##
**Simulation of localization failure with strain-gradient-enhanced damage mechanics.**
*(English)*
Zbl 1076.74050

The authors introduce the effect of local strain gradient into continuum damage mechanics and account for those characteristic behaviors of localization, which are difficult for estimation by conventional continuum mechanics methods. The most important point in strain gradient approach, which describes the basic behavior of localization failure area of materials, is its damaging function, including an internal length scale. The last can be used to express the scale effects of mechanical response of brittle rock mass.

The authors choose the local strain gradient in the Fleck-Hutchinson form, relating the strain and strain gradient to the displacement field, and adopt a scalar variable and a second-order tensor respectively for the isotropic and anisotropic damage models. First, it is shown that the general form of damage evolution law is inconvenient for numerical implementation. An alternative simple explicit formulation is proposed in the paper. By the definition of damage, the thermodynamic force conjugated to the damage variable is regarded as the energy-releasing rate in fracture mechanics. Therefore, the internal variable is determined explicitly in terms of the maximum difference between a threshold value and the energy release rate during the loading/unloading process. The damage evolution law is adopted in Carmeliet-de Borst form [J. Carmeliet and R. de Borst, Int. J. Solids Struct. 32, No. 8–9, 1149–1160 (1995; Zbl 0919.73224)].

Then, a second-order damage tensor in the Oda-Cowin form is used to describe the microdefect-induced anisotropy. To account for the strain gradient in implementation of finite element method (FEM), elements with C1 continuity are adopted instead of C0 continuous elements. The paper adopts the Xia-Hutchinson method [Z. C. Xia and J. W. Hutchinson, Int. J. Solids Struct. 31, No. 8, 1133–1148 (1994; Zbl 0945.74670)] to account for the first gradient of strain within the framework of brittle damage mechanics for rock-like materials. When regular rectangle elements are encountered, the coupling between finite difference method and conventional FEM is used to avoid large modification to the existing FEM code and to obtain relatively higher efficiency and reasonably good accuracy. Finally, application of the anisotropic model to the 3D nonlinear FEM analysis of Ertan arch dam is conducted, and the results of numerical simulation of failure behavior are compared with those obtained by geophysical model tests of the dam.

The authors choose the local strain gradient in the Fleck-Hutchinson form, relating the strain and strain gradient to the displacement field, and adopt a scalar variable and a second-order tensor respectively for the isotropic and anisotropic damage models. First, it is shown that the general form of damage evolution law is inconvenient for numerical implementation. An alternative simple explicit formulation is proposed in the paper. By the definition of damage, the thermodynamic force conjugated to the damage variable is regarded as the energy-releasing rate in fracture mechanics. Therefore, the internal variable is determined explicitly in terms of the maximum difference between a threshold value and the energy release rate during the loading/unloading process. The damage evolution law is adopted in Carmeliet-de Borst form [J. Carmeliet and R. de Borst, Int. J. Solids Struct. 32, No. 8–9, 1149–1160 (1995; Zbl 0919.73224)].

Then, a second-order damage tensor in the Oda-Cowin form is used to describe the microdefect-induced anisotropy. To account for the strain gradient in implementation of finite element method (FEM), elements with C1 continuity are adopted instead of C0 continuous elements. The paper adopts the Xia-Hutchinson method [Z. C. Xia and J. W. Hutchinson, Int. J. Solids Struct. 31, No. 8, 1133–1148 (1994; Zbl 0945.74670)] to account for the first gradient of strain within the framework of brittle damage mechanics for rock-like materials. When regular rectangle elements are encountered, the coupling between finite difference method and conventional FEM is used to avoid large modification to the existing FEM code and to obtain relatively higher efficiency and reasonably good accuracy. Finally, application of the anisotropic model to the 3D nonlinear FEM analysis of Ertan arch dam is conducted, and the results of numerical simulation of failure behavior are compared with those obtained by geophysical model tests of the dam.

Reviewer: I. A. Parinov (Rostov-na-Donu)

### MSC:

74R20 | Anelastic fracture and damage |

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

74S20 | Finite difference methods applied to problems in solid mechanics |

74L10 | Soil and rock mechanics |

PDFBibTeX
XMLCite

\textit{W. Zhou} et al., Int. J. Numer. Anal. Methods Geomech. 26, No. 8, 793--813 (2002; Zbl 1076.74050)

Full Text:
DOI

### References:

[1] | Rudnicki, Journal of the Mechanics and Physics of Solids 23 pp 371– (1975) |

[2] | Ottosen, International Journal of Solids and Structures 27 pp 401– (1991) |

[3] | Benallal, Archives of Mechanics 44 pp 15– (1992) |

[4] | Comi, Mechanics of Cohesive-Frictional Materials 4 pp 17– (1999) |

[5] | Fremond, International Journal of Solids and Structures 33 pp 1083– (1996) |

[6] | Muhlhaus, Geotechnique 37 pp 217– (1987) · doi:10.1680/geot.1987.37.3.271 |

