考虑相变状态及温度效应的冻结砂土非关联弹塑性本构模型

doi:10.3724/1000-6915.jrme.2024.0748

Abstract
Figure/Table
References (0)
Related Citation (15)

Download: PDF (854 KB) HTML (1 KB)
Export: BibTeX | EndNote (RIS)

Abstract Frost heave is the main factor affecting the mechanical properties of frozen soil，which will lead to more significant strain softening and dilatancy characteristics of frozen soil during the shear process. To accurately describe the mechanical characteristics of frozen soil，this paper introduces the concept of phase transition state concept to describe the dilatancy of frozen soil，solving the problem of difficult to unify the softening and dilatancy characteristic point displacement laws using only critical state theory. Furthermore，by combining the fractional order differentiation，the non-associated plastic flow law of stress-strain of frozen soil is described，avoiding the additional construction of plastic potential surfaces. The mechanical response of frozen soil is described by introducing the yield function considering the influence of temperature. Based on the hardening rule considering phase transformation and the yield condition considering temperature effect，an improved elasto-plastic constitutive model is proposed to evaluate the mechanical properties of frozen sand. The model unifies the temperature effect，phase transition state，critical state and non-associated plastic flow in the theoretical framework of elasto-plastic mechanics，and all model parameters have clear physical meaning. Finally，by simulating the stress-strain behavior of frozen sand，and comparing with the experimental data，the results show that the constitutive model can effectively capture the phase transition and strain softening law of frozen sand at different temperatures and confining pressures.

Key words： soil mechanics critical stress phase transition state temperature effect non-associated plastic flow frozen sand

	Service

	E-mail this article
	Add to my bookshelf
	Add to citation manager
	E-mail Alert
	RSS
	Articles by authors
	WANG Yuke1
	2
	3
	JIANG Rui1
	2
	3
	GUO Chengchao4
	WANG Fuming1
	2
	3
	4

Cite this article:

WANG Yuke1,2,3, et al. Non-associated elasto-plastic constitutive model of frozen sand considering the influence of phase transition state and temperature[J]. , 2025, 44(4): 1026-1039.

URL:

https://rockmech.whrsm.ac.cn/EN/10.3724/1000-6915.jrme.2024.0748 OR https://rockmech.whrsm.ac.cn/EN/Y2025/V44/I4/1026

[19]	路德春，金辰逸，梁靖宇，等. 考虑状态相关的砂土非正交弹塑性本构模型[J]. 岩土工程学报，2023，45(2)：221-231.(LU Dechun，JIN Chenyi，LIANG Jingyu，et al. State-dependent non-orthogonal elastoplastic constitutive model for sand[J]. Chinese Journal of Geotechnical Engineering，2023，45(2)：221-231.(in Chinese))
[14]	梁靖宇，杜修力，路德春，等. 特征应力空间中土的分数阶临界状态模型[J]. 岩土工程学报，2019，41(3)：581-587.(LIANG Jingyu，DU Xiuli，LU Dechun，et al. Fractional-order critical state model for soils in characteristic stress space[J]. Chinese Journal of Geotechnical Engineering，2019，41(3)：581-587.(in Chinese))
[1]	马巍，王大雁. 中国冻土力学研究50 a回顾与展望[J]. 岩土工程学报，2012，34(4)：625-640.(MA Wei，WANG Dayan. Studies on frozen soil mechanics in China in past 50 years and their prospect[J]. Chinese Journal of Geotechnical Engineering，2012，34(4)：625-640.(in Chinese))
[20]	LAI Y，LIAO M，HU K. A constitutive model of frozen saline sandy soil based on energy dissipation theory[J]. International Journal of Plasticity，2016，78：84-113.
[15]	LIANG J，LU D，ZHOU X，et al. Non-orthogonal elastoplastic constitutive model with the critical state for clay[J]. Computers and Geotechnics，2019，116：103200.
