ИДЕНТИФИКАЦИЯ КОЭФФИЦИЕНТОВ ИНТЕНСИВНОСТИ НАПРЯЖЕНИЙ, T-НАПРЯЖЕНИЙ И КОЭФФИЦИЕНТОВ РЕГУЛЯРНЫХ СЛАГАЕМЫХ ВЫСОКОГО ПОРЯДКА В РАЗЛОЖЕНИИ УИЛЬЯМСА С ПОМОЩЬЮ МОЛЕКУЛЯРНО-ДИНАМИЧЕСКОГО МОДЕЛИРОВАНИЯ

Степанова, Л.В.; Белова, О.Н.

doi:10.15593/perm.mech/2023.2.06

ИДЕНТИФИКАЦИЯ КОЭФФИЦИЕНТОВ ИНТЕНСИВНОСТИ НАПРЯЖЕНИЙ, T-НАПРЯЖЕНИЙ И КОЭФФИЦИЕНТОВ РЕГУЛЯРНЫХ СЛАГАЕМЫХ ВЫСОКОГО ПОРЯДКА В РАЗЛОЖЕНИИ УИЛЬЯМСА С ПОМОЩЬЮ МОЛЕКУЛЯРНО-ДИНАМИЧЕСКОГО МОДЕЛИРОВАНИЯ

Л.В. Степанова, О.Н. Белова

Сведения об авторах

Л.В. Степанова
Самарский национальный исследовательский университет имени академика С.П. Королева, Самара, Россия
О.Н. Белова
Самарский национальный исследовательский университет имени академика С.П. Королева, Самара, Россия

https://doi.org/10.15593/perm.mech/2023.2.06

Описано опыт применения метода молекулярной динамики для анализа поля напряжений в медной пластине с одним боковым горизонтальным и наклонным надрезами. Моделирование, основанное на методе молекулярной динамики и реализованное в классическом коде молекулярной динамики Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), направлено на оценку параметров континуальной механики хрупкого разрушения, таких как коэффициенты интенсивности напряжений, T-напряжения и коэффициенты слагаемых более высокого порядка ряда Макса Уильямса, аппроксимирующего поле напряжений вблизи острия надреза для нормального разрыва (мода I), поперечного сдвига (мода II) и смешанного нагружения образца с одним боковым надрезом в условиях линейно-упругого деформирования. Ключевой целью исследования является сравнение континуального и атомистического подходов для оценки полей напряжений вблизи устья надреза на примере одной из наиболее распространенных конфигураций образцов с трещинами и надрезами. Коэффициенты интенсивности напряжений, T-напряжения и коэффициенты регулярных слагаемых более высокого порядка в асимптотическом представлении Уильямса механических величин в медной пластине с одним боковым надрезом при нормальном отрыве и смешанном нагружении оцениваются с помощью дискретного подхода (атомистического моделирования) и континуального подхода (метода конечных элементов). Выполнен широкий класс молекулярно-динамических вычислений в LAMMPS. Атомистические значения коэффициентов интенсивности напряжений и коэффициентов неособых слагаемых более высокого порядка ряда Уильямса вычислены и сопоставлены со значениями, полученными из конечно-элементных решений. Показано, что континуальная теория разрушения успешно описывает поля вблизи вершины трещины даже при чрезвычайно ограниченном поле сингулярных напряжений, составляющем всего несколько нанометров. Угловые распределения компонент напряжений извлекаются из атомистического моделирования и сравниваются с угловыми распределениями напряжений континуальной механики линейного упругого разрушения. Сравнение показывает хорошее соответствие между двумя подходами.

Ключевые слова

молекулярно-динамическое моделированиекоэффициент интенсивности напряженийT-напряжениякоэффициенты регулярных слагаемых более высокого порядка ряда М. Уильямсасмешанное нагружениеклассическая механика разрушения

Identification of stress intensity factors, T-stresses and higher-order coefficients of regular terms in the Williams series expansion through molecular dynamics simulations

Molecular dynamics (MD) approach and finite element analysis are enforced for the investigating the stress – strain fields in the proximity of the notch tip in a copper plate with single horizontal and inclined edge notches. The MD simulation embodied in a classical molecular dynamics program Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is aimed at evaluating conventional continuum linear elastic fracture mechanics key parameters, precisely stress intensity factors (SIF), T-stresses and generalized stress intensity factors (higher order factors) of the Max Williams power series expansion (WE) of the stress field adjacent with the notch tip for pure tensile (Mode I), pure shear (Mode II) and mixed mode (combinations of Mode I and Mode II) loadings of the notched specimen in linear elastic isotropic media. The paramount intent of the research is the comparability of continuum and atomistic procedures for the appraisement of the near notch tip fields exploiting the exemplification of one of the widespread cracked configurations. SIFs, T-stresses and higher order amplitude coefficients of the WE for the single-edge notched Cu plane under Mode I and Mixed Mode loadings are estimated by atomistic and finite element modelling. The wide class of the MD computations in LAMMPS is effectuated. The atomistic values of SIFs and amplitude factors of higher order terms of the WE are correlated with the quantities gained from the numeric solutions obtained by finite element method. It is elucidated that the continuum fracture theory properly characterizes failure and the near notch tip fields even at tremendously limited distances of only few nanometers. The angular stress distributions found from atomic modeling are restored and correlated with the angular behaviours of the stresses obtained from continuum linear elastic fracture theory. The juxtaposition is shown to be in reasonable agreement between two approaches.

