research article explicit dynamic dda method considering

Research ArticleExplicit Dynamic DDA Method consideringDynamic Contact Force

Jian Zhao12 Ming Xiao12 Juntao Chen12 and Dongdong Li12

1State Key Laboratory of Water Resources and Hydropower Engineering Science Wuhan University Wuhan Hubei 430072 China2Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering Ministry of Education Wuhan UniversityWuhan Hubei 430072 China

Correspondence should be addressed to Ming Xiao mxiao57163com

Received 16 September 2016 Revised 22 November 2016 Accepted 28 November 2016

Academic Editor Vadim V Silberschmidt

Copyright copy 2016 Jian Zhao et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

Thediscontinuous deformation analysismethod proposed byShi [1] is a new and efficient numerical method for discontin-uous rock mass The DDA method can efficiently simulatemutual contact between blocks accounting for the impact ofgeological joints on block movement The DDA method iswidely applied in geotechnical engineering

Currently dynamic issues such as impact explosionand earthquake are very common in geotechnical engineer-ing Compared with static loading dynamic loading oftenchanges over time The inertial force of structure caused byacceleration cannot be ignored compared with the loadingitself and the degree of deformation caused by dynamic load-ing is often larger and the failure mode is often more com-plex Much research work focuses on the study of dynamicproblems using the DDA method Pei et al [2] simulated theprocess of a bridge pierrsquos dynamic response after being hitby an instable rock mass Ma et al [3] simulated the failureprocess of a Brazilian disc under dynamic loading Ning et al[4] simulated the phenomenon of a blast holersquos expansion the

failure of a rockmass and the formation of a rock pile Zhanget al [5] simulated the law of propagation and attenuation ofstress wave caused by blasting in a jointed rock mass Ning[6 7] simulated the failure process of a continuous mediumwith artificial joints and applied it to blasting engineeringWu [8 9] simulated the failure process of the Chiu-fen-erh-shan slope and the Tsaoling slope under seismic loading Shi[10] studied the rockfall and collapse of the Yubabuna tunnelunder seismic loading Kong et al [11] studied the stability of aconcrete-faced rockfill damunder seismic loadingMoreoverthe DDAmethod has been applied in the dynamic analysis ofa bridge [12] retaining wall [13] towel [14] and levee [15]

Dynamic contact [16 17] is one of the main issues studiedin the dynamic DDA method and the research focuses onthe dynamic contact detection and the calculation of dynamiccontact force From the aspect of contact detection thecontactmodes between blocks change rapidly under dynamicloading and contact and separation can occur repeatedlyA large number of contact predictions and judgments areneeded and an open-close iteration is adopted in the conven-tional DDA method to distinguish the contact modes (fixed

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 7431245 12 pageshttpdxdoiorg10115520167431245

2 Shock and Vibration

sliding and separation modes) In the process of open-closeiteration the contact modes are assumed in advance and cor-rected for those contact pairs that violate the contact criteriaObviously open-close iteration does not have strict mathe-matical convergence criteria and the scheme cannot guaran-tee that the iteration is always convergent [18]Meanwhile thetime consumed during the open-close iteration is very long Ifthe number of time steps of the dynamic analysis is very largeopen-close iteration would consumemuch time and decreasethe efficiency of calculation

From the aspect of dynamic contact force short embed-ding distance is allowed in the conventional DDA methodand contact force is the product of embedding distance andcontact stiffness The determination of the contact stiffnessvalue heavily relies on the engineerrsquos experience and does nothave a specific rule [19] Meanwhile under dynamic loadingthe dynamic contact force between blocks is related to theimpact velocity the stress wave velocity andYoungrsquosmodulus[20]The product of embedding distance and contact stiffnessin the conventional DDAmethod cannot reflect the dynamicfeature If the dynamic contact force cannot be calculatedprecisely the corresponding block displacement cannot beaccurate either

The current numerical calculation methods aiming atthe dynamic contact problems are the penalty method [2122] and the Lagrange multiplier method [23 24] and theirimproved versions [25 26] The product of embeddingdistance and contact stiffness is regarded as the contact forcein the penalty method and it cannot represent the precisevalue of the dynamic contact force The contact force is setas the unknown in the Lagrange multiplier method to meetthe contact conditions precisely but it increases the degree offreedom of the simultaneous equations and the difficulty ofsolving the simultaneous equations Meanwhile all of thesemethods adopt the open-close iteration to obtain the contactmodes of the contact pairs which consumes considerabletime and decreases the efficiency of calculation Aiming atthe dynamic response of cracks Liu [27 28] proposed thedynamic contact force method which has been applied ingeotechnical engineering The explicit scheme is adopted inthe algorithm and the displacement at the next time step canbe expressed with the displacement and the contact force atthe current time step The dynamic solution of displacementcan be obtained with the relationship between displacementand contact force Due to the explicit scheme the algorithmcan greatly reduce the calculation time and computer mem-ory consumed The algorithm can easily reach convergenceand is suitable for analyzing large-scale complex dynamiccontact problemsHowever the contact pairs of the algorithmare independent and have no connection with each otherwhich violates the contact conditions in the DDA methodThe contact pairs in the block system in the DDA methodcan influence each other and the modification of one contactpairrsquos mode may lead to the modification of another contactpairrsquos mode From this perspective the dynamic contact forcemethod cannot precisely simulate the contact mode of thecomplex block system Moreover there are other numericalmethods such as the impulse model method [29] the initialdisplacement method [30] and the dynamic contact model

method [31]Thesemethods aremainly applied in the analysisof a crackrsquos dynamic response and are rarely applied ingeotechnical engineering

This paper proposes the explicit dynamic DDA methodconsidering dynamic contact force based on the previousresearch which aims at solving the problems of low efficiencyof dynamic contact detection and the simulation of dynamiccontact force in the conventional DDA method Becausethere are many factors that can affect the dynamic contactforce and it is difficult to determine the calculation formulaof the dynamic contact force with the reference of Liursquosdynamic contact force method dynamic contact force is setas an unknown in this paper The explicit expression of blockdisplacement containing dynamic contact force is deducedand the contact constraint equations of a block system areobtained based on the contact relationship between displace-ment and dynamic contact forceThe simultaneous equationsare solved to obtain the corresponding displacement anddynamic contact forceThere is no need to conduct the open-close iteration repeatedly to obtain the contact modes andthe dynamic contact force can be precisely calculated Due tothe explicit scheme it can reduce the calculation time and thecomputer memory consumed The calculating efficiency isgreatly improved and it can be applied in large-scale geotech-nical engineering containing complex dynamic contact prob-lems

2 The Simultaneous Equations of the ExplicitDynamic DDA Method consideringDynamic Contact Force

The conventional DDA method is based on the principle ofminimum potential energyThe potential energy of the blocksystem is calculated and integrated and the displacementcan be obtained by minimizing the overall potential energywhich is the same as for the finite element method Howeverthe inertia force potential energy generated by unbalancedforce and the contact potential energy generated by the con-tact between blocks must be calculated in the conventionalDDA method while this is not needed in the finite elementmethod In this section the simultaneous equations of theexplicit dynamic DDA method are deduced from the twoaspects of inertia force potential energy and contact potentialenergy

21 Dynamic Contact Force Simulated by Point LoadingA block system is composed of many single blocks andcontact force exists between blocksThe overall simultaneousequations of a block system can be obtained by integrating thesimultaneous equations of every single block In the followingdeduction contact force is set as an unknown Assume thatthe block 119894 contacts the adjacent blocks the contact effectbetween the block 119894 and the adjacent blocks can be regardedas the point loading which is illustrated in Figure 1

The contact force applied on block 119894 as point loading canbe classified into two categories The first category is activecontact force 1198771198941 as illustrated in Figure 1 and it is appliedto the contact point of block 119894 itself The second category is

Shock and Vibration 3

Ωk

Ωj

Ωi

Rk1 RYk1

RYi1

RXk1

RXi1

Ri1

Ri2

RYi2

RYj2

RXj2

Ri2

RXi2

Figure 1 Contact model of blocks

passive contact force 1198771198942 as illustrated in Figure 1 and it isapplied to the contact line of block 119894 itself From the principleof interacting force the passive contact force 1198771198942 applied toblock 119894 and the active contact force 1198771198952 applied to block 119895 area pair of interacting forces thus the relationship between thetwo interacting forces can be expressed as1198771198942 = minus1198771198952 (1)

