双向循环载荷作用下饱和粘土的边界面模型
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摘要
嵌入式海洋工程结构在风、浪、流等海洋环境下,产生的循环载荷会传至周围的海洋土,使得海洋土不断处于双向的压缩-拉伸状态。部分该类结构的承载性能由其在循环载荷作用下产生的拉伸变形决定,因此如何能够准确地模拟双向循环载荷作用下嵌入式海洋工程结构的承载能力和工作性能具有重要的意义。现有考虑循环载荷作用的弹塑性模型,往往采用类似剑桥模型的椭圆形屈服面,海洋土体一般为天然固结状态,已有研究表明该类屈服面在土体处于拉伸状态时,因弹性区域偏大,导致计算精度较差。为了解决椭圆屈服面因拉伸部分的弹性区域过大,使得塑性变形偏小的不合理性。本文结合基于空间滑动面破坏准则的变换应力法和作者新近提出的硬化准则,建立了能考虑土体在双向循环载荷作用下尤其处于拉伸状态时动力特性的边界面模型。将模型数值计算的结果与试验结果比较,验证了该模型的合理性。
为了实现模型的数值计算,需要选取一种高效、精确的应力积分算法(本构积分算法),使得计算精度高、收敛速度快、资源利用率大。本文基于回映应力积分算法,推导了三种该模型的应力更新算法公式,编写了相应的Fortran语言程序,实现了模型的数值计算。并且在同一工况中,相同模型参数下,运用不同积分算法对模型数值计算的结果进行对比和数据分析,考察不同算法对该模型计算结果的影响。同时,与试验结果进行对比,验证了回映应力积分算法的准确性和合理性。
然而,目前该弹塑性模型只在常规三轴应力条件下得到了验证,实际工程中土体所处的应力状态往往为三维应力状态,为了将修正模型从三轴试验的轴对称状态推广至真三轴应力状态,以期运用该模型分析嵌入式海洋工程结构与海洋土体的耦合动力响应,本文利用变换应力法实现了模型的三维化,并给出了相应的公式推导和数值实现方法。
The embedded offshore structures truly transmit the cyclic loading caused by winds, waves and currents to the surrounding marine soils. The marine soils are subjected to a two-way cyclic loading so that the soils involve both compression and extension states. The destruction of parts of the embedded offshore structures is due to extension deformation produced during the cycle loading. Therefore, it is significance to accurately simulate the dynamic characteristics between the soils and the embedded offshore structures.The current elastoplastic constitutive models always adopt an elliptic yield surface similar to the one in the Cam-Clay model. However, several studies have demonstrated that these kinds of models which adopt an elliptic yield surface are not suitable for predicting the characteristics of the soil under the extension state, due to its over-predicted elastic region. In this paper, combining the transformed stress tensors which are based on the spatially mobilized plane (SMP) yield criterion with the new hardening rule by the authors, a new bounding-surface plasticity model is proposed for the two-way cyclic behaviors of the saturated soils. The predicted results by the model show a good agreement with the experimental data both for the monotonic tests and the two-way cyclic triaxial tests.
In order to apply the revised elastoplastic constitutive models proposed by the author the program are developed for the implementation of the models into a language software FORTRAN by using the return mapping algorithm. In the same condition and model parameters, different stress integration are used for the constitutive model to verify the accuracy. The influences of the algorithms to the model are also checked.
However, the elasticplastic model is only demonstrated under triaxial condition at present. The stress state of soils is generally in the three-dimensional state . In order to analysis the dynamic responses produced by the embedded ocean offshore structures and the marine soils, the model is extended to the three-dimensional model by the transformed stress method.