Keywords

molecular dynamics simulationatomistic and continuum stress intensity factorsT-stressesgeneralized stress intensity factorshigher-order terms of the Williams series expansionmixed mode loadingclassical fracture mechanics

Литература

1.Ritchie R.O., Liu D. Introduction to fracture mechanics // Elsevier, 2021. Scholar
2.Breakdown of Continuum Fracture Mechanics at the Nanoscale / T. Shimada, K. Ouchi, Y. Chihara, T. Kitamura // Sci Rep. – 2015. – No 8596. DOI: 10.1038/srep08596 Scholar
3.Hello G. Derivation of complete crack-tip stress expansions from Westergaard-Sanford solutions // International Journal of Solids and Structures. – 2018. – Vol. 144–145. – Р. 265–275. DOI: 10.1016/j.ijsolstr.2018.05.012 Scholar
4.Karihaloo B.L., Xiao Q.Z. Accurate determination of the coefficients of elastic crack tip asymptotic field by a hybrid crack element with p-adaptivity // Engineering Fracture Mechanics. – 2001. – Vol. 68, no. 15. – P. 1609–1630. DOI: 10.1016/S0013-7944(01)00063-7 Scholar
5.Benito deCelis, Argon Ali S. Molecular dynamics simulation of crack tip processes in alpha-iron and copper // Journal of Applied Physics. – 1983. – Vol. 54. – P. 4864. DOI: 10.1063/1.332796. Scholar
6.Buehler M.J., Gao H.J., Huang Y. Atomistic and continuum studies of stress and strain fields near a rapidly propagating crack in a harmonic lattice // Theoretical and Applied Fracture Mechanics. – 2004 – Vol. 41. – Р. 21–42. DOI: 10.1016/j.tafmec.2003.11.022 Scholar
7.Cheng S.H., Sun C.T. Applicability of continuum fracture mechanics in atomistic systems // Proc. ASME. IMECE2011. – 2011. – No 8. – Р. 283–288. DOI: 10.1115/IMECE2011-63478 Scholar
8.Wilson M.A., Grutzik S.J., Chandross M. Continuum stress intensity factors from atomistic fracture simulations // Comput. Methods Appl. Mech. Engrg. – 2019. – No. 354. – P. 732–749. DOI: 10.1016/j.cma.2019.05.050 Scholar
9.Roy S., Roy A. A computational investigation of length-scale effects in the fracture behaviour of a graphene sheet using the atomistic J-integral // Engineering Fracture Mechanics. – 2019. – No. 207. – P. 165–180. DOI: 10.1016/j.engfracmech.2018.12.012 Scholar
10.Tsai J.-L., Tzeng S.-H., Tzou Y.-J. Characterizing the fracture parameters of a graphene sheet using atomistic simulation and continuum mechanics // International Journal of Solids and Structures. – 2010. – No. 47. – P. 503–509. DOI: 10.1016/j.ijsolstr.2009.10.017 Scholar
11.Jin Y., Yuan F.G. Nanoscopic modelling of fracture of 2D graphene systems // Journal of Nanoscience and Nanotechnology. – 2005. – Vol. 5, no. 4. – P. 601–608. DOI: 10.1166/jnn.2005.071 Scholar
12.Crack front propagation and fracture in a graphene sheet: a molecular-dynamics study on parallel computers / A. Omeltchenko, J. Yu, R.K. Kalia, P. Vashishta // Physical Review Letters. – 1997. – Vol. 78. – P. 2148–2151. DOI: 10.1103/PhysRevLett.78.2148 Scholar
13.Mechanical responses of pristine and defective C3N nanosheets studied by molecular dynamics simulations / A.H.N. Shirazi, R. Abadi, M. Izadifar, N. Alajlan, T. Rabczuk // Computational Materials Science. – 2018. – Vol. 147. – P. 316–321. DOI: 10.1016/j.commatsci.2018.01.058 Scholar
14.Fracture toughness and crack propagation behavior of nanoscale beryllium oxide graphene-like structures: A molecular dynamics simulation analysis / M.Z. Dehaghani, A.H. Mashhadzadeh, A. Salmankhani, Z. Karami, S. Habibzadeh, M.R. Ganjali, M.R. Saeb // Engineering Fracture Mechanics. – 2020. – Vol. 235. – P. 107194. DOI: 10.1016/j.engfracmech.2020.107194 Scholar
15.Fracture mechanics of polycrystalline beryllium oxide nanosheets: A theoretical basis / M.Z. Dehaghani A., Salmankhani, A.H. Mashhadzadeh, S. Habibzadeh, O. Abida, M.R. Saeb // Engineering Fracture Mechanics. – 2021. – Vol. 244. – P. 107552. DOI: 10.1016/j.engfracmech.2021.107552 Scholar
