岩石应力–应变滞后现象的定量研究
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摘要
从应力–应变滞后回线"X"形的切线模量出发,探讨岩石滞后的定量描述。高应力作用下岩石的应力–应变滞后回线的形状是尖叶状的,反映在正弦波应力作用下应变的相位在加、卸载反转点处发生改变;滞后回线的切线模量和泊松比呈不对称的"X"形,且在加、卸载反转点处是不连续的。这种岩石在应力–应变图中出现的滞后回线是由于循环荷载时应力和应变的相位不同步造成的。理论计算和实验结果表明,岩石应力–应变滞后回线切线模量"X"形切线模量的夹角θ′与岩石的耗散角有确定性关系,θ′同样也反映岩石的滞后程度,它与岩石的耗散角在衡量岩石的滞后方面具有同等价值。
Hysteresis loop of rock subjected to cyclic loading is the phenomenon of nonclassical nonlinearity. The purpose of this paper is to discuss the quantitative expression of hysteretic behaviors in rocks by introducing a measurement of X-shaped tangent modulus of stress-strain hysteresis loop. The stress-strain hysteresis loop is cusped,which is the consequence of phase change of strain at the reverse point of loading and unloading under the applied external sinusoidal force. Tangent modulus and tangent Poisson′s ratio are in asymmetrical X-shaped as the function of axial strain under cyclic loading,and they both are discontinuous at the reverse point of loading and unloading. Hysteresis observed in rock′s stress-strain diagram under cyclic loading condition is the consequence of phase difference between the stress and its correspondent strain. Theoretical calculation,based on the above observed fact and a series of uniaxial stress cycling measurements for sandstone and marble,demonstrates the deterministic relationship between inclination of X-shaped tangent modulus of hysteresis loopθ′ and its dissipation angle. Inclination of X-shaped tangent modulus of hysteresis loop θ ′ can be used to reflect nonlinear stress-strain hysteretic degree in rock and rock-like materials.
Hysteresis loop of rock subjected to cyclic loading is the phenomenon of nonclassical nonlinearity. The purpose of this paper is to discuss the quantitative expression of hysteretic behaviors in rocks by introducing a measurement of X-shaped tangent modulus of stress-strain hysteresis loop. The stress-strain hysteresis loop is cusped,which is the consequence of phase change of strain at the reverse point of loading and unloading under the applied external sinusoidal force. Tangent modulus and tangent Poisson′s ratio are in asymmetrical X-shaped as the function of axial strain under cyclic loading,and they both are discontinuous at the reverse point of loading and unloading. Hysteresis observed in rock′s stress-strain diagram under cyclic loading condition is the consequence of phase difference between the stress and its correspondent strain. Theoretical calculation,based on the above observed fact and a series of uniaxial stress cycling measurements for sandstone and marble,demonstrates the deterministic relationship between inclination of X-shaped tangent modulus of hysteresis loopθ′ and its dissipation angle. Inclination of X-shaped tangent modulus of hysteresis loop θ ′ can be used to reflect nonlinear stress-strain hysteretic degree in rock and rock-like materials.
引文
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[19]任建喜,葛修润.单轴压缩岩石损伤演化细观机制及其本构模型研究[J].岩石力学与工程学报,2001,20(4):425–431.(REN Jianxi,GE Xiurun.Study on rock meso-damage evolution law and its constitutive model under uniaxial compression loading[J].Chinese Journal of Rock Mechanics and Engineering,2001,20(4):425–431.(in Chinese))
[20]TAO Z Y,MO H H.An experimental study and analysis of the behaviour of rock under cyclic loading[J].International Journal of Rock Mechanics and Mining Science,1990,27(1):51–56.
[21]LI N,ZHANG P,CHEN Y,et al.Fatigue properties of cracked,saturated and frozen sandstone samples under cyclic loading[J].International Journal of Rock Mechanics and Mining Sciences,2003,40(1):145–150.
