Numerical analyses of pillar behavior with variation in yield criterion,dilatancy, rock heterogeneity and length to width ratio
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摘要
With recent advances in numerical modeling, design of underground structures increasingly relies on numerical modeling-based analysis approaches. While modeling tools like the discrete element method(DEM) and the combined finite-discrete element method(FDEM) are useful for investigating small-scale damage processes, continuum models remain the primary practical tool for most field-scale problems.The results obtained from such models are significantly dependent on the selection of an appropriate yield criterion and dilation angle. Towards improving its capabilities in handling mining-related problems, the authors have previously developed a new yield criterion(called progressive S-shaped criterion). The focus of the current study is to demonstrate its use in modeling rock pillars through a comparative analysis against four other yield criteria. In addition to the progressive S-shaped criterion,only one out of the four other criteria predicted a trend in strength consistent with an empirical pillar strength database compiled from the literature. Given the closely-knit relationship between yield criteria and dilation angle in controlling the overall damage process, a separate comparison was conducted using a mobilized dilation model, a zero degree dilation angle and a constant non-zero dilation angle. This study also investigates the impact of meso-scale heterogeneity in mechanical properties on the overall model response by assigning probability distributions to the input parameters. The comparisons revealed that an isotropic model using a combination of progressive S-shaped criterion and mobilized dilation angle model is sufficient in capturing the behaviors of rock pillars. Subsequently, the pillar model was used to assess the effect of L/W(length/width) ratio on the peak strength.
With recent advances in numerical modeling, design of underground structures increasingly relies on numerical modeling-based analysis approaches. While modeling tools like the discrete element method(DEM) and the combined finite-discrete element method(FDEM) are useful for investigating small-scale damage processes, continuum models remain the primary practical tool for most field-scale problems.The results obtained from such models are significantly dependent on the selection of an appropriate yield criterion and dilation angle. Towards improving its capabilities in handling mining-related problems, the authors have previously developed a new yield criterion(called progressive S-shaped criterion). The focus of the current study is to demonstrate its use in modeling rock pillars through a comparative analysis against four other yield criteria. In addition to the progressive S-shaped criterion,only one out of the four other criteria predicted a trend in strength consistent with an empirical pillar strength database compiled from the literature. Given the closely-knit relationship between yield criteria and dilation angle in controlling the overall damage process, a separate comparison was conducted using a mobilized dilation model, a zero degree dilation angle and a constant non-zero dilation angle. This study also investigates the impact of meso-scale heterogeneity in mechanical properties on the overall model response by assigning probability distributions to the input parameters. The comparisons revealed that an isotropic model using a combination of progressive S-shaped criterion and mobilized dilation angle model is sufficient in capturing the behaviors of rock pillars. Subsequently, the pillar model was used to assess the effect of L/W(length/width) ratio on the peak strength.
With recent advances in numerical modeling, design of underground structures increasingly relies on numerical modeling-based analysis approaches. While modeling tools like the discrete element method(DEM) and the combined finite-discrete element method(FDEM) are useful for investigating small-scale damage processes, continuum models remain the primary practical tool for most field-scale problems.The results obtained from such models are significantly dependent on the selection of an appropriate yield criterion and dilation angle. Towards improving its capabilities in handling mining-related problems, the authors have previously developed a new yield criterion(called progressive S-shaped criterion). The focus of the current study is to demonstrate its use in modeling rock pillars through a comparative analysis against four other yield criteria. In addition to the progressive S-shaped criterion,only one out of the four other criteria predicted a trend in strength consistent with an empirical pillar strength database compiled from the literature. Given the closely-knit relationship between yield criteria and dilation angle in controlling the overall damage process, a separate comparison was conducted using a mobilized dilation model, a zero degree dilation angle and a constant non-zero dilation angle. This study also investigates the impact of meso-scale heterogeneity in mechanical properties on the overall model response by assigning probability distributions to the input parameters. The comparisons revealed that an isotropic model using a combination of progressive S-shaped criterion and mobilized dilation angle model is sufficient in capturing the behaviors of rock pillars. Subsequently, the pillar model was used to assess the effect of L/W(length/width) ratio on the peak strength.
