Equilibrium internal fractures in elastic bodies supported by thin flexible coatings
Objective. In this paper, the authors study problems of a plane strain of elastic bodies containing internal rectilinear fractures. In each case, the margins of the considered areas are supported by thin flexible coatings. The first part of the paper is devoted to the problem of an infinite elastic...
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| Format: | Article |
| Language: | Russian |
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Dagestan State Technical University
2020-10-01
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| Series: | Вестник Дагестанского государственного технического университета: Технические науки |
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| Online Access: | https://vestnik.dgtu.ru/jour/article/view/847 |
| _version_ | 1797700885132869632 |
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| author | B. V. Sobol E. V. Rashidova P. V. Vasiliev A. I. Novikova |
| author_facet | B. V. Sobol E. V. Rashidova P. V. Vasiliev A. I. Novikova |
| author_sort | B. V. Sobol |
| collection | DOAJ |
| description | Objective. In this paper, the authors study problems of a plane strain of elastic bodies containing internal rectilinear fractures. In each case, the margins of the considered areas are supported by thin flexible coatings. The first part of the paper is devoted to the problem of an infinite elastic wedge, the faces of which are free from the outside and reinforced with a thin flexible material, and the bisector contains a rectilinear fracture with regular forces applied to the margins, and to the study of the stress concentration at the fracture vertices. In the second part of the paper, the authors consider the problem of an equilibrium radial internal fracture in the cross-section of a round pipe. The inner surface of the pipe experiences hydrostatic pressure; the outer surface is reinforced with a thin flexible coating. The purpose of the study in each of the presented tasks is to determine the values of the influence factor. Methods. Both problems are united by a single approach, in which the presence of a coating is modeled mathematically, using special marginal conditions obtained based on an asymptotic analysis of the exact solution for a strip or ring flexible coating of small relative thickness. In the first issue, the singular integral equation is derived using the Mellin transform, which allows proceeding to the solution of a system of ordinary differential equations and obtaining a singular integral equation relative to the derivative of the discontinuity function of the first kind with a Cauchy kernel. In the second issue, discontinuous solutions are constructed using the Fourier series, resulting in a singular integral equation of a similar structure. Previously, similar ideas were successfully implemented by the authors in the study of the problem of the equilibrium state of a strip with a coating weakened by an internal transverse fracture under arbitrary conditions on the lower edge of the strip. Conclusion. Singular integral equations for the considered problems are obtained. The collocation method is used to construct solutions of singular integral equations for various combinations of geometric and physical characteristics of issues. In all the considered cases, the values of the influence factor were calculated. The analysis of changes in the influence factor depending on various combinations of geometric parameters and mechanical characteristics of problems is carried out. It is noted that with increasing rigidity of the coating and increasing its thickness, the values of the influence factor decrease; the increase in the value of the influence factor is provided by approaching the fracture to the body margin and increasing its relative length. |
| first_indexed | 2024-03-12T04:27:58Z |
| format | Article |
| id | doaj.art-7cbc3bffa8164a3286635c6437e23d13 |
| institution | Directory Open Access Journal |
| issn | 2073-6185 2542-095X |
| language | Russian |
| last_indexed | 2024-03-12T04:27:58Z |
| publishDate | 2020-10-01 |
| publisher | Dagestan State Technical University |
| record_format | Article |
| series | Вестник Дагестанского государственного технического университета: Технические науки |
| spelling | doaj.art-7cbc3bffa8164a3286635c6437e23d132023-09-03T10:15:21ZrusDagestan State Technical UniversityВестник Дагестанского государственного технического университета: Технические науки2073-61852542-095X2020-10-0147311112110.21822/2073-6185-2020-47-3-111-121599Equilibrium internal fractures in elastic bodies supported by thin flexible coatingsB. V. Sobol0E. V. Rashidova1P. V. Vasiliev2A. I. Novikova3Донской государственный технический университетДонской государственный технический университетДонской государственный технический университетДонской государственный технический университетObjective. In this paper, the authors study problems of a plane strain of elastic bodies containing internal rectilinear fractures. In each case, the margins of the considered areas are supported by thin flexible coatings. The first part of the paper is devoted to the problem of an infinite elastic wedge, the faces of which are free from the outside and reinforced with a thin flexible material, and the bisector contains a rectilinear fracture with regular forces applied to the margins, and to the study of the stress concentration at the fracture vertices. In the second part of the paper, the authors consider the problem of an equilibrium radial internal fracture in the cross-section of a round pipe. The inner surface of the pipe experiences hydrostatic pressure; the outer surface is reinforced with a thin flexible coating. The purpose of the study in each of the presented tasks is to determine the values of the influence factor. Methods. Both problems are united by a single approach, in which the presence of a coating is modeled mathematically, using special marginal conditions obtained based on an asymptotic analysis of the exact solution for a strip or ring flexible coating of small relative thickness. In the first issue, the singular integral equation is derived using the Mellin transform, which allows proceeding to the solution of a system of ordinary differential equations and obtaining a singular integral equation relative to the derivative of the discontinuity function of the first kind with a Cauchy kernel. In the second issue, discontinuous solutions are constructed using the Fourier series, resulting in a singular integral equation of a similar structure. Previously, similar ideas were successfully implemented by the authors in the study of the problem of the equilibrium state of a strip with a coating weakened by an internal transverse fracture under arbitrary conditions on the lower edge of the strip. Conclusion. Singular integral equations for the considered problems are obtained. The collocation method is used to construct solutions of singular integral equations for various combinations of geometric and physical characteristics of issues. In all the considered cases, the values of the influence factor were calculated. The analysis of changes in the influence factor depending on various combinations of geometric parameters and mechanical characteristics of problems is carried out. It is noted that with increasing rigidity of the coating and increasing its thickness, the values of the influence factor decrease; the increase in the value of the influence factor is provided by approaching the fracture to the body margin and increasing its relative length.https://vestnik.dgtu.ru/jour/article/view/847трещинабесконечный упругий клинупругое кольцотонкое покрытиетригонометрические рядыинтегральное преобразование меллинаметод разрывных решенийкоэффициент интенсивности напряженийфактор влиянияметод коллокации |
| spellingShingle | B. V. Sobol E. V. Rashidova P. V. Vasiliev A. I. Novikova Equilibrium internal fractures in elastic bodies supported by thin flexible coatings Вестник Дагестанского государственного технического университета: Технические науки трещина бесконечный упругий клин упругое кольцо тонкое покрытие тригонометрические ряды интегральное преобразование меллина метод разрывных решений коэффициент интенсивности напряжений фактор влияния метод коллокации |
| title | Equilibrium internal fractures in elastic bodies supported by thin flexible coatings |
| title_full | Equilibrium internal fractures in elastic bodies supported by thin flexible coatings |
| title_fullStr | Equilibrium internal fractures in elastic bodies supported by thin flexible coatings |
| title_full_unstemmed | Equilibrium internal fractures in elastic bodies supported by thin flexible coatings |
| title_short | Equilibrium internal fractures in elastic bodies supported by thin flexible coatings |
| title_sort | equilibrium internal fractures in elastic bodies supported by thin flexible coatings |
| topic | трещина бесконечный упругий клин упругое кольцо тонкое покрытие тригонометрические ряды интегральное преобразование меллина метод разрывных решений коэффициент интенсивности напряжений фактор влияния метод коллокации |
| url | https://vestnik.dgtu.ru/jour/article/view/847 |
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