Analytical solution of the equilibrium of the transverse crack in a composite elastic plane
UDC
539.3Abstract
The problem of stress concentration in the vicinity of the crack tip of finite length perpendicularly located to the interface between two elastic half-planes was reviewed. The usage of integral discontinuous solutions method enabled us to reduce the problem to solving a singular integral equation of the first kind with Cauchy kernel. We reviewed the limiting case of the problem when the half-plane without the crack is absent and the boundary of the body under review is free from stress. The solution of the integral equation is constructed in the form of an asymptotic expansion with respect to the small parameter characterizing the relative distance between the crack and the interface. In order to determine the effectiveness of obtained expansions we construct the solution of the same problem by collocation method. It was executed a comparison of the results with the known ones, previously obtained by numerical methods. The values of the normal stress intensity factor in the vicinity of the crack tip for various combinations of geometrical and physical parameters of the problem were obtained.
Keywords:
crack, half-plane, stress intensity factor, influence factorFunding information
Исследование выполнено при финансовой поддержке РФФИ (14-08-00142_а).
References
- Zak A.R., Williams M.L. Crack point singularities at a biomaterial interface // Journal of Applied Mechanics. 1963. Vol. 30. P. 142-143.
- Cook T.S., Erdogan F. Stress in bounded material with a crack perpendicular to the interface // Int. J. Engng. Sci. 1972. Vol. 10. P. 677-697.
- Erdogan F., Biricikoglu B. Two bonded half plane with a crack through the interface // Int. J. Engng. Sci. 1973. Vol. 11. P. 745-766.
- Bogy D.B. On the plane elastic problem of a loaded crack terminating a material interface // Int. J. Fract. 1971. Vol. 38. P. 911-918.
- Chen S.H., Wang T.C., Kao-Walter S. A crack perpendicular to the bimaterial interface in finite solid // Int. J. Solids Struct. 2003. Vol. 40. P. 2731-2755.
- Макарян В.С., Чилингарян Г.С. Напряженное состояние составной упругой плоскости с трещиной, перпендикулярной к границе раздела материалов // Mechanics. Proceedings of National Academy of Sciences of Armenia. 2008. No. 61(2). P. 3-12. [Makaryan V.S., Chilingaryan G.S. Napryazhennoe sostoyanie sostavnoy uprugoy ploskosti s treshchinoy, perpendikulyarnoy k granitse razdela materialov [The stress state of the composite elastic plane with a crack perpendicular to the interface between materials]. Mechanics. Proc. of National Academy of Sciences of Armenia, 2008, no. 61(2), pp. 3-12. (In Russian)]
- Александров В.М., Мхитарян С.М. Контактные задачи для тел с тонкими покрытиями и прослойками. М.: Наука, 1983. 488 с. [Aleksandrov V.M., Mkhitaryan S.M. Kontaktnye zadachi dlya tel s tonkimi pokrytiyami i prosloykami [Contact problems for bodies with thin coatings and layers]. Moscow: Nauka Publ., 1983, 488 p. (In Russian)]
- Мхитарян С.М. О контакте между бесконечным стрингером и упругой полубесконечной пластиной с вертикальной трещиной. Актуальные проблемы механики сплошной среды: Труды III межд. конф., Ереван, 2012. Т. 1. С. 74-78. [Mkhitaryan S.M. O kontakte mezhdu beskonechnym stringerom i uprugoy polubeskonechnoy plastinoy s vertikal'noy treshchinoy [On contact between the infinite stringer and semi-infinite elastic plate with a vertical crack]. In Sbornik trudov III mezhd. konf. "Aktual'nye problemy mekhaniki sploshnoy sredy", Erevan, 2012. vol. 1. [Proc. of the III Int. Conf. 'Topical problems of continuum mechanics', Yerevan, 2012] pp. 74-78. (In Russian)]
- Мураками Ю. Справочник по коэффициентам интенсивности напряжений. М.: Мир, 1990. 556 с. [Murakami Yu. Spravochnik po koeffitsientam intensivnosti napryazheniy [The reference of coefficients of stress intensity]. Moscow, Mir Publ., 1990, 556 p.]
- Александров В.М., Сметанин Б.И., Соболь Б.В. Тонкие концентраторы напряжений в упругих телах. М.: Физматлит, 1993. 224 c. [Aleksandrov V.M., Smetanin B.I., Sobol' B.V. Tonkie kontsentratory napryazheniy v uprugikh telakh [Thin stress concentrators in elastic bodies]. Moscow, Fizmatlit Publ., 1993, 224 p. (In Russian)]
- Соболь Б.В., Краснощеков А.А. Влияниe тонкой накладки на коэффициент интенсивности напряжений в задачах о поперечной трещине в полуплоскости и полосе // Вестник ДГТУ. 2013. № 5/6 (74). С. 25-35. [Sobol' B.V., Krasnoshchekov A.A. Vliyanie tonkoy nakladki na koeffitsient intensivnosti napryazheniy v zadachakh o poperechnoy treshchine v poluploskosti i polose [Thin plate effect on stress intensity factor in problems on transverse crack in half-plane and stripe]. Vestnik DSTU [Proc. of Dagestan State Technical University], 2013, no. 5/6(74), pp. 25-35. (In Russian)]
- Саврук М.П. Двумерные задачи упругости для тел с трещинами. Киев: Наук. думка, 1981. 324 с. [Savruk M.P. Dvumernye zadachi uprugosti dlya tel s treshchinami [Two-dimensional problems of elasticity for bodies with cracks]. Kiev, Naukova Dumka Publ., 1981, 324 p. (In Russian)]
- Градштейн И.С., Рыжик И.М. Таблицы интегралов, сумм, рядов и произведений. М.: Наука, 1971. 1108 с. [Gradshtejn I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products]. Moscow: Nauka, 1971. 1108 p. (In Russian)]
Downloads
Downloads
Dates
Submitted
Accepted
Published
How to Cite
License
Copyright (c) 2014 Соболь Б.В., Рашидова Е.В., Борисова Е.В.
This work is licensed under a Creative Commons Attribution 4.0 International License.
