280 Arctangent eL

e1 kPa ¼ 3.4 · Cc þ 0.48 remolded soils ð54 cases; SD ¼ 0.19;R2 ¼ 0.96Þ ð16Þ

A limited set of nine cases was identified to compare natural soils with the same soil after remolding (from Mitchell 1956; Mesri et al. 1975; Burland 1990; Hong et al. 2006; Wesley 2009). Structured natural soils pack at higher void ratio enatural1 kPa ¼ 1.6 · eremolded1 kPa and compress more Cnaturalc ¼ 1.7 · Cremoldedc than the remolded counterparts.

The link between void ratio at low stress and compressibility in logðσ 0Þ models emerges in other models too. For example, a mea- sure of compressibility in the hyperbolic model σ 0c=ðeL − eHÞ is linked to the asymptotic low-stress void ratio eL as [Fig. 2(b)]

eL ¼ 15.8 � σ 0c

1 kPa

eL − eH �−0.4

natural clays ð23 cases;SD ¼ 0.4;R2 ¼ 0.84Þ ð17Þ

Parameter Invertibility

Parameters were selected by least-square fitting the models to the data, min½Σðemi − epi Þ2�. The error surface about the optimal param- eter set is explored by varying one parameter at a time to assess the invertibility of each parameter. As anticipated, limited data at very low or very high stress results in poor convergence for eL or eH, respectively.

Model parameters can be constrained with a priori data, for ex- ample, mineralogy to bound eL, and geological data to restrict eH. Furthermore, the β-exponent is bounded (Table 1), and the low- stress void ratio eL and compressibility are correlated (Fig. 2). These constraints and correlations help to identify a self-consistent set of fitting parameters; furthermore, they suggest an effective model complexity lower than the four unknowns involved in these models.

Void ratio in sedimentary basins: Compressibility models [Eqs. (1)–(3), (9)–(12)] can be integrated to compute void ratio trends versus depth. The nonlinear decrease in void ratio with depth observed in sedimentary basins (e.g., data reported in Aplin et al. 1995) can be properly matched with all models reviewed here.

Tangent stiffness: Small-strain versus large-strain. Numerical solutions that use a tangent formulation involve the tangent constrained modulus M

M ¼ ∂σ 0

∂ε ¼ −∂σ 0 ∂e ð1þ eÞ ð18Þ

The void ratio versus stress models discussed earlier can be ap- plied to isotropic loading conditions by replacing σ 0 → p 0 and e → v ¼ 1þ e. Then, the tangent bulk stiffness K becomes

K ¼ dp 0

dεv ¼ −dp

0

dv v ð19Þ

Equations obtained for either M or K for all compressibility models [Eqs. (1)–(3) and (9)–(12)] can be found in the Supplemen- tal Data.

The tangent constrained modulus M or bulk modulus K com- puted using Eqs. (18) and (19) (Supplemental Data) are fundamen- tally different from the small strain stiffness measured at the same e − σ 0 state. The tangent stiffness is a mathematical concept that reveals the instantaneous rate of fabric change during a large strain test. By contrast, a small-strain perturbation (e.g., shear wave propagation) is a constant fabric measurement of stiffness and is determined by contact deformation. Therefore, the magnitude of tangent stiffness and the trends reported in Fig. 1 should not be associated to small-stress stiffness values.

Settlement computation: Estimation of yield stress: Standard settlement analyses assume recompression (er1 kPa, Cr) and normal compression (en1 kPa, Cc) segments before and after the yield stress σ 0y. Several ad hoc methods have been proposed to determine the yield stress or preconsolidation pressure (Casagrande 1936; Janbu 1969; Pacheco 1970; Sallfors 1975; Butterfield 1979; Becker et al. 1987; Oikawa 1987; Jose et al. 1989; Sridharan et al. 1991; Onitsuka et al. 1995; Grozic et al. 2003; Clementino 2005; Boone 2010; Ku and Mayne 2013). Single function models, such as those compiled and augmented here, can capture the complete compres- sion response and streamline computations.

Table 1. Fitting Parameters (Refer to Data Presented in Fig. 1)

Model Parameters Montmorillonite

0.001 N [Fig. 1(a)] Bothkennar [Fig. 1(b)]

Modified Terzaghi

eL 36.5 2.05 eH 0.20 0.52 ec 52 5.2 Cc 21.5 1.6

Power eL 35.1 2.03 eH 0.40 0.63 β 1.0 2.0

σ 0c (kPa) 48 700 Exponential eL 38.5 2.00

eH 1.40 0.72 β 0.55 1.0

σ 0c (kPa) 70 400 Hyperbolic eL 36.0 1.98

eH 0.40 0.65 β 0.9 1.32

σ 0c (kPa) 43 280 Arctangent eL 36.5 1.95

eH 0.30 0.72 β 0.7 1.3

σ 0c (kPa) 42 250