WANG Zhe-chao1, 2, QIAO Li-ping3, LI Shu-cai1, LIN Chun-jin1
1 . Research Center of Geotechnical and Structural Engineering, Shandong University, Jinan 250061, China ; 2. Department of Civil Engineering, University of Calgary, Calgary, Alberta T2N1N4, Canada; 3. Department of Engineering Mechanics, Shandong University, Jinan 250061, China
Abstract： Soil creep is such a process that the deformation of soil develops with time under a state of constant stress. Following the tradition of continuum mechanics, the time-dependent creep rate is always employed in the investigations of the soil creep. However, in practices, soils undergo complex loading-unloading cycles and are allowed to creep on the stages at different stress and strain levels. In this complex situation, the initial states (or configurations) of the creep stages are different and thus the traditional continuum mechanics is incapable in describing the soil creep. Different from the traditional continuum mechanics, the internal-variable theory is based on the concept of internal variable, which describes the internal structure of materials. In this study, an internal-variable creep model is proposed, in which the creep rate of soils is dependent on not only applied stress, but also irreversible strain, which is adopted as internal variable for soil deformation. A series of laboratory tests have been performed to verify the proposed creep model. The parametric relation between the proposed internal-variable and the traditional power law creep models is derived. A method to accelerate creep tests is proposed based on the proposed creep model. It is found that the proposed internal-variable creep model is capable in describing the soil creep under complex conditions.
王者超,乔丽苹,李术才,林春金,. 土的内变量蠕变模型研究[J]. 岩土工程学报, 2011, 33(10): 1569-1575.
WANG Zhe-chao, QIAO Li-ping, LI Shu-cai, LIN Chun-jin . An internal-variable creep model for soils. Chinese J. Geot. Eng., 2011, 33(10): 1569-1575.
4 讨 论 4.1 土的内变量蠕变模型与传统蠕变模型参数关系 由于塑性应变与时间无关，所以： ，于是式（ 3 ）可写为 。 (7) 图 11 冰渍土分级加载蠕变试验数据和模型预测对比 Fig. 11 Comparison between test data and model prediction for stepwise creep tests 图 12 冰渍土多阶段蠕变试验数据和模型预测对比 Fig. 12 Comparison between test data and model prediction for multiple stage creep tests 式（ 7 ）对时间积分有 ， (8) 式中， 为蠕变初始状态下土的最大不可恢复应变。式（ 8 ）对时间求导可得传统幂函数时间硬化蠕变模型表达式： 。 (9) 4.2 加快蠕变试验方法 土蠕变性质研究的一个重要目的是预测土的长期力学性质。在常规蠕变试验中，一般采取简单加载方式获得土的蠕变性质。在这些试验中，土样被加载到指定应力水平，然后试验持续进行。基于目前的技术水平，这种方式开展的蠕变试验一般不超过几年，而这与多数与土有关建筑物的设计寿命是不能相比的。因此提出一个可以加快蠕变试验的方法无疑是有重要意义的 [8, 15] 。 根据本文的讨论，土的蠕变性质取决于施加外力的大小和土的不可恢复应变。而土的不可恢复应变可通过蠕变过程或塑性变形过程产生。相比蠕变过程，塑性变形过程可在较短时间内获得较大的不可恢复变形。以图 8 中冰渍土的蠕变试验为例，如果采用常规蠕变试验，需要很长时间才能获得偏应力水平为 360 kPa 下后面两阶段（ 360-2 和 360-3 ）的蠕变数据。而本文试验中，通过加卸载循环，试样的不可恢复应变在较短时间内得以增加，从而得到了土的长期蠕变行为。根据式（ 9 ）可以得到阶段 360-2 和 360-3 的等效蠕变时间，即在常规蠕变试验中所对应时间。经过计算，在常规试验中到达阶段 360-2 和 360-3 起点所需时间分别为 28176 和 63875 h 。图 13 给出了这两个阶段在常规蠕变试验曲线上的位置。从图中不难看出，采用加快蠕变试验可以在较短时间内获得土的长期蠕变行为。 图 13 冰渍土各蠕变阶段在常规蠕变曲线上位置 Fig. 13 Positions of creep stages on creep curve of till obtained in .. conventional triaxial creep tests 5 结论与展望 （ 1 ）提出采用不可恢复应变作为土变形的内变量。 （ 2 ）开展了复杂条件下土的蠕变试验，具体讨论了不可恢复应变和应力对土蠕变性质的影响。 （ 3 ）提出了土的内变量蠕变模型，并利用试验数据进行了验证。 （ 4 ）讨论了土的内变量蠕变模型与传统时间硬化幂函数蠕变模型参数间的关系，提出了一种加快蠕变试验的方法。 然而，土的蠕变性质与土的种类、土所处应力应变状态和加载速率等因素均有密切关系。目前为止，本文模型只是在三轴正常固结黏土蠕变试验中得到了验证。因此，该模型是否适用于其他情况则有待于进一步验证。 [1] SINGH A, MITCHELL J K. General stress-strain-time function for soils[J]. Journal of Soil Mechanics & Foundations Division, ASCE, 1968, 94 (SM1): 21 – 46. [2] BISHOP A W, LOVENBURY H T. Creep characteristics of two undisturbed clays[C]// Proceedings of 7th International Conference of Soil Mechanics and Foundation Engineering, 1969, 1 : 29 – 37. [3] MESRI G, GODLEWSKI P M. Time and stress compressibility interrelationship[J]. Journal of the Geotechnical Engineering Division, ASCE, 1977, 103 (5): 417 – 430. [4] 卢萍珍 , 曾 静 , 盛 谦 . 软黏土蠕变试验及其经验模型研究 [J]. 岩土力学 , 2008, 29 (4): 1041 – 1045. (LU Ping-zhen, ZENG Jing, SHENG Qian. Creep tests on soft clay and its empirical models[J]. Rock and Soil Mechanics, 2008, 29 (4): 1041 – 1045. (in Chinese)) [5] 王者超 , 乔丽苹 . 土蠕变性质及其模型研究综述与讨论 [J]. 岩土力学 , 2011, 32 (8): 2251 – 2260. (WANG Zhe-chao, QIAO Li-ping. A review and discussion of creep behaviour of soil and its models[J]. Rock and soil Mechanics, 2011, 32 (8): 2251 – 2260 . (in Chinese) ) [6] YIN J H, GRAHAM J. Equivalent times and one-dimensional elastic viscoplastic modelling of time-dependent stress-strain behaviour of clays[J]. Canadian Geotechnical Journal, 1994, 31 (1): 45 – 52. [7] WANG Z. Soil creep behaviour: laboratory testing and numerical modeling[D]. Calgary: University of Calgary, 2010. [8] 孙 钧 . 岩土材料流变及其工程应用 [M]. 北京 : 中国建筑工业出版社 , 1999. (SUN Jun. Rheological behavior of geomaterials and its engineering applications[M]. Beijing: China Architecture and Building Press, 1999. (in Chinese)) [9] RICE J R. Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity [J]. Journal of the Mechanics and Physics of Solids, 1971, 19 (6): 433 – 455. [10] COLLINS I F, HOULSBY G T. Application of thermomechanical principles to the modelling of geotechnical materials[J]. Proceedings of Royal Society of London, Series A, 1997, 453 : 1975 – 2001. [11] 黄茂松 , 钟辉虹 , 李永盛 . 天然状态结构性软黏土的边界面弹塑性模型 [J]. 水利学报 , 2003, 34 (12): 47 – 52. (HUANG Mao-song, ZHONG Hui-hong, LI Yong-sheng. Elastoplastic boundary surface model for natural soft clay with structural damage[J]. Journal of Hydraulic Engineering, 2003, 34 (12): 47 – 52. (in Chinese)) [12] 陈敬虞 , 龚晓南 , 邓亚虹 . 基于内变量理论的岩土材料本 构关系研究 [J]. 浙江大学学报（理学版） , 2008, 35 (3): 355 – 360. (CHEN Jing-yu, GONG Xiao-nan, DENG Ya-hong. Research on the constitutive relation of geomaterials based on the theory of internal variable[J]. Journal of Zhejiang University (Science Edition), 2008, 35 (3): 335 – 360. (in Chinese)) [13] DESAI C S, ZHANG D. Viscoplastic model for geological materials with generalized flow rule[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1987, 11 (6): 603 – 620. [14] 李兴照 , 黄茂松 , 王录民 . 流变性软黏土的弹塑性边界面本构方程 [J]. 岩石力学与工程学报 , 2006, 26 (7): 1393 – 1401. (LI Xing-zhao, HUANG Mao-song, WANG Lu-min. Bounding surface elasto-viscoplastic constitutive model for rheological behaviors of soft clays[J]. Chinese Journal of Rock Mechanics and Engineering, 2006, 26 (7): 1393 – 1401. (in Chinese)) [15] MITCHELL J K, SOGA K. Fundamentals of soil behavior [M]. 