Abstract
We study the scaling of strength and toughness in function of temperature, loading rate and system size, to investigate the difference between tensile failure and fracture failure. Molecular simulation is used to estimate the failure of intact and cracked bodies while varying temperature, strain rate and system size over many orders of magnitude, making it possible to identify scaling laws. Two materials are considered: an idealized toy model, for which a scaling law can be derived analytically, and a realistic molecular model of graphene. The results show that strength and toughness follow very similar scalings with temperature and loading rate, but differ markedly regarding the scaling with system size. Strength scales with the number of atoms whereas toughness scales with the number of cracks. It means that intermediate situations of moderate stress concentrations (e.g., notch) can exhibit not obvious size scaling, in-between those of strength and toughness. Following a theoretical analysis of failure as a thermally activated process, we could rationalize the observed scaling and formulate a general rate–temperature–size equivalence. The scaling law of the toy model can be derived rigorously but is not representative of real materials because of a force discontinuity in the potential. A more representative scaling law, valid for graphene, is proposed with a different exponent.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
\(\varDelta E_{TS}\) is the integration of the repelling force to exert on the atoms of the failing bond to bring it to failure, while maintaining all other atoms at static equilibrium. By linearity of the bond network, this repelling force depends linearly on the bond length, hence the quadratic energy barrier.
References
Allen MP, Tildesley DJ (1989) Computer simulation of liquids. Oxford University Press, Oxford
Anderson TL (2005) Fracture mechanics: fundamentals and applications. CRC Press, Boca Raton
Argon A (1979) Plastic deformation in metallic glasses. Acta Metall 27(1):47–58. https://doi.org/10.1016/0001-6160(79)90055-5. http://linkinghub.elsevier.com/retrieve/pii/0001616079900555
Bazant ZP, Chen EP (1997) Scaling of structural failure. Appl Mech Rev 50(10):593. https://doi.org/10.1115/1.3101672. http://appliedmechanicsreviews.asmedigitalcollection.asme.org/article.aspx?articleid=1396002
Belytschko T, Xiao SP, Schatz GC, Ruoff RS (2002) Atomistic simulations of nanotube fracture. Phys Rev B 65(23):235430. https://doi.org/10.1103/PhysRevB.65.235430
Brenner DW (1990) Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys Rev B 42(15):9458–9471. https://doi.org/10.1103/PhysRevB.46.1948.2. http://link.aps.org/doi/10.1103/PhysRevB.42.9458
Brochard L, Hantal G, Laubie H, Ulm FJ, Pellenq RJM (2015) Capturing material toughness by molecular simulation: accounting for large yielding effects and limits. Int J Fract 194(2):149–167. https://doi.org/10.1007/s10704-015-0045-y
Brochard L, Tejada IG, Sab K (2016) From yield to fracture, failure initiation captured by molecular simulation. J Mech Phys Solids 95:632–646. https://doi.org/10.1016/j.jmps.2016.05.005. http://www.sciencedirect.com/science/article/pii/S0022509616300424. http://linkinghub.elsevier.com/retrieve/pii/S0022509616300424
Carpinteri A (1994) Scaling laws and renormalization groups for strength and toughness of disordered materials. Int J Solids Struct 31(3):291–302. https://doi.org/10.1016/0020-7683(94)90107-4. http://linkinghub.elsevier.com/retrieve/pii/0020768394901074
Carpinteri A, Pugno N (2005) Are scaling laws on strength of solids related to mechanics or to geometry? Nat Mater 4(6):421–423. https://doi.org/10.1038/nmat1408
Dewapriya MAN, Rajapakse RKND, Phani AS (2014) Atomistic and continuum modelling of temperature-dependent fracture of graphene. Int J Fract 187(2):199–212. https://doi.org/10.1007/s10704-014-9931-y