16.Fracture behavior of SiGe nanosheets: Mechanics of monocrystalline vs. polycrystalline structure / M.Z. Dehaghani, M.E. Safa, F. Yousefi, A. Salmankhani Z., Karami, A. Dadrasi, A.H. Mashhadzadeh, F.J. Stadler, M.R. Saeb // Engineering Fracture Mechanics. – 2021. – Vol. 251. – P. 107782. DOI: 10.1016/j.engfracmech.2021.107782 Scholar
17.Theoretical examination of the fracture behavior of BC3 polycrystalline nanosheets: Effect of crack size and temperature / A. Dadrasi, A. Albooyeh, S. Fooladpanjeh, A. Salmankhani, A.H. Mashhadzadeh, M.R. Saeb // Mechanics of Materials. – 2022. – Vol. 165. – P. 104158. DOI: 10.1016/j.mechmat.2021.104158 Scholar
18.A theoretical insight into the fracture behavior of the edge-cracked polycrystalline BC3 polycrystalline nanosheets / A. Dadrasi, S. Fooladpanjeh, A. Albooyeh A., Salmankhani, A.H. Mashhadzadeh, M.R. Saeb // Computational Materials Science. – 2021. – Vol. 192. – P. 110345. DOI: 10.1016/j.commatsci.2021.110345 Scholar
19.Applying molecular dynamics simulation to take the fingerprint of polycrystalline SiC nanosheets / F. Molaei, M.Z. Dehaghani, A. Salmankhani, S. Fooladpanjeh, S.M. Sajadi, M.E. Safa, O. Abida, S. Habibzadeh, A.H. Mashhadzadeh, M.R. Saeb // Computational Materials Science. – 2021. – Vol. 200. – P. 110770. DOI: 10.1016/j.commatsci.2021.110770 Scholar
20.Molaei F. Molecular dynamics simulation of edge crack in single crystalline alpha quartz // Journal of Molecular Graphics and Modelling. – 2022. – Vol. 111. – P. 108085. DOI: 10/1016/j.jmgm.2021.108085 Scholar
21.Fracture Toughness Estimation of Single-crystal Aluminum at Nanoscale / W. Velilla-Diaz, L. Ricardo, A. Palencia, H.R. Zambrano // Nanomaterials. – 2021. – Vol. 11, no. 3. – P. 689. DOI: 10.3390/nano11030680 Scholar
22.Le M.-Q. Molecular dynamics study of the fracture of single layer buckled silicon monosulfide and germanium selenide // Archives of mechanics. – 2022. – Vol. 74. – P. 1–10. DOI: 10.244423/aom.3871 Scholar
23.Nguen H.-T., Le M.-Q., Nguen V.-T. Mode-I stress intensity factors of silicene, AIN, and SiC hexagonal sheets // Material Research Express. – 2018. – Vol. 5, no. 6. – P. 065025. DOI: 10.1088/2053-1591/aac807 Scholar
24.A coupled/continuum study of graphene fracture / M. Xu, A. Tabarraei, J.T. Paci, J. Oswald, T. Belytschko // International Journal of Fracture. – 2012. – Vol. 173. – P. 163–173. DOI: 10.1007/s10704-011-9675-x Scholar
25.Le M.Q., Batra R.C. Mode-I stress intensity factor in single layer graphene sheets // Computational Materials Science. – 2016. – Vol. 118. – P. 251–258. DOI: 10.1016/j.commatsci.2016.03.027 Scholar
26.Gallo P. Some Considerations on Stress Intensity Factor at Atomic Scale // In: Gdoutos E., Konsta-Gdoutos M. (eds) Proceedings of the Third International Conference on Theoretical, Applied and Experimental Mechanics. ICTAEM 2020. Structural Integrity. – 2020. – No. 16. DOI: 10.1007/978-3-030-47883-4_57 Scholar
27.Mai N.T., Choi S.T. Atomic-scale mutual integrals for mixed-mode fracture: Abnormal fracture toughness of grain boundaries in graphene // International Journal of Solids and Structures. – 2018. – No. 138. – P. 205–216. DOI: 10.1016/j.ijsolstr.2018.01.013 Scholar
28.Stepanova L.V., Bronnikov S.A. Molecular Dynamics Modeling of Crack Propagation // Journal of Physics: Conference Series. – 2019. – Vol. 1368, no. 4. – P. 042039. DOI: 10.1088/1742-6596/1368/4/042039 Scholar
29.Stepanova L.V., 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. – No. 109. – P. 102691. DOI: 10.1016/j.tafmec.2020.102691 Scholar
30.Stepanova L., Belova O. Stress intensity factors of Continuum fracture mechanics at the nanoscale // Procedia Structural Integrity. – 2022. – Vol. 37. – P. 900–907. Scholar
31.Stepanova L.V., Belova O.N. A molecular dynamics simulation analysis of mixed mode crack growth // AIP Conference Proceedings. – 2021. – No. 2371. – P. 020012. DOI: 10.1063/5.0059574. Scholar
32.Chen Z., Wang H., Liu G.-R. Fatigue crack propagation in carbon steel using RVE based model. – 2021. arXiv:2103.15399v1. Scholar
