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Kinetic Energy Dissipation and Convergence Criterion of Discontinuous Deformations Analysis (DDA) for Geotechnical Engineering

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Abstract

The discontinuous deformation analysis (DDA) is a numerical method for modeling discontinuous deformation behaviour of jointed rocks. In this paper, two basic problems are discussed related to kinetic energy dissipation and the convergence criterion for the DDA method when it is applied to geotechnical engineering. In view of the fact that the deformation and progressive failure can be treated as a quasi-static process with low kinetic energy dissipation rates, this paper introduces a viscous damping component to absorb discrete blocks’ kinetic energy, establishes the global equations of motion of the discrete block system that take damping effects into account, investigates the energy dissipation mechanism when solving a static or quasi-static problem, and defines the convergence criteria of displacement, kinetic energy and unbalanced force for DDA solutions when the system arrives at a stable state.

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Abbreviations

\( x_{0} ,y_{0} \) :

Mass center coordinates of block

\( \Updelta u_{0} ,\Updelta v_{0} \) :

x- and y-translation increments of block

\( \Updelta \gamma_{0} \) :

Rotation angle increment of block around its mass center

\( \Updelta \varepsilon_{x} ,\Updelta \varepsilon_{y} \) :

Increments of normal strains of block

\( \Updelta \gamma_{xy} \) :

Increment of shear strain of block

e :

Index of block

\( \Updelta {\mathbf{D}} \) :

Vector of unknown variables of block

T :

Shape function matrix of block

L :

Differential operator matrix

\( \Updelta {\mathbf{u}},{\mathbf{u}} \) :

Displacement increment vector and total displacement vector

\( \Updelta {\mathbf{\varepsilon }},\Updelta {\varvec{\sigma}} \) :

Strain increment matrix and stress increment matrix

E :

Elastic constant matrix

\( \Uppi_{\text{p}} \) :

Total potential energy

\( \Uppi_{\varepsilon } ,\Uppi_{{\sigma_{0} }} ,\Uppi_{\text{i}} \) :

Elastic strain energy, initial stress potential energy and inertia energy

\( \Uppi_{\text{s}} ,\Uppi_{\text{b}} ,\Uppi_{\text{r}} \) :

Surface force potential energy, body force potential energy and viscous force potential energy

\( \rho ,\gamma \) :

Mass density and unit weight

\( E,\upsilon \) :

Young’s modulus and Poisson’s ratio

\( \phi ,C,T \) :

Angle of friction, cohesion and tensile strength

\( {\mathbf{b}},{\varvec{\sigma}}_{0} ,{\mathbf{p}} \) :

Body force, initial stress and surface force vectors

\( {\mathbf{M}},{\mathbf{K}},{\mathbf{f}} \) :

Mass matrix, stiffness matrix and load matrix

\( {\dot{\mathbf{D}}},{\ddot{\mathbf{D}}} \) :

Velocity and acceleration vectors

\( {\mathbf{V}}_{0} \) :

Initial velocity vector

\( \Updelta t \) :

Time increment

\( i \) :

Index of time step

\( n \) :

Number of time step

\( j \) :

Index of vertices of block

\( l \) :

Number of vertices of block

\( s \) :

Area of block

\( M \) :

Mass of block

\( {\mathbf{x}}_{j} \) :

Coordinate of jth vertex

\( \mu ,\mu_{\text{c}} \) :

Damping coefficient, critical damping coefficient

\( {\mathbf{r}} \) :

Viscous force vector

\( {\mathbf{q}} \) :

Unbalanced force vector

\( {\mathbf{F}}_{\text{ext}} \) :

External load vector

\( \Uppi_{\text{ext}} \) :

Potential energy of external loads

\( \kappa \) :

Kinetic energy

\( \zeta_{\text{d}} \) :

Convergence criterion for displacement

\( \zeta_{\text{v}} \) :

Convergence criterion for kinetic energy

\( \zeta_{\text{f}} \) :

Convergence criterion for unbalanced force

\( \delta \) :

User-specified convergence accuracy

\( d,\dot{d},\ddot{d} \) :

Displacement, velocity and acceleration of SDOF system

\( \xi ,\bar{\xi } \) :

Damping ratio and equivalent damping ratio

\( \omega \) :

Undamped frequency of vibration

\( k \) :

Stiffness of SDOF system

\( k_{\text{d}} \) :

Deformation stiffness of block

\( k_{\text{n}} ,k_{\text{s}} \) :

Normal spring stiffness and shear spring stiffness

\( p \) :

Uniformly distributed pressure

\( g_{2} \) :

Maximum displacement ratio in DDA program

K01:

