zldmЗаводская лаборатория. Диагностика материаловIndustrial laboratory. Diagnostics of materials1028-68612588-0187ООО «Издательство «ТЕСТ-ЗЛ»10.26896/1028-6861-2026-92-7-60-67zldm-2888Research ArticleМЕХАНИКА МАТЕРИАЛОВ: ПРОЧНОСТЬ, РЕСУРС, БЕЗОПАСНОСТЬMECHANICAL TESTING METHODSМодифицированная диаграмма трещиностойкости трехмерных тел с вырезамиModified failure assessment diagram of three-dimensional solids with notchesМатвиенкоЮ. Г.MatvienkoY. G.

Юрий Григорьевич Матвиенко

101000, Москва, Малый Харитоньевский пер., д. 4

Yury G. Matvienko

4, Malyi Kharitonievskii per., Moscow, 101000

ygmatvienko@gmail.comИнститут машиноведения им. А. А. Благонравова РАНРоссияBlagonravov Mechanical Engineering Research Institute of the Russian Academy of SciencesRussian Federation2026260720269276067Copyright © Матвиенко Ю.Г., 20262026Матвиенко Ю.Г.Matvienko Y.G.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.zldm.ru/jour/article/view/2888

Предложена модифицированная диаграмма трещиностойкости трехмерного тела с вырезом, учитывающая параметры стеснения деформаций в терминах несингулярных Txx- и Tzz-напряжений в зоне разрушения перед фронтом вершины выреза. Для описания диаграмм трещиностойкости сформулирован локальный критерий разрушения, основанный на среднем максимальном тангенциальном напряжении в зоне процесса разрушения. При этом среднее максимальное тангенциальное напряжение равно локальной прочности в зоне разрушения, которая рассчитывается на основе критерия пластичности Хубера – Мизеса с учетом несингулярных Txx- and Tzz-напряжений. Для отражения комбинированного эффекта параметров стеснения деформаций предложено использовать комплексный параметр стеснения деформаций у фронта вершины выреза трехмерного тела (Tz-параметр), отражающий стеснение деформаций как перпендикулярно, так и вдоль фронта вершины выреза посредством компонент несингулярных Txx- и Tzz-напряжений. Установлено значительное влияние теоретического коэффициента концентрации напряжений, стеснения деформаций вдоль фронта вершины надреза и Tz-параметра стеснения деформаций на диаграммы трещиностойкости трехмерного тела с вырезом. Диаграммы трещиностойкости сдвигаются по оси абсцисс с увеличением теоретического коэффициента концентрации напряжений. Кроме того, диаграммы трещиностойкости трехмерного тела расположены между диаграммами, соответствующими условиям плоского напряженного состояния и плоской деформации. В случае неограниченного теоретического коэффициента концентрации напряжений (Kt → ∞) диаграммы трещиностойкости тела с вырезом трансформируются в диаграммы трещиностойкости тела с трещиной.

The paper deals with a modified notch failure assessment diagram taking into account in-plane and out-of-plane constraint parameters in terms of nonsingular Txx- and Tzz-stresses in the fracture process zone ahead of the notch tip front. A local failure criterion based on an average maximum tangential stress in the fracture process zone has been proposed to describe notch failure assessment curves. The average maximum tangential stress is assumed to be equal to the local strength in the fracture process zone which is calculated by means of the Huber – Mises plasticity criterion taking into account nonsingular Txx- and Tzz-stresses. To reflect the combined effect of the constraint parameters, it is proposed to use the Tz-parameter, which is a complex parameter of constraint at the notch tip front of a three-dimensional solids and reflects the constraint both perpendicular and along the notch tip front through the components of non-singular Txx- and Tzz-stresses, respectively. It was established that there is the significant effect of the elastic stress concentration factor, out-of-plane notch tip constraints and Tz-parameter on notch failure assessment curves in the case of three-dimensional solids. Notch failure assessment curves shift along the abscissa axis as the theoretical stress concentration factor increases. Besides, notch failure assessment curves for three-dimensional solids are located between the curves corresponding to the conditions of plane stress and plane strain. In the case of an unlimited theoretical stress concentration factor, the notch failure assessment curves for a notched solid transform into the failure assessment curves for a solid with the crack.