33.Subramanian A.K., Sun C.T. Continuum interpretation of virial stress in molecular simulations // International Journal of Solids and Structures. – 2008. – No. 45. – P. 4340–4346. DOI: 10.1016/j.ijsolstr.2008.03.016 Scholar
34.Calculation of stress in atomistic simulation / J.A. Zimmerman, E.B. Webb III, J.J. Hoyt, R.E. Jones, P.A. Klein, D.J. Bammann // Modelling and Simulation in Materials Science and Engineering. – 2004. – Vol. 12, iss. 4. – P. S319–S332. DOI: 10.1088/0965-0393/12/4/S03 Scholar
35.Liu B., Qiu X. How to compute the atomic stress objectively // Journal of Computational and Theoretical Nanoscience. – 2008. – Vol. 6, no. 5. – P. 1081–1089. DOI: arXiv:0810.0803v2 Scholar
36.Yang J., Komvopoulos K. A stress analysis method for molecular dynamics systems. // International Journal of Solids and Structures. – 2020. – Vol. 193–194. – P. 98–105. DOI: 10.1016/j.ijsolstr.2020.02.003 Scholar
37.Gasemi A., Gao W. A method to apply Piola-Kirchhoff stress in molecular statics simulations // Computational Materials Science. – 2021. – Vol. 195. – P. 110496. DOI: 10.1016/j.commatsci.2021.110496 Scholar
38.Investigation of crack tip dislocation emission in aluminum using multiscale molecular dynamics simulation and continuum modelling / V.I. Yamakov, D.H. Warner, R.J. Zamora, E. Saether, W.A. Curtin, E.N. Glassgen // Journal of the Mechanics and Physics of Solids. – 2014. – Vol. 65. – P. 35–53. DOI: 10/1016/j.jmps.2013.12.009 Scholar
39.Nikravesh Y., Sameti A.R., Jkkoei A.R. An atomistic-continuum multiscale analysis for heterogeneous nanomaterials and its application in nanoporous gold foam // Applied Mathematical Modelling. – 2022. DOI: 10/1016/j.apm.2022.02.029 Scholar
40.Ganesh K.V., Patra P.K., Travis K.P. Multiscale modelling of impact through molecular dynamics and smooth particle hydrodynamics // Physica A. – 2022. – Vol. 593. – P. 126903. DOI: 10/1016/j.physa.2022.126903 Scholar
41.A predictive discrete-continuum multiscale model of plasticity with quantified uncertainty / J. Tan, U. Villa, N. Shamsaei, S. Shao, H.M. Zbib, D. Faghihi // International Journal of Plasticity. – 2021. – Vol. 138. – P. 102935. DOI: 10/1016/j.ijplas.2021.102935 Scholar
42.Multiscale Concurrent Atomistic (CAC) modeling of multicomponent alloys / K. Chu, A. Diaz, Y. Chen, T. Zhu, D.L. McDowell // Computational Materials Science. – 2022. – Vol. 201. – P. 110873. DOI: 10/1016/j.commatsci.2021.110873 Scholar
43.Hello G., Tahar M.B., Roeland J.-M. Analytical determination of coefficients in crack-tip stress expansions for a finite crack in an infinite plane medium // International Journal of Solids and Structures. – 2012. – Vol. 49. – P. 556–566. DOI: 10.1016/j.ijsolstr.2011.10.024 Scholar
44.Nejati M., Ghouli S., Ayatollahi M.R. Crack tip asymptotic fields in anisotropic planes: Importance of higher order terms // Applied Mathematical Modelling. – 2021. – Vol. 91. – P. 837–862. DOI: 10.1016/j.apm.2020.09.025 Scholar
45.Elastic stress analysis of blunt V-notches under mixed mode loading by considering higher order terms / A.M. Mirzaei, M.R. Ayatolahi, B. Bahrami, F. Berto // Applied Mathematical Modelling. – 2020. – Vol. 78. – P. 665–684. DOI: 10.1016/j.apm.2019.09.049 Scholar
46.Vivekanandan A., Ramesh K. Study of interaction effects of asymmetric cracks under biaxial loading using digital photoelasticity // Theoretical and Applied Fracture Mechanics. – 2019. – Vol. 99. – P. 104–117. DOI: 10.1016/j.tafmec.2018.11.011 Scholar
47.Степанова Л.В. Экспериментальное и конечно-элементное определение коэффициентов многопараметрического асимптотического разложения М. Уильямса у вершины трещины в линейно-упругом изотропном материале. Часть I // Вестник Пермского национального исследовательского политехнического университета. Механика. – 2020. – No 4. – С. 237–249. DOI: 10.15593/perm.mech/2020.4.20 Scholar
