An energy approach to Modified Cam-Clay plasticity and damage modeling of cohesive soils

  • Kane C. BennettEmail author
Research Paper


An alternative approach to Modified Cam-Clay (MCC) critical state plasticity coupled with damage is proposed. The provided hyper-elastoplastic/damage framework is motivated by a desire to ensure thermodynamic consistency of model predictions, and is shown to satisfy the principle of maximum plastic dissipation, enabling enforcement of the plastic dissipation (and Clausius–Planck) inequality. A small strain Eshelby-like stress is derived as being energy-conjugate to the plastic strain rate, and relation to the damage-energy release rate is exploited to pose a coupled damage/yield MCC criteria (MCC-D). Relations between volumetric damage and porosity changes are examined, along with a method for estimating the damage state from bulk moduli measurements. The model is shown to simulate well existing high pressure compression measurements of Boulder clay, and connections between damage-related model parameters and behavior exemplifying different microstructural states is examined through numerical experiments.


Critical state plasticity Cohesive soil Consolidation Damage Eshelby stress Modified Cam-Clay 



The author is grateful for research funding from the Stanford Gere Research Fellowship, and for valuable input from Professor Ronaldo I. Borja during the original conception and formation of the ideas behind this work while at Stanford University.


  1. 1.
    Abou-Chakra Guéry A, Cormery F, Shao JF, Kondo D (2008) A micromechanical model of elastoplastic and damage behavior of a cohesive geomaterial. Int J Solids Struct 45:1406–1429zbMATHCrossRefGoogle Scholar
  2. 2.
    Alizadeh A, Gatmiri B (2017) An elasto–plastic damage model for argillaceous geomaterials. Appl Clay Sci 135:82–94CrossRefGoogle Scholar
  3. 3.
    Arson C, Pereira J-M (2012) Influence of damage on pore size distribution and permeability of rocks. Int J Numer Anal Method Geomech 37:810–831CrossRefGoogle Scholar
  4. 4.
    Arson C, Vanorio T (2015) Chemomechanical evolution of pore space in carbonate microstructures upon dissolution: linking pore geometry to bulk elasticity. J Geophys Res Solid Earth 120:6878–6894. CrossRefGoogle Scholar
  5. 5.
    Bennett KC, Berla LA, Nix WD, Borja RI (2015) Instrumented nanoindentation and 3D mechanistic modeling of a shale at multiple scales. Acta Geotech 10(1):1–14CrossRefGoogle Scholar
  6. 6.
    Bennett KC, Regueiro RA, Borja RI (2016) Finite strain elastoplasticity for materials capable of undergoing plastic volume change. Int J Plast 77:214–245. CrossRefGoogle Scholar
  7. 7.
    Bennett KC, Luscher DJ, Buechler MA, Yeager JD (2018) A micromechanical framework and modified self-consistent homogenization scheme for the thermoelasticity of porous bonded particle assemblies. Int J Solids Struct 139:224–237CrossRefGoogle Scholar
  8. 8.
    Bennett KC, Luscher DJ (2019) Effective thermoelasticity of polymer-bonded particle composites with imperfect interfaces and thermally expansive interphases. J Elast 136(1):55–85MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bennett KC, Borja RI (2018) Hyper-elastoplastic/damage modeling of rock with application to porous limestone. Int J Solids Struct 143:218–231CrossRefGoogle Scholar
  10. 10.
    Bennett KC, Regueiro RA, Luscher DJ (2018) Anisotropic finite hyper-elastoplasticity of geomaterials with Drucker–Prager/Cap type constitutive model formulation. Int J Plast. CrossRefGoogle Scholar
  11. 11.
    Bikong C, Hoxha D, Shao JF (2015) A micro-macro model for time-dependent behavior of clayey rocks due to anisotropic propagation of microcracks. Int J Plast 69:73–88CrossRefGoogle Scholar
  12. 12.
    Bignonnet F, Dormieux L, Kondo D (2016) A micro-mechanical model for the plasticity of porous granular media and link with the Cam clay model. Int J Plast 79:259–274CrossRefGoogle Scholar
  13. 13.
    Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164zbMATHCrossRefGoogle Scholar
  14. 14.
    Borja RI (2006) On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int J Solids Struct 43(6):1764–1786zbMATHCrossRefGoogle Scholar
  15. 15.
    Borja R (2013) Plasticity modeling & computation. Springer, BerlinzbMATHGoogle Scholar
  16. 16.
