ЭФФЕКТЫ СЛАГАЕМЫХ ВЫСШЕГО ПОРЯДКА МАЛОСТИ В ОБОБЩЕННЫХ РЯДАХ ДЛЯ ПРЕДСТАВЛЕНИЯ ПОЛЕЙ, АССОЦИИРОВАННЫХ С ВЕРШИНОЙ ТРЕЩИНЫ В АНИЗОТРОПНЫХ СРЕДАХ. ЧАСТЬ 1. СРАВНЕНИЕ ТОЧНОГО И АСИМПТОТИЧЕСКОГО РЕШЕНИЙ ОПРЕДЕЛЕНИЯ НАПРЯЖЕНИЙ У ВЕРШИНЫ ТРЕЩИНЫ

Степанова, Л.В.; Мушанкова, К.А.

doi:10.15593/perm.mech/2025.1.05

ЭФФЕКТЫ СЛАГАЕМЫХ ВЫСШЕГО ПОРЯДКА МАЛОСТИ В ОБОБЩЕННЫХ РЯДАХ ДЛЯ ПРЕДСТАВЛЕНИЯ ПОЛЕЙ, АССОЦИИРОВАННЫХ С ВЕРШИНОЙ ТРЕЩИНЫ В АНИЗОТРОПНЫХ СРЕДАХ. ЧАСТЬ 1. СРАВНЕНИЕ ТОЧНОГО И АСИМПТОТИЧЕСКОГО РЕШЕНИЙ ОПРЕДЕЛЕНИЯ НАПРЯЖЕНИЙ У ВЕРШИНЫ ТРЕЩИНЫ

Л.В. Степанова, К.А. Мушанкова

Сведения об авторах

Л.В. Степанова
Самарский национальный исследовательский университет имени академика С.П. Королева, Самара, Российская Федерация
К.А. Мушанкова
Самарский национальный исследовательский университет имени академика С.П. Королева, Самара, Российская Федерация

https://doi.org/10.15593/perm.mech/2025.1.05

Приводится анализ обобщенных на случай анизотропных линейно упругих сред асимптотических рядов, представляющих поля перемещений, деформаций и напряжений вблизи кончика острой трещины в анизотропных средах. Ключевой целью исследования является оценка механических полей в ближайшей окрестности вершины трещины в анизотропных материалах с простейшей кубической сингонией упругих свойств среды. Асимптотические ряды построены на основании классической теории упругости анизотропного тела. С помощью анализа рядов, ассоциированных с полями у вершины трещины, показано, что высшие приближения с коэффициентами, получившими название обобщенных коэффициентов интенсивности напряжений (ОКИН), оказывают значительное влияние на целостное представление поля напряжений при увеличении расстояния от кончика трещины. На примере плоской задачи для бесконечной анизотропной плоскости с кубической симметрией свойств (с различными тензорами упругих модулей, имеющими три независимых элемента) при различной ориентации трещины по отношению к осям симметрии упругих свойств показано, что в обобщенных рядах следует сохранять помимо первых двух слагаемых (содержащих коэффициенты интенсивности напряжений (КИН) и Т-напряжения) члены более высоких порядков малости. Построены угловые (θ-распределения) зависимости компонент тензора напряжений в плоской задаче для двух различных расстояний от острия трещины, с принятием во внимание различное количество слагаемых ряда. Простое визуальное сопоставление и анализ зависимостей компонент тензора от полярного угла, выстроенных с учетом различного числа слагаемых ряда, ясно указывают на необходимость удержания высших приближений ряда. В целях расширения области, в которой справедливо решение в рядах, необходимо сохранять большее количество слагаемых. Все вычисления проведены для реальных материалов, упругие постоянные которых определены с помощью метода молекулярной динамики для монокристаллических веществ с гранецентрированной кубической (ГЦК) решеткой.

Ключевые слова

анизотропные средытеория упругости анизотропного телаортотропные материалыкубическая сингонияасимптотическое решениевлияние высших приближениймолекулярная динамика.

