International Journal of Fracture

, Volume 211, Issue 1–2, pp 13–42 | Cite as

A coupled thermo-mechanical bond-based peridynamics for simulating thermal cracking in rocks

  • Yunteng Wang
  • Xiaoping ZhouEmail author
  • Miaomiao Kou
Original Paper


A coupled thermo-mechanical bond-based peridynamical (TM-BB-PD) method is developed to simulate thermal cracking processes in rocks. The coupled thermo-mechanical model consists of two parts. In the first part, temperature distribution of the system is modeled based on the heat conduction equation. In the second part, the mechanical deformation caused by temperature change is calculated to investigate thermal fracture problems. The multi-rate explicit time integration scheme is proposed to overcome the multi-scale time problem in coupled thermo-mechanical systems. Two benchmark examples, i.e., steady-state heat conduction and transient heat conduction with deformation problem, are performed to illustrate the correctness and accuracy of the proposed coupled numerical method in dealing with thermo-mechanical problems. Moreover, two kinds of numerical convergence for peridynamics, i.e., m- and \(\delta \)-convergences, are tested. The thermal cracking behaviors in rocks are also investigated using the proposed coupled numerical method. The present numerical results are in good agreement with the previous numerical and experimental data. Effects of PD material point distributions and nonlocal ratios on thermal cracking patterns are also studied. It can be found from the numerical results that thermal crack growth paths do not increases with changes of PD material point spacing when the nonlocal ratio is larger than 4. The present numerical results also indicate that thermal crack growth paths are slightly affected by the arrangements of PD material points. Moreover, influences of thermal expansion coefficients and inhomogeneous properties on thermal cracking patterns are investigated, and the corresponding thermal fracture mechanism is analyzed in simulations. Finally, a LdB granite specimen with a borehole in the heated experiment is taken as an application example to examine applicability and usefulness of the proposed numerical method. Numerical results are in good agreement with the previous experimental and numerical results. Meanwhile, it can be found from the numerical results that the coupled TM-BB-PD has the capacity to capture phenomena of temperature jumps across cracks, which cannot be captured in the previous numerical simulations.


Coupled thermo-mechanical model Bond-based peridynamics Numerical convergence Thermal cracking process Multi-rate explicit time integration scheme 



Authors would like to thank Dr. Shujun Peng from School of Naval Architecture, Ocean and Civil Engineering in Shanghai Jiao Tong University for helpful discussions. The present work is partially carried out with financial support from the National Natural Science Foundation of China (Grant Nos. 51325903 and 51679017), Natural Basic Research Program 973 of China (Grant No. 2014CB046903).


  1. Abdalla H (2006) Concrete cover requirements for FRP reinforced members in hot climates. Compos Struct 73:61–69CrossRefGoogle Scholar
  2. Agwai A (2011) A peridynamic approach for coupled fields. Ph. D. Dissertation, University of Arizona, Tucson, Arizona, USGoogle Scholar
  3. Amani J, Oterkus E, Areias P, Zi G, Nguyen-Thoi T, Rabczuk T (2016) A non-ordinary state-based peridynamics formulation for thermoplastic fracture. Int J Impact Eng 87:83–94CrossRefGoogle Scholar
  4. Bobaru F, Duangpanya M (2010) The peridynamic formulation for transient heat conduction. Int J Heat Mass Transf 53(19):4047–4059CrossRefGoogle Scholar
  5. Bobaru F, Duangpanya M (2012) A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities. J Comput Phys 231(7):2764–2785CrossRefGoogle Scholar