4th ed. New Jersey: John Wiley & Sons, 2005. [16] LADE P V, LIU C T. Experimental study of drained creep behavior of sand[J]. Journal of Engineering Mechanics, 1998, 124 (8): 912 – 920. [17] ABAQUS Inc. ABAQUS Documentation[M]. Providence: Rhode Island, 2006. [18] MESRI G, CASTRO A. <>Cα/<>Cc concept and<> K0 during secondary compression[J]. Journal of Geotechnical Engineering, 1987, 113 (3): 230 – 247. [19] KUHN M R. Micromechanical aspects of soil creep[D]. Berkeley: University of California at Berkeley, 1987. [20] BOWMAN E T, SOGA K. Creep, ageing and microstructural change in dense granular materials[J]. Soils and Foundations, 2003, 43 (4): 107 – 117. [21] 王者超 , 李术才 . 高应力下颗粒材料一维力学特性研究 II: 蠕变性质 [J]. 岩土力学 , 2010, 31 (11): 3392 – 3396. (WANG Zhe-chao, LI Shu-cai. One dimensional mechanical behavior of granular materials at high stresses, Part II: Creep behavior[J]. Rock and Soil Mechanics, 2011, 31 (11): 3392 – 3396. (in Chinese)) [22] ADACHI T, OKANO M. A constitutive equation for normally consolidated clay[J]. Soils and Foundations, 1974, 14 (4): 55 – 73. [23] 袁 静 , 龚晓南 , 益德清 . 岩土流变模型的比较研究 [J]. 岩石力学与工程学报 , 2001, 20 (6): 772 – 779. (YUAN Jing, GONG Xiao-nan, YI De-qing. Comparison study on rheological constitutive models[J]. Chinese Journal of Rock Mechanics and Engineering, 2001, 20 (6): 772 – 779. (in Chinese))
图 11 冰渍土分级加载蠕变试验数据和模型预测对比 Fig. 11 Comparison between test data and model prediction for stepwise creep tests 
图 12 冰渍土多阶段蠕变试验数据和模型预测对比 Fig. 12 Comparison between test data and model prediction for multiple stage creep tests 式（ 7 ）对时间积分有 
图 13 冰渍土各蠕变阶段在常规蠕变曲线上位置 Fig. 13 Positions of creep stages on creep curve of till obtained in .. conventional triaxial creep tests 5 结论与展望 （ 1 ）提出采用不可恢复应变作为土变形的内变量。 （ 2 ）开展了复杂条件下土的蠕变试验，具体讨论了不可恢复应变和应力对土蠕变性质的影响。 （ 3 ）提出了土的内变量蠕变模型，并利用试验数据进行了验证。 （ 4 ）讨论了土的内变量蠕变模型与传统时间硬化幂函数蠕变模型参数间的关系，提出了一种加快蠕变试验的方法。 然而，土的蠕变性质与土的种类、土所处应力应变状态和加载速率等因素均有密切关系。目前为止，本文模型只是在三轴正常固结黏土蠕变试验中得到了验证。因此，该模型是否适用于其他情况则有待于进一步验证。 [1] SINGH A, MITCHELL J K. General stress-strain-time function for soils[J]. Journal of Soil Mechanics & Foundations Division, ASCE, 1968, 94 (SM1): 21 – 46. [2] BISHOP A W, LOVENBURY H T. Creep characteristics of two undisturbed clays[C]// Proceedings of 7th International Conference of Soil Mechanics and Foundation Engineering, 1969, 1 : 29 – 37. [3] MESRI G, GODLEWSKI P M. Time and stress compressibility interrelationship[J]. Journal of the Geotechnical Engineering Division, ASCE, 1977, 103 (5): 417 – 430. [4] 卢萍珍 , 曾 静 , 盛 谦 . 软黏土蠕变试验及其经验模型研究 [J]. 岩土力学 , 2008, 29 (4): 1041 – 1045. (LU Ping-zhen, ZENG Jing, SHENG Qian. Creep tests on soft clay and its empirical models[J]. Rock and Soil Mechanics, 2008, 29 (4): 1041 – 1045. (in Chinese)) [5] 王者超 , 乔丽苹 . 土蠕变性质及其模型研究综述与讨论 [J]. 岩土力学 , 2011, 32 (8): 2251 – 2260. (WANG Zhe-chao, QIAO Li-ping. A review and discussion of creep behaviour of soil and its models[J]. Rock and soil Mechanics, 2011, 32 (8): 2251 – 2260 . (in Chinese) ) [6] YIN J H, GRAHAM J. Equivalent times and one-dimensional elastic viscoplastic modelling of time-dependent stress-strain behaviour of clays[J]. Canadian Geotechnical Journal, 1994, 31 (1): 45 – 52. [7] WANG Z. Soil creep behaviour: laboratory testing and numerical modeling[D]. Calgary: University of Calgary, 2010. [8] 孙 钧 . 