локальный критерий разрушениядиаграмма трещиностойкости тела с вырезомTz-параметр стеснения деформаций у фронта вершины вырезаTxx- и Tzz-напряженияlocal failure criterionnotch failure assessment diagramTz-constraint parameter ahead of the notch tip frontTxx- and Tzz-stressesРабота выполнена за счет средств Государственного задания, код (шифр) научной темы 3-13, присвоенной учредителем (организацией) FFGU-2024-0020.ReferencesZerbst U., Ainsworth R. A., Schwalbe K.-H. Basic principles of analytical flaw assessment methods / Int. J. Pressure Vessels Piping. 2000. Vol. 77. P. 855 – 867. DOI: 10.1016/s0308-0161(01)00008Zerbst U., Ainsworth R. A., Schwalbe K.-H. Basic principles of analytical flaw assessment methods / Int. J. Pressure Vessels Piping. 2000. Vol. 77. P. 855 – 867. DOI: 10.1016/s0308-0161(01)00008Webster S., Bannister A. Structural integrity assessment procedure for Europe — of the SINTAP program overview / Eng. Fract. Mech. 2000. Vol. 67. P. 481 – 514. DOI: 10.1016/s0013-7944(00)00070-9Webster S., Bannister A. Structural integrity assessment procedure for Europe — of the SINTAP program overview / Eng. Fract. Mech. 2000. Vol. 67. P. 481 – 514. DOI: 10.1016/s0013-7944(00)00070-9Ainsworth R. A., Bannister A. C., Zerbst U. An overview of the European flaw assessment procedure SINTAP and its validation / Int. J. Pressure Vessels Piping. 2000. Vol. 77. P. 869 – 876. DOI: 10.1016/s0308-0161(01)00009-6Ainsworth R. A., Bannister A. C., Zerbst U. An overview of the European flaw assessment procedure SINTAP and its validation / Int. J. Pressure Vessels Piping. 2000. Vol. 77. P. 869 – 876. DOI: 10.1016/s0308-0161(01)00009-6Fuentes J. D., Cicero S., Ibáñez-Gutiérrez F. T., Procopio I. On the use of British standard 7910 option 1 failure assessment diagram to non-metallic materials / Fatigue Fract. Eng. Mater. Struct. 2018. Vol. 41. P. 146 – 158. DOI: 10.1111/ffe.12668Fuentes J. D., Cicero S., Ibáñez-Gutiérrez F. T., Procopio I. On the use of British standard 7910 option 1 failure assessment diagram to non-metallic materials / Fatigue Fract. Eng. Mater. Struct. 2018. Vol. 41. P. 146 – 158. DOI: 10.1111/ffe.12668Guo W. Three-dimensional analyses of plastic constraint for through-thickness cracked bodies / Eng. Fract. Mech. 1999. Vol. 62. P. 383 – 407. DOI: 10.1016/s0013-7944(98)00102-7Guo W. Three-dimensional analyses of plastic constraint for through-thickness cracked bodies / Eng. Fract. Mech. 1999. Vol. 62. P. 383 – 407. DOI: 10.1016/s0013-7944(98)00102-7Ainsworth R. A., O’Dowd N. P. Constraint in the failure assessment diagram approach for fracture assessment / J. Pressure Vessel Technol. 1995. Vol. 117. P. 260 – 267. DOI: 10.1115/1.2842125Ainsworth R. A., O’Dowd N. P. Constraint in the failure assessment diagram approach for fracture assessment / J. Pressure Vessel Technol. 1995. Vol. 117. P. 260 – 267. DOI: 10.1115/1.2842125Burdekin F. M., Xu W. G. Effects of biaxial loading and residual stresses on constraint / Int. J. Pressure Vessels Piping. 2003. Vol. 80. P. 755 – 773. DOI: 10.1016/j.ijpvp.2003.01.001Burdekin F. M., Xu W. G. Effects of biaxial loading and residual stresses on constraint / Int. J. Pressure Vessels Piping. 