平台巴西劈裂试验确定岩石抗拉强度的理论分析
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摘要
为了完善平台巴西劈裂试验测定岩石抗拉强度的理论基础,基于二维弹性理论,建立对弦载荷下的平台巴西劈裂力学模型,采用应力场叠加法求得圆盘内的应力近似解析解。该应力理论解与有限元数值解一致,证明理论求解的合理性。在此基础上,对比研究了平台加载角对圆盘内应力大小及应力集中程度的影响,其结果表明:平台加载角越大,圆盘加载处的应力集中程度和压拉应力比都急剧降低,而圆盘内的拉应力值和拉伸区却只有一定程度减小;过大或过小的平台加载角都不利于平台巴西试样发生中心拉伸劈裂破坏,其最优平台加载角在20°~30°之间。最后,依据Griffith强度破坏准则,推得岩石抗拉强度的理论计算公式,其结果与已有抗拉强度经验公式及试验所得抗拉强度相符较好。
This study aims to improve the fundamental of theory of the flattened Brazilian disk test for determining the tensile strength of rocks. On the basis of the two-dimensional elasticity theory, a mechanical model is proposed with the chordally opposing distributed compressive loads. And the approximate analytical solution of stress components inside the Brazilian disk is obtained with the superposition method of stresses. The theoretical solutions are validated to be rational through comparing with the numerical results from the finite element method. By using the solutions, the effect of flattened loading angles on the value and the concentration degree of stress inside the disk is researched contrastively. The results show that the concentration degree of stress near loading point and the ratio of compressive stress to tensile stress inside the disk quickly decrease, but the tensile stress value and tensile region slightly reduce with the increasing of flattened loading angle. Further, it is found that both too large or too small loading angle is detrimental to the central tensile splitting failure of specimen in the standard Brazilian disk test, and the optimal flattened loading angles are between 20° and 30°. Finally, according to the Griffith strength failure criterion, the formula for tensile strength of rocks is theoretically derived. And the theoretical tensile strength values obtained in this paper are in good agreement with the existing experimental ones.
This study aims to improve the fundamental of theory of the flattened Brazilian disk test for determining the tensile strength of rocks. On the basis of the two-dimensional elasticity theory, a mechanical model is proposed with the chordally opposing distributed compressive loads. And the approximate analytical solution of stress components inside the Brazilian disk is obtained with the superposition method of stresses. The theoretical solutions are validated to be rational through comparing with the numerical results from the finite element method. By using the solutions, the effect of flattened loading angles on the value and the concentration degree of stress inside the disk is researched contrastively. The results show that the concentration degree of stress near loading point and the ratio of compressive stress to tensile stress inside the disk quickly decrease, but the tensile stress value and tensile region slightly reduce with the increasing of flattened loading angle. Further, it is found that both too large or too small loading angle is detrimental to the central tensile splitting failure of specimen in the standard Brazilian disk test, and the optimal flattened loading angles are between 20° and 30°. Finally, according to the Griffith strength failure criterion, the formula for tensile strength of rocks is theoretically derived. And the theoretical tensile strength values obtained in this paper are in good agreement with the existing experimental ones.
引文
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[2]SATOCH Y.Position and load of failure in Brazilian test a numerical analysis by Griffith criterion[J].Journal of the Society of Materials Science,Japan,1987,36(410):1219-1224.
[3]王启智,贾学明.用平台巴西圆盘试样确定脆性岩石的弹性模量,拉伸强度和断裂韧度——第一部分:解析和数值结果[J].岩石力学与工程学报,2002,21(9):1285-1289.WANG Qi-zhi,JIA Xue-ming.Determination of elastic modulus tensile strength and fracture toughness of brittle rocks by using flattened Brazilian disk specimen—Part I:Analytical and numerical results[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(9):1285-1289.
[4]MA C-C,HUNG K-M.Exact full-field analysis of strain and displacement for circular disks subjected to partially distributed compressions[J].International Journal of Mechanical Sciences,2008,50(2):275-292.
[5]YU Y,ZHANG J,ZHANG J.A modified Brazilian disk tension test[J].International Journal of Rock Mechanics and Mining Sciences,2009,46(2):421-425.
[6]MARKIDES C F,PAZIS D,KOURKOULIS S.Closed full-field solutions for stresses and displacements in the Brazilian disk under distributed radial load[J].International Journal of Rock Mechanics and Mining Sciences,2010,47(2):227-237.
[7]ERARSLAN N,LIANG Z Z,WILLIAMS D J.Experimental and numerical studies on determination of indirect tensile strength of rocks[J].Rock Mechanics and Rock Engineering,2012,45(5):739-751.
[8]ERARSLAN N,WILLIAMS D J.Experimental,numerical and analytical studies on tensile strength of rocks[J].International Journal of Rock Mechanics and Mining Sciences,2012,49:21-30.
[9]KOURKOULIS S,MARKIDES C F,CHATZISTERGOS P.The Brazilian disc under parabolically varying load:theoretical and experimental study of the displacement field[J].International Journal of Solids and Structures,2012,49(7):959-972.
