白云岩受压声学特性及其在裂缝研究中的应用
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摘要
通过实验手段研究了白云岩在三轴和单轴加载过程中的声波波速、幅度、频谱特性的变化特征.结果发现:(1)白云岩在受压变形过程中,声波波速的变化较好地反映了岩石内部裂缝的闭合、产生、扩展和贯穿等变化.总体而言,横波波速比纵波能更好地预测裂缝的产生,而纵波对裂缝的非稳定发展或贯穿有较敏感的反映.(2)随着轴压的增大,纵、横波波形的末端都有散射波的出现,且横波波形末端的"鱼尾"状散射波信号更为明显(此时应力点为极限强度的60%左右),预示着岩石内部裂缝的产生和稳定扩展.(3)频谱曲线也很好地反映了岩石内部结构变形情况.随着岩石受力的增大,频谱曲线上的振幅都呈现增大趋势,标志着岩石的压实阶段;当频谱曲线上低频段较高频段活跃时,标志着裂缝的产生;更有甚者,低频还会取代高频成为主频.(4)在岩石受压裂缝闭合阶段,首波振幅和频谱主振幅都呈现上升趋势;而在裂缝不稳定扩展阶段,主振幅表现出比首波振幅低的上升趋势;振幅曲线达到峰值后都存在一突降拐点,预示着岩石的贯通破坏.研究对于地层岩石的动态长期监测和工程岩体的稳定性评价都具有重要的理论参考价值.
The variation characteristics of acoustic velocity, amplitude and spectral characteristics of dolomite under triaxial and uniaxial loading are studied by means of the experimental method. The following conclusions can be drawn.(1) During the process of compression deformation of the dolomite, the change of acoustic wave velocity well reflects the closure, generation, expansion and penetration of cracks in the rock. In general, the shear acoustic wave velocity is better than the longitudinal acoustis wave to predict the generation of cracks, while the longitudinal acoustic wave has a more sensitive reflection on the unstable expansion of the cracks.(2) With the axial load increases, the scattered wave signal appears at the end of the longitudinal acoustic wave and transverse wave pattern. And the "fishtail" scattered wave signal at the end of the shear waveform is more obvious(at this point, the axial stress reaches 60% of the limit strength), which indicates the generation and stability expansion of the crack in the rock.(3) The spectral curve is also a good reflection to the deformation of the internal structure in the dolomite rock. As the axial load increases, the amplitude of the spectrum curve shows an increasing trend, which indicates the compaction stage of the rock; when the low frequency signal is more active than the high frequency one in frequency spectrum curve, it marks the occurrence of the crack; what is more, low frequency will replace the high frequency to become the main frequency.(4) In the stage of crack closure, both the first arrival wave amplitude and the principal amplitude tend to increase; while in the stage of unsteady expansion of the crack,the principal amplitude tends to increase more slowly than the first arrival wave amplitude; after the amplitude curve reaches the peak there is a sudden drop point, which indicates the destruction of the rock. The variable characteristic of acoustic propagation displayed during the process of rock pressed makes a good sense to the prediction of rock crack and the evaluation of rock stability. This study has important theoretical reference value for the dynamic long-term monitoring to the rock deformation and the stability evaluation of engineering rock mass.
The variation characteristics of acoustic velocity, amplitude and spectral characteristics of dolomite under triaxial and uniaxial loading are studied by means of the experimental method. The following conclusions can be drawn.(1) During the process of compression deformation of the dolomite, the change of acoustic wave velocity well reflects the closure, generation, expansion and penetration of cracks in the rock. In general, the shear acoustic wave velocity is better than the longitudinal acoustis wave to predict the generation of cracks, while the longitudinal acoustic wave has a more sensitive reflection on the unstable expansion of the cracks.(2) With the axial load increases, the scattered wave signal appears at the end of the longitudinal acoustic wave and transverse wave pattern. And the "fishtail" scattered wave signal at the end of the shear waveform is more obvious(at this point, the axial stress reaches 60% of the limit strength), which indicates the generation and stability expansion of the crack in the rock.(3) The spectral curve is also a good reflection to the deformation of the internal structure in the dolomite rock. As the axial load increases, the amplitude of the spectrum curve shows an increasing trend, which indicates the compaction stage of the rock; when the low frequency signal is more active than the high frequency one in frequency spectrum curve, it marks the occurrence of the crack; what is more, low frequency will replace the high frequency to become the main frequency.(4) In the stage of crack closure, both the first arrival wave amplitude and the principal amplitude tend to increase; while in the stage of unsteady expansion of the crack,the principal amplitude tends to increase more slowly than the first arrival wave amplitude; after the amplitude curve reaches the peak there is a sudden drop point, which indicates the destruction of the rock. The variable characteristic of acoustic propagation displayed during the process of rock pressed makes a good sense to the prediction of rock crack and the evaluation of rock stability. This study has important theoretical reference value for the dynamic long-term monitoring to the rock deformation and the stability evaluation of engineering rock mass.
