Extended TTS Model for Thermal and Mechanical Creep of Clay and Sand

In the e-TTS model, the stress-strain relationship for thermoelastic coupling is obtained from a function for the elastic potential energy density, ${\omega }_e$, to guarantee that the model is thermodynamically acceptablewhere ${\epsilon }_{ij}^e$ = elastic strain tensor; and ${\pi }_{ij}$ = elastic stress tensor.

Following Zheng and Cheng (2017), thermoelastic behavior of granular media (with Hertzian contact) can be represented bywhere $B=B_0\exp (B_1{\rho }_{\mathrm{d}})$; $B_0$, $B_1$, and $\xi$ = material parameters; $\mathrm{\Delta }T=T-T_0$ = temperature change; $T_0$ = reference temperature; $3K_e{\beta }_T\mathrm{\Delta }T$ = additional isotropic thermal loading to restrain the thermoelastic expansion induced by the temperature change; $K_e$ = secant elastic bulk modulus of the solid skeleton, the formulation of which is shown in the Appendix; ${\beta }_T$ = linear thermal expansion coefficient for the soil skeleton; ${\epsilon }_v^e$ and ${\epsilon }_s^e=\sqrt{e_{ij}^e{e}_{ij}^e}$ = volumetric and 2nd invariant of deviator elastic strain tensor, respectively; and $e_{ij}$ = components of the deviatoric strain tensor.

Zhang and Cheng (2017) established a generalized effective stress composition for saturated soils using the granular solid hydrodynamics approachwhere the first term on the right side of the effective stress equation is elastic stress, ${\pi }_{ij}$, which is the most important components of the effective stress; the partial stress ${\sigma }_{ij}^{bw}$ = bound effective stress (due to disjoining pressures between bound water and clay particles); ${\sigma }_{ij}^{v\alpha }$ = viscous stress associated with dissipative flows of material viscosity; and ${\sigma }_{ij}^g$ = granular effective stress, associated with kinetic energy fluctuation at the granular scale. Considering only the first term effective stress for saturated thermoelastic soils can then be expressed as

The total strain rate can be divided into the elastic strain rate and the inelastic strain rate

In classical plasticity theory, plastic strain is adopted to describe the inelastic strain for granular materials (where $d_t{\epsilon }_{ij}^D$ represents the plastic strain rate). The variation of plastic strain is described by the plastic potential function. In the TTS model, the variation of inelastic strain is attributable to the energy dissipation according to GSH theory. The total energy dissipation rate $R$ can be expressed as the product of dissipative forces and flowswhere ${\pi }_{ij}$ = dissipative forces corresponding to the effective stresses; $T_g$ = granular temperature; $d_t{\epsilon }_{ij}$ = total strain rates; $Y_{ij}$ = corresponding flows comprising the plastic strain rates; $I_g$ = the rate of conversion of granular (local) entropy; and ${\sigma }_{ij}^{vs}$ = viscous stresses of the soil skeleton.

The dissipative flows can be expressed as linear functions of the dissipative forces based on Onsager’s reciprocal principle. This well-known principle has been used to solve coupled fluid and heat flows in soil mechanics (Mitchell and Soga 2005). The TTS model describes incremental constitutive behavior as follows:where ${\lambda }_{ijkl}$ and $\gamma$ = migration coefficients.

It can be theoretically shown that the nonelastic strain rate ($d_t{\epsilon }_{ij}^D$) is the same as $Y_{ij}$ (Zhang and Cheng 2017). Zhang and Cheng (2017) propose the following expression for the dissipative flow $Y_{ij}$ (or plastic strain rate, $d_t{\epsilon }_{ij}^D$) to describe the transient elasticity of granular materialwhere ${\epsilon }_{ij}^e$, ${\epsilon }_{ij}^h$ = tensors of elastic and hysteretic strains tensors; and the parameter $a$ controls rate dependence in material behavior.

The hysteretic strains are additional state parameters that improve model predictions of hysteretic stress-strain response under cyclic loading and enable the model to describe locked-in elastic strains (Coussy 1995; Collins 2005; Yan and Li 2011).