Effects of Terms of High Order in Synthesized Polynomial Series Approximation for Fields Associated with the Crack Tip in Anisotropic Media. Part 1. Comparison of Exact and Asymptotic Solutions for Determining Stresses at the Crack Tip

The article presents the analysis of asymptotic series expansions generalized to the case of anisotropic linearly elastic media representing fields of displacements, strains and stresses around the tip of an acute crack in anisotropic media. We study anisotropic materials with the simplest cubic symmetry and fields in the proximate neighbourhood of the crack tip. The asymptotic series are constructed on the essential principles of the classical theory of elasticity of an anisotropic body. By analyzing the series associated with the fields at the crack tip, it is shown that higher approximations with coefficients called generalized stress intensity factors have an essential impact on the accurate and relevant representation of the stress field to expand the zone of the asymptotic solution. Using the example of a plane problem for an infinite anisotropic plane with the cubic symmetry of properties (with different tensors of elastic modules having three independent elements) with different crack orientations relative to the axes of the symmetry of elastic properties, it is shown that in the generalized series, in addition to the first two terms (containing stress concentration factors and T-stresses), terms of higher orders of smallness should be preserved. The circumferential θ-approximations of the stress components at various spans from the crack tip are assembled with the retention of a different quantity of series terms to obtain sufficiently accurate approximations. A comparison of the dependencies of the tensor components on the polar angle obtained taking into account the different quantity of series terms clearly indicates the need to retain the higher approximations of the series. In order to expand the area in which the solution in the series is valid, it is necessary to preserve a larger number of terms. All computations were performed for real materials which elastic constants were determined using the molecular dynamics method for single-crystal substances with a face-centered cubic (FCC) lattice.

Keywords

anisotropic mediaelasticity theory for anisotropic mediaorthotropic mediacubic syngonyasymptotic solutioneffect of higher order termsmolecular dynamics.

