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Weakly nonlinear fracture mechanics: experiments and theory

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Abstract

Material failure occurs at the small scales in the immediate vicinity of the tip of a crack. Due to its generally microscopic size and the typically high crack propagation velocity, direct observation of the dynamic behavior in this highly deformed region has been prohibitively difficult. Here we present direct measurements of the deformation surrounding the tip of dynamic mode I cracks propagating in brittle elastomers at velocities ranging from 0.2 to 0.8C s . Both the detailed fracture dynamics and fractography of these materials are identical to that of standard brittle amorphous materials such as soda-lime glass. These measurements demonstrate how Linear Elastic Fracture Mechanics (LEFM) breaks down near the tip of a crack. This breakdown is quantitatively described by extending LEFM to the weakly nonlinear regime, by considering nonlinear elastic constitutive laws up to second order in the displacement-gradients. The theory predicts that, at scales within a dynamic lengthscale ℓ nl from the tip of a single crack, significant logr displacements and 1/r displacement-gradient contributions arise, and provides excellent quantitative agreement with the measured near-tip deformation. As ℓ nl is consistent with lengthscales that appear in crack tip instabilities, this “weakly nonlinear fracture mechanics” framework may serve as a springboard for the development of a comprehensive theory of fracture dynamics.

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References

  • Abraham FF (2001) The atomic dynamics of fracture. J Mech Phys Solids 49: 2095–2111

    Article  MATH  ADS  Google Scholar 

  • Abraham FF, Brodbeck D, Rafey RA, Rudge WE (1994) Instability dynamics of fracture—a computer simulation investigation. Phys Rev Lett 73: 272–275

    Article  PubMed  ADS  Google Scholar 

  • Abraham FF et al (2002) Simulating materials failure by using up to one billion atoms and the world’s fastest computer: brittle fracture. Proc Natl Acad Sci USA 99: 5777–5782

    Article  PubMed  ADS  CAS  Google Scholar 

  • Adda-Bedia M (2005) Brittle fracture dynamics with arbitrary paths III the branching instability under general loading. J Mech Phys Solids 53: 227–248

    Article  MATH  ADS  Google Scholar 

  • Afek I, Bouchbinder E, Katzav E, Mathiesen J, Procaccia I (2005) Void formation and roughening in slow fracture. Phys Rev E 71: 066127

    Article  ADS  CAS  Google Scholar 

  • Aranson IS, Kalatsky VA, Vinokur VM (2000) Continuum field description of crack propagation. Phys Rev Lett 85: 118–121

    Article  PubMed  ADS  CAS  Google Scholar 

  • Astrom J, Timomen J (1996) Crack bifurcations in a strained lattice. Phys Rev B 54: 9585

    Article  ADS  Google Scholar 

  • Bazant ZP, Novak D (2000) Probabilistic nonlocal theory for quasibrittle fracture initiation and size effect I: theory. J Eng Mech Asce 126: 166–174

    Article  Google Scholar 

  • Bouchbinder E (2009) Dynamic crack tip equation of motion: high-speed oscillatory instability. Phys Rev Lett 103: 164301

    Article  PubMed  ADS  CAS  Google Scholar 

  • Bouchbinder E, Livne A, Fineberg J (2008) Weakly nonlinear theory of dynamic fracture. Phys Rev Lett 101: 264302

    Article  PubMed  ADS  CAS  Google Scholar 

  • Bouchbinder E, Livne A, Fineberg J (2009) The 1/r singularity in weakly nonlinear fracture mechanics. J Mech Phys Solids 57: 1568–1577

    Article  MathSciNet  ADS  Google Scholar 

  • Bouchbinder E, Lo TS (2008) Elastic nonlinearities in a one-dimensional model of fracture. Phys Rev E 78: 056105

    Article  MathSciNet  ADS  CAS  Google Scholar 

  • Bouchbinder E, Mathiesen J, Procaccia I (2004) Roughening of fracture surfaces: the role of plastic deformation. Phys Rev Lett 92: 245505

    Article  PubMed  ADS  CAS  Google Scholar 

  • Bouchbinder E, Mathiesen J, Procaccia I (2005) Branching instabilities in rapid fracture: dynamics and geometry. Phy Rev E 71: 056118

    Article  MathSciNet  ADS  CAS  Google Scholar 

  • Bouchbinder E, Procaccia I (2005) Non-universality in micro-branching instabilities in rapid fracture. Phys Rev E 72: 055103

    Article  MathSciNet  ADS  CAS  Google Scholar 

  • Bouchbinder E, Procaccia I (2007) Oscillatory instability in two-dimensional dynamic fracture. Phys Rev Lett 98: 124302

    Article  PubMed  ADS  CAS  Google Scholar 

  • Bouchbinder E, Pomyalov A, Procaccia I (2006) Dissipative visco-plastic deformation in dynamic fracture: tip blunting and velocity selection. Phys Rev Lett 97: 134301

