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Cohesive Dynamics and Brittle Fracture

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An Erratum to this article was published on 16 August 2016

Abstract

We formulate a nonlocal cohesive model for calculating the deformation inside a cracking body. In this model a set of physical properties including elastic and softening behavior are assigned to each point in the medium. We work within the small deformation setting and use the peridynamic formulation. Here strains are calculated as difference quotients. The constitutive relation is given by a nonlocal cohesive law relating force to strain. At each instant of the evolution we identify a process zone where strains lie above a threshold value. Perturbation analysis shows that jump discontinuities within the process zone can become unstable and grow. We derive an explicit inequality that shows that the size of the process zone is controlled by the ratio given by the length scale of nonlocal interaction divided by the characteristic dimension of the sample. The process zone is shown to concentrate on a set of zero volume in the limit where the length scale of nonlocal interaction vanishes with respect to the size of the domain. In this limit the dynamic evolution is seen to have bounded linear elastic energy and Griffith surface energy. The limit dynamics corresponds to the simultaneous evolution of linear elastic displacement and the fracture set across which the displacement is discontinuous. We conclude illustrating how aspects of the approach developed here can be applied to limits of dynamics associated with other energies that \(\varGamma\)-converge to the Griffith fracture energy.

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References

  1. Agwai, A., Guven, I., Madenci, E.: Predicting crack propagation with peridynamics: a comparative study. Int. J. Fract. 171, 65–78 (2011)

    Article  MATH  Google Scholar 

  2. Alicandro, R., Focardi, M., Gelli, M.S.: Finite-difference approximation of energies in fracture mechanics. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 23, 671–709 (2000)

    MathSciNet  MATH  Google Scholar 

  3. Ambrosio, L., Brades, A.: Energies in SBV and variational models in fracture mechanics. In: Cioranescu, D., Damlamian, A., Donato, P. (eds.) Homogenization and Applications to Materials Science 9, pp. 1–22. Gakkotosho, Tokyo (1997)

    Google Scholar 

  4. Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via \(\varGamma\)-convergence. Commun. Pure Appl. Math. XLIII, 999–1036 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ambrosio, L., Coscia, A., Dal Maso, G.: Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139, 201–238 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2006)

    Book  MATH  Google Scholar 

  7. Barenblatt, G.I.: The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech. 7, 55–129 (1962)

    Article  MathSciNet  Google Scholar 

  8. Bažant, Z.P., Planas, J.: Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press, Boca Raton (1998)

    Google Scholar 

  9. Bellettini, G., Coscia, A., Dal Maso, G.: Compactness and lower semicontinuity properties in SBD(\(\varOmega\)). Math. Z. 228, 337–351 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bellido, J.C., Morra-Corral, C., Pedregal, P.: Hyperelasticity as a \(\varGamma\)-limit of peridynamics when the horizon goes to zero. Calc. Var. Partial Differ. Equ. 54, 1643–1670 (2015). doi:10.1007/s00526-015-0839-9

    Article  MathSciNet  MATH  Google Scholar 

  11. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45, 601–620 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Belytschko, T., Gracie, R., Ventura, G.: A review of the extended/generalized finite element methods for material modelling. Model. Simul. Mater. Sci. Eng. 17, 043001 (2009)

    Article  ADS  Google Scholar 

  13. Bobaru, F., Hu, W.: The meaning, selection, and use of the Peridynamic horizon and its relation to crack branching in brittle materials. Int. J. Fract. 176, 215–222 (2012)

    Article  Google Scholar 

  14. Borden, M., Verhoosel, C., Scott, M., Hughes, T., Landis, C.: A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217–220, 77–95 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bouchbinder, E., Fineberg, J., Marder, M.: Dynamics of simple cracks. Annu. Rev. Condens. Matter Phys. 1, 371–395 (2010)

    Article  ADS  Google Scholar 

  16. Bourdin, B., Francfort, G., Marigo, J.-J.: The variational approach to fracture. J. Elast. 91, 5–148 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bourdin, B., Larsen, C., Richardson, C.: A time-discrete model for dynamic fracture based on crack regularization. Int. J. Fract. 168, 133–143 (2011)