2003. Vol. 80. P. 755 – 773. DOI: 10.1016/j.ijpvp.2003.01.001Neimitz A. Modification of Dugdale model to include the work hardening and in- and out-of-plane constraints / Eng. Fract. Mech. 2004. Vol. 71. P. 1585 – 1600. DOI: 10.1016/s0013-7944(03)00212-1Neimitz A. Modification of Dugdale model to include the work hardening and in- and out-of-plane constraints / Eng. Fract. Mech. 2004. Vol. 71. P. 1585 – 1600. DOI: 10.1016/s0013-7944(03)00212-1Matvienko Yu. G. Models and criteria of fracture mechanics. — Moscow: Fizmatlit, 2006. — 328 p. [in Russian].Matvienko Yu. G. Models and criteria of fracture mechanics. — Moscow: Fizmatlit, 2006. — 328 p. [in Russian].Matvienko Yu. G. Two-parametric fracture mechanics. — Moscow: Fizmatlit, 2020. — 208 p. [in Russian].Matvienko Yu. G. Two-parametric fracture mechanics. — Moscow: Fizmatlit, 2020. — 208 p. [in Russian].Sherry A. H., Wilkes M. A., Beardsmore D. W., Lidbury D. P. G. Material constraint parameters for the assessment of shallow defects in structural components. Part I. Parameter solutions / Eng. Fract. Mech. 2005. Vol. 72. P. 2373 – 2395. DOI: 10.1016/j.engfracmech.2004.12.009Sherry A. H., Wilkes M. A., Beardsmore D. W., Lidbury D. P. G. Material constraint parameters for the assessment of shallow defects in structural components. Part I. Parameter solutions / Eng. Fract. Mech. 2005. Vol. 72. P. 2373 – 2395. DOI: 10.1016/j.engfracmech.2004.12.009Adib H., Jallouf S., Schmitt C., et al. Evaluation of the effect of corrosion defects on the structural integrity of X52 gas pipelines using the SINTAP procedure and notch theory / Int. J. Pressure Vessels Piping. 2007. Vol. 84. P. 123 – 131. DOI: 10.1016/j.ijpvp.2006.10.005Adib H., Jallouf S., Schmitt C., et al. Evaluation of the effect of corrosion defects on the structural integrity of X52 gas pipelines using the SINTAP procedure and notch theory / Int. J. Pressure Vessels Piping. 2007. Vol. 84. P. 123 – 131. DOI: 10.1016/j.ijpvp.2006.10.005Pluvinage G. Analysis of failure emanating from a notch / Mater. Sci. 2020. Vol. 55. No. 6. P. 781 – 793. DOI: 10.1007/s11003-020-00371-8Pluvinage G. Analysis of failure emanating from a notch / Mater. Sci. 2020. Vol. 55. No. 6. P. 781 – 793. DOI: 10.1007/s11003-020-00371-8Cicero S., Madrazo V., Carrascal I. A, Cicero R. Assessment of notched structural components using failure assessment diagrams and the theory of critical distances / Eng. Fract. Mech. 2011. Vol. 78. P. 2809 – 2825. DOI: 10.1016/j.engfracmech.2011.08.009Cicero S., Madrazo V., Carrascal I. A, Cicero R. Assessment of notched structural components using failure assessment diagrams and the theory of critical distances / Eng. Fract. Mech. 2011. Vol. 78. P. 2809 – 2825. DOI: 10.1016/j.engfracmech.2011.08.009Cicero S. Assessment of structural materials containing notch-type defects: a comprehensive validation of the FAD-TCD methodology on metallic and non-metallic materials / Theor. Appl. Fract. Mech. 2024. Vol. 133. Part B. Art. 104612. DOI: 10.1016/j.tafmec.2024.104612Cicero S. Assessment of structural materials containing notch-type defects: a comprehensive validation of the FAD-TCD methodology on metallic and non-metallic materials / Theor. Appl. Fract. Mech. 2024. Vol. 133. Part B. Art. 104612. DOI: 10.1016/j.tafmec.2024.104612Huang X. Constraint-corrected failure assessment diagrams based on crack tip plastic zone which is the functions of Txx- and Tzz-stresses / Eng. Fract. Mech. 2024. Vol. 309. Art. 110403. DOI: 10.1016/j.engfracmech.2024.110403Huang X. Constraint-corrected failure