引文
1 Rudajev V,Vilhelm J,Lokajicek T.Laboratory studies of acoustic emission prior to uniaxial compressive rock failure.International Journal of Rock Mechanics&Mining Sciences,2000,37(4):699-7042 Thaml LG,Liu H,Tang CA,et a1.On tension failure of 2-D rock specimens and associated acoustic emission.Rock Mechanics&Rock Engineering,2005,38(1):1-19
3 Ganne P,Vervoort A,Wevess M.Quantification of pre-break brittle damage correlation between acoustic emission and observed mierofraeture.International Journal of Rock Mechanics&Mining Sciences,2007,44(5):720-729
4裴建良,刘建锋,左建平等.基于声发射定位的自然裂隙动态演化过程研究.岩石力学与工程学报,2013,32(4):696-704(Pei Jianliang,Liu Jianfeng,Zuo Jianping,et al.Investigation on dynamic evolution process of natural fractures based on acoustic emission position.Chinese Journal of Rock Mechanics and Engineering,2013,32(4):696-704(in Chinese))
5杨永杰,王德超,郭明福等.基于三轴压缩声发射试验的岩石损伤特征研究.岩石力学与工程学报,2014,33(1):98-104(Yang Yongjie,Wang Dechao,Guo Mingfu,et al.Study of rock damage characteristics based on acoustic emission tests under triaxial compression.Chinese Journal of Rock Mechanics and Engineering,2014,33(1):98-104(in Chinese))
6赵星光,李鹏飞,马利科等.循环加、卸载条件下北山深部花岗岩损伤与扩容特性.岩石力学与工程学报,2014,33(9):1740-1748(Zhao Xingguang,Li Pengfei,Ma Like,et al.Damage and dilation characteristics of deep granite at Beishan under cyclic loadingunloading conditions.Chinese Journal of Rock Mechanics and Engineering,2014,33(9):1740-1748(in Chinese))
7赵小平,陈淑芬.基于声发射振幅分布的裂隙岩体破坏演化过程.岩石力学与工程学报,2015,34(增1):3012-3017(Zhao Xiaoping,Chen Shufen.Failure evolution prosess of fractured rockmass based on amplitude of acoustic emission.Chinese Journal of Rock Mechanics and Engineering,2015,34(S1):3012-3017(in Chinese))
8 Xie HP,Liu JF,Ju Y,et al.Fractal property of spatial distribution of acoustic emissions during the failure process of bedded rock salt.International Journal of Rock Mechanics&Mining Sciences,2011,48(8):1344-1351
9武晋文,赵阳升,万志军等.高温均匀压力花岗岩热破裂声发射特性实验研究.煤炭学报,2012,37(7):1111-1117(Wu Jinwen,Zhao Yangsheng,Wan Zhijun,et al.Experimental study of acoustic emission of granite due to thermal cracking under high temperature and isostatic stress.Journal of China Coal Society,2012,37(7):1111-1117(in Chinese))
10曾鹏,刘阳军,纪洪广等.单轴压缩下粗砂岩临界破坏的多频段声发射耦合判据和前兆识别特征.岩土工程学报,2017,39(3):509-
517(Zeng Peng,Liu Yangjun,Ji Hongguang,et al.Coupling criteria and precursor identification characteristics of multi-band acoustic emission of gritstone fracture under uniaxial compression.Chinese Journal of Geotechnical Engineering,2017,39(3):509-517(in Chinese))
11刘建坡,徐世达,李元辉等.预制孔岩石破坏过程中的声发射时空演化特征研究.岩石力学与工程学报,2012,31(12):2538-2547(Liu Jianpo,Xu Shida,Li Yuanhui,et al.Studies of AE timespace evolution characteristics during failure process of rock specimens with prefabricated holes.Chinese Journal of Rock Mechanics and Engineering,2012,31(12):2538-2547(in Chinese))
12 Kim J,Lee K,Cho W,et al.A comparative evaluation of stressstrain and acoustic emission methods for quantitative damage assessments of brittle rock.Rock Mechanics and Rock Engineering,2015,48(2):495-508
13 Zhang Z,Zhang R,Xie H,et al.Differences in the acoustic emission characteristics of rock salt compared with granite and marble during the damage evolution process.Environmental Earth Sciences,2015,73(11):6987-6999