[2]	郭利娜. 冻土理论研究进展[J]. 水利水电技术(中英文)，2019，50(3)：145-154.(GUO Lina. Research progress of frozen soil theory[J]. Water Resources and Hydropower Engineering，2019，50(3)：145-154.(in Chinese))
[21]	董晓丽，赵成刚，张卫华. 考虑相变状态的较密实饱和砂土弹塑性模型[J]. 工程力学，2017，34(1)：51-57.(DONG Xiaoli，ZHAO Chenggang，ZHANG Weihua. The saturated dense sand elastic-plastic model considering phase transition state[J]. Engineering Mechanics，2017，34(1)：51-57.(in Chinese))
[16]	孙增春，刘汉龙，肖杨. 砂-粉混合料的分数阶塑性本构模型[J]. 岩土工程学报，2024，46(8)：1 596-1 604.(SUN Zengchun，LIU Hanlong，XIAO Yang. Fractional-order plasticity model of sand-silt mixtures[J]. Chinese Journal of Geotechnical Engineering，2024，46(8)：1 596-1 604.(in Chinese))
[3]	李顺群，高凌霞，柴寿喜. 冻土力学性质影响因素的显著性和交互作用研究[J]. 岩土力学，2012，33(4)：1 173-1 177.(LI Shunqun，GAO Lingxia，CHAI Shouxi. Significance and interaction of factors on mechanical properties of frozen soil[J]. Rock and Soil Mechanics，2012，33(4)：1 173-1 177.(in Chinese))
[22]	DE Z，LIU E，LIU X，et al. A new strength criterion for frozen soils considering the influence of temperature and coarse-grained contents[J]. Cold Regions Science and Technology，2017，143：1-12.
[17]	汪成贵，束善治，肖杨，等. 考虑钙质砂颗粒破碎的分数阶边界面本构模型[J]. 岩土工程学报，2023，45(6)：1 162-1 170.(WANG Chenggui，SHU Shanzhi，XIAO Yang，et al. Fractional-order bounding surface model considering breakage of calcareous sand [J]. Chinese Journal of Geotechnical Engineering，2023，45(6)：1 162-1 170.(in Chinese))
[4]	刘勤龙，李旭，姚兆明，等. 冻土强度特性及其主控因素综述[J]. 冰川冻土，2023，45(3)：1 092-1 104.(LIU Qinlong，LI Xu，YAO Zhaoming，et al. Strength characteristics of frozen soil and the main controlling factors：a review[J]. Journal of Glaciology and Geocryology，2023，45(3)：1 092-1 104.(in Chinese))
[23]	HUECKEL T，FRANCOIS B，LALOUI L. Explaining thermal failure in saturated clays[J]. Géotechnique，2009，59(3)：197-212.
[18]	路德春，韩佳月，梁靖宇，等. 横观各向同性黏土的非正交弹塑性本构模型[J]. 岩石力学与工程学报，2020，39(4)：793-803.(LU Dechun，HAN Jiayue，LIANG Jingyu，et al. Non-orthogonal elastoplastic constitutive model of transversely isotropic clay[J]. Chinese Journal of Rock Mechanics and Engineering，2020，39(4)：793-803.(in Chinese))
[5]	刘昉，戚园春，侯庆志，等. 冻土条件下无黏性土堤漫顶溃决试验研究[J]. 水利水电技术(中英文)，2021，52(12)：146-156.(LIU Fang，QI Yuanchun，HOU Qingzhi，et al. Experimental study on overtopping failure of non-cohesive embankment under frozen soil condition[J]. Water Resources and Hydropower Engineering，2021，52(12)：146-156.(in Chinese))
[24]	ROSCOE K H，SCHOFIELD A N，THURAIRAJAH A. Yielding of clays in states wetter than critical[J]. Géotechnique，1963，13(3)：211-240.
