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Dimensionless characterization of the non-linear soil consolidation problem of Davis and Raymond. Extended models and universal curves

G. García-Ros
G. García-Ros
, 
I. Alhama
I. Alhama
 e 
F. Alhama
F. Alhama
   | 21 giu 2019
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Pubblicato online: 21 giu 2019
Pagine: 61 - 78
Ricevuto: 20 feb 2019
Accettato: 13 apr 2019
DOI: https://doi.org/10.2478/AMNS.2019.1.00008
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© 2019 G. García-Ros et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Average degree of settlement evolution for the extended Davis and Raymond model with both non-constant cv and 1+e and variable dz.
Average degree of settlement evolution for the extended Davis and Raymond model with both non-constant cv and 1+e and variable dz.

Fig. 2

Ratio τo,s/τo,σ′ as a function of σ′f/σ′o for the extended model of Davis and Raymond with both non-constant cv and 1+e and variable dz.
Ratio τo,s/τo,σ′ as a function of σ′f/σ′o for the extended model of Davis and Raymond with both non-constant cv and 1+e and variable dz.

Fig. 3

Evolution of average degree of pressure dissipation for the extended Davis and Raymond model with both non-constant cv and 1+e and variable dz.
Evolution of average degree of pressure dissipation for the extended Davis and Raymond model with both non-constant cv and 1+e and variable dz.

Verification of the dimensionless groups for the extended Davis and Raymond model with both non-constant cv and 1+e and variable dz.

Case Ko (m/yr) eo Ic σ′o (N/m2) Ho (m) σ′f (N/m2) cvo (m2/yr) τo,σ′ (yr) τo,s (yr) π1σ′ π2σ′ π1s
01 0.02 1.5 0.45 30000 1 60000 0.783 0.4941 0.4328 0.967 2.0 0.847
02 0.04 1.5 0.45 15000 1 30000 0.783 0.4941 0.4328 0.967 2.0 0.847
03 0.02 0.25 0.1125 30000 1 60000 1.566 0.4941 0.4328 0.967 2.0 0.847
04 0.04 1.5 0.45 60000 2 120000 3.133 0.4941 0.4328 0.967 2.0 0.847
05 0.03 1 0.3 25000 1.5 50000 1.175 0.926 0.811 0.967 2.0 0.847
06 0.02 1.5 0.45 30000 1 120000 0.783 0.5501 0.4328 1.077 4.0 0.847
07 0.02 1.5 0.45 30000 2 120000 0.783 2.2004 1.7312 1.077 4.0 0.847
08 0.02 1.5 0.45 30000 1 240000 0.783 0.6001 0.4328 1.175 8.0 0.847
09 0.02 1.5 0.45 30000 1 480000 0.783 0.6444 0.4328 262 16.0 0.847

Governing equations for the different variants of the Davis and Raymond model

Pressure Settlement
Davis and Raymond cv∂2σ′∂z2−1σ′∂σ′∂z2=∂σ′∂t $\begin{array}{} \displaystyle {{\rm{c}}_{\rm{v}}}\left\{ {\frac{{{\partial ^2}{\rm{\sigma '}}}}{{\partial {{\rm{z}}^2}}} - \frac{1}{{{\rm{\sigma '}}}}{{\left( {\frac{{\partial {\rm{\sigma '}}}}{{\partial {\rm{z}}}}} \right)}^2}} \right\} = {\rm{\;}}\frac{{\partial {\rm{\sigma '}}}}{{\partial {\rm{t}}}} \end{array}$ cv∂2ζ∂z2=∂ζ∂t $\begin{array}{} \displaystyle {{\rm{c}}_{\rm{v}}}\frac{{{\partial ^2\zeta}}}{{\partial {{\rm{z}}^2}}} = {\rm{\;}}\frac{{\partial \zeta}}{{\rm{\partial t}}} \end{array}$
Davis and Raymond 1+e ≠ constant

the assumption of variable dz adds dz=dzo1+e1+eo $\begin{array}{} \displaystyle d{\rm{z}} = d{{\rm{z}}_{\rm{o}}}\frac{{1 + {\rm{e}}}}{{1 + {{\rm{e}}_{\rm{o}}}}} \end{array}$