Dynamic damping parameter in DDA program

References

  • Bao HR, Hatzor YH, Huang X (2012) A new viscous boundary condition in the two-dimensional discontinuous deformation analysis method for wave propagation Problems. Rock Mech Rock Eng 45:919–928

    Google Scholar 

  • Beyabanaki SAR, Mikola RG, Hatami K (2008) Three-dimensional deformation analysis (3-D DDA) using a new contact resolution algorithm. Comput Geotech 35:346–356

    Article  Google Scholar 

  • Beyabanaki SAR, Jafari A, Biabanaki SOR, Yeung MR (2009) Nodal-based three-dimensional deformation analysis (3-D DDA). Comput Geotech 36:359–372

    Article  Google Scholar 

  • Cai YE, Liang GP, Shi GH, Cook NGW (1996) Studying impact problem by using LDDA method. In: Proceedings of the first international forum on discontinuous deformation analysis (DDA) and simulations of discontinuous media, Berkeley, pp 288–294

  • Chen GQ, Miki S, Ohnishi Y (1997) Development of the interactive visualization system for DDA. In: Proceedings of the 9th international conference on computer methods and advances in geomechanics, Wuhan, pp 495–500

  • Cheng YM, Zhang YH (2000) Rigid body rotation and block internal discretization in DDA analysis. Int J Numer Anal Methods Geomech 24:567–578

    Article  Google Scholar 

  • Chern JC, Koo CY, Chen S (1990) Development of second order displacement function for DDA and manifold method. In: Working forum on the manifold method of material analysis, Vicksburg, pp 183–202

  • Clatworthy D, Scheele F (1999) A method of sub-meshing in discontinuous deformation. In: Proceedings of the third international conference on analysis of discontinuous deformation from theory to practice, Colorado, pp 85–94

  • Clough RW, Penzien J (2003) Dynamics of structures, 3rd edn. Computers and Structures, Inc., Berkeley

  • Doolin DM, Sitar N (2001) DDAML-discontinuous deformation analysis markup language. Int J Rock Mech Min Sci 38:467–474

    Article  Google Scholar 

  • Doolin DM, Sitar N (2004) Time integration in discontinuous deformation analysis. ASCE J Eng Mech 130:249–258

    Article  Google Scholar 

  • Grayeli R, Mortazavi A (2006) Discontinuous deformation analysis with second-order finite element meshed block. Int J Numer Anal Methods Geomech 30:1545–1561

    Article  Google Scholar 

  • Hatzor YH, Arzi AA, Zaslavsky Y, Shapira A (2004) Dynamic stability analysis of jointed rock slopes using DDA method: King Herod’s Palace, Masada, Israel. Int J Rock Mech Min Sci 41:813–832

    Article  Google Scholar 

  • Hsiung SM (2001) Discontinuous deformation analysis (DDA) with nth order polynomial displacement functions. In: Proceedings of the 38th US rock mechanics symposium, Washington DC, pp 1437–1444

  • Jiang QH, Yeung MR (2004) A model of point-to-face contact for three-dimensional discontinuous deformation analysis. Rock Mech Rock Eng 37:95–116

    Article  Google Scholar 

  • Jiao YY, Zhang XL, Zhao J, Liu QS (2007) Viscous boundary of DDA for modeling stress wave propagation in jointed rock. Int J Rock Mech Min Sci 44:1070–1076

    Article  Google Scholar 

  • Jiao YY, Zhang XL, Zhao J (2012) Two-dimensional DDA constitutive model for simulating rock fragmentation. ASCE J Eng Mech 138(2):199–209

    Article  Google Scholar 

  • Jing LR (1998) Formulation of discontinuous deformation analysis (DDA)—an implicit discrete element model for block systems. Eng Geol 49:371–381

    Article  Google Scholar 

  • Jing LR, Ma Y, Fang Z (2001) Modeling of fluid flow and solid deformation for fractured rocks with discontinuous deformation analysis (DDA) method. Int J Rock Mech Min Sci 38:343–355

    Article  Google Scholar 

  • Ke TC (1996) Artificial joint-based DDA. In: Proceedings of the first international forum on discontinuous deformation analysis (DDA) and simulations of discontinuous media, Berkeley, pp 326–333

  • Kim YI, Amadei B, Pan E (1999) Modeling the effect of water, excavation sequence and rock reinforcement with discontinuous deformation analysis. Int J Rock Mech Min Sci 36:949–970

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Liang GP, Wang CG (1996) LDDA on the high speed catenary-pantograph system dynamics. In: Proceedings of the first international forum on discontinuous deformation analysis (DDA) and simulations of discontinuous media, Berkeley, pp 350–356