引文
Alejano LR, Alonso E. Considerations of the dilatancy angle in rocks and rock masses. International Journal of Rock Mechanics and Mining Sciences2005;42(4):481-507.
Babcock C. Critique of pillar design equations from 1833 to 1990. Denver, CO:U.S.Department of the Interior, Bureau of Mines; 1994. IC 9398.
Bahrani N, Kaiser PK, Valley B. Distinct element method simulation of an analogue for a highly interlocked, non-persistently jointed rockmass. International Journal of Rock Mechanics and Mining Sciences 2014;71:117-30.
Castro L, McCreath D, Kaiser PK. Rockmass strength determination from breakouts in tunnels and boreholes. In:Proceedings of the 8th ISRM congress. Tokyo,Japan:International Society for Rock Mechanics and Rock Engineering; 1995.p. 531-6.
Chugh YP, Abbasi B. A numerical analysis of a four-way coal mine intersection with primary and secondary supports. In:Proceedings of the 12th ISRM congress.Beijing, China:International Society for Rock Mechanics and Rock Engineering;2012. p. 363-8.
Diederichs MS. Rock fracture and collapse under low confinement conditions. Rock Mechanics and Rock Engineering 2003;36:339-81.
Diederichs MS. The 2003 Canadian Geotechnical Colloquium:mechanistic interpretation and practical application of damage and spalling prediction criteria for deep tunnelling. Canadian Geotechnical Journal 2005;44(9):1082-116.
Diederichs MS, Martin CD. Measurement of spalling parameters from laboratory testing. In:Rock mechanics in civil and environmental engineering. London:Taylor&Francis Group; 2010. p. 323-6.
Dolinar DR, Esterhuizen GS. Evaluation of the effect of length on the strength of slender pillars in limestone mines using numerical modeling. In:Proceedings of the 26th international conference on ground control mining. Morgantown, WV:West Virginia University; 2007. p. 304-13.
Edelbro C. Numerical modelling of observed fallouts in hard rock masses using an instantaneous cohesion-softening friction-hardening model. Tunnelling and Underground Space Technology 2009;24(4):398-409.
Edelbro C. Different approaches for simulating brittle failure in two hard rock mass cases:a parametric study. Rock Mechanics and Rock Engineering 2010;43(2):151-65.
Esterhuizen GS. Combined point estimate and Monte Carlo techniques for the analysis of wedge failure in rock slopes. In:Statics and dynamic consideration in rock engineering. Rotterdam:A.A. Balkema; 1990. p. 125-32.
Farahmand K, Diederichs MS. A calibrated synthetic rock mass(SRM)model for simulating crack growth in granitic rock considering grain scale heterogeneity of polycrystalline rock. In:Proceedings of the 49th U.S. rock mechanics/geomechanics symposium. San Francisco, California:American Rock Mechanics Association; 2015. ARMA-2015-430.
Ghazvinian E, Diederichs MS, Quey R. 3D random Voronoi grain-based models for simulation for brittle rock damage and fabric-guided micro-fracturing. Journal of Rock Mechanics and Geotechnical Engineering 2014;6(6):506-21.
Grau RH, Krog RB, Robertson SB. Maximizing the ventilation of large-opening mines. In:Proceedings of the 11th US/North American mine ventilation symposium. PA:University Park; 2006. p. 1-7.
Guo S, Qi S, Zou Y, Zheng B. Numerical studies on the failure process of heterogeneous brittle rocks or rock-like materials under uniaxial compression. Materials2017;10(4):378.
Hajiabdolmajid V, Kaiser PK, Martin CD. Modelling brittle failure of rock. International Journal of Rock Mechanics and Mining Sciences 2002;39(6):731-41.
Hedley DGF, Grant F. Stope and pillar design for the Elliot Lake uranium mines. The Canadian Mining and Metallurgical Bulletin 1972;65:1165-86.
Hoek E, Carranza-Torres C, Corkum B. Hoek-Brown failure criterion-2002 edition. In:Proceedings of NARMS-TAC conference, Toronto, Canada; 2002.p. 267-73.
Hsu S, Nelson PP. Material spatial variability and slope stability for weak rock masses. Journal of Geotechnical and Geoenvironmental Engineering2006;132(2):183-93.
