Evaluation of Three Weight Functions for Nonlocal Regularization of Sand Models
Publication: International Journal of Geomechanics
Volume 24, Issue 7
Abstract
Nonlocal regularization is frequently used to resolve the mesh dependency issue that is caused by strain softening in finite-element (FE) simulations. Some or all variables that affect strain softening are assumed to depend on the local, neighboring, or both in this method. The weight function is the main component of a regularization method. There are three widely used weight functions, which include the Gaussian distribution (GD), Galavi and Schweiger (GS), and over-nonlocal (ON) functions. All of them could alleviate or eliminate the mesh dependency in simple boundary value problems (BVPs), such as plane strain compression; the evaluation of their performance in real-world BVPs is rare. A detailed comparison of these functions has been carried out based on an anisotropic sand model that accounts for the evolution of anisotropy. The increment of void ratio is assumed nonlocal. All functions give mesh-independent force–displacement relationships in drained and undrained plane strain compression tests. The shear band thickness shows a small variation when the mesh size is smaller than the internal length. None could eliminate the mesh dependency of shear band orientation. The GS method is the most efficient in eliminating the mesh dependency in the strip footing problem. The ON method could give excessive overpredictions of the volume expansion around strip footings, which leads to unrealistic low reaction forces on strip footings at large deformations. All three weight functions give mesh-independent results for the earth pressure that acts on a retaining wall.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All code, models, and data generated or used in this study are available from the corresponding author upon reasonable request.
Acknowledgments
The support of the International Exchanges grants of the Royal Society (IES\R1\201132; IEC\NSFC\223020) is acknowledged.
References
Aifantis, E. C. 1984. “Microscopic processes and macroscopic response.” In Mechanics of engineering materials, edited by C. S. Desai, and R. H. Gallagher, 1–22. New York: Wiley.
Alsaleh, M. I., G. Z. Voyiadjis, and K. A. Alshibli. 2006. “Modelling strain localization in granular materials using micropolar theory: Mathematical formulations.” Int. J. Numer. Anal. Methods Geomech. 30 (15): 1501–1524. https://doi.org/10.1002/nag.533.
Alshibli, K. A., M. I. Alsaleh, and G. Z. Voyiadjis. 2006. “Modelling strain localization in granular materials using micropolar theory: Numerical implementation and verification.” Int. J. Numer. Anal. Methods Geomech. 30: 1525–1544. https://doi.org/10.1002/nag.534.
Bažant, Z. P., and P. G. Gambarova. 1984. “Crack shear in concrete: Crack band microflane model.” J. Struct. Eng. 110 (9): 2015–2035. https://doi.org/10.1061/(ASCE)0733-9445(1984)110:9(2015).
Bazant, Z. P., and M. Jirasek. 2002. “Nonlocal integral formulations of plasticity and damage: Survey of progress.” J. Eng. Mech. 128 (11): 1119–1149. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:11(1119).
Cui, W., X. Wu, D. M. Potts, and L. Zdravković. 2023. “Nonlocal strain regularisation for critical state models with volumetric hardening.” Comput. Geotech. 157: 105350. https://doi.org/10.1016/j.compgeo.2023.105350.
de Borst, R., and H.-B. Mühlhaus. 1992. “Gradient-dependent plasticity: Formulation and algorithmic aspects.” Int. J. Numer. Methods Eng. 35 (3): 521–539. https://doi.org/10.1002/nme.1620350307.
de Borst, R., L. J. Sluys, H.-B. Muhlhaus, and J. Pamin. 1993. “Fundamental issues in finite element analyses of localization of deformation.” Eng. Comput. 10 (2): 99–121. https://doi.org/10.1108/eb023897.
Dorgan, R. J., and G. Z. Voyiadjis. 2003. “Nonlocal dislocation based plasticity incorporating gradients of hardening.” Mech. Mater. 35 (8): 721–732. https://doi.org/10.1016/S0167-6636(02)00202-8.
Eringen, A. C. 1972. “Nonlocal polar elastic continua.” Int. J. Eng. Sci. 10 (1): 1–16. https://doi.org/10.1016/0020-7225(72)90070-5.
Eringen, A. C., and B. S. Kim. 1974. “Stress concentration at the tip of crack.” Mech. Res. Commun. 1: 233–237. https://doi.org/10.1016/0093-6413(74)90070-6.
