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An Improved Discontinuous Deformation Analysis to Solve Numerical Creep Problem in Shear Direction

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Abstract

Discontinuous deformation analysis (DDA) has been widely accepted recently for both static and dynamic problems. When studying block sliding cases, an unphysical phenomenon “numerical creep” (an additional small and finite sliding displacement in each step) yet may be observed. The accumulated creep displacement brings error to the stability and runout distance analysis. Therefore, a modification for DDA is proposed to overcome this numerical creep. The contact evolving in a sliding block case is first investigated and the cause to the numerical creep is deliberately illustrated. Then a modified open-close iteration (OCI) process is proposed to avoid such unphysical phenomenon, in which a new shear force evaluation method is introduced for the locked contacts that transformed from sliding state. Subsequently, the improved DDA with the modified OCI is checked by several block sliding examples under gravity and time-dependent dynamic forces. The comparing results suggest that the simulating accuracy of DDA is significantly improved.

Highlights

  • Illustrated reason of the so-called “numerical creep” in Discontinuous Deformation Analysis (DDA).

  • Improved DDA with modified open-close iteration (OCI) that avoids the numerical creep.

  • Verified improved DDA through cases about block stability and sliding.

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Abbreviations

u 0, v 0, r 0 :

Rigid body motion terms of a block

ε x, ε y, γ xy :

Constant strain terms of a block

D i :

Deformation matrix of block i

U = (u, v):

Displacement at any point P (x, y)

T i (x, y):

Displacement transformation matrix

x 0, y 0 :

Coordinates of block centroid

N :

Number of blocks in a blocky system

D :

Displacement term in a blocky system

\(\dot{{\varvec{D}}}\) :

Velocity term in a blocky system

\(\ddot{{\varvec{D}}}\) :

Acceleration term in a blocky system

M :

Mass term of a blocky system

C :

Damping term of a blocky system

K :

Stiffness term of a blocky system

F :

External force subjected by a system

γ :

Parameter for velocity weighting

β :

Parameter for acceleration weighting

n :

Calculation time steps

\(\hat{K}\) :

Equivalent stiffness matrix

\(\hat{\varvec{F}}\) :

Equivalent load vector

K ij (i, j = 1, 2, …, N):

6 × 6 Stiffness submatrices

F i (i = 1, 2, …, N):

6 × 1 Load vector

c :

Cohesion strength

σ t :

Tension strength

k n :

Stiffness of normal spring

k τ :

Stiffness of shear spring

d n :

Penetration distance in normal direction

d τ :

Penetration distance in shear direction

φ :

Friction angle

S :

Effective contact area

F n :

Normal contact force

F τ :

Shear contact force

s 0 :

Shear displacement in the previous loop

v 0 :

Shear velocity in the previous loop

s 1 :

Shear displacement in the current loop

v 1 :

Shear velocity in the current loop

Δt′ :

Time increment of the sliding stage

f :

Static friction

F e :

Constant equivalent force in the whole step

I :

Impulse

r :

Ratio

m :

Mass of a rigid sliding block

a y :

Yield acceleration

θ :

Initial sliding time

a t :

Acceleration along slope

d t :

Downslope displacement

References

  • Bao H, Zhao Z (2012) The vertex-to-vertex contact analysis in the two-dimensional discontinuous deformation analysis. Adv Eng Softw 45:1–10

    Article  Google Scholar 

  • Beyabanaki SAR, Jafari A, Yeung MR (2010) High-order three-dimensional discontinuous deformation analysis (3-D DDA). Int J Numer Methods Biomed Eng 26:1522–1547

    Article  Google Scholar 

  • Chen K-T, Wu J-H (2018) Simulating the failure process of the Xinmo landslide using discontinuous deformation analysis. Eng Geol 239:269–281. https://doi.org/10.1016/j.enggeo.2018.04.002

    Article  Google Scholar 

  • Chen G, Zheng L, Zhang Y, Wu J (2013) Numerical simulation in rockfall analysis: a close comparison of 2-D and 3-D DDA. Rock Mech Rock Eng 46:527–541. https://doi.org/10.1007/s00603-012-0360-9