Hudyma MR. Rib pillar design in open stope mining[MSc Thesis]. The University of British Columbia; 1988.
Iannacchione AT. Analysis of pillar design practices and techniques for US limestone mines. Transactions of the Institution of Mining&Metallurgy(Section A)1999;108:A152-60.
Itasca FLAC. 3D version 5.0:theory and background. Minneapolis, Minnesota, USA:Itasca Consulting Group; 2016.
Jaeger JC, Cook NGW. Fundamentals of rock mechanics. 3rd ed. London:Chapman&Hall; 1979.
Jing L, Stephansson O. Fundamental of discrete element methods for rock engineering:theory and application. Amsterdam, Netherland:Elsevier; 2007.
Kaiser PK, Kim B, Bewich RP, Valley B. Rock mass strength at depth and implication for pillar design. Mining Technology 2011; 120(3):170-9.
Kaiser PK, Kim B. Characterization of strength of intact brittle rock considering confinement-dependent failure processes. Rock Mechanics and Rock Engineering 2015;48(1):107-19.
Krauland N, Soder PE. Determining pillar strength from pillar failure observation.Engineering and Mining Journal 1987;188(8):34-40.
Lan H, Martin CD, Hu B. Effect of heterogeneity on brittle rock on micromechanical extensile behavior during compression testing. Journal of Geophysical Research:Solid Earth 2010;115(B1):14. https://doi.org/10.1029/2009JB006496.
Langford JC, Diederichs MS. Quantifying uncertainty in Hoek-Brown intact strength envelope. International Journal of Rock Mechanics and Mining Sciences2015;74:91-102.
Lisjak A, Grasselli G. A review of discrete modeling techniques for fracturing processes in discontinuous rock masses. Journal of Rock Mechanics and Geotechnical Engineering 2014;6(4):301-14.
Mark C. Pillar design methods for longwall mining. U.S. Department of the Interior,Bureau of Mines; 1990. p. 53. IC 9247.
Mark C, Chase FE. Analysis of retreat mining pillar stability(ARMPS). In:Proceedings of new technology for ground control in retreat mining. U.S. Department of Health and Human Services, National Institute for Occupational Safety and Health(NIOSH); 1997. p. 17-34. IC 9446.
Martin CD, Chandler NA. The progressive fracture of Lac Du Bonnet granite. International Journal of Rock Mechanics and Mining Sciences&Geomechanics Abstracts 1994;31(6):643-59.
Martin CD. Seventeenth Canadian Geotechnical Colloquium:the effect of cohesion loss and stress path on brittle rock strength. Canadian Geotechnical Journal1997;34(5):698-725.
Martin CD, Kaiser PK, McCreath DR. Hoek-Brown parameters for predicting the depth of brittle failure around tunnels. Canadian Geotechnical Journal1999;36(1):136-51.
Martin CD, Maybee WG. The strength of hard rock pillar. International Journal of Rock Mechanics and Mining Sciences 2000;37(8):1239-46.
Mayer JM, Stead D. Exploration into the causes of uncertainty in UDEC grain boundary models. Computers and Geotechnics 2017;82:110-23.
Mortazavi A, Hassani FP, Shabani M. A numerical investigation of rock pillar failure mechanism in underground openings. Computers and Geotechnics 2009;36(5):619-97.
Munjiza A. The combined finite-discrete element method. John Wiley and Sons,Inc.; 2004.
Nicksiar M, Martin CD. Evaluation of methods for determining crack initiation in compression tests on low-porosity rocks. Rock Mechanics and Rock Engineering2012;45(4):607-17.
Patton FD. Multiple modes of shear failure in rock. In:Proceedings of the 1st congress ISRM. Lisbon:ISRM; 1966. p. 509-13.
Pelli F, Kaiser PK, Morgenstrern NR. An interpretation of ground movements recorded during construction of the Donkin-Morien tunnel. Canadian Geotechnical Journal 1991;28(2):239-54.
Reinsalu E. Stochastic approach to room and pillar failure in oil shale mining.Proceedings of the Estonian Academy of Sciences 2000;6(3):207-16.
Renani HR, Martin CD. Modeling the progressive failure of hard rock. Tunnelling and Underground Space Technology 2018;74:71-81.