assessment diagrams based on crack tip plastic zone which is the functions of Txx- and Tzz-stresses / Eng. Fract. Mech. 2024. Vol. 309. Art. 110403. DOI: 10.1016/j.engfracmech.2024.110403Yusof F. Three-dimensional constraint-based failure assessment diagrams / Theor. Appl. Fract. Mech. 2023. Vol. 125. Part B. Art. 103889. DOI: 10.1016/j.tafmec.2023.103889Yusof F. Three-dimensional constraint-based failure assessment diagrams / Theor. Appl. Fract. Mech. 2023. Vol. 125. Part B. Art. 103889. DOI: 10.1016/j.tafmec.2023.103889Neuber H. Theory of notch stresses: principles for exact calculation of strength with reference to structural form and material. — Berlin: Springer-Verlag, 1958. — 180 p.Neuber H. Theory of notch stresses: principles for exact calculation of strength with reference to structural form and material. — Berlin: Springer-Verlag, 1958. — 180 p.Novozhilov V. V. On a necessary and sufficient criterion for brittle strength / J. Appl. Math. Mech. 1969. Vol. 33. P. 201 – 210. DOI: 10.1016/0021-8928(69)90025-2Novozhilov V. V. On a necessary and sufficient criterion for brittle strength / J. Appl. Math. Mech. 1969. Vol. 33. P. 201 – 210. DOI: 10.1016/0021-8928(69)90025-2Dyskin A. V. Crack growth criteria incorporating non-singular stresses: size effect in apparent fracture toughness / Int. J. Fract. 1997. Vol. 83. P. 191 – 206. DOI: 10.1023/a:1007304015524Dyskin A. V. Crack growth criteria incorporating non-singular stresses: size effect in apparent fracture toughness / Int. J. Fract. 1997. Vol. 83. P. 191 – 206. DOI: 10.1023/a:1007304015524Fenghui W. Prediction of intrinsic fracture toughness for brittle materials from the apparent toughness of notched-crack specimen / J. Mater. Sci. 2000. Vol. 35. P. 2543 – 2546. DOI: 10.1023/a:1004746508509Fenghui W. Prediction of intrinsic fracture toughness for brittle materials from the apparent toughness of notched-crack specimen / J. Mater. Sci. 2000. Vol. 35. P. 2543 – 2546. DOI: 10.1023/a:1004746508509Kim J. H., Kim D. H., Moon S. I. Evaluation of static and dynamic fracture toughness using apparent fracture toughness of notched specimens / Mater. Sci. Eng. A. 2004. Vols. 387 – 389. P. 381 – 384. DOI: 10.1016/j.msea.2004.01.134Kim J. H., Kim D. H., Moon S. I. Evaluation of static and dynamic fracture toughness using apparent fracture toughness of notched specimens / Mater. Sci. Eng. A. 2004. Vols. 387 – 389. P. 381 – 384. DOI: 10.1016/j.msea.2004.01.134Matvienko Yu. G. Local fracture criterion to describe failure assessment diagrams for a body with a crack/notch / Int. J. Fract. 2003. Vol. 124. P. 107 – 112. DOI: 10.1023/b:frac.0000018229.08344.96Matvienko Yu. G. Local fracture criterion to describe failure assessment diagrams for a body with a crack/notch / Int. J. Fract. 2003. Vol. 124. P. 107 – 112. DOI: 10.1023/b:frac.0000018229.08344.96Matvienko Yu. G. Erratum: Local fracture criterion to describe failure assessment diagrams for a body with a crack/ notch / Int. J. Fract. 2005. Vol. 131. P. 309. DOI: 10.1007/s10704-004-5300-6Matvienko Yu. G. Erratum: Local fracture criterion to describe failure assessment diagrams for a body with a crack/ notch / Int. J. Fract. 