    Article  Google Scholar 

  • Cheng YM, Zhang YH (2000) Rigid body rotation and block internal discretization in DDA analysis. Int J Numer Anal Meth Geomech 24:567–578. https://doi.org/10.1002/(Sici)1096-9853(200005)24:6%3c567::Aid-Nag83%3e3.0.Co;2-N

    Article  Google Scholar 

  • Do TN, Wu JH (2020) Verifying discontinuous deformation analysis simulations of the jointed rock mass behavior of shallow twin mountain tunnels. Int J Rock Mech Min Sci 2020:130. https://doi.org/10.1016/j.ijrmms.2020.104322

    Article  Google Scholar 

  • Fan H, He S (2015) An angle-based method dealing with vertex–vertex contact in the two-dimensional discontinuous deformation analysis (DDA). Rock Mech Rock Eng 48:2031–2043

    Article  Google Scholar 

  • Fan H, Zheng H, Zhao J (2017) Discontinuous deformation analysis based on strain-rotation decomposition. Int J Rock Mech Min Sci 92:19–29

    Article  Google Scholar 

  • Fan X et al (2018) What we have learned from the 2008 Wenchuan earthquake and its aftermath: a decade of research and challenges. Eng Geol 241:25–32

    Article  Google Scholar 

  • Fu XD, Sheng Q, Zhang YH, Chen J (2016) Investigation of highly efficient algorithms for solving linear equations in the discontinuous deformation analysis method. Int J Numer Anal Meth Geomech 40:469–486. https://doi.org/10.1002/nag.2407

    Article  Google Scholar 

  • Fu X, Sheng Q, Zhang Y, Chen J (2017a) Time-frequency analysis of seismic wave propagation across a rock mass using the discontinuous deformation analysis method. Int J Geomech 17:04017024

    Article  Google Scholar 

  • Fu X, Sheng Q, Zhang Y, Chen J, Zhang S, Zhang Z (2017b) Computation of the safety factor for slope stability using discontinuous deformation analysis and the vector sum method. Comput Geotech 92:68–76

    Article  Google Scholar 

  • Grayeli R, Hatami K (2008) Implementation of the finite element method in the three-dimensional discontinuous deformation analysis (3D-DDA). Int J Numer Anal Meth Geomech 32:1883–1902

    Article  Google Scholar 

  • Hatzor YH, Feintuch A (2001) The validity of dynamic block displacement prediction using DDA. Int J Rock Mech Min Sci 38:599–606. https://doi.org/10.1016/S1365-1609(01)00026-0

    Article  Google Scholar 

  • Jiang W, Zheng H (2015) An efficient remedy for the false volume expansion of DDA when simulating large rotation. Comput Geotech 70:18–23. https://doi.org/10.1016/j.compgeo.2015.07.008

    Article  Google Scholar 

  • Jiang H, Wang L, Li L, Guo Z (2014) Safety evaluation of an ancient masonry seawall structure with modified DDA method. Comput Geotech 55:277–289

    Article  Google Scholar 

  • Jiao Y, Zhang X, Zhao J, Liu Q (2007) Viscous boundary of DDA for modeling stress wave propagation in jointed rock. Int J Rock Mech Min 7:1070–1076

    Article  Google Scholar 

  • Kaidi S, Rouainia M, Ouahsine A (2012) Stability of breakwaters under hydrodynamic loading using a coupled DDA/FEM approach. Ocean Eng 55:62–70

    Article  Google Scholar 

  • Kamai R, Hatzor YH (2008) Numerical analysis of block stone displacements in ancient masonry structures: a new method to estimate historic ground motions. Int J Numer Anal Meth Geomech 32:1321–1340

    Article  Google Scholar 

  • Ke T-C (1993) Simulated testing of two dimensional heterogeneous and discontinuous rock masses using discontinuous deformation analysis. University of California, Berkeley

  • Koo C, Chern J-C (1998) Modification of the DDA method for rigid block problems. Int J Rock Mech Min 35:683–693

    Article  Google Scholar 

  • Lin CT (1995) Extensions to the discontinuous deformation analysis for jointed rock masses and other blocky systems. University of Colorado

  • Liu G, Li J (2019) Research on the effect of tree barriers on rockfall using a three-dimensional discontinuous deformation analysis method. Int J Comput Methods 17:1950046

