Skip to main content
SpringerLink
Log in

A three-dimensional damage-based contact element model for simulating the interfacial behaviors of rocks and its validation and applications

  • Original Article
  • Published:
Geomechanics and Geophysics for Geo-Energy and Geo-Resources Aims and scope Submit manuscript

Abstract

To describe the mechanical behavior of interfaces existing in rocks, a three-dimensional contact element model with a certain thickness was proposed that could simultaneously consider the heterogeneity, damage evolution and state variation of interfaces. The Weibull distribution function was employed to describe the rock heterogeneity. The damage evolution of the contact element was analyzed by using the equivalent damage method. Two criteria—the Mohr–Coulomb criterion and the maximum tensile stress criterion—were employed to assess the damage occurrence. Once the element was identified as being completely damaged, the state variation of the contact element was determined. Three types of element states—contact without sliding, contact with sliding, and separation—were taken into consideration. The model was then implemented and programmed into the finite element method and validated with experimental measurements and theoretical solutions for three classical rock mechanical problems, including one static problem (i.e., a direct shear test) and two dynamic problems (the seismic responses of rock joints and split-Hopkinson pressure bar tests), in an attempt to examine the applicability of the proposed contact element model. The results demonstrated that the contact element model can effectively model the rock interfacial problems and can therefore help solve problems in discontinuous rock masses.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Abbreviations

l, w, t :

Length, width and thickness of the contact element

σcr, σtr :

Residual compressive and tensile strength

σ1, σ3 :

Maximum principal stress and minimum principal stress

σc, σt :

Uniaxial compressive and tensile strength

\(\bar{\varepsilon }\) :

Equivalent principal strain

D :

Damage variable

εc0, εt0 :

Strains at σc and σt

εcu, εtu :

Ultimate compressive and tensile strain

f c :

Element shear stress

Φ :

Frictional angle

λc, λt :

Ratio of σc to σcr and ratio of σt to σtr

\(\left\{ \delta \right\}^{\text{e}}\) :

Element displacement

u, v, w :

Nodal displacement in the x, y and z directions

Δu, Δv, Δw :

Displacement difference between the upper and lower surfaces of contact element in the x, y and z directions

N1, N2, N3, N4 :

Shape functions

γxz, γxz, εn :

Shear strains and normal strain

{σ0}:

Initial stress

G, E :

Shear modulus and elastic modulus

[K]e, [K]:

Element stiffness matrix and total stiffness matrix

λs, λn :

Shear and normal stiffness

u, f :

Nodal displacement vector and nodal loading vector

σ 0 :

Adhesive strength

fs, fd :

Static and dynamic friction coefficient

σn, τs :

Normal stress and shear stress

α, α0 :

Parameters in the Weibull distribution function

m :

Heterogeneity index

[S]:

Elastic matrix

T p :

Transmission coefficient

Z :

Seismic impedance

Ω :

Angular frequency

References

  • Bahaaddini M, Sharrock G, Hebblewhite BK (2013) Numerical direct shear tests to model the shear behaviour of rock joints. Comput Geotech 51(Supplement C):101–115

    Google Scholar 

  • Bandis SC, Lumsden AC, Barton NR (1983) Fundamentals of rock joint deformation. Int J Rock Mech Min Sci Geomech Abstract 20(6):249–268

    Google Scholar 

  • Bao H, Zhao Z (2010) An alternative scheme for the corner–corner contact in the two-dimensional discontinuous deformation analysis. Adv Eng Softw 41(2):206–212

    MATH  Google Scholar 

  • Barton N, Bandis S, Bakhtar K (1985) Strength, deformation and conductivity coupling of rock joints. Int J Rock Mech Min Sci Geomech Abstract 22(3):121–140

    Google Scholar 

  • Bayram YB, Nied HF (2000) Enriched finite element-penalty function method for modeling interface cracks with contact. Eng Fract Mech 65(5):541–557

    Google Scholar 

  • Byerlee J (1978) Friction of rocks. Pure Appl Geophys PAGEOPH 116(4–5):615–626

    Google Scholar 

  • Cai YE, He T, Wang R (2000) Numerical simulation of dynamic process of the Tangshan earthquake by a new method—LDDA. Pure appl Geophys 157:2083–2104

