Cam-Clay plasticity. III: Extension of the infinitesimal model to include finite strains. (English) Zbl 0959.74010

[For part I see the first author and S. R. Lee, ibid. 78, No. 1, 49-72 (1990; Zbl 0718.73034); for part II see the first author, ibid. 88, No. 2, 225-240 (1991; Zbl 0746.73008).]
Summary: The infinitesimal version of the modified Cam-Clay model of critical state soil mechanics is reformulated to include finite deformation effects. Central to the formulation are the choice of a hardening law that is appropriate for cases involving large plastic volumetric strains, and the use of a class of two-invariant stored energy functions appropriate for Cam-Clay-type models that includes, as special cases, the constant elastic shear modulus approximation and the pressure-dependent shear modulus elastic model. The analytical model is cast within the framework of finite deformation theory based on a multiplicative decomposition of the deformation gradient. For the fully elasto-plastic case, return mapping is done implicitiy in the space defined by the invariants of elastic logarithmic principal stretches, which requires the solution at each stress point of no more than three simultaneous nonlinear equations. A finite element analysis of a strip footing problem is presented to illustrate a prototype example where finite deformation effects significantly impact the predicted response.


74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
74L10 Soil and rock mechanics
74S05 Finite element methods applied to problems in solid mechanics
Full Text: DOI