The active contact force1198771198941 can be decomposed into1198771198831198941in the direction of 119909 and 1198771198841198941 in the direction of 119910 and thecontact potential energy generated by active contact forceapplied to block 119894 can be expressed as

Π119901 = minus (11987711988311989411199061 + 1198771198841198941V1) = minus (1199061 V1) 11987711988311989411198771198841198941= minus [119863119894]119879 [1198791198941 (119909 119910)]119879 11987711988311989411198771198841198941 (2)

where [119863119894] is the displacement of block 119894 and [1198791198941(119909 119910)] is thedisplacement function at the contact point of block 119894 Takingthe derivative of minusΠ119901 on displacement [119863119894] a 6times1 submatrixof point loading can be obtained and expressed as

[1198791198941 (119909 119910)]119879 11987711988311989411198771198841198941 = [1198791198941 (119909 119910)]119879 1198771198941 997888rarr [119877119894] (3)

Meanwhile the point loading submatrix correspondingto passive contact force 1198771198942 can be expressed as

[1198791198942 (119909 119910)]119879 11987711988311989421198771198841198942 = [1198791198942 (119909 119910)]119879 1198771198942 997888rarr [119877119894] (4)

To unify the context all of the equations are deducedwith active contact force With reference to (1) (4) can beexpressed as[1198791198942 (119909 119910)]119879 1198771198942 = minus [1198791198942 (119909 119910)]119879 1198771198952 997888rarr [119877119894] (5)

After the contact submatrix of block 119894 is calculatedthe remaining submatrices including elastic submatrix ini-tial stress submatrix loading submatrix and displacement-constraint submatrix except the inertia force submatrix ofblock 119894 are calculated and integrated into the stiffness matrix[119870119894] and the loading matrix 119865119894 With reference to (4) and(5) the simultaneous equations of block 119894 can be expressed as[119870119894] 119880119894 = 119865119894 + 119877119894 (6)

Assume that the number of blocks of the whole blocksystem is 119873119887 A similar deduction can be made for everyblock and the simultaneous equations can be expressed as[119870119894] 119880119894 = 119865119894 + 119877119894 (119894 = 1 119873119887) (7)

Integrating (7) of all blocks the overall simultaneous equa-tions of the block system can be expressed as[119870] 119880 = 119865 + 119877 (8)

All of the contact force in (8) is taken as active contact forceand inertia force is not taken into consideration

22 Explicit Expression of Block Displacement ContainingDynamic Contact Force Substituting the inertia force sub-matrix [119872] into (8) the overall simultaneous equations ofthe block system considering inertia force can be expressed as[119872] + [119870] 119880 = 119865 + 119877 (9)

where [119872] is the overall massmatrix of the block system afterbeing integrated and the mass submatrix of each single block[119872119894] can be expressed as[119872119894] = 120588∬

Ω119894

[119879119894]119879 [119879119894] 119889119909 119889119910 (119894 = 1 sdot sdot sdot 119873119887) (10)

where 120588 is the density of a block and [119879119894] is the displacementfunction

The acceleration at every time step can be assumed asa constant in the conventional DDA method and can beexpressed as (119905) = (119905 minus Δ119905)

= 2Δ1199052 119880 (119905) minus 2Δ119905 (119905 minus Δ119905) (11)

The implicit scheme is adopted to solve (9) in the con-ventional DDAmethod and (9) needs to be iteratively solvedat every time step Due to the large degree of freedom of theoverall simultaneous equations a large amount of disk spaceneeds to be occupied and much time needs to be consumedduring the process of calculation The contact modes areunknown at the beginning of every time step and open-close iteration needs to be conducted repeatedly to determinethe contact modes Thus the overall simultaneous equationsneed to be solved repeatedly at every time step Obviouslythe implicit scheme leads to low calculation efficiency whilesolving the large-scale dynamicDDA simultaneous equations


containing complex contact problems which needs to beimproved

In this paper instead of taking (11) to approximate theacceleration at every time step the central difference method[32] is adopted The central difference method is the explicitalgorithm and solving the equilibrium equations iterativelyis not needed with the explicit algorithm Its calculationspeed is very fast and it demands less computation memorywhich leads to high computational efficiency The centraldifference method is very suitable for solving large-scalegeotechnical problems with dynamic contact problems andthe acceleration can be expressed as (119905)

= 1Δ1199052 (119880 (119905 minus Δ119905) minus 2 119880 (119905) + 119880 (119905 + Δ119905)) (12)

Substituting (12) into (9) the discretization of the overallsimultaneous equations in the time domain can be expressedas [119872] 1Δ1199052 (119880 (119905 minus Δ119905) minus 2 119880 (119905) + 119880 (119905 + Δ119905))+ [119870] 119880 (119905) = 119865 (119905) + 119877 (119905) (13)

The displacement at time step 119905 + Δ119905 can be obtained using(13) and can be expressed as119880 (119905 + Δ119905) = 119880 (119905 + Δ119905) + (Δ119905)2 [119872]minus1 119877 (119905) 119880 (119905 + Δ119905) = 2 119880 (119905) minus 119880 (119905 minus Δ119905)+ (Δ119905)2 [119872]minus1 (119865 (119905) minus [119870] 119880 (119905)) (14)

It can be seen from (14) that the displacement 119880(119905 + Δ119905)at time step 119905 + Δ119905 is determined by the displacement at timesteps 119905 and 119905minusΔ119905 which is the distinctive feature of the explicitalgorithm [32] From (14) the displacement 119880(119905+Δ119905) at timestep 119905 +Δ119905 consists of two parts [18] the predictable displace-ment 119880(119905 + Δ119905) without considering dynamic contact forceand the amended displacement (Δ119905)2[119872]minus1119877(119905) caused bydynamic contact force The predictable displacement 119880(119905 +Δ119905) is only related to the displacement and the external loadat time steps 119905 and 119905 minus Δ119905 and is a known quantity Thus thedynamic contact force 119877(119905) needs to be solved to obtain thedisplacement at time step 119905 + Δ1199053 Contact Constraint Equations of

the Block System

In the conventional DDA method open-close iteration isadopted to detect contact modes and it is a test-and-errorscheme The scheme cannot guarantee that the iteration isalways convergent In this paper after the explicit expressionof block displacement containing dynamic contact force isdeduced the contact constraint between the dynamic contactforce and the block displacement [33 34] is introduced toform the contact constraint equations which can play the roleof open-close iteration

Ωi

Ωj

dnRn120575n

n

Figure 2 Model of normal contact constraint

31 Normal Contact Constraint From the aspect of normalcontact constraint the contact mode of the contact pair maybe open or embedding The relationship between displace-ment and dynamic contact force can be illustrated as inFigure 2

When the contact mode is embedding (15) is met andexpressed as 119889119899 = 0119877119899 gt 0 (15)

When the contact mode is open (16) is met and expressed as119889119899 gt 0119877119899 = 0 (16)

where 119889119899 is the normal relative distance and 119877119899 is the normalcontact force According to (15) and (16) (17) can be expressedas follows

min (119889119899 119877119899) = 0 (17)

Normal relative distance 119889119899 can be expressed as119889119899 = 120575119899 + ([119879119895] 119863119895 minus [119879119894] 119863119894) (18)

where 120575119899 is the initial spacing of the contact pair at the begin-ning of the time step 119863119895 and 119863119894 are the displacements ofthe two contacted blocks respectively [119879119895] and [119879119894] are thedisplacementmatrices of the two nearest points of the contactpair at the beginning of the time step and is the normalvector which points towards the inner part of the block

The contact force in (14) is presented in the global coordi-nate system but the normal contact force in (17) is presentedin the local coordinate system Thus the coordinate trans-form is needed between the two groups of contact force in (14)and (17) and the coordinate transform can be expressed as119877119899 = 119877 (19)


Ωi

Ωj

d120591

R120591120591

Figure 3 Model of tangential contact constraint

32 Tangential Contact Constraint From the aspect of tan-gential contact constraint the contact mode of the contactpair may be sliding or fixed The relationship between dis-placement and dynamic contact force can be illustrated as inFigure 3

When the contact mode is sliding (20) is met andexpressed as 119889120591 = 0120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816 gt 0 (20)

When the contact mode is fixed (21) is met and expressed as10038161003816100381610038161198891205911003816100381610038161003816 gt 0120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816 = 0 (21)

where 119889120591 is the tangential relative distance 119877119899 is the normalcontact force 119877120591 is the tangential contact force and 120583 is thefriction coefficient According to (20) and (21) (22) can beexpressed as

min (10038161003816100381610038161198891205911003816100381610038161003816 120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816) = 0 (22)