14 Nur A.Effects of stress on velocity anisotropy in rocks with cracks.Journal of Geophysical Research,1971,76(8):2022-2034
15 Le Pichon A.Analysis of infrasonic waves generated by the rupture of the Sumatra Mw=9.0 in the Indian ocean.Geophysical Research Abstracts,2005
16 Chou HT,Chang YL,Zhang SC.Acoustic signals and geophone response of rainfall-induced debris flows.Journal of the Chinese Institute of Engineers,2013,36(3):335-347
17 Ashraf MA,Carl HS,Chandra SR.Effect of cementation on ultrasonic velocities in sandstones.Geophysics,2007,72(2):E53-E58
18尤明庆,苏承东,李小双.损伤岩石试样的力学特性与纵波速度关系研究.岩石力学与工程学报,2008,27(3):458-467(You Mingqing,Su Chengdong,Li Xiaoshuang.Study on relation between mechanical properties and longitudinal wave velocities for damaged rock samples.Chinese Journal of Rock Mechanics and Engineering,2008,27(3):458-467(in Chinese))
19刘维国,单钰铭,刘荣和.砂岩扩容过程中超声波衰减的实验研究.成都理工大学学报(自然科学版),2006,33(6):611-616(Liu Weiguo,Shan Yuming,Liu Ronghe.Study on ultrasonic attenuation in the process of rock dilatancy by amplitude spectrum ratio.Journal of Chengdu University of Technology(Science and Technology),2006,33(6):611-616(in Chinese))
20陈乔,刘向君,刘洪等.层理性页岩地层超声波透射实验.天然气工业,2013,33(8):140-144(Chen Qiao,Liu Xiangjun,Liu Hong,et al.An experimental study of ultrasonic penetration througu bedding shale reservoir.Natural Gas Industry,2013,33(8):140-144(in Chinese))
21熊健,梁利喜,刘向君等.川南地区龙马溪组页岩岩石声波透射实验研究.地下空间与工程学报,2014,10(5):1071-1077(Xiong Jian,Liang Lixi,Liu Xiangjun,et al.Experimental study on acoustic penetration through the Longmaxi formation shale rock in south region of Sichuan Basin.Chinese Journal of Underground Space and Engineering,2014,10(5):1071-1077(in Chinese))
22 King MS.Elastic wave propagation in and permeability for rocks with multiple parallel fractures.International Journal of Rock Mechanics and Mining Sciences,2002,39(8):1033-1043
23 Petru?álek M,Vilhelm J,Rudajev V,et al.Determination of the anisotropy of elastic waves monitored by a sparse sensor network.International Journal of Rock Mechanics and Mining Sciences,2013,60:208-216
24 Chow TM,Meglis IL,Young RP.Progressive microcrack development in tests on Lac du Bonnet granite-II.Ultrasonic tomographic imaging.International Journal of Rock Mechanics and Mining Sciences&Geomechanics Abstracts,1995,32(8):751-761
25 Ougier-Simonin A,Fortin J,Gu′eguen Y,et al.Cracksin glass under triaxial conditions.International Journal of Engineering Science,2011,49(1):105-121
26张国凯,李海波,夏祥等.岩石波速与损伤演化规律研究.岩石力学与工程学报,2015,34(11):2270-2277(Zhang Guokai,Li Haibo,Xia Xiang,et al.Wave velocity and damage development of rock.Chinese Journal of Rock Mechanics and Engineering,2015,34(11):2270-2277(in Chinese))
27张国凯,李海波,夏祥等.单轴加载条件下花岗岩声发射及波传播特性研究.岩石力学与工程学报,2017,36(5):1133-1144(Zhang Guokai,Li Haibo,Xia Xiang,et al.Experiment study on acoustic emission and wave propagation in granite under uniaxial compression.Chinese Journal of Rock Mechanics and Engineering,2017,36(5):1133-1144(in Chinese))