    Google Scholar 

  • Chaboche JL (1988) Continuum damage mechanics: part I—general concepts. J Appl Mech 55:59–64

    Google Scholar 

  • Chai SB, Li JC, Rong LF, Li NN (2017) Theoretical study for induced seismic wave propagation across rock masses during underground exploitation. Geomech Geophys Geo-energy Geo-resour 3(2):95–105

    Google Scholar 

  • Chen X, Li JC, Cai MF, Zou Y, Zhao J (2015) Experimental study on wave propagation across a rock joint with rough surface. Rock Mech Rock Eng 48(6):2225–2234

    Google Scholar 

  • Chen HR, Qing SQ, Xue L, Yang BC, Zhang K, Wu XW (2017) Characterization of brittle failure of rock and limitation of Weibull distribution. Prog Geophys 32(5):2200–2206 (In Chinese)

    Google Scholar 

  • Cheng H, Zhou XP (2018) New technique for frictional contact on crack slip in the extended finite-element method framework. J Eng Mech 144(8):04018059

    Google Scholar 

  • Chow CL, Wang J (1987) An anisotropic theory of elasticity for continuum damage mechanics. Int J Frac 33:3–16

    Google Scholar 

  • Cundall PA (1971) A computer model for simulating progressive, large scale movements in blocky rock systems. In: Proceedings of international symposium on rock fracture, II-8, vol 1. International Society for Rock Mechanics, Nancy, France

  • Deb D, Das KC (2010) Extended finite element method for the analysis of discontinuities in rock masses. Geotech Geolog Eng 28(5):643–659

    Google Scholar 

  • Deng XF, Zhu JB, Chen SG, Zhao J (2012) Some fundamental issues and verification of 3DEC in modeling wave propagation in jointed rock masses. Rock Mech Rock Eng 45:943–951

    Google Scholar 

  • Desai CS, Zaman MM, Lightner JG, Siriwardane HJ (1984) Thin-layer element for interfaces and joints. Int J Numer Anal Methods Geomech 8(1):19–43

    Google Scholar 

  • Duarte MT, Liu HY, Kou SQ, Lindqvist PA, Miskovsky K (2005) Microstructural modeling approach applied to rock material. J Mater Eng Perform 14(1):104–111

    Google Scholar 

  • Fang Z, Harrison JP (2002) Application of a local degradation model to the analysis of brittle fracture of laboratory scale rock specimens under triaxial conditions. Int J Rock Mech Min Sci 39(4):459–476

    Google Scholar 

  • Gehle C, Kutter HK (2003) Breakage and shear behaviour of intermittent rock joints. Int J Rock Mech Min Sci 40(5):687–700

    Google Scholar 

  • Goodman RE, Taylor RL, Brekke TL (1968) A model for the mechanics of jointed rocks. J Soil Mech Found Div 94

  • Grange S, Forquin P, Mencacci S, Hild F (2008) On the dynamic fragmentation of two limestones using edge on impact test. Int J Impact Eng 35(9):977–991

    Google Scholar 

  • He L, Tian Q, Zhao Z, Zhao X, Zhang Q, Zhao J (2018) Rock slope stability and stabilization analysis using the coupled DDA and FEM method: NDDA approach. Int J Geomech 18(6):04018044

    Google Scholar 

  • Horn HM, Deere DU (1962) Frictional characteristics of minerals. Geotechnique 12(4):319–335

    Google Scholar 

  • Jaeger JC, Cook NG, Zimmerman R (2009) Fundamental of rock mechanics. Wiley, New York

    Google Scholar 

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

    Google Scholar 

  • Ju Y, Xie H, Zheng Z, Lu J, Mao L, Gao F, Peng RD (2014) Visualization of the complex structure and stress field inside rock by means of 3D printing technology. Chin Sci Bull 59(36):5354–5365

    Google Scholar 

  • Ju Y, Wang L, Xie H, Ma G, Zheng Z, Mao L (2017) Visualization and transparentization of the structure and stress field of aggregated geomaterials through 3D printing and photoelastic techniques. Rock Mech Rock Eng 50(6):1383–1407