为了实现模型的数值计算,需要选取一种高效、精确的应力积分算法(本构积分算法),使得计算精度高、收敛速度快、资源利用率大。本文基于回映应力积分算法,推导了三种该模型的应力更新算法公式,编写了相应的Fortran语言程序,实现了模型的数值计算。并且在同一工况中,相同模型参数下,运用不同积分算法对模型数值计算的结果进行对比和数据分析,考察不同算法对该模型计算结果的影响。同时,与试验结果进行对比,验证了回映应力积分算法的准确性和合理性。
然而,目前该弹塑性模型只在常规三轴应力条件下得到了验证,实际工程中土体所处的应力状态往往为三维应力状态,为了将修正模型从三轴试验的轴对称状态推广至真三轴应力状态,以期运用该模型分析嵌入式海洋工程结构与海洋土体的耦合动力响应,本文利用变换应力法实现了模型的三维化,并给出了相应的公式推导和数值实现方法。
The embedded offshore structures truly transmit the cyclic loading caused by winds, waves and currents to the surrounding marine soils. The marine soils are subjected to a two-way cyclic loading so that the soils involve both compression and extension states. The destruction of parts of the embedded offshore structures is due to extension deformation produced during the cycle loading. Therefore, it is significance to accurately simulate the dynamic characteristics between the soils and the embedded offshore structures.The current elastoplastic constitutive models always adopt an elliptic yield surface similar to the one in the Cam-Clay model. However, several studies have demonstrated that these kinds of models which adopt an elliptic yield surface are not suitable for predicting the characteristics of the soil under the extension state, due to its over-predicted elastic region. In this paper, combining the transformed stress tensors which are based on the spatially mobilized plane (SMP) yield criterion with the new hardening rule by the authors, a new bounding-surface plasticity model is proposed for the two-way cyclic behaviors of the saturated soils. The predicted results by the model show a good agreement with the experimental data both for the monotonic tests and the two-way cyclic triaxial tests.
In order to apply the revised elastoplastic constitutive models proposed by the author the program are developed for the implementation of the models into a language software FORTRAN by using the return mapping algorithm. In the same condition and model parameters, different stress integration are used for the constitutive model to verify the accuracy. The influences of the algorithms to the model are also checked.
However, the elasticplastic model is only demonstrated under triaxial condition at present. The stress state of soils is generally in the three-dimensional state . In order to analysis the dynamic responses produced by the embedded ocean offshore structures and the marine soils, the model is extended to the three-dimensional model by the transformed stress method.
引文
[1] Iwan D.W. On a class of models for the yielding behaviour of continuous and composite systems[J]. Journal of Applied Mechanics, 1967, 34:612–617.
[2] Mroz Z., Norris V.A., Zienkiewicz O.C. An anisotropic hardening model for soils and its application to cyclic loading[J]. International Journal For Numerical and Analytical Methods in Geomechenics, 1978, 2(3): 203-221.
[3] Mroz Z., Norris V.A., Zienkiewicz O.C. An anisotropic critical state model for soils subject to cyclic loading[J]. Geotechnique, 1981, 31(4): 451-469.
[4] Dafalias Y.F. and Popov E.P. A model of non-linearly hardening materials for complex loading[J]. Acta Mechanica, 1975.21, 173–192.
[5] Dafalias Y.F. and Hemnann L.R. Bounding surface formulation of soil plasticity. [C]// Soil Mechanics-Transient and Cyclic Loads. New York, John Wiley and Sons, 1980:253-282
[6] Dafalias Y.F. and Herrmann L.R. Bounding surface plasticity. I: Mathematical foundation and hypoplasticity[J]. Journal of Engineering Mechanics, 1986, 112(9): 966-987.
[7] Dafalias Y.F. and Herrmann L.R. Bounding surface plasticity. II: Application to isotropic cohesive Soils[J]. Journal of Engineering Mechanics, 1986, 112(12): 1292-1318.
[8] Crouch R.S., Wolf J.P., Dafalias Y.F. Unified critical-state bounding-surface plasticity model for soil[J]. Journal of Engineering Mechanics 1994, 120(11): 2251-2270.
[9] Crouch R.S. and Wolf J.P. Unified 3D critical state bounding-surface plasticity model for soils incorporating continuous plastic loading under cyclic paths. Part I: constitutive relations[J]. International Journal for Numerical and Analytical Methods in Geomechanics,1994,18:735-758.
[10] Crouch R.S. and Wolf J.P. Unified 3D critical state bounding-surface plasticity model for soils incorporating continuous plastic loading under cyclic paths. Part II: calibration and simulations[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1994, 18:759-784.
[11] Li T. and Meissner H. Two-surface plasticity model for cyclic undrained behavior of clays[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2002, 128(7): 613-626.
[12] Yu H.S., Khong C., Wang J. A unified plasticity model for cyclic behavior of clay and sand[J]. Mechanics Research Communications, 2007,34: 97-114.
[13]黄茂松,刘明,柳艳华.循环荷载下软黏土的各向异性边界面模型[J].水利学报,2009,40(2):188-193.
[14] Liang R.Y. and Ma F. Anisotropic plasticity model for undrained cyclic behavior of clays. I: Theory[J]. Journal of Geotechnical Engineering, 1992, 118(2): 229-245.
