- Email: [email protected]
On modelling the compressibility of unsaturated fine-grained soils
HOSTED BY
Available online at www.sciencedirect.com
ScienceDirect Soils and Foundations xxx (2017) xxx–xxx www.elsevier.com/locate/sandf
On modelling the compressibility of unsaturated fine-grained soils Dong Tang a,b, Eshan Ganju c, Shahedur Rahman c, Dianqing Li a,⇑, Monica Prezzi c, Rodrigo Salgado c a
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan 430072, PR China b Hunan Lishui Hydro & Power Co., Ltd., Changsha 410004, PR China c Lyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907-1284, USA Received 9 October 2016; received in revised form 12 June 2017; accepted 4 July 2017
Abstract An empirical expression linking the slope k of the normal compression line (NCL) for unsaturated soils to the effective degree of saturation Se is studied in this technical note. The parameters in this empirical expression are intrinsic soil parameter a, the slope of the normal compression line (NCL) at full saturation, k100%, and the slope of the normal compression line at the driest state achievable by a soil, kd (a state at which only adsorbed water remains in the pores). The determination of k at any other level of saturation usually requires a considerable amount of test data. This technical note explores the relationship between regression parameter a and the particle size, quantified by the D10 of the soil, based on test data for five different soils taken from the literature. A regression equation is developed that relates a to D10. Based on the proposed regression equation, the slope of the normal compression line for unsaturated finegrained soils at any degree of saturation can be estimated when the test data are limited. Ó 2017 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Unsaturated soil; Effective stress; Volume change behavior; Normal compression line (NCL); Particle size
1. Introduction The principles of soil mechanics are well understood and widely accepted for saturated soils (Salgado, 2008; Terzaghi, 1943). However, those of unsaturated soils are not understood to the same degree. This technical note focuses on the response of the normal and isotropic compression of unsaturated soils. This response is expressed in saturated soil mechanics using the normal compression line (NCL), which is defined as a straight line in the plane of specific volume t versus the natural logarithm of mean effective stress p0 .
Peer review under responsibility of The Japanese Geotechnical Society. ⇑ Corresponding author. E-mail address: [email protected] (D. Li).
t ¼ N k ln
p0 pr
ð1Þ
where pr is a reference stress used for normalization (usually set to 1 kPa); N is the specific volume at the chosen reference stress; and k is the slope of the normal compression line, observed to be a constant for a specific soil. Ideally, in formulating an effective stress-based formulation for unsaturated soil mechanics, the NCL should be unique in the t vs ln p0 space, with p0 defined in a way that takes into account the degree of saturation of the soil and the resulting suction (Sheng, 2011a). Lu and Likos (2006) have, in essence, separated the sources of the interparticle forces that can be determined (total stress, pore water pressure, and air pressure) from those that cannot be measured or separately determined (surface tension, double layer forces, and van der Waals forces). These forces should all satisfy the governing equations, starting
https://doi.org/10.1016/j.sandf.2017.08.020 0038-0806/Ó 2017 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
2
D. Tang et al. / Soils and Foundations xxx (2017) xxx–xxx
with equilibrium in a static problem. They are interrelated, depending on the same intrinsic soil variables, and are enabled by each other. For example, surface tension only exists because of the coexistence of air and water; and thus, the difference between air and water pressures is directly related to it. Since surface tension, double-layer, van der Waals, and any other electro-chemical forces cannot be measured, a definition of effective stress accounting for suction (the difference between air and water pressures enabled by these forces) that recovers the principle of effective stress for fully saturated and practically dry soil is desirable. This definition requires a parameter in order to reflect the complex interaction between these forces. To the degree that these interactions cannot be accurately captured, nonuniqueness in certain constitutive relationships may occur, including on the normal compression line (NCL) in space. The main objectives of this technical note are to study the problem of volume change in unsaturated soils due to 1-D or isotropic compression and to assess how the slope k of the normal compression line changes with the degree of saturation. Firstly, the proper selection of a stress variable (effective stress) is discussed as the use of ad-hoc stress variables can result in non-uniqueness for the normal compression line (NCL) in the t – ln p0 space. Then, the expression proposed by Zhou et al. (2012a), used to model the k of unsaturated soil as a function of the degree of saturation (in the form of the effective degree of saturation (Alonso et al., 2010)), is introduced. Subsequently, data available from the literature on the effect of the degree of saturation on k are provided. Lastly, the effect of the particle size on k is considered and a connection is made between an intrinsic parameter a (used to model the variation in k with the degree of saturation) and the soil particle size distribution. 