Variational inequality‐based framework of discontinuous deformation analysis

Abstract

Summary For modeling discrete particle‐block systems, a new framework of discontinuous deformation analysis is established on the basis of finite‐dimensional variational inequality. The presented method takes into account the contacts, the rolling resistance, and the tensile resistance of cemented interface among particles and blocks using the corresponding variational or quasivariational inequalities. The new formulation avoids using the artificial springs that are usually indispensable in many conventional methods dealing with similar discrete problems and conveniently integrates the rigid circle particles, the nonrigid ring particles, and the arbitrary shape blocks into a uniform framework. The proposed discontinuous deformation analysis approach is further coupled with the finite element method using a node‐based composite contact matrix and several simple transformation matrices to solve practical problems. A particle/block‐based composite contact matrix is constructed to further broaden the application of the proposed method. The accuracy, robustness, and capability of the presented method are demonstrated with examples.

A static discrete element method with discontinuous deformation analysis

“…It has been employed to examine the contact precision of distinct blocks in numerical modeling. Numerical results regarding contact forces obtained by the classic DDA and an improved DDA are compared in the work of Fan et al In the work of Meng et al, the example has been implemented to validate a second‐order cone programming formulation of DDA.…”

“…The analytical solution of F 1 and F 2 are 1 MN and 7 MN, respectively. For dynamic formulations used in the work of Fan et al, the time step was set as 0.001 seconds and 100 calculation steps were used to converge the final state. In contrast, the static DEM takes merely one calculation step to obtain the results.…”

“…Abbreviations: DDA, discontinuous deformation analysis; DDA‐VI, variational inequality‐based DDA; DDA‐P, second‐order cone programming–based DDA . Note: 100 calculation steps are employed for DDA‐VI and DDA‐P while one calculation step is used in this study.…”

“…The artificial springs are completely avoided, and the accuracy is demonstrated. The approach is further developed to account for more general block shapes such as spheres, cylinders, and cones . In addition, the second‐order cone formulation of DDA is proposed to tackle the problem .…”

Modified predictor‐corrector solution approach for efficient discontinuous deformation analysis of jointed rock masses

“…In previous studies, the penalty function method, Lagrange multiplier method, augmented Lagrange multiplier method, linear complementary approach, the dual form DDA, variational inequality theory, and variable metric method have been implemented in DDA to solve the contact problems. These constraints become more complex when nonlinear contact constitutive models are involved.…”