Tangential relative distance 119889120591 can be expressed as119889120591 = ([119879119895] 119863119895 minus [119879119894] 119863119894) (23)

where 119863119895 and 119863119894 are the displacements of the two con-tacted blocks respectively [119879119895] and [119879119894] are the displacementmatrices of the two nearest points of the contact pair at thebeginning of the time step and is the tangential vectorwhich rotates from the normal vector clockwise

Similarly the coordinate transform is needed between thetangential contact force 119877120591 in the local coordinate systemand the contact force 119877 in the global coordinate system Thecoordinate transform can be expressed as119877120591 = 119877 (24)

33 Smoothing Contact Constraint Equations Each contactpair is jointly controlled by the normal contact constraintand tangential contact constraint According to (17) and (22)

Sliding

Sliding

Fixed

d120591

R120591

120583Rn

minus120583Rn

Figure 4 Frictional constraint condition

the contact constraint equations of each contact pair can beexpressed as

min (119889119899 119877119899) = 0 (25a)

min (10038161003816100381610038161198891205911003816100381610038161003816 120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816) = 0 (25b)

Equations (25a) and (25b) are not smooth equations andthe numerical method is not suitable to solve the equationsThus it is necessary to deduce the smoothing approximationexpression [35] of the contact constraint equations Equation(25a) can be smoothed and expressed asminus1205761 ln [exp(minus1198891198991205761 ) + exp(minus1198771198991205761 )] = 0 (26)

where 1205761 is the approximation factorWhen 1205761 approaches 0+the degree of approximation is extremely high

The symbol of absolute value is contained in (25b) andit is not convenient to get (25b) smooth via the conventionalmethod Equation (25b) presents the friction criterion and itcan be illustrated as in Figure 4

Several functions can be considered to approximate (25b)and the most representative functions can be the followingthree functions whose value ranges from minus1 to 1

(1) Square root function 1198661(Δ) = ΔradicΔ2 + 120576(2) Arctangent function 1198662(Δ) = (2120587)arctan(Δ120576)(3) Exponential function 1198663(Δ) = 1 minus 2(119890Δ120576 + 1)The expression forms of the square root function and the

exponential function are much more complicated than thatof the arctangent function In Section 34 the approximatedfunction will be substituted to calculate the dynamic contactforce In order to reduce the computational difficulty thearctangent function is chosen to approximate (25b) Equation(25b) can be smoothed using the arctangent function [36] andit can be expressed as119877120591 minus 2120587120583119877119899 arctan(1198891205911205762 ) = 0 (27)


where 1205762 is the approximation factorWhen 1205762 approaches 0+the degree of approximation is extremely high The curves ofthe approximated step functions with 1205762 being 60times10minus8 areillustrated in Figure 5

In general the allowable embedding distance 119889120591 is rathershort and the magnitude of the horizontal coordinate Δ canapproach 10minus7 The approximation factor 1205762 can be reducedto further improve the accuracy Although the value of 1205762 issmall 119889120591 and 1205762 can be calculated within the existing compu-tational capabilities thus the operability and accuracy can beensured

According to (26) and (27) the approximation expressionof (25a) and (25b) can be expressed as

minus1205761 ln [exp(minus1198891198991205761 ) + exp(minus1198771198991205761 )] = 0 (28a)

119877120591 minus 2120587120583119877119899 arctan(1198891205911205762 ) = 0 (28b)

As seen in (18) and (23) the normal relative distance 119889119899and the tangential relative distance 119889120591 can be expressed withblock displacement while block displacement 119863 can beexpressed with dynamic contact force 119877 as in (14) Thus(28a) and (28b) contain only the displacement 119863 as anunknown

Because there are many contact pairs in the block sys-tem every contact pair needs to meet convergence criteriasynchronously and they are related to each other Thus it isnecessary to solve the corresponding (28a) and (28b) for eachcontact pair simultaneously Assume that the number of con-tact pairs is 119873119888 contact constraint equations 119867(119877119899 119877120591)can be expressed as119867(119877119899 119877120591)

= (minus1205761ln[exp(minus1198891198941198991205761 ) + exp(minus1198771198941198991205761 )]119877119894120591 minus 2120587120583119877119894119899arctan(1198891198941205911205762 ) )

119894=1

119894=119873119888= 0(29)

34 Solving Contact Constraint Equations Using NewtonrsquosMethod The convergence rate of the Newton method forsolving nonlinear equations is fast and can achieve a squareconvergence rate moreover it has a more stable and morerobust calculation convergence [37] Thus Newtonrsquos methodis adopted to solve the contact constraint equations of theblock system 119867(119877119899 119877120591) Let (119877119899 119877120591) be 119877 and thecontact constraint equations can be expressed as 119867(119877)Because the contact constraint equations are very complexthe gradient on dynamic contact force 119877 cannot be cal-culated via the analytic method easily Thus it is necessaryto approximate the gradient using the numerical methodCurrently the difference quotient is widely used to approx-imate the gradient 119881(119877) and the calculation procedure isas follows

minus12 minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10 12minus10minus08minus06minus04minus02

000204060810

R120591120583

Rn

d12059110minus7

Figure 5 Approximated function curve of the step function

(1) Let 119896 be 0 and specify the initial dynamic contact force119877(0) Specify the termination factor 120576 gt 0(2) Calculate the difference quotient119881119896 (119877119896) = 119867(119877119896 + ℎ119896 119890) minus 119867(119877119896)ℎ119896 (30)

where ℎ119896 controls the accuracy of the calculation and 119890 isthe unit vector in 119899-dimensional Euclid space(3) Solve the following linear equation119881119896 (119877119896) Δ119877119896 + 119867119896 (119877119896) = 0 (31)(4)Calculate the error value 119892(119877119896) = 119867(119877119896)119879119867(119877119896)Check if the termination condition 119892(119877119896) le 1205761015840 is verified Ifso stop iterating and set 119877119896 as the optimal solution of thedynamic contact force If not conduct the following step(5) Set time step 119896 as 119896+1 calculate 119877119896+1 = 119877119896+Δ119877119896and return to step (2)(6)After determining the optimal solution of the dynamiccontact force 119877 substitute 119877 into (14) and the optimalsolution of block displacement 119863 can be obtained

4 Boundary Setting

The research area of many dynamic problems is a finitedomain such as the case of impact of the block on the bar inSection 51The boundary of the model can be set as the fixedboundary or the free boundary Wave reflection occurs whenthe wave reaches the fixed or free boundary

The research area of many other dynamic problems is aninfinite domain or semi-infinite domain such as the case ofblasting which is illustrated in Figure 6 The stress wave isexpected to propagate freely in the study area Meanwhilethe actual numerical model used for calculation cannot beinfinitely large and the research area can only be a finitedomain If the boundary is not modified wave reflectionoccurs when the wave reaches the boundary which contra-dicts the real facts [38] Currently a viscous boundary is


p

Viscousboundary

Figure 6 DDA model under blasting loading

widely used to absorb the energy of the wave propagating tothe boundary to avoid wave reflection [39]

The specificmethod of simulating the viscous boundary isto apply normal viscous force 119891119899 and tangential viscous force119891119904 at the boundary of the numerical model119891119899 = 120588119888119901V119899119891119904 = 120588119888119904V119904 (32)

where 120588 is the density of the block 119888119901 and 119888119904 are thepropagation velocity of the P wave and S wave in the blockmedium respectively and V119899 and V119904 are the normal andtangential velocity of the blocks at the boundary respectively

Based on the theory of wave propagation 119888119901 and 119888119904 can bedetermined by solving the following equations

119888119901 = radic 119864 (1 minus ])120588 (1 + ]) (1 minus 2]) 119888119904 = radic 1198642120588 (1 + ])

(33)

where119864 ] and 120588 are Youngrsquos modulus and Poissonrsquos ratio anddensity respectively

In fact a viscous boundary is more effective for theinternal source problems such as blasting It cannot fullysimulate earthquakes and other extraneous source problems[40] such as the case of the tunnel under seismic loading inSection 52