The e-TTS model accounts for interactions between the macrolevel and microlevel of particle or granular behavior by a double-entropy approach. In any irreversible process the total real entropy represents the irrecoverable deformations at the macro level, while the granular entropy is attributable to the energy dissipative mechanisms of soils at the microscopic level, such as sliding, rolling, and collision of particles, leading to a change in the kinetic energy and elastic potential energy. The energy evolution and dissipation at the microscopic level is similar to molecular motion and can be described through a granular entropy density $s_g$ ($=b{\rho }_d{T}_g$ where $b$ = constant; and ${\rho }_d$ = dry mass density) and its conjugate variable, $T_g$ granular temperature (Haff 1983; Jiang and Liu 2009). The e-TTS model assumes that granular entropy increases according to the following equationwhere $R_g/T_g$ = granular entropy production rate; $T_g$ can be understood as the dissipative force of the granular fluctuation; $I_g$ = corresponding dissipative flow; and $I_g{T}_g$ = energy dissipation rate induced by the granular fluctuation.

In addition, it should be noted that the reorganization of soil particles can be activated by mechanical and/or thermal loading, such that the strain rate and/or rate of temperature change can be understood as dissipative forces for granular fluctuation, the “dissipative flows” corresponding to $d_t{\epsilon }_{ij}$ and $d_{t}T$ are denoted as ${\sigma }_{ij}^g$ and $M$, respectively. Hence, the granular entropy can be expressed as follows:with dissipation flow ${\sigma }_{ij}^g$

In order to describe the thermal-compaction of granular materials (sands, glass beads, etc.), the dissipative flow $M$ in e-TTS model can be expressed as a linear function of the rate of temperature change following the Onsager’s reciprocal principle again, shown in [Eq. 12(a)].

In clays, this universal thermal-compaction mechanism is also present in addition to the conversion process from bound water to free water during temperature elevation, which is already modeled in the original TTS model (Zhang and Cheng 2016, 2017). So the dissipative flow $M$ for clays has to be generalized into [Eq. 12(b)], in which the first term represents the effect of bound water to free water conversion and the second term represents the effect of thermal compaction mechanism of soil skeleton. This is similar to the concept used in the MIT-SR elasto-visco-plastic model for clay behavior (Yuan and Whittle 2018, 2021)where ${\theta }_t$ = migration coefficient for thermal-compaction of soil skeleton; ${\psi }_g$ controls the direction of temperature change for clay: ${\psi }_g>0$ for heating and ${\psi }_g=0$ for cooling, ${\pi }_{kk}$ = mean effective stress; ${\varphi }_{bw}(=V_{bw}/V)$ and $\varphi$ = bound water porosity and total porosity, respectively; and ${\alpha }_{bf}$ controls the conversion rate of bound water to free water. Zymnis et al. (2019) describe experimental methods to measure ${\alpha }_{bf}$.

Finally, the evolution of granular temperature can be derived as follows:where ${\eta }_g$, ${\zeta }_g$, $\gamma$, ${\psi }_g$, and ${\theta }_t$ = constant migration coefficients.

According to [Eq. 12(b)], there are two distinct mechanisms that relate changes in granular temperature to the temperature of the soil. The first involves the conversion from bound to free water, previously discussed by Zymnis et al. (2018) while the second, described by the migration coefficient, ${\theta }_t$, describes thermally induced skeleton compaction within the granular materials. In addition, there is still one more thermoelastic related physical mechanism, characterized by the term of $d_t{\epsilon }_{kk}$ in Eq. (13). If the temperature of soil is changed, this term becomes temperature dependent following Eqs. (4) and (8). This physical mechanism refers to the thermoelastic induced volumetric change, which is not purely thermal expansion/contraction any more according to the Eq. (8). This mechanism plays a less important role in complex thermal-mechanical coupled problems of both clays and sands, compared with the aforementioned two distinct mechanisms.

It is should be noted that there are other thermally-induced transport phenomena in soils including: (1) thermo-osmosis, and (2) thermal-filtration, which refers to the influence of a pressure gradient on heat flow (Mitchell and Soga 2005). These coupled processes have been discussed by Zhang and Cheng (2017), but are not defined within the constitutive law at the REV level in this paper.

The thermal creep or cyclic volumetric deformations are generally measured under 1D (oedometric) conditions, where there are two independent components of stress and strain. The Appendix summarizes the e-TTS model formulation for saturated soils in triaxial space. These relations define the effective stress-strain-temperature-time relations of soils at the elemental/point level and are used to simulate the thermal creep and cyclic thermomechanical response described in detail subsequently.

Within the e-TTS formulation, the inelastic strain is strain-rate dependent (Appendix) while the equations describing the evolution of granular temperature and inelastic strain are influenced by thermal effects [Eq. (13) in the Appendix]. Thus, the model simulates thermal creep behavior as a function of both strain rate and temperature.