[19]	路德春，金辰逸，梁靖宇，等. 考虑状态相关的砂土非正交弹塑性本构模型[J]. 岩土工程学报，2023，45(2)：221-231.(LU Dechun，JIN Chenyi，LIANG Jingyu，et al. State-dependent non-orthogonal elastoplastic constitutive model for sand[J]. Chinese Journal of Geotechnical Engineering，2023，45(2)：221-231.(in Chinese))
[6]	齐吉琳，马巍. 冻土的力学性质及研究现状[J]. 岩土力学，2010，31(1)：133-143.(QI Jilin，MA Wei. State-of-art of research on mechanical properties of frozen soils[J]. Rock and Soil Mechanics，2010，31(1)：133-143.(in Chinese))
[25]	LI X，WANG Y. Linear representation of steady-state line for sand[J]. Journal of Geotechnical and Geoenvironmental Engineering，1998，124(12)：1 215-1 217.
[20]	LAI Y，LIAO M，HU K. A constitutive model of frozen saline sandy soil based on energy dissipation theory[J]. International Journal of Plasticity，2016，78：84-113.
[7]	CHAMBERLAIN E，GROVES C，PERHAM J. The mechanical behavior of frozen earth materials under high pressure triaxial test conditions[J]. Geotechnique，1972，22(3)：469-483.
[26]	LI X，DAFALIAS Y F. Dilatancy for cohesionless soils[J]. Géotechnique，2000，50(4)：449-460.
[21]	董晓丽，赵成刚，张卫华. 考虑相变状态的较密实饱和砂土弹塑性模型[J]. 工程力学，2017，34(1)：51-57.(DONG Xiaoli，ZHAO Chenggang，ZHANG Weihua. The saturated dense sand elastic-plastic model considering phase transition state[J]. Engineering Mechanics，2017，34(1)：51-57.(in Chinese))
[8]	马付龙，刘恩龙，王丹，等. 高温高含冰量冻结砂土的三轴压缩力学特性[J]. 哈尔滨工业大学学报，2024，56(7)：124-131.(MA Fulong，LIU Enlong，WANG Dan，et al. Experimental study on triaxial compression mechanical characteristics of warm and ice-rich frozen sand[J]. Journal of Harbin Institute of Technology，2024，56(7)：124-131.(in Chinese))
[27]	BEEN K，JEFFERIES M G. A state parameter for sands[J]. Géotechnique，1985，35(2)：99-112.
[22]	DE Z，LIU E，LIU X，et al. A new strength criterion for frozen soils considering the influence of temperature and coarse-grained contents[J]. Cold Regions Science and Technology，2017，143：1-12.
[9]	XU X，WANG Y，BAI R，et al. Comparative studies on mechanical behavior of frozen natural saline silty sand and frozen desalted silty sand[J]. Cold Regions Science and Technology，2016，132：81-88.
[28]	刘振亚，刘建坤，李旭，等. 毛细黏聚与冰胶结作用对非饱和粉质黏土冻结强度及变形特性的影响[J]. 岩石力学与工程学报，2018，37(6)：1 551-1 559.(LIU Zhenya，LIU Jiankun，LI Xu，et al. Effect of capillary cohesion and ice cementation on strength and deformation of unsaturated frozen silty clay[J]. Chinese Journal of Rock Mechanics and Engineering，2018，37(6)：1 551-1 559.(in Chinese))
[23]	HUECKEL T，FRANCOIS B，LALOUI L. Explaining thermal failure in saturated clays[J]. Géotechnique，2009，59(3)：197-212.
[10]	雷乐乐，王大雁，李栋伟，等. 考虑应力水平影响的冻结黏土变形特性[J]. 岩石力学与工程学报，2021，40(增1)：2 905-2 912.(LEI Lele，WANG Dayan，LI Dongwei，et al. Deformation characteristics of frozen clay with considering the influence of mean principal stress[J]. Chinese Journal of Rock Mechanics and Engineering，2021，40(Supp.1)：2 905-2 912.(in Chinese))
[29]	姚仰平，侯伟，罗汀. 土的统一硬化模型[J]. 岩石力学与工程学报，2009，28(10)：2 135-2 151.(YAO Yangping，HOU Wei，LUO Ting. Unified hardening model for soils[J]. Chinese Journal of Rock Mechanics and Engineering，2009，28(10)：2 135-2 151.(in Chinese))
[24]	ROSCOE K H，SCHOFIELD A N，THURAIRAJAH A. Yielding of clays in states wetter than critical[J]. Géotechnique，1963，13(3)：211-240.
