Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks

The fracture mechanical behaviour of thin-walled structures with cracks is highly significant for structural strength design, safety and reliability analysis, and defect evaluation. In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of...
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Autor*in:	Feng Yue [verfasserIn] Ziyan Wu [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2021

Schlagwörter:	crack thin-walled structures fracture mechanical parameters crack initiation angle plastic zone

Übergeordnetes Werk:	In: Metals - MDPI AG, 2012, 11(2021), 4, p 592
Übergeordnetes Werk:	volume:11 ; year:2021 ; number:4, p 592

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.3390/met11040592

Katalog-ID:	DOAJ057723265

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520			\|a The fracture mechanical behaviour of thin-walled structures with cracks is highly significant for structural strength design, safety and reliability analysis, and defect evaluation. In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of thin-walled cylindrical shells with cracks are investigated. First, based on the J-integral and displacement extrapolation methods, the stress intensity factors of thin-walled cylindrical shells with circumferential cracks and compound cracks are studied using linear elastic fracture mechanics, respectively. Second, based on the theory of maximum circumferential tensile stress of compound cracks, the number of singular elements at a crack tip is varied to determine the node of the element corresponding to the maximum circumferential tensile stress, and the initiation angle for a compound crack is predicted. Third, based on the J-integral theory, the size of the plastic zone and J-integral of a thin-walled cylindrical shell with a circumferential crack are analysed, using elastic-plastic fracture mechanics. The results show that the stress in front of a crack tip does not increase after reaching the yield strength and enters the stage of plastic development, and the predicted initiation angle of an oblique crack mainly depends on its original inclination angle. The conclusions have theoretical and engineering significance for the selection of the fracture criteria and determination of the failure modes of thin-walled structures with cracks.
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allfields	10.3390/met11040592 doi (DE-627)DOAJ057723265 (DE-599)DOAJ360753cb27d5490eb210a3c8af64987a DE-627 ger DE-627 rakwb eng TN1-997 Feng Yue verfasserin aut Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The fracture mechanical behaviour of thin-walled structures with cracks is highly significant for structural strength design, safety and reliability analysis, and defect evaluation. In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of thin-walled cylindrical shells with cracks are investigated. First, based on the J-integral and displacement extrapolation methods, the stress intensity factors of thin-walled cylindrical shells with circumferential cracks and compound cracks are studied using linear elastic fracture mechanics, respectively. Second, based on the theory of maximum circumferential tensile stress of compound cracks, the number of singular elements at a crack tip is varied to determine the node of the element corresponding to the maximum circumferential tensile stress, and the initiation angle for a compound crack is predicted. Third, based on the J-integral theory, the size of the plastic zone and J-integral of a thin-walled cylindrical shell with a circumferential crack are analysed, using elastic-plastic fracture mechanics. The results show that the stress in front of a crack tip does not increase after reaching the yield strength and enters the stage of plastic development, and the predicted initiation angle of an oblique crack mainly depends on its original inclination angle. The conclusions have theoretical and engineering significance for the selection of the fracture criteria and determination of the failure modes of thin-walled structures with cracks. crack thin-walled structures fracture mechanical parameters crack initiation angle plastic zone Mining engineering. Metallurgy Ziyan Wu verfasserin aut In Metals MDPI AG, 2012 11(2021), 4, p 592 (DE-627)718627172 (DE-600)2662252-X 20754701 nnns volume:11 year:2021 number:4, p 592 https://doi.org/10.3390/met11040592 kostenfrei https://doaj.org/article/360753cb27d5490eb210a3c8af64987a kostenfrei https://www.mdpi.com/2075-4701/11/4/592 kostenfrei https://doaj.org/toc/2075-4701 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 11 2021 4, p 592