    Article  MATH  Google Scholar 

  18. Braides, A.: Approximation of Free Discontinuity Problems. Lecture Notes in Mathematics, vol. 1694. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  19. Braides, A.: Discrete approximation of functionals with jumps and creases. In: Homogenization. Naples, 2001. Gakuto Internat. Ser. Math. Sci. Appl., vol. 18, pp. 147–153. Gakkotosho, Tokyo (2003)

    Google Scholar 

  20. Braides, A.: Local Minimization, Variational Evolution and \(\varGamma\)-Convergence. Lecture Notes in Mathematics, vol. 2094. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  21. Braides, A., Gelli, M.S.: Limits of discrete systems with long-range interactions. J. Convex Anal. 9, 363–399 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Buehler, M.J., Abraham, F.F., Gao, H.: Hyperelasticity governs dynamic fracture at a critical length scale. Nature 426, 141–146 (2003)

    Article  ADS  Google Scholar 

  23. Cox, B.N., Yang, Q.D.: In quest of virtual tests for structural composites. Science 314, 1102–1107 (2006)

    Article  ADS  Google Scholar 

  24. Driver, B.: Analysis Tools with Applications. Springer, Berlin (2003). E-book

    Google Scholar 

  25. Du, Q., Gunzburger, M., Lehoucq, R., Zhou, K.: Analysis of the volume-constrained peridynamic Navier equation of linear elasticity. J. Elast. 113, 193–217 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Duarte, C.A., Hamzeh, O.N., Liszka, T.J., Tworzydlo, W.W.: A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Comput. Methods Appl. Mech. Eng. 190, 2227–2262 (2001)

    Article  ADS  MATH  Google Scholar 

  27. Dugdale, D.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960)

    Article  ADS  Google Scholar 

  28. Dyal, K., Bhattacharya, K.: Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54, 1811–1842 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Emmrich, E., Puhst, D.: Well-posedness of the peridynamic model with Lipschitz continuous pairwise force function. Commun. Math. Sci. 11, 1039–1049 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Emmrich, E., Weckner, O.: On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5, 851–864 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. Am. Math. Soc., Providence (2010)

    MATH  Google Scholar 

  32. Falk, M., Needleman, A., Rice, J.R.: A critical evaluation of cohesive zone models of dynamic fracture. J. Phys. IV 11, 43–50 (2001)

    Google Scholar 

  33. Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)

    MATH  Google Scholar 

  34. Foster, J., Silling, S.A., Chen, W.: An energy based failure criterion for use with peridynamic states. Int. J. Multiscale Comput. Eng. 9, 675–688 (2011)

    Article  Google Scholar 

  35. Francfort, G., Larsen, C.: Existence and convergence for quasi-static evolution in brittle fracture. Commun. Pure Appl. Math. 56, 1465–1500 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Francfort, G., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Freund, L.B.: Dynamic Fracture Mechanics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  38. Gerstle, W., Sau, N., Silling, S.: Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237, 1250–1258 (2007)

    Article  Google Scholar 

  39. Giacomini, A.: Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Differ. Equ. 22, 129–172 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Gobbino, M.: Finite difference approximation of the Mumford-Shah functional. Commun. Pure Appl. Math. 51, 197–228 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. Gobbino, M., Mora, M.G.: Finite difference approximation of free discontinuity problems. Proc. R. Soc. Edinb., Sect. A, Math. 131, 567–595 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ha, Y.D., Bobaru, F.: Studies of dynamic crack propagation and crack branching with peridynamics. Int. J. Fract. 162, 229–244 (2010)

    Article  MATH  Google Scholar 

  43. Hanche-Olsen, B., Holden, H.: The Kolomogorov-Riesz compactness theorem. Expo. Math. 28, 385–394 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Hillerborg, A., Modeer, M., Petersson, P.E.: Analysis of crack formation and crack growth by means of fracture mechanics and finite elements. Cem. Concr. Res. 6, 731–781 (1976)

    Article  Google Scholar 

  45. Larsen, C.J., Ortner, C., Suli, E.: Existence of solutions to a regularized model of dynamic fracture. Math. Models Methods Appl. Sci. 20, 1021–1048 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lipton, R.: Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elast. 117, 21–50 (2014). doi:10.1007/s10659-013-9463-0