    Borja RI, Choo J (2016) Cam-clay plasticity, part VIII: a constitutive framework for porous materials with evolving internal structure. Comput Method Appl Mech Eng 309:653–679MathSciNetCrossRefGoogle Scholar
  17. 17.
    Borja RI, Lee SR (1990) Cam-clay plasticity, part 1: implicit integration of elasto-plastic constitutive equations. Comput Method Appl Mech Eng 78:49–72zbMATHCrossRefGoogle Scholar
  18. 18.
    Borja RI, Tamagnini C (1998) Cam-clay plasticity part III: extension of the infintesimal model to include finite strains. Comput Meth Appl Mech Eng 155:73–95zbMATHCrossRefGoogle Scholar
  19. 19.
    Bowen RM (1982) Compressible porous media models by use of the theory of mixtures. Int J Eng Sci 20(6):697–735zbMATHCrossRefGoogle Scholar
  20. 20.
    Bryant EC, Sun W (2019) A micromorphic critical state plasticity model for capturing the size-dependent anisotropic effect of shale, clay, and mudstone. Comput Method Appl Mech Eng 354:56–95CrossRefGoogle Scholar
  21. 21.
    Cattoni E, Tamagnini C (2018) On the seismic response of a propped r.c. diaphragm wall in a saturated clay. Acta Geotechnica. CrossRefGoogle Scholar
  22. 22.
    Chang CS, Bennett KC (2015) Micromechanical modeling for the deformation of sand with non-coaxiality between the stress and material axes. J Eng Mech 143(1):C4015001Google Scholar
  23. 23.
    Chazallon C, Hicher Y (1998) A constitutive model coupling elastoplasticity and damage for cohesive-frictional materials. Mech Cohes Frict Mater 3:41–63CrossRefGoogle Scholar
  24. 24.
    Chen RP, Zhu S, Hong PY, Cheng W, Cui YJ (2019) A two-surface plasticity model for cyclic behavior of saturated clay. Acta Geotech 14(2):279–293CrossRefGoogle Scholar
  25. 25.
    Choo J, Borja RI (2015) Stabilized mixed finite elements for deformable porous media with double porosity. Comput Methods Appl Mech Eng 293:0–23MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Clayton JD (2006) Continuum multiscale modeling of finite deformation plasticity and anisotropic damage in polycrystals. Theor Appl Fract Mech 45:163–185CrossRefGoogle Scholar
  27. 27.
    Clayton JD, Tonge AL (2015) A nonlinear anisotropic elastic–inelastic constitutive model for polycrystalline ceramics and minerals with application to boron carbide. Int J Solids Struct 64:191–207CrossRefGoogle Scholar
  28. 28.
    Coleman B, Gurtin M (1967) Thermodynamics with internal variables. J Chem Phys 47:597–613CrossRefGoogle Scholar
  29. 29.
    Coleman B, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Ration Mech Anal 13(1):167–168MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Dafalias YF (1986) An anisotropic critical state soil plasticity model. Mech Res Commun 13(6):341–347zbMATHCrossRefGoogle Scholar
  31. 31.
    de Souza Neto E, Peric D, Owen D (2008) Computational methods for plasticity. Wiley, West SusssexCrossRefGoogle Scholar
  32. 32.
    Desbois G, Urai JL, Schuck B, Hoehne N, Oelker A, Bésuelle P, Viggiani G, Schmatz J, Klaver J (2017) Deformation mechanisms and resealing of damage zones in experimentally deformed cemented and un-cemented clay-rich geomaterials, at low bulk strain. In: EGU General Assembly Conference Abstracts, vol 19, p 4447Google Scholar
  33. 33.
    Delage P (2010) A microstructure approach to the sensitivity and compressibility of some Eastern Canada sensitive clays. Géotechnique 60(5):353–368CrossRefGoogle Scholar
  34. 34.
    Einav I, Houlsby GT, Nguyen GD (2007) Coupled damage and plasticity models derived from energy and dissipation potentials. Int J Solids Struct 44:2487–2508zbMATHCrossRefGoogle Scholar
  35. 35.
    Fossum AF, Fredrich JT (2000) Cap plasticity models and compactive and dilatant pre-failure deformation. In: Proc fourth North American rock mechanics symposium, NARMS, 2000, Seattle, Washington, pp 1169–1176Google Scholar
  36. 36.
    Grinfeld M (1991) Thermodynamic methods in the theory of heterogeneous system. Wiley, New YorkGoogle Scholar
  37. 37.
