A linearly-independent higher-order extended numerical manifold method and its application to multiple crack growth simulation

The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants.Therefore,higherorder NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy.In addition,it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics.Thus,some other enriched local approximations are introduced to model the stress singularity at the crack tip.Generally,higher-order NMM,especially first-order NMM wherein local approximations are first-order polynomials,has the linear dependence problems as other partition of unit(PUM)based numerical methods does.To overcome this problem,an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method(FEM),which has no linear dependence issue.Meanwhile,the stresses at the nodes of mathematical mesh(the nodal stresses in FEM)are continuous and the degrees of freedom defined on the physical patches are physically meaningful.Next,the extended NMM is employed to solve multiple crack propagation problems.It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation.Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM.The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions.Thus the effectiveness and correctness of the developed NMM have been validated.

Engineering rock mechanics practices in the underground powerhouse at JinpingⅠhydropower station

Based on the analyses of data obtained from the underground powerhouse at Jinping I hydropower station, a comprehensive review of engineering rock mechanics practice in the underground powerhouse is first conducted. The distribution of strata, lithology, and initial geo-stress, the excavation process and corresponding rock mass support measures, the deformation and failure characteristics of the surrounding rock mass, the stress characteristics of anchorage structures in the cavern complex, and numerical simulations of surrounding rock mass stability and anchor support performance are presented. The results indicate that the underground powerhouse of Jinping I hydropower station is characterized by high to extremely high geo-stresses during rock excavation. Excessive surrounding rock mass deformation and high stress of anchorage structures, surrounding rock mass unloading damage, and local cracking failure of surrounding rock masses, etc., are mainly caused by rock mass excavation. Deformations of surrounding rock masses and stresses in anchorage structures here are larger than those found elsewhere： 20% of extensometers in the main powerhouse record more than 50 mm with the maximum at around 250 mm observed in the downstream sidewall of the transformer hall. There are about 25% of the anchor bolts having recorded stresses of more than 200 MPa. Jinping I hydropower plant is the first to have an underground powerhouse construction conducted in host rocks under extremely high geo-stress conditions, with the ratio of rock mass strength to geo-stress of less than 2.0. The results can provide a reference to underground powerhouse construction in similar geological conditions.