非饱和砂土及黏土的水-力耦合双屈服面模型

李舰; 刘凯; 尹振宇; 崔玉军; 殷建华

doi:10.11779/CJGE202001008

非饱和砂土及黏土的水-力耦合双屈服面模型 English Version

李舰^1,,
刘凯²,
尹振宇^2, ,,
崔玉军³,
殷建华²

1.
北京交通大学土木建筑工程学院,北京100044
2.
香港理工大学土木与环境工程学系,香港
3.
法国国立路桥大学,巴黎

基金项目:

中央高校基本科研业务费专项资金项目 2019JBM083

详细信息

作者简介:
李舰（1985— ）,男,讲师,主要从事非饱和土本构模型及其数值计算方面的研究工作。E-mail：jianli@bjtu.edu.cn

通讯作者:
尹振宇, E-mail：zhenyu.yin@polyu.edu.hk; zhenyu.yin@gmail.com

中图分类号: TU43
计量
- 文章访问数: 402
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- PDF下载量: 409
出版历程
- 收稿日期: 2018-08-20
- 网络出版日期: 2022-12-07
- 刊出日期: 2019-12-31

Hydro-mechanical double-yield-surface model for unsaturated sand and clay

1.
School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
2.
Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, China
3.
Ecole des Ponts ParisTech, Laboratoire Navier/CERMES, Paris, France

摘要

摘要: 使用精细的本构模型来同时分析非饱和砂土和黏土水力耦合特性依旧是一个挑战。提出了一种包含加载湿陷及剪切滑移双重塑性机制的水-力耦合模型。力学模型中选取Bishop应力作为应力变量,采用孔隙比和有效饱和度作为状态变量。模型中给出了孔隙比-平均土骨架应力的半对数坐标系内依赖于有效饱和度的临界状态线的表达式,其与剪切滑动屈服面的非关联流动准则的结合保证了模型可以更合理地预测非饱和土（包括砂土和黏土）的剪胀和剪缩性质。通过模拟粉砂和高岭土的三轴试验,对该模型预测非饱和土主要特征的能力进行了分析。
- 剪胀性 /
- 有效饱和度 /
- 临界状态 /
- 水-力耦合 /
- 非饱和土
Abstract: Hydro-mechanical coupling behavior analysis with a sophisticated model for both unsaturated sand and clay is still a challenge. In this study, a hydro-mechanical coupling model with two plastic deformation mechanisms, i.e., loading collapse and shear sliding, for unsaturated soils is formulated using the Bishop’s stress as the stress variable and using the void ratio and the effective degree of saturation as the state variables. An expression for the critical state line related to the effective degree of saturation in the void ratio-soil skeleton stress semilog plane is explicitly implemented, which is combined with a non-associated flow rule for the shear sliding yield surface to guarantee the satisfactory simulation of the dilation or contraction during shear for unsaturated soils including sand and clay. The predictive capability of the model to reproduce the main features of unsaturated soil behavior is analyzed by simulating the triaxial tests on silty sand and kaolin.
- dilatancy /
- effective degree of saturation /
- critical state /
- hydro-mechanical coupling /
- unsaturated soil

HTML全文

图 1 在 $p^{'}$ -q平面上加载湿陷边界面、加载湿陷当前应力面.和剪切滑动屈服面

Figure 1. Loading collapse bounding surface, loading collapse current stress surface and shear-sliding yield surface on $p^{'}$ -q plane

下载: 全尺寸图片幻灯片

图 2 压实高岭土的临界状态线以及截距与有效饱和度间关系^[40]

Figure 2. Critical state lines, and relationship between intercept and effective degree of saturation of compacted Speswhite kaolin^[40]

下载: 全尺寸图片幻灯片

图 3 土持水特征曲线模型

Figure 3. Model for water-retention behaviors

下载: 全尺寸图片幻灯片

图 4 粉砂在不同基质吸力和净平均应力下排水三轴试验结果.与模拟结果对比

Figure 4. Comparison between measured and computed results from drained triaxial compression tests at different suctions and net confining stresses on silty sand

下载: 全尺寸图片幻灯片

图 5 高岭土在不同基质吸力和净平均应力下排水三轴试验结果与模拟结果对比

Figure 5. Comparison between measured and computed results from drained triaxial compression tests at different suctions and net confining stresses on kaolin

下载: 全尺寸图片幻灯片

表 1 粉砂和高岭土模型参数及初始状态参数表

Table 1 Values of model parameters and initial values of soil state for tests on silty sand and kaolin

