Dynamic analysis of discontinuous rock masses has been studied intensively in recent years, and Numerical Manifold Method (NMM) is one of the feasible methods in this field. However, conventional NMM often encounters difficulty in setting appropriate analysis parameters which can predict sliding between the blocks precisely. In this study, to enhance the robustness and the accuracy of NMM, the integration method of friction law is focused. The implicit updating scheme of friction force (return mapping algorithm) and Newton-Raphson iteration is newly introduced to NMM, and the performance of the proposed ‘fully implicit’ NMM is examined through the numerical examples.


Prediction of dynamic behaviors of rock masses induced by earthquakes, blasting, and excavation is still remaining important problem in rock engineering field, and a number of numerical methods has been proposed during these decades. The Numerical Manifold Method (NMM) is one of those methods originally proposed by Shi (1991). The NMM is formulated based on the principle of the potential energy minimization, the contact treatment by the penalty method, the finite cover approximation of the displacement field, and the implicit time integration (Newmark β method). Since the formulation enables to treat the material deformation and the material discontinuity in a uniform manner and to use large time increment, the NMM has been applied to many rock mechanics problems such as earthquake induced slope failure (Miki et al., 2013), crack propagation analysis (Zhang et al., 2010), and progressive failure of rock slopes (An et al., 2014). However, the NMM often encounters difficulty in setting proper numerical parameters (e.g. penalty coefficient, time increment) that can reproduce accurate solution in the analyses concerning dynamic sliding along the discontinuity. Although the prediction accuracy of the sliding strongly affects the global behavior of the block system, the essential countermeasure has not been established yet.

In this study, as a cause of the above drawback, the integration algorithm of the friction law is focused on. In the original NMM code, the friction force along a discontinuous interface is updated incrementally as the shear displacement progresses. Then, when the friction force reaches the friction strength, the friction force will be drawn back to the true strength at the ‘next’ time step (Fig. 1). This scheme induces the overestimation of the friction, and the error of sliding displacement is unavoidable. Additionally, since the amount of the error depends on both of the penalty value and the tangential displacement increment, the determination of the proper parameter set is difficult.

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