从断裂力学观点探讨 b 值的物理实质
ON THE PHYSICAL ESSENCE OF b VALUE FOR AE OF ROCK TESTS AND NATURAL EARTHQUAKES IN TERMS OF FRACTURE MECHANICS
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摘要: 岩石试验声发射及天然地震的 b 值到底反映了什么样的物理实质,是一个长期以来引起争论的问题.本文运用断裂力学的观点和方法,研究岩石试验试件内微裂纹系中各裂纹的扩展顺序及声发射能量,由此确定整个声发射序列,并进而确定 b 值.如果假设裂纹长度的分布密度函数为 p()=B-.经过力学分析和数学推导,可得到关系式 b=3/2.由此看出:b 值的物理实质是,它反映了介质的断裂构造状态————包括介质中裂纹系的空间分布及其它影响材料中裂纹扩展的物理量(如断裂韧度、摩擦系数等)的空间分布.这一结论与大多数实验及观测结果相符.我们的研究结果与茂木清夫观点有共同之处.茂木清夫认为:介质的不均匀性是决定 b 值的重要因素.介质的不均匀性一词固然也包括了介质的断裂构造状态在内,但是本文之观点比茂木更进一步说明了问题的实质.Abstract: The problem of physical essence of b value for AE of rock tests and natural earthquakes-has been a controversial topic during the past two decades. In the present paper the order and* energy of each microfracturing of the microcrack system existing in a rock specimen is discussed from the viewpoint of fracture mechanics, then the whole series of AE and subsequently the value of b can also be determinated. The order of microfracturing depends on the parameters Kei/Kcf where Kei is the effective stress intensity factor of the i-th microcrack and Kci is-the fracture toughness in the site of the i-th crack. The energy of each AE can be expressed as ilofli G1ds where G1 is energy release rate of nicrofracturing, lof and li are the original and final crack lengths respectively and i is the emanating efficiency for the i-th crack. If we assume that the distribution density function of microcrack length l is p(l)=Bl-r where B and are constants, then the expression b = 3/2 can be deduced. Hence we have come to the conclusion that the value of b mirrors essentially the crack-system-configuration of the material which means the distribution of the sizes, shapes and orientations of micro-cracks over the space as well as the distribution of the other concerning physical parameters such as fracture toughness, friction coeficient etc. Our conclusion is somewhat similar in character to Mo-gi's. He concluded that the heterogeneity of the material plays the most important role in determining the value of b. The term heterogeneity of the material of course covers the idea of crack-system-configuration, but we think that our view has a little bit deeper insight to the problem than that of Mogi's.
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[2] Mogi, K., Study of the elastic shocks caused by the fracture of heterogeneous materials and its relation to earthquake phenomena, Bull. Earthguake Res. Inst., 40, 125——173, 1962.
[3] Vinogradov, S. D., On the distribution of the number of fractures in dependence on the energy liberated by the distruction of rocks, Bull. Acad. Sci. USSR, Geoyhys. Ser. 12, 1292——1293, 1959.
[4] Scholz, C. H., The frequency——magnitude relation of microfracturing in rock and its relation to earthquakes,Bull. Scismol. Soc. Am., 58, 399——415, 1968.
[5] 李全林、于绿、郝柏林、陈锦标,地震频度——震级关系的时空扫描,2——6,地震出版社,1979,
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[7] Palaniswamy, K. and W. G. Knauss, On the problem of crack extension in brittle solids under general loading, Mechanics Today, Vol. 4, ed. S. Nemat——Nasser, Pergmen Press Inc. 1978.
[8] 尹祥础,固体力学,4"——471,地震出版社,1985,
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[11] Rudnicki, J. W,and H.Kanamori, Effects of fault Interaction on moment, stress drop and strain energy release, J. Geoyhys. Res., 86, B3, 1785——1793,1981.
[12] Rice, J. R., An examination of the fracture mechanics energy balance from the point of view of continuum mechanics, Proceeding of the 1——st international coiference of fracture, ed. T. Yokobori, 283——308, Japanese Soc. for Strength and Fracture, Sendai, Japan, 1966.
[13] Kanamori, H., and D. L. Anderson, Theoretical basis of some empirical relation in seismology, Bull. Seis.Soc. Am., 5, 1075——1095, 1976.
[14] Mogi, K., Regional variations in magnitude——frequency relation of earthquake, Budl. Earthgua友.Res. Inst.,45, 313——325, 1976.
[15] 茂木清夫,地震一その本性をさぐろ, 28——45,东京大学出版会,1981,
[16] 李纪汉、刘晓红、郝晋升、方亚如、蔡戴恩、耿乃光,华北地区五种岩石破裂前声发射b值的变化,地球物理学报(待发表).
[17] Hallbauer, D. K., H. Wagner and N. G. W. Cook, Some observations concerning the microscopic and mechanical behaviour of quartize specimens in stiff triaxial compression tests, Int. J. rock Mech. Min. Sci.and Ceomech. Abstr., 10, 713——726, 1973.
