Acta Geotechnica

, Volume 14, Issue 2, pp 295–311 | Cite as

Simulation of long-term thermo-mechanical response of clay using an advanced constitutive model

  • Despina M. ZymnisEmail author
  • Andrew J. Whittle
  • Xiaohui Cheng
Research Paper


There is extensive data to show that heating and cooling produces irrecoverable deformations in clays under fully drained conditions. The effects are most pronounced for normally and lightly overconsolidated clays that undergo significant compression. Most constitutive models have key limitations for predicting the thermo-mechanical response of clays through long-term (seasonal) cycles of heating and cooling. The Tsinghua ThermoSoil model (TTS; Zhang and Cheng in Int J Numer Anal Methods Geomech 41(4):527–554, 2017) presents a novel theoretical framework for simulating the coupled thermo-mechanical response of clays. The model uses a double-entropy approach to capture effects of energy dissipation at the microscopic particulate contact level on continuum behavior. This paper proposes a simple procedure for calibrating input parameters and illustrates this process using recent laboratory data for Geneva Clay (Di Donna and Laloui in Eng Geol 190:65–76, 2015). We then investigate capabilities of the TTS model in simulating familiar aspects of thermal consolidation of clays as well as the long-term, progressive accumulation of strains associated with seasonal heating and cooling processes for shallow geothermal systems installed in clays. The model predicts the existence of a long-term steady-state condition where there is no further accumulation of strain. This state depends on the consolidation stress and stress history but is independent of the imposed range of temperature, Tcyc. However, the value of Tcyc does affect the rate of accumulation of strain with thermal cycles. Simulations for normally consolidated Geneva Clay find steady-state strain conditions ranged from 2.0 to 3.7% accumulating within N = 10–50 thermal cycles.


Constitutive modeling Cyclic thermal loading Irrecoverable deformations Long-term response Thermo-mechanical response 

List of symbol

Lower case


TTS model input constant that controls rate effects


TTS model input constant related to cohesion


TTS model input constant related to the critical state friction angle


Void ratio


TTS model input constant that controls hysteretic strains


TTS model input constant that controls elastic strain evolution


TTS model input constant that controls elastic strain evolution and location of reload curve


TTS model input constant that controls the contribution of volumetric and deviatoric strains on granular temperature production


TTS model input constant that controls the rate of granular temperature production


TTS model input constant that controls the amount of thermal volumetric strains produced due to heating and cooling


Mean total stress


Mean effective stress


Shear stress


Water content

Upper case


TTS model input constant that controls the location of the VCL


TTS model input constant that controls the slope of the VCL


Compressibility index


Specific gravity of soil


Granular entropy conversion rate


In situ coefficient of earth pressure at rest


Secant elastic bulk modulus of the solid skeleton


TTS model input constant that controls the shift of the VCL due to increase in temperature


Slope of critical state line




Imposed range of temperature during cyclic heating and cooling


Granular temperature



Input constant for TTS model that controls the conversion of bound to free water during heating


Thermal expansion coefficient


Deviatoric strain


Volumetric strain


TTS model input constant that controls the coefficient of earth pressure at rest K0


Mass density


Dry density of soil medium


Horizontal effective stress


Mean effective stress


Preconsolidation pressure


Vertical effective stress




Elastic potential energy density function










Reference initial state


At temperature 20°C


Bound water


Free water


Normally consolidated




Soil skeleton or solid particles





Critical state line


Normally consolidated




Overconsolidation ratio


Specific surface area


Specimen Quality Designation


Tsinghua ThermoSoil Model


Virgin consolidation line



The Authors are grateful for the support provided by the Low Carbon Energy University Alliance (LCEUA), which enabled three-way collaborations with colleagues at Tsinghua University and the University of Cambridge. The first Author (DMZ) also received a Robert A. Brown, Onassis Foundation, Exponent and Martin Foundation Fellowships for her Ph.D. studies.