    Article  PubMed  ADS  CAS  Google Scholar 

  • Boudet JF, Ciliberto S, Steinberg V (1995) Experimental study of the instability of crack propagation in brittle materials. Europhys Lett 30: 337–342

    Article  ADS  CAS  Google Scholar 

  • Broberg KB (1999) Cracks and fracture. Academic Press, San Diego

    Google Scholar 

  • Buehler MJ, Abraham FF, Gao H (2003) Hyperelasticity governs dynamic fracture at a critical length scale. Nature 426: 141–146

    Article  PubMed  ADS  CAS  Google Scholar 

  • Buehler MJ, Gao H (2006) Dynamical fracture instabilities due to local hyperelasticity at crack tips. Nature 439: 307–310

    Article  PubMed  ADS  CAS  Google Scholar 

  • Celarie F, Prades S, Bonamy D et al (2006) Glass breaks like metal, but at the nanometer scale. Phys Rev Lett 90: 075504

    Article  ADS  CAS  Google Scholar 

  • Falk ML, Needleman A, Rice JR (2001) A critical evaluation of cohesive zone models of dynamic fracture. J Phys IV Colloq 11: 43–50

    Google Scholar 

  • Fineberg J, Marder M (1999) Instability in dynamic fracture. Phys Rep 313: 1–108

    Article  MathSciNet  ADS  CAS  Google Scholar 

  • Freund LB (1998) Dynamic fracture mechanics. Cambridge University Press, Cambridge

    Google Scholar 

  • Gao H (1996) A theory of local limiting speed in dynamic fracture. J Mech Phys Solids 44: 1453–1474

    Article  ADS  CAS  Google Scholar 

  • Gross SP, Fineberg J, Marder M, McCormick WD, Swinney HL (1993) Acoustic emissions from rapidly moving cracks. Phys Rev Lett 71: 3162–3165

    Article  PubMed  ADS  Google Scholar 

  • Gumbsch P (2001) Atomistic aspects of fractures. Zeitschrift Fur Angewandte Mathematik Und Mechanik 81: S137–S140

    Google Scholar 

  • Gumbsch P, Gao H (1999) Dislocations faster than the speed of sound. Science 283: 965–968

    Article  PubMed  ADS  CAS  Google Scholar 

  • Gumbsch P, Taeri-Baghbadrani S, Brunner D, Sigle W, Ruhle A (2001) Plasticity and an inverse brittle-to-ductile transition in strontium titanate. Phys Rev Lett 87: 085505

    Article  PubMed  ADS  CAS  Google Scholar 

  • Hakim V, Karma A (2005) Crack path prediction in anisotropic brittle materials. Phys Rev Lett 95: 235501

    Article  PubMed  ADS  CAS  Google Scholar 

  • Hauch JA, Marder MP (1998) Energy balance in dynamic fracture, investigated by a potential drop technique. Int J Fract 90: 133–151

    Article  CAS  Google Scholar 

  • Henry H, Levine H (2004) Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys Rev Lett 93: 105504

    Article  PubMed  ADS  CAS  Google Scholar 

  • Holland D, Marder M (1998) Ideal brittle fracture of silicon studied with molecular dynamics. Phys Rev Lett 80: 746–749

    Article  ADS  CAS  Google Scholar 

  • Holland D, Marder M (1999) Cracks and atoms. Adv Mat 11: 793–806

    Article  CAS  Google Scholar 

  • Holzapfel GA (2000) Nonlinear solid mechanics. Wiley, Chichester

    MATH  Google Scholar 

  • Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 87: 045501

    Article  PubMed  ADS  CAS  Google Scholar 

  • Karma A, Lobkovsky AE (2004) Unsteady crack motion and branching in a phase-field model of brittle fracture. Phys Rev Lett 92: 245510

    Article  PubMed  ADS  CAS  Google Scholar 

  • Katzav E, Adda-Bedia M, Arias R (2007) Theory of dynamic branching in brittle materials. Int J Fract 143: 245–271

    Article  Google Scholar 

  • Kessler DA (2000) Steady-state cracks in viscoelastic lattice models II. Phys Rev E 61: 2348–2360

    Article  ADS  CAS  Google Scholar 

  • Kessler DA, Levine H (1999) Arrested cracks in nonlinear lattice models of brittle fracture. Phys Rev E 60: 7569–7571

    Article  ADS  CAS  Google Scholar 

  • Klein PA, Foulk JW, Chen EP, Wimmer SA, Gao HJ (2001) Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods. Theor Appl Fract Mech 37: 99–166

    Article  Google Scholar 

  • Knowles JK, Sternberg E (1983) Large deformations near a tip of an interface-crack between two Neo-Hookean sheets. J Elast 13: 257–293

    Article  MATH  MathSciNet  Google Scholar 

  • Livne A, Cohen G, Fineberg J (2005) Universality and hysteretic dynamics in rapid fracture. Phys Rev Lett 94: 224301