Литература

1.Nejati, M. Crack tip asymptotic fields in anisotropic planes: Importance of higher order terms / M. Nejati, S. Ghouli, R. Aytollahi // Applied Mathematical Modelling. – 2021. – Vol. 91. – P. 837–862. DOI: 10.1016/j.apm.2020.09.025 Scholar
2.Лекницкий, С.Г. Теория упругости анизотропного тела / С.Г. Лекницкий. – М.: Наука, 1977. – 416 с. Scholar
3.Bowie, O.L. Central Crack in Plane Orthotropic Rectangular Sheet / O.L. Bowie, C.E. Freese // International Journal of Fracture. – 1972. – Vol. 8(1). – P. 49–57. DOI: 10.1007/BF00185197 Scholar
4.Sih, G.C. On cracks in rectilinearly anisotropic bodies / G.C. Sih, P.C. Paris, G.R. Irwin // International Journal of Fracture. – 1965. – Vol. 1(3). – P. 189–203. DOI: 10/1007/bf00186854 Scholar
5.Carloni, C. Biaxial Load Effect on Crack Initiation for Orthotropic Materials / Carloni C., A. Piva, E. Viola // Meccanica. – 2004. – Vol. 39. – P. 331–344. DOI: 10.1023/B:MECC.0000029363.67419.80 Scholar
6.Sadd, M.H. Elasticity. Theory, Applications and Numerics / M.H. Sadd. – Amsterdam: Academic Press, 2020. – 624 p. Scholar
7.Васильев, В.В. Новое решение задачи о трещине в растягиваемой ортотропной пластине / В.В. Васильев, С.А. Лурье, В.А. Салов // Известия Российской академии наук. Механика деформируемого твердого тела. – 2021. – № 6. – С. 23–32. DOI: 10.31857/S0572329921060167 Scholar
8.Stepanova, L.V. Coefficients of the Williams power expansion of the near crack stress field in continuum linear elastic fracture mechanics at the nanoscale / L.V. Stepanova, O.N. Belova // Theoretical and Applied Fracture Mechanics. – 2022. – Vol. 119. – 103298. DOI: 10.1016/j.tafmec.2022.103298 Scholar
9.Belova, O.N. Importance of the higher order terms of the Williams series expansions: Experimental Aspects and Finite Element Simulations / O.N. Belova, L.V. Stepanova // Procedia Structural Integrity. – 2022. – Vol. 39. – P. 770–785. DOI: 10.1016/j.prostr.2022.03.151 Scholar
10.Stepanova L., Bronnikov S. A computational study of the mixed-mode crack behavior by molecular dynamics method and the multi-parameter crack field description of classical fracture mechanics / L. Stepanova, S. Bronnikov // Theoretical and Applied Fracture Mechanics. – 2020. – Vol. 109. – 102691. DOI: 10.1016/j.tafmec.2020.102691 Scholar
11.In situ synchrotron radiation µCT indentation of cortical bone: Anisotropic crack propagation, local deformation, and fracture / M.P. Fernandez, J. Schwiedrzik, A Burki., F. Peyrin, J. Michler, P. Zysset, U. Wolfram // Acta Biomaterialia. – 2023. – Vol. 167. – P. 83–99. DOI: 10.1016/j.actbio.2023.04.038 Scholar
12.Stress intensity factor calculation of the cracks interacted by the oval-holes in anisotropic elastic solids under remote and non-uniform surface stresses / Q-h. Rao, C-c. Zhao, W. Yi, D. Sun, Z. Liu // Theoretical and Applied Fracture Mechanics. – 2022. – Vol. 121. – 103475. DOI: 10.1016/j.tafmec.2022.103475 Scholar
13.Ayatollahi, M.R. The finite element over-deterministic method to calculate the coefficients of crack tip asymptotic fields in anisotropic planes / M.R. Ayatollahi, M. Nejati, S. Ghouli // Engineering Fracture Mechanics. – 2020. – Vol. 231. – 106082. DOI: 10.1016/j.engfractmech.2020.106982 Scholar
14.Nejati, M. Crack tip asymptotic field and K-dominant region for anisotropic semi-circular bend specimen / M. Nejati, S. Ghouli, M.R. Ayatollahi // Theoretical and Applied Fracture Mechanics. – 2020. – Vol. 109. – 102640. DOI: 10.1016/j.tafmec.2020.102640 Scholar
