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This paper proposes an explicit dynamic DDA method considering dynamic contact force, which aims at solving the problems of low efficiency of dynamic contact detection and the simulation of dynamic contact force in the conventional DDA method. The mutual contact between blocks can be regarded as the application of point loading on a single block, and the corresponding contact submatrix can be calculated and the simultaneous equations of the block system can be integrated. The central difference method is adopted to deduce the explicit expression of block displacement containing dynamic contact force. With the relationship between displacement and dynamic contact force, contact constraint equations of a block system are obtained to calculate the dynamic contact force and the corresponding block displacement. The accuracy of the explicit dynamic DDA method is verified using two numerical cases. The calculation results show that the new DDA method can be applied in large-scale geotechnical engineering.

The discontinuous deformation analysis method proposed by Shi [

Currently, dynamic issues such as impact, explosion, and earthquake are very common in geotechnical engineering. Compared with static loading, dynamic loading often changes over time. The inertial force of structure caused by acceleration cannot be ignored compared with the loading itself, and the degree of deformation caused by dynamic loading is often larger and the failure mode is often more complex. Much research work focuses on the study of dynamic problems using the DDA method. Pei et al. [

Dynamic contact [

From the aspect of dynamic contact force, short embedding distance is allowed in the conventional DDA method, and contact force is the product of embedding distance and contact stiffness. The determination of the contact stiffness value heavily relies on the engineer’s experience and does not have a specific rule [

The current numerical calculation methods aiming at the dynamic contact problems are the penalty method [

This paper proposes the explicit dynamic DDA method considering dynamic contact force based on the previous research, which aims at solving the problems of low efficiency of dynamic contact detection and the simulation of dynamic contact force in the conventional DDA method. Because there are many factors that can affect the dynamic contact force and it is difficult to determine the calculation formula of the dynamic contact force, with the reference of Liu’s dynamic contact force method, dynamic contact force is set as an unknown in this paper. The explicit expression of block displacement containing dynamic contact force is deduced and the contact constraint equations of a block system are obtained based on the contact relationship between displacement and dynamic contact force. The simultaneous equations are solved to obtain the corresponding displacement and dynamic contact force. There is no need to conduct the open-close iteration repeatedly to obtain the contact modes, and the dynamic contact force can be precisely calculated. Due to the explicit scheme, it can reduce the calculation time and the computer memory consumed. The calculating efficiency is greatly improved, and it can be applied in large-scale geotechnical engineering containing complex dynamic contact problems.

The conventional DDA method is based on the principle of minimum potential energy. The potential energy of the block system is calculated and integrated, and the displacement can be obtained by minimizing the overall potential energy, which is the same as for the finite element method. However, the inertia force potential energy generated by unbalanced force and the contact potential energy generated by the contact between blocks must be calculated in the conventional DDA method, while this is not needed in the finite element method. In this section, the simultaneous equations of the explicit dynamic DDA method are deduced from the two aspects of inertia force potential energy and contact potential energy.

A block system is composed of many single blocks, and contact force exists between blocks. The overall simultaneous equations of a block system can be obtained by integrating the simultaneous equations of every single block. In the following deduction, contact force is set as an unknown. Assume that the block

Contact model of blocks.

The contact force applied on block

The active contact force

Meanwhile, the point loading submatrix corresponding to passive contact force

After the contact submatrix of block

Assume that the number of blocks of the whole block system is

Substituting the inertia force submatrix

The acceleration at every time step can be assumed as a constant in the conventional DDA method and can be expressed as

The implicit scheme is adopted to solve (

In this paper, instead of taking (

It can be seen from (

In the conventional DDA method, open-close iteration is adopted to detect contact modes, and it is a test-and-error scheme. The scheme cannot guarantee that the iteration is always convergent. In this paper, after the explicit expression of block displacement containing dynamic contact force is deduced, the contact constraint between the dynamic contact force and the block displacement [

From the aspect of normal contact constraint, the contact mode of the contact pair may be open or embedding. The relationship between displacement and dynamic contact force can be illustrated as in Figure

Model of normal contact constraint.

When the contact mode is embedding, (

Normal relative distance

The contact force in (

From the aspect of tangential contact constraint, the contact mode of the contact pair may be sliding or fixed. The relationship between displacement and dynamic contact force can be illustrated as in Figure

Model of tangential contact constraint.

When the contact mode is sliding, (

Tangential relative distance

Similarly, the coordinate transform is needed between the tangential contact force

Each contact pair is jointly controlled by the normal contact constraint and tangential contact constraint. According to (

Equations (

The symbol of absolute value is contained in (

Frictional constraint condition.

Several functions can be considered to approximate (

Square root function

Arctangent function

Exponential function

The expression forms of the square root function and the exponential function are much more complicated than that of the arctangent function. In Section

Approximated function curve of the step function.

In general, the allowable embedding distance ^{−7}. The approximation factor

According to (

As seen in (

Because there are many contact pairs in the block system, every contact pair needs to meet convergence criteria synchronously and they are related to each other. Thus, it is necessary to solve the corresponding (

The convergence rate of the Newton method for solving nonlinear equations is fast and can achieve a square convergence rate; moreover, it has a more stable and more robust calculation convergence [

The research area of many dynamic problems is a finite domain, such as the case of impact of the block on the bar in Section

The research area of many other dynamic problems is an infinite domain or semi-infinite domain, such as the case of blasting, which is illustrated in Figure

DDA model under blasting loading.