    Google Scholar 

  • Kachanov L (2013) Introduction to continuum damage mechanics, vol 10. Springer, Berlin

    MATH  Google Scholar 

  • Katona MG (1983) A simple contact–friction interface element with applications to buried culverts. Int J Numer Analy Meth Geomech 7(3):371–384

    MATH  Google Scholar 

  • Kawakata H, Cho A, Kiyama T, Yanagidani T, Kusunose K, Shimada M (1999) Three-dimensional observations of faulting process in Westerly granite under uniaxial and triaxial conditions by X-ray CT scan. Tectonophysics 313(3):293–305

    Google Scholar 

  • Lan H, Martin CD, Hu B (2010) Effect of heterogeneity of brittle rock on micromechanical extensile behavior during compression loading. J Geophys Res Solid Earth 115:B01202

    Google Scholar 

  • Li QM, Meng H (2003) Further investigation on the dynamic compressive strength enhancement of concrete-like materials based on split Hopkinson pressure bar tests. Int J Impact Eng 28(5):537–555

    Google Scholar 

  • Li G, Tang CA (2015) A statistical meso-damage mechanical method for modeling trans-scale progressive failure process of rock. Int J Rock Mech Min Sci 74:133–150

    Google Scholar 

  • Li LC, Tang CA, Li G, Wang SY, Liang ZZ, Zhang YB (2012) Numerical simulation of 3D hydraulic fracturing based on an improved flow-stress-damage model and a parallel FEM technique. Rock Mech Rock Eng 45(5):801–818

    Google Scholar 

  • Li JC, Li HB, Zhao J (2015a) An improved equivalent viscoelastic medium method for wave propagation across layered rock masses. Int J Rock Mech Min Sci 73(1):62–69

    Google Scholar 

  • Li W, Bai J, Cheng J, Peng S, Liu H (2015b) Determination of coal–rock interface strength by laboratory direct shear tests under constant normal load. Int J Rock Mech Mini Sci 77:60–67

    Google Scholar 

  • Li FX, Yang YM, Fan X, Xu B, Ju Y, Wang YL, Li Y, Chen JL (2018) Numerical analysis of the hydrofracturing behaviours and mechanisms of heterogeneous reservoir rock using the continuum-based discrete element method considering pre-existing fractures. Geomech Geophys Geo-Energy Geo-resour 4(4):383–401

    Google Scholar 

  • Li H, Deng JH, Yin JH, Zhu JB (2020) Modelling the shearing behaviour of joints using an improved shear box genesis approach in particle flow code (2D) and its validation. Geomech Geophys Geo-energy Geo-resour 6(1):15

    Google Scholar 

  • Liang ZZ, Tang CA, Li HX, Xu T, Zhang YB (2004) Numerical simulation of 3-d failure process in heterogeneous rocks. Int J Rock Mech Min Sci 41:323–328

    Google Scholar 

  • Liang ZZ, Xing H, Wang SY, Williams DJ, Tang CA (2012) A three-dimensional numerical investigation of the fracture of rock specimens containing a pre-existing surface flaw. Comput Geotech 45:19–33

    Google Scholar 

  • Liao ZY (2018) Numerical investigation on rock mechanical characteristic of rocks under dynamic loadings. Ph.D. thesis. Dalian University of Technology, China. (in Chinese)

  • Liao ZY, Zhu JB, Xia KW, Tang CA (2016a) Determination of dynamic compressive and tensile behavior of rocks from numerical tests of split Hopkinson pressure and tension bars. Rock Mech Rock Eng 49:3917–3934

    Google Scholar 

  • Liao ZY, Zhu JB, Tang CA (2016b) A numerical study on dynamic responses of underground tunnel in jointed rock masses. RocDyn-2, Suzhou, China

  • Lin CT, Amadei B, Jung J, Dwyer J (1996) Extensions of discontinuous deformation analysis for jointed rock mass. Int J Rock Mech Min Sci Geomech Abstr 33(7):671–694