[22]陈运平,席道瑛,薛彦伟.循环荷载下饱和岩石的应力–应变动态响应[J].石油地球物理勘探,2003,38(4):409–413.(CHEN Yunping,XI Daoying,XUE Yanwei.Stress-strain dynamic response of saturated rock under cyclic loading[J].Oil Geophysical Prospecting,2003,38(4):409–413.(in Chinese))
[23]温丹,施行觉.结合内时理论研究岩石衰减与应变振幅的关系[J].实验力学,2004,19(1):19–23.(WEN Dan,SHI Xingjue.A study on the relationship between attenuation and strain amplitude in rockusing endochronic theory[J].Journal of Experimental Mechanics,2004,19(1):19–23.(in Chinese))
[24]陈运平,席道瑛,薛彦伟.循环荷载下饱和岩石的滞后和衰减[J].地球物理学报,2004,47(4):672–679.(CHEN Yunping,XI Daoying,XUE Yanwei.Hysteresis and attenuation of saturated rocks under cyclic loading[J].Chinese Journal of Geophysics,2004,47(4):672–679.(in Chinese))
[25]席道瑛,陈运平,陶月赞,等.岩石的非线性弹性滞后特征[J].岩石力学与工程学报,2006,25(6):1086–1093.(XI Daoying,CHEN Yunping,TAO Yuezan,et al.Nonlinear elastic hysteretic characteristics in rocks[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(6):1086–1093.(in Chinese))
[26]包雪阳,施行觉.岩石非线性弹性的实验研究及其P-M模型的理论解释[J].岩石力学和工程学报,2004,23(20):3397–3404.(BAO Xueyang,SHI Xingjue.Experimental research and interpretion on nonlinear elasticity of rock using P-M model[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(20):3397–3404.(in Chinese))
[27]许江,王维忠,杨秀贵,等.细粒砂岩在循环加、卸载条件下变形实验[J].重庆大学学报(自然科学版),2004,27(12):60–62.(XU Jiang,WANG Weizhong,YANG Xiugui,et al.Experimental study on the deformation characteristics of fine-sandstone under the loading and unloading conditions[J].Journal of Chongqing University(Natural Science),2004,27(12):60–62.(in Chinese))
[28]包雪阳,施行觉,温丹,等.岩石模量与应变振幅关系的实验研究及理论解释[J].中国地震,2005,21(1):508–518.(BAO Xueyang,SHI Xingjue,WEN Dan,et al.Experimental study and interpretation on relation between modulus of rocks and strain amplitudes[J].Earthquake Research in China,2005,21(1):508–518.(in Chinese))
[29]席道瑛,刘小燕,张程远.由宏观滞回曲线分析岩石的微细观损伤[J].岩石力学与工程学报,2003,22(2):182–187.(XI Daoying,LIU Xiaoyan,ZHANG Chengyuan.Analysis of mirco-and meso-damage of rock by macro-hysteresis curve[J].Chinese Journal of Rock Mechanics and Engineering,2003,22(2),182–187.(in Chinese))
[30]许江,尹光志,王鸿,等.不同应力水平时砂岩滞回曲线演化的实验研究[J].重庆建筑大学学报,2006,28(2):40–42.(XU Jiang,YIN Guangzhi,WANG Hong,et al.Experimental research on the evolution of hysteresis curve of rock in different axial stress levels[J].Journal of Chongqing Jianzhu University,2006,28(2):40–42.(in Chinese))
[31]王鸿,许江,杨秀贵.循环载荷条件下岩石塑性滞回环的演化规律[J].重庆大学学报(自然科学版),2006,29(4):80–82.(WANG Hong,XU Jiang,YANG Xiugui.Evolvement of plastic-loop of fine-sandstone under the loading and unloading conditions[J].Journal of Chongqing University(Natural Science),2006,29(4):80–82.(in Chinese))
[32]陈运平,刘干斌,姚海林.岩石滞后非线性弹性模拟的研究[J].岩土力学,2006,27(3):341–347.(CHEN Yunping,LIU Ganbin,YAO Hailin.Study on simulation for hysteretic nonlinear elasticity of rock[J].Rock and Soil Mechanics,2006,27(3):341–347.(in Chinese))
[33]许江,鲜学福,王鸿,等.循环加、卸载条件下岩石类材料变形特性的实验研究[J].岩石力学与工程学报,2006,25(增1):3040–3045.(XU Jiang,XIAN Xuefu,WANG Hong,et al.Experimental study on rock deformation characteristics under cycling loading and unloading conditions[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(Supp.1):3040–3045.(in Chinese))
[34]席道瑛,唐杰,陶月赞.岩石中微细观缺陷对外力和温度响应概率的实验和模拟研究[J].岩石力学与工程学报,2007,26(1):9–14.(XI Daoying,TANG Jie,TAO Yuezan.Experiment and simulation study on response probability of mesoscopic structure to external force in rocks[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(1):9–14.(in Chinese))
[2]GUYER R A,JOHNSON P A.Nonlinear mesoscopic elasticity:evidence for a new class of materials[J].Physics Today,1999,52(4):30–36.
[3]TUTUNCU A N,PODIO A L,GREGORY A R,et.al.Nonlinear viscoelastic behavior of sedimentary rock,part I:effect of frequency and strain amplitude[J].Geophysics,1998,63(1):184–194.
[4]TUTUNCU A N,PODIO A L,GREGORY A R,et al.Nonlinear viscoelastic behavior of sedimentary rock,part II:hysteresis effects and influence of type of fluid on elastic moduli[J].Geophysics,1998,63(1):195–203.
