Journal of Elasticity

, Volume 124, Issue 2, pp 143–191 | Cite as

Cohesive Dynamics and Brittle Fracture

  • Robert LiptonEmail author


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.


Peridynamics Dynamic brittle fracture Fracture toughness Process zone \(\varGamma\)-Convergence 

Mathematics Subject Classification

34A34 74H55 74R10 



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.


  1. 1.
    Agwai, A., Guven, I., Madenci, E.: Predicting crack propagation with peridynamics: a comparative study. Int. J. Fract. 171, 65–78 (2011) CrossRefzbMATHGoogle Scholar
  2. 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) MathSciNetzbMATHGoogle Scholar
  3. 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. 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) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ambrosio, L., Coscia, A., Dal Maso, G.: Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139, 201–238 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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) CrossRefzbMATHGoogle Scholar
  7. 7.
    Barenblatt, G.I.: The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech. 7, 55–129 (1962) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bažant, Z.P., Planas, J.: Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press, Boca Raton (1998) Google Scholar
  9. 9.
    Bellettini, G., Coscia, A., Dal Maso, G.: Compactness and lower semicontinuity properties in SBD(\(\varOmega\)). Math. Z. 228, 337–351 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45, 601–620 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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) ADSCrossRefGoogle Scholar
  13. 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) CrossRefGoogle Scholar
  14. 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) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bouchbinder, E., Fineberg, J., Marder, M.: Dynamics of simple cracks. Annu. Rev. Condens. Matter Phys. 1, 371–395 (2010) ADSCrossRefGoogle Scholar
  16. 16.
    Bourdin, B., Francfort, G., Marigo, J.-J.: The variational approach to fracture. J. Elast. 91, 5–148 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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) CrossRefzbMATHGoogle Scholar
  18. 18.
    Braides, A.: Approximation of Free Discontinuity Problems. Lecture Notes in Mathematics, vol. 1694. Springer, Berlin (1998) CrossRefzbMATHGoogle Scholar
  19. 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. 20.
    Braides, A.: Local Minimization, Variational Evolution and \(\varGamma\)-Convergence. Lecture Notes in Mathematics, vol. 2094. Springer, Berlin (2014) CrossRefzbMATHGoogle Scholar
  21. 21.
    Braides, A., Gelli, M.S.: Limits of discrete systems with long-range interactions. J. Convex Anal. 9, 363–399 (2002) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Buehler, M.J., Abraham, F.F., Gao, H.: Hyperelasticity governs dynamic fracture at a critical length scale. Nature 426, 141–146 (2003) ADSCrossRefGoogle Scholar
  23. 23.
    Cox, B.N., Yang, Q.D.: In quest of virtual tests for structural composites. Science 314, 1102–1107 (2006) ADSCrossRefGoogle Scholar
  24. 24.
    Driver, B.: Analysis Tools with Applications. Springer, Berlin (2003). E-book Google Scholar
  25. 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) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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) ADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Dugdale, D.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960) ADSCrossRefGoogle Scholar
  28. 28.
    Dyal, K., Bhattacharya, K.: Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54, 1811–1842 (2006) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Emmrich, E., Puhst, D.: Well-posedness of the peridynamic model with Lipschitz continuous pairwise force function. Commun. Math. Sci. 11, 1039–1049 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 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) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. Am. Math. Soc., Providence (2010) zbMATHGoogle Scholar
  32. 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. 33.
    Federer, H.: Geometric Measure Theory. Springer, Berlin (1969) zbMATHGoogle Scholar
  34. 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) CrossRefGoogle Scholar
  35. 35.
    Francfort, G., Larsen, C.: Existence and convergence for quasi-static evolution in brittle fracture. Commun. Pure Appl. Math. 56, 1465–1500 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Francfort, G., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Freund, L.B.: Dynamic Fracture Mechanics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge (1998) zbMATHGoogle Scholar
  38. 38.
    Gerstle, W., Sau, N., Silling, S.: Peridynamic modeling of concrete structures. Nucl. Eng. Des. 237, 1250–1258 (2007) CrossRefGoogle Scholar
  39. 39.
    Giacomini, A.: Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Differ. Equ. 22, 129–172 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Gobbino, M.: Finite difference approximation of the Mumford-Shah functional. Commun. Pure Appl. Math. 51, 197–228 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Gobbino, M., Mora, M.G.: Finite difference approximation of free discontinuity problems. Proc. R. Soc. Edinb., Sect. A, Math. 131, 567–595 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ha, Y.D., Bobaru, F.: Studies of dynamic crack propagation and crack branching with peridynamics. Int. J. Fract. 162, 229–244 (2010) CrossRefzbMATHGoogle Scholar
  43. 43.
    Hanche-Olsen, B., Holden, H.: The Kolomogorov-Riesz compactness theorem. Expo. Math. 28, 385–394 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  44. 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) CrossRefGoogle Scholar
  45. 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) MathSciNetCrossRefzbMATHGoogle Scholar
  46. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Lussardi, L., Negri, M.: Convergence of nonlocal finite element energies for fracture mechanics. Numer. Funct. Anal. Optim. 28, 83–109 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Marder, M.: Supersonic rupture of rubber. J. Mech. Phys. Solids 54, 491–532 (2006) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Marder, M., Gross, S.: Origin of crack tip instabilities. J. Mech. Phys. Solids 43, 1–48 (1995) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 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) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 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. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  53. 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) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 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) CrossRefzbMATHGoogle Scholar
  55. 55.
    Morgan, F.: Geometric Measure Theory, a Beginner’s Guide. Academic Press, San Diego (1995) zbMATHGoogle Scholar
  56. 56.
    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 17, 577–685 (1989) MathSciNetCrossRefzbMATHGoogle Scholar
  57. 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) ADSCrossRefzbMATHGoogle Scholar
  58. 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) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Schmidt, B., Fraternali, F., Ortiz, M.: Eigenfracture: an eigendeformation approach to variational fracture. Multiscale Model. Simul. 7, 1237–1266 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Silling, S.A., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526–1535 (2005) CrossRefGoogle Scholar
  62. 62.
    Silling, S.A., Bobaru, F.: Peridynamic modeling of membranes and fibers. Int. J. Non-Linear Mech. 40, 395–409 (2005) ADSCrossRefzbMATHGoogle Scholar
  63. 63.
    Silling, S.A., Lehoucq, R.: Convergence of peridynamics to classical elasticity theory. J. Elast. 93, 13–37 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Silling, S., Weckner, O., Askari, E., Bobaru, F.: Crack nucleation in a peridynamic solid. Int. J. Fract. 162, 219–227 (2010) CrossRefzbMATHGoogle Scholar
  66. 66.
    Slepyan, L.I.: Models and Phenomena in Fracture Mechanics. Springer, Berlin (2002) CrossRefzbMATHGoogle Scholar
  67. 67.
    Suquet, P.M.: Un espace fonctionnel pour les équations de la plasticité. Ann. Fac. Sci. Toulouse 1, 77–87 (1979) MathSciNetCrossRefzbMATHGoogle Scholar
  68. 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) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. 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) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Willis, J.R.: A comparison of the fracture criteria of Griffith and Barenblatt. J. Mech. Phys. Solids 15, 152–162 (1967) ADSGoogle Scholar
  71. 71.
    Xu, X.P., Needleman, A.: Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42, 1397–1434 (1994) ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUnited States

Personalised recommendations