    Google Scholar 

  • Liu JB, Yao L, Wang D (1994) A method for calculating the dynamic effect of dynamic and static friction on a contactable crack. Acta Mech Sin 26(4):494–502 (in Chinese)

    Google Scholar 

  • Liu XL, Han GF, Wang EZ, Wang SJ, Nawnit K (2018) Multiscale hierarchical analysis of rock mass and prediction of its mechanical and hydraulic properties. J Rock Mech Geotech Eng 10:694–702

    Google Scholar 

  • Lockner D, Byerlee JD, Kuksenko V, Ponomarev A, Sidorin A (1991) Quasi-static fault growth and shear fracture energy in granite. Nature 350:39–42

    Google Scholar 

  • Ma K, Liu GY, Guo LJ, Zhuang DY, Collins DS (2020) Deformation and stability of a discontinuity-controlled rock slope at Dagangshan hydropower station using three-dimensional discontinuous deformation analysis. Int J Rock Mech Min Sci 130:104313

    Google Scholar 

  • Mahabadi OK, Lisjak A, Munjiza A, Grasselli G (2012) Y-Geo: new combined finite-discrete element numerical code for geomechanical applications. Int J Geomech 12(6):676–688

    Google Scholar 

  • Munjiza A (2004) The combined finite-discrete element method. Wiley, Hoboken

    MATH  Google Scholar 

  • Nemat SN, Horii H (1982) Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst. J Geophys Res Solid Earth 87:6805–6822

    Google Scholar 

  • Olsson R, Barton N (2001) An improved model for hydromechanical coupling during shearing of rock joints. Int J Rock Mech Min Sci 38(3):317–329

    Google Scholar 

  • Pyrak-Nolte LG, Lr Myer, Cook NGW (1990) Transmission of seismic waves across single natural fractures. J Geophys Res Solid Earth 95(B6):8617–8638

    Google Scholar 

  • Ramana YV, Gogte BS (1989) Dependence of coefficient of sliding friction in rocks on lithology and mineral characteristics. Eng Geol 26(3):271–279

    Google Scholar 

  • Schellart WP (2000) Shear test results for cohesion and friction coefficients for different granular materials: scaling implications for their usage in analogue modelling. Tectonophysics 324(1):1–16

    Google Scholar 

  • Shi GH (1988) Discontinuous deformation analysis—a new model for the statics and dynamics of block systems. Ph.D. thesis, Univ. of California, Berkeley, CA

  • Shi F, Wang XL, Liu C, Liu H, Wu HG (2017) An XFEM-based method with reduction technique for modeling hydraulic fracture propagation in formations containing frictional natural fractures. Eng Frac Mech 173(64):90

    Google Scholar 

  • Siavelis M, Guiton M, Massin P, Moës N (2013) Large sliding contact along branched discontinuities with X-FEM. Comput Mech 52(1):201–219

    MathSciNet  MATH  Google Scholar 

  • Talesnick ML (2007) Determination of shear interface parameters between rock blocks for centrifuge modeling. Rock Mech Rock Eng 40(4):405–418

    Google Scholar 

  • Tang CA (1997) Numerical simulation of progressive rock failure and associated seismicity. Int J Rock Mech Min Sci 34(2):249–261

    MathSciNet  Google Scholar 

  • Tang SB, Tang CA (2015) Crack propagation and coalescence in quasi-brittle materials at high temperatures. Eng Fract Mech 134:404–432

    Google Scholar 

  • Tham LG, Liu H, Tang CA, Lee PKK, Tsui Y (2005) On tension failure of 2-D rock specimens and associated acoustic emission. Rock Mech Rock Eng 38(1):1–19

    Google Scholar 

  • Townend E, Thompson BD, Benson PM, Meredith PG, Baud P, Young RP (2008) Imaging compaction band propagation in Diemelstadt sandstone using acoustic emission locations. Geophys Res Lett 35:L15301

    Google Scholar 

  • Wang YN, Zhao MH, Li SH, Wang JG (2005) Stochastic structural model for rock and soil aggregates by continuum-based discrete element method. Eng Mater Sci 48:95–106