[11]	姚兆明，蹇膨远，郭梦圆. 冻结黏土分数阶损伤蠕变模型参数确定及验证[J]. 科学技术与工程，2023，23(31)：13 507-13 514.(YAO Zhaoming，JIAN Pengyuan，GUO Mengyuan. Parameter determination and verification of fractional damage creep model frozen of clay[J]. Science Technology and Engineering，2023，23(31)：13 507-13 514. (in Chinese))
[30]	YAO Y，SUN D，MATSUOKA H. A unified constitutive model for both clay and sand with hardening parameter independent on stress path[J]. Computers and Geotechnics，2008，35(2)：210-222.
[25]	LI X，WANG Y. Linear representation of steady-state line for sand[J]. Journal of Geotechnical and Geoenvironmental Engineering，1998，124(12)：1 215-1 217.
[12]	LAI Y，JIN L，CHANG X. Yield criterion and elasto-plastic damage constitutive model for frozen sandy soil[J]. International Journal of Plasticity，2009，25(6)：1 177-1 205.
[31]	张卫华，赵成刚，傅方. 饱和砂土相变状态边界面本构模型[J]. 岩土工程学报，2013，35(5)：930-939.(ZHANG Weihua，ZHAO Chenggan，FU Fang. Bounding-surface constitutive model for saturated sands based on phase transformation state[J]. Chinese Journal of Geotechnical Engineering，2013，35(5)：930-939.(in Chinese))
[26]	LI X，DAFALIAS Y F. Dilatancy for cohesionless soils[J]. Géotechnique，2000，50(4)：449-460.
[13]	WANG Y，LI J，YU X，et al. Study on fractional-order elastic-plastic constitutive model of river silt based on critical state theory[J]. Marine Georesources and Geotechnology，2024，42(1)：59-66.
[32]	赵春雷，赵成刚，张卫华. 饱和砂土基于相变状态的不排水本构模型[J]. 工程力学，2015，32(12)：68-76.(ZHAO Chunlei，ZHAO Chenggang，ZHANG Weihua. An undrained constitutive model of saturated sands based on the phase transformation[J]. Engineering Mechanics，2015，32(12)：68-76.(in Chinese))
[27]	BEEN K，JEFFERIES M G. A state parameter for sands[J]. Géotechnique，1985，35(2)：99-112.
[14]	梁靖宇，杜修力，路德春，等. 特征应力空间中土的分数阶临界状态模型[J]. 岩土工程学报，2019，41(3)：581-587.(LIANG Jingyu，DU Xiuli，LU Dechun，et al. Fractional-order critical state model for soils in characteristic stress space[J]. Chinese Journal of Geotechnical Engineering，2019，41(3)：581-587.(in Chinese))
[33]	孙逸飞，高玉峰，鞠雯. 分数阶塑性力学及其砂土本构模型[J]. 岩土工程学报，2018，40(8)：1 535-1 541.(SUN Yifei，GAO Yufeng，JU Wen. Fractional plasticity and its application in constitutive model for sands[J]. Chinese Journal of Geotechnical Engineering，2018，40(8)：1 535-1 541.(in Chinese))
[28]	刘振亚，刘建坤，李旭，等. 毛细黏聚与冰胶结作用对非饱和粉质黏土冻结强度及变形特性的影响[J]. 岩石力学与工程学报，2018，37(6)：1 551-1 559.(LIU Zhenya，LIU Jiankun，LI Xu，et al. Effect of capillary cohesion and ice cementation on strength and deformation of unsaturated frozen silty clay[J]. Chinese Journal of Rock Mechanics and Engineering，2018，37(6)：1 551-1 559.(in Chinese))
[15]	LIANG J，LU D，ZHOU X，et al. Non-orthogonal elastoplastic constitutive model with the critical state for clay[J]. Computers and Geotechnics，2019，116：103200.
