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The TTS model (Zhang and Cheng 2017) was developed to describe the thermo-hydro-mechanical coupling behavior of saturated soils based on a theory of nonequilibrium thermodynamics referred to as Granular Solid Hydrodynamics [(GSH) (Jiang and Liu 2003, 2007, 2009)]. The TTS model uses a hyperelastic relation to describe elasticity of granular media. Time-dependent inelastic behavior is attributable to the energy dissipation determined by Onsager’s reciprocal principle. A new intrinsic variable, granular entropy, replaces the plastic strain to describe the various irreversible granular-level processes of soils and to provide a more fundamental physical basis for characterizing the complex mechanical behavior of soils. The TTS model can describe major features of soil behavior including critical states, irreversible strains in undrained cyclic loading, and long-term thermal-mechanical response (Zhang and Cheng 2016, 2017). The analysis in this paper is based on an extension of the TTS formulation to unify the description of thermal creep, strain accumulation, and cyclic stabilization of clays and sands.Thermoelastic BehaviorIn the e-TTS model, the stress-strain relationship for thermoelastic coupling is obtained from a function for the elastic potential energy density, ωe, to guarantee that the model is thermodynamically acceptable (1) where εije = elastic strain tensor; and πij = elastic stress tensor.Following Zheng and Cheng (2017), thermoelastic behavior of granular media (with Hertzian contact) can be represented by (2) ωe=B(εve)1.5[25(εve)2+ξ(εse)2]+∫3KeβTΔTdεvewhere B=B0exp(B1ρd); B0, B1, and ξ = material parameters; ΔT=T−T0 = temperature change; T0 = reference temperature; 3KeβTΔT = additional isotropic thermal loading to restrain the thermoelastic expansion induced by the temperature change; Ke = secant elastic bulk modulus of the solid skeleton, the formulation of which is shown in the Appendix; βT = linear thermal expansion coefficient for the soil skeleton; εve and εse=eijeeije = volumetric and 2nd invariant of deviator elastic strain tensor, respectively; and eij = components of the deviatoric strain tensor.Zhang and Cheng (2017) established a generalized effective stress composition for saturated soils using the granular solid hydrodynamics approach (3) σij′=πij+σijbw+∑α=s,f,bσijvα+σijgwhere the first term on the right side of the effective stress equation is elastic stress, πij, which is the most important components of the effective stress; the partial stress σijbw = bound effective stress (due to disjoining pressures between bound water and clay particles); σijvα = viscous stress associated with dissipative flows of material viscosity; and σijg = 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 (4) σij′≈πij=1.4B(εve)2.5δij+1.5Bξ(εve)0.5(εse)2δij+3KeβT(T−T0)δij+2Bξ(εve)1.5eijeThermoviscoplastic BehaviorThe total strain rate can be divided into the elastic strain rate and the inelastic strain rate (5) In classical plasticity theory, plastic strain is adopted to describe the inelastic strain for granular materials (where dtεijD 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 flows (6) R=Yijπij+IgTg+σijvsdtεijwhere πij = dissipative forces corresponding to the effective stresses; Tg = granular temperature; dtεij = total strain rates; Yij = corresponding flows comprising the plastic strain rates; Ig = the rate of conversion of granular (local) entropy; and σijvs = 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: (7) [YijIg]=[λijkl00γ][πijTg]where λijkl and γ = migration coefficients.It can be theoretically shown that the nonelastic strain rate (dtεijD) is the same as Yij (Zhang and Cheng 2017). Zhang and Cheng (2017) propose the following expression for the dissipative flow Yij (or plastic strain rate, dtεijD) to describe the transient elasticity of granular material (8) Yij=dtεijD=λsTga(eije−eijh)+λvTga(εkke−εkkh)δijwhere εije, εijh = 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).New Formulation for Granular EntropyThe 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 sg (=bρdTg where b = constant; and ρd = dry mass density) and its conjugate variable, Tg granular temperature (Haff 1983; Jiang and Liu 2009). The e-TTS model assumes that granular entropy increases according to the following equation (9) where Rg/Tg = granular entropy production rate; Tg can be understood as the dissipative force of the granular fluctuation; Ig = corresponding dissipative flow; and IgTg = 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 dtεij and dtT are denoted as σijg and M, respectively. Hence, the granular entropy can be expressed as follows: (10) with dissipation flow σijg(11) σijg=ηgTgdteij+ζgTgdtεkkδijIn 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) (12a) (12b) M=[ψgπkkαbfϕbw3ρd(1−ϕ)+θtTg]dtTfor  claywhere θt = migration coefficient for thermal-compaction of soil skeleton; ψg controls the direction of temperature change for clay: ψg>0 for heating and ψg=0 for cooling, πkk = mean effective stress; ϕbw(=Vbw/V) and ϕ = bound water porosity and total porosity, respectively; and αbf controls the conversion rate of bound water to free water. Zymnis et al. (2019) describe experimental methods to measure αbf.Finally, the evolution of granular temperature can be derived as follows: (13) dtTg=ηg(dteij)2+ζg(dtεkk)2−γTgbρd+[ψgπkkαbfϕbw3ρd(1−ϕ)+θt](dtT)2where ηg, ζg, γ, ψg, and θt = 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, θt, 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 dtεkk 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 (dtT), 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 m1,0 to m6) 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 (B0, B1, ξ, a, h, m1, m2, m3, m4) can be calibrated by isotropic/1D compression and creep tests, while the thermal component parameters (βT, LT, αbf, m5, m6) are obtained from a thermal creep test. The initial elastic strains can be obtained from the initial consolidation stress state (p′, q, e). We assume the initial locked strain is zero and initial granular temperature, Tg=10−4.Table 1. e-TTS model parametersTable 1. e-TTS model parametersParameterSymbolPhysical meaningTodi clayBangkok sandB0kPaInterception of NCL10×1031.5×103B1m3/kgSlope of NCL1×10−37.7×10−3ξ—Related to K00.10.1a—Controls rate effects0.450.5H—Hysteretic strains (slope of unload curve)0.90.9m2—Controls elastic strain evolution and location of reload curve8×1046×105m2=ηg/bηg: controls contribution of dteij on σijgb: controls contribution of Tg on sgm3—Controls contribution of volumet ric and deviatoric strains on granular temperature production—6×105m3=ζg/bζg: controls contribution of dtεkk on σijgm4kgs−1°C−1Rate of granular temperature production50010m4=γ/bγ: dependency of energy dissipation rateβT°C−1Volumetric thermal expansion of soils3.5×10−58.1×10−6m6s−2°C−1Controls volume change of soil skeleton by temperature changes—10−5m6=θt/bθt: effect of thermal compaction on Tgαbf°C−1Affects conversion of bound water to free water during heating0.0237—m5s3m−2°C−1Controls amount of thermal volumetric strains by conversion of bound water2×10−5—m5=<ψg/b>ψg: rate of bound water changes on Tg0 (cooling)m1,0—Controls elastic strain evolution0.120.04m1=λv/λsλv: controls contribution of εve on Yijλs: controls contribution of εse on YijLT°C−1Controls temperature dependence of NCL0.010.01All model parameters listed in the Table 1 remain constant under temperature changes or gradients. It should be noted that m5 is constant for a heating process and m5=0 for cooling. The bound water content or bound water porosity ϕbw=Vbw/V changes with the temperature.



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