岩土材料流变及其工程应用 [M]. 北京 : 中国建筑工业出版社 , 1999. (SUN Jun. Rheological behavior of geomaterials and its engineering applications[M]. Beijing: China Architecture and Building Press, 1999. (in Chinese)) [9] RICE J R. Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity [J]. Journal of the Mechanics and Physics of Solids, 1971, 19 (6): 433 – 455. [10] COLLINS I F, HOULSBY G T. Application of thermomechanical principles to the modelling of geotechnical materials[J]. Proceedings of Royal Society of London, Series A, 1997, 453 : 1975 – 2001. [11] 黄茂松 , 钟辉虹 , 李永盛 . 天然状态结构性软黏土的边界面弹塑性模型 [J]. 水利学报 , 2003, 34 (12): 47 – 52. (HUANG Mao-song, ZHONG Hui-hong, LI Yong-sheng. Elastoplastic boundary surface model for natural soft clay with structural damage[J]. Journal of Hydraulic Engineering, 2003, 34 (12): 47 – 52. (in Chinese)) [12] 陈敬虞 , 龚晓南 , 邓亚虹 . 基于内变量理论的岩土材料本 构关系研究 [J]. 浙江大学学报（理学版） , 2008, 35 (3): 355 – 360. (CHEN Jing-yu, GONG Xiao-nan, DENG Ya-hong. Research on the constitutive relation of geomaterials based on the theory of internal variable[J]. Journal of Zhejiang University (Science Edition), 2008, 35 (3): 335 – 360. (in Chinese)) [13] DESAI C S, ZHANG D. Viscoplastic model for geological materials with generalized flow rule[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1987, 11 (6): 603 – 620. [14] 李兴照 , 黄茂松 , 王录民 . 流变性软黏土的弹塑性边界面本构方程 [J]. 岩石力学与工程学报 , 2006, 26 (7): 1393 – 1401. (LI Xing-zhao, HUANG Mao-song, WANG Lu-min. Bounding surface elasto-viscoplastic constitutive model for rheological behaviors of soft clays[J]. Chinese Journal of Rock Mechanics and Engineering, 2006, 26 (7): 1393 – 1401. (in Chinese)) [15] MITCHELL J K, SOGA K. Fundamentals of soil behavior [M]. 4th ed. New Jersey: John Wiley & Sons, 2005. [16] LADE P V, LIU C T. Experimental study of drained creep behavior of sand[J]. Journal of Engineering Mechanics, 1998, 124 (8): 912 – 920. [17] ABAQUS Inc. ABAQUS Documentation[M]. Providence: Rhode Island, 2006. [18] MESRI G, CASTRO A. <>Cα/<>Cc concept and<> K0 during secondary compression[J]. Journal of Geotechnical Engineering, 1987, 113 (3): 230 – 247. [19] KUHN M R. Micromechanical aspects of soil creep[D]. Berkeley: University of California at Berkeley, 1987. [20] BOWMAN E T, SOGA K. Creep, ageing and microstructural change in dense granular materials[J]. Soils and Foundations, 2003, 43 (4): 107 – 117. [21] 王者超 , 李术才 . 高应力下颗粒材料一维力学特性研究 II: 蠕变性质 [J]. 岩土力学 , 2010, 31 (11): 3392 – 3396. (WANG Zhe-chao, LI Shu-cai. One dimensional mechanical behavior of granular materials at high stresses, Part II: Creep behavior[J]. Rock and Soil Mechanics, 2011, 31 (11): 3392 – 3396. (in Chinese)) [22] ADACHI T, OKANO M. A constitutive equation for normally consolidated clay[J]. Soils and Foundations, 1974, 14 (4): 55 – 73. [23] 袁 静 , 龚晓南 , 益德清 . 岩土流变模型的比较研究 [J]. 岩石力学与工程学报 , 2001, 20 (6): 772 – 779. (YUAN Jing, GONG Xiao-nan, YI De-qing. Comparison study on rheological constitutive models[J]. Chinese Journal of Rock Mechanics and Engineering, 2001, 20 (6): 772 – 779. (in Chinese))