类型	$λ$	$κ$	$ν$	$b_{sw}$	$M$	$G_{p0}$
粉砂	0.019	0.003	0.3	5.5	1.523	2000
高岭土	0.110	0.011	0.3	3.0	0.793	40
类型	$Γ_{csat}$	$a_{Γ}$	$n_{G}$	$n_{p}$	$D$	$n_{pt}$
粉砂	0.560	0.140	1.0	2.5	1.0	30.0
高岭土	2.000	0.490	1.0	2.0	0.7	2.0
类型	$λ_{w}$	$κ_{w}$	$p_{a}$	$b_{ws}$	$S_{rres}$ /%
粉砂	0.04	0.001	1.0	3.2	45.5
高岭土	0.23	0.010	101.3	1.1	25.0
类型	$e_{0}$	$ρ_{d}$ /(g·cm^-3)	$S_{r0}$ /%	${p^{'}}_{c}^{c}$ /kPa	$s_{i}$ /kPa	$s_{d}$ /kPa
粉砂	0.347	2.00	75.0	450	1000	800
高岭土	1.206	1.20	60.1	210	350	300

下载: 导出CSV

参考文献(49)

[1]	陈正汉. 重塑非饱和黄土的变形、强度、屈服和水量变化特性[J]. 岩土工程学报, 1999, 21(1): 82-90. doi: 10.3321/j.issn:1000-4548.1999.01.018 CHEN Zheng-han. Deformation, strength, yield and moisture change of a remolded unsaturated loess[J]. Chinese Journal of Geotechnical Engineering, 1999, 21(1): 82-90. (in Chinese) doi: 10.3321/j.issn:1000-4548.1999.01.018
[2]	黄海, 陈正汉, 李刚. 非饱和土在p-s平面上的屈服轨迹及土-水特征曲线的探讨[J]. 岩土力学, 2000, 21(4): 316-321. doi: 10.3969/j.issn.1000-7598.2000.04.002 HUANG Hai, CHEN Zheng-han, LI Gang. A study on yield locus of unsaturated soil on p-s plane and soil-water characteristic curve[J]. Rock and Soil Mechanics, 2000, 21(4): 316-321. (in Chinese) doi: 10.3969/j.issn.1000-7598.2000.04.002
[3]	詹良通, 吴宏伟. 吸力对非饱和膨胀土抗剪强度及剪胀特性的影响[J]. 岩土工程学报, 2007, 29(1): 82-87. doi: 10.3321/j.issn:1000-4548.2007.01.013 ZHAN Liang-tong, NG Charles W W. Effect of suction on shear strength and dilatancy of an unsaturated expansive clay[J]. Chinese Journal of Geotechnical Engineering, 2007, 29(1): 82-87. (in Chinese) doi: 10.3321/j.issn:1000-4548.2007.01.013
[4]	姚仰平, 牛雷, 韩黎明, 等. 超固结非饱和土的试验研究[J]. 岩土力学, 2011, 32(6): 1601-1606. doi: 10.3969/j.issn.1000-7598.2011.06.001 YAO Yang-ping, NIU Lei, HAN Li-ming, et al. Experimental study of behaviors of overconsolidated unsaturated clays[J]. Rock and Soil Mechanics, 2011, 32(6): 1601-1606. (in Chinese) doi: 10.3969/j.issn.1000-7598.2011.06.001
[5]	陈存礼, 张登飞, 董玉柱, 等. 常含水率三轴条件下非饱和原状黄土的吸力和力学特性[J]. 岩土工程学报, 2014, 36(7): 1195-1202. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201407004.htm CHEN Cun-li, ZHANG Deng-fei, DONG Yu-zhu, et al. Suction and mechanical behaviours of unsaturated intact loess from constant water content triaxial tests[J]. Chinese Journal of Geotechnical Engineering, 2014, 36(7): 1195-1202. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201407004.htm
[6]	MA T, WEI C, WEI H. Hydraulic and mechanical behavior of unsaturated silt: experimental and theoretical characterization[J]. International Journal of Geomechanics, 2016, 16(6): D4015007. doi: 10.1061/(ASCE)GM.1943-5622.0000576
[7]	高登辉, 陈正汉, 郭楠, 等. 干密度和基质吸力对重塑非饱和黄土变形与强度特性的影响[J]. 岩石力学与工程学报, 2017, 36(3): 736-744. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201703023.htm GAO Deng-hui, CHEN Zheng-han, GUO Nan, et al. The influence of dry density and matric suction on the deformation and the strength characteristics of the remolded unsaturated loess soils[J]. Chinese Journal of Rock Mechanics and Engineering, 2017, 36(3): 736-744. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201703023.htm
[8]	徐筱, 赵成刚. 高吸力下黏性土的抗剪强度和体变特性[J]. 岩土力学, 2018, 39(5): 1598-1610. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201805007.htm XU Xiao, ZHAO Cheng-gang. Shear strength and volume change behavior of clay-rich soil at high suctions[J]. Rock and Soil Mechanics, 2018, 39(5): 1598-1610. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201805007.htm
