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**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 |

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

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