[10]MARKIDES C F,KOURKOULIS S.The stress field in a standardized Brazilian disc:The influence of the loading type acting on the actual contact length[J].Rock Mechanics and Rock Engineering,2012,45(2):145-158.
[11]KOURKOULIS S,MARKIDES C F,CHATZISTERGOS P.The standardized Brazilian disc test as a contact problem[J].International Journal of Rock Mechanics and Mining Sciences,2013,57:132-141.
[12]WANG Q Z,JIA X M,KOU S Q,et al.The flattened Brazilian disc specimen used for testing elastic modulus,tensile strength and fracture toughness of brittle rocks:analytical and numerical results[J].International Journal of Rock Mechanics and Mining Sciences,2004,41(2):245-253.
[13]WANG Q Z,XING L.Determination of fracture toughness KIC by using the flattened Brazilian disk specimen for rocks[J].Engineering fracture mechanics,1999,64(2):193-201.
[14]王启智,吴礼舟.用平台巴西圆盘试样确定脆性岩石的弹性模量,拉伸强度和断裂韧度——第二部分:试验结果[J].岩石力学与工程学报,2004,23(2):199-204.WANG Qi-zhi,WU Li-zhou.Determination of elastic modulus tensile strength and fracture toughness of brittle rocks by using flattened Brazilian disk specimen—Part II:Analytical and numerical results[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(2):199-204.
[15]尤明庆,苏承东.平台巴西圆盘劈裂和岩石抗拉强度的试验研究[J].岩石力学与工程学报,2004,23(18):3106-3122.YOU Ming-qing,SU Cheng-dong.Experimental study on split test with flattened disk and tensile strength of rock[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(18):3106-3112.
[16]尤明庆,苏承东.平台圆盘劈裂的理论和试验[J].岩石力学与工程学报,2004,23(1):170-174.YOU Ming-qing,SU Cheng-dong.Split test of flattened rock disk and related theory[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(1):170-174.
[17]于庆磊,唐春安,杨天鸿,等.平台中心角对岩石抗拉强度测定影响的数值分析[J].岩土力学,2008,29(12):3251-3255.YU Qing-lei,TANG Chun-an,YANG Tian-hong,et al.Numerical analysis of influence of central angle of flats on tensile strength of granite in split test with flattened disk[J].Rock and Soil Mechanics,2008,29(12):3251-3255.
[18]孟京京,曹平,张科,等.基于颗粒流的平台圆盘巴西劈裂和岩石抗拉强度[J].中南大学学报(自然科学版),2013,44(6):2499-2454.MENG Jing-jing,CAO Ping,ZHANG Ke,et al.Brazil split test of flattened disk and rock tensile strength using particle flow code[J].Journal of Central South University(Science and Technology),2013,44(6):2499-2454.
[19]喻勇,徐跃良.采用平台巴西圆盘试样测试岩石抗拉强度的方法[J].岩石力学与工程学报,2006,25(7):1457-1462.YU Yong,XU Yue-liang.Method to determine tensile of rock using flattened Brazilian disk[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(7):1457-1462.
[20]徐芝纶.弹性力学(第四版)[M].北京:高等教育出版社,2006:54-85.
[21]VAN CAUWELAERT F,ECKMANN B.Indirect tensile test applied to anisotropic materials[J].Materials and Structures,1994,27(1):54-60.
[22]JIANHONG Y,WU F,SUN J.Estimation of the tensile elastic modulus using Brazilian disc by applying diametrically opposed concentrated loads[J].International Journal of Rock Mechanics and Mining Sciences,2009,46(3):568-576.
[23]刘继国,曾亚武.岩石试件端面摩擦效应数值模拟研究[J].工程地质学报,2005,13(2):247-251.LIU Ji-guo,ZENG Ya-wu.Numerical simulation of the end frictional effect of rock specimens[J].Journal of Engineering Geology,2005,13(2):247-251.
[24]GRIFFITH A.The phenomena of rupture and flow in solids[J].Philosophical Transactions of the Royal Society of London:Series A,Containing Papers of a Mathematical or Physical Character,1921,221:163-198.
[2]SATOCH Y.Position and load of failure in Brazilian test a numerical analysis by Griffith criterion[J].Journal of the Society of Materials Science,Japan,1987,36(410):1219-1224.
[3]王启智,贾学明.用平台巴西圆盘试样确定脆性岩石的弹性模量,拉伸强度和断裂韧度——第一部分:解析和数值结果[J].岩石力学与工程学报,2002,21(9):1285-1289.WANG Qi-zhi,JIA Xue-ming.Determination of elastic modulus tensile strength and fracture toughness of brittle rocks by using flattened Brazilian disk specimen—Part I:Analytical and numerical results[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(9):1285-1289.
[4]MA C-C,HUNG K-M.Exact full-field analysis of strain and displacement for circular disks subjected to partially distributed compressions[J].International Journal of Mechanical Sciences,2008,50(2):275-292.
[5]YU Y,ZHANG J,ZHANG J.A modified Brazilian disk tension test[J].International Journal of Rock Mechanics and Mining Sciences,2009,46(2):421-425.