 cv constant		 cv1+e∂2σ′∂z2+Icσ′ln10−1+eσ′∂σ′∂z2=∂σ′∂t $\begin{array}{} \displaystyle {{\rm{c}}_{\rm{v}}}\left\{ {\left( {1 + {\rm{e}}} \right)\frac{{{\partial ^2}{\rm{\sigma '}}}}{{\partial {{\rm{z}}^2}}} + \left( {\frac{{{{\rm{I}}_{\rm{c}}}}}{{{\rm{\sigma '}}\ln \left( {10} \right)}} - \frac{{1 + {\rm{e}}}}{{{\rm{\sigma '}}}}} \right){{\left( {\frac{{\partial {\rm{\sigma '}}}}{{\partial {\rm{z}}}}} \right)}^2}} \right\} = {\rm{\;}}\frac{{\partial {\rm{\sigma '}}}}{{\partial {\rm{t}}}} \end{array}$ cv1+e∂2ζ∂z2−cv∂ζ∂z2=∂ζ∂t $\begin{array}{} \displaystyle {{\rm{c}}_{\rm{v}}}\left( {1 + {\rm{e}}} \right)\frac{{{\partial ^2 \zeta}}}{{\partial {{\rm{z}}^2}}} - {{\rm{c}}_{\rm{v}}}{\left( {\frac{{\partial\zeta}}{{\partial\rm{z}}}} \right)^2} = {\rm{\;}}\frac{{\partial \zeta}}{{\partial\rm{t}}} \end{array}$
Davis and Raymond 1+e ≠ constant

the assumption of variable dz adds dz=dzo1+e1+eo $\begin{array}{} \displaystyle d{\rm{z}} = d{{\rm{z}}_{\rm{o}}}\frac{{1 + {\rm{e}}}}{{1 + {{\rm{e}}_{\rm{o}}}}} \end{array}$

 cv constant		 cvo1+e21+eo∂2σ′∂z2−1σ′∂σ′∂z2=∂σ′∂t $\begin{array}{} \displaystyle {{\rm{c}}_{{\rm{vo}}}}\frac{{{{\left( {1 + {\rm{e}}} \right)}^2}}}{{1 + {{\rm{e}}_{\rm{o}}}}}\left\{ {\frac{{{\partial ^2}{\rm{\sigma '}}}}{{\partial {{\rm{z}}^2}}} - \frac{1}{{{\rm{\sigma '}}}}{{\left( {\frac{{\partial {\rm{\sigma '}}}}{{\partial {\rm{z}}}}} \right)}^2}} \right\} = {\rm{\;}}\frac{{\partial {\rm{\sigma '}}}}{{\partial {\rm{t}}}} \end{array}$ cvo1+e21+eo∂2ζ∂z2=∂ζ∂t $\begin{array}{} \displaystyle {{\rm{c}}_{{\rm{vo}}}}\frac{{{{\left( {1 + {\rm{e}}} \right)}^2}}}{{1 + {{\rm{e}}_{\rm{o}}}}}\frac{{{\partial ^2 \zeta}}}{{\partial {{\rm{z}}^2}}} = {\rm{\;}}\frac{{\partial \zeta}}{{\partial\rm{t}}} \end{array}$

Dimensionless groups that characterize the solutions for the different variants of the Davis and Raymond model