Fig. 2 shows the solving procedure of thermal cyclic strain for the e-TTS model with a known rate of temperature ($d_{t}T$), the current effective stress state and initial values of state variables. The rates of elastic strains, total strains, and mass conservation are updated using Eqs. (14), (16), and (19), and then the rates of granular temperature, hysteretic strains (locked entropy) are then updated using Eqs. (17) and (18). In the current implementation the key governing equation [granular entropy, Eq. (17)] is solved using Matlab (ode45 solver).

There are 14 parameters (material constants) in the extended TTS model. The relationships between migration coefficients and model input parameters ($b$ and $m_{1,0}$ to $m_6$) are presented in Table 1. These can be calibrated using conventional laboratory tests. Zymnis et al. (2018) illustrate the calibration procedure for clay, under the assumption of strain rate–independent behavior ($a=0.5$). In this work, the isothermal mechanical component parameters ($B_0$, $B_1$, $\xi$, $a$, $h$, $m_1$, $m_2$, $m_3$, $m_4$) can be calibrated by isotropic/1D compression and creep tests, while the thermal component parameters (${\beta }_T$, $L_T$, ${\alpha }_{bf}$, $m_5$, $m_6$) are obtained from a thermal creep test. The initial elastic strains can be obtained from the initial consolidation stress state ($p^{\prime }$, $q$, $e$). We assume the initial locked strain is zero and initial granular temperature, $T_g={10}^{-4}$.

All model parameters listed in the Table 1 remain constant under temperature changes or gradients. It should be noted that $m_5$ is constant for a heating process and $m_5=0$ for cooling. The bound water content or bound water porosity ${\varphi }_{bw}=V_{bw}/V$ changes with the temperature.

Fig. 3 compares the measured experimental results for creep due to thermal and isothermal phases of tests on reconstituted Todi clay reported by Burghignoli et al. (2000) with corresponding simulation results using the e-TTS model. The measured data show that isothermal creep is strongly influenced by stress history with contractive creep strains occurring for $\mathrm{OCRs}=1.0$, 4.0, and volumetric expansion at $\mathrm{OCR}=8$ [Fig. 3(a)]. The thermal cycle amplifies the magnitudes of compressive strains at $\mathrm{OCR}=1$ and expansive strains at $\mathrm{OCR}=8$ [Fig. 3(b)]. The e-TTS captures the measured effects of OCR for creep under isothermal conditions and the relative magnitudes of thermal creep strains associated with the imposed cycle of heating and cooling. The model tends to overestimate the measured compressive creep strains at $\mathrm{OCR}=1.0$ and underestimates expansive strains at $\mathrm{OCR}=8.0$ using the current calibration of model parameters. However, it should be noted that most existing thermomechanical constitutive models ignore the thermally activated creep (Hueckel and Borsetto 1990; Cui et al. 2000; Abuel-Naga et al. 2007; Di Donna and Laloui 2015). As a consequence, either the thermal deformation is significantly underestimated or the thermal creep is significantly overestimated to fit to the total creep strain. Increasing the heating time (or decreasing the heating rate) inevitably induces a larger difference between the thermal creep and isothermal creep strains.

Fig. 4(a) shows the experimental results and simulation results of the isotropic compressive curve for reconstituted Todi clay. A typical unloading/reloading hysteretic loop was reproduced by the e-TTS model. In the e-TTS model, the locked energy (i.e., hysteretic strain) represents the influence of the stress history on the isothermal/mechanical and thermal creep. In general, the magnitude of thermal creep is smaller in OC soils than in NC soils. The transition from compressive to expansive creep strains has been characterized in experimental studies at a critical OCR (Yao and Fang 2020). The critical OCR can also be used to estimate whether the thermal creep will induce contraction or expansion. Within the e-TTS model, the critical OCR is controlled by the unloading module through the parameter $h$ that describes hysteretic response in unloading and reloading [Eq. (18)].

Fig. 4(a) illustrates e-TTS model predictions of the compressive and swelling behavior at selected values of $h$. When the Todi clay was unloaded from normally consolidated state ($p^{\prime }=396\mathrm{kPa}$) to 98 and 49 kPa, the corresponding OCR values were 4 and 8, respectively. Once the soil is unloaded, the locked strain increases, accompanied by increasing OCR, while the elastic strain decreases due to the reduction in effective stress [$h=0.1$, 0.3, and 0.9 simulated by e-TTS, Fig. 4(b)]. The difference between elastic and locked strain transit from negative to positive, resulting in a plastic compressive or swelling strain [Eq. (16)]. Therefore, the e-TTS model demonstrates that lower values of the OCR result in higher values of locked strain, indicating that the thermal creep will generate compressive strains. Conversely, higher values of the OCR result in lower values of the locked strain, and thermal creep predicts swelling behavior.