  1. 1.
    Abuel-Naga HM, Bergado DT, Ramana GV, Grino L, Rujivipat P, They Y (2006) Experimental evaluation of engineering behavior of Bangkok clay under elevated temperature. ASCE J Geotech Geoenviron Eng 132(7):902–910CrossRefGoogle Scholar
  2. 2.
    Anderson DM, Low PF (1958) The density of water adsorbed by Li-, Na-, and K-bentonite. Soil Sci Soc Am Proc 22:99–103CrossRefGoogle Scholar
  3. 3.
    Baldi G, Hueckel T, Peano A, Pellegrini R (1991) Developments in modeling of thermo-hydro-geomechanical behaviour of Boom clay and clay-based buffer materials, report 13365/2 EN. Publications of the European Communities, LuxembourgGoogle Scholar
  4. 4.
    Booker JR, Savvidou C (1985) Consolidation around a point heat source. Int J Numer Anal Methods Geomech 9:173–184CrossRefGoogle Scholar
  5. 5.
    Brochard L, Honório T, Vandamme M, Bornert M, Peigney M (2017) Nanoscale origin of the thermomechanical behavior of clays. Acta Geotech 12:1261–1279CrossRefGoogle Scholar
  6. 6.
    Campanella RG, Mitchell JK (1968) Influence of temperature variations on soil behavior. ASCE J Soil Mech Found Eng Div 94(SM3):709–734Google Scholar
  7. 7.
    Cekerevac C (2003) Thermal effects on the mechanical behaviour of saturated clays: an experimental and constitutive study. Ph.D. thesis, École Polytechnique Federale de LausanneGoogle Scholar
  8. 8.
    Cekerevac C, Laloui L (2004) Experimental study of thermal effects on the mechanical behaviour of a clay. Int J Numer Anal Meth Geomech 28(3):209–228. CrossRefGoogle Scholar
  9. 9.
    Cui YJ, Sultan N, Delage P (2000) A thermomechanical model for saturated clays. Can Geotech J 37(3):607–620CrossRefGoogle Scholar
  10. 10.
    De Wit CT, Arens PL (1950) Moisture content and density of some clay minerals and some remarks on the hydration pattern of clay. In: Transition 4th international congress soil science, vol 2, pp 59–62Google Scholar
  11. 11.
    Delage P, Sultan N, Cui YJ (2000) On the thermal consolidation of Boom clay. Can Geotech J 37:343–354CrossRefGoogle Scholar
  12. 12.
    Di Donna A (2014). Thermo-mechanical aspects of energy piles. Ph.D. Thesis, École Polytechnique Federale de LausanneGoogle Scholar
  13. 13.
    Di Donna A, Laloui L (2015) Response of soil subjected to thermal cyclic loading: experimental and constitutive study. Eng Geol 190:65–76CrossRefGoogle Scholar
  14. 14.
    Fripiat JJ, Letellier M, Levitz P (1984) Interaction of water with clay surfaces. Philos Trans R Soc Lond A311:287–299CrossRefGoogle Scholar
  15. 15.
    Gens A (2003) The role of geotechnical engineering for nuclear energy utilization. In: Proceedings of 13th European conference on soil mechanics and geotechnical engineering, Prague 3, pp 25–67Google Scholar
  16. 16.
    Gens A (2010) Soil–environment interactions in geotechnical engineering. Géotechnique 60(1):3–74CrossRefGoogle Scholar
  17. 17.
    Gens A, Olivella S (2001) Numerical analysis of radioactive waste disposal. In: Schrefler BA (ed) Environmental geomechanics. Springer, Wien, pp 203–234CrossRefGoogle Scholar
  18. 18.
    Gens A, Olivella S (2005) “Numerical modelling in nuclear waste storage engineering. In: Proceedings of 11th IACMAG international conference, Turin 4, pp 555–570Google Scholar
  19. 19.
    Gens A, Garitte B, Olivella S, Vaunat J (2009) Applications of multiphysical geomechanics in underground nuclear waste storage. Eur J Environ Civil Eng 13(7–8):937–962CrossRefGoogle Scholar
  20. 20.
    Gens A, Sanchez M, LdoN Guimaraes, Alonso EE, Lloret A, Olivella S, Villar MV, Huertas F (2009) A full-scale in situ heating test for high-level nuclear waste disposal: observations, analysis and interpretation. Géotechnique 59(4):377–399CrossRefGoogle Scholar
  21. 21.
    Groot SRD, Mazur P (1962) Non-equilibrium thermodynamics. North-Holland Publication Co, AmsterdamzbMATHGoogle Scholar
  22. 22.
    Houlsby G, Puzrin AM (2007) Principles of hyperplasticity: an approach to plasticity theory based on thermodynamic principles. Springer, LondonGoogle Scholar
  23. 23.
    Hueckel TA (1992) Water-mineral interaction in hygromechanics of clays exposed to environmental loads: a mixture-theory approach. Can Geotech J 29:1071–1086CrossRefGoogle Scholar
  24. 24.
    Hueckel T, Baldi G (1990) Thermoplasticity of saturated clays: experimental constitutive study. J Geotech Eng ASCE 116(12):1768–1796Google Scholar
  25. 25.
    Hueckel T, Borsetto M (1990) Thermoplasticity of saturated soils and shales: constitutive equations. ASCE J Geotech Eng 116(2):1765–1777CrossRefGoogle Scholar
  26. 26.