15.Stepanova, L.V. Stress intensity factors, T-stresses and higher order coefficients of the Williams series expansion and their evaluation through molecular dynamics simulations / L.V. Stepanova, O.N. Belova // Mechanics of Advanced Materials and Structures. – 2023. – Vol. 30, iss. 19. – P. 3862–3884. DOI: 10.1080/15376494.2022.2084800 Scholar
16.On the anisotropy of shear fracture toughness in rocks / M. Nejati, B. Bahrami, M.R. Ayatollahi, T. Driesner // Theoretical and Applied Fracture Mechanics. – 2021. – Vol. 113. –102946. DOI: 10.1016/j.tafmec.2021.102946 Scholar
17.Bodaghi, F. Estimation of solidification cracking susceptibility in Al-Si-Cu alloy weld: effects of anisotropic permeability and deformation orientation / F. Bodaghi, M. Movahedi, A.H. Kokabi // Journal of Materials Research and Technology. – 2023. – Vol. 23. – P. 2351–2361. DOI: 10.1016/j.jmrt.2023.01.138 Scholar
18.Wang, C. An adaptive finite element method for crack propagation based on a multifunctional super singular element / C. Wang, X. Ping, X. Wang // International Journal of Mechanical Sciences. – 2023. – Vol. 247. – 108191. DOI: 10.1016/j.ijmecsci.2023.108191 Scholar
19.Khaji, Z. Examining the effect of crack initiation angle on fracture behavior of orthotropic materials under mixed-mode I/II loading / Z. Khaji, M. Fakoor // International Journal of Solids and Structures. – 2022. – Vol. 256. – 111952. DOI: 10.1016/j.ijsolstr.2022.111952 Scholar
20.Gailac, R. ELATE: an open-source online application for analysis and visualization of elastic tensors / R. Gailac, P. Pullumbi, F.-X. Coudert // Journal of Physics: Condensed Matter. – 2016. – Vol. 28. – 275201. DOI: 10.1088/0953-8984/28/27/275201 Scholar
21.Ran, Z. VELAS: An open-source toolbox for visualization and analysis of elastic anisotropy / Z. Ran, C. Zou, Z. Wei, H. Wang // Computer Physics Communications. – 2022. – Vol. 283. – 108540. DOI: 10.1016/j.cpc.2022.108540 Scholar
22.Thube, Y.S. A simple, robust novel Williams series-based FE-analytical hybrid technique for evaluation of SIFs and higher order coefficients / Y.S. Thube, T.P. Gotkhindi // Theoretical and Applied Fracture Mechanics. – 2023. – Vol. 127. – 104101. DOI: 10.1016/j.tafm.2023.104101 Scholar
23.Hu, B. Analysis of 2D elastic solid with multiple V-notches by a fast multipole BEM with multi-order asymptotic terms / B. Hu, C. Li, Z. Niu // Applied Mathematical Modelling. – 2023. – Vol. 118. – P. 762–779. DOI: 10.1016/j.apm.2023.02.008 Scholar
24.Determination of singular and higher non-singular stress for angular heterogeneous material notch / W. Pan, C. Cheng, F. Wang, Z. Hu, J. Li // Engineering Fracture Mechanics. – 2023. – Vol. 292. – 109592. DOI: 10.1016/j.engfractmech.2023.109592 Scholar
25.Novel boundary crack front elements with Williams' eigenexpansion properties for 3D crack analysis / Y. Zhong, G. Xie, L. Wang, K. Li, X. Wang, W. He, S. Wang // Archive of Applied Mechanics. – 2023. – Vol. 93. – P. 745–760. DOI: 10.1007/s00419-022-02296-x Scholar
26.Степанова, Л.В. Экспериментальное и конечно-элементное определение коэффициентов многопараметрического асимптотического разложения М. Уильямса у вершины трещины в линейно-упругом изотропном материале. Часть I / Л.В. Степанова // Вестник Пермского национального исследовательского политехнического университета. Механика. – 2020. – № 4. – С. 237–249. – DOI 10.15593/perm.mech/2020.4.20 Scholar
27.Степанова, Л.В. Экспериментальное и конечно-элементное определение коэффициентов многопараметрического асимптотического разложения М. Уильямса у вершины трещины в линейно-упругом изотропном материале. Часть II / Л.В. Степанова // Вестник Пермского национального исследовательского политехнического университета. Механика. – 2021. – № 1. – С. 72–85. DOI 10.15593/perm.mech/2021.1.08 Scholar