Pressure Settlement
Davis and Raymond π1=τo,σ′cvHo2π2=σ′fσ′o $\begin{array}{} \displaystyle {{\rm{\pi }}_1} = \frac{{\tau_{{\rm{o}},{\rm{\sigma '}}}{{\rm{c}}_{\rm{v}}}}}{{{\rm{H}}_{\rm{o}}^2}} \;\;\;\; {{\rm{\pi }}_2} = \frac{{{{{\rm{\sigma '}}}_{\rm{f}}}{\rm{\;}}}}{{{{{\rm{\sigma '}}}_{\rm{o}}}}} \end{array}$ π1=τo,scvHo2 $\begin{array}{} \displaystyle {{\rm{\pi }}_1} = \frac{{\tau_{{\rm{o}},{\rm{s}}}{{\rm{c}}_{\rm{v}}}}}{{{\rm{H}}_{\rm{o}}^2}} \end{array}$
Davis and Raymond 1+e ≠ constant cv constant dz constant and dz ≠ constant π1=τo,σ′cv1+eoHo2π2=σ′fσ′oπ3=Ic1+eo $\begin{array}{} \displaystyle {{\rm{\pi }}_1} = \frac{{\tau_{{\rm{o}},{\rm{\sigma '}}}{{\rm{c}}_{\rm{v}}}\left( {1 + {{\rm{e}}_{\rm{o}}}} \right)}}{{{\rm{H}}_{\rm{o}}^2}}\hspace{15pt} {{\rm{\pi }}_2} = \frac{{{{{\rm{\sigma '}}}_{\rm{f}}}}}{{{{{\rm{\sigma '}}}_{\rm{o}}}}}{\rm{\;}}{{\rm{\pi }}_3} = \frac{{{{\rm{I}}_{\rm{c}}}}}{{1 + {{\rm{e}}_{\rm{o}}}}} \end{array}$ π1=τo,scv1+eoHo2π2=HfHo $\begin{array}{} \displaystyle {{\rm{\pi }}_1} = \frac{{\tau_{{\rm{o}},{\rm{s}}}{{\rm{c}}_{\rm{v}}}\left( {1 + {{\rm{e}}_{\rm{o}}}} \right)}}{{{\rm{H}}_{\rm{o}}^2}}{{\rm{\pi }}_2} = \frac{{{{\rm{H}}_{\rm{f}}}}}{{{{\rm{H}}_{\rm{o}}}}} \end{array}$
Davis and Raymond 1+e ≠ constant cv ≠ constant dz constant π1=τo,σ′cvo1+eoHo2π2=σ′fσ′oπ3=Ic1+eo $\begin{array}{} \displaystyle {{\rm{\pi }}_1} = \frac{{\tau_{{\rm{o}},{\rm{\sigma '}}}{{\rm{c}}_{{\rm{vo}}}}\left( {1 + {{\rm{e}}_{\rm{o}}}} \right)}}{{{\rm{H}}_{\rm{o}}^2}}\hspace{15pt} {{\rm{\pi }}_2} = \frac{{{{{\rm{\sigma '}}}_{\rm{f}}}}}{{{{{\rm{\sigma '}}}_{\rm{o}}}}}{\rm{\;}}{{\rm{\pi }}_3} = \frac{{{{\rm{I}}_{\rm{c}}}}}{{1 + {{\rm{e}}_{\rm{o}}}}} \end{array}$ π1=τo,scvo1+eoHo2π2=HfHo $\begin{array}{} \displaystyle {{\rm{\pi }}_1} = \frac{{\tau_{{\rm{o}},{\rm{s}}}{{\rm{c}}_{{\rm{vo}}}}\left( {1 + {{\rm{e}}_{\rm{o}}}} \right)}}{{{\rm{H}}_{\rm{o}}^2}}\;\;\;{{\rm{\pi }}_2} = \frac{{{{\rm{H}}_{\rm{f}}}}}{{{{\rm{H}}_{\rm{o}}}}} \end{array}$
Davis and Raymond 1+e ≠ constant cv ≠ constant dz ≠ constant π1=τo,σ′cvo1+eoHo2π2=σ′fσ′o $\begin{array}{} \displaystyle {{\rm{\pi }}_1} = \frac{{\tau_{{\rm{o}},{\rm{\sigma '}}}{{\rm{c}}_{{\rm{vo}}}}\left( {1 + {{\rm{e}}_{\rm{o}}}} \right)}}{{{\rm{H}}_{\rm{o}}^2}}\hspace{15pt} {{\rm{\pi }}_2} = \frac{{{{{\rm{\sigma '}}}_{\rm{f}}}}}{{{{{\rm{\sigma '}}}_{\rm{o}}}}} \end{array}$ π1=τo,scvo1+eoHo2 $\begin{array}{} \displaystyle {{\rm{\pi }}_1} = \frac{{\tau_{{\rm{o}},{\rm{s}}}{{\rm{c}}_{{\rm{vo}}}}\left( {1 + {{\rm{e}}_{\rm{o}}}} \right)}}{{{\rm{H}}_{\rm{o}}^2}} \end{array}$
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