    Hueckel T, Pellegrini R, Del Olmo C (1998) A constitutive study of thermo-elasto-plasticity of deep carbonatic clays. Int J Numer Anal Meth Geomech 22(7):549–574CrossRefzbMATHGoogle Scholar
  27. 27.
    Jâky J (1944) A nyugalmi nyomâs tényezöje (The coefficient of earth pressure at rest). Magyar Mérnok és Epitész Egylet Közlönye (Journal for Society of Hungarian Architects and Engineers), October, pp 355–358Google Scholar
  28. 28.
    Jiang Y, Liu M (2007) From elasticity to hypoplasticity: dynamics of granular solids. Phys Rev Lett 99(10):105501CrossRefGoogle Scholar
  29. 29.
    Jiang Y, Liu M (2009) Granular solid hydrodynamics. Granular Matter 11(3):139–156CrossRefzbMATHGoogle Scholar
  30. 30.
    Ladd CC, DeGroot DJ (2003) Recommended practice for soft ground site characterization: arthur casagrande lecture. In: 12th panamerican conference on soil mechanics and geotechnical engineering, vol 1, pp 1–57Google Scholar
  31. 31.
    Laloui L, Francois B (2009) ACMEG-T: soil thermoplasticity model. J Eng Mech ASCE 135(9):932–944CrossRefGoogle Scholar
  32. 32.
    Laloui L, Moreni M, Vulliet L (2003) Behavior of a dual-purpose pile as foundation and heat exchanger. Can Geotech J 40(2):388–402. CrossRefGoogle Scholar
  33. 33.
    Laloui L, Nuth M, Vulliet L (2006) Experimental and numerical investigations of the behaviour of a heat exchanger pile. Int J Numer Anal Meth Geomech 30(8):763–781. CrossRefGoogle Scholar
  34. 34.
    Lambe TW (1960) The structure of compacted clay. ASCE J Soil Mech Found Eng Div 125:682–706Google Scholar
  35. 35.
    Leroueil S, Marques MES (1996) Importance of strain rate and temperature effects in geotechnical engineering. ASCE Meas Model Soil Behav 61:1–60Google Scholar
  36. 36.
    Mackenzie RC (1958) Density of water sorbed on montmorillonite. Nature 181:334CrossRefGoogle Scholar
  37. 37.
    Mitchell JK (1993) Fundamentals of soil behavior, 2nd edn. Wiley, New YorkGoogle Scholar
  38. 38.
    Mooney RW, Keenan AG, Wood LA (1952) Adsorption of water vapor on montmorillonite. J Am Chem Soc 74:1371–1374CrossRefGoogle Scholar
  39. 39.
    Morin R, Silva AJ (1984) The effects of high pressure and high temperature on some physical properties of ocean sediments. J Geophys Res 89(B1):511–526CrossRefGoogle Scholar
  40. 40.
    Norrish K (1954) The swelling of montmorillonite. Faraday Soc Discuss 18:120–134CrossRefGoogle Scholar
  41. 41.
    Paaswell RE (1967) Temperature effects on clay soil consolidation. J Soil Mech Found Eng Div ASCE 93(SM3):9–22Google Scholar
  42. 42.
    Panagiotidou AI (2017) Adaptation of granular solid hydrodynamics for modeling sand behavior. Ph.D. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  43. 43.
    Smith DW, Booker JR (1989) Boundary integral analysis of transient thermoelasticity. Int J Numer Anal Meth Geomech 13(3):283–302CrossRefzbMATHGoogle Scholar
  44. 44.
    Sposito G (1984) The surface chemistry of soils. Oxford University Press, OxfordGoogle Scholar
  45. 45.
    Sposito G, Prost R (1982) Structure of water adsorbed on smectites. Chem Rev 82(6):553–573CrossRefGoogle Scholar
  46. 46.
    Suter JL, Coveney PV, Greenwell HC, Thyveetil MA (2007) Large-scale molecular dynamics study of montmorillonite clay: emergence of undulatory fluctuations and determination of material properties. J Phys Chem C 111:8248–8259CrossRefGoogle Scholar
  47. 47.
    Zhang Z, Cheng X (2013) Simulation of nonisothermal consolidation of saturated soils based on a thermodynamic model. Sci World J. Google Scholar
  48. 48.
    Zhang Z, Cheng X (2017) A fully coupled THM model based on a non-equilibrium thermodynamic approach and its application. Int J Numer Anal Meth Geomech 41(4):527–554CrossRefGoogle Scholar
  49. 49.
    Zymnis DM (2016) Long-term ground response for borehole heat exchangers in clay. Ph.D. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  50. 50.
    Zymnis DM, Whittle AJ (2014) Numerical simulation of a shallow geothermal heating/cooling system. Proc ASCE GeoCongress GSP 234:2767–2776Google Scholar
  51. 51.
    Zymnis DM, Whittle AJ, Germaine JT (2019) Measurements of temperature-dependent bound water in clays. ASTM Geotech Test J. Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Despina M. Zymnis
    • 1
    Email author
  • Andrew J. Whittle
    • 2
  • Xiaohui Cheng
    • 3
  1. 1.Department of Civil and Environmental EngineeringMITCambridgeUSA
  2. 2.Edmund K. Turner Professor of Civil and Environmental EngineeringMITCambridgeUSA
  3. 3.Civil EngineeringTsinghua UniversityBeijingChina

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