28.An analytical stress field for bi-material V-notches with end hole: New solution and effects of higher order terms / S.K. Alavi, M.R. Ayatollahi, B. Bahrami, M. Nejati // Mathematics and Mechanics of Solids. – 2023. – Vol. 28(2): – P. 464–478. DOI: 10.1177/10812865221084311 Scholar
29.Mirzaei, A. Asymptotic stress field and the coefficients of singular and higher order terms for V-notches with end holes under mixed-mode loading / A. Mirzaei, M. Ayatollahi, B. Bahrami // International Journal of Solids and Structures. – 2019. – Vol. 172. – P. 51–69. DOI: 10.1016/j.ijsolstr.2019.05.011 Scholar
30.Improved stress and displacement fields around V-notches with end holes / B. Bahrami, M.R. Ayatollahi, A.M. Mirzaei, F. Berto // Engineering Fracture Mechanics. – 2019. – Vol. 217. – 106539. DOI: 10.1016/j.engfracmech.2019.106539 Scholar
31.Bahrami, B. Analysis of stresses and displacements in the vicinity of blunt V-notches by considering higher order terms / B. Bahrami, M.R. Ayatollahi, A.M. Mirzaei // Fatigue and Engineering Materials and Structures. – 2019. – Vol. 42(8). – P. 1760–1774. DOI: 10.1111/ffe.13016 Scholar
32.Melching, D. Advanced crack tip field characterization using conjugate work integrals / D. Melching, E. Breitbarth // International Journal of Fatigue. – 2023. – Vol. – 169. – 107501. DOI: 10.1016/j.jifatigue.2023.107501 Scholar
33.Матвиенко, Ю.Г. Основы физики и механики разрушения / Ю.Г. Матвиенко. – М.: Автономная некоммерческая организация «Издательство физико-математической литературы», 2022. – 144 с. Scholar
34.Матвиенко, Ю.Г. Двухпараметрический упругопластический критерий разрушения и скорректированная вязкость разрушения / Ю.Г. Матвиенко // Заводская лаборатория. Диагностика материалов. – 2022. – Т. 88, № 8. – С. 59–69. DOI: 10.26896/1028-6861-2022-88-8-59-69 Scholar
35.Matvienko, Yu.G. The two-parameter fracture criterion based on the J-A concept / Yu.G. Matvienko // Fatigue & Fracture of Engineering Materials & Structures. – 2023. – Vol. 46, no. 9. – P. 3261–3273. DOI 10.1111/ffe.14075 Scholar
36.Матвиенко, Ю.Г. Двухпараметрическая механика разрушения / Ю.Г. Матвиенко. – М.: Издательская фирма «Физико-математическая литература», 2020. – 210 с. Scholar
37.Baldi, A. Kinematic estimation of fracture mechanics parameter with automatic crack-tip identification / A Baldi., P.M. Santucci // Engineering Fracture Mechanics. – 2022. – Vol. 259. – 108082. DOI: 10.1016/j.engfracmech.2021.108082 Scholar
38.Crack Analysis Tool in Python-CrackPy / T. Strohmann, D. Melching, F. Paysan, A. Klein, E. Dietrich, G. Requena, E. Breitbarth. – 2022. DOI: 10.5281/zenodo.7323623 Scholar
39.The toughness of mechanical metamaterials / A.J.D. Shaikeea, H. Cui, M.O’ Masta, X.R. Zheng, V.S. Deshpande // Nat. Mater. – 2022. – Vol. 21. – P. 297–304. DOI: 10.1038/s41563-021-01182-1 Scholar
40.U-Net learning for the automatic identification of the sandstone crack tip position to determine mixed-mode stress intensity factors utilizing digital image correlation method / Y. Li, T. Ni, F. Zhang, Y. Li, J. Zuo, S. Zhao // Theoretical and Applied Fracture Mechanics. – 2023. – Vol. 127. – 104028. DOI: 10.1016/j.tafmec.2023.104028 Scholar
41.Stepanova, L.V. Atomistic and continuum ascertainment of the crack tip stress fields in anisotropic elastic cubic media / L.V. Stepanova, K.A. Mushankova // Theoretical and Applied Fracture Mechanics. – 2024. – Vol. 133. Part B. – 104613, DOI: 10.1016/j.tafmec.2024.104613 Scholar