  6. Breitenfeld MS, Geubelle PH, Weckner O, Silling SA (2014) Non-ordinary state-based peridynamic analysis of stationary crack problems. Comput Methods Appl Mech Eng 272:233–350CrossRefGoogle Scholar
  7. Cai CZ, Li GS, Huang ZW, Shen ZH, Tian SC, Wei JW (2014) Experimental study of the effect of liquid nitrogen cooling on rock pore structure. J Nat Gas Sci Eng 24:507–517CrossRefGoogle Scholar
  8. Carlson SR, Jansen DP, Young RP (1993) Thermally induced fracturing of Lac du bonnet granite. Report RP020AECL. Eng Seismol Lab, Queen’s Univ, Kingston, Canada, pp 1–13Google Scholar
  9. Chen YL, Ni J, Shao W, Azzam R (2012) Experimental study on the influence of temperature on the mechanical properties of granite under uniaxial compression and fatigue loading. Int J Rock Mech Min Sci 56(15):62–66Google Scholar
  10. Cheng Z, Zhang G, Wang Y, Bobaru F (2015) A peridynamic model for dynamic fracture in functionally graded materials. Compos Struct 133:529–546CrossRefGoogle Scholar
  11. Cheng Z, Liu Y, Zhao J, Feng H, Wu Y (2018) Numerical simulation of crack propagation and branching in functionally graded materials using peridynamic modeling. Eng Fract Mech 191:13–32. CrossRefGoogle Scholar
  12. Cruz CR, Gillen M (1980) Thermal expansion of Portland cement paste, mortar and concrete at high temperatures. Fire Mater 4(2):66–70CrossRefGoogle Scholar
  13. David C, Menendez B, Darot M (1999) Influence of stress-induced and thermal cracking on physical properties and microstructure of la peyratte granite. Int J Rock Mech Min Sci 36(4):433–448Google Scholar
  14. D’Antuono P, Morandini M (2017) Thermal shock response via weakly coupled peridynamic thermo-mechanics. Int J Solids Struct 129:74–89CrossRefGoogle Scholar
  15. Dipasquale D, Sarego G, Zaccariotto M, Galvanetto U (2016) Dependence of crack paths on the orientation of regular 2D peridynamic grids. Eng Fract Mech 160:248–263CrossRefGoogle Scholar
  16. Fan H, Bergel GL, Li S (2016) A hybrid peridynamics-SPH simulation of soil fragmentation by blast loads of buried explosive. Int J Impact Eng 87:14–27CrossRefGoogle Scholar
  17. Fan H, Li S (2017) A Peridynamics-SPH modeling and simulation of blast fragmentation of soil under buried explosive loads. Comput Methods Appl Mech Eng 318:349–381CrossRefGoogle Scholar
  18. Feng Y, Han K, Li C, Owen D (2008) Discrete thermal element modelling of heat conduction in particle systems: basic formulations. J Comput Phys 227:5072–5089CrossRefGoogle Scholar
  19. Foster JT, Silling SA, Chen WW (2010) Viscoplasticity using peridynamics. Int J Numer Methods Eng 81:1242–1258Google Scholar
  20. Ghajari M, Iannucci L, Curtis PA (2014) A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media. Comput Methods Appl Mech Eng 276:431–452CrossRefGoogle Scholar
  21. Ghassemi A (2012) A review of some rock mechanics issues in geothermal reservoir development. Geotech Geol Eng 30(3):647–664CrossRefGoogle Scholar
  22. Giannopoulos GI, Anifantis NK (2005) Thermal fracture interference: a two dimensional boundary element approach. Int J Fract 132(4):351–369CrossRefGoogle Scholar
  23. Gu X, Zhang Q, Xia X (2017) Voronoi-based peridynamics and cracking analysis with adaptive refinement. Int J Numer Methods Eng 112(13):2087–2109CrossRefGoogle Scholar
  24. Ha YD, Bobaru F (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162(1–2):229–244CrossRefGoogle Scholar
  25. Ha YD, Bobaru F (2011) Characteristics of dynamic brittle fracture captured with peridynamics. Eng Fract Mech 78:1156–1168CrossRefGoogle Scholar
  26. Ha YD, Lee J, Hong JW (2015) Fracturing patterns of rock-like materials in compression captured with peridynamics. Eng Fract Mech 144:176–193CrossRefGoogle Scholar
