On the Determination of Softening Parameters for Nonlocal Constitutive Models
Publication: Journal of Geotechnical and Geoenvironmental Engineering
Volume 150, Issue 12
Abstract
Several regularized formulations exist in the literature, such as nonlocal constitutive models, for the objective simulation of localized deformations in quasibrittle materials that prevent the well-known pathological dependence on the employed mesh. Although they usually require a significant discretization to represent the localization zone of a given material, characterized by some length scale linked to its microstructure, scaling techniques exist to derive representative parameters for full-scale simulation of geotechnical problems. However, the softening response observed in conventional triaxial or uniaxial compression tests is not solely a constitutive feature but also the result of a complex boundary value problem (BVP), where the specific localization pattern formed affects the global response of the experiment. Therefore, constitutive softening parameters should be backcalculated through the simulation of a given laboratory test as a BVP. However, as demonstrated in this paper, the cylindrical geometry of the samples in conventional tests hinders the onset of localization, resulting in a stiffer response that underestimates the resulting nominal softening rate. Within this context, the paper aimed to demonstrate the complexity and difficulty in the determination of a softening rate for nonlocal constitutive models from data of conventional triaxial tests performed on cylindrical specimens and to provide relevant insights toward a practical procedure for the determination of parameters in regularized simulations of quasibrittle geomaterials for practical applications.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
The first author has been supported by a Conahcyt scholarship.
References
Anand, L., O. Aslan, and S. A. Chester. 2012. “A large-deformation gradient theory for elastic plastic materials: Strain softening and regularization of shear bands.” Int. J. Plast. 30-31 (Mar): 116–143. https://doi.org/10.1016/j.ijplas.2011.10.002.
Bažant, Z. P., and M. Jirásek. 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.
Bažant, Z. P., and G. Pijaudier-Cabot. 1988. “Nonlocal continuum damage, localization instability and convergence.” J. Appl. Mech. 55 (2): 287–293. https://doi.org/10.1115/1.3173674.
Bentley Systems. 2022. Plaxis CONNECT Edition V22.01. Exton, PA: Bentley Systems.
Bobiński, J., and J. Tejchman. 2005. “Modelling of concrete behaviour with a non-local continuum damage approach.” Archiv. Hydro-Eng. Environ. Mech. 52 (3): 243–263.
Borja, R. I., X. Song, A. L. Rechenmacher, S. Abedi, and W. Wu. 2013. “Shear band in sand with spatially varying density.” J. Mech. Phys. Solids 61 (1): 219–234. https://doi.org/10.1016/j.jmps.2012.07.008.
Burland, J. B. 1990. “On the compressibility and shear strength of natural clays.” Géotechnique 40 (3): 329–378. https://doi.org/10.1680/geot.1990.40.3.329.
Chevalier, B., P. Villard, and G. Combe. 2011. “Investigation of load-transfer mechanisms in geotechnical earthstructures with thin fill platforms reinforced by rigid inclusions.” Int. J. Geomech. 11 (3): 239–250. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000083.
Collin, F., P. Kotronis, and B. Pardoen. 2021. Numerical modeling of multiphysics couplings and strain localization, 203–225. Hoboken, NJ: Wiley.
Cosserat, E., and F. Cosserat. 1909. Théorie des corps déformables. Paris: A. Hermann et fils.
Deb, A., J. H. Prevost, and B. Loret. 1996. “Adaptive meshing for dynamic strain localization.” Comput. Methods Appl. Mech. Eng. 137 (3–4): 285–306. https://doi.org/10.1016/S0045-7825(96)01068-7.
de Borst, R., L. J. Sluys, H.-B. Mühlhaus, 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.
Desrues, J., and G. Viggiani. 2004. “Strain localization in sand: An overview of the experimental results obtained in Grenoble using stereophotogrammetry.” Int. J. Numer. Anal. Methods Geomech. 28 (4): 279–321. https://doi.org/10.1002/nag.338.
Eringen, A. C. 1966. “Mechanics of micromorphic materials.” In Proc., 11th Int. Congress for Applied Mechanics, edited by H. Görtler, 131–138. Berlin: Springer. https://doi.org/10.1007/978-3-662-29364-5_12.
Eringen, A. C. 1981. “On nonlocal plasticity.” Int. J. Eng. Sci. 19 (12): 1461–1474. https://doi.org/10.1016/0020-7225(81)90072-0.
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.
Garcia, J. A. B., J. F. Rodríguez Rebolledo, D. V. dos Santos Mützenberg, B. Caicedo, and G. de Farias Neves Gitirana. 2021. “Experimental investigation of a load-transfer material for foundations reinforced by rigid inclusions.” J. Geotech. Geoenviron. Eng. 147 (10): 1–14. https://doi.org/10.1061/(ASCE)GT.1943-5606.0002649.