    Gurson A (1977) Continuum theory of ductile rupture by void nucleation and growth: part 1-yield criteria and flow rules for porous ductile media. J Eng Mat Tech 99:2–15CrossRefGoogle Scholar
  38. 38.
    Hackl K (1997) Generalized standard media and variational principles in classical and finite strain elastoplasticity. J Mech Phys Solids 45(5):667–688MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Hamiel Y, Lyakhovsky V, Agnon A (2004) Coupled evolution of damage and porosity in poroelastic media: theory and applications to deformation of porous rocks. Geophys J Int 156:701–713CrossRefGoogle Scholar
  40. 40.
    Hill R (1948) A variational principle of maximum plastic work in classical plasticity. Q J Mech Appl Math 1:18–28MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Holzapfel G (2000) Nonlinear solid mechanics. Wiley, West SussexzbMATHGoogle Scholar
  42. 42.
    Holzapfel G, Gasser T, Ogden R (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Houlsby GT, Puzrin AM (2000) A thermomechanical framework for constitutive models for rate-independent dissipative materials. Int J Plast 16(9):1017–1047zbMATHCrossRefGoogle Scholar
  44. 44.
    Jin W, Arson C (2019) Fluid-driven transition from damage to fracture in anisotropic porous media: a multi-scale XFEM approach. Acta Geotech. CrossRefGoogle Scholar
  45. 45.
    Ju WJ (1989) On energy based coupled elastoplastic damage theories: constitutive modeling and computational aspects. Int J Solids Struct 25(7):803–833zbMATHCrossRefGoogle Scholar
  46. 46.
    Ju JW, Yuan KY, Kuo AW (2012) Novel strain energy based coupled elastoplastic damage and healing models for geomaterials—part I: formulations. Int J Damag Mech 21:525–549CrossRefGoogle Scholar
  47. 47.
    Kachanov LM (1958) Time of the rupture process under the creep condition. Isz Akad Nauk SSSR Otd Techn Nauk 8:26–31Google Scholar
  48. 48.
    Krajcinovic D (1996) Damage mechanics. In: Achenbach JD, Budiansky D, Lauwerier HA, Saffman PG, van Wijngaarden L, Willis WR (eds) North-Holland series in applied mathematics and mechanics. Elsevier, AmsterdamGoogle Scholar
  49. 49.
    Lambe TW, Whitman RV (1969) Soil mechanics. Wiley, New YorkGoogle Scholar
  50. 50.
    Lei H, Wong H, Fabbri A, Bui TA, Limam A (2016) Some general remarks on hyperplasticity modelling and its extension to partially saturated soils. Z Angew Math Phys 67:64MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Lemaitre J (1985a) A continuous damage mechanics model for ductile fracture. J Eng Mat Tech 107:83–89CrossRefGoogle Scholar
  52. 52.
    Lemaitre J (1985b) Coupled elasto–plasticity and damage constitutive equations. Comput Method Appl Mech Eng 51:31–49zbMATHCrossRefGoogle Scholar
  53. 53.
    Lemaitre J, Chaboche JL (1990) Mechanics of solid materials. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  54. 54.
    Liu XS, Ning JG, Tan YL, Gu QH (2016) Damage constitutive model based on energy dissipation for intact rock subjected to cyclic loading. Int J Rock Miner Sci 85:27–32CrossRefGoogle Scholar
  55. 55.
    Lubliner J (1984) A maximum-dissipation principle in generalized plasticity. Acta Mech 52:225–237MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Lubliner J (1986) Normality rules in large deformation plasticity. Mech Mat 5:29–34CrossRefGoogle Scholar
  57. 57.
    Mandel J (1964) Contribution théorique à l’écrouissage et des lois de l’écoulement plastique. In: Proc 11th Int Cong Appl Mech, pp 502–509Google Scholar
  58. 58.
    Mavko G, Mukerji T (1998) Bounds on low-frequency seismic velocities in partially saturated rocks. Geophysics 63(3):918–924CrossRefGoogle Scholar
  59. 59.
    Miehe C, Gürses E (2007) A robust algorithm for configurational-force-driven brittle crack propagation with R-adaptive mesh alignment. Int J Numer Method Eng 72:127–155MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Mun W, McCartney J (2015) Compression mechanisms of unsaturated clay under high stresses. Can Geotech J 52(12):1676–1684CrossRefGoogle Scholar
  61. 61.
    Nguyen L, Fatahi B, Khabbaz H (2014) A constitutive model for cemented clays capturing cementation degradation. Int J Plast 56:1–18CrossRefGoogle Scholar
  62. 62.