The specific method of simulating the viscous boundary is to apply normal viscous force

Based on the theory of wave propagation,

In fact, a viscous boundary is more effective for the internal source problems, such as blasting. It cannot fully simulate earthquakes and other extraneous source problems [

Generally, the lateral boundaries are free-field boundaries to simulate earthquakes. Similar to (

Two cases are chosen to verify the explicit dynamic DDA method considering dynamic contact force proposed in this paper. The case of impact of the block on the bar can verify the correctness of the dynamic contact force of the block under impact dynamic loading. The case of the Xianglushan Tunnel under seismic loading can verify that the new DDA method can be applied in large-scale geotechnical engineering considering complex contact problems.

Suppose there is a block with an initial velocity ^{3}, length

DDA model of impact of a block on the bar.

The block and the bar can be seen as a collapsing block system. In the process of impact, the grids of the bar make contact with the moving block, and dynamic contact force is generated in this process. The analytical solution [

In the conventional DDA method, the determination of contact stiffness heavily relies on the engineer’s experience, and the contact stiffness affects the calculation accuracy greatly. Dynamic contact force is introduced in the dynamic DDA method in this paper, and the value of contact stiffness does not have an effect on calculation, but Young’s modulus still affects the calculation accuracy [

Dynamic contact force-time curve with different values of Young’s modulus.

The initial impact velocity affects the calculation accuracy greatly, and the value of dynamic contact force greatly varies for different initial impact velocities. The initial impact velocity is taken as

Dynamic contact force-time curve for different initial impact velocities.

According to the analysis of Figures

Xianglushan Tunnel is in the Yunnan Province of southwest China and is located in a strong earthquake zone with a magnitude of intensity level that reaches VII. The average depth of the tunnel is 700 m, and the maximum depth reaches 900 m. The diameter of the tunnel is 9.8 m, and the whole tunnel spans approximately 62.73 km with several faults. The tunnel is supported by supporting bolts. In the centralized zone of the faults, two groups of widely distributed and fully developed joints exist. The parameters of the joints are N30°~35°E/NW70°~80° and EW/N50°~60°. In this paper, the typical section where the two groups of joints intersect is studied. The DDA model of the Xianglushan Tunnel is illustrated in Figure

DDA model of Xianglushan Tunnel under seismic loading.

The size of the numerical DDA model is

Mechanical parameters of rock and joints.

Materials | Density (kg/m^{3}) | Young’s modulus (GPa) | Poisson’s ratio | Friction angle (°) | Cohesion (MPa) | Tensile strength (MPa) |
---|---|---|---|---|---|---|

Rock | 2850 | 5 | 0.29 | — | — | — |

Joints | — | — | — | 35 | 0.4 | 0 |

Because the underground tunnel is greatly affected by low frequency seismic waves [^{2}. The acceleration-time history curve of the seismic wave needs to be filtered and the baseline corrected. The seismic wave is input from the base of the model. The acceleration-time history curve of the KOBE wave is illustrated in Figure

Acceleration-time curve of the KOBE wave.

In this paper, the displacement-time history of four typical monitoring points 1, 2, 3, and 4 under seismic loading is recorded. The stability of surrounding rock of the tunnel without supporting bolts is studied [

Displacement-time curve of monitoring points without supporting bolts.

An implicit cylindric anchor bar element method is adopted for the implementation of the supporting bolts embedded in rock. The embedded supporting bolts are considered to improve the stiffness of the rock in the numerical model. Therefore the stiffness of the supporting bolts can be superimposed onto the stiffness of rock during numerical simulation. This method is detailed in [

The displacement-time history curves of the monitoring points with the supporting bolts are illustrated in Figure

Displacement-time curve of monitoring points with supporting bolts.

As can be seen in Figure

In this case, the joints are fully developed and the number of blocks is very large. The contact modes of the blocks are very complex and vary frequently under seismic loading. However, the open-close iteration is avoided in the process of calculation and the contact modes are detected by solving the contact constraint equations of the block system. The contact modes of every time step do not need to be revised repeatedly. As observed in Figures

Selection and pairing on contact pairs are often required before contact calculations can be made. When solving the static problems with the conventional static DDA, since the time step is very short, the contact modes of the contact pairs are not modified very frequently. In some continuous time step, the contact modes of the contact pairs can be the same and there is no need to conduct the selection and pairing on contact pairs. Whereas, when solving the dynamic problems with the dynamic DDA, since the dynamic loading acts on the blocks, the contact modes of the contact pairs can be modified very frequently. The repairing of the contact pairs is required at each time step, which increases the computational difficulty. Furthermore, when the number of blocks in the block system is very large and the reciprocating effect of the dynamic loading is very obvious, the selection and pairing of the contact pairs in the early stage can consume much time, which can reduce computational efficiency. Thus, the preselection and the prepairing of the contact pairs under the dynamic loading will be our research focus in the next step.

This paper proposed the explicit dynamic DDA method considering dynamic contact force to solve the problem of contact modes detection and dynamic contact force calculation. The calculation process of the explicit dynamic DDA method is deduced and is applied in two numerical cases. Several conclusions can be drawn from this study.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Key Basic Research and Development Plan (973) of China (Project no. 2015CB057904) and the National Natural Science Foundation of China (Project no. 51579191). This support is acknowledged and greatly appreciated.