A thermodynamic constitutive model for undrained monotonic and cyclic shear behavior of saturated soils

Abstract

Based on a non-equilibrium thermodynamic approach, this paper has established a constitutive model for saturated soils different from the classical elasto-plastic model and other thermodynamics-based models. It does not require concepts such as yield surface and flow rule. Instead, through a quantitative description of the energy in the reversible and irreversible processes in materials, the constitutive relations have been obtained theoretically. Using the elastic potential energy function, the stress is expressed as a function of elastic strain. By introducing the granular entropy and locked energy concepts, the non-elastic strain evolution is determined according to the non-equilibrium thermodynamics. The model can describe many basic mechanical properties of saturated soils under monotonic and cyclic shear loadings, such as the stress state boundary, the critical state and especially the cyclic mechanical behavior. Based on this model, the paper simulates the cyclic undrained triaxial shear tests for saturated clays and analyzes the impact of factors that include cyclic stress ratio, OCR value and cyclic stress frequency. The simulation results are compared with experimental results in order to validate the model.

Appendices

Appendix 1: Main parameters used in the proposed model

Parameter type	Elastic potential energy density function parameters			Migration coefficient		Hysteric parameters
	B 0/Pa	B 1/(m3/kg)	c/[-]	m 1/[−]	m 2/min−(1−2a)/a	h/[−]
	ξ/[−]	ζ/[−]		m 3/[−]	m 4/(kg/m3/min)	w/[−]
Hangzhou Clay	31.2	0.0123	0.0203	15	920.6	0.036
	0.276	0.0534		3 × 10−3	6 × 103	0.96
Kaolin Clay	2.35 × 106	0.00438	0.0282	4.44	1088	0.0205
	0.194	0.0445		–	1.4 × 104	0.96
MC Clay	6.59 × 107	0.00438	0.00754	51	691	0.01
	0.0426	0.0082		–	3.2 × 103	0.72

1. The two upper and lower values correspond to the upper and lower parameters for the corresponding clay. c is the cohesion-related parameter. B ₁, m ₂ and m ₃ determine the density-dependent behavior, the undrained critical strength and the non-elastic volumetric deformation of soil, respectively. m ₁ and m ₄ are related to the strain hardening/softening under undrained shear. In this paper, the parameter a is taken as 0.455.

Appendices 2: Model parameter calibration procedure

In this paper, the main model parameters include the following categories. The relevant parameters of the elastic potential energy density function (Eq. (17)) are B ₀, B ₁, c, ξ and ζ; the migration coefficients are m ₁, m ₂, m ₃ and m ₄ (see Eq. (24)); the hysteresis-related parameters are h and w; the rate-related parameter is a. The calibration processes of these parameters are briefly described below.

• 1. First, the critical state line in e–p ^′–q space (Fig. 7) is determined using two groups of undrained triaxial tests with constant strain rates, which can be used to determine the parameters B ₁. For a isotropically consolidated state with a confining pressure level of p′ ₀ as,

  $$ p_{0}^{{\prime }} = 0.6B_0\times\exp(B_1\times p_{d0})(\varepsilon_{v0}^{e} + c)^{0.5} \varepsilon_{v0}^{e2} + 0.8B_0\times\exp(B_1\times p_{d0})(\varepsilon_{v0}^{e} + c)^{1.5} \varepsilon_{v0}^{e} $$

  (31)

  where ρ _d0 and $\varepsilon_{v0}^{e}$ are the corresponding dry density and elastic volumetric strain, respectively. $\varepsilon_{v0}^{e}$ can be estimated by rebound test (i.e., the unloading of consolidation pressure). Then, using Eqs. (29 and 31), parameters B ₀ and c can be obtained.

From Eq. (25b), the ratio of the absolute value of undrained critical shear strength of triaxial compression to that of triaxial extension under the same axial strain rate is

$$ \left| {\frac{{q_{c} }}{{q_{ex} }}} \right| = \frac{{\xi + 6^{{{{ - 5} \mathord{\left/ {\vphantom {{ - 5} 6}} \right. \kern-0pt} 6}}} \varsigma }}{{\xi - 6^{{{{ - 5} \mathord{\left/ {\vphantom {{ - 5} 6}} \right. \kern-0pt} 6}}} \varsigma }} > 1 $$

(32)

According to the undrained triaxial compression test with a constant strain rate under the confining pressure p′ ₀, the critical effective stresses q _cr and p′ _cr can be obtained. Using triaxial tensile test with the same strain rate, the critical strength ratio of the compression path and extension path can also be obtained. Then, using Eqs. (25, 30 and 32), parameters ξ, ζ and the deviatoric elastic strain in critical state ($\left| {(\varepsilon_{d}^{e} )_{cr} } \right|$) can be calculated.

• 2. Parameter h can be determined according to h = 0.6$\left| {(\varepsilon_{d}^{e} )_{cr} } \right|$. Using Eq. (30c), the value of parameter m ₂ can be determined (axial strain rate d _t ε ₁ is given by tests). Keeping the calibrated parameters above unchanged, adjusting the values of m ₁ and m ₄ can adjust the stress–strain relation curve before reaching critical state. Accordingly, m ₁ and m ₄ can be calibrated. Equation (24) suggests that parameter m ₃ is related to isotropic consolidation, but have no direct relation to undrained shear processes (because d _t ε _v  = 0). The value of parameter m ₃ can thus be calibrated using the isotropic consolidation curve. When calibrating the consolidation curve for over-consolidated soil, the initial locked strain value should be adjusted to adjust the position of the pre-consolidation pressure.

• 3. Parameter a, related to rate dependency, can be calibrated using the critical strength values of triaxial undrained compression tests under different shear rates and the same confining pressure. This paper recommends that, for the most of cohesive soils, the a value be set to be about 0.45.