[15] Liang R.Y. and Ma F. Anisotropic plasticity model for undrained cyclic behavior of clays. II: Theory[J]. Journal of Geotechnical Engineering, 1992, 118(2): 246-265.
[16]胡存,刘海笑.适用于饱和粘土循环动力分析的新型边界面塑性模型[J].水利学报, 2011, 42(10): 1192-1200.
[17] Matsuoka H., Yao Y.P., Sun D.A. The Cam-clay models revised by the SMP criterion[J]. Soils and Foundations, 1999, 39(1): 81-95.
[18]孙德安,姚仰平,殷宗泽.基于SMP准则的双屈服面弹塑性模型的三维化[J].岩土工程学报, 1999, 21(5): 631-634.
[19]孙德安,姚仰平,殷宗泽.初始应力各向异性土的弹塑性模型[J].岩土力学, 2000, 21(3): 221-226.
[20]韩国城,连阵营,姚仰平.基于SMP准则的改进剑桥模型及其在基坑工程中的应用[J].大连理工大学学报, 2002, 42(1): 93-97.
[21]皱博,姚仰平,路德春.变换应力三维化方法在清华模型中的应用[J].岩石力学与工程学报, 2005, 24(23): 4303-4307.
[22]韩国城,连阵营,姚仰平.一个适用于深基坑开挖的三维各向异性模型[J].水利学报, 2002, 33(11): 14-19.
[23] Prevost J.H. Mathematical modeling of monotonic and cyclic undrained clay behavior[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1977, 1(2): 195-216.
[24] Prevost J.H. Anisotropic undrained stress-strain behavior of clays[J]. Journal of Geotechnical Engineering Division, ASCE, 1978, 104(GT8): 1075-1090.
[25] Prevost J.H. Plasticity theory for soil stress-strain behavior[J]. Journal of the Engineering Mechanics Division, ASCE, 1978, 104(EM5): 1177-1194.
[26] Krieg R.D. A practical two surface plasticity theory[J]. Journal of Application Mechanics, ASME, 1975, 42: 641-646.
[27] Mroz Z., Norris V.A., Zienkiewicz O.C. Application of an anisotropic hardening model in the analysis of elastoplastic deformation of soil. Geotechnique, 1979, 29(1): 1-34.
[28] Hashiguchi K. Two- and three-surface plasticity[C]. Proceedings of 5th International Conference on Numerical Methods in Geomechanics, Nagoya, 1985, (1): 285-292.
[29] Hashiguchi K., Chen Z.P., Tsutsumi S. and Sakajo S. Cyclic elastoplastic constitutive eqution of cohesive and noncohesive soils[C]. Proceedings of 6th International Symposium Numerical Analytical Methods in Geomechanics.
[30] Hashiguchi K. and Chen Z.P. Elastoplastic constitutive equation of soils with the subloading surface and the rotational hardening[J]. International Journal of Numerical Analytical Methods in Geomechanics, 1998,22(3): 197-227.
[31] Katti D.R. and Desai C.S. Modeling and testing of cohesive soil using disturbeb-state concept[J]. Journal of Geotechnical Engineering Division, ASCE, 1995,121(5): 648-658.
[32] Hashiguchi H. Subloading surface model in unconventional plasticity[J]. Journal of Solids Structure, 1989, 25(8): 917-945.
[33] Lee S.R. and Oh S. An anisotropic hardening constitutive model based on generalized isotropic hardening rule for modeling clay behavior[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1995, 19: 683-703.
[34] Anderson K.H., Pool J.H., Brown S.F. Cyclic and static laboratory tests on drammen clay[J]. Journal of the Geotechnical Engineering Division, ASCE, 1980(5), 106: 499-529.
[35] Byrne B.W. and Houlsby G.T. Experimental investigations of response of suction caissons to transient vertical loading[J]. Journal of Geotechnical and Environmental Engineering, 2002, 128(11): 926-939.
[36] Byrne B.W and Houlsby G.T. Experimental investigations of response of suction caissons to transient combine loading[J]. Journal of Geotechnical and Environmental Engineering, 2004, 130(3): 240-253.
[37] Sekiguchi H. and Ohta H. Induced anisotropy and time dependency in clay[A]. In :Proceedings of Specialty session 9th International Conference on Soil Mechanics and Foundation Engineering [C]. Tokyo, 1977: 229-238.