Generally the lateral boundaries are free-field boundariesto simulate earthquakes Similar to (32) the lateral bound-aries are applied using the following equations119891119909 = minus120588119888119901 (V119898119909 minus V119891119891119909 ) S + 119891119891119891119909 119891119910 = minus120588119888119904 (V119898119910 minus V119891119891119910 ) S + 119891119891119891119910 (34)

where V119898119909 and V119898119910 are the velocities of the block boundarygridpoint in 119909 and 119910 directions respectively V119891119891119909 and V119891119891119910 arethe velocities of the gridpoint in the free field in 119909 and 119910directions respectively and 119891119891119891119909 and 119891119891119891119910 are the force of thegridpoint in the free field in 119909 and 119910 directions respectively

5 Validation

Two cases are chosen to verify the explicit dynamic DDAmethod considering dynamic contact force proposed in this

0 M2

L3

L2

120588

M1 L1

Figure 7 DDA model of impact of a block on the bar

paperThe case of impact of the block on the bar can verify thecorrectness of the dynamic contact force of the block underimpact dynamic loadingThe case of the Xianglushan Tunnelunder seismic loading can verify that the new DDA methodcan be applied in large-scale geotechnical engineering con-sidering complex contact problems

51 Longitudinal Impact of a Block on the Bar Suppose thereis a blockwith an initial velocity V0 striking towards a bar withthe boundary fixed which is illustrated in Figure 7The blockhas the following properties mass 1198721 = 10 kg and length1198711 = 0075m The bar has the following properties mass1198722 = 15 kg density 120588 = 2700 kgm3 length 1198712 = 05mand 1198713 = 005m Gravity is not considered in the numericalmodel The artificial joints are set in the bar and the bar isdivided into 10 subblocks The value of the artificial jointsis large enough to avoid the subblocks being divorced fromeach other during the process of impact By setting differentinitial values of impact velocity and Youngrsquos modulus thedevelopment regularity of dynamic contact force caused byimpact of the block on the bar can be studied

The block and the bar can be seen as a collapsing blocksystem In the process of impact the grids of the bar makecontact with the moving block and dynamic contact forceis generated in this process The analytical solution [20] andthe numerical solution of the dynamic contact force can beobtained

In the conventional DDA method the determination ofcontact stiffness heavily relies on the engineerrsquos experienceand the contact stiffness affects the calculation accuracygreatly Dynamic contact force is introduced in the dynamicDDA method in this paper and the value of contact stiffnessdoes not have an effect on calculation but Youngrsquos modu-lus still affects the calculation accuracy [19] To study theinfluence of Youngrsquos modulus on dynamic contact force theinitial impact velocity is taken as V0 = 125ms and Youngrsquosmodulus is taken as 119864 = 056 times 107 10 times 107 225 times 107 and90 times 107 Pa Figure 8 illustrates the dynamic contact force-time curve with different values of Youngrsquos modulus thedevelopment regularity of the dynamic contact force can alsobe observed in Figure 8 The dynamic contact force reachesthe first peak in a rather short time period after impact andthen shows a declining trend Then the stress wave caused byimpacting propagates towards the right fixed boundary in thebar When the stress wave reaches the right fixed boundary itcan reflect and propagate back towards the left free boundaryThe dynamic contact force reaches the second peak when thestress wave reaches the left free boundary At this time thetime consumed is approximately 21198712119888 Then the dynamic


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3200

03

06

09

12

15

18

21

24

27

30

33

Dyn

amic

cont

act f

orce

(KN

)

Time (10minus3 s)

E = 90 times 107 Pa analytical solutionE = 90 times 107 Pa DDA solutionE = 225 times 107 Pa analytical solutionE = 225 times 107 Pa DDA solutionE = 10 times 107 Pa analytical solutionE = 10 times 107 Pa DDA solutionE = 056 times 107 Pa analytical solutionE = 056 times 107 Pa DDA solution

Figure 8 Dynamic contact force-time curve with different values ofYoungrsquos modulus

contact force decreases to 0 within a period time of 1198712119888where 1198712 is the length of the bar and 119888 is the propagationvelocity of the stress wave in the bar From the aspect of thevalue of the dynamic contact force the corresponding valueof contact force is larger when the value of Youngrsquos modulusis larger From the aspect of time period of impact the cor-responding time period of impact is shorter when the valueof Youngrsquos modulus is larger which reveals the rule that thestress wave propagates much faster in the bar with a largerYoungrsquos modulus [41] Meanwhile in Figure 8 the dotted linerepresents the DDA solution and the solid line representsthe analytical solution The analytical solution and the DDAsolution of the dynamic contact force are very similar fordifferent values of Youngrsquos modulus and the dynamic DDAmethod has high calculation precision

The initial impact velocity affects the calculation accuracygreatly and the value of dynamic contact force greatly variesfor different initial impact velocitiesThe initial impact veloc-ity is taken as V0 = 125 250 375 and 500ms Figure 9illustrates a dynamic contact force-time curve for differentinitial impact velocities As shown in Figure 9 the develop-ment regularity of the dynamic contact force is substantiallythe same Two peak values of dynamic contact force are pre-sented in the curve and the value finally eventually declinesto zero From the aspect of the value of the dynamic contactforce the corresponding value of contact force is larger whenthe value of initial impact velocity is larger and the peakvalue of dynamic contact force is basically proportional to theinitial impact velocity From the aspect of time period ofimpact the initial impact velocity has little effect on the

0 2 4 6 8 10 12 14 16 18 20 22 24 2600

04

08

12

16

20

24

28

32

36

40

44

Dyn

amic

cont

act f

orce

(KN

)

Time (10minus3 s)0 = 500ms analytical solution0 = 500ms DDA solution0 = 375ms analytical solution0 = 375ms DDA solution0 = 250ms analytical solution0 = 250ms DDA solution0 = 125ms analytical solution0 = 125ms DDA solution

Figure 9 Dynamic contact force-time curve for different initialimpact velocities

impact time and the development pace of the dynamic con-tact force for different impact velocities is basically the sameThe impact time is only related to the length of the bar and thepropagation velocity of the stress wave and has little relationto the initial impact velocity [41] Meanwhile the analyticalsolution and the DDA solution of the dynamic contact forceare very similar for different initial impact velocities and thedynamic DDA method has high calculation precision

According to the analysis of Figures 8 and 9 it can beconcluded that the explicit dynamic DDA method consid-ering dynamic contact force proposed in this paper can welldescribe the dynamic response of contact forces generated bydynamic loading It can alsowell reflect the propagation effectof a stress wave in the block

52 Stability Analysis of the Xianglushan Tunnel underSeismic Loading

521 Project Overview and Numerical Model XianglushanTunnel is in the Yunnan Province of southwest China andis located in a strong earthquake zone with a magnitude ofintensity level that reaches VII The average depth of thetunnel is 700m and the maximum depth reaches 900mThediameter of the tunnel is 98m and the whole tunnel spansapproximately 6273 km with several faults The tunnel issupported by supporting bolts In the centralized zone of thefaults two groups of widely distributed and fully developedjoints exist The parameters of the joints are N30∘sim35∘ENW70∘sim80∘ and EWN50∘sim60∘ In this paper the typicalsection where the two groups of joints intersect is studied


Table 1 Mechanical parameters of rock and joints

Materials Density (kgm3) Youngrsquos modulus (GPa) Poissonrsquos ratio Friction angle (∘) Cohesion (MPa) Tensile strength (MPa)Rock 2850 5 029 mdash mdash mdashJoints mdash mdash mdash 35 04 0

Viscousboundary

Free-fieldcouplingboundary

Supporting bolts

12

34

Figure 10 DDA model of Xianglushan Tunnel under seismicloading

The DDA model of the Xianglushan Tunnel is illustrated inFigure 10

The size of the numerical DDAmodel is 200times300m andthe depth of a typical section is 700m Because the depthof the tunnel is larger than the size of the tunnel and thenumber of blocks in the studied area is very large it is not veryconvenient to simulate all of the overlying strata above thetunnel In this case the numerical DDA model is simplifiedbeing intercepted at 100m above the tunnel The size of thetunnel is very small and the depth is very large a tunnelwith a large depth should be stable compared with a shallowtunnel The scope of the surrounding rock is more than 10times larger than the size of the tunnel and the size of thenumerical model canmeet the needs of the research [42]Theweight of the overlying strata is applied to the top of themodel[43] with the stress of 1468MPa Two groups of joints witha spacing of 3m are distributed in the model To facilitatedetection the four typical monitoring points 1 2 3 and 4are set around the tunnel The mechanical parameters of thesurrounding rock and joints are shown in Table 1