2. Stresses between soil particles Considering a representative elementary volume (REV) of a soil–water–air mixture, as done by Lu and Godt (2013), and working with the assumption made by Houlsby (1997) that the areas on which pore-air and pore-water pressures act are proportional to their volume fraction, the following effective stress expression can be derived using the macroscopic stresses: r0 ¼ ðrtotal ua Þ þ S e ðua uw Þ
ð2Þ
where r0 is the ratio of the summation of the soil-particle contact forces to the gross cross-sectional area of the soil; rtotal is the total external applied stress; ua is the pore-air pressure; uw is the pore-water pressure; and Se is the effective degree of saturation, defined by Alonso et al. (2010) as Se ¼
S r S res r 1 S res r
ð3Þ Sres r
where Sr is the degree of saturation and is the residual degree of saturation at which the pore water in the soil only exists in micro-pores or in the adsorbed state.
Effective stress, as defined for use in Eq. (2), is the interparticle force averaged over the gross surface area of the REV. The efficacy of the effective stress definition using Se is affected by the type of soil considered, particularly the particle size. Surface tension and electrochemical forces are all affected by saturation and the particle size. Important for reflection in the present work, Lu and Godt (2013) showed that the electric double-layer repulsive forces, van der Waals forces, and surface tension forces all decrease with the increasing particle size and are weak in sand and gravel. As these forces are very low in sand, regardless of the degree of saturation, the compressibility of sand is not significantly influenced by the degree of saturation, an observation supported by Cho and Santamarina (2001) for spherical glass beads. 3. Normal compression line (NCL) of unsaturated soils To extend Eq. (1) to the unsaturated state, two alternative assumptions have been proposed in the literature (Zhou et al., 2012a): (1) parameters N and/or k are assumed to be functions of suction s and (2) parameters N and/or k are assumed to be functions of the degree of saturation, as expressed by either Sr or Se. In most geotechnical tests, it is possible to control and measure suction s using the axis-translation technique (Hilf, 1956; Vanapalli et al., 2008; Zhang and Li, 2010; Tang et al., 2017). Therefore, researchers initially attempted to link N and/or k to s. This approach was widely used in early work on the constitutive modelling of unsaturated soils. Sheng (2011a,b) and Zhou et al. (2012a) pointed out that there are some limitations to this approach. Further investigation into the limitations of this approach is beyond the scope of this technical note. To overcome these limitations, Al-Badran and Schanz (2010, 2014) proposed the concept of ‘‘Sr – lines”, according to which N and k both increase with a decrease in Sr. Other researchers suggested linking only k to Sr (Sheng, 2011a) or Se (Zhou et al., 2012a,b), as, conceptually, parameter N should be a constant because it corresponds to the state at which the soil has experienced no consolidation at all (since it is equal to the value of the specific volume at a very small effective stress). The NCL in the t vs ln p0 space can be written with its slope as a function of the degree of saturation, as suggested by Zhou et al. (2012a). t ¼ N kðS e Þ ln
p0 pr
ð4Þ
The value of k varies from that at full saturation (k100%) to that at the driest state (kd). Zhou et al. (2012a) suggested the following form for the expression of k: kðS e Þ ¼ k100% ð1 S e Þa ðk100% kd Þ
ð5Þ
where a is a fitting parameter that controls the variation in k with Se. Zhou et al. (2012a) suggested that kd may be taken as the slope of the elastic compression line in the
Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
D. Tang et al. / Soils and Foundations xxx (2017) xxx–xxx
ð6Þ
MH CH ML ML ML Clay and silt division is 5 lm; sand and silt division is 75 lm. a
Clay (%)
90 47 44 32 22 10 48 56 48 39
Silt (%) Sand (%)
0 5 0 20 39 2.61 2.69 2.71 2.64 – 32 33 13 11 16 32 29 20 18 16 0.16 0.25 0.54 0.41 0.86 Speswhite Kaolin Fujinomori Clay Catalpo Clay Soil CL-1 Soil EDO-1
64 62 33 29 32
Percentage by weighta Gs PI PL LL D10 (lm)
Table 1 Summary of soil properties.