48.Степанова Л.В. Экспериментальное и конечно-элементное определение коэффициентов многопараметрического асимптотического разложения М. Уильямса у вершины трещины в линейно-упругом изотропном материале. Часть II // Вестник Пермского национального исследовательского политехнического университета. Механика. – 2021. – No 1. – С. 72–85. DOI: 10.15593/perm.mech/2021.1.08 Scholar
49.Li Y., Zheng K. Crack tip asymptotic field coefficients analyses based on the extended finite element method using over-deterministic displacement field fitting method // Theoretical and Applied Fracture Mechanics. – 2021. – Vol. 113. – P. 102971. DOI: 10.1016/j.tafmec.2021.102971 Scholar
50.Ayatollahi M.R., Nejati M., Ghouli S. The finite element over-deterministic method to calculate the coefficients of the crack tip asymptotic fields in anisotropic planes // Engineering Fracture Mechanics. – 2020. – Vol. 231. – P. 106982. DOI: 10.1016/j.engfracmech.2020.106982 Scholar
51.Determination of SIFs and T-stress using an over-deterministic method based on stress fields: Static and dynamic / C. Hou, Z. Wnag, X. Jim, X. Ji, X. Fan // Engineering Fracture Mechanics. – 2021. – Vol. 242. – P. 107455. DOI: 10.1016/j.engfracmech.2020.107455. Scholar
52.Patil P., Vyasarayani C.P, Ramji M. Linear least squares approach for evaluating crack tip fracture parameters using isochromatic and isoclinic data from digital photoelasticity // Optics and Lasers in Engineering. – 2017. – Vol. 93. – P. 182–194. DOI: 10.1016/j.optlaseng.2017.02.003 Scholar
53.Constantinescu A., Korsunsky A. Elasticity with Mathematica. An Introduction to continuum mechanics. – Cambridge: Cambridge University Press, 2007. – 267 p. Scholar
54.Nordmann J., Aßmus M., Altenbach H. Visualising elastic anisotropy: theoretical background and computational implementation // Continuum Mechanics and Thermodynamics. – 2018. – Vol. 30, no. 4. – P. 689–708. DOI: 10.1007/s00161-018-0635-9 Scholar
55.Gaillac R., Pullumbi P., Coudert F.-X. ELATE: an open – source online application for analysis and visualization of elastic tensors // J. Phys. Condensed Matter. – 2016. – Vol. 28, no. 27. – P. 275201. DOI: 10.1088/0953-8984/28/27/275201 Scholar
56.Daw M.S., Baskes M.I. Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals // Phys. Rev. – 1984. – Vol. B29 (12). – P. 6443. Scholar
57.The improvement of mechanical properties of conventional concretes using carbon nanoparticles using molecular dynamics simulations / L. Zhao, M.K.M. Nasuton, M. Hekmatifar, R. Sabetvand, P. Kamenkov, D. Toghraie, A. Alizadeh, T. Ghahari // Scientific Reports. – 2021. – Vol. 11. – P. 20265. DOI: 10.1038/s41598-021-99616-y Scholar
58.Stukowski A. Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool // Model. Simul. Mater. Sci. Eng. – 2010. – Vol. 18. – P. 015012. Scholar
59.Sanford R.J., Dally J.W. A general method for determining mixed mode stress intensity factors // Engineering Fracture Mechanics. – 1972. – Vol. 4. – P. 357–366. Scholar
60.Ramesh K., Gupta S., Kelkar A.A. Evaluation of stress filed parameters in fracture mechanics by photoelasticity- Revisited // Engineering Fracture Mechanics. – 1997. – Vol. 56, no. 3. – P. 25–45. Scholar
61.Stepanova L.V., Belova O.N. A molecular dynamics simulation analysis of mixed mode crack growth // AIP Conference Proceedings. – 2021. – No. 2371. – P. 020012. DOI: 10.1063/5.0059574. Scholar
62.Nirwal S., Katukam R. An approach for Coupling FEM&Molecular Dynamics // International Journal of Emerging Trends in Engineering Research. – 2015. – Vol. 3, no. 10. – P. 7–19. Scholar
63.Chakraborty S., Ghost S. A concurrent atomistic-crystal plasticity multiscale model for crack propagation in crystalline metallic materials // Computer methods in Applied Mechanics and Engineering. – 2021. – Vol. 379. – P. 113748. DOI: 10.1016/j.cma.2021.113748 Scholar
64.Yasbolaghi R., Khoei A.R. A continuum-atomistic multi-scale analysis of temperature field problems and its application in phononic nano-structures // Finite Elements in Analysis & Design. – 2022. – Vol. 198. – P. 103643. DOI: 10/1016/j.finel.2021.103643 Scholar