  27. Han F, Lubineau G, Azdoud Y, Askari A (2016a) A morphing approach to couple state-based peridynamics with classical continuum mechanics. Comput Methods Appl Mech Eng 301:336–358CrossRefGoogle Scholar
  28. Han F, Lubineau G, Azdoud Y (2016b) Adaptive coupling between damage mechanics and peridynamics: a route for objective simulation of material degradation up to complete failure. J Mech Phys Solids 94:453–472CrossRefGoogle Scholar
  29. Heuze FE (1983) High-temperature mechanical, physical and thermal properties of granitic rocks—a review. Int J Rock Mech Min Sci 20(1):3–10CrossRefGoogle Scholar
  30. Henke SF, Shanbhag S (2014) Mesh sensitivity in peridynamic simulations. Comput Phys Commun 185(1):181–193CrossRefGoogle Scholar
  31. Homand-Etienne F, Houpert R (1989) Thermally induced microcracking granites: characterization and analysis. Int J Rock Mech Min Sci 26(2):125–134CrossRefGoogle Scholar
  32. Huang D, Lu GD, Qiao PZ (2015) An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis. Int J Mech Sci 94–95:111–122CrossRefGoogle Scholar
  33. Huang X, Tang SB, Tang CA, Xie LM, Tao ZY (2017) Numerical simulation of cracking behavior in artificially designed rock models subjected to heating from a central borehole. Int J Rock Mech Min Sci 98:191–202CrossRefGoogle Scholar
  34. Jansen DP, Carlson SR, Young RP, Hutchins DA (1993) Ultrasonic imaging and acoustic emission monitoring of thermally induced microcracks in Lac du Bonnet granite. J Geophys Res 98(B12):22231–22243CrossRefGoogle Scholar
  35. Jackson R, Lau JSO, Annor A (1999) Mechanical, thermo-mechanical and joint properties of rock samples from the site of AECL’s underground research laboratory, Lac du bonnet, Manitoba. Can Geotech Conf, Winnipeg, Can Geotech Soc, pp 41–49Google Scholar
  36. Jiao YY, Zhang XL, Zhang HQ, Li HB, Yang SQ, Li JC (2015) A coupled thermo-mechanical discontinuum model for simulating rock cracking induced by temperature stresses. Comput Geotech 67:142–149CrossRefGoogle Scholar
  37. Kilic B, Madenci E (2009) Prediction of crack paths in a quenched glass plate by using peridynamic theory. Int J Fract 156(2):165–177CrossRefGoogle Scholar
  38. Kilic B, Madenci E (2010) An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theor Appl Fract Mech 53:194–204CrossRefGoogle Scholar
  39. Kilic B, Agwai A, Madenci E (2009) Peridynamic theory for progressive damage prediction in center-cracked composite laminates. Compos Struct 90:141–151CrossRefGoogle Scholar
  40. Kwon S, Cho W (2008) The influence of an excavation damaged zone on the thermalmechanical and hydro-mechanical behaviors of an underground excavation. Eng Geol 101:110–123CrossRefGoogle Scholar
  41. Lan H, Martin CD, Andersson JC (2013) Evolution of in situ rock mass damage induced by mechanical-thermal loading. Rock Mech Rock Eng 46:153–168CrossRefGoogle Scholar
  42. Lee J, Ha YD, Hong JW (2017a) Crack coalescence morphology in rock-like material under compression. Int J Fract 203(1–2):211–236CrossRefGoogle Scholar
  43. Lee J, Hong JW, Jung JW (2017b) The mechanism of fracture coalescence in pre-cracked rock-type material with three flaws. Eng Geol 223:31–47CrossRefGoogle Scholar
  44. Mahmutoglu Y (1998) Mechanical behaviour of cyclically heated fine grained rock. Rock Mech Rock Eng 31:169–179CrossRefGoogle Scholar
  45. Ngo M, Brancherie D, Ibrahimbegovic A (2014) Softening behavior of quasi-brittle material under full thermo-mechanical coupling condition: theoretical formulation and finite element implementation. Comput Methods Appl Mech Eng 281:1–28CrossRefGoogle Scholar
  46. Ni T, Zhu QZ, Zhao LY, Li PF (2017) Peridynamic simulation of fracture in quasi brittle solids using irregular finite element mesh. Eng Fract Mech. Google Scholar
  47. Oterkus S, Madenci E, Agwai A (2014a) Peridynamic thermal diffusion. J Comput Phys 265:71–96CrossRefGoogle Scholar