28张晓平,王思敬,刘泉声.一种测量岩石压缩裂纹扩展过程声波波速的连续测量方法.工程地质学报,2018,26(1):91-96(Zhang Xiaoping,Wang Sijing,Liu Quansheng.A continuing measure method for acoustic wave velocity measurement during cracking processes of rock.Journal of Engineering Geology,2018,26(1):91-96(in Chinese))
29 Yang SQ,Jing HW.Strength failure and crack coalescence behavior of brittle sandstone samples containing a single fissure under uniaxial compression.International Journal of Fracture,2011,168(2):227-250
30 Lokaj′??cek T,Goel RK,Rudajev V,et al.Assessment of velocity anisotropy in rocks.International Journal of Rock Mechanics and Mining Sciences,2013,57:142-152
31尹国庆,张辉,袁芳等.白云岩储层地质力学特征分析及在储层改造优化应用--以塔中东部为例.天然气地球科学,2015,26(7):1277-1288(Yin Guoqing,Zhang Hui,Yuan Fang,et al.Geomechanical characteristics of dolomite reservoir and its application in stimulation optimization:An example of east of Tazhong.Natural Gas Geoscience,2015,26(7):1277-1288(in Chinese))
32林皋.地下结构地震响应的计算模型.力学学报,2017,49(3):528-542(Lin Gao.A computational model for seismic response analysis of underground stauctures.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):528-542(in Chinese))
33何川,齐春,封坤等.基于D-P准则的盾构隧道围岩与衬砌结构相互作用分析.力学学报,2017,49(1):31-40(He Chuan,Qi Chun,Feng Kun,et al.Theoretical analysis of interaction between surrounding rocks and linging strcture of shield tunnel based on Drucker-Prager yield criteria.Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):31-40(in Chinese))
34文龙飞,王理想,田荣.动载下裂纹应力强度因子计算的改进型扩展有限元法.力学学报,2018,50(3):599-610(Wen Longfei,Wang Lixiang,Tian Rong.Accurate computation on dynamic SIFs using improved XFEM.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):599-610(in Chinese))
35万义钊,刘曰武,吴能友等.基于离散裂缝的多段压裂水平井数值试井模型及应用.力学学报,2018,50(1):147-156(Wan Yizhao,Liu Yuewu,Wu Nengyou,et al.A numerical well test model for multi-fractured horizontal wells based on discrete-fracture model and its application.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):147-156(in Chinese))
3 Ganne P,Vervoort A,Wevess M.Quantification of pre-break brittle damage correlation between acoustic emission and observed mierofraeture.International Journal of Rock Mechanics&Mining Sciences,2007,44(5):720-729
4裴建良,刘建锋,左建平等.基于声发射定位的自然裂隙动态演化过程研究.岩石力学与工程学报,2013,32(4):696-704(Pei Jianliang,Liu Jianfeng,Zuo Jianping,et al.Investigation on dynamic evolution process of natural fractures based on acoustic emission position.Chinese Journal of Rock Mechanics and Engineering,2013,32(4):696-704(in Chinese))
5杨永杰,王德超,郭明福等.基于三轴压缩声发射试验的岩石损伤特征研究.岩石力学与工程学报,2014,33(1):98-104(Yang Yongjie,Wang Dechao,Guo Mingfu,et al.Study of rock damage characteristics based on acoustic emission tests under triaxial compression.Chinese Journal of Rock Mechanics and Engineering,2014,33(1):98-104(in Chinese))
6赵星光,李鹏飞,马利科等.循环加、卸载条件下北山深部花岗岩损伤与扩容特性.岩石力学与工程学报,2014,33(9):1740-1748(Zhao Xingguang,Li Pengfei,Ma Like,et al.Damage and dilation characteristics of deep granite at Beishan under cyclic loadingunloading conditions.Chinese Journal of Rock Mechanics and Engineering,2014,33(9):1740-1748(in Chinese))
7赵小平,陈淑芬.基于声发射振幅分布的裂隙岩体破坏演化过程.岩石力学与工程学报,2015,34(增1):3012-3017(Zhao Xiaoping,Chen Shufen.Failure evolution prosess of fractured rockmass based on amplitude of acoustic emission.Chinese Journal of Rock Mechanics and Engineering,2015,34(S1):3012-3017(in Chinese))