    Google Scholar 

  • Wang ZL, Li YC, Wang JG (2007) A damage-softening statistical constitutive model considering rock residual strength. Comput Geosci 33(1):1–9

    Google Scholar 

  • Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18(3):293–297

    MATH  Google Scholar 

  • Wong TF, Wong RH, Chau KT, Tang CA (2006) Microcrack statistics, Weibull distribution and micromechanical modeling of compressive failure in rock. Mech Mater 38(7):664–681

    Google Scholar 

  • Wu JH, Juang CH, Lin HM (2005) Vertex-to-face contact searching algorithm for three-dimensional frictionless contact problems. Int J Numer Method Eng 63(6):876–897

    MATH  Google Scholar 

  • Xia YJ, Zhang CQ, Zhou H, Hou J, Su GS, Gao Y, Liu N, Singh HK (2020) Mechanical behavior of structurally reconstructed irregular columnar jointed rock mass using 3D printing. Eng Geol 268:105509

    Google Scholar 

  • Xie HP, Zhu JB, Zhou T, Zhang K, Zhou CT (2020) Conceptualization and preliminary study of engineering disturbed rock dynamics. Geomech Geophys Geo-Energy Geo-resour 6(2):34

    Google Scholar 

  • Xing HL, Makinouchi A (2003) Finite element modelling of frictional instability between deformable rocks. Int J Numer Anal Methods Geomech 27(12):1005–1025

    MATH  Google Scholar 

  • Xu XH, Ma SP, Xia MF, Ke FJ, Bai YL (2005) Synchronous multi-scale observations on rock damage and rupture. Theor Appl Fract Mech 44(2):146–156

    Google Scholar 

  • Yang SQ, Yang DS, Jing HW, Li YH, Wang SY (2011) An experimental study of the fracture coalescence behaviour of brittle sandstone specimens containing three fissures. Rock Mech Rock Eng 45(4):563–582

    Google Scholar 

  • Yang SQ, Ranjith PG, Huang YH, Yin PF, Jing HW, Gui YL, Yu QL (2015a) Experimental investigation on mechanical damage characteristics of sandstone under triaxial cyclic loading. Geophys J Int 201(2):662–682

    Google Scholar 

  • Yang YF, Li G, Liang ZZ, Tang CA (2015b) Numerical investigation on crack branching during collision for rock-like material. Theor Appl Fract Mech 76:35–49

    Google Scholar 

  • Yu Y, Wang EZ, Zhong JW, Liu XL, Li PH, Shi M, Zhang ZG (2014) Stability analysis of abutment slopes based on long-term monitoring and numerical simulation. Eng Geol 183:159–169

    Google Scholar 

  • Yu QL, Zhu WC, Ranjith PG, Shao SS (2018) Numerical simulation and interpretation of the grain size effect on rock strength. Geomech Geophys Geo-Energy Geo-resour 4(2):157–173

    Google Scholar 

  • Zencker U, Clos R (1999) Limiting conditions for compression testing of flat specimens in the split Hopkinson pressure bar. Exp Mech 39(4):343–348

    Google Scholar 

  • Zhang QB, Zhao J (2014) A review of dynamic experimental techniques and mechanical behaviour of rock materials. Rock Mech Rock Eng 47(4):1411–1478

    Google Scholar 

  • Zhang ZH, Deng JH, Zhu JB, Li LR (2018) An experimental investigation of failure mechanisms of jointed and intact marbles under compression based on quantitative analysis of waveforms from acoustic emissions. Rock Mech Rock Eng 51(7):2299–2307

    Google Scholar 

  • Zhao J, Li HB (2000) Experimental determination of dynamic tensile properties of a granite. Int J Rock Mech Min Sci 37(5):861–866

    Google Scholar 

  • Zhao XB, Zhu JB, Zhao J, Cai JG (2012) Study of wave attenuation across parallel fractures using propagator matrix method. J Numer Anal Methods Geomech 36:1264–1279

    Google Scholar 

  • Zheng F, Jiao YY, Leung YF, Zhu JB (2018) Algorithmic robustness for contact analysis of polyhedral blocks in discontinuous deformation analysis framework. Comput Geotech 104:288–301