[6]MARKIDES C F,PAZIS D,KOURKOULIS S.Closed full-field solutions for stresses and displacements in the Brazilian disk under distributed radial load[J].International Journal of Rock Mechanics and Mining Sciences,2010,47(2):227-237.
[7]ERARSLAN N,LIANG Z Z,WILLIAMS D J.Experimental and numerical studies on determination of indirect tensile strength of rocks[J].Rock Mechanics and Rock Engineering,2012,45(5):739-751.
[8]ERARSLAN N,WILLIAMS D J.Experimental,numerical and analytical studies on tensile strength of rocks[J].International Journal of Rock Mechanics and Mining Sciences,2012,49:21-30.
[9]KOURKOULIS S,MARKIDES C F,CHATZISTERGOS P.The Brazilian disc under parabolically varying load:theoretical and experimental study of the displacement field[J].International Journal of Solids and Structures,2012,49(7):959-972.
[10]MARKIDES C F,KOURKOULIS S.The stress field in a standardized Brazilian disc:The influence of the loading type acting on the actual contact length[J].Rock Mechanics and Rock Engineering,2012,45(2):145-158.
[11]KOURKOULIS S,MARKIDES C F,CHATZISTERGOS P.The standardized Brazilian disc test as a contact problem[J].International Journal of Rock Mechanics and Mining Sciences,2013,57:132-141.
[12]WANG Q Z,JIA X M,KOU S Q,et al.The flattened Brazilian disc specimen used for testing elastic modulus,tensile strength and fracture toughness of brittle rocks:analytical and numerical results[J].International Journal of Rock Mechanics and Mining Sciences,2004,41(2):245-253.
[13]WANG Q Z,XING L.Determination of fracture toughness KIC by using the flattened Brazilian disk specimen for rocks[J].Engineering fracture mechanics,1999,64(2):193-201.
[14]王启智,吴礼舟.用平台巴西圆盘试样确定脆性岩石的弹性模量,拉伸强度和断裂韧度——第二部分:试验结果[J].岩石力学与工程学报,2004,23(2):199-204.WANG Qi-zhi,WU Li-zhou.Determination of elastic modulus tensile strength and fracture toughness of brittle rocks by using flattened Brazilian disk specimen—Part II:Analytical and numerical results[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(2):199-204.
[15]尤明庆,苏承东.平台巴西圆盘劈裂和岩石抗拉强度的试验研究[J].岩石力学与工程学报,2004,23(18):3106-3122.YOU Ming-qing,SU Cheng-dong.Experimental study on split test with flattened disk and tensile strength of rock[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(18):3106-3112.
[16]尤明庆,苏承东.平台圆盘劈裂的理论和试验[J].岩石力学与工程学报,2004,23(1):170-174.YOU Ming-qing,SU Cheng-dong.Split test of flattened rock disk and related theory[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(1):170-174.
[17]于庆磊,唐春安,杨天鸿,等.平台中心角对岩石抗拉强度测定影响的数值分析[J].岩土力学,2008,29(12):3251-3255.YU Qing-lei,TANG Chun-an,YANG Tian-hong,et al.Numerical analysis of influence of central angle of flats on tensile strength of granite in split test with flattened disk[J].Rock and Soil Mechanics,2008,29(12):3251-3255.
[18]孟京京,曹平,张科,等.基于颗粒流的平台圆盘巴西劈裂和岩石抗拉强度[J].中南大学学报(自然科学版),2013,44(6):2499-2454.MENG Jing-jing,CAO Ping,ZHANG Ke,et al.Brazil split test of flattened disk and rock tensile strength using particle flow code[J].Journal of Central South University(Science and Technology),2013,44(6):2499-2454.
[19]喻勇,徐跃良.采用平台巴西圆盘试样测试岩石抗拉强度的方法[J].岩石力学与工程学报,2006,25(7):1457-1462.YU Yong,XU Yue-liang.Method to determine tensile of rock using flattened Brazilian disk[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(7):1457-1462.
[20]徐芝纶.弹性力学(第四版)[M].北京:高等教育出版社,2006:54-85.
[21]VAN CAUWELAERT F,ECKMANN B.Indirect tensile test applied to anisotropic materials[J].Materials and Structures,1994,27(1):54-60.
[22]JIANHONG Y,WU F,SUN J.Estimation of the tensile elastic modulus using Brazilian disc by applying diametrically opposed concentrated loads[J].International Journal of Rock Mechanics and Mining Sciences,2009,46(3):568-576.
[23]刘继国,曾亚武.岩石试件端面摩擦效应数值模拟研究[J].工程地质学报,2005,13(2):247-251.LIU Ji-guo,ZENG Ya-wu.Numerical simulation of the end frictional effect of rock specimens[J].Journal of Engineering Geology,2005,13(2):247-251.
[24]GRIFFITH A.The phenomena of rupture and flow in solids[J].Philosophical Transactions of the Royal Society of London:Series A,Containing Papers of a Mathematical or Physical Character,1921,221:163-198.