Thermal and mechanical creep of clay in hypoplasticity

Clays and clay soils or shales have attracted a lot of interest in a variety of applications, including the development of geothermal resources [1], energy foundations [2], oil exploration [3], energy storage [4], and the storage of nuclear waste [5]. The continued operation of ground source heat pump installations can lead to considerable long-term settlements, which could negatively affect the adjacent or underlying foundations [1]. Therefore, thermal volume change has been widely experimentally investigated in clays [4, 7, 8, 9]. Especially in [4] the authors studied the thermal and mechanical consolidation of saturated marine clays through laboratory element tests, where excess pore pressures were generated by heating samples at constant water content and then allowed to dissipate. In constant stress creep experiments described in [4] it was documented that thermal creep strains typically increased linearly with log time at rates controlled by the prevailing temperature. 
On the other hand, the mechanical time dependency of the stress–strain behaviour of soft soils, especially highly plastic clay, is generally too significant to be ignored [10, 11]. The constitutive modelling of the time-dependent stress–strain behaviour of soils has been an active area of research for five decades and has attracted much attention from the international geotechnical community in recent years as denoted in [12]. In [13] a visco-hypoplastic (VHP) model for normally and overconsolidated clays has been proposed. Probably the most salient feature of hypoplasticity itself is that loading and unloading can be described with only one equation as with the strain and stress rate denoted as  and , respectively. The elastic stiffness tensor is represented by ;  is the degree of nonlinearity and  is the flow rule (direction of hypoplastic strain). The last part of the equation expresses the time-dependent strain rate (i.e. viscous) with the material parameters  as the viscosity index and  being the compression index.  denotes the overconsolidation ratio. As may be observed, the model is not restricted solely to time-dependent clay materials, because  does not represent a singularity for the constitutive equation as in other hypoplastic models. The model has been extended in [14] to account for the small-strain stiffness and the mechanical behaviour under cyclic loading. It follows the critical state theory and incorporates a loading surface for the definition of , see Fig. 1A). Time-dependent one-dimensional behaviour of clays is in most cases explained by the isotache framework, which assumes a unique relation between effective stress, strain, and strain rate in compression, shown as loci of constant strain rate in  space, see Fig. 1B). The creep deformation at constant effective stress (; Fig. 1C)) corresponds to a decrease in strain rate of the soil (path A to B in Fig. 1C)). Consolidation stress history, represented by swelling along the path AC, causes a marked reduction in compressive creep rates at low  (CD), while expansive/dilative creep strains occur at higher  (CEE). 
Due to the temperature dependency of the compressibility of clays the isotachs may be considered as temperature dependent as well. As shown in [6] the isotache loci at a given strain rate for normally consolidated states are functions of strain rate and temperature. Increases in temperature cause additional compressive thermal creep strains for NC and lightly OC states (BB and DD in Fig.1C)) and augment the swelling strains at higher  (EE). 
In the framework of hypoplasticity, the non-isothermal behaviour has not gained much attention [15]. This work is devoted to the extension of the VHP model formulation to present a unified model for both thermal and viscous strains in clays. The model predictions are assessed through comparisons with existing laboratory experiments from [7] on two clays with measurements of both isothermal/mechanical creep and thermally induced creep strains, providing a thorough calibration scheme as well.

Clays and clay soils or shales have attracted a lot of interest in a variety of applications, including the development of geothermal resources [1], energy foundations [2], oil exploration [3], energy storage [4], and the storage of nuclear waste [5].The con tinued operation of ground source heat pump installations can lead to considerable long-term settlements, which could negatively affect the adjacent or underlying foundations [1].Therefore, thermal volume change has been widely experimentally investigated in clays [4,7,8,9].Especially in [4] the authors studied the thermal and mechanical consolidation of saturated marine clays through laboratory element tests, where excess pore pressures were generated by heating samples at constant water content and then allowed to dissipate.In constant stress creep experiments described in [4] it was documented that thermal creep strains typically increased linearly with log time at rates controlled by the prevailing temperature.
On the other hand, the mechanical time dependency of the stress-strain behaviour of soft soils, especially highly plastic clay, is generally too significant to be ignored [10,11].The constitutive modelling of the time-dependent stress-strain behaviour of soils has been an active area of research for five decades and has attracted much attention from the international geotechnical community in recent years as denoted in [12].In [13]    ) , with the strain and stress rate denoted as ̇ and ̇, respectively.The elastic stiffness tensor is represented by E;  is the degree of nonlinearity and  is the flow rule (direction of hypoplastic strain).The last part of the equation expresses the time-dependent strain rate (i.e.viscous) with the material parameters   as the viscosity index and  being the compression index. denotes the overconsolidation ratio.As may be observed, the model is not restricted solely to time-dependent clay materials, because   = 0 does not represent a singularity for the constitutive equation as in other hypoplastic models.The model has been extended in [14] to account for the small-strain stiffness and the mechanical behaviour under cyclic loading.It follows the critical state theory and incorporates a loading surface for the definition of , see Fig. 1A).Time-dependent one-dimensional behaviour of clays is in most cases explained by the isotache framework, which assumes a unique relation between effective stress, strain, and strain rate in compression, shown as loci of constant strain rate in  − log(′) space, see Fig. 1B).The creep deformation at constant effective stress ( 0 ′; Fig. 1C)) corresponds to a decrease in strain rate of the soil (path A to B′ in Fig. 1C)).Consolidation stress history, represented by swelling along the path A−C′, causes a marked reduction in compressive creep rates at low  (C′ −D′), while expansive/dilative creep strains occur at higher  (C′ −E′ −E′′).
Due to the temperature dependency of the compressibility of clays the isotachs may be considered as temperature dependent as well.As shown in [6] the isotache loci at a given strain rate for normally consolidated states are functions of strain rate and temperature.Increases in temperature cause additional compressive thermal creep strains for NC and lightly OC states (B′ −B′′ and D′ −D′′ in Fig. 1C)) and augment the swelling strains at higher  (E′ −E′′).
a visco-hypoplastic (VHP) model for normally and overconsolidated clays has been proposed.Probably the most salient feature of hypoplasticity itself is that loading and unloading can be described with only one equation as ̇= E: (̇− ̇ℎ  − ̇  ) = E: (̇− ‖̇‖ −    (

Figure 1 :
Figure 1: A) Critical state and loading surface; B) Oedometer simulations with different strain rates (isotachs); C) Unification of thermal and mechanical creep in 1D compression tests through isotache framework from [6].