[38]胡存,刘海笑.考虑循环载荷下饱和粘土软化的损伤边界面模型[J].岩土力学(已录用).
[39] Stipho A.S.A. Experimental and theoretical investigation of the behavior of anisotropically consolidated kaolin[D]. Cardiff, University of Cardiff, 1978.
[40] DíAZ-RODRíGUEZ J.A., Moreno P., Salinas G. Undrainde shear behavior of Mexico city sediments during and after cyclic loading[J]. Canadian Geotechnical Journal, 1989, 26:(1):159-162.
[41] Simo J.C., Ortiz M. A unified approach to finite deformation elastoplastic analysis based on the use of hyper-elastic constitutive equations[J]. Computer Methods in Applied Mechanics and Engineering, 1985, 49: 221-245.
[42] Simo J.C., Taylor R.L. Consistent tangent operators for rate-independent elasto-plasticity[J]. Computer Methods in Applied Mechanics and Engineering, 1985, 48: 101-118.
[43] Simo J.C., Taylor R.L. Return mapping algorithm for plane stress elastoplasticity[J]. International Journal for Numerical Methods in Engineering, 1986, 22: 649-670.
[44] Simo J.C., Ju J.W., Pister K.S., Taylor R.L. An assessment of cap model: consistent return algorithms and rate-dependent extension[J]. Journal of Engineering Mechanics, ASCE, 1988, 114: 191-218.
[45] Simo J.C., Kennedy J.G., Govindjee S. Non-smooth multi-surface plasticity and viscoplasticity, Loading/unloadingconditions and numerical algorithms[J]. Computer Methods in Applied Mechanics and Engineering, 1988, 26: 2161-2185.
[46] Simo J.C., Hughes T.J.R. On the varitational foundations of assumed strain methods[J]. Journal of Applied Mechanics, 1986, 53(1): 51-55.
[47] Hughes T.J.R., Liu W.K. Implicit-explicit finite elements in transient analysis: 153stability theory[J]. Journal of Applied Mechanics, 1978, 45(2): 371-375.
[48] Oh S., Lee S.R. Formulation of implicit stress intergration and consistent tangent modulus for an anisotropic hardening constitutive model[J]. Computer Methods in Applied Mechanics and Engineering, 2001, 191: 255-27.
[49]甄文战.岩土材料变形局部化问题理论及数值分析研究[D].上海,上海大学, 2010.
[50] Oritz M., Popov E.P. Accuracy and stability of intergration algorithms for elastoplastic constitutive equations[J]. International Journal for Numerical Methods in Engrg, 1985, 21: 1561-1576.
[51] Majid T.M., Rung P. On intergration of a cyclic soil plasticity model[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2001, 25: 525-549.
[52] Moran B., Ortiz M., Shih C.F. Formulation of implicit finite-element methods for multiplicative finite deformation plasticity[J]. International Journal for Numerical Methods in Engineering, 1990, 29(3): 483-514.
[53] Zienkiewicz O.C., Pande G.N., Some useful forms of isotropic yield surface for soil and rock mechanics[A]. Finite Elements in Geomechnaics(Edited by Gudehus G )[C]. Wiley, London, 1977, 179-190.
[2] Mroz Z., Norris V.A., Zienkiewicz O.C. An anisotropic hardening model for soils and its application to cyclic loading[J]. International Journal For Numerical and Analytical Methods in Geomechenics, 1978, 2(3): 203-221.
[3] Mroz Z., Norris V.A., Zienkiewicz O.C. An anisotropic critical state model for soils subject to cyclic loading[J]. Geotechnique, 1981, 31(4): 451-469.
[4] Dafalias Y.F. and Popov E.P. A model of non-linearly hardening materials for complex loading[J]. Acta Mechanica, 1975.21, 173–192.
[5] Dafalias Y.F. and Hemnann L.R. Bounding surface formulation of soil plasticity. [C]// Soil Mechanics-Transient and Cyclic Loads. New York, John Wiley and Sons, 1980:253-282
[6] Dafalias Y.F. and Herrmann L.R. Bounding surface plasticity. I: Mathematical foundation and hypoplasticity[J]. Journal of Engineering Mechanics, 1986, 112(9): 966-987.
[7] Dafalias Y.F. and Herrmann L.R. Bounding surface plasticity. II: Application to isotropic cohesive Soils[J]. Journal of Engineering Mechanics, 1986, 112(12): 1292-1318.
