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

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## Abstract

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, *T*_{cyc}. However, the value of *T*_{cyc} 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.

## Keywords

Constitutive modeling Cyclic thermal loading Irrecoverable deformations Long-term response Thermo-mechanical response## List of symbol

## Lower case

*a*TTS model input constant that controls rate effects

*c*TTS model input constant related to cohesion

*c*′TTS model input constant related to the critical state friction angle

*e*Void ratio

*h*TTS model input constant that controls hysteretic strains

*m*_{1}TTS model input constant that controls elastic strain evolution

*m*_{2}TTS model input constant that controls elastic strain evolution and location of reload curve

*m*_{3}TTS model input constant that controls the contribution of volumetric and deviatoric strains on granular temperature production

*m*_{4}TTS model input constant that controls the rate of granular temperature production

*m*_{5}TTS model input constant that controls the amount of thermal volumetric strains produced due to heating and cooling

*p*Mean total stress

*p*′Mean effective stress

*q*Shear stress

*w*Water content

## Upper case

*B*_{0}TTS model input constant that controls the location of the VCL

*B*_{1}TTS model input constant that controls the slope of the VCL

- C
_{c} Compressibility index

- G
_{s} Specific gravity of soil

- \(\dot{I}_{\text{g}}\)
Granular entropy conversion rate

*K*_{0}In situ coefficient of earth pressure at rest

*K*_{e}Secant elastic bulk modulus of the solid skeleton

*L*_{T}TTS model input constant that controls the shift of the VCL due to increase in temperature

*M*Slope of critical state line

*T*Temperature

*T*_{cyc}Imposed range of temperature during cyclic heating and cooling

*T*_{g}Granular temperature

## Greek

*α*_{bf}Input constant for TTS model that controls the conversion of bound to free water during heating

*β*Thermal expansion coefficient

*ε*_{s}Deviatoric strain

*ε*_{v}Volumetric strain

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

*K*_{0}*ρ*Mass density

*ρ*_{d}Dry density of soil medium

*σ*_{h}′Horizontal effective stress

*σ*_{oct}′Mean effective stress

*σ*_{p}′Preconsolidation pressure

*σ*_{v}′Vertical effective stress

*φ*Porosity

*ω*_{e}Elastic potential energy density function

## Superscripts

- D
Irreversible

- e
Elastic

- h
Hysteretic

## Subscripts

- 0
Reference initial state

- 20
At temperature 20°C

- bw
Bound water

- fw
Free water

- NC
Normally consolidated

- OC
Overconsolidated

- s
Soil skeleton or solid particles

- w
Water

## Abbreviations

- CSL
Critical state line

- NC
Normally consolidated

- OC
Overconsolidated

- OCR
Overconsolidation ratio

- SSA
Specific surface area

- SQD
Specimen Quality Designation

- TTS
Tsinghua ThermoSoil Model

- VCL
Virgin consolidation line

## Notes

### Acknowledgements

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.

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