  48. Oterkus S, Madenci E, Agwai A (2014b) Fully coupled peridynamic thermomechanics. J Mech Phys Solids 64:1–23CrossRefGoogle Scholar
  49. Oterkus S, Madenci E, Oterkus E (2017) Fully coupled poroelastic peridynamic formulation for fluid-filled fractures. Eng Geol 225:19–28CrossRefGoogle Scholar
  50. Rabczuk T, Ren H (2017) A peridynamics formulation for quasi-static fracture and contact in rock. Eng Geol 225:42–48CrossRefGoogle Scholar
  51. Ren H, Zhuang X, Rabczuk T (2016) Dual-horizon peridynamic. Int J Numer Methods Eng 108(12):1451–1476CrossRefGoogle Scholar
  52. Ren H, Zhuang X, Rabczuk T (2017) Dual-horizon peridynamics: a stable solution to varying horizons. Comput Methods Appl Mech Eng 318:762–782CrossRefGoogle Scholar
  53. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209CrossRefGoogle Scholar
  54. Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17):1526–1535CrossRefGoogle Scholar
  55. Silling SA, Epton M, Weckne O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88:151–184CrossRefGoogle Scholar
  56. Silling SA, Lehoucq RB (2010) Peridynamic theory of solid mechanics. Adv Appl Mech 44:73–168CrossRefGoogle Scholar
  57. Shen B, Kim H, Park E, Kim T, Wuttke M, Rinne M, Backers T, Stephansson O (2013) Multi-region boundary element analysis for coupled thermal-fracturing processes in geomaterials. Rock Mech Rock Eng 46(1):135–151CrossRefGoogle Scholar
  58. Shojaei A, Mudric T, Zaccariotto M, Galvanetto U (2016) A coupled meshless finite point/Peridynamic method for 2D dynamic fracture analysis. Int J Mech Sci 119:419–431CrossRefGoogle Scholar
  59. Shojaei A, Zaccariotto M, Galvanetto U (2017a) Coupling of 2D discretized peridynamics with a meshless method based on classical elasticity using switching of nodal behaviour. Eng Comput 34(5):1334–1366CrossRefGoogle Scholar
  60. Shojaei A, Mossaiby F, Zaccariotto M, Galvanetto U (2017b) The meshless finite point method for transient elastodynamic problems. Acta Mech 228(10):3581–3593CrossRefGoogle Scholar
  61. Tang SB, Tang CA, Zhu WC, Wang SH, Yu QL (2006) Numerical investigation on rock failure process induced by thermal stress. Chin J Rock Mech Eng 25(10):2071–2078Google Scholar
  62. Tang SB, Zhang H, Tang CA, Liu HY (2016) Numerical model for the cracking behavior of heterogeneous brittle solids subjected to thermal shock. Int J Solids Struct 80:520–531CrossRefGoogle Scholar
  63. Tavallali A, Vervoort A (2010) Effect of layer orientation on the failure of layered sandstone under Brazilian test conditions. Int J Rock Mech Min Sci 47(2):313–322CrossRefGoogle Scholar
  64. Tomac I, Gutierrez M (2015) Formulation and implementation of coupled forced heat convection and heat conduction in DEM. Acta Geotech 10(4):421–433CrossRefGoogle Scholar
  65. Tupek MR, Rimoli JJ, Radovitzky R (2013) An approach for incorporating classical continuum damage models in state-based peridynamics. Comput Methods Appl Mech Eng 263(24):20–26CrossRefGoogle Scholar
  66. Vervoort A, Min K, Konietzky H, Cho J, Debecker B, Dinh Q, Fruhwirt T, Tavallali A (2014) Failure of transversely isotropic rock under Brazilian test condition. Int J Rock Mech Min Sci 70:343–352Google Scholar
  67. Vishal V, Pradhan SP, Singh TN (2011) Tensile strength of rock under elevated temperature. Geotech Geol Eng 29:1127–1133CrossRefGoogle Scholar
  68. Wang Y, Zhou X, Xu X (2016) Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics. Eng Fract Mech 163:248–273CrossRefGoogle Scholar
  69. Wang Y, Zhou X, Shou Y (2017) The modeling of crack propagation and coalescence in rocks under uniaxial compression using the novel conjugated bond-based peridynamics. Int J Mech Sci 128–129:614–643CrossRefGoogle Scholar
  70. Wang Y, Zhou X, Wang Y, Shou Y (2018) A 3-D conjugated bond-pair-based peridynamic formulation for initiation and propagation of cracks in brittle solids. Int J Solids Struct 134:89–115CrossRefGoogle Scholar
