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

- 41 Downloads

## Abstract

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.

## Keywords

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

### Acknowledgements

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.

## References

- 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.Alizadeh A, Gatmiri B (2017) An elasto–plastic damage model for argillaceous geomaterials. Appl Clay Sci 135:82–94CrossRefGoogle Scholar
- 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.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. https://doi.org/10.1002/2015JB012087 CrossRefGoogle Scholar
- 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.Bennett KC, Regueiro RA, Borja RI (2016) Finite strain elastoplasticity for materials capable of undergoing plastic volume change. Int J Plast 77:214–245. https://doi.org/10.1016/j.ijplas.2015.10.007 CrossRefGoogle Scholar
- 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.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.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.Bennett KC, Regueiro RA, Luscher DJ (2018) Anisotropic finite hyper-elastoplasticity of geomaterials with Drucker–Prager/Cap type constitutive model formulation. Int J Plast. https://doi.org/10.1016/j.ijplas.2018.11.010 CrossRefGoogle Scholar
- 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.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.Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164zbMATHCrossRefGoogle Scholar
- 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.Borja R (2013) Plasticity modeling & computation. Springer, BerlinzbMATHGoogle Scholar
- 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.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.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.Bowen RM (1982) Compressible porous media models by use of the theory of mixtures. Int J Eng Sci 20(6):697–735zbMATHCrossRefGoogle Scholar
- 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.Cattoni E, Tamagnini C (2018) On the seismic response of a propped r.c. diaphragm wall in a saturated clay. Acta Geotechnica. https://doi.org/10.1007/s11440-019-00771-4 CrossRefGoogle Scholar
- 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.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.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.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.Clayton JD (2006) Continuum multiscale modeling of finite deformation plasticity and anisotropic damage in polycrystals. Theor Appl Fract Mech 45:163–185CrossRefGoogle Scholar
- 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.Coleman B, Gurtin M (1967) Thermodynamics with internal variables. J Chem Phys 47:597–613CrossRefGoogle Scholar
- 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.Dafalias YF (1986) An anisotropic critical state soil plasticity model. Mech Res Commun 13(6):341–347zbMATHCrossRefGoogle Scholar
- 31.de Souza Neto E, Peric D, Owen D (2008) Computational methods for plasticity. Wiley, West SusssexCrossRefGoogle Scholar
- 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.Delage P (2010) A microstructure approach to the sensitivity and compressibility of some Eastern Canada sensitive clays. Géotechnique 60(5):353–368CrossRefGoogle Scholar
- 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.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.Grinfeld M (1991) Thermodynamic methods in the theory of heterogeneous system. Wiley, New YorkGoogle Scholar
- 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.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.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.Hill R (1948) A variational principle of maximum plastic work in classical plasticity. Q J Mech Appl Math 1:18–28MathSciNetzbMATHCrossRefGoogle Scholar
- 41.Holzapfel G (2000) Nonlinear solid mechanics. Wiley, West SussexzbMATHGoogle Scholar
- 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.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.Jin W, Arson C (2019) Fluid-driven transition from damage to fracture in anisotropic porous media: a multi-scale XFEM approach. Acta Geotech. https://doi.org/10.1007/s11440-019-00813-x CrossRefGoogle Scholar
- 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.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.Kachanov LM (1958) Time of the rupture process under the creep condition. Isz Akad Nauk SSSR Otd Techn Nauk 8:26–31Google Scholar
- 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.Lambe TW, Whitman RV (1969) Soil mechanics. Wiley, New YorkGoogle Scholar
- 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.Lemaitre J (1985a) A continuous damage mechanics model for ductile fracture. J Eng Mat Tech 107:83–89CrossRefGoogle Scholar
- 52.Lemaitre J (1985b) Coupled elasto–plasticity and damage constitutive equations. Comput Method Appl Mech Eng 51:31–49zbMATHCrossRefGoogle Scholar
- 53.Lemaitre J, Chaboche JL (1990) Mechanics of solid materials. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
- 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.Lubliner J (1984) A maximum-dissipation principle in generalized plasticity. Acta Mech 52:225–237MathSciNetzbMATHCrossRefGoogle Scholar
- 56.Lubliner J (1986) Normality rules in large deformation plasticity. Mech Mat 5:29–34CrossRefGoogle Scholar
- 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.Mavko G, Mukerji T (1998) Bounds on low-frequency seismic velocities in partially saturated rocks. Geophysics 63(3):918–924CrossRefGoogle Scholar
- 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.Mun W, McCartney J (2015) Compression mechanisms of unsaturated clay under high stresses. Can Geotech J 52(12):1676–1684CrossRefGoogle Scholar
- 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.Rakotomanana L (2004) A geometric approach to thermodynamics of dissipating continua. Birkhäser, BostonzbMATHCrossRefGoogle Scholar
- 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.Roscoe KH, Schofield AN, Wroth CP (1958) On the yielding of soils. Géotechnique 8(1):22–53CrossRefGoogle Scholar
- 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.Schofield A, Wroth P (1968) Critical state soil mechanics, vol 310. McGraw-Hill, LondonGoogle Scholar
- 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.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.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.Simo J, Hughes T (1998) Computational inelasticity. Springer, BerlinzbMATHGoogle Scholar
- 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.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.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.Terzaghi K (1943) Theoretical soil mechanics. Wiley, New YorkCrossRefGoogle Scholar
- 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.Truesdell C, Toupin R (1960) The classical field theories. Springer, BerlinCrossRefGoogle Scholar
- 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.von Mises R (1928) Mechanik der plastischen formänderung von krinstallen. Math Mech 8:161–185zbMATHGoogle Scholar
- 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.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. https://doi.org/10.1016/j.cma.2019.04.037 MathSciNetCrossRefGoogle Scholar
- 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.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.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