3D DDA vs. analytical solutions for dynamic sliding of a tetrahedral wedge

Introduction
In this research, the validity of the three dimensional Discontinuous Deformation Analysis (3D-DDA)1 is examined using analytical solutions for three dimensional problems involving two different failure modes: 1) dynamic sliding of a single block on an inclined plane, and 2) dynamic sliding of a tetrahedral wedge simultaneously on two faces.
From the early nineties, researchers in the DDA community have documented the accuracy of the original two-dimensional method (2D-DDA) by performing validation studies with respect to analytical solutions, by comparison with results of other numerical techniques, and from laboratory and field data. A paper by MacLaughlin et al.2 contains a summary of nearly 100 published quantitative validation studies. With respect to dynamic loading and response, several works [e.g. Refs. 3, 4] have calibrated 2D-DDA results with respect to the Newmark
method5 and the Goodman and Seed6 solution. Wartman et al.7 investigated the analytical implementation of the Newmark method and Goodman and Seed solution with laboratory tests, using physical tests of a block sliding on a tilting and shaking table. Tsesarsky et al.8 used Wartman’s data to explore the validity of the 2D-DDA results for dynamic loading.
As expected from numerical forward modelling analysis, the input parameters, such as the contact spring stiffness, the boundary conditions, and time interval, have a decisive influence on the accuracy of the output results.
Recently the validity and accuracy of 3D-DDA has been explored, yet only preliminary or partial work on this subject has been published to date.9–14 The reason may be due to the difficulty in developing a complete contact theory that governs the interaction of many 3D blocks.14 Considering 3D validations, Shi1 reports very high accuracy for two examples of block sliding modelled with 3D-DDA, subjected to gravitational load only. Moosavi et al.11 compare 3D-DDA results for dynamic block displacement with an analytical solution. Yeung
et al.13 validate the wedge stability analysis method using physical models and field case histories, and report a good agreement between physical and numerical results in terms of both the effective failure mode and the block displacement history, although no quantitative comparison between 3D-DDA and lab test results is reported.
In this study, an independent mathematical solution for dynamic block sliding in 3D is developed based on the vector analysis (VA) formulation presented by Goodman and Shi.15
The developed 3D solution employs static formulation of the force balance on the block at each time-step, according to the assumed sliding mode. The incremental sliding force or acceleration thus calculated is integrated numerically twice to yield the three displacement components (x,y,z) versus time t. 16 We first compare the developed VA and existing Newmark solutions, and then proceed with the developed 3D VA solution in validation of 3D-DDA.1