[34]	刘彦芝，杨艳，党云贵. Caputo型分数阶q-微分方程的初值问题[J]. 华中师范大学学报：自然科学版，2022，56(2)：240-244.(LIU Yanzhi，YANG Yan，DANG Yungui. Initial value problem of fractional q-differential equation with Caputo derivative[J]. Journal of Central China Normal University：Natural Science，2022，56(2)：240-244. (in Chinese))
[29]	姚仰平，侯伟，罗汀. 土的统一硬化模型[J]. 岩石力学与工程学报，2009，28(10)：2 135-2 151.(YAO Yangping，HOU Wei，LUO Ting. Unified hardening model for soils[J]. Chinese Journal of Rock Mechanics and Engineering，2009，28(10)：2 135-2 151.(in Chinese))
[16]	孙增春，刘汉龙，肖杨. 砂-粉混合料的分数阶塑性本构模型[J]. 岩土工程学报，2024，46(8)：1 596-1 604.(SUN Zengchun，LIU Hanlong，XIAO Yang. Fractional-order plasticity model of sand-silt mixtures[J]. Chinese Journal of Geotechnical Engineering，2024，46(8)：1 596-1 604.(in Chinese))
[35]	ZHOU X，LU D，DU X，et al. A 3D non-orthogonal plastic damage model for concrete[J]. Computer Methods in Applied Mechanics and Engineering，2020，360：112716.
[30]	YAO Y，SUN D，MATSUOKA H. A unified constitutive model for both clay and sand with hardening parameter independent on stress path[J]. Computers and Geotechnics，2008，35(2)：210-222.
[17]	汪成贵，束善治，肖杨，等. 考虑钙质砂颗粒破碎的分数阶边界面本构模型[J]. 岩土工程学报，2023，45(6)：1 162-1 170.(WANG Chenggui，SHU Shanzhi，XIAO Yang，et al. Fractional-order bounding surface model considering breakage of calcareous sand [J]. Chinese Journal of Geotechnical Engineering，2023，45(6)：1 162-1 170.(in Chinese))
[36]	ROSCOE K，BURLAND J. On the generalised stress-strain behaviour of ‘wet’ clay[J]. Engineering Plasticity，1968：535-609.
[31]	张卫华，赵成刚，傅方. 饱和砂土相变状态边界面本构模型[J]. 岩土工程学报，2013，35(5)：930-939.(ZHANG Weihua，ZHAO Chenggan，FU Fang. Bounding-surface constitutive model for saturated sands based on phase transformation state[J]. Chinese Journal of Geotechnical Engineering，2013，35(5)：930-939.(in Chinese))
[18]	路德春，韩佳月，梁靖宇，等. 横观各向同性黏土的非正交弹塑性本构模型[J]. 岩石力学与工程学报，2020，39(4)：793-803.(LU Dechun，HAN Jiayue，LIANG Jingyu，et al. Non-orthogonal elastoplastic constitutive model of transversely isotropic clay[J]. Chinese Journal of Rock Mechanics and Engineering，2020，39(4)：793-803.(in Chinese))
[37]	ZHOU X，LU D，ZHANG Y，et al. An open-source unconstrained stress updating algorithm for the modified Cam-clay model[J]. Computer Methods in Applied Mechanics and Engineering，2022，390：114356.
[32]	赵春雷，赵成刚，张卫华. 饱和砂土基于相变状态的不排水本构模型[J]. 工程力学，2015，32(12)：68-76.(ZHAO Chunlei，ZHAO Chenggang，ZHANG Weihua. An undrained constitutive model of saturated sands based on the phase transformation[J]. Engineering Mechanics，2015，32(12)：68-76.(in Chinese))
[19]	路德春，金辰逸，梁靖宇，等. 考虑状态相关的砂土非正交弹塑性本构模型[J]. 岩土工程学报，2023，45(2)：221-231.(LU Dechun，JIN Chenyi，LIANG Jingyu，et al. State-dependent non-orthogonal elastoplastic constitutive model for sand[J]. Chinese Journal of Geotechnical Engineering，2023，45(2)：221-231.(in Chinese))
[38]	ZHOU X，SHI A，LU D，et al. A return mapping algorithm based on the hyper dual step derivative approximation for elastoplastic models[J]. Computer Methods in Applied Mechanics and Engineering，2023，417：116418.