Dugdale D (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104. https://doi.org/10.1016/0022-5096(60)90013-2
Dumitrica T, Hua M, Yakobson BI (2006) Symmetry-, time-, and temperature-dependent strength of carbon nanotubes. Proc Natl Acad Sci 103(16):6105–6109. https://doi.org/10.1073/pnas.0600945103
Flores KM, Dauskardt RH (1999) Enhanced toughness due to stable crack tip damage zones in bulk metallic glass. Scr Mater 41(9):937–943. https://doi.org/10.1016/S1359-6462(99)00243-2. http://linkinghub.elsevier.com/retrieve/pii/S1359646299002432
Frenkel D, Smit B (2002) Undestanding molecular simulation: from algorithms to applications, 2nd edn. Academic Press, New York
Johnson WL, Samwer K (2005) A universal criterion for plastic yielding of metallic glasses with a (T/Tg)\(^{\wedge }\)(2/3) temperature dependence. Phys Rev Lett 95(19):195501. https://doi.org/10.1103/PhysRevLett.95.195501
Karihaloo BL, Wang J, Grzybowski M (1996) Doubly periodic arrays of bridged cracks and short fibre-reinforced cementitious composites. J Mech Phys Solids 44(10):1565–1586. https://doi.org/10.1016/0022-5096(96)00053-1. http://linkinghub.elsevier.com/retrieve/pii/0022509696000531
Khare R, Mielke SL, Paci JT, Zhang S, Ballarini R, Schatz GC, Belytschko T (2007) Coupled quantum mechanical/molecular mechanical modeling of the fracture of defective carbon nanotubes and graphene sheets. Phys Rev B 75(7):075412. https://doi.org/10.1103/PhysRevB.75.075412
Lee C, Wei X, Kysar JW, Hone J (2008) Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321(5887):385–388. https://doi.org/10.1126/science.1157996. http://www.ncbi.nlm.nih.gov/pubmed/18635798. http://www.sciencemag.org/cgi/doi/10.1126/science.1157996,47749150628
Legoll F, Luskin M, Moeckel R (2007) Non-ergodicity of the NoséHoover thermostatted harmonic oscillator. Arch Ration Mech Anal 184(3):449–463. https://doi.org/10.1007/s00205-006-0029-1
Leguillon D (2002) Strength or toughness? A criterion for crack onset at a notch. Eur J Mech A/Solids 21(1):61–72. https://doi.org/10.1016/S0997-7538(01)01184-6
Liu F, Ming P, Li J (2007) Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys Rev B 76(6):064120. https://doi.org/10.1103/PhysRevB.76.064120
Marder M, Gross S (1994) Origin of crack tip instabilities. J Mech Phys Solids 43(1):1–48. https://doi.org/10.1016/0022-5096(94)00060-I. http://arxiv.org/abs/chao-dyn/9410009,9410009
Moura MJB, Marder M (2013) Tearing of free-standing graphene. Phys Rev E 88(3):032405. https://doi.org/10.1103/PhysRevE.88.032405
Novozhilov V (1969) On a necessary and sufficient criterion for brittle strength. J Appl Math Mech 33(2):201–210. https://doi.org/10.1016/0021-8928(69)90025-2. http://linkinghub.elsevier.com/retrieve/pii/0021892869900252
Omeltchenko A, Yu J, Kalia RK, Vashishta P (1997) Crack front propagation and fracture in a graphite sheet: a molecular-dynamics study on parallel computers. Phys Rev Lett 78(11):2148–2151. https://doi.org/10.1103/PhysRevLett.78.2148
Pechenik L, Levine H, Da Kessler (2002) Steady-state mode I cracks in a viscoelastic triangular lattice. J Mech Phys Solids 50(3):583–613. https://doi.org/10.1016/S0022-5096(01)00061-8. http://arxiv.org/abs/cond-mat/0002314. http://linkinghub.elsevier.com/retrieve/pii/S0022509601000618,0002314
Petrov VA, Orlov AN (1975) Contribution of thermal fluctuations to the scattering and the gauge effect of longevity. Int J Fract 11(5):881–886. https://doi.org/10.1007/BF00012904
Petrov VA, Orlov AN (1976) Statistical kinetics of thermally activated fracture. Int J Fract 12(2):231–238. https://doi.org/10.1007/BF00036980
Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117(1):1–19. https://doi.org/10.1006/jcph.1995.1039. http://linkinghub.elsevier.com/retrieve/pii/S002199918571039X
Ponson L, Bonamy D, Barbier L (2006) Cleaved surface of<math display=“inline”><mrow><mi>i</mi><mtext></mtext><mi mathvariant=“normal”>Al</mi><mi mathvariant=“normal”>Pd</mi><mi mathvariant=“normal”>Mn</mi></mrow></math> quasicrystals: influence of the local temperature elevation at. Phys Rev B 74(18):184205 https://doi.org/10.1103/PhysRevB.74.184205