2005. Vol. 131. P. 309. DOI: 10.1007/s10704-004-5300-6Pluvinage G. Fracture and fatigue emanating from stress concentrators. — Dordrecht: Kluwer Academic Publishers, 2003. — 233 p.Pluvinage G. Fracture and fatigue emanating from stress concentrators. — Dordrecht: Kluwer Academic Publishers, 2003. — 233 p.Taylor D. Predicting the fracture strength of ceramic materials using the theory of critical distances / Eng. Fract. Mech. 2004. Vol. 71. P. 2407 – 2416. DOI: 10.1016/j.engfracmech.2004.01.002Taylor D. Predicting the fracture strength of ceramic materials using the theory of critical distances / Eng. Fract. Mech. 2004. Vol. 71. P. 2407 – 2416. DOI: 10.1016/j.engfracmech.2004.01.002Jumei Lu, Jianhui Liu, Fei Zhou, et al. Fatigue life prediction of notched components based on energy gradient using reconstructed critical distance theory / Fatigue Fract. Eng. Mater. Struct. 2024. Vol. 47. P. 4228 – 4247. DOI: 10.1111/ffe.14432Jumei Lu, Jianhui Liu, Fei Zhou, et al. Fatigue life prediction of notched components based on energy gradient using reconstructed critical distance theory / Fatigue Fract. Eng. Mater. Struct. 2024. Vol. 47. P. 4228 – 4247. DOI: 10.1111/ffe.14432Creager M., Paris P. C. Elastic field equations for blunt cracks with reference to stress corrosion cracking / Int. J. Fract. 1967. Vol. 3. P. 247 – 252. DOI: 10.1007/BF00182890Creager M., Paris P. C. Elastic field equations for blunt cracks with reference to stress corrosion cracking / Int. J. Fract. 1967. Vol. 3. P. 247 – 252. DOI: 10.1007/BF00182890Nakamura T., Parks D. M. Determination of elastic T-stress along three-dimensional crack front an interaction integral / Int. J. Solids Struct. 1992. Vol. 29. P. 1597 – 1611. DOI: 10.1016/0020-7683(92)90011-hNakamura T., Parks D. M. Determination of elastic T-stress along three-dimensional crack front an interaction integral / Int. J. Solids Struct. 1992. Vol. 29. P. 1597 – 1611. DOI: 10.1016/0020-7683(92)90011-hPokrovskii A. M., Matvienko Yu. G., Egranov M. P. Prediction of the durability of a plate with a through crack taking into account biaxial constraints of deformations along the front of a normal rupture crack / Industr. Lab. Mater. Diagn. 2023. Vol. 89. No. 9. P. 53 – 63 [in Russian]. DOI: 10.26896/1028-6861-2023-89-9-53-63Pokrovskii A. M., Matvienko Yu. G., Egranov M. P. Prediction of the durability of a plate with a through crack taking into account biaxial constraints of deformations along the front of a normal rupture crack / Industr. Lab. Mater. Diagn. 2023. Vol. 89. No. 9. P. 53 – 63 [in Russian]. DOI: 10.26896/1028-6861-2023-89-9-53-63Matvienko Yu. G, Pokrovskii A. M. Two-dimensional crack-tip constraint in the crack path prediction for a mixed mode I/II loading conditions / Eng. Fract. Mech. 2025. Vol. 323. Art. 111218. DOI: 10.1016/j.engfracmech.2025.111218Matvienko Yu. G, Pokrovskii A. M. Two-dimensional crack-tip constraint in the crack path prediction for a mixed mode I/II loading conditions / Eng. Fract. Mech. 2025. Vol. 323. Art. 111218. DOI: 10.1016/j.engfracmech.2025.111218Smith E. Fracture initiation at the root of a blunt flaw: description in terms of Kr – Lr failure assessment curves / Int. J. Pressure Vessels Piping. 1999. Vol. 76. P. 799 – 800. DOI: 10.1016/s0308-0161(99)00048-4Smith E. Fracture initiation at the root of a blunt flaw: description in terms of Kr – Lr failure assessment curves / Int. J. Pressure Vessels Piping. 1999. Vol. 76. P. 799 – 800. DOI: 10.1016/s0308-0161(99)00048-4

The authors declare that there are no conflicts of interest present.