    Rakotomanana L (2004) A geometric approach to thermodynamics of dissipating continua. Birkhäser, BostonzbMATHCrossRefGoogle Scholar
  63. 63.
    Ricard Y, Bercovici D (2003) Two-phase damage theory and crustal rock failure: the theoretical ‘void’ limit, and the prediction of experimental data. Geophys J Int 155:1057–1064CrossRefGoogle Scholar
  64. 64.
    Roscoe KH, Schofield AN, Wroth CP (1958) On the yielding of soils. Géotechnique 8(1):22–53CrossRefGoogle Scholar
  65. 65.
    Salari MR, Saeb S, Willam KJ, Patchet SJ, Carrasco RC (1998) A coupled elastoplastic damage model for geomaterials. Comput Method Appl Mech Eng 193:2625–2643zbMATHCrossRefGoogle Scholar
  66. 66.
    Schofield A, Wroth P (1968) Critical state soil mechanics, vol 310. McGraw-Hill, LondonGoogle Scholar
  67. 67.
    Shen WQ, Shao JF, Kondo D, Gatmiri B (2012) A micro–macro model for clayey rocks with a plastic compressible porous matrix. Int J Plast 36:64–85CrossRefGoogle Scholar
  68. 68.
    Shen WQ, Shao JF (2016) An incremental micro–macro model for porous geomaterials with double porosity and inclusion. Int J Plast 83:37–54CrossRefGoogle Scholar
  69. 69.
    Simo J (1988) A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: part I. Continuum formulation. Comput Method Appl M 66:199–219zbMATHCrossRefGoogle Scholar
  70. 70.
    Simo J, Hughes T (1998) Computational inelasticity. Springer, BerlinzbMATHGoogle Scholar
  71. 71.
    Simo J, Meschke G (1993) A new class of algorithms for classical plasticity extended to finite strains. Applications to geomaterials. Comput Mech 11:253–278MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Sone H, Zoback M (2013) Mechanical properties of shale-gas reservoir rocks—part 1: static and dynamic elastic properties and anisotropy. Can Geotech J 78(5):D381–D392Google Scholar
  73. 73.
    Stavropoulou E, Andò E, Tengattini A, Briffaut M, Dufour F, Atkins D, Armand G (2019) Liquid water uptake in unconfined Callovo Oxfordian clay-rock studied with neutron and X-ray imaging. Acta Geotech 14(1):19–33CrossRefGoogle Scholar
  74. 74.
    Terzaghi K (1943) Theoretical soil mechanics. Wiley, New YorkCrossRefGoogle Scholar
  75. 75.
    Truesdell C, Noll W (1965) The non-linear field theories of mechanics. In: Flügge S (ed) Handbuch der Physik III/3. Springer, BerlinGoogle Scholar
  76. 76.
    Truesdell C, Toupin R (1960) The classical field theories. Springer, BerlinCrossRefGoogle Scholar
  77. 77.
    Vanorio T, Ebert Y, Grombacher D (2014) What laboratory-induced dissolution tell us about natural diagenetic trends of carbonate rocks. Geol Soc Lond Spec Publ 406(1):311–329CrossRefGoogle Scholar
  78. 78.
    von Mises R (1928) Mechanik der plastischen formänderung von krinstallen. Math Mech 8:161–185zbMATHGoogle Scholar
  79. 79.
    Wang K, Sun W-C (2016) A semi-implicit discrete-continuum coupling method for porous media based on the effective stress principle at finite strain. Comput Methods Appl Mech Eng 304:546–583MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Zhang Q, Choo J, Borja RI (2019) On the preferential flow patterns induced by transverse isotropy and non-Darcy flow in double porosity media. Comput Method Appl Mech Eng 353:570–592. MathSciNetCrossRefGoogle Scholar
  81. 81.
    Zhu C, Arson C (2012) A model of damage and healing coupling halite thermo-mechanical behavior to microstructure evolution. Geotech Geol Eng (Special Issue: Thermohydromechanical behavior or soils and energy geostructures)Google Scholar
  82. 82.
    Zhu Q, Kondo D, Shao J, Penesee V (2007) Micromechanical modelling of anisotropic damage in brittle rocks and application. Int J Rock Mech Miner 45:467–477CrossRefGoogle Scholar
  83. 83.
    Zhu QZ, Shao JF, Mainguy M (2010) A micromechanics-based elastoplastic damage model for granular materials at low confining pressure. Int J Plast 26:586–602zbMATHCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Fluid Dynamics and Solid Mechanics Group (T-3), Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA

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