[5]MCCALL K R,GUYER R A.Equation of state and wave propagation in hysteretic nonlinear elastic materials[J].Journal of Geophysical Research,1994,99(B12):23887–23897.
[6]GUYER R A,MCCALL K R,BOITNOITT G N,et al.Quantitative implementation of Preisach-Mayergoyz space to find static and dynamic elastic moduli in rock[J].Journal of Geophysical Research,1997,102(B3):5281–5293.
[7]XU H,DAY S M,MINSTER J B.Model for nonlinear wave propagation derived from rock hysteresis measurement[J].Journal of Geophysical Research,1998,103(B3):29915–29926.
[8]XU H,DAY S M,MINSTER J B.Hysteresis and two-dimensional nonlinear wave propagation in Berea Sandstone[J].Journal of Geophysical Research,2000,105(B3):6163–6175.
[9]DAY S M,MINSTER J B.Numerical simulation of attenuated wavefields using a Pade approximant method[J].Geophysical Journal of the Royal Astronomic Sociaty,1984,78(1):105–118.
[10]BRENNAN B J,STACEY F D.Frequency dependence of elasticity of rock—test of seismic velocity dispersion[J].Nature,1977,268:220–222.
[11]VAN DEN ABEELE K E A,CARMELIET J,JOHNSON P A,et.al.Influence of water saturation on the nonlinear elastic mesoscopic response in earth materials and the implications to the mechanism of nonlinearity[J].Journal of Geophysical Research,2002,107(B6):4–11.
[12]MAYERGOYZ I D,ANDREI P.Preisach modeling of clockwise hysteresis and its application to front propagation problems[J].Journal of Applied Physics,2002,91(10):7645–7647.
[13]HIRSEKORN S,DELSANTO P P.On the universality of nonclassical nonlinear phenomena and their classification[J].Applied Physics Letters,2004,84(8):1413–1415.
[14]SCALERANDI M,NOBILI M,GRIFFA A S,et al.Preisach-Mayergoyz approach to fatigue-induced irreversibility[J].Physical Review B,2006,73(9):092103.
[15]葛修润.周期荷载作用下岩石大型三轴实验的变形和强度特性研究[J].岩土力学,1987,8(2):11–19.(GE Xiurun.Study on deformation and strength behavior of the large-sized triaxial rock sample under cyclic loading[J].Rock and Soil Mechanics,1987,8(2):11–19.(in Chinese))
[16]莫海鸿.岩石的循环实验及本构关系的研究[J].岩石力学与工程学报,1988,7(3):215–224.(MO Haihong.Investigation of cyclic loading tests and constitutive relation of rock[J].Chinese Journal of Rock Mechanics and Engineering,1988,7(3):215–224.(in Chinese))
[17]葛修润,蒋宇,卢允德,等.周期荷载作用下岩石疲劳变形特性实验研究[J].岩石力学与工程学报,2003,22(10):1581–1585.(GE Xiurun,JIANG Yu,LU Yunde,et al.Testing study on fatigue deformation law of rock under cyclic loading[J].Chinese Journal of Rock Mechanics and Engineering,2003,22(10):1581–1585.(in Chinese))
[18]章清叙,葛修润,黄铭,等.周期荷载作用下红砂岩三轴疲劳变形特性实验研究[J].岩石力学与工程学报,2006,25(3):473–478.(ZHANG Qingxu,GE Xiurun,HUANG Ming,et al.Testing study on fatigue deformation law of red-sandstone under triaxial compression with cyclic loading[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(3):473–478.(in Chinese))
[19]任建喜,葛修润.单轴压缩岩石损伤演化细观机制及其本构模型研究[J].岩石力学与工程学报,2001,20(4):425–431.(REN Jianxi,GE Xiurun.Study on rock meso-damage evolution law and its constitutive model under uniaxial compression loading[J].Chinese Journal of Rock Mechanics and Engineering,2001,20(4):425–431.(in Chinese))
[20]TAO Z Y,MO H H.An experimental study and analysis of the behaviour of rock under cyclic loading[J].International Journal of Rock Mechanics and Mining Science,1990,27(1):51–56.
[21]LI N,ZHANG P,CHEN Y,et al.Fatigue properties of cracked,saturated and frozen sandstone samples under cyclic loading[J].International Journal of Rock Mechanics and Mining Sciences,2003,40(1):145–150.