[7] | De Borst, Journal of Numerical Methods in Engineering 35 pp 521– (1992) |

[8] | Ba?ant, RILEM Materials and Structures 16 pp 155– (1983) |

[9] | Ba?ant, Journal of Engineering Mechanics 110 pp 1666– (1984) |

[10] | Ba?ant, Journal of Engineering Mechanics 116 pp 2485– (1990) |

[11] | Fracture Size Effect in Concrete and Other Quasi-brittle Materials. CRC Press LLC: Florida, U.S.A., 1998. |

[12] | Peerlings, International Journal for Numerical Methods in Engineering 39 pp 3391– (1996a) |

[13] | Peerlings, European Journal of Mechanics Part A/Solids 15 pp 937– (1996b) |

[14] | Peerlings, Mechanics of Cohesive-Frictional Materials 3 pp 323– (1998) |

[15] | Geers, Comparative Methods in Applied Mechanics and Engineering 160 pp 133– (1998a) |

[16] | Geers, International Journal of Solids and Structures 36 pp 2557– (1998b) |

[17] | Kuhl Ellen, Computational Materials Sciences 16 pp 176– (2000) |

[18] | Tvergaard, International Journal of Solids and Structures 32 pp 1063– (1995) |

[19] | Theorie des Corps Deformables. A. Hermann& files: Paris, 1909. |

[20] | Toupin, Archive for Rational Mechanics and Analysis 11 pp 385– (1962) |

[21] | Mindlin, Archive for Rational Mechanics and Analytic 16 pp 51– (1964) |

[22] | Mindlin, International Journal on Solids and Structures 28 pp 845– (1965) |

[23] | Aifantis, Transactions of ASME Journal of Engineering Materials and Technics 106 pp 326– (1984a) |

[24] | Aifantis, International Journal of Engineering Science 22 pp 961– (1984b) |

[25] | Muhlhaus, International Journal Solids and Structures 28 pp 845– (1991) |

[26] | Fleck, Journal of Mechanics and Physics of Solids 41 pp 1825– (1993) |

[27] | Strain gradient plasticity. In Advances in Applied Mechanics, Vol. 33, (eds). Academic Press: New York, 1997; 295-361. · Zbl 0894.73031 |

[28] | Fleck, Acta Metallurgica et Materialia 42 pp 475– (1994) |

[29] | Gao, Naturwissenschaftlen 86 pp 507– (1999) |

[30] | Studies on the process of rock burst and failure. Doctoral Thesis, Tsinghua University, China, Beijing, 1999. |

[31] | Xu Songlin, Chinese Journal of Geotechnical Engineering 23 pp 296– (2001) |

[32] | Simo, International Journal of Solids and Structures 23 pp 821– (1987) |

[33] | Shu, Journal of the Mechanics and Physics of Solids 47 pp 297– (1999) |

[34] | Stolken, Acta Materialia 46 pp 5109– (1998) |

[35] | Nix, Metallic Transactions A 20A pp 2217– (1989) |

[36] | De Guzman, Materials Research Symposium Proceedings 308 pp 613– (1993) · doi:10.1557/PROC-308-613 |

[37] | Stelmashenko, Acta Metallurgica et Materialia 41 pp 2855– (1993) |

[38] | Ma, Journal of Materials and Researches 10 pp 853– (1995) |

[39] | Poole, Scripta Metallurgica et Materialia 34 pp 559– (1996) · doi:10.1016/1359-6462(95)00524-2 |

[40] | McElhaney, Journal of Materials and Researches 13 pp 1300– (1998) |

[41] | Nix, Journal of the Mechanics and Physics of Solids 46 pp 411– (1998) |

[42] | Carmeliet, International Journal of Solids and Structures 32 pp 1149– (1995) |

[43] | Tensorial nature of damage measuring internal variables. In Proceedings of IUTAM Symposium on Physical Nonlinearities in Structural Analysis. (eds). Springer: Senlis, France, 1981; 140-155. · doi:10.1007/978-3-642-81582-9_20 |

[44] | Carol, ASCE Journal of Engineering Mechanics 117 pp 2429– (1991) |

[45] | Lemaiter, European Journal of Mechanics A/Solids 19 pp 187– (2000) |

[46] | Shao, Mechanisms of Materials 32 pp 607– (2000) |

[47] | Oda, Solids and Foundations 24 pp 27– (1984) · doi:10.3208/sandf1972.24.3_27 |

[48] | Cowin, Mechanisms of Materials 4 pp 137– (1985) |

[49] | Swoboda, International Journal of Solids and Structures 36 pp 1719– (1999) |

[50] | A fracture damage model for jointed rock masses and its application to the stability analysis of dam abutments. International Conference on Constitutive Laws for Engineering Materials, Chongqing, China, 1989; 8. |

[51] | The Finite Element Method, Chapter 19. McGraw-Hill: UK, 1977; 500-526. |

[52] | Xia, Journal of the Mechanics and Physics of Solids 44 pp 1621– (1996) |

[53] | Wei, Journal of the Mechanics and Physics of Solids 45 pp 1253– (1997) |

[54] | Zervos, International Journal of Solids and Structures 38 pp 5081– (2001) |

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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.