the size of molecules

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http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0001482
http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0001482
http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0001482
http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0001482
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mailto:[email protected]
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crushing, and creep (Barden 1965; Mesri and Godlewski 1977; Mesri and Castro 1987). Past history loses relevance regardless of the natural or remolded origin of specimens (Terzaghi and Peck 1948; Chilingar and Knight 1960; Skempton 1969; Burland 1990; Hong et al. 2012).

Soil Compressibility Models

Classical e- logðσ 0Þ compressibility models and new functions are reviewed in this section. In all cases, the models are generalized to satisfy asymptotic void ratios eL as σ 0 → 0, and eH as σ 0 → ∞.

Semi-Logarithmic e- log σ 0 Models

The classical linear equation in logðσ 0Þ is the most common model used in geotechnical engineering (Terzaghi and Peck 1948; Schofield and Wroth 1968)

e ¼ eref − Cc log � σ 0

σ 0ref

� ð1Þ

where the void ratio eref corresponds to effective stress σ 0 ¼ σ 0ref . The normalization stress σ 0ref is selected a priori for normalization, such as σ 0ref ¼ 1 kPa, and it is not a model parameter. Therefore, this function has two model parameters: eref and Cc; it fits normally consolidated soil data at intermediate stress levels, but predicts e → ∞ as σ 0z → 0 and e < 0 as σ 0z → ∞.

The lower soil compressibility at high stress requires higher or- der terms, such as the cubic polynomial suggested by Burland (1990) for remolded soil data

e ¼ eref − α · log � σ 0

σ 0ref

�

þ β · � log

� σ 0

σ 0ref

�� 3

ðstress range 10 kPa < σ 0 < 10 MPaÞ

ð2Þ in terms of three model parameters eref , α, and β. In addition, the asymptotic void ratio eL at low stress can be imposed as a plateau.

Alternatively, the classical semi-logarithmic Terzaghi model can be modified to satisfy asymptotic conditions: e → eL at low stress σ 0 → 0, and e → eH at high stress σ 0 → ∞

e ¼ ec − Cc log �

1 kPa σ 0 þ σ 0L

þ 1 kPa σ 0H

�−1 ð3Þ

where parameters ec and Cc determine the central trend, and void ratio asymptotes eL and eH define stresses σ 0L and σ

0 H

σ 0H ¼ 10ðec−eHÞ=Cc · kPa when σ 0 → ∞ ð4Þ

σ 0L ¼ σ 0H

10ðeL−eHÞ=Cc − 1 when σ 0 → 0 ð5Þ

The generalized Terzaghi model in Eq. (3) involves four model parameters of clear physical meaning.

Models in Terms of e − σ 0β Power function: From gases to soils. Loosely packed small grains bear resemblance to the notion of a gas. Boyle-Mariotte’s law ignores the size of molecules, and concludes that pressure and volume are inversely related

PV ¼ constant → V ¼ a=P ð6Þ where α = constant. van der Waals corrected this expression to take into consideration the size of molecules and rewrote Boyle’s equation in terms of the contractible volume V 0, i.e., the total volume Vt minus the volume excluded Vex by the molecules, V 0 ¼ Vt − Vex. He also considered intermolecular interactions and the additional stress due to uncompensated attraction at the boundaries.

Following a parallel analysis, and taking into consideration electrical attraction and repulsion in fine-grained soils (σA−σR), a plausible equation for soil compressibility becomes:

ðσ 0 þ σA − σRÞðVt − VsÞ ¼ α ð7Þ If the volume of solids as Vs is assumed constant, this equation can be written in terms of void ratio e ¼ ðVt − VsÞ=Vs

e ¼ α 0

σ 0 þ σA − σR ð8Þ