spelling	10.3390/met11040592 doi (DE-627)DOAJ057723265 (DE-599)DOAJ360753cb27d5490eb210a3c8af64987a DE-627 ger DE-627 rakwb eng TN1-997 Feng Yue verfasserin aut Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The fracture mechanical behaviour of thin-walled structures with cracks is highly significant for structural strength design, safety and reliability analysis, and defect evaluation. In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of thin-walled cylindrical shells with cracks are investigated. First, based on the J-integral and displacement extrapolation methods, the stress intensity factors of thin-walled cylindrical shells with circumferential cracks and compound cracks are studied using linear elastic fracture mechanics, respectively. Second, based on the theory of maximum circumferential tensile stress of compound cracks, the number of singular elements at a crack tip is varied to determine the node of the element corresponding to the maximum circumferential tensile stress, and the initiation angle for a compound crack is predicted. Third, based on the J-integral theory, the size of the plastic zone and J-integral of a thin-walled cylindrical shell with a circumferential crack are analysed, using elastic-plastic fracture mechanics. The results show that the stress in front of a crack tip does not increase after reaching the yield strength and enters the stage of plastic development, and the predicted initiation angle of an oblique crack mainly depends on its original inclination angle. The conclusions have theoretical and engineering significance for the selection of the fracture criteria and determination of the failure modes of thin-walled structures with cracks. crack thin-walled structures fracture mechanical parameters crack initiation angle plastic zone Mining engineering. Metallurgy Ziyan Wu verfasserin aut In Metals MDPI AG, 2012 11(2021), 4, p 592 (DE-627)718627172 (DE-600)2662252-X 20754701 nnns volume:11 year:2021 number:4, p 592 https://doi.org/10.3390/met11040592 kostenfrei https://doaj.org/article/360753cb27d5490eb210a3c8af64987a kostenfrei https://www.mdpi.com/2075-4701/11/4/592 kostenfrei https://doaj.org/toc/2075-4701 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 11 2021 4, p 592
allfields_unstemmed	10.3390/met11040592 doi (DE-627)DOAJ057723265 (DE-599)DOAJ360753cb27d5490eb210a3c8af64987a DE-627 ger DE-627 rakwb eng TN1-997 Feng Yue verfasserin aut Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The fracture mechanical behaviour of thin-walled structures with cracks is highly significant for structural strength design, safety and reliability analysis, and defect evaluation. In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of thin-walled cylindrical shells with cracks are investigated. First, based on the J-integral and displacement extrapolation methods, the stress intensity factors of thin-walled cylindrical shells with circumferential cracks and compound cracks are studied using linear elastic fracture mechanics, respectively. Second, based on the theory of maximum circumferential tensile stress of compound cracks, the number of singular elements at a crack tip is varied to determine the node of the element corresponding to the maximum circumferential tensile stress, and the initiation angle for a compound crack is predicted. Third, based on the J-integral theory, the size of the plastic zone and J-integral of a thin-walled cylindrical shell with a circumferential crack are analysed, using elastic-plastic fracture mechanics. The results show that the stress in front of a crack tip does not increase after reaching the yield strength and enters the stage of plastic development, and the predicted initiation angle of an oblique crack mainly depends on its original inclination angle. The conclusions have theoretical and engineering significance for the selection of the fracture criteria and determination of the failure modes of thin-walled structures with cracks. crack thin-walled structures fracture mechanical parameters crack initiation angle plastic zone Mining engineering. Metallurgy Ziyan Wu verfasserin aut In Metals MDPI AG, 2012 11(2021), 4, p 592 (DE-627)718627172 (DE-600)2662252-X 20754701 nnns volume:11 year:2021 number:4, p 592 https://doi.org/10.3390/met11040592 kostenfrei https://doaj.org/article/360753cb27d5490eb210a3c8af64987a kostenfrei https://www.mdpi.com/2075-4701/11/4/592 kostenfrei https://doaj.org/toc/2075-4701 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 11 2021 4, p 592
allfieldsGer	10.3390/met11040592 doi (DE-627)DOAJ057723265 (DE-599)DOAJ360753cb27d5490eb210a3c8af64987a DE-627 ger DE-627 rakwb eng TN1-997 Feng Yue verfasserin aut Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The fracture mechanical behaviour of thin-walled structures with cracks is highly significant for structural strength design, safety and reliability analysis, and defect evaluation. In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of thin-walled cylindrical shells with cracks are investigated. First, based on the J-integral and displacement extrapolation methods, the stress intensity factors of thin-walled cylindrical shells with circumferential cracks and compound cracks are studied using linear elastic fracture mechanics, respectively. Second, based on the theory of maximum circumferential tensile stress of compound cracks, the number of singular elements at a crack tip is varied to determine the node of the element corresponding to the maximum circumferential tensile stress, and the initiation angle for a compound crack is predicted. Third, based on the J-integral theory, the size of the plastic zone and J-integral of a thin-walled cylindrical shell with a circumferential crack are analysed, using elastic-plastic fracture mechanics. The results show that the stress in front of a crack tip does not increase after reaching the yield strength and enters the stage of plastic