[33]	孙逸飞，高玉峰，鞠雯. 分数阶塑性力学及其砂土本构模型[J]. 岩土工程学报，2018，40(8)：1 535-1 541.(SUN Yifei，GAO Yufeng，JU Wen. Fractional plasticity and its application in constitutive model for sands[J]. Chinese Journal of Geotechnical Engineering，2018，40(8)：1 535-1 541.(in Chinese))
[20]	LAI Y，LIAO M，HU K. A constitutive model of frozen saline sandy soil based on energy dissipation theory[J]. International Journal of Plasticity，2016，78：84-113.
[39]	SU C，LU D，ZHOU X，et al. An implicit stress update algorithm for the plastic nonlocal damage model of concrete[J]. Computer Methods in Applied Mechanics and Engineering，2023，414：116189.
[34]	刘彦芝，杨艳，党云贵. Caputo型分数阶q-微分方程的初值问题[J]. 华中师范大学学报：自然科学版，2022，56(2)：240-244.(LIU Yanzhi，YANG Yan，DANG Yungui. Initial value problem of fractional q-differential equation with Caputo derivative[J]. Journal of Central China Normal University：Natural Science，2022，56(2)：240-244. (in Chinese))
[21]	董晓丽，赵成刚，张卫华. 考虑相变状态的较密实饱和砂土弹塑性模型[J]. 工程力学，2017，34(1)：51-57.(DONG Xiaoli，ZHAO Chenggang，ZHANG Weihua. The saturated dense sand elastic-plastic model considering phase transition state[J]. Engineering Mechanics，2017，34(1)：51-57.(in Chinese))
[40]	LU D，ZHANG Y，ZHOU X，et al. An efficient nonlocal integral method based on the octree algorithm[J]. Computers and Geotechnics，2024，176：106796.
[35]	ZHOU X，LU D，DU X，et al. A 3D non-orthogonal plastic damage model for concrete[J]. Computer Methods in Applied Mechanics and Engineering，2020，360：112716.
[22]	DE Z，LIU E，LIU X，et al. A new strength criterion for frozen soils considering the influence of temperature and coarse-grained contents[J]. Cold Regions Science and Technology，2017，143：1-12.
[36]	ROSCOE K，BURLAND J. On the generalised stress-strain behaviour of ‘wet’ clay[J]. Engineering Plasticity，1968：535-609.
[23]	HUECKEL T，FRANCOIS B，LALOUI L. Explaining thermal failure in saturated clays[J]. Géotechnique，2009，59(3)：197-212.
[37]	ZHOU X，LU D，ZHANG Y，et al. An open-source unconstrained stress updating algorithm for the modified Cam-clay model[J]. Computer Methods in Applied Mechanics and Engineering，2022，390：114356.
[24]	ROSCOE K H，SCHOFIELD A N，THURAIRAJAH A. Yielding of clays in states wetter than critical[J]. Géotechnique，1963，13(3)：211-240.
[38]	ZHOU X，SHI A，LU D，et al. A return mapping algorithm based on the hyper dual step derivative approximation for elastoplastic models[J]. Computer Methods in Applied Mechanics and Engineering，2023，417：116418.
[25]	LI X，WANG Y. Linear representation of steady-state line for sand[J]. Journal of Geotechnical and Geoenvironmental Engineering，1998，124(12)：1 215-1 217.
[39]	SU C，LU D，ZHOU X，et al. An implicit stress update algorithm for the plastic nonlocal damage model of concrete[J]. Computer Methods in Applied Mechanics and Engineering，2023，414：116189.
[26]	LI X，DAFALIAS Y F. Dilatancy for cohesionless soils[J]. Géotechnique，2000，50(4)：449-460.
[40]	LU D，ZHANG Y，ZHOU X，et al. An efficient nonlocal integral method based on the octree algorithm[J]. Computers and Geotechnics，2024，176：106796.