Ryder JA, Ozbay MU. A methodology for designing pillar layouts for shallow mining.In:International symposium on static and dynamic considerations in rock engineering. Swaziland:ISRM; 1990.
Schmertmann JH, Osterberg JO. An experimental study of the development of cohesion and friction with axial strain in saturated cohesive soils. Research conference on shear strength of cohesive soils. 1960. p. 643-94. Boulder, CO,USA.
Sinha S, Chugh YP. An evaluation of roof support plans at two coal mines in Illinois using numerical models. International Journal of Rock Mechanics and Mining Sciences 2016;82:1-9.
Sinha S, Walton G. Insight into hard rock pillar behavior from numerical simulation using a progressive S-shaped failure criterion. In:Proceedings of 51st US rockmechanics/geomechanics symposium. San Francisco, USA:American Rock Mechanics Association; 2017.
Sinha S, Walton G. A progressive S-shaped yield criterion and its application to the study of rock pillar behavior. International Journal of Rock Mechanics and Mining Sciences 2018;105:98-109.
Sjoberg J. Failure modes and pillar behavior in the Zinkgruvan mine. In:Proceedings of the 33rd U.S. rock mechanics symposium. American Rock Mechanics Association; 1992. p. 491-500.
Tang C. Numerical simulation of progressive rock failure and associated seismicity. International Journal of Rock Mechanics and Mining Sciences1997;34(2):249-61.
Valley B, Kim B, Suorineni FT, Bahrani N, Bewick RP. Influence of confinement dependent failure processes on rock mass strength at depth. In:Qian Q, Zhou Y,editors. ISRM 12th international congress on rock mechanics. London:Springer;2011. p. 855-60.
Vermeer PA, De Borst R. Non-associated plasticity for soils, concrete and rock.Heron 1984;29(3):1-64.
Wagner H. Determination of the complete load-deformation characteristics of coal pillars. In:Proceedings of the congress of the international society for rock mechanics. Denver, CO, USA:IRSM; 1974. p. 1067-81.
Wagner H. Pillar design in coal mines. Journal of the South African Institute of Mining&Metallurgy 1980;80(1):37-45.
Walton G. Improving continuum models for excavations in rock masses under high stress through an enhanced understanding of post-yield dilatancy[PhD Thesis].Kingston, Canada:Queen's University; 2014.
Walton G, Diederichs MS, Punkkinen A. The influence of constitutive model selection on predicted stresses and yield in deep mine pillars-a case study at the Creighton mine, Sudbury, Canada. Geomechanics and Tunneling 2015;8(5):441-9.
Walton G, Diederichs MS. A new model for the dilation of brittle rocks based on laboratory compression test data with separate treatment of dilatancy mobilization and decay. Geotechnical and Geological Engineering2015a;33(3):661-79.
Walton G, Diederichs MS. A mine shaft case study on the accurate prediction of yield and displacements in stressed ground using lab-derived material properties. Tunnelling and Underground Space Technology 2015b;49:98-113.
Walton G, Diederichs MS, Punkkinen A, Whitmore J. Back analysis of a pillar monitoring experiment at 2.4 km depth in the Sudbury Basin, Canada. International Journal of Rock Mechanics and Mining Sciences 2016;85:33-51.
Walton G, Hedayat A, Kim E, Labrie D. Post-yield strength and dilatancy evolution across the brittle-ductile transition in Indiana limestone. Rock Mechanics and Rock Engineering 2017;50(7):1691-710.
Wilson AH. The stability of underground workings in the soft rocks of coal measures. Geotechnical and Geological Engineering 1983; 1(2):91-187.
Yan C, Zheng H, Sun G, Ge X. Combined finite-discrete element method for simulation of hydraulic fracturing. Rock Mechanics and Rock Engineering2016;49(4):1389-410.
Zhao XG, Cai M. A mobilized dilation angle model for rocks. International Journal of Rock Mechanics and Mining Sciences 2010;47(3):368-84.
Zhao XG, Cai M, Cai M. Considerations of rock dilation on modeling failure and deformation of hard rocks-a case study of the mine-by test tunnel in Canada.Journal of Rock Mechanics and Geotechnical Engineering 2010;2(4):338-49.