    Article  Google Scholar 

  • Liu S-G, Li Z-J, Zhang H, Wu W, Zhong G-H, Lou S (2018) A 3-D DDA damage analysis of brick masonry buildings under the impact of boulders in mountainous areas. J Mount Sci 15:657–671

    Article  Google Scholar 

  • MacLaughlin MM (1997) Discontinuous deformation analysis of the kinematics of landslides. University of California, Berkeley

  • Mikola RG, Sitar N (2014) 3D simulation of tsunami wave induced by rock slope failure using coupled DDA-SPH. In: 48th US rock mechanics/geomechanics symposium, 2014. American Rock Mechanics Association

  • Mortazavi A, Katsabanis P (2001) Modelling burden size and strata dip effects on the surface blasting process. Int J Rock Mech Min 38:481–498

    Article  Google Scholar 

  • Ning Y, Yang J, An X, Ma G (2010) Simulation of blast induced crater in jointed rock mass by discontinuous deformation analysis method. Front Architect Civ Eng China 4:223–232

    Article  Google Scholar 

  • Ohnishi Y, Yamamukai K, Chen G (1996) Application of DDA in rock fall analysis. In: 2nd North American rock mechanics symposium, 1996. American Rock Mechanics Association

  • O’Sullivan C, Bray JD (2003) Modified shear spring formulation for discontinuous deformation analysis of particulate media. J Eng Mech-ASCE 129:830–834. https://doi.org/10.1061/(Asce)0733-9399(2003)129:7(830)

    Article  Google Scholar 

  • Peng X, Yu P, Zhang Y, Chen G (2018) Applying modified discontinuous deformation analysis to assess the dynamic response of sites containing discontinuities. Eng Geol 246:349–360. https://doi.org/10.1016/j.enggeo.2018.10.011

    Article  Google Scholar 

  • Peng X et al (2019a) Parallel computing of three-dimensional discontinuous deformation analysis based on OpenMP. Comput Geotech 106:304–313. https://doi.org/10.1016/j.compgeo.2018.11.016

    Article  Google Scholar 

  • Peng X, Chen G, Yu P, Zhang Y, Wang J (2019b) Improvement of joint definition and determination in three-dimensional discontinuous deformation analysis. Comput Geotech 110:148–160. https://doi.org/10.1016/j.compgeo.2019.02.016

    Article  Google Scholar 

  • Peng X, Yu P, Chen G, Xia M, Zhang Y (2019c) Development of a coupled DDA–SPH method and its application to dynamic simulation of landslides involving solid-fluid interaction. Rock Mech Rock Eng 53:113–131. https://doi.org/10.1007/s00603-019-01900-x

    Article  Google Scholar 

  • Peng X, Yu P, Chen G, Xia M, Zhang Y (2020) CPU-accelerated explicit discontinuous deformation analysis and its application to landslide analysis. Appl Math Model 77:216–234. https://doi.org/10.1016/j.apm.2019.07.028

    Article  Google Scholar 

  • Shi GH (1988) Discontinuous deformation analysis: a new numerical model for the statics and dynamics of block systems. PhD thesis, Berkeley: University of California

  • Shi GH (2001) Three dimensional discontinuous deformation analysis. In: Paper presented at the The 38th U.S. symposium on rock mechanics (USRMS), Washington, D.C.

  • Song Y, Huang D, Cen D (2016) Numerical modelling of the 2008 Wenchuan earthquake-triggered Daguangbao landslide using a velocity and displacement dependent friction Law. Eng Geol 215:50–68

    Article  Google Scholar 

  • Song YX, Huang D, Zeng B (2017) GPU-based parallel computation for discontinuous deformation analysis (DDA) method and its application to modelling earthquake-induced landslide. Comput Geotech 86:80–94. https://doi.org/10.1016/j.compgeo.2017.01.001

    Article  Google Scholar 

  • Tsesarsky M, Hatzor YH (2006) Tunnel roof deflection in blocky rock masses as a function of joint spacing and friction–a parametric study using discontinuous deformation analysis (DDA). Tunn Undergr Sp Tech 21:29–45

    Article  Google Scholar 

  • Tsesarsky M, Hatzor YH, Sitar N (2005) Dynamic displacement of a block on an inclined plane: analytical, experimental and DDA results. Rock Mech Rock Eng 38:153–167. https://doi.org/10.1007/s00603-004-0043-2