522 Seismic Loading Because the underground tunnel isgreatly affected by low frequency seismic waves [44 45]and the KOBE wave contains a rich low frequency bandthe KOBE wave is chosen as the input seismic wave witha duration of 20 s and a peak acceleration of 817ms2 Theacceleration-time history curve of the seismic wave needsto be filtered and the baseline corrected The seismic waveis input from the base of the model The acceleration-timehistory curve of the KOBE wave is illustrated in Figure 11

523 Calculation Results In this paper the displacement-time history of four typical monitoring points 1 2 3 and 4under seismic loading is recorded The stability of surround-ing rock of the tunnel without supporting bolts is studied[46 47] and the displacement-time history curve of the

0 2 4 6 8 10 12 14 16 18 20

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Acce

lera

tion

(ms

2)

Figure 11 Acceleration-time curve of the KOBE wave

monitoring points without supporting bolts is illustrated inFigure 12 Because the size of the tunnel is not very largethe displacement-time history curves of monitoring points1 2 and 4 are basically coincident and the displacement ofmonitoring point 3 is slightly larger than the remaining threemonitoring points As observed in Figure 12 without thesupporting bolts the maximum instantaneous displacementof the monitoring points can reach 014m and the permanentdisplacement after the earthquake can reach 0014m

An implicit cylindric anchor bar element method isadopted for the implementation of the supporting boltsembedded in rock The embedded supporting bolts are con-sidered to improve the stiffness of the rock in the numericalmodel Therefore the stiffness of the supporting bolts can besuperimposed onto the stiffness of rock during numericalsimulation This method is detailed in [48]

The displacement-time history curves of the monitoringpoints with the supporting bolts are illustrated in Figure 13The displacement-time history curves of monitoring points1 and 2 are basically coincident As observed in Figure 13with the supporting bolts the maximum instantaneous dis-placement of the monitoring points can reach 0056m andthe permanent displacement after the earthquake can reach00068m The deformation of the tunnel has been reducedand the stability of the tunnel has been improved significantlycompared with the case without supporting bolts

As can be seen in Figure 13 the displacement of surround-ing rock of theXianglushanTunnel is small and the anchoringeffect on the surrounding rock is obvious Since the depth ofthe tunnel is very large the self-stability of the tunnel canbe very good The two groups of joints existing in the sur-rounding rock also did not form the blocks which are proneto collapse Thus the Xianglushan Tunnel itself can maintainthe stability of surrounding rock with the supporting bolts


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus004minus002

000002004006008010012014

Disp

lace

men

t (m

)

Time (s)

12

34

Figure 12 Displacement-time curve of monitoring points withoutsupporting bolts

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus0008

0000

0008

0016

0024

0032

0040

0048

0056

Time (s)

Disp

lace

men

t (m

)

12

34

Figure 13 Displacement-time curve of monitoring points withsupporting bolts

In this case the joints are fully developed and the numberof blocks is very large The contact modes of the blocks arevery complex and vary frequently under seismic loadingHowever the open-close iteration is avoided in the process ofcalculation and the contact modes are detected by solving thecontact constraint equations of the block systemThe contactmodes of every time step do not need to be revised repeatedlyAs observed in Figures 12 and 13 the results can well reflectthe displacement and the deformation of the tunnel underseismic loading and the calculation result is in accordancewith the real case Thus we can draw the conclusion that theexplicit dynamic DDA method considering dynamic contactforce can be applied in the dynamic response analysis of large-scale geotechnical engineering

6 Discussion

Selection and pairing on contact pairs are often requiredbefore contact calculations can be made When solving thestatic problems with the conventional static DDA since thetime step is very short the contact modes of the contact pairsare not modified very frequently In some continuous timestep the contact modes of the contact pairs can be the sameand there is no need to conduct the selection and pairing oncontact pairs Whereas when solving the dynamic problemswith the dynamicDDA since the dynamic loading acts on theblocks the contactmodes of the contact pairs can bemodifiedvery frequentlyThe repairing of the contact pairs is required ateach time step which increases the computational difficultyFurthermore when the number of blocks in the block systemis very large and the reciprocating effect of the dynamic load-ing is very obvious the selection and pairing of the contactpairs in the early stage can consume much time which canreduce computational efficiency Thus the preselection andthe prepairing of the contact pairs under the dynamic loadingwill be our research focus in the next step

7 Conclusion

This paper proposed the explicit dynamic DDA methodconsidering dynamic contact force to solve the problem ofcontact modes detection and dynamic contact force calcu-lation The calculation process of the explicit dynamic DDAmethod is deduced and is applied in two numerical casesSeveral conclusions can be drawn from this study(1) Compared with the conventional DDA methodwhich adopts block displacement as the unknown for analy-sis the dynamic contact force is adopted to solve the problemThe improvement can fully take into account the actual situ-ation instead of taking the dynamic contact force as merelyproportional to the embedding distance This new methodcan simulate the dynamic contact force between blocks in amore accurate way(2) Based on the contact constraint between the blockdisplacement and the dynamic contact force the contactconstraint equations of the block system are deduced whichreplaces the role of the open-close iteration in the conven-tional DDA method The contact constraint equations canreflect the nonlinear relation of contact constraints and cantake into account the interaction between different contactpairsThe contact modes can be obtained once without beingrevised repeatedly(3) The accuracy of the explicit dynamic DDA methodis verified by using two numerical cases In the case of theimpact of a block on a bar taking different values of Youngrsquosmodulus and initial impact velocity the results can wellreflect the development regularity of the dynamic contactforce A high degree of agreement exists between the analyt-ical solution and the numerical solution of the dynamic con-tact force In the case of stability analysis of the XianglushanTunnel under seismic loading the contact modes of blocksunder seismic loading are very complex The results canreflect the secular deformation behavior of the tunnel underseismic loading Thus the explicit dynamic DDA method


considering dynamic contact force is still applicable topractical engineering and can be applied in the dynamicresponse analysis of large-scale geotechnical engineering

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Key BasicResearch and Development Plan (973) of China (Project no2015CB057904) and theNationalNatural Science Foundationof China (Project no 51579191)This support is acknowledgedand greatly appreciated

References

[1] G H Shi Discontinuous Deformation Analysis-A New Numer-ical Model for the Statics and Dynamics of Block SystemsDepartment of Civil Engineering University of CaliforniaBerkeley Calif USA 1988

[2] X Pei RHuang and S Li ldquoStudy of dynamic response of bridgepier shocked by falling rock induced by intensive earthquakerdquoChinese Journal of Rock Mechanics and Engineering vol 30 no2 pp 3995ndash4001 2011

[3] JMa X Zhang Y Jiao andH Zhang ldquoNumerical simulation offailure of brazilian disc specimen under dynamic loading usingdiscontinuous deformation analysis methodrdquo Chinese Journalof Rock Mechanics and Engineering vol 34 no 9 pp 1805ndash18142015

[4] Y-J Ning J Yang and P-W Chen ldquoNumerical simulation ofrock blasting in jointed rock mass by DDA methodrdquo Rock ampSoil Mechanics vol 31 no 7 pp 2259ndash2263 2010

[5] X-L Zhang Y-Y Jiao Q-S Liu and B-L Liu ldquoNumericalstudy on effect of joints on blasting wave propagation in rockmassrdquo Rock and Soil Mechanics vol 29 no 3 pp 717ndash721 2008

[6] Y Ning J Yang X An and G Ma ldquoModelling rock fracturingand blast-induced rock mass failure via advanced discretisationwithin the discontinuous deformation analysis frameworkrdquoComputers and Geotechnics vol 38 no 1 pp 40ndash49 2011

[7] Y Ning J Yang G Ma and P Chen ldquoModelling rock blastingconsidering explosion gas penetration using discontinuousdeformation analysisrdquo Rock Mechanics amp Rock Engineering vol44 no 4 pp 483ndash490 2011

[8] J-H Wu ldquoSeismic landslide simulations in discontinuousdeformation analysisrdquo Computers amp Geotechnics vol 37 no 5pp 594ndash601 2010

[9] J-H Wu and C-H Chen ldquoApplication of DDA to simu-late characteristics of the Tsaoling landsliderdquo Computers andGeotechnics vol 38 no 5 pp 741ndash750 2011

[10] G-H Shi Applications of Discontinuous Deformation Analysis(DDA) to Rock Engineering Springer Berlin Germany 2007

[11] X-J Kong J Liu and G-C Han ldquoDynamic failure test andnumerical simulation of model concrete-faced rockfill damrdquoChinese Journal of Geotechnical Engineering vol 25 no 1 p 262003