  • Lin CT, Amadei B, Jung J, Dwyer J (1996) Extensions of discontinuous deformation analysis for jointed rock masses. Int J Rock Mech Min Sci 33:671–694

    Article  Google Scholar 

  • Ma MY, Zaman M, Zhou JH (1996) Discontinuous deformation analysis using the third order displacement function. In: Proceedings of the first international forum on discontinuous deformation analysis (DDA) and simulations of discontinuous media, Berkeley, pp 383–394

  • MacLaughlin MM, Doolin DM (2006) Review of validation of the discontinuous deformation analysis (DDA) method. Int J Numer Anal Methods Geomech 30:271–305

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Ning YJ, Yang J, Ma GW, Chen PW (2011) Modelling rock blasting considering explosion gas penetration using discontinuous deformation analysis. Rock Mech Rock Eng 44:483–490

    Article  Google Scholar 

  • Ohnishi Y, Yamamukai K, Chen GH (1996) Application of DDA in rock-fall analysis, In: Proceedings of the 2nd North American rock mechanics symposium, Quebec, pp 2031–2037

  • Ohnishi Y, Nishiyama S, Sasaki T (2006) Development and application of discontinuous deformation analysis. In: The 4th Asian rock mechanics symposium, Singapore, pp 59–70

  • Pearce CJ, Thavalingam A, Liao Z, Bicanic N (2000) Computational aspects of the discontinuous deformation analysis framework for modelling concrete fracture. Eng Fract Mech 65:283–298

    Article  Google Scholar 

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

  • Shi GH (2001) Theory and examples of three dimensional discontinuous deformation analysis. In: Proceedings of the 2nd Asian rock mechanics symposium, Beijing, pp 27–32

  • Shi GH, Goodman RE (1989) Generalization of two-dimensional discontinuous deformation analysis for forward modelling. Int J Numer Anal Methods Geomech 13:359–380

    Article  Google Scholar 

  • Thomas PA, Bray JD (1999) Capturing nonspherical shape of granular media with disk clusters. ASCE J Geotech Geoenviron Eng 125:169–178

    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(2):153–167

    Article  Google Scholar 

  • Wang CY, Chuang CC, Sheng J (1996) Time integration theories for the DDA method with finite element meshes. In: Proceedings of the first international forum on discontinuous deformation analysis (DDA) and simulations of discontinuous media, Berkeley, pp 263–287

  • Wartman J, Bray JD, Seed RB (2003) Inclined plane studies of the Newmark sliding block procedures. J Geotech Geoenviron Eng ASCE 129(8):673–684

    Article  Google Scholar 

  • Wu JH, Juang CH, Lin HM (2005) Vertex-to-face contact searching algorithm for three-dimensional frictionless contact problems. Int J Numer Methods Eng 63:876–897

    Article  Google Scholar 

  • Yeung MR, Jiang QH, Sun N (2007) A model of edge-to-edge contact for three-dimensional discontinuous deformation analysis. Comput Geotech 34:175–186

    Article  Google Scholar 

  • Zhang YJ (2006) Equivalent model and numerical analysis and laboratory test for jointed rock masses. Chin J Geotech Eng 28:29–32

    Google Scholar 

  • Zheng H, Jiang W (2009) Discontinuous deformation analysis based on complementary theory. Sci China Ser E Technol Sci 52:2547–2554

    Article  Google Scholar 

Download references

Acknowledgments

The authors thank Dr. L. Jing and Prof. Y. Hatzor for their insightful suggestions and valuable comments in improving this present study. The work reported in this paper has received financial support from the National Natural Science Foundation of China (Projects Nos. NSFC 51079110) and Natural Basic Research Program of China (973 Program No. 2010CB732005). This support is gratefully acknowledged.

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Authors and Affiliations

  1. School of Civil Engineering, Wuhan University, Wuhan, 430072, People’s Republic of China

    Qinghui Jiang

  2. State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, 430072, People’s Republic of China

    Qinghui Jiang, Yifeng Chen & Chuangbing Zhou

  3. Civil Engineering Department, California State Polytechnic University, Pomona, CA, 91768, USA

    Man-chu Ronald Yeung

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  1. Qinghui Jiang
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  2. Yifeng Chen
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Correspondence to Qinghui Jiang.

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Jiang, Q., Chen, Y., Zhou, C. et al. Kinetic Energy Dissipation and Convergence Criterion of Discontinuous Deformations Analysis (DDA) for Geotechnical Engineering. Rock Mech Rock Eng 46, 1443–1460 (2013). https://doi.org/10.1007/s00603-012-0356-5

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