The e-TTS model reproduces a series of thermal isotaches for NC Todi clay to describe the isothermal and thermal creep. The magnitude of isothermal creep is controlled by the viscous parameter, $a$ [Eq. (16)], which describes the effects of strain rate on the preconsolidation pressure, and in turn influences the location of the normal consolidation line. Smaller values of $a$ increase the magnitude of the creep strains [Fig. 5(a)]. The magnitude of thermal creep is controlled by the viscous parameter, $a$, and thermal parameter $L_T$ [Eq. (16)], which describes the combining effects of strain rate and temperature on the pre-consolidation pressure. Smaller values of $a$ and/or larger values of $L_T$ increase the magnitude of the thermal creep strains [Fig. 5(a)]. Fig. 5(b) shows the effects of heating rate and magnitude of temperature change on thermal creep. For a NC clay specimen consolidated to 200 kPa [Point A; Fig. 5(a)], isothermal creep at constant effective produces 0.34% compressive strain over a period of 3,000 mins [A-B Fig. 5(b)]. A temperature increase, $\mathrm{\Delta }T=10°\mathrm{C}$, applied over the same time period increases the creep strain (A-C; 0.58%) at the same effective stress state, while $\mathrm{\Delta }T=20°\mathrm{C}$, produces 0.87% compressive strain (A-D) over the same time period. This result clearly shows the role of strain rate and temperature on isotache states of NC clays, where the thermal creep depends on the viscous parameters $\alpha$ and $L_T$ [$m_1$ in Eq. (16)]. In summary, increasing the magnitude of temperature rise induces a higher thermal creep. The greater the magnitude of temperature rise, the larger the contribution of thermal creep to the total deformation.

The e-TTS model considers the influences of porosity and effective stress on the initial inelastic strain (Appendix) and temperature-induced skeleton compaction through evolution of the granular temperature, $T_g$ [Eq. (131)]. Sittidumrong et al. (2019) carried out cyclic heating-cooling tests and creep tests on Bangkok sand under 1D conditions in modified oedometer cell. Their experimental program comprised pairs of tests (done in parallel) on saturated specimens of reconstituted sand prepared at the same water content. One specimen in each pair was subjected to conventional incremental loading under isothermal conditions ($T=28°\mathrm{C}$), while the second was subject a series of 100 thermal cycles ($T=28°\mathrm{C}–50°\mathrm{C}$) each with 1 hr duration at the imposed levels of effective stress. Heating rates were selected to ensure there was no development of pore water pressure in the specimens. The authors then report thermal creep by subtraction of the isothermal creep strains assuming identical behavior for each pair of test specimens. They present results at four selected formation densities (${\gamma }_d=17.7$, 16.5, 15.7, and $15.0\mathrm{kN}/{\mathrm{m}}^3$) and three levels of vertical effective stress (${\sigma }_v^{\prime }=2.01$, 3.21, and 4.01 MPa).

Fig. 6 shows that the computed and measured thermal creep strains for the first five temperature cycles of Bangkok sand at two different initial density states [$D_r=30\%$, 70% in Figs. 5(a and b), respectively], using a single set of e-TTS model input parameters (Table 1). The e-TTS model presents thermal swelling strain at first two elevated temperature, but the accumulative thermal strain after two temperature cycles tends to be compressive, which shows the same tendency as the measured accumulative strain. The e-TTS model reproduces the thermally induced compaction phenomenon for sand after a single temperature cycle and the increase of accumulative strain with cycles of heating and cooling.

Fig. 7 summarizes the predicted and measured accumulation of isothermal and thermal creep strains over 100 (1 h) temperature cycles for the Bangkok sand. The e-TTS model shows very good agreement between the computed and measured thermal creep strains. As expected, the accumulated thermal creep strains increase with the level of effective stress and reduce with increasing relative density.

The e-TTS model predicts that strain accumulation approaches a definite limit for each initial state. This behavior can be related to the underlying elastic and hysteretic/locked strains. Fig. 8 shows the convergence of these state parameters over 60 thermal cycles (${\epsilon }_v^e-{\epsilon }_v^h=0$) for the Bangkok sand at $D_r=30\%$ and ${\sigma }_v^{\prime }=3.21\mathrm{MPa}$.