[8] Crouch R.S., Wolf J.P., Dafalias Y.F. Unified critical-state bounding-surface plasticity model for soil[J]. Journal of Engineering Mechanics 1994, 120(11): 2251-2270.
[9] Crouch R.S. and Wolf J.P. Unified 3D critical state bounding-surface plasticity model for soils incorporating continuous plastic loading under cyclic paths. Part I: constitutive relations[J]. International Journal for Numerical and Analytical Methods in Geomechanics,1994,18:735-758.
[10] Crouch R.S. and Wolf J.P. Unified 3D critical state bounding-surface plasticity model for soils incorporating continuous plastic loading under cyclic paths. Part II: calibration and simulations[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1994, 18:759-784.
[11] Li T. and Meissner H. Two-surface plasticity model for cyclic undrained behavior of clays[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2002, 128(7): 613-626.
[12] Yu H.S., Khong C., Wang J. A unified plasticity model for cyclic behavior of clay and sand[J]. Mechanics Research Communications, 2007,34: 97-114.
[13]黄茂松,刘明,柳艳华.循环荷载下软黏土的各向异性边界面模型[J].水利学报,2009,40(2):188-193.
[14] Liang R.Y. and Ma F. Anisotropic plasticity model for undrained cyclic behavior of clays. I: Theory[J]. Journal of Geotechnical Engineering, 1992, 118(2): 229-245.
[15] Liang R.Y. and Ma F. Anisotropic plasticity model for undrained cyclic behavior of clays. II: Theory[J]. Journal of Geotechnical Engineering, 1992, 118(2): 246-265.
[16]胡存,刘海笑.适用于饱和粘土循环动力分析的新型边界面塑性模型[J].水利学报, 2011, 42(10): 1192-1200.
[17] Matsuoka H., Yao Y.P., Sun D.A. The Cam-clay models revised by the SMP criterion[J]. Soils and Foundations, 1999, 39(1): 81-95.
[18]孙德安,姚仰平,殷宗泽.基于SMP准则的双屈服面弹塑性模型的三维化[J].岩土工程学报, 1999, 21(5): 631-634.
[19]孙德安,姚仰平,殷宗泽.初始应力各向异性土的弹塑性模型[J].岩土力学, 2000, 21(3): 221-226.
[20]韩国城,连阵营,姚仰平.基于SMP准则的改进剑桥模型及其在基坑工程中的应用[J].大连理工大学学报, 2002, 42(1): 93-97.
[21]皱博,姚仰平,路德春.变换应力三维化方法在清华模型中的应用[J].岩石力学与工程学报, 2005, 24(23): 4303-4307.
[22]韩国城,连阵营,姚仰平.一个适用于深基坑开挖的三维各向异性模型[J].水利学报, 2002, 33(11): 14-19.
[23] Prevost J.H. Mathematical modeling of monotonic and cyclic undrained clay behavior[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1977, 1(2): 195-216.
[24] Prevost J.H. Anisotropic undrained stress-strain behavior of clays[J]. Journal of Geotechnical Engineering Division, ASCE, 1978, 104(GT8): 1075-1090.
[25] Prevost J.H. Plasticity theory for soil stress-strain behavior[J]. Journal of the Engineering Mechanics Division, ASCE, 1978, 104(EM5): 1177-1194.
[26] Krieg R.D. A practical two surface plasticity theory[J]. Journal of Application Mechanics, ASME, 1975, 42: 641-646.
[27] Mroz Z., Norris V.A., Zienkiewicz O.C. Application of an anisotropic hardening model in the analysis of elastoplastic deformation of soil. Geotechnique, 1979, 29(1): 1-34.
[28] Hashiguchi K. Two- and three-surface plasticity[C]. Proceedings of 5th International Conference on Numerical Methods in Geomechanics, Nagoya, 1985, (1): 285-292.
[29] Hashiguchi K., Chen Z.P., Tsutsumi S. and Sakajo S. Cyclic elastoplastic constitutive eqution of cohesive and noncohesive soils[C]. Proceedings of 6th International Symposium Numerical Analytical Methods in Geomechanics.
[30] Hashiguchi K. and Chen Z.P. Elastoplastic constitutive equation of soils with the subloading surface and the rotational hardening[J]. International Journal of Numerical Analytical Methods in Geomechanics, 1998,22(3): 197-227.
[31] Katti D.R. and Desai C.S. Modeling and testing of cohesive soil using disturbeb-state concept[J]. Journal of Geotechnical Engineering Division, ASCE, 1995,121(5): 648-658.