1.Ritchie R.O., Liu D. Introduction to fracture mechanics. Elsevier, 2021. Scholar
2.Shimada T., Ouchi K., Chihara Y., Kitamura T. Breakdown of Continuum Fracture Mechanics at the Nanoscale. Sci Rep., 2015, no. 8596. https://doi.org/doi: 10.1038/srep08596. Scholar
3.Hello, G. Derivation of complete crack-tip stress expansions from Westergaard-Sanford solutions. International Journal of Solids and Structures, 2018, no. 144-145, pp. 265-275. http://doi.org/10.1016/j.ijsolstr.2018.05.012. Scholar
4.Karihaloo, B.L., Xiao, Q.Z. Accurate determination of the coefficients of elastic crack tip asymptotic field by a hybrid crack element with p-adaptivity. Engineering Fracture Mechanics, 2001, vol. 68, no. 15, pp. 1609-1630. https://doi.org/10.1016/S0013-7944(01)00063-7. Scholar
5.Benito deCelis, Argon Ali S. Molecular dynamics simulation of crack tip processes in alpha-iron and copper. Journal of Applied Physics, 1983, vol. 54, 4864. https://doi.org/10.1063/1.332796. Scholar
6.Buehler M.J., Gao H.J., Huang Y. Atomistic and continuum studies of stress and strain fields near a rapidly propagating crack in a harmonic lattice. Theoretical and Applied Fracture Mechanics, 2004, vol. 41, pp. 21-42. https://doi.org/10.1016/j.tafmec.2003.11.022. Scholar
7.Cheng S.H., Sun C.T. Applicability of continuum fracture mechanics in atomistic systems. Proc. ASME. IMECE2011, 2011, no. 8, pp. 283-288. https://doi.org/10.1115/IMECE2011-63478. Scholar
8.Wilson, M.A., Grutzik, S.J., Chandross, M. Continuum stress intensity factors from atomistic fracture simulations. Comput. Methods Appl. Mech. Engrg., 2019, no. 354, pp. 732-749. https://doi.org/10.1016/j.cma.2019.05.050. Scholar
9.Roy, S., Roy A. A computational investigation of length-scale effects in the fracture behaviour of a graphene sheet using the atomistic J-integral. Engineering Fracture Mechanics, 2019, no. 207, pp. 165-180. https://doi.org/10.1016/j.engfracmech.2018.12.012. Scholar
10.Tsai, J.-L., Tzeng, S.-H., Tzou, Y.-J. Characterizing the fracture parameters of a graphene sheet using atomistic simulation and continuum mechanics. International Journal of Solids and Structures, 2010, no. 47, pp. 503-509. https://doi.org/10.1016/j.ijsolstr.2009.10.017. Scholar
11.Jin Y., Yuan F.G. Nanoscopic modelling of fracture of 2D graphene systems. Journal of Nanoscience and Nanotechnology, 2005, vol. 5, no. 4, pp. 601-608. https://doi.org/10.1166/jnn.2005.071. Scholar
12.Omeltchenko A., Yu J., Kalia R.K., Vashishta P. Crack front propagation and fracture in a graphene sheet: a molecular-dynamics study on parallel computers. Physical Review Letters, 1997, vol. 78. pp. 2148-2151, https://doi.org/10.1103/PhysRevLett.78.2148. Scholar
13.Shirazi A.H.N., Abadi R., Izadifar M., Alajlan N., Rabczuk T. Mechanical responses of pristine and defective C3N nanosheets studied by molecular dynamics simulations. Computational Materials Science, 2018, vol. 147, pp. 316-321. https://doi.org/10.1016/j.commatsci.2018.01.058. Scholar
14.Dehaghani M.Z., Mashhadzadeh A.H., Salmankhani A., Karami Z., Habibzadeh S., Ganjali M.R., Saeb M.R. Fracture toughness and crack propagation behavior of nanoscale beryllium oxide graphene-like structures: A molecular dynamics simulation analysis. Engineering Fracture Mechanics, 2020, vol. 235, 107194. https://doi.org/10.1016/j.engfracmech.2020.107194. Scholar
15.Dehaghani M.Z., Salmankhani A., Mashhadzadeh A.H., Habibzadeh S., Abida O., Saeb M.R. Fracture mechanics of polycrystalline beryllium oxide nanosheets: A theoretical basis. Engineering Fracture Mechanics, 2021, vol. 244, 107552. 10.1016/j.engfracmech.2021.107552. Scholar
16.Dehaghani M.Z.,Safa M.E., Yousefi F., Salmankhani A.,Karami Z., Dadrasi A., Mashhadzadeh A.H., Stadler F.J., Saeb M.R. Fracture behavior of SiGe nanosheets: Mechanics of monocrystalline vs. polycrystalline structure. Engineering Fracture Mechanics, 2021, vol. 251. 107782. https://doi.org/10.1016/j.engfracmech.2021.107782. Scholar
17.Dadrasi A., Albooyeh A., Fooladpanjeh S., Salmankhani A., Mashhadzadeh A.H., Saeb M.R. Theoretical examination of the fracture behavior of BC3 polycrystalline nanosheets: Effect of crack size and temperature. Mechanics of Materials, 2022, vol. 165, 104158. https://doi.org/10.1016/j.mechmat.2021.104158. Scholar
18.Dadrasi A., Fooladpanjeh S., Albooyeh A., Salmankhani A., Mashhadzadeh A.H., Saeb M.R. A theoretical insight into the fracture behavior of the edge-cracked polycrystalline BC3 polycrystalline nanosheets. Computational Materials Science, 2021, vol. 192, 110345, https://doi.org/10.1016/j.commatsci.2021.110345. Scholar
19.Molaei F., Dehaghani M.Z., Salmankhani A., Fooladpanjeh S., Sajadi S.M., Safa M.E., Abida O., Habibzadeh S., Mashhadzadeh A.H., Saeb M.R. Applying molecular dynamics simulation to take the fingerprint of polycrystalline SiC nanosheets. Computational Materials Science, 2021, vol. 200, 110770. https://doi.org/10.1016/j.commatsci.2021.110770. Scholar
20.Molaei F. Molecular dynamics simulation of edge crack in single crystalline alpha quartz. Journal of Molecular Graphics and Modelling. 2022, vol. 111, 108085. https://doi.org/10/1016/j.jmgm.2021.108085. Scholar