[22]陈运平,席道瑛,薛彦伟.循环荷载下饱和岩石的应力–应变动态响应[J].石油地球物理勘探,2003,38(4):409–413.(CHEN Yunping,XI Daoying,XUE Yanwei.Stress-strain dynamic response of saturated rock under cyclic loading[J].Oil Geophysical Prospecting,2003,38(4):409–413.(in Chinese))
[23]温丹,施行觉.结合内时理论研究岩石衰减与应变振幅的关系[J].实验力学,2004,19(1):19–23.(WEN Dan,SHI Xingjue.A study on the relationship between attenuation and strain amplitude in rockusing endochronic theory[J].Journal of Experimental Mechanics,2004,19(1):19–23.(in Chinese))
[24]陈运平,席道瑛,薛彦伟.循环荷载下饱和岩石的滞后和衰减[J].地球物理学报,2004,47(4):672–679.(CHEN Yunping,XI Daoying,XUE Yanwei.Hysteresis and attenuation of saturated rocks under cyclic loading[J].Chinese Journal of Geophysics,2004,47(4):672–679.(in Chinese))
[25]席道瑛,陈运平,陶月赞,等.岩石的非线性弹性滞后特征[J].岩石力学与工程学报,2006,25(6):1086–1093.(XI Daoying,CHEN Yunping,TAO Yuezan,et al.Nonlinear elastic hysteretic characteristics in rocks[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(6):1086–1093.(in Chinese))
[26]包雪阳,施行觉.岩石非线性弹性的实验研究及其P-M模型的理论解释[J].岩石力学和工程学报,2004,23(20):3397–3404.(BAO Xueyang,SHI Xingjue.Experimental research and interpretion on nonlinear elasticity of rock using P-M model[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(20):3397–3404.(in Chinese))
[27]许江,王维忠,杨秀贵,等.细粒砂岩在循环加、卸载条件下变形实验[J].重庆大学学报(自然科学版),2004,27(12):60–62.(XU Jiang,WANG Weizhong,YANG Xiugui,et al.Experimental study on the deformation characteristics of fine-sandstone under the loading and unloading conditions[J].Journal of Chongqing University(Natural Science),2004,27(12):60–62.(in Chinese))
[28]包雪阳,施行觉,温丹,等.岩石模量与应变振幅关系的实验研究及理论解释[J].中国地震,2005,21(1):508–518.(BAO Xueyang,SHI Xingjue,WEN Dan,et al.Experimental study and interpretation on relation between modulus of rocks and strain amplitudes[J].Earthquake Research in China,2005,21(1):508–518.(in Chinese))
[29]席道瑛,刘小燕,张程远.由宏观滞回曲线分析岩石的微细观损伤[J].岩石力学与工程学报,2003,22(2):182–187.(XI Daoying,LIU Xiaoyan,ZHANG Chengyuan.Analysis of mirco-and meso-damage of rock by macro-hysteresis curve[J].Chinese Journal of Rock Mechanics and Engineering,2003,22(2),182–187.(in Chinese))
[30]许江,尹光志,王鸿,等.不同应力水平时砂岩滞回曲线演化的实验研究[J].重庆建筑大学学报,2006,28(2):40–42.(XU Jiang,YIN Guangzhi,WANG Hong,et al.Experimental research on the evolution of hysteresis curve of rock in different axial stress levels[J].Journal of Chongqing Jianzhu University,2006,28(2):40–42.(in Chinese))
[31]王鸿,许江,杨秀贵.循环载荷条件下岩石塑性滞回环的演化规律[J].重庆大学学报(自然科学版),2006,29(4):80–82.(WANG Hong,XU Jiang,YANG Xiugui.Evolvement of plastic-loop of fine-sandstone under the loading and unloading conditions[J].Journal of Chongqing University(Natural Science),2006,29(4):80–82.(in Chinese))
[32]陈运平,刘干斌,姚海林.岩石滞后非线性弹性模拟的研究[J].岩土力学,2006,27(3):341–347.(CHEN Yunping,LIU Ganbin,YAO Hailin.Study on simulation for hysteretic nonlinear elasticity of rock[J].Rock and Soil Mechanics,2006,27(3):341–347.(in Chinese))
[33]许江,鲜学福,王鸿,等.循环加、卸载条件下岩石类材料变形特性的实验研究[J].岩石力学与工程学报,2006,25(增1):3040–3045.(XU Jiang,XIAN Xuefu,WANG Hong,et al.Experimental study on rock deformation characteristics under cycling loading and unloading conditions[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(Supp.1):3040–3045.(in Chinese))
[34]席道瑛,唐杰,陶月赞.岩石中微细观缺陷对外力和温度响应概率的实验和模拟研究[J].岩石力学与工程学报,2007,26(1):9–14.(XI Daoying,TANG Jie,TAO Yuezan.Experiment and simulation study on response probability of mesoscopic structure to external force in rocks[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(1):9–14.(in Chinese))
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