  71. Wanne TS, Young RP (2008) Bonded-particle modeling of thermally fractured granite. Int J Rock Mech Min Sci 45(5):789–799CrossRefGoogle Scholar
  72. Weckner O, Abeyaratne R (2005) The effect of long-range forces on the dynamics of a bar. J Mech Phys Solids 53:705–728CrossRefGoogle Scholar
  73. Weckner O, Emmrich E (2005) Numerical simulation of the dynamics of a non-local inhomogeneous, infinite bar. J Comput Appl Mech 6:311–319Google Scholar
  74. Wei CH, Zhu WC, Yu QL, Xu T, Jeon S (2015) Numerical simulation of excavation damaged zone under coupled thermal-mechanical conditions with varying mechanical parameters. Int J Rock Mech Min Sci 75:169–181Google Scholar
  75. Wisetsaen S, Walsri C, Fuenkajorn K (2015) Effects of loading rate and temperature on tensile strength and deformation of rock salt. Int J Rock Mech Min Sci 73:10–14Google Scholar
  76. Xia M, Zhao C, Hobbs BE (2014) Particle simulation of thermally- induced rock damage with consideration of temperature-dependent elastic modulus and strength. Comput Geotech 55(1):461–473CrossRefGoogle Scholar
  77. Xu X, Gao F, Shen X, Xie H (2008) Mechanical characteristics and microcosmic mechanism of granite under temperature loads. J China Univ Min Technol 18:413–441CrossRefGoogle Scholar
  78. Yaghoobi A, Chorzepa MG (2015) Meshless modeling framework for fiber reinforced concrete structures. Comput Struct 161:43–54CrossRefGoogle Scholar
  79. Yaghoobi A, Chorzepa MG (2017) Fracture analysis of fiber reinforced concrete structures in the micropolar peridynamic analysis framework. Eng Fract Mech 169:238–250CrossRefGoogle Scholar
  80. Yan C, Zheng H (2017) A coupled thermo-mechanical model based on the combined finite-discrete element method for simulating thermal cracking of rocks. Int J Rock Mech Min Sci 91:170–178Google Scholar
  81. Zhang ZX, Yu J, Kou SQ, Lindqvist P (2001) Effects of high temperatures on dynamic rock fracture. Int J Rock Mech Min Sci 38(2):211–225CrossRefGoogle Scholar
  82. Zhang G, Le Q, Loghin A, Subramaniyan A, Bobaru F (2016) Validation of a peridynamic model for fatigue cracking. Eng Fract Mech 162:76–94CrossRefGoogle Scholar
  83. Zhang H, Qiao PZ (2017) An extended state-based peridynamic model for damage growth prediction of bimaterial structures under thermomechanical loading. Eng Fract Mech. Google Scholar
  84. Zhu QZ, Ni T (2017) Peridynamic formulations enriched with bond rotation effects. Int J Eng Sci 121:118–129CrossRefGoogle Scholar
  85. Zhou XP, Wang YT, Xu XM (2016) Numerical simulation of initiation, propagation and coalescence of cracks using the non-ordinary state-based peridynamics. Int J Fract 201(2):213–234CrossRefGoogle Scholar
  86. Zhou XP, Wang YT (2016) Numerical simulation of crack propagation and coalescence in pre-cracked rock-like Brazilian disks using the non-ordinary state-based peridynamics. Int J Rock Mech Min Sci 89:235–249Google Scholar
  87. Zhou XP, Wang YT, Shou YD, Kou MM (2018) A novel conjugated bond linear elastic model in bond-based peridynamics for fracture problems under dynamic loads. Eng Fract Mech 188:151–183. CrossRefGoogle Scholar
  88. Zhou XP, Bi J (2018) Numerical simulation of thermal cracking in rocks based on general particle dynamics. J Eng Mech 144(1):04017156. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Coal Mine Disaster Dynamics and ControlChongqing UniversityChongqingPeople’s Republic of China
  2. 2.Institute of Geology and Underground Engineering, School of Civil EngineeringChongqing UniversityChongqingPeople’s Republic of China
  3. 3.Key Laboratory of New Technology for Construction of Cities in Mountain AreaChongqing UniversityChongqingPeople’s Republic of China

Personalised recommendations