[9]	陈皓, 吕海波, 陈正汉, 等. 考虑温度影响的高庙子膨润土强度与变形特性试验研究[J]. 岩石力学与工程学报, 2018, 37(8): 1962-1979. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201808019.htm CHEN Hao, LÜ Hai-bo, CHEN Zheng-han, et al. Strength and volume change of buffer material under high temperature and pressure[J]. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(8): 1962-1979. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201808019.htm
[10]	朱青青, 苗强强, 陈正汉, 等. 非饱和含黏砂土的弹塑性剪胀特性研究[J]. 岩土工程学报, 2018, 40(增刊1): 65-72. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC2018S1012.htm ZHU Qing-qing, MIAO Qiang-qiang, CHEN Zheng-han, et al. Elastoplastic dilatancy relations of unsaturated clayey sand[J]. Chinese Journal of Geotechnical Engineering, 2018, 40(S1): 65-72. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC2018S1012.htm
[11]	BISHOP A W. The principle of effective stress[J]. Teknisk Ukeblad, 1959, 39: 113-143.
[12]	GALLIPOLI D, GENS A, SHARMA R, et al. An elasto-plastic model for unsaturated soil incorporating the effects of suction and degree of saturation on mechanical behaviour[J]. Géotechnique, 2003, 53(1): 123-135. doi: 10.1680/geot.2003.53.1.123
[13]	WHEELER S J, SHARMA R S, BUISSON M S R. Coupling of hydraulic hysteresis and stress-strain behaviour in unsaturated soils[J]. Géotechnique, 2003, 53(1): 41-54. doi: 10.1680/geot.2003.53.1.41
[14]	FREDLUND D G, MORGENSTERN N R. Stress state variables for unsaturated soils[J]. Journal of Geotechnical Engineering, 1977, 103(GT5): 447-466.
[15]	TARATINO A, MONGIOVÕÁ L, BOSCO G. An experimental investigation on the independent isotropic stress variables for unsaturated soils[J]. Géotechnique, 2000, 50(3): 275-282. doi: 10.1680/geot.2000.50.3.275
[16]	张龙, 陈正汉, 周凤玺, 等. 非饱和土应力状态变量试验验证研究[J]. 岩土工程学报, 2017, 39(2): 380-384. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201702029.htm ZHANG Long, CHEN Zheng-han, ZHOU Feng-xi, et al. Test verification of stress state variables for unsaturated soils[J]. Chinese Journal of Geotechnical Engineering, 2017, 39(2): 380-384. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201702029.htm
[17]	张龙, 陈正汉, 周凤玺, 等. 从变形、水量变化和抗剪强度三个方面验证非饱和土的两个应力状态变量[J]. 岩土工程学报, 2017, 39(5): 906-915. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201705021.htm ZHANG Long, CHEN Zheng-han, ZHOU Feng-xi, et al. Verification of rationality of two stress state variables of unsaturated soil from deformation, moisture change and strength[J]. Chinese Journal of Rock Mechanics and Engineering, 2017, 39(2): 380-384. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201705021.htm
[18]	ZHOU A N, SHENG D, SLOAN S W, et al. Interpretation of unsaturated soil behaviour in the stress-saturation space I: volume change and water retention behaviour[J]. Computers and Geotechnics, 2012, 43: 178-187. doi: 10.1016/j.compgeo.2012.04.010
[19]	ZHANG F, IKARIYA T. A new model for unsaturated soil using skeleton stress and degree of saturation as state variables[J]. Soils and Foundations, 2011, 51(1): 67-81. doi: 10.3208/sandf.51.67
[20]	ALONSO E E, GENS A, JOSA A. A constitutive model for partially saturated soils[J]. Géotechnique, 1990, 40(3): 405-430. doi: 10.1680/geot.1990.40.3.405
[21]	孙德安. 非饱和土的水力和力学特性及其弹塑性描述[J]. 岩土力学, 2009, 30(11): 3217-3231. doi: 10.3969/j.issn.1000-7598.2009.11.001 SUN De-an. Hydro-mechanical behaviours of unsaturated soils and their elastoplastic modelling[J]. Rock and Soil Mechanics, 2009, 30(11): 3217-3231. (in Chinese) doi: 10.3969/j.issn.1000-7598.2009.11.001
[22]	马田田, 韦昌富, 陈盼, 等. 非饱和土毛细滞回与变形耦合弹塑性本构模型[J]. 岩土力学, 2012, 33(11): 3263-3270. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201211011.htm MA Tian-tian, WEI Chang-fu, CHEN Pan, et al. An elastoplastic constitutive model of unsaturated soils with capillary hysteresis and deformation coupling[J]. Rock and Soil Mechanics, 2012, 33(11): 3263-3270. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201211011.htm