Gens, A. 2013. “On the hydromechanical behaviour of argillaceous hard soils-weak rocks.” In Vol. 4 of Proc., 15th European Conf. on Soil Mechanics and Geotechnical Engineering—Geotechnics of Hard Soils—Weak Rocks, edited by A. Anagnostopoulos, M. Pachakis, and C. Tsatsanifos, 71–118. Athens, Greece: IOS Press. https://doi.org/10.3233/978-1-61499-199-1-71.
Gens, A., I. Carol, and E. E. Alonso. 1990. “A constitutive model for rock joints formulation and numerical implementation.” Comput. Geotech. 9 (1–2): 3–20. https://doi.org/10.1016/0266-352X(90)90026-R.
González Acosta, J. L., M. A. Mánica, P. J. Vardon, M. A. Hicks, and A. Gens. 2024. “A nonlocal material point method for the simulation of large deformation problems in brittle soils.” Comput. Geotech. 172 (Aug): 106424. https://doi.org/10.1016/j.compgeo.2024.106424.
Gutierrez, M., and I. Vardoulakis. 2007. “Energy dissipation and post-bifurcation behaviour of granular soils.” Int. J. Numer. Anal. Methods Geomech. 31 (3): 435–455. https://doi.org/10.1002/nag.588.
Khoei, A. R., and K. Karimi. 2008. “An enriched-FEM model for simulation of localization phenomenon in Cosserat continuum theory.” Comput. Mater. Sci. 44 (2): 733–749. https://doi.org/10.1016/j.commatsci.2008.05.019.
Laurich, B., J. L. Urai, G. Desbois, C. Vollmer, and C. Nussbaum. 2014. “Microstructural evolution of an incipient fault zone in Opalinus clay: Insights from an optical and electron microscopic study of ion-beam polished samples from the main fault in the Mt-Terri Underground Research Laboratory.” J. Struct. Geol. 67 (Oct): 107–128. https://doi.org/10.1016/j.jsg.2014.07.014.
Loret, B., and J. H. Prevost. 1990. “Dynamic strain localization in elasto-(visco-)plastic solids, Part 1. General formulation and one-dimensional examples.” Comput. Methods Appl. Mech. Eng. 83 (3): 247–273. https://doi.org/10.1016/0045-7825(90)90073-U.
Mánica, M. A. 2018. “Analysis of underground excavations in argillaceous hard soils—Weak rocks.” Ph.D. thesis, Dept. of Civil and Environmental Engineering, Technical Univ. of Catalonia.
Mánica, M. A., M. O. Ciantia, and A. Gens. 2020. “On the stability of underground caves in calcareous rocks due to long-term weathering.” Rock Mech. Rock Eng. 53 (9): 3885–3901. https://doi.org/10.1007/s00603-020-02142-y.
Mánica, M. A., A. Gens, J. Vaunat, G. Armand, and M. N. Vu. 2022a. “Numerical simulation of underground excavations in an indurated clay using non-local regularisation. Part 1: Formulation and base case.” Géotechnique 72 (12): 1092–1112. https://doi.org/10.1680/jgeot.20.P.246.
Mánica, M. A., A. Gens, J. Vaunat, G. Armand, and M. N. Vu. 2022b. “Numerical simulation of underground excavations in an indurated clay using non-local regularisation. Part 2: Sensitivity analysis.” Géotechnique 72 (12): 1113–1128. https://doi.org/10.1680/jgeot.20.p.247.
Mánica, M. A., A. Gens, J. Vaunat, and D. F. Ruiz. 2018. “Nonlocal plasticity modelling of strain localisation in stiff clays.” Comput. Geotech. 103 (Nov): 138–150. https://doi.org/10.1016/j.compgeo.2018.07.008.
Menon, S., and X. Song. 2021. “A computational periporomechanics model for localized failure in unsaturated porous media.” Comput. Methods Appl. Mech. Eng. 384 (Oct): 113932. https://doi.org/10.1016/j.cma.2021.113932.
Monforte, L., M. O. Ciantia, J. M. Carbonell, M. Arroyo, and A. Gens. 2019. “A stable mesh-independent approach for numerical modelling of structured soils at large strains.” Comput. Geotech. 116 (Dec): 103215. https://doi.org/10.1016/j.compgeo.2019.103215.
Monforte, L., A. Gens, M. Arroyo, M. Mánica, and J. M. Carbonell. 2021. “Analysis of cone penetration in brittle liquefiable soils.” Comput. Geotech. 134 (Jun): 104123. https://doi.org/10.1016/j.compgeo.2021.104123.
Mozaffari, M., W. Liu, and M. Ghafghazi. 2022. “Influence of specimen nonuniformity and end restraint conditions on drained triaxial compression test results in sand.” Can. Geotech. J. 59 (8): 1414–1426. https://doi.org/10.1139/cgj-2021-0505.
