Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 7480937, 9 pages

https://doi.org/10.1155/2017/7480937

## Study on the Damping of the Discontinuous Deformation Analysis Based on Two-Block Model

^{1}School of Civil Engineering, Wuhan University, Wuhan 430072, China^{2}School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China^{3}Key Laboratory of Geotechnical Mechanics and Engineering of the Ministry of Water Resources, Yangtze River Scientific Research Institute, Wuhan 430010, China

Correspondence should be addressed to Qinghui Jiang

Received 24 April 2017; Accepted 21 June 2017; Published 18 July 2017

Academic Editor: Michele Betti

Copyright © 2017 Xixia Feng 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

The deformation and failure of rock mass is a process of energy dissipation; the damping of DDA is a very important and basic problem. The correctness and effectiveness of DDA rely on the appropriate values of the numeric controlling parameters like time interval, spring stiffness, and assumed maximum displacement ratio , and the contact using the penalty method is the core content of DDA. A mechanical model of two contact blocks loaded with the normal force acting along one side of block boundary is established to study the DDA damping problem, which involves the contact and eliminates the influence of some numeric control parameters (e.g., ). Based on the Newmark method and the theory of DDA, the motion equations of two-block system can be established, and then the relationship of some numeric control parameters and the influence of damping can be obtained. The algorithmic damping increases with the increasing of time interval. Given a very small time interval, the spring stiffness may have no obvious effect on the algorithmic damping. The numerical results reveal that the essence of time interval influencing the open-close iteration is the fact that the algorithmic damping is mainly controlled by time interval.

#### 1. Introduction

The strong earthquakes can destroy rock mass and induce an abundance of landslides, which cause potentially serious threat to both lives and properties. So it is very necessary to study the seismic response of rock mass. Generally, the numerical methods are widely used for dealing with the problem. Furthermore, as the unfavorable geological structure of rock masses, the discontinue-based methods are more suitable for studying the problem.

The discontinuous deformation analysis (DDA), proposed by Shi and Goodman [1, 2], can be used to simulate the discontinuous deformation behavior of jointed rock masses. The correctness and effectiveness of DDA have been verified by MacLaughlin and Doolin [3]. The DDA method is established on the block kinetics; the block dynamic analysis of DDA has been confirmed by many scholars [4–6]. Meanwhile, the DDA method has been widely applied to the seismic response of rock masses. Zhang et al. [7] investigated the seismic response analysis of large underground caverns with the modified DDA method. Zhang et al. [8] studied the run-out analysis of the Daguangbao landslides subjected to near-fault multidirection earthquake forces. Zhang et al. [9] have verified the mobility of earthquake-induced landslide by DDA method. Wu and Chen [10] simulated the kinematic behavior of sliding blocks of rock in the earthquake-induced Tsaoling landslide using seismic discontinuous deformation analysis (DDA). These studies show that DDA method has great potential in the analysis of seismic response of rock masses.

It is generally recognized that the deformation and failure of rock mass is a process of energy dissipation, which should be highlighted in the dynamic problem of rock mass, such as earthquakes. Since the damping research of DDA is the basis of the seismic response, the damping characteristics should be investigated when using DDA to simulate the earthquake-induced dynamic problems of rock mass. In original DDA program, kinetic damping was introduced to discount the initial velocity of each time step. In static analysis the initial velocity is set to be zero for each time step, while the initial velocity for the dynamic analyses has not been discounted. It should be noted that the value of kinetic damping plays a key role in the dynamic analysis of rock mass. Hatzor et al. [11] and Tsesarsky et al. [5] obtained that 2% kinetic damping was the correct number for the dynamic analysis of a single block on an inclined subjected to dynamic loading. However, the significance of the kinetic damping is still not clear, and the value of initial velocity discounted is hard to choose; therefore, some researchers have introduced self-adaptive damping [12] and viscous damping [13, 14] to address these problems. Lin and Xie [15] also analyzed the performance of Newmark time integration with kinetic damping.