8 Xie HP,Liu JF,Ju Y,et al.Fractal property of spatial distribution of acoustic emissions during the failure process of bedded rock salt.International Journal of Rock Mechanics&Mining Sciences,2011,48(8):1344-1351
9武晋文,赵阳升,万志军等.高温均匀压力花岗岩热破裂声发射特性实验研究.煤炭学报,2012,37(7):1111-1117(Wu Jinwen,Zhao Yangsheng,Wan Zhijun,et al.Experimental study of acoustic emission of granite due to thermal cracking under high temperature and isostatic stress.Journal of China Coal Society,2012,37(7):1111-1117(in Chinese))
10曾鹏,刘阳军,纪洪广等.单轴压缩下粗砂岩临界破坏的多频段声发射耦合判据和前兆识别特征.岩土工程学报,2017,39(3):509-
517(Zeng Peng,Liu Yangjun,Ji Hongguang,et al.Coupling criteria and precursor identification characteristics of multi-band acoustic emission of gritstone fracture under uniaxial compression.Chinese Journal of Geotechnical Engineering,2017,39(3):509-517(in Chinese))
11刘建坡,徐世达,李元辉等.预制孔岩石破坏过程中的声发射时空演化特征研究.岩石力学与工程学报,2012,31(12):2538-2547(Liu Jianpo,Xu Shida,Li Yuanhui,et al.Studies of AE timespace evolution characteristics during failure process of rock specimens with prefabricated holes.Chinese Journal of Rock Mechanics and Engineering,2012,31(12):2538-2547(in Chinese))
12 Kim J,Lee K,Cho W,et al.A comparative evaluation of stressstrain and acoustic emission methods for quantitative damage assessments of brittle rock.Rock Mechanics and Rock Engineering,2015,48(2):495-508
13 Zhang Z,Zhang R,Xie H,et al.Differences in the acoustic emission characteristics of rock salt compared with granite and marble during the damage evolution process.Environmental Earth Sciences,2015,73(11):6987-6999
14 Nur A.Effects of stress on velocity anisotropy in rocks with cracks.Journal of Geophysical Research,1971,76(8):2022-2034
15 Le Pichon A.Analysis of infrasonic waves generated by the rupture of the Sumatra Mw=9.0 in the Indian ocean.Geophysical Research Abstracts,2005
16 Chou HT,Chang YL,Zhang SC.Acoustic signals and geophone response of rainfall-induced debris flows.Journal of the Chinese Institute of Engineers,2013,36(3):335-347
17 Ashraf MA,Carl HS,Chandra SR.Effect of cementation on ultrasonic velocities in sandstones.Geophysics,2007,72(2):E53-E58
18尤明庆,苏承东,李小双.损伤岩石试样的力学特性与纵波速度关系研究.岩石力学与工程学报,2008,27(3):458-467(You Mingqing,Su Chengdong,Li Xiaoshuang.Study on relation between mechanical properties and longitudinal wave velocities for damaged rock samples.Chinese Journal of Rock Mechanics and Engineering,2008,27(3):458-467(in Chinese))
19刘维国,单钰铭,刘荣和.砂岩扩容过程中超声波衰减的实验研究.成都理工大学学报(自然科学版),2006,33(6):611-616(Liu Weiguo,Shan Yuming,Liu Ronghe.Study on ultrasonic attenuation in the process of rock dilatancy by amplitude spectrum ratio.Journal of Chengdu University of Technology(Science and Technology),2006,33(6):611-616(in Chinese))
20陈乔,刘向君,刘洪等.层理性页岩地层超声波透射实验.天然气工业,2013,33(8):140-144(Chen Qiao,Liu Xiangjun,Liu Hong,et al.An experimental study of ultrasonic penetration througu bedding shale reservoir.Natural Gas Industry,2013,33(8):140-144(in Chinese))
21熊健,梁利喜,刘向君等.川南地区龙马溪组页岩岩石声波透射实验研究.地下空间与工程学报,2014,10(5):1071-1077(Xiong Jian,Liang Lixi,Liu Xiangjun,et al.Experimental study on acoustic penetration through the Longmaxi formation shale rock in south region of Sichuan Basin.Chinese Journal of Underground Space and Engineering,2014,10(5):1071-1077(in Chinese))