[1] Atkinson, J. H., Foundations and Slopes: An Introduction to Applications of Critical State Soil Mechanics (1981), Halsted Press: Halsted Press New York
[2] Roscoe, K. H.; Burland, J. H., On the generalized stress-strain behavior of ‘wet’ clay, (Heyman, J.; Leckie, F. A., Engineering Plasticity (1968), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 535-609 · Zbl 0233.73047
[3] Schofield, A.; Wroth, P., Critical State Soil Mechanics (1968), McGraw-Hill: McGraw-Hill New York
[4] Borja, R. I.; Lee, S. R., Cam-Clay plasticity, Part I: Implicit integration of elasto-plastic constitutive relations, Comput. Methods Appl. Mech. Engrg., 78, 49-72 (1990) · Zbl 0718.73034
[5] Borja, R. I., Cam-Clay plasticity, Part II: Implicit integration of constitutive equation based on a nonlinear elastic stress predictor, Comput. Methods Appl. Mech. Engrg., 88, 225-240 (1991) · Zbl 0746.73008
[6] Britto, A. M.; Gunn, M. J., Critical State Soil Mechanics via Finite Elements (1987), John Wiley and Sons: John Wiley and Sons New York · Zbl 0704.73080
[7] Gens, A.; Potts, D. M., Critical state models in computational geomechanics, Engrg. Comput., 5, 178-197 (1988)
[8] Ortiz, M.; Simo, J. C., An analysis of a new class of integration algorithms for elastoplastic constitutive relations, Int. J. Numer. Methods Engrg., 23, 353-366 (1986) · Zbl 0585.73058
[9] Potts, D. M.; Ganendra, D., An evaluation of substepping and implicit stress point algorithms, Comput. Methods Appl. Mech. Engrg., 119, 341-354 (1994) · Zbl 0852.73068
[10] Simo, J. C.; Kennedy, J. G.; Govindjee, S., Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms, Int. J. Numer. Methods Engrg., 26, 2161-2185 (1988) · Zbl 0661.73058
[11] Butterfield, R., A natural compression law for soils, Géotechnique, 29, 469-480 (1979)
[12] Hashiguchi, K.; Ueno, M., Elasto-plastic constitutive laws of granular materials, (Murayama, S.; Schofield, A. N., Constitutive Equations of Soils, Proc. Ninth Int. Conf. Soil Mech. Found. Engrg.. Constitutive Equations of Soils, Proc. Ninth Int. Conf. Soil Mech. Found. Engrg., Specialty Session, 9 (1977)), 73-82, Tokyo
[13] Hashiguchi, K., On the linear relations of \(V\)-ln \(p\) and ln \(v\)-ln \(p\) for isotropic consolidation of soils, Int. J. Numer. Analyt. Methods Geomech., 19, 367-376 (1995) · Zbl 0821.73055
[14] Al Tabbaa, A., Permeability and stress-strain response of speswhite kaolin, (Ph.D. Thesis (1987), Cambridge University)
[15] Stallebrass, S. E., Modeling the effect of recent stress history on the deformation of overconsolidated soils, (Ph.D. Thesis (1990), Univ. of London)
[16] Houlsby, G. T.; Wroth, C. P.; Wood, D. M., Predictions of the results of laboratory tests on a clay using a critical state model, (Gudehus; Darve; Vardoulakis, Proc. Internat. Workshop on Constitutive Relations for Soils. Proc. Internat. Workshop on Constitutive Relations for Soils, Grenoble (1984), Balkema: Balkema Rotterdam)
[17] Zytynski, M.; Randolph, M. K.; Nova, R.; Wroth, C. P., On modeling the unloading-reloading behaviour of soils, Int. J. Numer. Anal. Methods Geomech., 2, 87-93 (1978)
[18] Simo, J. C.; Meschke, G., A new class of algorithms for classical plasticity extended to finite strains. Application to geomaterials, Comput. Mech., 11, 253-278 (1993) · Zbl 0814.73030
[19] Lambe, T. W.; Whitman, R. V., Soil Mechanics (1969), Wiley: Wiley New York
[20] Houlsby, G. T., The use of a variable shear modulus in elastic-plastic models for clays, Comput. Geotech., 1, 3-13 (1985)
[21] Lade, P. V.; Nelson, R. B., Modelling the elastic behaviour of granular materials, Int. J. Numer. Anal. Methods Geomech., 11, 521-542 (1987)
[22] Loret, B., On the choice of elastic parameters for sand, Int. J. Numer. Anal. Methods Geomech., 9, 285-292 (1985)
[23] Simo, J. C.; Hughes, T. J.R., Elastoplasticity and viscoplasticity—Computational Aspects, (Springer Ser. Appl. Math. (1989), Springer: Springer Berlin) · Zbl 0934.74003
[24] Mandel, J., Thermodynamics and plasticity, (Domingers, J. J.Delgado; Nina, N. R.; Whitelaw, J. H., Foundations of Continuum Thermodynamics (1974), Macmillan: Macmillan London), 283-304
[25] Lee, E. H., Elastic-plastic deformations at finite strains, ASME J. Appl. Mech., 36, 1-6 (1968) · Zbl 0179.55603
[26] Hughes, T. J.R., Numerical implementation of constitutive models: Rate independent deviatoric plasticity, (Nemat-Nasser, S.; Asaro, R.; Hegemier, G., Theoretical Foundations for Large-Scale Computations of Nonlinear Material Behaviour (1984), Martinus Nijhoff: Martinus Nijhoff The Netherlands)
[27] Needleman, A.; Tvegaard, V., Finite element analysis of localization plasticity, (Oden, J. T.; Carey, G. F., Finite Elements, Vol. V: Special Problems in Solid Mechanics (1984), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ)
[28] Argyris, J. H.; Doltsinis, J. St., On the large strain inelastic analysis in natural formulation—Part I: Quasistatic problems, Comput. Methods Appl. Mech. Engrg., 20, 213-252 (1979) · Zbl 0437.73065
[29] Hill, R., The Mathematical Theory of Plasticity (1950), Clarendon: Clarendon Oxford · Zbl 0041.10802
[30] Green, A. E.; Naghdi, P. M., A general theory of an elastic-plastic continuum, Arch. Rat. Mech. Anal., 18, 251-281 (1965) · Zbl 0133.17701
[31] Green, A. E.; McInnis, B. C., Generalized hypo-elasticity, (Proc. Roy. Soc. Edinburgh, A57 (1967)), 220-230 · Zbl 0149.43201
[32] Nagtegaal, J. C.; de Jong, J. E., Some aspects of non-isotropic work-hardening in finite strain plasticity, (Lee, E. H.; Mallet, R. L., Plasticity of Metals at Finite Strains, Proc. Research Workshop (1981), Stanford University), 65-102
[33] Simo, J. C.; Taylor, R. L., Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms, Comput. Methods Appl. Mech. Engrg., 85, 273-310 (1991) · Zbl 0764.73104
[34] Simo, J. C., Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory, Comput. Methods Appl. Mech. Engrg., 99, 61-112 (1992) · Zbl 0764.73089
[35] Borja, R. I.; Alarcón, E., A mathematical framework for finite strain elasto-plastic consolidation, Part 1: Balance laws, variational formulation, and linearization, Comput. Methods Appl. Mech. Engrg., 122, 145-171 (1995) · Zbl 0852.73051
[36] Marsden, J. E.; Hughes, T. J.R., Mathematical Foundations of Elasticity (1983), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0545.73031
[37] Ortiz, M.; Martin, J. B., Symmetry-preserving return mapping algorithms and incrementally extremal paths: a unification of concepts, Int. J. Numer. Methods Engrg., 28, 1839-1853 (1989) · Zbl 0704.73030
[38] Calabresi, G.; Rampello, S.; Callisto, L., The leaning tower of Pisa. Geotechnical characterization of the tower’s subsoil within the framework of the critical state theory, (Research Report (January 1993), Dipartimento di Ingeneria Structturale e Geotecnica, Université degli Studi di Roma ‘La Sapienza’)
[39] Dennis, J. E.; Moré, J. J., Quasi-Newton methods, motivation and theory, SIAM Rev., 19, 1, 46-89 (1977) · Zbl 0356.65041
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