Galavi, V., and H. F. Schweiger. 2010. “Nonlocal multilaminate model for strain softening analysis.” Int. J. Geomech. 10 (1): 30–44. https://doi.org/10.1061/(ASCE)1532-3641(2010)10:1(30).
Gao, Z., X. Li, and D. Lu. 2022. “Nonlocal regularization of an anisotropic critical state model for sand.” Acta Geotech. 17 (2): 427–439. https://doi.org/10.1007/s11440-021-01236-3.
Gao, Z., D. Lu, and X. Du. 2020. “Bearing capacity and failure mechanism of strip footings on anisotropic sand.” J. Eng. Mech. 146 (8): 04020081. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001814.
Gao, Z., and J. Zhao. 2013. “Strain localization and fabric evolution in sand.” Int. J. Solids Struct. 50: 3634–3648. https://doi.org/10.1016/j.ijsolstr.2013.07.005.
Gao, Z., J. Zhao, and X. Li. 2021. “The deformation and failure of strip footings on anisotropic cohesionless sloping grounds.” Int. J. Numer. Anal. Methods Geomech. 45 (10): 1526–1545. https://doi.org/10.1002/nag.3212.
Gao, Z., J. Zhao, X.-S. Li, and Y. F. Dafalias. 2014. “A critical state sand plasticity model accounting for fabric evolution.” Int. J. Numer. Anal. Methods Geomech. 38 (4): 370–390. https://doi.org/10.1002/nag.2211.
Guo, N., and J. Zhao. 2016. “Multiscale insights into classical geomechanics problems.” Int. J. Numer. Anal. Methods Geomech. 40 (3): 367–390. https://doi.org/10.1002/nag.2406.
Guo, P., and D. F. E. Stolle. 2013. “Coupled analysis of bifurcation and shear band in saturated soils.” Soils Found. 53 (4): 525–539. https://doi.org/10.1016/j.sandf.2013.06.005.
Higo, Y. 2004. “Instability and strain localization analysis of water-saturated clay by elasto-viscoplastic constitutive models.” Ph.D. thesis, Dept. of Engineering, Kyoto Univ.
Kimura, T., O. Kusakabe, and K. Saitoh. 1985. “Geotechnical model tests of bearing capacity problems in a centrifuge.” Géotechnique 35 (1): 33–45. https://doi.org/10.1680/geot.1985.35.1.33.
Lazari, M., L. Sanavia, and B. A. Schrefler. 2015. “Local and non-local elasto-viscoplasticity in strain localization analysis of multiphase geomaterials.” Int. J. Numer. Anal. Methods Geomech. 39 (14): 1570–1592. https://doi.org/10.1002/nag.2408.
Li, X. S., and Y. F. Dafalias. 2012. “Anisotropic critical state theory: Role of fabric.” J. Eng. Mech. 138 (3): 263–275.
Lu, D., J. Liang, X. Du, C. Ma, and Z. Gao. 2019a. “Fractional elastoplastic constitutive model for soils based on a novel 3D fractional plastic flow rule.” Comput. Geotech. 105: 277–290. https://doi.org/10.1016/j.compgeo.2018.10.004.
Lu, D., Y. Zhang, X. Zhou, C. Su, Z. Gao, and X. Du. 2023. “A robust stress update algorithm for elastoplastic models without analytical derivation of the consistent tangent operator and loading/unloading estimation.” Int. J. Numer. Anal. Methods Geomech. 47: 1022–1050. https://doi.org/10.1002/nag.3503.
Lu, D., X. Zhou, X. Du, and G. Wang. 2019b. “A 3D fractional elastoplastic constitutive model for concrete material.” Int. J. Solids Struct. 165: 160–175. https://doi.org/10.1016/j.ijsolstr.2019.02.004.
Lü, X., J.-P. Bardet, and M. Huang. 2009. “Numerical solutions of strain localization with nonlocal softening plasticity.” Comput. Methods Appl. Mech. Eng. 198 (47–48): 3702–3711. https://doi.org/10.1016/j.cma.2009.08.002.
Mallikarachchi, H. 2019. “Constitutive modelling of shear localisation in saturated dilative sand.” Ph.D. thesis, Dept. of Engineering, Univ. of Cambridge.
Mallikarachchi, H., and K. Soga. 2020. “A two-scale constitutive framework for modelling localised deformation in saturated dilative hardening material.” Int. J. Numer. Anal. Methods Geomech. 44 (14): 1958–1982. https://doi.org/10.1002/nag.3115.
