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**Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bonds.**
*(English)*
Zbl 0974.74008

Summary: A virtual internal bond (VIB) model with randomized cohesive interactions between material particles is proposed for the integration of continuum models with cohesive surfaces und atomistic models with interatomic bonding. This approach differs from an atomistic model in that a phenomenological “cohesive force law” is assumed to act between “material particles” which are not necessarily atoms; it also differs from a cohesive surface model in that, rather than imposing a cohesive law along a prescribed set of discrete surfaces, a randomized network of cohesive bonds is statistically incorporated into the constitutive law of the material via the Cauchy-Born rule, i.e. by equating the strain energy function on the continuum level to the potential energy stored in the cohesive bonds due to an imposed deformation. This work is motivated by the notion that materials exhibit multiscale cohesive behaviors ranging from interatomic bonding to macroscopic ductile failure.

It is shown that the linear elastic behavior of the VIB model is isotropic and obeys the Cauchy relation; the instantaneous elastic properties under equibiaxial stretching are transversely isotropic, with all the in-plane components of the material tangent moduli vanishing in the cohesive stress limit; the instantaneous properties under equitriaxial stretching are isotropic with a finite strain modulus. We demonstrate through two preliminary numerical examples that the VIB model can be applied in direct simulation of crack growth without a presumed fracture criterion.

It is shown that the linear elastic behavior of the VIB model is isotropic and obeys the Cauchy relation; the instantaneous elastic properties under equibiaxial stretching are transversely isotropic, with all the in-plane components of the material tangent moduli vanishing in the cohesive stress limit; the instantaneous properties under equitriaxial stretching are isotropic with a finite strain modulus. We demonstrate through two preliminary numerical examples that the VIB model can be applied in direct simulation of crack growth without a presumed fracture criterion.

### MSC:

74A45 | Theories of fracture and damage |

74R10 | Brittle fracture |

74B20 | Nonlinear elasticity |

74S05 | Finite element methods applied to problems in solid mechanics |

74A60 | Micromechanical theories |

### Keywords:

virtual internal bond model; cohesive force law; crack bifurcation; crack propagation; finite strain; crack arrest; finite element method; Cauchy-Born rule; strain energy; potential energy; interatomic bonding; equibiaxial stretching; equitriaxial stretching; crack growth
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\textit{H. Gao} and \textit{P. Klein}, J. Mech. Phys. Solids 46, No. 2, 187--218 (1998; Zbl 0974.74008)

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### References:

[1] | Abraham, F.F.; Brodbeck, D.; Rafey, R.A.; Rudge, W.E., Instability dynamics of fracture: a computer simulation investigation, Phys. rev. lett., 73, 272-275, (1994) |

[2] | Abraham, F.F., On the transition from brittle to plastic failure in breaking a nanocrystal under tension (NUT), Europhys. lett., 38, 103-106, (1997) |

[3] | Barenblatt, G.I., The formation of equilibrium cracks during brittle fracture: general ideas and hypotheses, axially symmetric cracks, Appl. math. mech. (PMM), 23, 622-636, (1959) · Zbl 0095.39202 |

[4] | Bassani, J.L.; Vitek, V.; Alber, E.S., Atomic-level elastic properties of interfaces and their relation to continua, Acta metall. mater., 40, S307-S320, (1992) |

[5] | Born, M.; Huang, K., Dynamical theories of crystal lattices, (1956), Clarendon Oxford |

[6] | Camacho, G.T.; Ortiz, M., Computational modeling of impact damage in brittle materials, Int. J. solids struct., 33, 2899-2938, (1996) · Zbl 0929.74101 |

[7] | Cheung, K.S.; Yip, S., A molecular dynamics simulation of crack tip extension: the brittle-to-ductile transition, Modeling simul. mater. sci. eng., 2, 856-892, (1994) |

[8] | Dugdale, D.S., Yielding of steel sheets containing slits, Journal of mechanics and physics of solids, 8, 100-104, (1960) |

[9] | Freund, L.B., Dynamic fracture mechanics, (1990), Cambridge University Press New York · Zbl 0712.73072 |