1.Nejati M., Ghouli S., Aytollahi M.R. Crack tip asymptotic fields in anisotropic planes: Importance of higher order terms. Applied Mathematical Modelling, 2021, vol. 91, pp. 837-862. DOI:10.1016/j.apm.2020.09.025 Scholar
2.Lekhnitskii S.G. Theory of Elasticity of an Anisotropic Body. M, Mir, 1981, 431 p. Scholar
3.Bowie O.L., Freese C.E. Central Crack in Plane Orthotropic Rectangular Sheet. International Journal of Fracture, 1972, vol. 8(1), pp. 49-57. DOI: 10.1007/BF00185197 Scholar
4.Sih G.C., Paris P.C., Irwin G.R. On cracks in rectilinearly anisotropic bodies. International Journal of Fracture, 1965, vol. 1(3), pp. 189-203. DOI: 10/1007/bf00186854 Scholar
5.Carloni C., Piva A., Viola E. Biaxial Load Effect on Crack Initiation for Orthotropic Materials. Meccanica, 2004, vol. 39, pp. 331-344. DOI: 10.1023/B:MECC.0000029363.67419.80 Scholar
6.Sadd M.H. Elasticity. Theory, Applications and Numerics. Amsterdam: Academic Press, 2020, 624 p. Scholar
7.Vasil'ev V.V., Lurie S.A., Salov V.A. New solution to the problem of a crack in an orthotropic plate under tension. Mechanics of Solids, 2021, vol. 56, no. 6, pp. 902-910. DOI: 10.31857/S0572329921060167 Scholar
8.Stepanova L.V., Belova O.N. Coefficients of the Williams power expansion of the near crack stress field in continuum linear elastic fracture mechanics at the nanoscale. Theoretical and Applied Fracture Mechanics, 2022, vol. 119, 103298. DOI:10.1016/j.tafmec.2022.103298 Scholar
9.Belova O.N., Stepanova L.V. Importance of the higher order terms of the Williams series expansions: Experimental Aspects and Finite Element Simulations. Procedia Structural Integrity, 2022, vol. 39, pp. 770-785. DOI:10.1016/j.prostr.2022.03.151 Scholar
10.Stepanova L., Bronnikov S. A computational study of the mixed-mode crack behavior by molecular dynamics method and the multi-parameter crack field description of classical fracture mechanics. Theoretical and Applied Fracture Mechanics, 2020, vol. 109, 102691. DOI:10.1016/j.tafmec.2020.102691 Scholar
11.Fernandez M.P., Schwiedrzik J., Burki A., Peyrin F., Michler J., Zysset P., Wolfram U. In situ synchrotron radiation μCT indentation of cortical bone: Anisotropic crack propagation, local deformation, and fracture. Acta Biomaterialia, 2023, vol. 167, pp. 83-99. DOI:10.1016/j.actbio.2023.04.038 Scholar
12.Rao Q-h., Zhao C.-c., Yi W., Sun D., Liu Z. Stress intensity factor calculation of the cracks interacted by the oval-holes in anisotropic elastic solids under remote and non-uniform surface stresses. Theoretical and Applied Fracture Mechanics, 2022, vol. 121, 103475. DOI:10.1016/j.tafmec.2022.103475 Scholar
13.Ayatollahi M.R., Nejati M., Ghouli S. The finite element over-deterministic method to calculate the coefficients of crack tip asymptotic fields in anisotropic planes. Engineering Fracture Mechanics, 2020, vol. 231, 106082. DOI:10.1016/j.engfracmech.2020.106982 Scholar
14.Nejati M., Ghouli S., Ayatollahi M.R. Crack tip asymptotic field and K-dominant region for anisotropic semi-circular bend specimen. Theoretical and Applied Fracture Mechanics, 2020, vol. 109, 102640. DOI:10.1016/j.tafmec.2020.102640 Scholar
15.Stepanova L.V., Belova O.N. Stress intensity factors, T-stresses and higher order coefficients of the Williams series expansion and their evaluation through molecular dynamics simulations. Mechanics of Advanced Materials and Structures, 2023, vol. 30, Issue 19, pp. 3862-3884. DOI: 10.1080/15376494.2022.2084800 Scholar
16.Nejati M., Bahrami B., Ayatollahi M.R., Driesner T. On the anisotropy of shear fracture toughness in rocks. Theoretical and Applied Fracture Mechanics, 2021, vol. 113, 102946. DOI:10.1016/j.tafmec.2021.102946 Scholar
17.Bodaghi F., Movahedi M., Kokabi A.H. Estimation of solidification cracking susceptibility in Al-Si-Cu alloy weld: effects of anisotropic permeability and deformation orientation. Journal of Materials Research and Technology, 2023, vol. 23, pp. 2351-2361. DOI:10.1016/j.jmrt.2023.01.138 Scholar
18.Wang C., Ping X., Wang X. An adaptive finite element method for crack propagation based on a multifunctional super singular element. International Journal of Mechanical Sciences, 2023, vol. 247, 108191. DOI:10.1016/j.ijmecsci.2023.108191 Scholar
19.Khaji Z., Fakoor M. Examining the effect of crack initiation angle on fracture behavior of orthotropic materials under mixed-mode I/II loading. International Journal of Solids and Structures, 2022, vol. 256, 111952. DOI: 10.1016/j.ijsolstr.2022.111952 Scholar
20.Gailac R., Pullumbi P., Coudert F.-X. ELATE: an open-source online application for analysis and visualization of elastic tensors. Journal of Physics: Condensed Matter, 2016, vol. 28, 275201. DOI: 10.1088/0953-8984/28/27/275201 Scholar