Regel’ VR, Slutsker AI, Tomashevski ÉE (1972) The kinetic nature of the strength of solids. Sov Phys Uspekhi 15(1):45–65. https://doi.org/10.1070/PU1972v015n01ABEH004945. http://stacks.iop.org/0038-5670/15/i=1/a=R03?key=crossref.123f17f2adf8d374f849d4f9578b0ef4
Rice JR, Levy N (1969) Local heating by plastic deformation at a crack tip. In: Argon AS (ed) Physics of strength and plasticity. MIT Press, Cambridge, pp 277–293
Schuh C, Hufnagel T, Ramamurty U (2007) Mechanical behavior of amorphous alloys. Acta Mater 55(12):4067–4109. https://doi.org/10.1016/j.actamat.2007.01.052. http://linkinghub.elsevier.com/retrieve/pii/S135964540700122X
Shenderova OA, Brenner DW, Omeltchenko A, Su X, Yang LH (2000) Atomistic modeling of the fracture of polycrystalline diamond. Phys Rev B 61(6):3877–3888. https://doi.org/10.1103/PhysRevB.61.3877. http://link.aps.org/doi/10.1103/PhysRevB.61.3877. http://prb.aps.org/abstract/PRB/v61/i6/p3877_1
Slepyan LI (2002) Models and phenomena in fracture mechanics. Foundations of engineering mechanics. Springer, Berlin. https://doi.org/10.1007/978-3-540-48010-5
Slutsker AI (2005) Atomic-level fluctuation mechanism of the fracture of solids (computer simulation studies). Phys Solid State 47(5):801. https://doi.org/10.1134/1.1924836
Thomson R (1986) Physics of fracture. In: Ehrenreich H, Turnbull D (eds) Solid state physics, vol 39. Academic Press, pp 1–129. https://doi.org/10.1016/S0081-1947(08)60368-9. http://linkinghub.elsevier.com/retrieve/pii/S0081194708603689
Thomson R, Zhou SJ, Carlsson AE, Tewary VK (1992) Lattice imperfections studied by use of lattice Greens functions. Phys Rev B 46(17):10613–10622. https://doi.org/10.1103/PhysRevB.46.10613
Wang G, Chan K, Xu X, Wang W (2008) Instability of crack propagation in brittle bulk metallic glass. Acta Mater 56(19):5845–5860. https://doi.org/10.1016/j.actamat.2008.08.005. http://linkinghub.elsevier.com/retrieve/pii/S1359645408005636
Wei C, Cho K, Srivastava D (2003) Tensile strength of carbon nanotubes under realistic temperature and strain rate. Phys Rev B 67(11):115407. https://doi.org/10.1103/PhysRevB.67.115407. http://arxiv.org/abs/cond-mat/0202513
Yazdani H, Hatami K (2015) Failure criterion for graphene in biaxial loading a molecular dynamics study. Model Simul Mater Sci Eng 23(6):065004 https://doi.org/10.1088/0965-0393/23/6/065004. http://stacks.iop.org/0965-0393/23/i=6/a=065004?key=crossref.66817f771850338a194c2b8253b568c9
Zehnder AT, Rosakis AJ (1991) On the temperature distribution at the vicinity of dynamically propagating cracks in 4340 steel. J Mech Phys Solids 39(3):385–415. https://doi.org/10.1016/0022-5096(91)90019-K. http://linkinghub.elsevier.com/retrieve/pii/002250969190019K
Zhang T, Li X, Kadkhodaei S, Gao H (2012) Flaw insensitive fracture in nanocrystalline graphene. Nano Lett 12(9):4605–4610. https://doi.org/10.1021/nl301908b
Zhang T, Li X, Gao H (2015) Fracture of graphene: a review. Int J Fract 196(1–2):1–31. https://doi.org/10.1007/s10704-015-0039-9
Zhu T, Li J, Samanta A, Leach A, Gall K (2008) Temperature and strain-rate dependence of surface dislocation nucleation. Phys Rev Lett 100(2):025502. https://doi.org/10.1103/PhysRevLett.100.025502
Zhurkov SN (1984) Kinetic concept of the strength of solids. Int J Fract 26(4):295–307. https://doi.org/10.1007/BF00962961
Acknowledgements
We gratefully acknowledge funding from the Labex MMCD provided by the national program Investments for the Future of the French National Research Agency (ANR-11-LABX-022-01)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brochard, L., Souguir, S. & Sab, K. Scaling of brittle failure: strength versus toughness. Int J Fract 210, 153–166 (2018). https://doi.org/10.1007/s10704-018-0268-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-018-0268-9