[18] Fonseka, G. M., S. A. F. Murrell and P. Barnes, Scanning electron microscope and acoustic emission studies of crack development in rocks, Int. J. Rock Mech. Min. Sci. and Ceomecfr. Abstr., 22, 273——289, 1985.
[19] linauss, W. G., The mechanics of polymer fracture, slpplied Mech. Reu., 26, 1——17, 1973.
[20] 尹祥础、郑天愉,地震孕育过程的流变模式,中国科学,(B)1982, 10:922——930.
[21] 郑天愉、尹祥础,断层的亚临界扩展和地震孕育过程,科学通报,21, 1325——1328, 1983,
[22] Aki, K., A probabilistic synthesis of precursory phenomena, Earthquake Prediction, An International Review,ed. D. W. Simpson and P. G. Richards, A. G. U., Washington, D. C., 1981.
[23] Aki, K., Tlie use of a physical model of fault mechanics for earthquake prediction, On Continental Seismicity and Earthquake Prediction, Seismological Press, Beijing, 1982.[1] Gutenberg, B. and C. F. Richer, Seismicity of the earth, 102——107, Princeton Univ. Press, 1954.
[2] Mogi, K., Study of the elastic shocks caused by the fracture of heterogeneous materials and its relation to earthquake phenomena, Bull. Earthguake Res. Inst., 40, 125——173, 1962.
[3] Vinogradov, S. D., On the distribution of the number of fractures in dependence on the energy liberated by the distruction of rocks, Bull. Acad. Sci. USSR, Geoyhys. Ser. 12, 1292——1293, 1959.
[4] Scholz, C. H., The frequency——magnitude relation of microfracturing in rock and its relation to earthquakes,Bull. Scismol. Soc. Am., 58, 399——415, 1968.
[5] 李全林、于绿、郝柏林、陈锦标,地震频度——震级关系的时空扫描,2——6,地震出版社,1979,
[6] 丁文镜,b值预报的物理基础,地震学报,2, 378——387, 1980,
[7] Palaniswamy, K. and W. G. Knauss, On the problem of crack extension in brittle solids under general loading, Mechanics Today, Vol. 4, ed. S. Nemat——Nasser, Pergmen Press Inc. 1978.
[8] 尹祥础,固体力学,4"——471,地震出版社,1985,
[9] 郑哲故,连续介质力学与断裂,力学进展,12,133——140, 1982,
[10] Erdogan, F., On stress distribution in plates with colliniar cut under arbitrary loads, Proceeding of the 4——th U. S. National Congress on Applived Mechanics, 547——553, ed. R. M. Rosenbery, ASME, Berkeley,Calif. 1962.
[11] Rudnicki, J. W,and H.Kanamori, Effects of fault Interaction on moment, stress drop and strain energy release, J. Geoyhys. Res., 86, B3, 1785——1793,1981.
[12] Rice, J. R., An examination of the fracture mechanics energy balance from the point of view of continuum mechanics, Proceeding of the 1——st international coiference of fracture, ed. T. Yokobori, 283——308, Japanese Soc. for Strength and Fracture, Sendai, Japan, 1966.
[13] Kanamori, H., and D. L. Anderson, Theoretical basis of some empirical relation in seismology, Bull. Seis.Soc. Am., 5, 1075——1095, 1976.
[14] Mogi, K., Regional variations in magnitude——frequency relation of earthquake, Budl. Earthgua友.Res. Inst.,45, 313——325, 1976.
[15] 茂木清夫,地震一その本性をさぐろ, 28——45,东京大学出版会,1981,
[16] 李纪汉、刘晓红、郝晋升、方亚如、蔡戴恩、耿乃光,华北地区五种岩石破裂前声发射b值的变化,地球物理学报(待发表).
[17] Hallbauer, D. K., H. Wagner and N. G. W. Cook, Some observations concerning the microscopic and mechanical behaviour of quartize specimens in stiff triaxial compression tests, Int. J. rock Mech. Min. Sci.and Ceomech. Abstr., 10, 713——726, 1973.
[18] Fonseka, G. M., S. A. F. Murrell and P. Barnes, Scanning electron microscope and acoustic emission studies of crack development in rocks, Int. J. Rock Mech. Min. Sci. and Ceomecfr. Abstr., 22, 273——289, 1985.
[19] linauss, W. G., The mechanics of polymer fracture, slpplied Mech. Reu., 26, 1——17, 1973.
[20] 尹祥础、郑天愉,地震孕育过程的流变模式,中国科学,(B)1982, 10:922——930.
[21] 郑天愉、尹祥础,断层的亚临界扩展和地震孕育过程,科学通报,21, 1325——1328, 1983,
[22] Aki, K., A probabilistic synthesis of precursory phenomena, Earthquake Prediction, An International Review,ed. D. W. Simpson and P. G. Richards, A. G. U., Washington, D. C., 1981.
[23] Aki, K., Tlie use of a physical model of fault mechanics for earthquake prediction, On Continental Seismicity and Earthquake Prediction, Seismological Press, Beijing, 1982.
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