To quantify the influence of the degree of saturation and the particle size on k, test data were collected from the literature. To ensure the quality of the test data, some requirements were necessary: (1) detailed soil characterization, especially the grain size distribution; (2) onedimensional compression data for both saturated and unsaturated states; (3) void ratio e, the net stress, the matric suction, and the degree of saturation data for each point on the NCL; and (4) sufficient amount of data so as to ensure reliability. Data satisfying these requirements are not widely available in the literature. Data for five soils – Speswhite Kaolin (Wheeler and Sivakumar, 1995), Fujinomori Clay (Kikumoto et al., 2010; Li et al., 2009), Catalpo Clay (Honda, 2000), Soil CL-1 (Burland et al., 2003; Jotisankasa, 2005), and Soil EDO-1 (Barrera, 2002) – satisfied these requirements adequately and were used in this study. The properties of these soils are given in Table 1 and the grain size distributions are presented in Fig. 1. A regression analysis of the test data using Eq. (5) is presented graphically in Fig. 2 and in tabular form in Table 2. For the regression analysis, the values for k100% and kd of each soil were obtained from experimental data available in the literature (1-D or isotropic compression). A reasonably good fit of Eq. (5) to the experimental data indicates a strong correlation of k with Se. The shape of the regression curves is governed by three parameters: (1) a, (2) k100%, and (3) kd. For example, Catalpo Clay has a k100% of 0.108 and a kd of 0.008, while EDO-1 has a k100% of 0.073 and a kd of 0.011. Therefore, Catalpo Clay has a wider compressibility range, which may be attributed to the fact that the average particle size of Catalpo Clay is smaller than that of EDO-1. While k100% and kd control the range of k, the slope of the k vs. Se curves is controlled by parameter a. An a value of 1 leads to a linear relationship between k and Se. The regression analysis of the soils presented in this technical note yielded a values in the range of 1.49 (for Speswhite Kaolin) to 2.7 (for EDO-1). For soils with limited compressibility
USCS
4. Effect of saturation on slope of normal compression line
Isotropic 1-D 1-D Isotropic 1-D
Type of consolidation test
Source
where PI is the plasticity index and Gs is the specific gravity of the soil. The effects of saturation on the slope of the normal compression line and the determination of fitting parameter a are discussed next. It should be noted here that while k100% may be obtained from an empirical expression similar to Eq. (6), the value of k100% was obtained in this technical note from 1-D and isotropic compression test data. Eq. (6) is provided only as an alternative method for obtaining k100% when sufficient experimental data are not available.
Soil
k100% ¼ 0:217 PI Gs
Wheeler and Sivakumar (1995) Kikumoto et al. (2010) and Li et al. (2009) Honda (2000) Burland et al. (2003) and Jotisankasa (2005) Barrera (2002)
absence of experimental data. The slope k100% of the NCL at full saturation must be obtained experimentally, but may also be estimated from correlations with index properties available in the literature (Wroth and Wood, 1978).
3
Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
4
D. Tang et al. / Soils and Foundations xxx (2017) xxx–xxx
note, D10 was used as the representative particle size for two main reasons: (1) D10 has routinely been used in geotechnical research to develop practical correlations between the grain size distribution and the hydromechanical properties of soil and (2) D10 is conceptually a representative particle size because fines tend to control the hydro-mechanical response of the soil. Parameters k100% and kd are unique for each soil and can be acquired from consolidation test data. The fitting parameter a appearing in Eq. (5) should also be unique for each soil, and a simple regression analysis of a vs D10 yields the following expression:
100
60
40
Speswhite Kaolin Fujinomori Clay Catalpo Clay Soil CL-1 Soil EDO-1
20 Sand
Silt
Clay 0 0.1
1
10
100
1000
10000
Particle size (μm)
Fig. 1. Grain size distribution of soils.