Mühlhaus, H.-B. 1986. “Scherfugenanalyse bei Granularem material im rahmen der cosserat-theorie.” Ingenieur-Arch. 56 (5): 389–399. https://doi.org/10.1007/BF02570619.
Nübel, K., and W. Huang. 2004. “A study of localized deformation pattern in granular media.” Comput. Methods Appl. Mech. Eng. 193 (27–29): 2719–2743. https://doi.org/10.1016/j.cma.2003.10.020.
Oka, F., T. Adachi, and A. Yashima. 1995. “A strain localization analysis using a viscoplastic softening model for clay.” Int. J. Plast. 11 (5): 523–545. https://doi.org/10.1016/S0749-6419(95)00020-8.
Okochi, Y., and F. Tatsuoka. 1984. “Some factors affecting K0-values of sand measured in triaxial cell.” Soils Found. 24 (3): 52–68. https://doi.org/10.3208/sandf1972.24.3_52.
Singh, V., S. Stanier, B. Bienen, and M. F. Randolph. 2021. “Modelling the behaviour of sensitive clays experiencing large deformations using non-local regularisation techniques.” Comput. Geotech. 133: 104025. https://doi.org/10.1016/j.compgeo.2021.104025.
Summersgill, F. C., S. Kontoe, and D. M. Potts. 2017. “On the use of nonlocal regularisation in slope stability problems.” Comput. Geotech. 82: 187–200. https://doi.org/10.1016/j.compgeo.2016.10.016.
Tang, H., Z. Hu, and X. Li. 2013. “Three-dimensional pressure-dependent elastoplastic Cosserat continuum model and finite element simulation of strain localization.” Int. J. Appl. Mech. 05 (03): 1350030. https://doi.org/10.1142/S1758825113500300.
Vermeer, P. A., and R. B. J. Brinkgreve. 1994. “A new effective non-local strain measure for softening plasticity.” In Localization and bifurcation theory for soil and rocks, edited by R. Chambon, J. Desrues, and I. Vardoulakis, 89–100. Rotterdam, Netherlands: Balkema.
Wang, W. M., L. J. Sluys, and R. de Borst. 1997. “Viscoplasticity for instabilities due to strain softening and strain-rate softening.” Int. J. Numer. Methods Eng. 40 (20): 3839–3864. https://doi.org/10.1002/(SICI)1097-0207(19971030)40:20<3839::AID-NME245>3.0.CO;2-6.
Widulinski, Ł, J. Tejchman, J. Kozicki, and D. Lesniewska. 2011. “Discrete simulations of shear zone patterning in sand in earth pressure problems of a retaining wall.” Int. J. Solids Struct. 48: 1191–1209. https://doi.org/10.1016/j.ijsolstr.2011.01.005.
Xue, D., X. Lü, M. Huang, and K.-W. Lim. 2022. “Nonlocal regularized numerical analyses for passive failure of tunnel head in strain-softening soils.” Comput. Geotech. 148: 104834. https://doi.org/10.1016/j.compgeo.2022.104834.
Yin, Z.-Y., C. S. Chang, M. Karstunen, and P.-Y. Hicher. 2010. “An anisotropic elastic–viscoplastic model for soft clays.” Int. J. Solids Struct. 47 (5): 665–677. https://doi.org/10.1016/j.ijsolstr.2009.11.004.
Zhao, J., D. Sheng, M. Rouainia, and S. W. Sloan. 2005. “Explicit stress integration of complex soil models.” Int. J. Numer. Anal. Methods Geomech. 29: 1209–1229. https://doi.org/10.1002/nag.456.
Zhou, X., D. Lu, C. Su, Z. Gao, and X. Du. 2022a. “An unconstrained stress updating algorithm with the line search method for elastoplastic soil models.” Comput. Geotech. 143: 104592. https://doi.org/10.1016/j.compgeo.2021.104592.
Zhou, X., D. Lu, Y. Zhang, X. Du, and T. Rabczuk. 2022b. “An open-source unconstrained stress updating algorithm for the modified Cam-clay model.” Comput. Methods Appl. Mech. Eng. 390: 114356. https://doi.org/10.1016/j.cma.2021.114356.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 24 • Issue 7 • July 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: May 9, 2023
Accepted: Jan 13, 2024
Published online: Apr 24, 2024
Published in print: Jul 1, 2024
Discussion open until: Sep 24, 2024
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.