Mathematical Problems in Engineering

Volume 2018, Article ID 4079035, 18 pages

https://doi.org/10.1155/2018/4079035

## Discontinuous Deformation Analysis Based on Wachspress Interpolation Function for Detailed Stress Distribution

^{1}Hubei Key Laboratory of Disaster Prevention and Mitigation, China Three Gorges University, Yichang, Hubei 443002, China^{2}State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China

Correspondence should be addressed to W. Jiang; moc.361@noiliewgnaij

Received 5 April 2018; Accepted 1 June 2018; Published 26 June 2018

Academic Editor: Petr Krysl

Copyright © 2018 W. Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the original discontinuous deformation analysis (DDA) method, the complete first-order displacement function is used to describe block movement and deformation, which induce constant stress and strain throughout the block. To achieve a more detailed stress distribution, Wachspress interpolation displacement function is employed to express the displacement of blocks in DDA, and the interactions between blocks are still governed under the original DDA. Displacements of the vertexes of all blocks constitute new freedom vectors, and the stiffness and force matrix formulations are derived again. In the new formulation, Wachspress interpolation ensures that the edges of the blocks are straight; therefore, contact detection can be processed based on the original DDA. Several classical examples are analyzed. The results show that the new formulation obtains similar configurations as the original DDA but provides more detailed and continuous stress distributions within block element.

#### 1. Introduction

The discontinuous deformation analysis (DDA) method, which was proposed by Shi [1] in 1988, represents a novel method of analyzing the dynamic mechanical behavior of block systems under large displacement. As a theoretically rigorous numerical method, the DDA has been a focus of the discontinuous computing field since it was first proposed, and it has been used to address diverse problems in geotechnical engineering, such as landslide simulations [2], rock slope stability assessments [3], rockfall path tracking [4], blasting effect evaluation [5], stability analysis of tunnels [6], analysis of the shear creep deformation of structural planes [7], simulation of the fracturing process initiated by hydraulic pressure [8], crack propagation modeling [9], and seismic wave propagation modeling [10].

After numerous enhancements, the DDA has matured to solve more complicated problems. Solutions have been proposed to restrain false volume expansion when simulating large rotation, such as adopting the second-order approximation of and in the shape function [11], executing a postadjustment once the open-close iteration converges [12], introducing a new displacement variable based on the trigonometric function transformations [13] and accumulating the total strain components in fixed local frames [14]. To avoid the introduction of virtual springs, which are commonly used in classical DDA, the DDA has been successfully reconfigured using the Lagrange multiplier method [15], the complementarity theory [16], and the variational inequality theory [17]. Recently, a dual-form DDA has been proposed by Zheng [18], in which the contact forces instead of the block displacements are utilized as the basic variables and the compatibility iteration for the quasi-variational inequality guarantees the convergence and solution efficiency. To improve the efficiency of disk-based DDA, Beyabanaki and Bagtzoglou [19] presented a new contact model for nonrigid disks, in which disk-disk and disk-boundary contacts are transformed into point-to-line contacts. To simulate progressive failure, Jiang et al. [20] introduced a viscous damping component to absorb the kinetic energy of discrete blocks and defined convergence criteria for DDA solutions, which provided more objective standards for the application of the DDA.

Considering that 3D analyses are always preferred for practical engineering problems, many researchers are working on expanding the original 2D DDA to a 3D version. When the blocks have been identified via 3D discontinuities cutting [21], detecting contact rapidly and reliably appears to be the major barrier to developing an efficient 3D DDA. Contact algorithms, such as the incision body method [22], fast common plane method [23], and main plane method [24], have already been suggested. Recently, Shi [25] suggested a new contact theory named the* E(A,B)* algorithm, and it provides a more robust method of eliminating the greatest obstacle in 3D DDA programming.