development, and the predicted initiation angle of an oblique crack mainly depends on its original inclination angle. The conclusions have theoretical and engineering significance for the selection of the fracture criteria and determination of the failure modes of thin-walled structures with cracks. crack thin-walled structures fracture mechanical parameters crack initiation angle plastic zone Mining engineering. Metallurgy Ziyan Wu verfasserin aut In Metals MDPI AG, 2012 11(2021), 4, p 592 (DE-627)718627172 (DE-600)2662252-X 20754701 nnns volume:11 year:2021 number:4, p 592 https://doi.org/10.3390/met11040592 kostenfrei https://doaj.org/article/360753cb27d5490eb210a3c8af64987a kostenfrei https://www.mdpi.com/2075-4701/11/4/592 kostenfrei https://doaj.org/toc/2075-4701 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 11 2021 4, p 592
allfieldsSound	10.3390/met11040592 doi (DE-627)DOAJ057723265 (DE-599)DOAJ360753cb27d5490eb210a3c8af64987a DE-627 ger DE-627 rakwb eng TN1-997 Feng Yue verfasserin aut Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The fracture mechanical behaviour of thin-walled structures with cracks is highly significant for structural strength design, safety and reliability analysis, and defect evaluation. In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of thin-walled cylindrical shells with cracks are investigated. First, based on the J-integral and displacement extrapolation methods, the stress intensity factors of thin-walled cylindrical shells with circumferential cracks and compound cracks are studied using linear elastic fracture mechanics, respectively. Second, based on the theory of maximum circumferential tensile stress of compound cracks, the number of singular elements at a crack tip is varied to determine the node of the element corresponding to the maximum circumferential tensile stress, and the initiation angle for a compound crack is predicted. Third, based on the J-integral theory, the size of the plastic zone and J-integral of a thin-walled cylindrical shell with a circumferential crack are analysed, using elastic-plastic fracture mechanics. The results show that the stress in front of a crack tip does not increase after reaching the yield strength and enters the stage of plastic development, and the predicted initiation angle of an oblique crack mainly depends on its original inclination angle. The conclusions have theoretical and engineering significance for the selection of the fracture criteria and determination of the failure modes of thin-walled structures with cracks. crack thin-walled structures fracture mechanical parameters crack initiation angle plastic zone Mining engineering. Metallurgy Ziyan Wu verfasserin aut In Metals MDPI AG, 2012 11(2021), 4, p 592 (DE-627)718627172 (DE-600)2662252-X 20754701 nnns volume:11 year:2021 number:4, p 592 https://doi.org/10.3390/met11040592 kostenfrei https://doaj.org/article/360753cb27d5490eb210a3c8af64987a kostenfrei https://www.mdpi.com/2075-4701/11/4/592 kostenfrei https://doaj.org/toc/2075-4701 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 11 2021 4, p 592
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source	In Metals 11(2021), 4, p 592 volume:11 year:2021 number:4, p 592
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authorswithroles_txt_mv	Feng Yue @@aut@@ Ziyan Wu @@aut@@
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In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of thin-walled cylindrical shells with cracks are investigated. First, based on the J-integral and displacement extrapolation methods, the stress intensity factors of thin-walled cylindrical shells with circumferential cracks and compound cracks are studied using linear elastic fracture mechanics, respectively. Second, based on the theory of maximum circumferential tensile stress of compound cracks, the number of singular elements at a crack tip is varied to determine the node of the element corresponding to the maximum circumferential tensile stress, and the initiation angle for a compound crack is predicted. Third, based on the J-integral theory, the size of the plastic zone and J-integral of a thin-walled cylindrical shell with a circumferential crack are analysed, using elastic-plastic fracture mechanics. 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callnumber-first	T - Technology
author	Feng Yue
spellingShingle	Feng Yue misc TN1-997 misc crack misc thin-walled structures misc fracture mechanical parameters misc crack initiation angle misc plastic zone misc Mining engineering. Metallurgy Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks
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topic_title	TN1-997 Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks crack thin-walled structures fracture mechanical parameters crack initiation angle plastic zone
topic	misc TN1-997 misc crack misc thin-walled structures misc fracture mechanical parameters misc crack initiation angle misc plastic zone misc Mining engineering. Metallurgy
topic_unstemmed	misc TN1-997 misc crack misc thin-walled structures misc fracture mechanical parameters misc crack initiation angle misc plastic zone misc Mining engineering. Metallurgy
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title	Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks
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title_full	Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks
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title_sort	fracture mechanical analysis of thin-walled cylindrical shells with cracks
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title_auth	Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks
abstract	The fracture mechanical behaviour of thin-walled structures with cracks is highly significant for structural strength design, safety and reliability analysis, and defect evaluation. In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of thin-walled cylindrical shells with cracks are investigated. First, based on the J-integral and displacement extrapolation methods, the stress intensity factors of thin-walled cylindrical shells with circumferential cracks and compound cracks are studied using linear elastic fracture mechanics, respectively. Second, based on the theory of maximum circumferential tensile stress of compound cracks, the number of singular elements at a crack tip is varied to determine the node of the element corresponding to the maximum circumferential tensile stress, and the initiation angle for a compound crack is predicted. Third, based on the J-integral theory, the size of the plastic zone and J-integral of a thin-walled cylindrical shell with a circumferential crack are analysed, using elastic-plastic fracture mechanics. The results show that the stress in front of a crack tip does not increase after reaching the yield strength and enters the stage of plastic development, and the predicted initiation angle of an oblique crack mainly depends on its original inclination angle. The conclusions have theoretical and engineering significance for the selection of the fracture criteria and determination of the failure modes of thin-walled structures with cracks.
abstractGer	The fracture mechanical behaviour of thin-walled structures with cracks is highly significant for structural strength design, safety and reliability analysis, and defect evaluation. In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of thin-walled cylindrical shells with cracks are investigated. First, based on the J-integral and displacement extrapolation methods, the stress intensity factors of thin-walled cylindrical shells with circumferential cracks and compound cracks are studied using linear elastic fracture mechanics, respectively. Second, based on the theory of maximum circumferential tensile stress of compound cracks, the number of singular elements at a crack tip is varied to determine the node of the element corresponding to the maximum circumferential tensile stress, and the initiation angle for a compound crack is predicted. Third, based on the J-integral theory, the size of the plastic zone and J-integral of a thin-walled cylindrical shell with a circumferential crack are analysed, using elastic-plastic fracture mechanics. The results show that the stress in front of a crack tip does not increase after reaching the yield strength and enters the stage of plastic development, and the predicted initiation angle of an oblique crack mainly depends on its original inclination angle. The conclusions have theoretical and engineering significance for the selection of the fracture criteria and determination of the failure modes of thin-walled structures with cracks.
abstract_unstemmed	The fracture mechanical behaviour of thin-walled structures with cracks is highly significant for structural strength design, safety and reliability analysis, and defect evaluation. In this study, the effects of various factors on the fracture parameters, crack initiation angles and plastic zones of thin-walled cylindrical shells with cracks are investigated. First, based on the J-integral and displacement extrapolation methods, the stress intensity factors of thin-walled cylindrical shells with circumferential cracks and compound cracks are studied using linear elastic fracture mechanics, respectively. Second, based on the theory of maximum circumferential tensile stress of compound cracks, the number of singular elements at a crack tip is varied to determine the node of the element corresponding to the maximum circumferential tensile stress, and the initiation angle for a compound crack is predicted. Third, based on the J-integral theory, the size of the plastic zone and J-integral of a thin-walled cylindrical shell with a circumferential crack are analysed, using elastic-plastic fracture mechanics. The results show that the stress in front of a crack tip does not increase after reaching the yield strength and enters the stage of plastic development, and the predicted initiation angle of an oblique crack mainly depends on its original inclination angle. The conclusions have theoretical and engineering significance for the selection of the fracture criteria and determination of the failure modes of thin-walled structures with cracks.
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title_short	Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks
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Fracture Mechanical Analysis of Thin-Walled Cylindrical Shells with Cracks

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