[10] | Gao, H., Surface roughening and branching instabilities in dynamic fracture, Journal of mechanics and physics of solids, 41, 457-486, (1993) |

[11] | Gao, H., A theory of local limiting speed in dynamic fracture, Journal of mechanics and physics of solids, 44, 1453-1474, (1996) |

[12] | Gao, H., Elastic waves in a hyperelastic solid near its plane strain equibiaxial cohesive limit, Phil. mag. lett., 76, 307-314, (1997) |

[13] | Hill, R., On the elasticity and stability of perfect crystals at finite strain, (), 225-240 · Zbl 0305.73028 |

[14] | Huang, K., On the atomic theory of elasticity, (), 178-194 · Zbl 0038.13802 |

[15] | Kad, B.K.; Dao, M.; Asaro, R.J., Numerical simulations of stress-strain behavior in two-phase α2 + γ lamellar tial alloys, Mat. sci. eng., A192/A193, 97-103, (1995) |

[16] | Knowles, J.K., The finite anti-phase shear field near the tip of a crack for a class of incompressible solid, J. elast., 13, 257-293, (1977) |

[17] | Knowles, J.K.; Sternberg, E., Large deformation near a tip of an interfacial crack between two neo-Hookean sheets, J. elast., 13, 257-293, (1983) · Zbl 0546.73079 |

[18] | Kohlhoff, S.; Gumbsch, P.; Fischmeister, H.F., Crack propagation in BCC crystals studied with a combined finite-element and atomistic model, Phil. mag. A, 64, 851-878, (1991) |

[19] | Kunin, I.A., Elastic media with microstructure, (1982), Springer-Verlag New York · Zbl 0167.54401 |

[20] | Lee, E.H., Elastic-plastic deformations at finite strain, J. appl. mech., 36, 1-6, (1969) · Zbl 0179.55603 |

[21] | Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0545.73031 |

[22] | Milstein, F., Review: theoretical elastic behaviour at large strains, J. mat. sci., 15, 1071-1084, (1980) |

[23] | Mindlin, R.D., Elasticity, piezoelectricity and crystal lattice dynamics, J. elast., 2, 217-282, (1972) |

[24] | Ogden, R.W., Non-linear elastic deformations, (1984), John Wiley and Sons New York · Zbl 0541.73044 |

[25] | Peirce, D.; Asaro, R.J.; Needleman, A., Material rate dependence and localized deformation in crystalline solids, Acta metall. mater., 31, 1951-1976, (1983) |

[26] | Simo, J.C., On the computational significance of the intermediate configuration and the hyperelastic stress relations in finite deformation elastoplasticity, Mechanics of materials, 4, 439-451, (1985) |

[27] | Stakgold, I., The Cauchy relations in a molecular theory of elasticity, Q. appl. mech., 8, 169-186, (1950) · Zbl 0037.42901 |

[28] | Tadmor, E.B.; Ortiz, M.; Phillips, R., Quasicontinuum analysis of defects in solids, Phil. mag. A, 73, 1529-1563, (1996) |

[29] | Truskinovsky, L., Fracture as a phase transformation, (), 322-332 |

[30] | Wallace, D.C., Thermodynamics of crystals, (1972), John Wiley and Sons New York |

[31] | Weiner, J.H., Hellmann-Feynman theorem, elastic moduli, and the Cauchy relation, Phys. rev. B, 24, 845-848, (1981) |

[32] | Weiner, J.H., Statistical mechanics of elasticity, (1983), John Wiley and Sons New York · Zbl 0616.73034 |

[33] | Willis, J.R., A comparison of the fracture criteria of griffith and Barenblatt, Journal of mechanics and physics of solids, 15, 151-162, (1967) |

[34] | Xu, X.-P.; Needleman, A., Numerical simulations of fast crack growth in brittle solids, Journal of mechanics and physics of solids, 42, 1397-1434, (1994) · Zbl 0825.73579 |

[35] | Zhou, S.J.; Beazley, D.M.; Lomdahl, P.S.; Holian, B.L., Large-scale molecular dynamics simulations of three-dimensional ductile failure, Phys. rev. lett., 78, 479-482, (1994) |

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