    Article  Google Scholar 

  • Wang W, Chen G, Zhang Y, Zheng L, Zhang H (2017) Dynamic simulation of landslide dam behavior considering kinematic characteristics using a coupled DDA-SPH method. Eng Anal Bound Elem 80:172–183

    Article  Google Scholar 

  • Wang W, Yin K, Chen G-Q, Chai B, Han Z, Zhou J (2019a) Practical application of the coupled DDA-SPH method in dynamic modeling for the formation of landslide dam. Landslides 2019:1–12

    Google Scholar 

  • Wang X, Wu W, Zhu H, Lin J-S, Zhang H (2019b) Contact detection between polygonal blocks based on a novel multi-cover system for discontinuous deformation analysis. Comput Geotech 111:56–65

    Article  Google Scholar 

  • Wu JH (2010) Seismic landslide simulations in discontinuous deformation analysis. Comput Geotech 37:594–601. https://doi.org/10.1016/j.compgeo.2010.03.007

    Article  Google Scholar 

  • Wu J-H, Chen C-H (2011) Application of DDA to simulate characteristics of the Tsaoling landslide. Comput Geotech 38:741–750

    Article  Google Scholar 

  • Wu J, Lin H (2013) Improvement of open-close iteration in DDA. In: Chen G, Ohnishi Y, Zheng L, Sasaki T (eds) Frontiers of discontinuous numerical methods and practical simulations in engineering and disaster prevention—proceedings of the 11th Int. conf. on analysis of discontinuous deformation, ICADD 2013. CRC Press, Balkema, pp 185–191

  • Wu J-H, Ohnishi Y, Nishiyama S (2004) Simulation of the mechanical behavior of inclined jointed rock masses during tunnel construction using discontinuous deformation analysis (DDA). Int J Rock Mech Min 41:731–743

    Article  Google Scholar 

  • Wu JH, Ohnishi Y, Nishiyama S (2005) A development of the discontinuous deformation analysis for rock fall analysis. Int J Numer Anal Meth Geomech 29:971–988

    Article  Google Scholar 

  • Wu W, Zhu H, Zhuang X, Ma G, Cai Y (2014) A multi-shell cover algorithm for contact detection in the three dimensional discontinuous deformation analysis. Theor Appl Fract Mech 72:136–149. https://doi.org/10.1016/j.tafmec.2014.03.004

    Article  Google Scholar 

  • Wu JH, Do TN, Chen CH, Wang GH (2017) New geometric restriction for the displacement-constraint points in discontinuous deformation analysis. Int J Geomech 2017:17. https://doi.org/10.1061/(Asce)Gm.1943-5622.0000648

    Article  Google Scholar 

  • Wu W, Wang X, Zhu H, Shou K-J, Lin J-S, Zhang H (2020a) Improvements in DDA program for rockslides with local in-circle contact method and modified open-close iteration. Eng Geol 2020:265. https://doi.org/10.1016/j.enggeo.2019.105433

    Article  Google Scholar 

  • Wu W, Wang X, Zhu H, Shou K-J, Lin J-S, Zhang H (2020b) Improvements on DDA program for rockslides with local in-circle contact method and modified open-close iteration. Eng Geol 2020:265

    Google Scholar 

  • Xiao YF, Miao QH, Huang M, Wang Y, Xue J (2017) Parallel computing of discontinuous deformation analysis based on graphics processing unit. Int J Geomech 17:E4016010. https://doi.org/10.1061/(Asce)Gm.1943-5622.0000717

    Article  Google Scholar 

  • Yeung M-cR (1991) Application of Shi's discontinuous deformation analysis to the study of rock behavior. In: University of California, Berkeley

  • Yu P, Zhang Y, Peng X, Chen G, Zhao JX (2019a) Distributed-spring edge-to-edge contact model for two-dimensional discontinuous deformation analysis. Rock Mech Rock Eng 53:365–382. https://doi.org/10.1007/s00603-019-01917-2

    Article  Google Scholar 

  • Yu P, Zhang Y, Peng X, Wang J, Chen G, Zhao JX (2019b) Evaluation of impact force of rock landslides acting on structures using discontinuous deformation analysis. Comput Geotech 2019:114. https://doi.org/10.1016/j.compgeo.2019b.103137

