2000). Napriazhonnoie sostoianie gornogo massiva s vyrobotkoy vblizi dnevnoj poverchnosti. Deformacija i razrushenije materialov s defectami i dinamicheskie javlenija в gornych porodach i vyrabotkacg: Sb. nauch. tr. X Mezhdunar. nauch. Shkoly (g. Alushta, 18-24 sent. 2000 g.). (s. 60–62). Simferopol.
Kaloerov, S. A., & Avdiushina, E. V. (2004). Napriazhonnoie sostoianie gornogo massiva s vyrobotkami vblizi zagrugennoj dnevnoj poverchnosti. Naukoni praci Donetskogo natsionalnoho tekhnichnoho universytetu. Serija girnycho-electromechanichna, 83, 129–134.
Kaloerov, S. A., & Vakulenko, S. V. (2004). Reshenie tsyklicheskoi zadachi dlia plastinki s otverstiiami i treshinami i yeio prilozhenie v gornom dele. Visnyk Doneckogo universytety. Seria A: Pryrodnychi nauki, 1, 37–42.
Kaloerov, S. A., Avdushina, E. V., & Myronenko, A. B. (2013). Kontsentratsiia napriazheniy v mnogosviaznych izotropnych plastinkach. Doneck: Doneck. nac. un-t.
Kravets, V. (2018). Napruzheno-deformovanyi stan ploshchyny z periodychnoiu systemoiu otvoriv z kraiovymy trishchynamy abo smuhamy plastychnosti. Suchasni problemy mekhaniky i matematyki: Materialy Mizhnar. nauk. konf. (m. Lviv, 22-25 trav. 2018 r.). (Т. 2, s. 44–47). Lviv: IPPMM.
Mushelishvili, N. I. (1966). Nekotorye osnovnye zadachi matematicheskoi teorii uprugosti. Moskva: Nauka.
Onyshko, L. Y. Varyvoda, I. I., & Ponomarenko, O. M. (2011). Doslidzhennia dynamichnoi concentracii napruzhen na kraiu kolovoho otvoru za dii na nioho neosesymetrychnoho navantazhennja. Naukovyj visnyk LNUBMBT im. S. Z. Gzhytskoho, 13(4), 106–114.
Protsenko, V., & Ukrainets, N. (2018). Analiz napruzheno-deformovanoho stanu pivprostoru z neskinchennoiu tsylindrychnoiu porozhnynoiu. Suchasni problemy mekhaniky i matematyki: Materialy Mizhnar. nauk. konf. (m. Lviv, 22-25 trav. 2018 r.). (T. 2, s. 85-87). Lviv: IPPMM.
Savin, H. N. (1968). Rospredelenie napriagenij okolo otverstij. Kyev: Nayk. dumka.
Shopa, T. (2018). Doslidzhennia dynamichnoi povedinky ortotropnych plastyn z otvoramy ta vkliuchenniamy. Suchasni problemy mekhaniky i matematyki: Materialy Mizhnar. nauk. konf. (m. Lviv, 22-25 trav. 2018 r.). (T. 2, s. 168-169). Lviv: IPPMM.
Slobodian, M., & Tsurkan, M. (2018). Roztiah plastyny z kruhovym otvorom ta dvoma radialnymy trishchynamy z urakhyvanniam plastychnych zon poblyzu ikh vershyn. Suchasni problemy mekhaniky i matematyki: Materialy Mizhnar. nauk. konf. (m. Lviv, 22-25 trav. 2018 r.). (T. 2, s. 96-98). Lviv: IPPMM.
Solar, T., & Maksymovych, O. (2018). Rehuliaryzatsiia formuly obernennia peretvorennja Laplasa stosovno vyznachennia koncentratsii dynamichnych napruzhen u plastynkach z otvoramy. Suchasni problemy mekhaniky i matematyki: Materialy Mizhnar. nauk. konf. (m. Lviv, 22-25 trav. 2018 r.). (T. 2, s. 161–163). Lviv: IPPMM.
Sovremennyie problemу koncentracii napriazhenij: Tr. Mezhdynar. nauch. conf., posviach. 75-letiyu akademika NAN Ukrainy A. S. Kosmodamianskogo. (1998). Donestk.
Suchasni problemy mekhaniky і matematyky: Pratsi Mizhnar. nauk. konf., prysviach. 90-richhiu vid dnia narodzhennia akademika NAN Ukrainy I. S. Pidstryhacha (m. Lviv, 22-25 trav. 2018 r.). (2018). Lviv: JPPMM.
Ufliand, I. S. (1968). Integralnye preobrazovanija v zadachah teorij uprugosti. Leningrad: Nauka.
Vakulenko, S. V., & Kaloerov, S. A. (2002). Priblizhonyi metod opredeleniia napriazhonnogo sostoyaniya mnogosviaznoi izotropnoi poluploskosti s otverstiiami i treshchinami. Teoreticheskaia i prikladnaia mekhanika, 35, 65-76.
