Abstract：Geomaterials can exhibit dilatancy and softening behavior in engineering practice. When simulating the strain-softening characteristics as well as the accompanied shear bands with the standard finite element(FE) method，the predicted results suffer from strong mesh dependency，which greatly affects the accuracy of the results and may even result in convergence difficulties. In this paper，a novel FE algorithm of nonlocal strain regularisation is proposed based on the critical state theory for soils. Compared to the conventional constitutive integration scheme used in a FE analysis，the proposed algorithm in this study is capable of effectively solving the mesh dependency and convergence problem when modelling the strain-softening behaviour of geomaterials. Meanwhile，the presented algorithm can adequately incorporate the nonlocal strains into the integration process ensuring its application to all critical state type models such as the modified Cam-clay model. The implementation procedures of the proposed method in a finite element program were explained in detail by taking the unified hardening model for clay and sand(CSUH) as an example. The effectiveness of the method was subsequently verified through a series of numerical analyses of biaxial compression tests. The numerical results become almost unchanged with the mesh discretization once the adopted mesh is sufficiently fine，while the associated iterative process exhibits robust performance.
姚仰平，武孝天，崔文杰. 基于岩土材料临界状态理论的有限元非局部软化算法——以CSUH模型为例[J]. 岩石力学与工程学报, 2023, 42(7): 1759-1766.
YAO Yangping，WU Xiaotian，CUI Wenjie. A finite element nonlocal strain-softening algorithm based on the critical state theory for geomaterials：a case study with csuh model. , 2023, 42(7): 1759-1766.
MÁNICA M A，GENS A，VAUNAT J，et al. Nonlocal plasticity modelling of strain localisation in stiff clays[J]. Computers and Geotechnics，2018，103：138-150.
[17]
POTTS D M，ZDRAVKOVI? L. Finite element analysis in geotechnical engineering：Theory[M]. London，UK：Thomas Telford，1999：1-440.
[18]
POTTS D M，ZDRAVKOVI? L. Finite element analysis in geotechnical engineering：application[M]. London，U K：Thomas Telford，2001：1-427.
[5]
BA?ANT Z P，BELYTSCHKO T B. Continuum theory for strain-softening[J]. Journal of Engineering Mechanics，1985，110(12)：1 666-1 692.
[1]
黄茂松，贾苍琴，钱建固. 岩土材料应变局部化的有限元分析方法[J]. 计算力学学报，2007，24(4)：465-471.(HUANG Maosong，JIA Cangqin，QIAN Jiangu. Strain localization problems in geomaterials using finite elements[J]. Chinese Journal of Computational Mechanics，2007，24(4)：465-471.(in Chinese))
[11]
SUMMERSGILL F，KONTOE S，POTTS D M. On the use of nonlocal regularisation in slope stability problems[J]. Computers and Geotechnics，2017，82：187-200.
[15]
MALLIKARACHCHI H，SOGA K. Post-localisation analysis of drained and undrained dense sand with a nonlocal critical state model[J]. Computers and Geotechnics，2020，124：1-13.
[20]
POTTS D M，GENS A. A critical assessment of methods of correcting for drift from the yield surface in elasto-plastic finite element analysis[J]. International Journal for Numerical and Analytical Methods in Geomechanics，1985，9(2)：149-159.
[21]
NAKAI T，HINOKIO M. A simple elastoplastic model for normally and overconsolidated soils with unified material parameters[J]. Soils and Foundations，2004，44(2)：53-70.
[3]
曲 勰，黄茂松，吕玺琳. 基于非局部Mohr-Coulomb模型的土体渐进破坏分析[J]. 岩土工程学报，2013，35(3)：253-530.(QU Xie，HUANG Maosong，LÜ Xilin. Progressive failure of soils based on non-local Mohr-Coulomb models[J]. Chinese Journal of Geotechnical Engineering，2013，35(3)：253-530.(in Chinese))
[7]
SUMMERSGILL F，KONTOE S，POTTS D M. Critical assessment of nonlocal strain-softening methods in biaxial compression[J]. International Journal of Geomechanics，2017，17(7)：04017006.
[8]
VERMEER P A，BRINKGREVE R B J. A new effective nonlocal strain measure for softening plasticity[C]// Localization and Bifurcation Theory for Soils and Rocks. Rotterdam：Balkema，1994：89-100.
[13]
CONTE E，SILVESTRI F，TRONCONE A. Stability analysis of slopes in soils with strain-softening behaviour[J]. Computers and Geotechnics，2010，37(5)：710-722.
[9]
GALAVI V，SCHWEIGER H F. Nonlocal multilaminate model for strain softening analysis[J]. International Journal of Geomechanics，2010，10(1)：30-44.
[19]
YAO Y P，HOU W，ZHOU A N. UH model：three-dimensional unified hardening model for overconsolidated clays[J]. Géotechnique，2009，59(5)：451-469.
[6]
BA?ANT Z P，JIRÁSEK M. Nonlocal integral formulations of plasticity and damage：Survey of progress[J]. Journal of Engineering Mechanics，2002，128(11)：1 119-1 149.
[16]
YAO Y P，LIU L，LUO T，et al. Unified hardening(UH) model for clays and sands[J]. Computers and Geotechnics，2019，110：326-343.
[4]
武守信，魏吉瑞，杨舒蔚. 基于能量等效原理的应变局部化分析：II. 有限元解法[J]. 力学学报，2017，49(4)：880-893.(Analysis of strain localization by energy equivalence：II. Finite element solution[J]. Chinese Journal of Theoretical and Applied Mechanics，2017，49(4)：880-893.(in Chinese))
[14]
MÁNICA M A，GENS A，VAUNAT J，et al. Numerical simulation of underground excavations in an indurated clay using non-local regularisation. Part 1：formulation and base case[J]. Géotechnique，2022，72(12)：1 092-1 112.
[2]
吕玺琳，黄茂松. 基于非局部塑性模型的应变局部化理论分析及数值模拟[J]. 计算力学学报，2011，28(5)：743-748.(LÜ Xilin，HUANG Maosong. Theoretical analysis and numerical simulation of strain localization in nonlocal plasticity model[J]. Chinese Journal of Computational Mechanics，2011，28(5)：743-748.(in Chinese))
[12]
SUMMERSGILL F， KONTOE S，POTTS D M. Stabilisation of excavated slopes in strain-softening materials with piles[J]. Géotechnique，2018，68(7)：626-639.
2023, Vol. 42 