Algorithmic damping is caused by the integration scheme of inertia force, which is a controlling factor of the convergence of open-close iteration. Doolin and Sitar [16] studied the time integration of DDA and noted that the algorithmic damping may be important considering the penalty formulation. Bao et al. [17] obtained an equivalent damping ratio from the algorithmic damping in DDA using the cantilever beam model. Overall, the damping researches of DDA are always in progress.

It is well known that the numeric controlling parameters (e.g., assumed maximum displacement ratio , time interval) in DDA are hard to choose; the correctness and effectiveness of DDA depend on the appropriate values of the these numeric control parameters. Moreover, the contact using the penalty method is the core content of DDA, and the open-close iteration convergence is the highlight. Over the years, researches of DDA damping that involve the contact and eliminate the influence of some numeric control parameters (e.g., ) have been rarely reported. In this study, a mechanical model of two contact blocks loaded with the normal force acting along one side of block boundary is established to study the aforementioned problems. Based on the mechanical model, the total equation motion is established easily and intuitively, and the influence of damping is analyzed. The results show that the essence of time interval influencing the open-close iteration is the algorithmic damping mainly controlled by the time interval.

#### 2. Equations of Motion of a Discrete Block System

The DDA method is in essence a numerical method to solve the motion and deformation of multiple rock blocks. Starting from the definitions of displacement functions, the equations of motion of the blocks are established firstly using potential energy minimization. The direct time-integration scheme is adopted to solve the equations of motion.

##### 2.1. Displacement Functions of Deformable Blocks

The DDA method selects the rigid body displacements and the strains of block elements as the basic unknown variables to solve for solutions. Assuming that an arbitrarily shaped polygonal block has uniform stress and strain, an incremental expression for the unknown variables of the block is written aswhere are the translation displacement increments at the point in and directions, is the rotation angle increment of the block around its mass center , and are the increments of normal and shear strains of the block.

Displacement increments of an arbitrary point in the block can be represented aswhere

##### 2.2. Equations of Motion of the Block System Based on the Newmark Method

For a discrete model, the total potential energy of the block system is the sum of all individual blocks’ potential energy. According to the minimum potential principle, the equation of motion of the block system in matrix form [13] can be expressed aswhere is the acceleration vector and , , and are the global mass matrix, stiffness matrix, and load matrix, respectively.

A direct time-integration scheme is used for solving (4), in which the acceleration and the velocity within each time step are taken as the following equations of the Newmark assumption [18]:While denoting , , and , it can be regarded that , , and are the initial velocity vector, the initial acceleration vector, and the displacement increment vector of the block system at the beginning of the current time step, respectively. The kinetic energy dissipation of DDA block system depends on the value of , which is the dynamic controlling parameter. Therefore, is introduced to the Newmark assumption; we haveSubstituting (6) into (4), the motion equation of the block system of the current time step can be derived aswhere

##### 2.3. Algorithmic Damping of the Newmark Method

For analysis of the stability and accuracy of Newmark method, taking , the following equation can be obtained based on (4)-(5): where sampling frequency and undamped frequency of vibration .

The solutions of the characteristic equation about (10) arewhere The real solution must have oscillation in the case of small damping, and the stability must be not infinitely growing; thuswhen is not restricted, (13) should be met, and then , , .

So the conditions of the unconditionally stable Newmark method areIt is thus obvious that [13, 14, 16, 18] both the average acceleration method (, ) and the constant acceleration method (, ) meet the unconditionally stable condition, while the linear acceleration method (, ) and the central difference method (, ) are the conditionally stable method.

In general, the accuracy of Newmark method depends on the time interval, the physical parameters of the system, and the loading condition. Here, we only analyze some widely used unconditionally stable integration schemes, to obtain an idea of the accuracy of these methods evaluated based on (11) . Figure 1 shows the algorithmic damping of the Newmark method versus sampling frequency .