21.Ran Z., Zou C., Wei Z., Wang H. VELAS: An open-source toolbox for visualization and analysis of elastic anisotropy. Computer Physics Communications, 2022, vol. 283, 108540. DOI:10.1016/j.cpc.2022.108540 Scholar
22.Thube Y.S., Gotkhindi T.P. A simple, robust novel Williams series-based FE-analytical hybrid technique for evaluation of SIFs and higher order coefficients. Theoretical and Applied Fracture Mechanics, 2023, vol. 127, 104101. DOI:10.1016/j.tafmec.2023.104101 Scholar
23.Hu B., Li C., Niu Z. Analysis of 2-D elastic solid with multiple V-notches by a fast multipole BEM with multi-order asymptotic terms. Applied Mathematical Modelling, 2023, vol. 118, pp. 762-779. DOI:10.1016/j.apm.2023.02.008 Scholar
24.Pan W., Cheng C., Wang F., Hu Z., Li J. Determination of singular and higher non-singular stress for angular heterogeneous material notch. Engineering Fracture Mechanics, 2023, vol. 292, 109592. DOI:10.1016/j.engfracmech.2023.109592 Scholar
25.Zhong Y., Xie G., Wang L., Li K., Wang X., He W., Wang S. Novel boundary crack front elements with Williams' eigenexpansion properties for 3D crack analysis. Archive of Applied Mechanics, 2023, vol.93, pp. 745-760. DOI: 10.1007/s00419-022-02296-x Scholar
26.Stepanova L.V. Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part I. PNRPU Mechanics Bulletin, 2020, no. 4, pp. 237-249. DOI: 10.15593/perm.mech/2020.4.20 Scholar
27.Stepanova L.V. Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part II. PNRPU Mechanics Bulletin, 2021, no. 1, pp. 72-85. DOI: 10.15593/perm.mech/2021.1.08 Scholar
28.Alavi SK, Ayatollahi MR, Bahrami B, Nejati M. An analytical stress field for bi-material V-notches with end hole: New solution and effects of higher order terms. Mathematics and Mechanics of Solids, 2023, vol. 28(2), pp. 464-478. DOI: 10.1177/10812865221084311 Scholar
29.Mirzaei A., Ayatollahi M., Bahrami B. Asymptotic stress field and the coefficients of singular and higher order terms for V-notches with end holes under mixed-mode loading. International Journal of Solids and Structures, 2019, vol. 172, pp. 51-69. DOI: 10.1016/j.ijsolstr.2019.05.011 Scholar
30.Bahrami B., Ayatollahi M.R., Mirzaei A.M., Berto F. Improved stress and displacement fields around V-notches with end holes. Engineering Fracture Mechanics, 2019, vol. 217, 106539. DOI: 10.1016/j.engfracmech.2019.106539 Scholar
31.Bahrami B., Ayatollahi M.R., Mirzaei A.M. Analysis of stresses and displacements in the vicinity of blunt V-notches by considering higher order terms. Fatigue and Engineering Materials and Structures, 2019, vol. 42(8), pp. 1760-1774. DOI: 10.1111/ffe.13016 Scholar
32.Melching D., Breitbarth E. Advanced crack tip field characterization using conjugate work integrals. International Journal of Fatigue, 2023, vol. 169, 107501. DOI: 10.1016/j.ijfatigue.2023.107501 Scholar
33.Matvienko Yu.G. Basics of physics and mechanics of fracture. M., «Fizmatlit», 2022, 144 p. Scholar
34.Matvienko Yu.G. Two-parameter elastic-plastic fracture criterion and corrected fracture toughness. Industrial Laboratory. Materials Diagnostics, 2022, vol. 88, no 8, pp. 59-69. DOI 10.26896/1028-6861-2022-88-8-59-69. Scholar
35.Matvienko, Yu. G. The two-parameter fracture criterion based on the J-A concept. Fatigue & Fracture of Engineering Materials & Structures, 2023, vol. 46, no. 9, pp. 3261-3273. - DOI 10.1111/ffe.14075. Scholar
36.Matvienko Yu.G. Two-parameter fracture mechanics. M., Publishing company "Physico-mathematical literature", 2020, 210 p. Scholar
37.Baldi A., Santucci P.M. Kinematic estimation of fracture mechanics parameter with automatic crack-tip identification. Engineering Fracture Mechanics, 2022, vol. 259, 108082. DOI: 10.1016/j.engfracmech.2021.108082 Scholar
38.Strohmann T., Melching D., Paysan F., Klein A., Dietrich E., Requena G., Breitbarth E. Crack Analysis Tool in Python-CrackPy, - 2022. DOI: 10.5281/zenodo.7323623. Scholar
39.Shaikeea A.J.D., Cui H., O'Masta, M., Zheng X.R., Deshpande V.S. The toughness of mechanical metamaterials. Nat. Mater, 2022, vol. 21, pp. 297-304. https://doi.org/10.1038/s41563-021-01182-1 Scholar
40.Li Y., Ni T., Zhang F., Li Y., Zuo J., Zhao S. U-Net learning for the automatic identification of the sandstone crack tip position to determine mixed-mode stress intensity factors utilizing digital image correlation method. Theoretical and Applied Fracture Mechanics, 2023, vol. 127, 104028. DOI: 10.1016/j.tafmec.2023.104028 Scholar
41.Stepanova L.V., Mushankova K.A. Atomistic and continuum ascertainment of the crack tip stress fields in anisotropic elastic cubic media. Theoretical and Applied Fracture Mechanics, 2024, vol. 133, Part B, 104613, https://doi.org/10.1016/j.tafmec.2024.104613. Scholar

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Опубликована: 23.04.2025
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