Babcock C. Critique of pillar design equations from 1833 to 1990. Denver, CO:U.S.Department of the Interior, Bureau of Mines; 1994. IC 9398.
Bahrani N, Kaiser PK, Valley B. Distinct element method simulation of an analogue for a highly interlocked, non-persistently jointed rockmass. International Journal of Rock Mechanics and Mining Sciences 2014;71:117-30.
Castro L, McCreath D, Kaiser PK. Rockmass strength determination from breakouts in tunnels and boreholes. In:Proceedings of the 8th ISRM congress. Tokyo,Japan:International Society for Rock Mechanics and Rock Engineering; 1995.p. 531-6.
Chugh YP, Abbasi B. A numerical analysis of a four-way coal mine intersection with primary and secondary supports. In:Proceedings of the 12th ISRM congress.Beijing, China:International Society for Rock Mechanics and Rock Engineering;2012. p. 363-8.
Diederichs MS. Rock fracture and collapse under low confinement conditions. Rock Mechanics and Rock Engineering 2003;36:339-81.
Diederichs MS. The 2003 Canadian Geotechnical Colloquium:mechanistic interpretation and practical application of damage and spalling prediction criteria for deep tunnelling. Canadian Geotechnical Journal 2005;44(9):1082-116.
Diederichs MS, Martin CD. Measurement of spalling parameters from laboratory testing. In:Rock mechanics in civil and environmental engineering. London:Taylor&Francis Group; 2010. p. 323-6.
Dolinar DR, Esterhuizen GS. Evaluation of the effect of length on the strength of slender pillars in limestone mines using numerical modeling. In:Proceedings of the 26th international conference on ground control mining. Morgantown, WV:West Virginia University; 2007. p. 304-13.
Edelbro C. Numerical modelling of observed fallouts in hard rock masses using an instantaneous cohesion-softening friction-hardening model. Tunnelling and Underground Space Technology 2009;24(4):398-409.
Edelbro C. Different approaches for simulating brittle failure in two hard rock mass cases:a parametric study. Rock Mechanics and Rock Engineering 2010;43(2):151-65.
Esterhuizen GS. Combined point estimate and Monte Carlo techniques for the analysis of wedge failure in rock slopes. In:Statics and dynamic consideration in rock engineering. Rotterdam:A.A. Balkema; 1990. p. 125-32.
Farahmand K, Diederichs MS. A calibrated synthetic rock mass(SRM)model for simulating crack growth in granitic rock considering grain scale heterogeneity of polycrystalline rock. In:Proceedings of the 49th U.S. rock mechanics/geomechanics symposium. San Francisco, California:American Rock Mechanics Association; 2015. ARMA-2015-430.
Ghazvinian E, Diederichs MS, Quey R. 3D random Voronoi grain-based models for simulation for brittle rock damage and fabric-guided micro-fracturing. Journal of Rock Mechanics and Geotechnical Engineering 2014;6(6):506-21.
Grau RH, Krog RB, Robertson SB. Maximizing the ventilation of large-opening mines. In:Proceedings of the 11th US/North American mine ventilation symposium. PA:University Park; 2006. p. 1-7.
Guo S, Qi S, Zou Y, Zheng B. Numerical studies on the failure process of heterogeneous brittle rocks or rock-like materials under uniaxial compression. Materials2017;10(4):378.
Hajiabdolmajid V, Kaiser PK, Martin CD. Modelling brittle failure of rock. International Journal of Rock Mechanics and Mining Sciences 2002;39(6):731-41.
Hedley DGF, Grant F. Stope and pillar design for the Elliot Lake uranium mines. The Canadian Mining and Metallurgical Bulletin 1972;65:1165-86.
Hoek E, Carranza-Torres C, Corkum B. Hoek-Brown failure criterion-2002 edition. In:Proceedings of NARMS-TAC conference, Toronto, Canada; 2002.p. 267-73.
Hsu S, Nelson PP. Material spatial variability and slope stability for weak rock masses. Journal of Geotechnical and Geoenvironmental Engineering2006;132(2):183-93.