[23]	LI J, YIN Z Y, CUI Y J, et al. Work input analysis for soils with double porosity and application to the hydro-mechanical modeling of unsaturated expansive clays[J]. Canadian Geotechnical Journal, 2017, 54(2): 173-187. doi: 10.1139/cgj-2015-0574
[24]	YAO Y P, NIU L, CUI W J. Unified hardening (UH) model for overconsolidated unsaturated soils[J]. Canadian Geotechnical Journal, 2014, 51(7): 810-821.
[25]	LI X S. Thermodynamics-based constitutive framework for unsaturated soils: 2 a basic triaxial model[J]. Géotechnique, 2007, 57(5): 423-435.
[26]	卢再华, 陈正汉, 曹继东. 原状膨胀土的强度变形特性及其本构模型研究[J]. 岩土力学, 2001, 22(3): 339-342. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200103023.htm LU Zai-hua, CHEN Zheng-han, CAO Ji-dong. A study on the strength and deformation characteristics and the constitutive model of natural expansive soils[J]. Rock and Soil Mechanics, 2001, 22(3): 339-342. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200103023.htm
[27]	卢再华, 陈正汉. 非饱和原状膨胀土的弹塑性损伤本构模型研究[J]. 岩土工程学报, 2003, 25(4): 422-426. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC200304008.htm LU Zai-hua, CHEN Zheng-han. An elastoplastic damage constitutive model of unsaturated undisturbed expansive soil[J]. Chinese Journal of Geotechnical Engineering, 2003, 25(4): 422-426. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC200304008.htm
[28]	CHIU C F, NG C W W. A state-dependent elasto-plastic model for saturated and unsaturated soils[J]. Géotechnique, 2003, 53(9): 809-829.
[29]	殷宗泽. 一个土体的双屈服面应力-应变模型[J]. 岩土工程学报, 1988, 10(4): 64-71. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC198804006.htm YIN Zong-ze. A stress-strain model of soil with double yield surfaces[J]. Chinese Journal of Geotechnical Engineering, 1988, 10(4): 64-71. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC198804006.htm
[30]	YIN Z Y, XU Q, HICHER P Y. A simple critical-state-based double-yield-surface model for clay behavior under complex loading[J]. Acta Geotechnica, 2013, 8(5): 509-523.
[31]	ALONSO E E, PEREIRA M J, VAUNAT J, et al. A microstructurally-based effective stress for unsaturated soils[J]. Géotechnique, 2010, 60(12): 913-925.
[32]	KHALILI N, ZARGARBASHI S. Influence of hydraulic hysteresis on effective stress in unsaturated soils[J]. Géotechnique, 2010, 60(9): 729-734.
[33]	LU N, GODT J W, WU D T. A closed-form equation for effective stress in unsaturated soil[J]. Water Resources Research, 2010, 46(5): W05515.
[34]	陈正汉, 郭楠. 非饱和土与特殊土力学及工程应用研究的新进展[J]. 岩土力学, 2019, 40(1): 1-54. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201901002.htm CHEN Zheng-han, GUO Nan. New developments of mechanics and application for unsaturated soils and special soils[J]. Rock and Soil Mechanics, 2019, 40(1): 1-54. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201901002.htm
[35]	CHRISTENSON H K. Capillary condensation due to van der Waals attraction in wet slits[J]. Physical Review Letters, 1994, 73(13): 1821-1824.
[36]	TULLER M, OR D, DUDLEY L M. Adsorption and capillary condensation in porous media: liquid retention and interfacial configurations in angular pores[J]. Water Resources Research, 1999, 35(7): 1949-1964.
[37]	TULLER M, OR D. Water films and scaling of soil characteristic curves at low water contents[J]. Water Resources Research, 2005, 41(9): W09403.
[38]	KONRAD J M, LEBEAU M. Capillary-based effective stress formulation for predicting shear strength of unsaturated soils[J]. Canadian Geotechnical Journal, 2015, 52(12): 2067-2076.