21.Velilla-Diaz W., Ricardo L., Palencia A., Zambrano H.R. Fracture Toughness Estimation of Single-crystal Aluminum at Nanoscale. Nanomaterials, 2021, vol. 11(3), 689. https://doi:10.3390/nano11030680 Scholar
22.Le M.-Q. Molecular dynamics study of the fracture of single layer buckled silicon monosulfide and germanium selenide. Archives of mechanics, 2022, vol. 74, pp. 1-10. https://doi.org/10.244423/aom.3871. Scholar
23.Nguen H.-T., Le M.-Q., Nguen V.-T. Mode-I stress intensity factors of silicene, AIN, and SiC hexagonal sheets. Material Research Express, 2018, vol. 5, no. 6, 065025. http://dx.doi.org/10.1088/2053-1591/aac807. Scholar
24.Xu M., Tabarraei A., Paci J.T., Oswald J., Belytschko T. A coupled/continuum study of graphene fracture. International Journal of Fracture, 2012, vol. 173, pp. 163-173. https://doi.org/10.1007/s10704-011-9675-x. Scholar
25.Le M.Q., Batra R.C. Mode-I stress intensity factor in single layer graphene sheets. Computational Materials Science, 2016, vol. 118, pp. 251-258. https://doi.org/10.1016/j.commatsci.2016.03.027. Scholar
26.Gallo, P. Some Considerations on Stress Intensity Factor at Atomic Scale. In: Gdoutos E., Konsta-Gdoutos M. (eds) Proceedings of the Third International Conference on Theoretical, Applied and Experimental Mechanics. ICTAEM 2020. Structural Integrity, 2020, no. 16. https://doi.org/10.1007/978-3-030-47883-4_57. Scholar
27.Mai, N.T., Choi, S.T. Atomic-scale mutual integrals for mixed-mode fracture: Abnormal fracture toughness of grain boundaries in graphene. International Journal of Solids and Structures, 2018, no. 138, pp. 205-216, https://doi.org/10.1016/j.ijsolstr.2018.01.013 Scholar
28.Stepanova, L.V., Bronnikov, S.A. Molecular Dynamics Modeling of Crack Propagation. Journal of Physics: Conference Series, 2019, no. 1368(4), 042039. https://doi.org/doi:10.1088/1742-6596/1368/4/042039. Scholar
29.Stepanova, L.V., 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, no.109, pp. 102691. https://doi.org/10.1016/j.tafmec.2020.102691. Scholar
30.Stepanova L., Belova O. Stress intensity factors of Continuum fracture mechanics at the nanoscale. Procedia Structural Integrity, 2022, vol. 37. pp. 900-907. Scholar
31.Stepanova, L.V., Belova, O.N. A molecular dynamics simulation analysis of mixed mode crack growth. AIP Conference Proceedings, 2021, no. 2371, 020012. https://doi.org/10.1063/5.0059574. Scholar
32.Chen Z., Wang H., Liu G.-R. Fatigue crack propagation in carbon steel using RVE based model. 2021, arXiv:2103.15399v1. Scholar
33.Subramanian, A.K., Sun, C.T. Continuum interpretation of virial stress in molecular simulations. International Journal of Solids and Structures, 2008, no. 45, pp. 4340-4346. https://doi.org/10.1016/j.ijsolstr.2008.03.016. Scholar
34.Zimmerman, J.A., Webb III, E.B., Hoyt, J.J., Jones R.E., Klein, P.A., Bammann, D.J. Calculation of stress in atomistic simulation. Modelling and Simulation in Materials Science and Engineering, 2004, vol. 12, Iss. 4, pp. S319–S332. https://doi.org/10.1088/0965-0393/12/4/S03. Scholar
35.Liu, B., Qiu, X., How to compute the atomic stress objectively. Journal of Computational and Theoretical Nanoscience, 2008, vol. 6, no. 5, pp. 1081-1089. https://doi.org/arXiv:0810.0803v2. Scholar
36.Yang, J., Komvopoulos, K., A stress analysis method for molecular dynamics systems. International Journal of Solids and Structures, 2020, vol. 193-194, pp. 98-105. https://doi.org/10.1016/j.ijsolstr.2020.02.003. Scholar
37.Gasemi, A., Gao, W., A method to apply Piola-Kirchhoff stress in molecular statics simulations. Computational Materials Science, 2021, vol. 195, 110496. https://doi.org/10.1016/j.commatsci.2021.110496. Scholar
38.Yamakov, V.I., Warner, D.H., Zamora, R.J., Saether, E., Curtin, W.A., Glassgen, E.N. Investigation of crack tip dislocation emission in aluminum using multiscale molecular dynamics simulation and continuum modelling. Journal of the Mechanics and Physics of Solids, 2014, vol. 65. pp. 35-53. http://doi.org/10/1016/j.jmps.2013.12.009. Scholar
39.Nikravesh, Y., Sameti, A.R., Jkkoei, A.R. An atomistic-continuum multiscale analysis for heterogeneous nanomaterials and its application in nanoporous gold foam. Applied Mathematical Modelling, 2022, http://doi.org/10/1016/j.apm.2022.02.029. Scholar
40.Ganesh, K.V., Patra, P.K., Travis, K.P. Multiscale modelling of impact through molecular dynamics and smooth particle hydrodynamics. Physica A, 2022, vol. 593. 126903. http://doi.org/10/1016/j.physa.2022.126903. Scholar
41.Tan, J., Villa, U., Shamsaei, N., Shao, S., Zbib, H., M., Faghihi, D. A predictive discrete-continuum multiscale model of plasticity with quantified uncertainty. International Journal of Plasticity, 2021, vol. 138. 102935. http://doi.org/10/1016/j.ijplas.2021.102935. Scholar
42.Chu, K., Diaz, A., Chen, Y., Zhu, T., McDowell, D.L. Multiscale Concurrent Atomistic (CAC) modeling of multicomponent alloys. Computational Materials Science, 2022, vol. 201, 110873. http://doi.org/10.1016/j.commatsci.2021.110873. Scholar