0.14 0.12 0.10
λ
0.08 0.06 Speswhite Kaolin Fujinomori Clay Catalpo Clay Soil CL-1 Soil EDO-1
0.04 0.02 0.00
a ¼ 2:77ðD10 Þ
0:35
ð7Þ
Fig. 3 shows how a varies with D10 and the trend line that best describes the relation between these two variables. As can be seen in this figure, a increases with an increase in D10. In addition to the trend line, the lines corresponding to the 95% confidence intervals are also presented in Fig. 3. While more data are required to further refine the relationship between a and D10, the use of Eqs. (5) and (7) makes it possible to estimate the value of k of the unsaturated soils for which there are limited or no test data. To verify the effectiveness of Eqs. (5) and (7) in the estimation of the value of k, the isotropic consolidation test data for Pearl Clay (Sun et al., 2007) are used. The D10 of Pearl Clay is 0.9 lm, the plasticity index is 22%, the 3.0 2.8
0
10
20
30
40
50
60
70
80
90
100
Se (%) Fig. 2. Regression analysis of experimental data.
data, such as EDO-1, the assessment of the a values becomes subject to some degree of uncertainty outside the available data range. A more complete dataset would yield a more accurate estimate of the a value. 5. Effects of particle size on slope of normal compression line Since k is influenced by the saturation and the particle size of the soil, at a specific saturation it can be viewed primarily as a function of the particle size. In this technical
Fitting parameter α
Passing (%)
80
2.6
α =2.77 ( D10 )
2.4
R 2 = 0.97
0.35
2.2 2.0 1.8
α from test data
1.6
Fitting curve for α 95% confidence interval
1.4 1.2 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Particle size parameter - D10 (μm) Fig. 3. Fitting parameter a versus D10.
Table 2 Regression results of a for five different soils. Soil
Speswhite Kaolin Fujinomori Clay Catalpo Clay Soil CL-1 Soil EDO-1
Soil properties
Regression results
D10 (lm)
k100%
kd
Fitting parameter a
R2
0.16 0.25 0.54 0.41 0.86
0.112 0.104 0.108 0.094 0.073
0.02 0.01 0.008 0.015 0.011
1.49 1.64 2.12 2.08 2.70
0.95 0.95 0.85 0.88 0.92
Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
D. Tang et al. / Soils and Foundations xxx (2017) xxx–xxx
5
0.14
Acknowledgements
0.12
This work was supported by the National Natural Science Foundation of China (Project Nos. 51329901, 51579190, 51528901) and the National Science Fund for Distinguished Young Scholars (Project No. 51225903). The authors wish to thank Prof. Adrian Russell, Prof. Tom Schanz, Prof. Zijun Cao, and Dr. Michinori Honda for their generous help in providing test data or valuable discussions.
0.10
λ
0.08 0.06 Data
λ100% from Eq. (6) and α from Eq. (7)
0.04
λ100% from test data and α from Eq. (7)
References
0.02 0.00 50
55
60
65
70
75
80
Se (%) Fig. 4. Comparison between values of k calculated from test data and from empirical equations for Pearl Clay (Sun et al., 2007).
specific gravity is 2.71, and k100% = 0.121. The values for k as a function of Se for Pearl Clay were calculated using Eq. (5) in two ways: (A) a was obtained from Eq. (7) and k100% was obtained from the saturated consolidation test data and (B) a was obtained from Eq. (7) and k100% was estimated using Eq. (6). Eq. (7) gave a value of 2.71 for a; using Eq. (6), k100% was estimated to be 0.130. For both cases, kd was assumed to be equal to the slope of the saturated recompression line, 0.012, as suggested by Zhou et al. (2012a). Fig. 4 compares the estimated k from the two cases against the test data from Sun et al. (2007). From Fig. 4, the estimated values for k in both cases (A) and (B) lie above, but close to, the values obtained from the test data, with the results from case (A) performing slightly better than those from case B. The comparison shows that the combined use of Eqs. (5) and (7) can produce estimates of the value of k that are reasonably accurate even when the test data are limited. 6. Conclusions The slope k of the normal compression line (NCL) for unsaturated soils was assumed to depend on the particle size and on the effective degree of saturation. This relationship was explored for five different soils. Effective particle size D10 was found to be the best variable for representing the dependence of a on the particle size. Regression analyses of the test data for these five soils led to an expression that allows for the estimation of the slope of the NCL k for unsaturated soils for any degree of saturation. The developed expression between a and D10 was compared against the data available in the literature, producing reasonable results. Further refinement of the expression proposed between a and D10 will become possible as more complete experimental data sets become available in the literature.