22 King MS.Elastic wave propagation in and permeability for rocks with multiple parallel fractures.International Journal of Rock Mechanics and Mining Sciences,2002,39(8):1033-1043
23 Petru?álek M,Vilhelm J,Rudajev V,et al.Determination of the anisotropy of elastic waves monitored by a sparse sensor network.International Journal of Rock Mechanics and Mining Sciences,2013,60:208-216
24 Chow TM,Meglis IL,Young RP.Progressive microcrack development in tests on Lac du Bonnet granite-II.Ultrasonic tomographic imaging.International Journal of Rock Mechanics and Mining Sciences&Geomechanics Abstracts,1995,32(8):751-761
25 Ougier-Simonin A,Fortin J,Gu′eguen Y,et al.Cracksin glass under triaxial conditions.International Journal of Engineering Science,2011,49(1):105-121
26张国凯,李海波,夏祥等.岩石波速与损伤演化规律研究.岩石力学与工程学报,2015,34(11):2270-2277(Zhang Guokai,Li Haibo,Xia Xiang,et al.Wave velocity and damage development of rock.Chinese Journal of Rock Mechanics and Engineering,2015,34(11):2270-2277(in Chinese))
27张国凯,李海波,夏祥等.单轴加载条件下花岗岩声发射及波传播特性研究.岩石力学与工程学报,2017,36(5):1133-1144(Zhang Guokai,Li Haibo,Xia Xiang,et al.Experiment study on acoustic emission and wave propagation in granite under uniaxial compression.Chinese Journal of Rock Mechanics and Engineering,2017,36(5):1133-1144(in Chinese))
28张晓平,王思敬,刘泉声.一种测量岩石压缩裂纹扩展过程声波波速的连续测量方法.工程地质学报,2018,26(1):91-96(Zhang Xiaoping,Wang Sijing,Liu Quansheng.A continuing measure method for acoustic wave velocity measurement during cracking processes of rock.Journal of Engineering Geology,2018,26(1):91-96(in Chinese))
29 Yang SQ,Jing HW.Strength failure and crack coalescence behavior of brittle sandstone samples containing a single fissure under uniaxial compression.International Journal of Fracture,2011,168(2):227-250
30 Lokaj′??cek T,Goel RK,Rudajev V,et al.Assessment of velocity anisotropy in rocks.International Journal of Rock Mechanics and Mining Sciences,2013,57:142-152
31尹国庆,张辉,袁芳等.白云岩储层地质力学特征分析及在储层改造优化应用--以塔中东部为例.天然气地球科学,2015,26(7):1277-1288(Yin Guoqing,Zhang Hui,Yuan Fang,et al.Geomechanical characteristics of dolomite reservoir and its application in stimulation optimization:An example of east of Tazhong.Natural Gas Geoscience,2015,26(7):1277-1288(in Chinese))
32林皋.地下结构地震响应的计算模型.力学学报,2017,49(3):528-542(Lin Gao.A computational model for seismic response analysis of underground stauctures.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):528-542(in Chinese))
33何川,齐春,封坤等.基于D-P准则的盾构隧道围岩与衬砌结构相互作用分析.力学学报,2017,49(1):31-40(He Chuan,Qi Chun,Feng Kun,et al.Theoretical analysis of interaction between surrounding rocks and linging strcture of shield tunnel based on Drucker-Prager yield criteria.Chinese Journal of Theoretical and Applied Mechanics,2017,49(1):31-40(in Chinese))
34文龙飞,王理想,田荣.动载下裂纹应力强度因子计算的改进型扩展有限元法.力学学报,2018,50(3):599-610(Wen Longfei,Wang Lixiang,Tian Rong.Accurate computation on dynamic SIFs using improved XFEM.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):599-610(in Chinese))
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