    Google Scholar 

  • Zhou T, Zhu JB (2018) Identification of a suitable 3D printing material for mimicking brittle and hard rocks and its brittleness enhancements. Rock Mech Rock Eng 51(3):765–777

    Google Scholar 

  • Zhou T, Zhu JB, Ju Y, Xie HP (2019) Volumetric fracturing behavior of 3D printed artificial rocks containing single and double 3D internal flaws under static uniaxial compression. Eng Fract Mech 205:190–204

    Google Scholar 

  • Zhou T, Zhu JB, Xie HP (2020) Mechanical and volumetric fracturing behavior of three-dimensional printing rock-like samples under dynamic loading. Rock Mech Rock Eng. https://doi.org/10.1007/s00603-020-02084-5

    Article  Google Scholar 

  • Zhu BF (1979) The finite element method theory and applications. China Water&Power Press, Beijing (in Chinese)

    Google Scholar 

  • Zhu WC, Tang CA (2006) Numerical simulation of Brazilian disk rock failure under static and dynamic loading. Int J Rock Mech Min Sci 43(2):236–252

    Google Scholar 

  • Zhu JB, Zhao J (2013) Obliquely incident wave propagation across rock joints with virtual wave source method. J Appl Geophy 88:23–30

    Google Scholar 

  • Zhu JB, Perino A, Zhao GF, Barla G, Li JC, Ma GW, Zhao J (2011) Seismic response of a single and a set of filled joints of viscoelastic deformational behaviour. Geophy J Int 186:1315–1330

    Google Scholar 

  • Zhu JB, Liao ZY, Tang CA (2016) Numerical SHPB tests of rocks under combined static and dynamic loading conditions with application to dynamic behavior of rocks under In situ stresses. Rock Mech Rock Eng 49:3935–3946

    Google Scholar 

  • Zhu JB, Zhou T, Liao ZY, Sun L, Li XB, Chen R (2018) Replication of internal defects and investigation of mechanical and fracture behaviour of rocks using 3D printing and 3D numerical methods in combination with X-ray computerized tomography. Int J Rock Mech Min Sci 106:198–212

    Google Scholar 

  • Zhu JB, Ren M, Liao ZY (2020) Wave propagation and diffraction through non-persistent rock joints: an analytical and numerical study. Int J Rock Mech Min Sci 132:104362

    Google Scholar 

  • Zienkiewicz OC (1977) The finite element method. McGraw-Hill, New York

    MATH  Google Scholar 

  • Zienkiewicz OC, Pande GN (1977) Time dependent multi-laminate model of rocks—a numerical study of deformation and failure of rock masses. Int J Numer Anal Meth Geomech 1:219–247

    Google Scholar 

Download references

Acknowledgements

We thank L. Zhang from Dalian University of Technology for providing laboratory measurements. This research was financially supported by the Department of Science and Technology of Guangdong Province (No. 2019ZT08G315) and the National Natural Science Foundation of China (Nos. 51627804 and 51904189).

Author information

Authors and Affiliations

  1. Guangdong Provincial Key Laboratory of Deep Earth Sciences and Geothermal Energy Exploitation and Utilization, Institute of Deep Earth Sciences and Green Energy, College of Civil and Transportation Engineering, Shenzhen University, Shenzhen, 518060, China

    Zhiyi Liao & Jianbo Zhu

  2. State Key Laboratory of Hydraulic Engineering Simulation and Safety, School of Civil Engineering, Tianjin University, Tianjin, China

    Meng Ren & Jianbo Zhu

  3. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China

    Zhiyi Liao & Chunan Tang

Authors
  1. Zhiyi Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Meng Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chunan Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jianbo Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianbo Zhu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liao, Z., Ren, M., Tang, C. et al. A three-dimensional damage-based contact element model for simulating the interfacial behaviors of rocks and its validation and applications. Geomech. Geophys. Geo-energ. Geo-resour. 6, 45 (2020). https://doi.org/10.1007/s40948-020-00171-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40948-020-00171-z

Keywords

Search

Navigation