[39]	ZHOU A N, HUANG R Q, SHENG D. Capillary water retention curve and shear strength of unsaturated soils[J]. Canadian Geotechnical Journal, 2016, 53(6): 974-987.
[40]	SIVAKUMAR V. A Critical State Framework for Unsaturated Soil[D]. Sheffield: University of Sheffield, 1993.
[41]	WEI C F, DEWOOLKAR M M. Formulation of capillary hysteresis with internal state variables[J]. Water Resources Research, 2006, 42(7): W074051-16.
[42]	胡冉, 陈益峰, 周创兵. 基于孔隙分布的变形土土水特征曲线模型[J]. 岩土工程学报, 2013, 35(8): 1451-1462. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201308013.htm HU Ran, CHEN Yi-feng, ZHOU Chuang-bing. A water retention curve model for deformable soils based on pore size distribution[J]. Chinese Journal of Geotechnical Engineering, 2013, 35(8): 1451-1462. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201308013.htm
[43]	张俊然, 许强, 孙德安. 多次干湿循环后土-水特征曲线的模拟[J]. 岩土力学, 2014, 35(3): 689-695. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201403013.htm ZHANG Jun-ran, XU Qiang, SUN De-an. Simulation of soil-water characteristic curves during drying and wetting cycles[J]. Rock and Soil Mechanics, 2014, 35(3): 689-695. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201403013.htm
[44]	刘艳, 赵成刚, 李舰, 等. 相间交界面对非饱和土应力状态的影响[J]. 力学学报, 2017, 49(2): 335-343. https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB201702010.htm LIU Yan, ZHAO Cheng-gang, LI Jian, et al. The influence of interfaces on the stress state in unsaturated soils[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(2): 335-343. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-LXXB201702010.htm
[45]	SHENG D, SLOAN S W, GENS A, et al. Finite element formulation and algorithms for unsaturated soils: part I theory[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2003, 27(9): 745-765.
[46]	李锡夔, 范益群. 非饱和土变形及渗流过程的有限元分析[J]. 岩土工程学报, 1998, 20(4): 20-24. LI Xi-kui, FAN Yi-qun. Finite element analysis of deformation and seepage process in unsaturated soils[J]. Chinese Journal of Geotechnical Engineering, 1998, 20(4): 20-24. (in Chinese)
[47]	李舰, 赵成刚, 刘艳, 等. 适用于膨胀性非饱和土的边界面模型的数值实现[J]. 岩石力学与工程学报, 2017, 36(10): 215-226. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201710024.htm LI Jian, ZHAO Cheng-gang, LIU Yan, et al. Numerical implementation of a bounding surface model for unsaturated expansive clays[J]. Chinese Journal of Rock Mechanics and Engineering, 2017, 36(10): 215-226. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201710024.htm
[48]	李舰, 王鹏月, 海路, 等. 膨胀性非饱和土本构模型的隐式和显式积分算法的比较[J]. 岩石力学与工程学报, 2018, 37(7): 1731-1740. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201807017.htm LI Jian, WANG Peng-yue, HAI Lu, et al. Implicit and explicit integration schemes of a constitutive model for unsaturated expansive clays[J]. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(7): 1731-1740. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201807017.htm
[49]	RAMPINO C, MANCUSO C, VINALE F. Experimental behaviour and modelling of an unsaturated compacted soil[J]. Canadian Geotechnical Journal, 2000, 37(4): 748-763.

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期刊类型引用(5)

1.	李晓强，梁靖宇，路德春，苗金波，杜修力. 非饱和土的非正交弹塑性本构模型. 中国科学:技术科学. 2022(07): 1048-1064 . 百度学术
2.	刘祎，蔡国庆，李舰，赵成刚. 非饱和土热–水–力全耦合本构模型及其验证. 岩土工程学报. 2021(03): 547-555 . 本站查看
3.	王壹敏，陈志敏，孙胜旗，赵运铎，张常书. 基于邓肯-张模型的低液限粉质黏土-砂的强度规律. 科学技术与工程. 2021(08): 3252-3257 . 百度学术
4.	胡小荣，蔡晓锋，瞿鹏. 基于坐标平移法的正常固结非饱和土三剪弹塑性本构模型. 应用数学和力学. 2021(08): 813-831 . 百度学术
5.	杨光昌，白冰，刘洋，陈佩佩. 描述饱和砂土剪切特性的一个热力学本构模型. 哈尔滨工业大学学报. 2021(11): 93-100 . 百度学术