Mühlhaus, H.-B., and E. C. Alfantis. 1991. “A variational principle for gradient plasticity.” Int. J. Solids Struct. 28 (7): 845–857. https://doi.org/10.1016/0020-7683(91)90004-Y.
Ortiz, M., and J. J. Quigley. 1991. “Adaptive mesh refinement in strain localization problems.” Comput. Methods Appl. Mech. Eng. 90 (1–3): 781–804. https://doi.org/10.1016/0045-7825(91)90184-8.
Pietruszczak, S., and Z. Mroz. 1981. “Finite element analysis of deformation of strain-softening materials.” Int. J. Numer. Methods Eng. 17 (3): 327–334. https://doi.org/10.1002/nme.1620170303.
Pijaudier-Cabot, G., and Z. P. Bažant. 1987. “Nonlocal damage theory.” J. Eng. Mech. 113 (10): 1512–1533. https://doi.org/10.1061/(ASCE)0733-9399(1987)113:10(1512.
Prevost, J. H., and B. Loret. 1990. “Dynamic strain localization in elasto-(visco-)plastic solids, part 2. Plane strain examples.” Comput. Methods Appl. Mech. Eng. 83 (3): 275–294. https://doi.org/10.1016/0045-7825(90)90074-V.
Schlosser, F., H. M. Jacobsen, and I. Juran. 1984. “Le renforcement des sols.” Revue Française de Géotechnique 29: 7–33. https://doi.org/10.1051/geotech/1984029007.
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 (May): 104025. https://doi.org/10.1016/j.compgeo.2021.104025.
Sluys, L. J., and R. de Borst. 1992. “Wave propagation and localization in a rate-dependent cracked medium-model formulation and one-dimensional examples.” Int. J. Solids Struct. 29 (23): 2945–2958. https://doi.org/10.1016/0020-7683(92)90151-I.
Song, X., and N. Khalili. 2019. “A peridynamics model for strain localization analysis of geomaterials.” Int. J. Numer. Anal. Methods Geomech. 43 (1): 77–96. https://doi.org/10.1002/nag.2854.
Summersgill, F. C., S. Kontoe, and D. M. Potts. 2017a. “Critical assessment of nonlocal strain-softening methods in biaxial compression.” Int. J. Geomech. 17 (7): 1–14. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000852.
Summersgill, F. C., S. Kontoe, and D. M. Potts. 2017b. “On the use of nonlocal regularisation in slope stability problems.” Comput. Geotech. 82 (Feb): 187–200. https://doi.org/10.1016/j.compgeo.2016.10.016.
Summersgill, F. C., S. Kontoe, and D. M. Potts. 2018. “Stabilisation of excavated slopes in strain-softening materials with piles.” Géotechnique 68 (7): 626–639. https://doi.org/10.1680/jgeot.17.P.096.
Tejchman, J., and E. Bauer. 1996. “Numerical simulation of shear band formation with a polar hypoplastic constitutive model.” Comput. Geotech. 19 (3): 221–244. https://doi.org/10.1016/0266-352X(96)00004-3.
Tejchman, J., and E. Bauer. 2005. “Fe-simulations of a direct and a true simple shear test within a polar hypoplasticity.” Comput. Geotech. 32 (1): 1–16. https://doi.org/10.1016/j.compgeo.2004.11.004.
Tejchman, J., and W. Wu. 1993. “Numerical study on patterning of shear bands in a Cosserat continuum.” Acta Mech. 99 (1): 61–74. https://doi.org/10.1007/BF01177235.
van Eekelen, H. A. M. 1980. “Isotropic yield surfaces in three dimensions for use in soil mechanics.” Int. J. Numer. Anal. Methods Geomech. 4 (1): 89–101. https://doi.org/10.1002/nag.1610040107.
Vermeer, P. A., and R. B. J. Brinkgreve. 1994. “A new effective non-local strain measure for softening plasticity.” In Localisation and bifurcation theory for soils and rock, edited by R. Chambon, J. Desrues, and I. Vardoulakis, 89–100. Rotterdam, Netherlands: A.A. Balkema.
Zienkiewicz, O. C., M. Huang, and M. Pastor. 1995. “Localization problems in plasticity using finite elements with adaptive remeshing.” Int. J. Numer. Anal. Methods Geomech. 19 (2): 127–148. https://doi.org/10.1002/nag.1610190205.
Information & Authors
Information
Published In
Journal of Geotechnical and Geoenvironmental Engineering
Volume 150 • Issue 12 • December 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: Nov 16, 2023
Accepted: Jul 23, 2024
Published online: Oct 12, 2024
Published in print: Dec 1, 2024
Discussion open until: Mar 12, 2025
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.