    Article  MathSciNet  MATH  Google Scholar 

  47. Lussardi, L., Negri, M.: Convergence of nonlocal finite element energies for fracture mechanics. Numer. Funct. Anal. Optim. 28, 83–109 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  48. Marder, M.: Supersonic rupture of rubber. J. Mech. Phys. Solids 54, 491–532 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Marigo, J.-J., Truskinovsky, L.: Initiation and propagation of fracture in the models of Griffith and Barenblatt. Contin. Mech. Thermodyn. 16, 391–409 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Matthies, H., Strang, G., Christiansen, E.: The saddle point of a differential program. In: Energy Methods in Finite Element Analysis. Wiley, New York (1979)

    Google Scholar 

  52. Mengesha, T., Du, Q.: Nonlocal constrained value problems for a linear peridynamic Navier equation. J. Elast. 116, 27–51 (2014). doi:10.1007/s10659-013-9456-z

    Article  MathSciNet  MATH  Google Scholar 

  53. Miehe, C., Hofacker, M., Welschinger, F.: A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Comput. Methods Appl. Mech. Eng. 199, 2765–2778 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Möes, N., Delbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131–150 (1999)

    Article  MATH  Google Scholar 

  55. Morgan, F.: Geometric Measure Theory, a Beginner’s Guide. Academic Press, San Diego (1995)

    MATH  Google Scholar 

  56. Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 17, 577–685 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  57. Oh, E.S., Walton, J.R., Slattery, J.C.: A theory of fracture based upon an extension of continuum mechanics to the nanoscale. J. Appl. Mech. 73, 792–798 (2006)

    Article  ADS  MATH  Google Scholar 

  58. Remmers, J.J.C., de Borst, R., Needleman, A.: The simulation of dynamic crack propagation using the cohesive segments method. J. Mech. Phys. Solids 56, 70–92 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. Schmidt, B., Fraternali, F., Ortiz, M.: Eigenfracture: an eigendeformation approach to variational fracture. Multiscale Model. Simul. 7, 1237–1266 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  60. Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  61. Silling, S.A., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526–1535 (2005)

    Article  Google Scholar 

  62. Silling, S.A., Bobaru, F.: Peridynamic modeling of membranes and fibers. Int. J. Non-Linear Mech. 40, 395–409 (2005)

    Article  ADS  MATH  Google Scholar 

  63. Silling, S.A., Lehoucq, R.: Convergence of peridynamics to classical elasticity theory. J. Elast. 93, 13–37 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  64. Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  65. Silling, S., Weckner, O., Askari, E., Bobaru, F.: Crack nucleation in a peridynamic solid. Int. J. Fract. 162, 219–227 (2010)

    Article  MATH  Google Scholar 

  66. Slepyan, L.I.: Models and Phenomena in Fracture Mechanics. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  67. Suquet, P.M.: Un espace fonctionnel pour les équations de la plasticité. Ann. Fac. Sci. Toulouse 1, 77–87 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  68. Weckner, O., Abeyaratne, R.: The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53, 705–728 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. Wheeler, M.F., Wick, T., Wollner, W.: An augmented-Lagrangian method for the phase-field approach for pressurized fractures. Comput. Methods Appl. Mech. Eng. 271, 69–85 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. Willis, J.R.: A comparison of the fracture criteria of Griffith and Barenblatt. J. Mech. Phys. Solids 15, 152–162 (1967)

    ADS  Google Scholar 

  71. Xu, X.P., Needleman, A.: Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42, 1397–1434 (1994)

    Article  ADS  MATH  Google Scholar 

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Acknowledgements

The author would like to thank Stewart Silling, Richard Lehoucq and Florin Bobaru for stimulating and fruitful discussions. This research is supported by NSF grant DMS-1211066, AFOSR grant FA9550-05-0008, and NSF EPSCOR Cooperative Agreement No. EPS-1003897 with additional support from the Louisiana Board of Regents.

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  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, United States

    Robert Lipton

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Lipton, R. Cohesive Dynamics and Brittle Fracture. J Elast 124, 143–191 (2016). https://doi.org/10.1007/s10659-015-9564-z

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