In the original DDA, the first-order displacement function is adopted to describe the block movement and deformation, which induce constant stress and strain throughout the block. More exact stress distribution within block element is beneficial to the analysis of practical problems, and several approaches for determining such distributions have been attempted. One method is to assemble small blocks to form a larger block by adopting artificial joints and subblocks [26, 27]. When the subblocking method is used, the constraints between subblocks must be accurately imposed for reasonably reflecting the deformation of an actual block, which is difficult in practical applications. Another approach is to use high-order displacement functions in the DDA [1]. The second-order and third-order displacement functions have been successfully implemented [28, 29], and the ability to model complicated stress and strain fields have been verified. If high-order displacement functions are used, then the edges of a block are likely to be transformed to curves. Thus, the dissection of edges should be executed to exactly impose contact conditions, which is an additional troublesome job. In addition, scholars have added finite element meshes in blocks so that stress variations within the blocks can be evaluated [30, 31]. Considering that the finite element method (FEM) is familiar to researchers, this process is well understood. However, when the deformation and displacement of one block are illustrated by inner finite elements [32], the stress and strain fields are usually discontinuous within the block, which appears to represent a drawback compared with the adoption of high-order displacement functions in the DDA. To some degree, these attempts have successfully acquired more complicated stress and strain fields than the original DDA; however, extra tasks are likely involved or advantages of the original DDA may be lost.

Over the past two decades, the polygonal finite element method (PFEM) has achieved remarkable progress in theory and application [33, 34], and it provides new techniques for attaining a more detailed stress distribution of blocks in DDA. Some interpolations, e.g., Wachspress interpolation [35], Laplace interpolation [36], Sibson interpolation [37], and Maximum entropy interpolation [38], were proposed for constructing displacement functions in the PFEM. All displacement functions based on these interpolations have the ability to produce straight edges on polygons undergoing deformation. To begin with, Wachspress interpolation that originated from projective geometry defines the shape function at point using areas of triangular composed of point and the vertices. The gradient of the shape function can be derived directly, which facilities the calculation of strain and stiffness matrices for elastic analysis. Because the shape function is defined in the Cartesian coordinate system, the numerical integration could be performed on the physical elements. Then, both Sibson and Laplace interpolation are natural neighbor-based interpolations, which define the shape function at point using the information of Voronoi cells or Voronoi edges. For a better implementation, the isoparametric mapping and reference elements are usually involved in the computation [39]. Furthermore, enlightened by the Shannon entropy problem [40] in information theory, maximum entropy interpolation proposed by Sukumar [38] derives the shape function at point from a constrained optimization problem. Calculating the shape function requires solving an optimization problem by numerical iteration, which means that the subsequent computation is more complicated. By comparison, DDA users will have an easier access to Wachspress interpolation for little prior knowledge is involved, and Wachspress interpolation has a better performance being enrolled in the DDA codes. Therefore, Wachspress interpolation function is adopted in this paper to govern the displacement of each block in DDA. After contact detection in Cartesian coordinate system, the contact constraints between blocks are enforced as the original DDA. The displacements of vertexes of blocks constitute new freedom vectors and formulations for the modified DDA are derived. The C++ computer codes are developed for the new formulation and are used to solve several examples. The ability of the proposed scheme to obtain more detailed stress distributions within block elements is well verified by the calculated results.

#### 2. Displacement Function of Blocks Based on Wachspress Interpolation

In the DDA method, the deformations and large displacements represent the accumulation of incremental displacements and deformations over many time steps. The movement and deformation of the th block within one time step are originally defined by six independent variables as follows:where and are the incremental displacement components of the centroid point of the th block in the horizontal and vertical directions, respectively; is the incremental rigid rotation angle around point ; and represents the incremental strains based on a small deformation assumption. The incremental displacement at any point in the th block can be related to : where is called the displacement transformation matrix of the block.Expression (2) has been verified to be the complete first-order displacement function [1]; therefore, each block presents constant stress and strain, which creates an obstacle to acquiring a more detailed stress distribution. Because most of the blocks in the DDA are polygons, Wachspress interpolation has great potential to be used developing a new displacement function for blocks. When Wachspress interpolation is employed to express the displacement of one block, the incremental displacements and of the vertexes form a new freedom vector of the block. Assuming that the th block contains vertexes within the current time step, could be rewritten as follows: where and are the incremental displacement components of the th vertex in the horizontal and vertical directions.

Wachspress interpolation requires that the shape function satisfies certain basic principles. For example, the five vertexes of a pentagon are arranged counterclockwise (shown in Figure 1). For the vertex , the edges including are called adjacent edges of , such as and , and the other edges are called opposite edges of , such as , , and .