[12] M Y Ma A D E Pan M T Luan et al ldquoSeismic analysisof stone arch bridges using discontinue deformation analysisrdquo

in Proceedings of the 11th World Conference on EarthquakeEngineering pp 1ndash8 June 1996

[13] S Nishiyama Y Ohnishi T Yanagawa et al ldquoStudy on stabilityof retaining wall of masonry type by using discontinuousdeformation analysisrdquo in Proceedings of the ISRM InternationalSymposium 3rd ARMS Y Ohnishi and K Aoki Eds pp 1221ndash1226 Mill Press Kyoto Japan 2004

[14] R Kamai and Y H Hatzor ldquoNumerical analysis of block stonedisplacements in ancient masonry structures a new methodto estimate historic ground motionsrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 32 no11 pp 1321ndash1340 2008

[15] C A Tang and H Y Lu ldquoThe DDD method based oncombination of RFPA and DDA Frontiers of DiscontinuousNumerical Methods and Practical Simulations in EngineeringandDisaster Preventionrdquo in Proceedings of the 11th InternationalConference on Analysis of Discontinuous Deformation (ICADDrsquo11) pp 105ndash112 Fukuoka Japan August 2013

[16] S A R Beyabanaki B Ferdosi and SMohammadi ldquoValidationof dynamic block displacement analysis and modification ofedge-to-edge contact constraints in 3-D DDArdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 46 no 7pp 1223ndash1234 2009

[17] S A R Beyabanaki R G Mikola and K Hatami ldquoThree-dimensional discontinuous deformation analysis (3-D DDA)using a new contact resolution algorithmrdquo Computers ampGeotechnics vol 35 no 3 pp 346ndash356 2008

[18] L-H Zhang T-Y Liu Q-B Li and T Chen ldquoModifieddynamic contact force method under reciprocating loadrdquo Engi-neering Mechanics vol 31 no 7 pp 8ndash14 2014

[19] W Jiang and H Zheng ldquoInfluences of artificial parameters inDDA methodrdquo Rock and Soil Mechanics vol 28 no 12 pp2603ndash2606 2007

[20] X T Yu Impact Dynamics Tsinghua University Press 2011[21] N Hu ldquoA solution method for dynamic contact problemsrdquo

Computers and Structures vol 63 no 6 pp 1053ndash1063 1997[22] Y Kanto and G Yagawa ldquoA dynamic contact buckling analysis

by the penalty finite element methodrdquo International Journal forNumerical Methods in Engineering vol 29 no 4 pp 755ndash7741990

[23] Y Kanto and G Yagawa ldquoDynamic contact buckling analysisby the penalty finite element methodrdquo International Journal forNumerical Methods in Engineering vol 29 no 4 pp 755ndash7741990

[24] A B Chaudhary and K-J Bathe ldquoA solution method for staticand dynamic analysis of three-dimensional contact problemswith frictionrdquo Computers and Structures vol 24 no 6 pp 855ndash873 1986

[25] J C Simo and T A Laursen ldquoAn augmented lagrangiantreatment of contact problems involving frictionrdquo Computers ampStructures An International Journal vol 42 no 1 pp 97ndash1161992

[26] T A Laursen and V Chawla ldquoDesign of energy conservingalgorithms for frictionless dynamic contact problemsrdquo Interna-tional Journal for Numerical Methods in Engineering vol 40 no5 pp 863ndash886 1997

[27] J Liu S Liu and X Du ldquoA method for the analysis of dynamicresponse of structure containing non-smooth contactable inter-facesrdquo Acta Mechanica Sinic vol 15 no 1 pp 63ndash72 1999

[28] L Shu L Jingbo S Jiaguang and C Yujian ldquoA methodfor analyzing three-dimensional dynamic contact problems in


visco-elastic media with kinetic and static frictionrdquo Computersand Structures vol 81 no 24-25 pp 2383ndash2394 2003

[29] Zhou Jing and Ni Hangen ldquoResponse of cracked structuresduring earthquakerdquo Earthquake Engineering amp EngineeringVibration vol 6 no 2 pp 52ndash58 1986

[30] Q Li H Zhou and G Lin ldquoDynamic fracture analysis of crackat bimaterial interfacerdquo Earthquake Engineering amp EngineeringVibration no 4 1990

[31] T JianRen Dynamic Contact Damping and Its Application inEngineering Dalian University 1987

[32] G Augusti ldquoDynamics of structures theory and applications toearthquake engineeringrdquoMeccanica vol 31 no 6 pp 719ndash7201996

[33] H Zheng and W Jiang ldquoDiscontinuous deformation analysisbased on complementary theoryrdquo Science in China Series ETechnological Sciences vol 52 no 9 pp 2547ndash2554 2009

[34] H Zheng and X Li ldquoMixed linear complementarity formu-lation of discontinuous deformation analysisrdquo InternationalJournal of Rock Mechanics ampMining Sciences vol 75 pp 23ndash322015

[35] H-W Zhang S-Y He and X-S Li ldquoNon-interior smoothingalgorithm for frictional contact problemsrdquoAppliedMathematicsamp Mechanics vol 25 no 1 pp 47ndash58 2004

[36] W Peter and G Zavarise Computational Contact MechanicsWiley Online Library 2002

[37] D PeterNewtonMethods for Nonlinear Problems Science Press2006

[38] Y Zhang X Fu and Q Sheng ldquoModification of the discontinu-ous deformation analysis method and its application to seismicresponse analysis of large underground cavernsrdquoTunnelling andUnderground Space Technology vol 40 pp 241ndash250 2014

[39] H Bao Y H Hatzor and X Huang ldquoA new viscous boundarycondition in the two-dimensional discontinuous deformationanalysismethod for wave propagation problemsrdquoRockMechan-ics and Rock Engineering vol 45 no 5 pp 919ndash928 2012

[40] Z Cui Q Sheng and X Leng ldquoControl effect of a largegeological discontinuity on the seismic response and stability ofunderground rock caverns a case study of the baihetan 1 surgechamberrdquo Rock Mechanics and Rock Engineering vol 49 no 6pp 2099ndash2114 2016

[41] J F DoyleWave Propagation in Structure Science Press 2013[42] S G Qian X L Tang and L J Zhang ldquoDiscussion on

calculation limits of 3D subterranean tunnel under dynamicresponserdquo Nonferrous Metals Design vol 37 pp 40ndash43 2010

[43] Z Cui Q Sheng and X Leng ldquoEffects of a controlling geolog-ical discontinuity on the seismic stability of an undergroundcavern subjected to near-fault ground motionsrdquo Bulletin ofEngineering Geology and the Environment pp 1ndash18 2016

[44] H-B Li X-D Ma W Shao X Xia H-C Dai and K-Q XiaoldquoInfluence of the earthquake parameters on the displacement ofrock cavernrdquo Chinese Journal of Rock Mechanics and Engineer-ing vol 24 pp 4627ndash4634 2005

[45] Z Cui Q Sheng X Leng and J Chen ldquoSeismic response andstability of underground rock caverns a case study of Baihetanunderground cavern complexrdquo Journal of the Chinese Instituteof Engineers vol 39 no 1 2016

[46] S M Hsiung and G H Shi ldquoSimulation of earthquake effectson underground excavations usingDiscontinuousDeformationAnalysis (DDA)rdquo in Proceedings of theThe 38th US Symposiumon Rock Mechanics (USRMS rsquo01) ARMA-01-1413 WashingtonDC USA July 2001

[47] Y H Hatzor A A Arzi Y Zaslavsky and A Shapira ldquoDynamicstability analysis of jointed rock slopes using the DDAmethodking Herodrsquos Palace Masada Israelrdquo International Journal ofRock Mechanics amp Mining Sciences vol 41 no 5 pp 813ndash8322004

[48] Q-H Jiang and D-X Feng ldquoModeling of rockbolts in three-dimensional discontinuous deformation analysisrdquoRock and SoilMechanics vol 22 no 2 pp 176ndash178 2001

International Journal of

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation



DistributedSensor Networks



sliding and separation modes) In the process of open-closeiteration the contact modes are assumed in advance and cor-rected for those contact pairs that violate the contact criteriaObviously open-close iteration does not have strict mathe-matical convergence criteria and the scheme cannot guaran-tee that the iteration is always convergent [18]Meanwhile thetime consumed during the open-close iteration is very long Ifthe number of time steps of the dynamic analysis is very largeopen-close iteration would consumemuch time and decreasethe efficiency of calculation