43.Hello, G., Tahar, M.B., Roeland, J.-M. Analytical determination of coefficients in crack-tip stress expansions for a finite crack in an infinite plane medium. International Journal of Solids and Structures, 2012, vol. 49, pp. 556-566. http://doi.org/10.1016/j.ijsolstr.2011.10.024. Scholar
44.Nejati M., Ghouli S., Ayatollahi M.R. Crack tip asymptotic fields in anisotropic planes: Importance of higher order terms. Applied Mathematical Modelling, 2021, vol. 91, pp. 837-862. https://doi.org/10.1016/j.apm.2020.09.025 Scholar
45.Mirzaei, A.M., Ayatolahi, M.R., Bahrami, B., Berto, F. Elastic stress analysis of blunt V-notches under mixed mode loading by considering higher order terms. Applied Mathematical Modelling, 2020, vol. 78, pp. 665-684. https://doi.org/10.1016/j.apm.2019.09.049. Scholar
46.Vivekanandan, A., Ramesh, K. Study of interaction effects of asymmetric cracks under biaxial loading using digital photoelasticity. Theoretical and Applied Fracture Mechanics, 2019, vol. 99, pp. 104-117. https://doi.org/10.1016/j.tafmec.2018.11.011. Scholar
47.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. https://doi.org/10.15593/perm.mech/2020.4.20. Scholar
48.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. https://doi.org/10.15593/perm.mech/2021.1.08. Scholar
49.Li, Y., Zheng, K. Crack tip asymptotic field coefficients analyses based on the extended finite element method using over-deterministic displacement field fitting method. Theoretical and Applied Fracture Mechanics, 2021, vol. 113, 102971. https://doi.org/10.1016/j.tafmec.2021.102971. Scholar
50.Ayatollahi, M.R., Nejati, M., Ghouli, S. The finite element over-deterministic method to calculate the coefficients of the crack tip asymptotic fields in anisotropic planes. Engineering Fracture Mechanics, 2020, vol. 231, 106982. https://doi.org/10.1016/j.engfracmech.2020.106982. Scholar
51.Hou, C., Wnag, Z., Jim, X., Ji, X., Fan, X. Determination of SIFs and T-stress using an over-deterministic method based on stress fields: Static and dynamic. Engineering Fracture Mechanics. 2021, vol. 242, 107455. https://doi.org/10.1016/j.engfracmech.2020.107455. Scholar
52.Patil, P., Vyasarayani, C.P, Ramji, M. Linear least squares approach for evaluating crack tip fracture parameters using isochromatic and isoclinic data from digital photoelasticity. Optics and Lasers in Engineering, 2017, vol. 93, pp. 182-194. https://doi.org/10.1016/j.optlaseng.2017.02.003. Scholar
53.Constantinescu A., Korsunsky A. Elasticity with Mathematica. An Introduction to continuum mechanics. Cambridge: Cambridge University Press, 2007. 267 p. Scholar
54.Nordmann J., Aßmus M., Altenbach H. Visualising elastic anisotropy: theoretical background and computational implementation. Continuum Mechanics and Thermodynamics, 2018, vol. 30(4). pp. 689-708. https://doi.org/10.1007/s00161-018-0635-9. Scholar
55.Gaillac R., Pullumbi P., Coudert F.-X. ELATE: an open –source online application for analysis and visualization of elastic tensors. J. Phys. Condensed Matter, 2016, vol. 28 (27), 275201. https://doi.org/10.1088/0953-8984/28/27/275201 Scholar
56.Daw, M.S., Baskes, M.I. Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev., 1984, B29 (12). 6443. Scholar
57.Zhao L., Nasuton M.K.M., Hekmatifar M., Sabetvand R., Kamenkov P., Toghraie D., Alizadeh A., Ghahari T. The improvement of mechanical properties of conventional concretes using carbon nanoparticles using molecular dynamics simulations. Scientific Reports, 2021. vol. 11, 20265. https://doi.org/10.1038/s41598-021-99616-y. Scholar
58.A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool, Model. Simul. Mater. Sci. Eng. 2010, vol. 18, 015012; OVITO website: https://www.ovito.org Scholar
59.Sanford R.J., Dally J.W. A general method for determining mixed mode stress intensity factors. Engineering Fracture Mechanics, 1972, vol.4, pp. 357-366. Scholar
60.Ramesh, K., Gupta, S., Kelkar, A.A. Evaluation of stress filed parameters in fracture mechanics by photoelasticity- Revisited. Engineering Fracture Mechanics, 1997, vol. 56, no. 3, pp. 25-45. Scholar
61.Stepanova, L.V., Belova, O.N. A molecular dynamics simulation analysis of mixed mode crack growth. AIP Conference Proceedings. 2021. vol. 2371, 020012. https://doi.org/10.1063/5.0059574. Scholar
62.Nirwal, S., Katukam, R. An approach for Coupling FEM&Molecular Dynamics. International Journal of Emerging Trends in Engineering Research, 2015, vol. 3, no. 10, pp. 7-19. Scholar
63.Chakraborty, S., Ghost, S. A concurrent atomistic-crystal plasticity multiscale model for crack propagation in crystalline metallic materials. Computer methods in Applied Mechanics and Engineering, 2021, vol. 379, 113748. https://doi.org/10.1016/j.cma.2021.113748 Scholar
64.Yasbolaghi R., Khoei A.R. A continuum-atomistic multi-scale analysis of temperature field problems and its application in phononic nano-structures. Finite Elements in Analysis & Design, 2022, vol. 198, 103643. https://doi.org/10/1016/j.finel.2021.103643. Scholar

Другие статьи выпуска

Весь выпуск (№ 2, 2023)

Скачать PDF

Информация о статье

Опубликована: 23.06.2023
В выпуске: 23.06.2023
Выпуск: № 2 (2023)