    Article  Google Scholar 

  • Zhang Y, Chen G, Zheng L, Li Y, Wu J (2013) Effects of near-fault seismic loadings on run-out of large-scale landslide: a case study. Eng Geol 166:216–236

    Article  Google Scholar 

  • Zhang Y, Xu Q, Chen G, Zhao JX, Zheng L (2014) Extension of discontinuous deformation analysis and application in cohesive-frictional slope analysis. Int J Rock Mech Min 70:533–545

    Article  Google Scholar 

  • Zhang H et al (2015a) A new discontinuous model for three dimensional analysis of fluid-solid interaction behavior. In: International symposium on geomechanics from micro to macro, IS-Cambridge 2014. Taylor and Francis-Balkema

  • Zhang Y et al (2015b) DDA validation of the mobility of earthquake-induced landslides. Eng Geol 194:38–51

    Article  Google Scholar 

  • Zhang H et al (2016) A new DDA model for kinematic analyses of rockslides on complex 3-D terrain. Bull Eng Geol Env 77:555–571. https://doi.org/10.1007/s10064-016-0971-6

    Article  Google Scholar 

  • Zhang H, Liu SG, Zheng L, Zhu HH, Zhuang XY, Zhang YB, Wu YQ (2018) Method for resolving contact indeterminacy in three-dimensional discontinuous deformation analysis. Int J Geomech 18:04018130. https://doi.org/10.1061/(asce)gm.1943-5622.0001259.?2018

    Article  Google Scholar 

  • Zhang Y, Wang J, Zhao JX, Li X, Liu J, Chen G, Yu P (2019) Multi-spring edge-to-edge contact model in discontinuous deformation analysis and its application in tensile failure behavior of joint rocks. Rock Mech Rock Eng 2019:5

    Google Scholar 

  • Zhang N, Li X, Lin XC (2020) A frictional spring and cohesive contact model for accurate simulation of contact forces in numerical manifold method. Int J Numer Meth Eng 121:2369–2397. https://doi.org/10.1002/nme.6311

    Article  Google Scholar 

  • Zhang Y, Xiang C, Yu P, Zhao L, Zhao J X, Fu H (2022) Investigation of permanent displacements of near-fault seismic slopes by a general sliding block model. Landslides 19(1):187–197. https://doi.org/10.1007/s10346-021-01736-z

    Article  Google Scholar 

  • Zhu H, Wu W, Chen J, Ma G, Liu X, Zhuang X (2016) Integration of three dimensional discontinuous deformation analysis (DDA) with binocular photogrammetry for stability analysis of tunnels in blocky rockmass. Tunn Undergr Sp Tech 51:30–40

    Article  Google Scholar 

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Acknowledgements

This study has received financial support from the National Natural Science Foundation of China (52108344); Science & Technology Department of Sichuan Province (2021YJ0390, 2020YFH0017, 2021YFS0321); the Second Tibetan Plateau Scientific Expedition and Research Program (STEP) (2019QZKK0906), Hebei Key Laboratory of Earthquake Disaster Prevention and Risk Assessment (FZ213202) and the Fundamental Research Funds for the Central Universities. The financial supports are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Geotechnical Engineering, School of Civil Engineering, Southwest Jiaotong University, Chengdu, 610031, China

    Pengcheng Yu, Yingbin Zhang, Dejian Li, Haiying Fu & Yin Cheng

  2. MOE Key Laboratory of Deep Earth Science and Engineering, Institute for Disaster Management and Reconstruction, Sichuan University, Chengdu, 610065, China

    Xinyan Peng

  3. Hebei Key Laboratory of Earthquake Disaster Prevention and Risk Assessment, Sanhe, 065201, China

    Pengcheng Yu

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  1. Pengcheng Yu
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  2. Xinyan Peng
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  3. Yingbin Zhang
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  4. Dejian Li
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Correspondence to Yingbin Zhang.

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Yu, P., Peng, X., Zhang, Y. et al. An Improved Discontinuous Deformation Analysis to Solve Numerical Creep Problem in Shear Direction. Rock Mech Rock Eng 55, 3107–3127 (2022). https://doi.org/10.1007/s00603-022-02798-8

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