From the aspect of dynamic contact force short embed-ding distance is allowed in the conventional DDA methodand contact force is the product of embedding distance andcontact stiffness The determination of the contact stiffnessvalue heavily relies on the engineerrsquos experience and does nothave a specific rule [19] Meanwhile under dynamic loadingthe dynamic contact force between blocks is related to theimpact velocity the stress wave velocity andYoungrsquosmodulus[20]The product of embedding distance and contact stiffnessin the conventional DDAmethod cannot reflect the dynamicfeature If the dynamic contact force cannot be calculatedprecisely the corresponding block displacement cannot beaccurate either

The current numerical calculation methods aiming atthe dynamic contact problems are the penalty method [2122] and the Lagrange multiplier method [23 24] and theirimproved versions [25 26] The product of embeddingdistance and contact stiffness is regarded as the contact forcein the penalty method and it cannot represent the precisevalue of the dynamic contact force The contact force is setas the unknown in the Lagrange multiplier method to meetthe contact conditions precisely but it increases the degree offreedom of the simultaneous equations and the difficulty ofsolving the simultaneous equations Meanwhile all of thesemethods adopt the open-close iteration to obtain the contactmodes of the contact pairs which consumes considerabletime and decreases the efficiency of calculation Aiming atthe dynamic response of cracks Liu [27 28] proposed thedynamic contact force method which has been applied ingeotechnical engineering The explicit scheme is adopted inthe algorithm and the displacement at the next time step canbe expressed with the displacement and the contact force atthe current time step The dynamic solution of displacementcan be obtained with the relationship between displacementand contact force Due to the explicit scheme the algorithmcan greatly reduce the calculation time and computer mem-ory consumed The algorithm can easily reach convergenceand is suitable for analyzing large-scale complex dynamiccontact problemsHowever the contact pairs of the algorithmare independent and have no connection with each otherwhich violates the contact conditions in the DDA methodThe contact pairs in the block system in the DDA methodcan influence each other and the modification of one contactpairrsquos mode may lead to the modification of another contactpairrsquos mode From this perspective the dynamic contact forcemethod cannot precisely simulate the contact mode of thecomplex block system Moreover there are other numericalmethods such as the impulse model method [29] the initialdisplacement method [30] and the dynamic contact model

method [31]Thesemethods aremainly applied in the analysisof a crackrsquos dynamic response and are rarely applied ingeotechnical engineering

This paper proposes the explicit dynamic DDA methodconsidering dynamic contact force based on the previousresearch which aims at solving the problems of low efficiencyof dynamic contact detection and the simulation of dynamiccontact force in the conventional DDA method Becausethere are many factors that can affect the dynamic contactforce and it is difficult to determine the calculation formulaof the dynamic contact force with the reference of Liursquosdynamic contact force method dynamic contact force is setas an unknown in this paper The explicit expression of blockdisplacement containing dynamic contact force is deducedand the contact constraint equations of a block system areobtained based on the contact relationship between displace-ment and dynamic contact forceThe simultaneous equationsare solved to obtain the corresponding displacement anddynamic contact forceThere is no need to conduct the open-close iteration repeatedly to obtain the contact modes andthe dynamic contact force can be precisely calculated Due tothe explicit scheme it can reduce the calculation time and thecomputer memory consumed The calculating efficiency isgreatly improved and it can be applied in large-scale geotech-nical engineering containing complex dynamic contact prob-lems

2 The Simultaneous Equations of the ExplicitDynamic DDA Method consideringDynamic Contact Force

The conventional DDA method is based on the principle ofminimum potential energyThe potential energy of the blocksystem is calculated and integrated and the displacementcan be obtained by minimizing the overall potential energywhich is the same as for the finite element method Howeverthe inertia force potential energy generated by unbalancedforce and the contact potential energy generated by the con-tact between blocks must be calculated in the conventionalDDA method while this is not needed in the finite elementmethod In this section the simultaneous equations of theexplicit dynamic DDA method are deduced from the twoaspects of inertia force potential energy and contact potentialenergy

21 Dynamic Contact Force Simulated by Point LoadingA block system is composed of many single blocks andcontact force exists between blocksThe overall simultaneousequations of a block system can be obtained by integrating thesimultaneous equations of every single block In the followingdeduction contact force is set as an unknown Assume thatthe block 119894 contacts the adjacent blocks the contact effectbetween the block 119894 and the adjacent blocks can be regardedas the point loading which is illustrated in Figure 1

The contact force applied on block 119894 as point loading canbe classified into two categories The first category is activecontact force 1198771198941 as illustrated in Figure 1 and it is appliedto the contact point of block 119894 itself The second category is


Ωk

Ωj

Ωi

Rk1 RYk1

RYi1

RXk1

RXi1

Ri1

Ri2

RYi2

RYj2

RXj2

Ri2

RXi2






[1198791198941 (119909 119910)]119879 11987711988311989411198771198841198941 = [1198791198941 (119909 119910)]119879 1198771198941 997888rarr [119877119894] (3)


[1198791198942 (119909 119910)]119879 11987711988311989421198771198841198942 = [1198791198942 (119909 119910)]119879 1198771198942 997888rarr [119877119894] (4)








Ω119894

[119879119894]119879 [119879119894] 119889119909 119889119910 (119894 = 1 sdot sdot sdot 119873119887) (10)



= 2Δ1199052 119880 (119905) minus 2Δ119905 (119905 minus Δ119905) (11)





= 1Δ1199052 (119880 (119905 minus Δ119905) minus 2 119880 (119905) + 119880 (119905 + Δ119905)) (12)




the Block System


Ωi

Ωj

dnRn120575n

n






min (119889119899 119877119899) = 0 (17)





Ωi

Ωj

d120591

R120591120591






min (10038161003816100381610038161198891205911003816100381610038161003816 120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816) = 0 (22)





Sliding

Sliding

Fixed

d120591

R120591

120583Rn

minus120583Rn



min (119889119899 119877119899) = 0 (25a)

min (10038161003816100381610038161198891205911003816100381610038161003816 120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816) = 0 (25b)












119877120591 minus 2120587120583119877119899 arctan(1198891205911205762 ) = 0 (28b)




119894=1

119894=119873119888= 0(29)



000204060810

R120591120583

Rn

d12059110minus7




4 Boundary Setting




p

Viscousboundary







(33)





5 Validation


0 M2

L3

L2

120588

M1 L1







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3200

03

06

09

12

15

18

21

24

27

30

33

Dyn

amic

cont

act f

orce

(KN

)

Time (10minus3 s)





0 2 4 6 8 10 12 14 16 18 20 22 24 2600

04

08

12

16

20

24

28

32

36

40

44

Dyn

amic

cont

act f

orce

(KN

)










Viscousboundary


Supporting bolts

12

34






0 2 4 6 8 10 12 14 16 18 20

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Acce

lera

tion

(ms

2)







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus004minus002

000002004006008010012014

Disp

lace

men

t (m

)

Time (s)

12

34


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus0008

0000

0008

0016

0024

0032

0040

0048

0056

Time (s)

Disp

lace

men

t (m

)

12

34



6 Discussion


7 Conclusion




Competing Interests


Acknowledgments


References






















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Ωk

Ωj

Ωi

Rk1 RYk1

RYi1

RXk1

RXi1

Ri1

Ri2

RYi2

RYj2

RXj2

Ri2

RXi2






[1198791198941 (119909 119910)]119879 11987711988311989411198771198841198941 = [1198791198941 (119909 119910)]119879 1198771198941 997888rarr [119877119894] (3)


[1198791198942 (119909 119910)]119879 11987711988311989421198771198841198942 = [1198791198942 (119909 119910)]119879 1198771198942 997888rarr [119877119894] (4)








Ω119894

[119879119894]119879 [119879119894] 119889119909 119889119910 (119894 = 1 sdot sdot sdot 119873119887) (10)



= 2Δ1199052 119880 (119905) minus 2Δ119905 (119905 minus Δ119905) (11)





= 1Δ1199052 (119880 (119905 minus Δ119905) minus 2 119880 (119905) + 119880 (119905 + Δ119905)) (12)




the Block System


Ωi

Ωj

dnRn120575n

n






min (119889119899 119877119899) = 0 (17)





Ωi

Ωj

d120591

R120591120591






min (10038161003816100381610038161198891205911003816100381610038161003816 120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816) = 0 (22)





