the characteristic stress
This inverse relationship between void ratio e and effective stress σ 0 can be generalized as a four-parameter inverse power func- tion that accommodates the two void ratio asymptotes eL and eH
e ¼ eH þ ðeL − eHÞ � σ 0 þ σ 0c
σ 0c
�−β ð9Þ
When the applied effective stress equals the characteristic effective stress σ 0 ¼ σc and β ¼ 1, the predicted void ratio is the average of the asymptotes e ¼ ðeL þ eHÞ=2. Higher β-exponents cause higher early compressibility at lower stresses. Power- type equations have been suggested in the past (Hansen 1969; Butterfield 1979; Juárez-Badillo 1981; Houlsby and Wroth 1991; Pestana and Whittle 1995).
Exponential: The main characteristic of exponential functions y ¼ expðxÞ is that the rate of change dy=dx is defined by the cur- rent state. Sigmoidal and Gompertz functions are special examples (Gompertz 1825; Gregory et al. 2006). The four-parameter Gompertz function can be expressed in terms of stress and void ratio, and adapted to satisfy eL and eH as follows:
e ¼ eH þ ðeL − eHÞ · exp− � σ 0 σ 0c
� β
ð10Þ
When the exponent β ¼ 1, the simpler three-paremeter expo- nential expression is obtained (Cargill 1984); it predicts that the soil will experience 63% of the volume change eL − eH when the applied effective stress equals the characteristic effective stress σ 0 ¼ σ 0c.
Hyperbolic: The hyperbolic model is extensively used in geo- mechanics to capture the prepeak deviatoric stress versus strain data (Kondner 1963; Duncan and Chang 1970). This model has two parameters: one defines the initial rate of change dy=dxjo, and the other provides the asymptotic value of y as x → ∞. The model can be adapted to capture compressibility data in terms of σ 0 − e. The generalized four-parameter hyperbolic model is
e ¼ eL − ðeL − eHÞ 1 1þ �σ 0cσ 0�β
ð11Þ
The simpler hyperbolic model (β ¼ 1) predicts that the void ra- tio will reach the intermediate void ratio e ¼ ðeL þ eHÞ=2when the applied effective stress equals the characteristic stress σ 0 ¼ σ 0c. Structuration and yield stress can be captured with higher values of the β-exponent.
© ASCE 06016003-2 J. Geotech. Geoenviron. Eng.
J. Geotech. Geoenviron. Eng., 06016003
D ow
nl oa
de d
fr om
a sc
el ib
ra ry
.o rg
b y
Po rt
la nd
S ta
te U
ni ve
rs ity
o n
05 /0
1/ 16
. C op
yr ig
ht A
SC E
. F or
p er
so na
l u se
o nl
y; a
ll ri
gh ts
r es
er ve
d.
Arctangent: Other functions that provide S-shaped trends can be adapted to satisfy asymptotic conditions relevant to soil compres- sion data. For example, the arctangent function can be generalized to include the power of the stress σ 0β in order to fit more brittle soil responses
e ¼ eL þ 2
π ðeL − eHÞ arctan
� − � σ 0
σ 0c
� β �
ð12Þ
in terms of four model parameters eL, eH, the characteristic stress σ 0c, and the β-exponent. For β ¼ 1, the void ratio reaches e ¼ ðeL þ eHÞ=2 when the applied effective stress equals the character- istic stress σ 0 ¼ σ 0c.