Hudyma MR. Rib pillar design in open stope mining[MSc Thesis]. The University of British Columbia; 1988.
Iannacchione AT. Analysis of pillar design practices and techniques for US limestone mines. Transactions of the Institution of Mining&Metallurgy(Section A)1999;108:A152-60.
Itasca FLAC. 3D version 5.0:theory and background. Minneapolis, Minnesota, USA:Itasca Consulting Group; 2016.
Jaeger JC, Cook NGW. Fundamentals of rock mechanics. 3rd ed. London:Chapman&Hall; 1979.
Jing L, Stephansson O. Fundamental of discrete element methods for rock engineering:theory and application. Amsterdam, Netherland:Elsevier; 2007.
Kaiser PK, Kim B, Bewich RP, Valley B. Rock mass strength at depth and implication for pillar design. Mining Technology 2011; 120(3):170-9.
Kaiser PK, Kim B. Characterization of strength of intact brittle rock considering confinement-dependent failure processes. Rock Mechanics and Rock Engineering 2015;48(1):107-19.
Krauland N, Soder PE. Determining pillar strength from pillar failure observation.Engineering and Mining Journal 1987;188(8):34-40.
Lan H, Martin CD, Hu B. Effect of heterogeneity on brittle rock on micromechanical extensile behavior during compression testing. Journal of Geophysical Research:Solid Earth 2010;115(B1):14. https://doi.org/10.1029/2009JB006496.
Langford JC, Diederichs MS. Quantifying uncertainty in Hoek-Brown intact strength envelope. International Journal of Rock Mechanics and Mining Sciences2015;74:91-102.
Lisjak A, Grasselli G. A review of discrete modeling techniques for fracturing processes in discontinuous rock masses. Journal of Rock Mechanics and Geotechnical Engineering 2014;6(4):301-14.
Mark C. Pillar design methods for longwall mining. U.S. Department of the Interior,Bureau of Mines; 1990. p. 53. IC 9247.
Mark C, Chase FE. Analysis of retreat mining pillar stability(ARMPS). In:Proceedings of new technology for ground control in retreat mining. U.S. Department of Health and Human Services, National Institute for Occupational Safety and Health(NIOSH); 1997. p. 17-34. IC 9446.
Martin CD, Chandler NA. The progressive fracture of Lac Du Bonnet granite. International Journal of Rock Mechanics and Mining Sciences&Geomechanics Abstracts 1994;31(6):643-59.
Martin CD. Seventeenth Canadian Geotechnical Colloquium:the effect of cohesion loss and stress path on brittle rock strength. Canadian Geotechnical Journal1997;34(5):698-725.
Martin CD, Kaiser PK, McCreath DR. Hoek-Brown parameters for predicting the depth of brittle failure around tunnels. Canadian Geotechnical Journal1999;36(1):136-51.
Martin CD, Maybee WG. The strength of hard rock pillar. International Journal of Rock Mechanics and Mining Sciences 2000;37(8):1239-46.
Mayer JM, Stead D. Exploration into the causes of uncertainty in UDEC grain boundary models. Computers and Geotechnics 2017;82:110-23.
Mortazavi A, Hassani FP, Shabani M. A numerical investigation of rock pillar failure mechanism in underground openings. Computers and Geotechnics 2009;36(5):619-97.
Munjiza A. The combined finite-discrete element method. John Wiley and Sons,Inc.; 2004.
Nicksiar M, Martin CD. Evaluation of methods for determining crack initiation in compression tests on low-porosity rocks. Rock Mechanics and Rock Engineering2012;45(4):607-17.
Patton FD. Multiple modes of shear failure in rock. In:Proceedings of the 1st congress ISRM. Lisbon:ISRM; 1966. p. 509-13.
Pelli F, Kaiser PK, Morgenstrern NR. An interpretation of ground movements recorded during construction of the Donkin-Morien tunnel. Canadian Geotechnical Journal 1991;28(2):239-54.
Reinsalu E. Stochastic approach to room and pillar failure in oil shale mining.Proceedings of the Estonian Academy of Sciences 2000;6(3):207-16.
Renani HR, Martin CD. Modeling the progressive failure of hard rock. Tunnelling and Underground Space Technology 2018;74:71-81.