Sliding

Sliding

Fixed

d120591

R120591

120583Rn

minus120583Rn



min (119889119899 119877119899) = 0 (25a)

min (10038161003816100381610038161198891205911003816100381610038161003816 120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816) = 0 (25b)












119877120591 minus 2120587120583119877119899 arctan(1198891205911205762 ) = 0 (28b)




119894=1

119894=119873119888= 0(29)



000204060810

R120591120583

Rn

d12059110minus7




4 Boundary Setting




p

Viscousboundary







(33)





5 Validation


0 M2

L3

L2

120588

M1 L1







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3200

03

06

09

12

15

18

21

24

27

30

33

Dyn

amic

cont

act f

orce

(KN

)

Time (10minus3 s)





0 2 4 6 8 10 12 14 16 18 20 22 24 2600

04

08

12

16

20

24

28

32

36

40

44

Dyn

amic

cont

act f

orce

(KN

)










Viscousboundary


Supporting bolts

12

34






0 2 4 6 8 10 12 14 16 18 20

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Acce

lera

tion

(ms

2)







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus004minus002

000002004006008010012014

Disp

lace

men

t (m

)

Time (s)

12

34


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus0008

0000

0008

0016

0024

0032

0040

0048

0056

Time (s)

Disp

lace

men

t (m

)

12

34



6 Discussion


7 Conclusion




Competing Interests


Acknowledgments


References






















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












= 1Δ1199052 (119880 (119905 minus Δ119905) minus 2 119880 (119905) + 119880 (119905 + Δ119905)) (12)




the Block System


Ωi

Ωj

dnRn120575n

n






min (119889119899 119877119899) = 0 (17)





Ωi

Ωj

d120591

R120591120591






min (10038161003816100381610038161198891205911003816100381610038161003816 120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816) = 0 (22)





Sliding

Sliding

Fixed

d120591

R120591

120583Rn

minus120583Rn



min (119889119899 119877119899) = 0 (25a)

min (10038161003816100381610038161198891205911003816100381610038161003816 120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816) = 0 (25b)












119877120591 minus 2120587120583119877119899 arctan(1198891205911205762 ) = 0 (28b)




119894=1

119894=119873119888= 0(29)



000204060810

R120591120583

Rn

d12059110minus7




4 Boundary Setting




p

Viscousboundary







(33)





5 Validation


0 M2

L3

L2

120588

M1 L1







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3200

03

06

09

12

15

18

21

24

27

30

33

Dyn

amic

cont

act f

orce

(KN

)

Time (10minus3 s)





0 2 4 6 8 10 12 14 16 18 20 22 24 2600

04

08

12

16

20

24

28

32

36

40

44

Dyn

amic

cont

act f

orce

(KN

)










Viscousboundary


Supporting bolts

12

34






0 2 4 6 8 10 12 14 16 18 20

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Acce

lera

tion

(ms

2)







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus004minus002

000002004006008010012014

Disp

lace

men

t (m

)

Time (s)

12

34


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus0008

0000

0008

0016

0024

0032

0040

0048

0056

Time (s)

Disp

lace

men

t (m

)

12

34



6 Discussion


7 Conclusion




Competing Interests


Acknowledgments


References






















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Ωi

Ωj

d120591

R120591120591






min (10038161003816100381610038161198891205911003816100381610038161003816 120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816) = 0 (22)





Sliding

Sliding

Fixed

d120591

R120591

120583Rn

minus120583Rn



min (119889119899 119877119899) = 0 (25a)

min (10038161003816100381610038161198891205911003816100381610038161003816 120583119877119899 minus 10038161003816100381610038161198771205911003816100381610038161003816) = 0 (25b)












119877120591 minus 2120587120583119877119899 arctan(1198891205911205762 ) = 0 (28b)




119894=1

119894=119873119888= 0(29)



000204060810

R120591120583

Rn

d12059110minus7




4 Boundary Setting




p

Viscousboundary







(33)





5 Validation


0 M2

L3

L2

120588

M1 L1







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3200

03

06

09

12

15

18

21

24

27

30

33

Dyn

amic

cont

act f

orce

(KN

)

Time (10minus3 s)





0 2 4 6 8 10 12 14 16 18 20 22 24 2600

04

08

12

16

20

24

28

32

36

40

44

Dyn

amic

cont

act f

orce

(KN

)










Viscousboundary


Supporting bolts

12

34






0 2 4 6 8 10 12 14 16 18 20

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Acce

lera

tion

(ms

2)







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus004minus002

000002004006008010012014

Disp

lace

men

t (m

)

Time (s)

12

34


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus0008

0000

0008

0016

0024

0032

0040

0048

0056

Time (s)

Disp

lace

men

t (m

)

12

34



6 Discussion


7 Conclusion




Competing Interests


Acknowledgments


References






















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119877120591 minus 2120587120583119877119899 arctan(1198891205911205762 ) = 0 (28b)




119894=1

119894=119873119888= 0(29)



000204060810

R120591120583

Rn

d12059110minus7




4 Boundary Setting




p

Viscousboundary







(33)





5 Validation


0 M2

L3

L2

120588

M1 L1







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3200

03

06

09

12

15

18

21

24

27

30

33

Dyn

amic

cont

act f

orce

(KN

)

Time (10minus3 s)





0 2 4 6 8 10 12 14 16 18 20 22 24 2600

04

08

12

16

20

24

28

32

36

40

44

Dyn

amic

cont

act f

orce

(KN

)










Viscousboundary


Supporting bolts

12

34






0 2 4 6 8 10 12 14 16 18 20

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Acce

lera

tion

(ms

2)







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus004minus002

000002004006008010012014

Disp

lace

men

t (m

)

Time (s)

12

34


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus0008

0000

0008

0016

0024

0032

0040

0048

0056

Time (s)

Disp

lace

men

t (m

)

12

34



6 Discussion


7 Conclusion




Competing Interests


Acknowledgments


References






















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










p

Viscousboundary







(33)





5 Validation


0 M2

L3

L2

120588

M1 L1







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3200

03

06

09

12

15

18

21

24

27

30

33

Dyn

amic

cont

act f

orce

(KN

)

Time (10minus3 s)





0 2 4 6 8 10 12 14 16 18 20 22 24 2600

04

08

12

16

20

24

28

32

36

40

44

Dyn

amic

cont

act f

orce

(KN

)










Viscousboundary


Supporting bolts

12

34






0 2 4 6 8 10 12 14 16 18 20

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Acce

lera

tion

(ms

2)







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus004minus002

000002004006008010012014

Disp

lace

men

t (m

)

Time (s)

12

34


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus0008

0000

0008

0016

0024

0032

0040

0048

0056

Time (s)

Disp

lace

men

t (m

)

12

34



6 Discussion


7 Conclusion




Competing Interests


Acknowledgments


References






















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3200

03

06

09

12

15

18

21

24

27

30

33

Dyn

amic

cont

act f

orce

(KN

)

Time (10minus3 s)





0 2 4 6 8 10 12 14 16 18 20 22 24 2600

04

08

12

16

20

24

28

32

36

40

44

Dyn

amic

cont

act f

orce

(KN

)










Viscousboundary


Supporting bolts

12

34






0 2 4 6 8 10 12 14 16 18 20

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Acce

lera

tion

(ms

2)







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus004minus002

000002004006008010012014

Disp

lace

men

t (m

)

Time (s)

12

34


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus0008

0000

0008

0016

0024

0032

0040

0048

0056

Time (s)

Disp

lace

men

t (m

)

12

34



6 Discussion


7 Conclusion




Competing Interests


Acknowledgments


References






















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Viscousboundary


Supporting bolts

12

34






0 2 4 6 8 10 12 14 16 18 20

minus8

minus6

minus4

minus2

0

2

4

6

Time (s)

Acce

lera

tion

(ms

2)







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus004minus002

000002004006008010012014

Disp

lace

men

t (m

)

Time (s)

12

34


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus0008

0000

0008

0016

0024

0032

0040

0048

0056

Time (s)

Disp

lace

men

t (m

)

12

34



6 Discussion


7 Conclusion




Competing Interests


Acknowledgments


References






















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus004minus002

000002004006008010012014

Disp

lace

men

t (m

)

Time (s)

12

34


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

minus0008

0000

0008

0016

0024

0032

0040

0048

0056

Time (s)

Disp

lace

men

t (m

)

12

34



6 Discussion


7 Conclusion




Competing Interests


Acknowledgments


References






















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Competing Interests


Acknowledgments


References






















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article explicit dynamic dda method considering

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