[32] Hashiguchi H. Subloading surface model in unconventional plasticity[J]. Journal of Solids Structure, 1989, 25(8): 917-945.
[33] Lee S.R. and Oh S. An anisotropic hardening constitutive model based on generalized isotropic hardening rule for modeling clay behavior[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1995, 19: 683-703.
[34] Anderson K.H., Pool J.H., Brown S.F. Cyclic and static laboratory tests on drammen clay[J]. Journal of the Geotechnical Engineering Division, ASCE, 1980(5), 106: 499-529.
[35] Byrne B.W. and Houlsby G.T. Experimental investigations of response of suction caissons to transient vertical loading[J]. Journal of Geotechnical and Environmental Engineering, 2002, 128(11): 926-939.
[36] Byrne B.W and Houlsby G.T. Experimental investigations of response of suction caissons to transient combine loading[J]. Journal of Geotechnical and Environmental Engineering, 2004, 130(3): 240-253.
[37] Sekiguchi H. and Ohta H. Induced anisotropy and time dependency in clay[A]. In :Proceedings of Specialty session 9th International Conference on Soil Mechanics and Foundation Engineering [C]. Tokyo, 1977: 229-238.
[38]胡存,刘海笑.考虑循环载荷下饱和粘土软化的损伤边界面模型[J].岩土力学(已录用).
[39] Stipho A.S.A. Experimental and theoretical investigation of the behavior of anisotropically consolidated kaolin[D]. Cardiff, University of Cardiff, 1978.
[40] DíAZ-RODRíGUEZ J.A., Moreno P., Salinas G. Undrainde shear behavior of Mexico city sediments during and after cyclic loading[J]. Canadian Geotechnical Journal, 1989, 26:(1):159-162.
[41] Simo J.C., Ortiz M. A unified approach to finite deformation elastoplastic analysis based on the use of hyper-elastic constitutive equations[J]. Computer Methods in Applied Mechanics and Engineering, 1985, 49: 221-245.
[42] Simo J.C., Taylor R.L. Consistent tangent operators for rate-independent elasto-plasticity[J]. Computer Methods in Applied Mechanics and Engineering, 1985, 48: 101-118.
[43] Simo J.C., Taylor R.L. Return mapping algorithm for plane stress elastoplasticity[J]. International Journal for Numerical Methods in Engineering, 1986, 22: 649-670.
[44] Simo J.C., Ju J.W., Pister K.S., Taylor R.L. An assessment of cap model: consistent return algorithms and rate-dependent extension[J]. Journal of Engineering Mechanics, ASCE, 1988, 114: 191-218.
[45] Simo J.C., Kennedy J.G., Govindjee S. Non-smooth multi-surface plasticity and viscoplasticity, Loading/unloadingconditions and numerical algorithms[J]. Computer Methods in Applied Mechanics and Engineering, 1988, 26: 2161-2185.
[46] Simo J.C., Hughes T.J.R. On the varitational foundations of assumed strain methods[J]. Journal of Applied Mechanics, 1986, 53(1): 51-55.
[47] Hughes T.J.R., Liu W.K. Implicit-explicit finite elements in transient analysis: 153stability theory[J]. Journal of Applied Mechanics, 1978, 45(2): 371-375.
[48] Oh S., Lee S.R. Formulation of implicit stress intergration and consistent tangent modulus for an anisotropic hardening constitutive model[J]. Computer Methods in Applied Mechanics and Engineering, 2001, 191: 255-27.
[49]甄文战.岩土材料变形局部化问题理论及数值分析研究[D].上海,上海大学, 2010.
[50] Oritz M., Popov E.P. Accuracy and stability of intergration algorithms for elastoplastic constitutive equations[J]. International Journal for Numerical Methods in Engrg, 1985, 21: 1561-1576.
[51] Majid T.M., Rung P. On intergration of a cyclic soil plasticity model[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2001, 25: 525-549.
[52] Moran B., Ortiz M., Shih C.F. Formulation of implicit finite-element methods for multiplicative finite deformation plasticity[J]. International Journal for Numerical Methods in Engineering, 1990, 29(3): 483-514.
[53] Zienkiewicz O.C., Pande G.N., Some useful forms of isotropic yield surface for soil and rock mechanics[A]. Finite Elements in Geomechnaics(Edited by Gudehus G )[C]. Wiley, London, 1977, 179-190.