    Article  PubMed  ADS  CAS  Google Scholar 

  • Livne A, Ben-David O, Fineberg J (2007) Oscillations in rapid fracture. Phys Rev Lett 98: 124301

    Article  PubMed  ADS  CAS  Google Scholar 

  • Livne A, Bouchbinder E, Fineberg J (2008) Breakdown of linear elastic fracture mechanics near the tip of a rapid crack. Phys Rev Lett 101: 264301

    Article  PubMed  ADS  CAS  Google Scholar 

  • Lobkovsky AE, Langer JS (1998) Dynamic ductile to brittle transition in a one-dimensional model of viscoplasticity. Phys Rev E 58: 1568–1576

    Article  ADS  CAS  Google Scholar 

  • Lu J, Ravi-Chandar K (1999) Inelastic deformation and localization in polycarbonate under tension. Int J Solids Struct 36: 391–425

    Article  MATH  Google Scholar 

  • Lu Z et al (2005) Dynamics of wing cracks and nanoscale damage in glass. Phys Rev Lett 95: 135501

    Article  PubMed  ADS  CAS  Google Scholar 

  • Marder M, Gross S (1995) Origin of crack tip instabilities. J Mech Phys Solids 43: 1–48

    Article  MATH  MathSciNet  ADS  CAS  Google Scholar 

  • Miller O, Freund LB, Needleman A (1999) Energy dissipation in dynamic fracture of brittle materials. Model Simul Mater Sci Eng. 7: 573–586

    Article  ADS  CAS  Google Scholar 

  • Needleman A, Tvergaard V (2000) Numerical modeling of the ductile-brittle transition. Int J Fract 101: 73–97

    Article  CAS  Google Scholar 

  • Pardoen T, Hutchinson JW (2000) An extended model for void growth and coalescence. J Mech Phys Solids 48: 2467–2512

    Article  MATH  ADS  Google Scholar 

  • Perez R, Gumbsch P (2000) Directional anisotropy in the cleavage fracture of silicon. Phys Rev Lett 84: 5347–5350

    Article  PubMed  ADS  CAS  Google Scholar 

  • Ravi-Chandar K, Yang B (1997) On the role of microcracks in the dynamic fracture of brittle materials. J Mech Phys Solids 45: 535–563

    Article  ADS  CAS  Google Scholar 

  • Riecke H (1999) Localized structures in pattern-forming systems. In: Golubitsky DLM, Strogatz S (eds) Pattern formation in continuous and coupled systems. Springer, New York, p 215

    Google Scholar 

  • Rice JR (1974) Limitations to the small scale yielding approximation for crack tip plasticity. J Mech Phys Solids 22: 17–26

    Article  ADS  Google Scholar 

  • Rountree CL et al (2002) Atomistic aspects of crack propagation in brittle materials: Multimillion atom molecular dynamics simulations. Annu Rev Mater Res 32: 377–400

    Article  CAS  Google Scholar 

  • Sharon E, Cohen G, Fineberg J (2002) Crack front waves and the dynamics of a rapidly moving crack. Phys Rev Lett 88: 085503

    Article  PubMed  ADS  CAS  Google Scholar 

  • Sharon E, Fineberg J (1998) Universal features of the microbranching instability in dynamic fracture. Philos Mag B 78: 243–251

    ADS  CAS  Google Scholar 

  • Sharon E, Fineberg J (1999) Confrming the continuum theory of dynamic brittle fracture for fast cracks. Nature 397: 333–335

    Article  ADS  CAS  Google Scholar 

  • Siegmund T, Fleck NA, Needleman A (1997) Dynamic crack growth across an interface. Int J Fract 85: 381–402

    Article  Google Scholar 

  • Slepyan L (1981) Dynamics of a crack in a lattice. Sov Phys Dokl 26: 538–540

    MATH  ADS  Google Scholar 

  • Spatschek R, Hartmann M, Brener E, Muller-Krumbhaar H, Kassner K (2006) Phase field modeling of fast crack propagation. Phys Rev Lett 96: 015502

    Article  PubMed  ADS  CAS  Google Scholar 

  • Williams ML (1957) On the stress distribution at the base of a stationary crack. J Appl Mech 24: 109–114

    MATH  MathSciNet  Google Scholar 

  • Xu XP, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42: 1397–1437

    Article  MATH  ADS  Google Scholar 

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Author notes

  1. Eran Bouchbinder

    Present address: Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100, Israel

Authors and Affiliations

  1. Racah Institute of Physics, Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Eran Bouchbinder, Ariel Livne & Jay Fineberg

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  1. Eran Bouchbinder
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  2. Ariel Livne
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  3. Jay Fineberg
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Correspondence to Jay Fineberg.

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Bouchbinder, E., Livne, A. & Fineberg, J. Weakly nonlinear fracture mechanics: experiments and theory. Int J Fract 162, 3–20 (2010). https://doi.org/10.1007/s10704-009-9427-3

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  • DOI: https://doi.org/10.1007/s10704-009-9427-3

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