Ryder JA, Ozbay MU. A methodology for designing pillar layouts for shallow mining.In:International symposium on static and dynamic considerations in rock engineering. Swaziland:ISRM; 1990.
Schmertmann JH, Osterberg JO. An experimental study of the development of cohesion and friction with axial strain in saturated cohesive soils. Research conference on shear strength of cohesive soils. 1960. p. 643-94. Boulder, CO,USA.
Sinha S, Chugh YP. An evaluation of roof support plans at two coal mines in Illinois using numerical models. International Journal of Rock Mechanics and Mining Sciences 2016;82:1-9.
Sinha S, Walton G. Insight into hard rock pillar behavior from numerical simulation using a progressive S-shaped failure criterion. In:Proceedings of 51st US rockmechanics/geomechanics symposium. San Francisco, USA:American Rock Mechanics Association; 2017.
Sinha S, Walton G. A progressive S-shaped yield criterion and its application to the study of rock pillar behavior. International Journal of Rock Mechanics and Mining Sciences 2018;105:98-109.
Sjoberg J. Failure modes and pillar behavior in the Zinkgruvan mine. In:Proceedings of the 33rd U.S. rock mechanics symposium. American Rock Mechanics Association; 1992. p. 491-500.
Tang C. Numerical simulation of progressive rock failure and associated seismicity. International Journal of Rock Mechanics and Mining Sciences1997;34(2):249-61.
Valley B, Kim B, Suorineni FT, Bahrani N, Bewick RP. Influence of confinement dependent failure processes on rock mass strength at depth. In:Qian Q, Zhou Y,editors. ISRM 12th international congress on rock mechanics. London:Springer;2011. p. 855-60.
Vermeer PA, De Borst R. Non-associated plasticity for soils, concrete and rock.Heron 1984;29(3):1-64.
Wagner H. Determination of the complete load-deformation characteristics of coal pillars. In:Proceedings of the congress of the international society for rock mechanics. Denver, CO, USA:IRSM; 1974. p. 1067-81.
Wagner H. Pillar design in coal mines. Journal of the South African Institute of Mining&Metallurgy 1980;80(1):37-45.
Walton G. Improving continuum models for excavations in rock masses under high stress through an enhanced understanding of post-yield dilatancy[PhD Thesis].Kingston, Canada:Queen's University; 2014.
Walton G, Diederichs MS, Punkkinen A. The influence of constitutive model selection on predicted stresses and yield in deep mine pillars-a case study at the Creighton mine, Sudbury, Canada. Geomechanics and Tunneling 2015;8(5):441-9.
Walton G, Diederichs MS. A new model for the dilation of brittle rocks based on laboratory compression test data with separate treatment of dilatancy mobilization and decay. Geotechnical and Geological Engineering2015a;33(3):661-79.
Walton G, Diederichs MS. A mine shaft case study on the accurate prediction of yield and displacements in stressed ground using lab-derived material properties. Tunnelling and Underground Space Technology 2015b;49:98-113.
Walton G, Diederichs MS, Punkkinen A, Whitmore J. Back analysis of a pillar monitoring experiment at 2.4 km depth in the Sudbury Basin, Canada. International Journal of Rock Mechanics and Mining Sciences 2016;85:33-51.
Walton G, Hedayat A, Kim E, Labrie D. Post-yield strength and dilatancy evolution across the brittle-ductile transition in Indiana limestone. Rock Mechanics and Rock Engineering 2017;50(7):1691-710.
Wilson AH. The stability of underground workings in the soft rocks of coal measures. Geotechnical and Geological Engineering 1983; 1(2):91-187.
Yan C, Zheng H, Sun G, Ge X. Combined finite-discrete element method for simulation of hydraulic fracturing. Rock Mechanics and Rock Engineering2016;49(4):1389-410.
Zhao XG, Cai M. A mobilized dilation angle model for rocks. International Journal of Rock Mechanics and Mining Sciences 2010;47(3):368-84.
Zhao XG, Cai M, Cai M. Considerations of rock dilation on modeling failure and deformation of hard rocks-a case study of the mine-by test tunnel in Canada.Journal of Rock Mechanics and Geotechnical Engineering 2010;2(4):338-49.