Finite elements with embedded strong discontinuities for the modeling of failure in solids.

*(English)*Zbl 1194.74431Summary: This paper presents new finite elements that incorporate strong discontinuities with linear interpolations of the displacement jumps for the modeling of failure in solids. The cases of interest are characterized by a localized cohesive law along a propagating discontinuity (e.g. a crack), with this propagation occurring in a general finite element mesh without remeshing. Plane problems are considered in the infinitesimal deformation range. The new elements are constructed by enhancing the strains of existing finite elements (including general displacement based, mixed, assumed and enhanced strain elements) with a series of strain modes that depend on the proper enhanced parameters local to the element. These strain modes are designed by identifying the strain fields to be captured exactly, including the rigid body motions of the two parts of a splitting element for a fully softened discontinuity, and the relative stretching of these parts for a linear tangential sliding of the discontinuity. This procedure accounts for the discrete kinematics of the underlying finite element and assures the lack of stress locking in general quadrilateral elements for linearly separating discontinuities, that is, spurious transfers of stresses through the discontinuity are avoided. The equations for the enhanced parameters are constructed by imposing the local equilibrium between the stresses in the bulk of the element and the tractions driving the aforementioned cohesive law, with the proper equilibrium operators to account for the linear kinematics of the discontinuity. Given the locality of all these considerations, the enhanced parameters can be eliminated by their static condensation at the element level, resulting in an efficient implementation of the resulting methods and involving minor modifications of an existing finite element code. A series of numerical tests and more general representative numerical simulations are presented to illustrate the performance of the new elements.

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\textit{C. Linder} and \textit{F. Armero}, Int. J. Numer. Methods Eng. 72, No. 12, 1391--1433 (2007; Zbl 1194.74431)

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##### References:

[1] | Armero, International Journal of Solids and Structures 33 pp 2863– (1996) |

[2] | Carter, International Journal for Numerical Methods in Engineering 47 pp 229– (2000) |

[3] | Bittencourt, Engineering Fracture Mechanics 55 pp 321– (1996) |

[4] | Ortiz, Computer Methods in Applied Mechanics and Engineering 90 pp 781– (1991) |

[5] | Marusich, International Journal for Numerical Methods in Engineering 38 pp 3675– (1995) |

[6] | Xu, Journal of the Mechanics and Physics of Solids 42 pp 1397– (1994) |

[7] | Camacho, International Journal of Solids and Structures 33 pp 2899– (1996) |

[8] | Ortiz, International Journal for Numerical Methods in Engineering 44 pp 1267– (1999) |

[9] | Dvorkin, International Journal for Numerical Methods in Engineering 30 pp 541– (1990) |

[10] | Simo, Computational Mechanics 12 pp 277– (1993) |

[11] | . Recent advances in the analysis and numerical simulation of strain localization in inelastic solids. Proceedings of the 4th Computational Plasticity Conference, Barcelona, Spain, 1995. |

[12] | Oliver, International Journal for Numerical Methods in Engineering 39 pp 3575– (1996) |

[13] | Steinmann, Mechanics of Cohesive-Frictional Materials 4 pp 133– (1999) |

[14] | Mosler, International Journal for Numerical Methods in Engineering 57 pp 1553– (2003) |

[15] | Belytschko, International Journal for Numerical Methods in Engineering 45 pp 601– (1999) |

[16] | Moës, International Journal for Numerical Methods in Engineering 46 pp 131– (1999) |

[17] | Wells, International Journal for Numerical Methods in Engineering 50 pp 2667– (2001) |

[18] | Wells, International Journal for Numerical Methods in Engineering 53 pp 1235– (2002) |

[19] | Hansbo, Computer Methods in Applied Mechanics and Engineering 193 pp 3523– (2004) · Zbl 1068.74076 |

[20] | Oliver, Computer Methods in Applied Mechanics and Engineering 195 pp 4732– (2006) |

[21] | Armero, Mechanics of Cohesive-Frictional Materials 4 pp 101– (1999) · Zbl 0968.74062 |

[22] | Armero, Computer Methods in Applied Mechanics and Engineering 191 pp 181– (2001) |

[23] | Callari, Computer Methods in Applied Mechanics and Engineering 191 pp 4371– (2002) |

[24] | Callari, Computer Methods in Applied Mechanics and Engineering 193 pp 2941– (2004) |

[25] | Ehrlich, Computational Mechanics 35 pp 237– (2005) |

[26] | Armero, Computer Methods in Applied Mechanics and Engineering 195 pp 1283– (2006) |

[27] | Borja, Computer Methods in Applied Mechanics and Engineering 190 pp 2555– (2001) |

[28] | Manzoli, Computers and Structures 84 pp 742– (2006) |

[29] | Bolzon, Computational Mechanics 27 pp 463– (2001) |

[30] | Alfaiate, International Journal of Solids and Structures 40 pp 5799– (2003) |

[31] | Simo, International Journal for Numerical Methods in Engineering 29 pp 1595– (1990) |

[32] | , . Stability and uniqueness of strain-softening computations. In Europe–US Symposium on Finite Element Methods, Nonlinear Problems, , (eds). Springer: Berlin, 1986; 119–142. |

[33] | Oliver, Computer Methods in Applied Mechanics and Engineering 193 pp 2987– (2004) |

[34] | . Finite elements with higher-order interpolations of strong discontinuities. Report No. UCB/SEMM-2006/02, University of California at Berkeley, 2006. |

[35] | Crack growth and development of fracture zones in plain concrete and similar materials. Report No. TVBM-1006, Division of Building Materials, University of Lund, Lund, Sweden, 1981. |

[36] | Rots, Heron 30 pp 1– (1985) |

[37] | Reinhardt, Heron 29 pp 1– (1984) |

[38] | . Mixed-mode crack propagation in mortar and concrete. Report No. 81-13, Department of Structural Engineering, Cornell University, Ithaca, NY, 1982. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.