[27]	BEEN K，JEFFERIES M G. A state parameter for sands[J]. Géotechnique，1985，35(2)：99-112.
[28]	刘振亚，刘建坤，李旭，等. 毛细黏聚与冰胶结作用对非饱和粉质黏土冻结强度及变形特性的影响[J]. 岩石力学与工程学报，2018，37(6)：1 551-1 559.(LIU Zhenya，LIU Jiankun，LI Xu，et al. Effect of capillary cohesion and ice cementation on strength and deformation of unsaturated frozen silty clay[J]. Chinese Journal of Rock Mechanics and Engineering，2018，37(6)：1 551-1 559.(in Chinese))
[29]	姚仰平，侯伟，罗汀. 土的统一硬化模型[J]. 岩石力学与工程学报，2009，28(10)：2 135-2 151.(YAO Yangping，HOU Wei，LUO Ting. Unified hardening model for soils[J]. Chinese Journal of Rock Mechanics and Engineering，2009，28(10)：2 135-2 151.(in Chinese))
[30]	YAO Y，SUN D，MATSUOKA H. A unified constitutive model for both clay and sand with hardening parameter independent on stress path[J]. Computers and Geotechnics，2008，35(2)：210-222.
[31]	张卫华，赵成刚，傅方. 饱和砂土相变状态边界面本构模型[J]. 岩土工程学报，2013，35(5)：930-939.(ZHANG Weihua，ZHAO Chenggan，FU Fang. Bounding-surface constitutive model for saturated sands based on phase transformation state[J]. Chinese Journal of Geotechnical Engineering，2013，35(5)：930-939.(in Chinese))
[32]	赵春雷，赵成刚，张卫华. 饱和砂土基于相变状态的不排水本构模型[J]. 工程力学，2015，32(12)：68-76.(ZHAO Chunlei，ZHAO Chenggang，ZHANG Weihua. An undrained constitutive model of saturated sands based on the phase transformation[J]. Engineering Mechanics，2015，32(12)：68-76.(in Chinese))
[33]	孙逸飞，高玉峰，鞠雯. 分数阶塑性力学及其砂土本构模型[J]. 岩土工程学报，2018，40(8)：1 535-1 541.(SUN Yifei，GAO Yufeng，JU Wen. Fractional plasticity and its application in constitutive model for sands[J]. Chinese Journal of Geotechnical Engineering，2018，40(8)：1 535-1 541.(in Chinese))
[34]	刘彦芝，杨艳，党云贵. Caputo型分数阶q-微分方程的初值问题[J]. 华中师范大学学报：自然科学版，2022，56(2)：240-244.(LIU Yanzhi，YANG Yan，DANG Yungui. Initial value problem of fractional q-differential equation with Caputo derivative[J]. Journal of Central China Normal University：Natural Science，2022，56(2)：240-244. (in Chinese))
[35]	ZHOU X，LU D，DU X，et al. A 3D non-orthogonal plastic damage model for concrete[J]. Computer Methods in Applied Mechanics and Engineering，2020，360：112716.
[36]	ROSCOE K，BURLAND J. On the generalised stress-strain behaviour of ‘wet’ clay[J]. Engineering Plasticity，1968：535-609.
[37]	ZHOU X，LU D，ZHANG Y，et al. An open-source unconstrained stress updating algorithm for the modified Cam-clay model[J]. Computer Methods in Applied Mechanics and Engineering，2022，390：114356.
[38]	ZHOU X，SHI A，LU D，et al. A return mapping algorithm based on the hyper dual step derivative approximation for elastoplastic models[J]. Computer Methods in Applied Mechanics and Engineering，2023，417：116418.
[39]	SU C，LU D，ZHOU X，et al. An implicit stress update algorithm for the plastic nonlocal damage model of concrete[J]. Computer Methods in Applied Mechanics and Engineering，2023，414：116189.
[40]	LU D，ZHANG Y，ZHOU X，et al. An efficient nonlocal integral method based on the octree algorithm[J]. Computers and Geotechnics，2024，176：106796.