Al-Badran, Y., Schanz, T., 2014. Modelling the compaction curve of finegrained soils. Soils Found. 54, 426–438. https://doi.org/10.1016/j. sandf.2014.04.011. Al-Badran, Y., Schanz, T., 2010. Yielding surface model of volume change characteristics for unsaturated fine grained soils. In: Buzzi, O., Fityus, S., Sheng, D. (Eds.), Unsaturated Soils: Theoretical and Numerical Advances in Unsaturated Soil Mechanics – II. CRC Press. Alonso, E.E., Olivella, S., Vaunat, J., Pereira, J.M., 2010. A microstructurally based effective stress for unsaturated soils. Ge´otechnique 60, 913–925. https://doi.org/10.1680/geot.8.P.002. Barrera, M., 2002. Estudio experimental del comportamiento hidromeca´nico de suelos colapsables. Ph. D. thesis. Universitat Polite`cnica de Catalunya (In Spanish). Burland, J.B., Cunningham, M.R., Dineen, K., Ridley, A.M., Dineen, K., Burland, J.B., 2003. The mechanical behaviour of a reconstituted unsaturated silty clay. Ge´otechnique 53, 183–194. https://doi.org/ 10.1680/geot.2003.53.2.183. Cho, G.C., Santamarina, J.C., 2001. Unsaturated particulate materials— particle-level studies. J. Geotech. Geoenvironmental Eng. 127, 84–96. https://doi.org/10.1061/(ASCE)1090-0241(2001) 127:1(84). Hilf, J., 1956. An investigation of pore water pressure in compacted cohesive soils. Ph. D. thesis, University of Colorado. Honda, M., 2000. Research on the prediction methods for the mechanical behaviors of unsaturated soils. Ph. D. thesis, Kobe University (in Japanese). Houlsby, G.T., 1997. The work input to an unsaturated granular material. Ge´otechnique 47, 193–196. https://doi.org/10.1680/geot.1997.47.1.193. Jotisankasa, A., 2005. Collapse behaviour of a compacted silty clay. Ph. D. thesis. Imperial College London. Kikumoto, M., Kyokawa, H., Nakai, T., Shahin, M.H., 2010. A simple elasto-plastic model for unsaturated soils and interpretations of collapse and compaction behaviours. In: Alonso, E.E. (Ed.), Unsaturated Soils. CRC Press, pp. 849–855. https://doi.org/10.1201/b10526131. Li, J.-Z., Peng, F.-L., Xu, L., 2009. One-dimensional viscous behavior of clay and its constitutive modeling. Int. J. Geomech. 9, 43–51. https:// doi.org/10.1061/(ASCE)1532-3641(2009)9:2(43). Lu, N., Godt, J.W., 2013. Hillslope Hydrology and Stability. Cambridge University Press. Cambridge. http://dx.doi.org/10.1017/ CBO9781139108164. Lu, N., Likos, W.J.W., 2006. Suction stress characteristic curve for unsaturated soil. J. Geotech. Geoenvironmental Eng. 132, 131–142. https://doi.org/10.1061/(ASCE)1090-0241(2006)132:2(131). Salgado, R., 2008. The Engineering of Foundations, 1st ed. McGraw-Hill, New York. http://dx.doi.org/10.1017/CBO9781107415324.004. Sheng, D., 2011a. Constitutive modelling of unsaturated soils: Discussion of fundamental principles. In: Alonso, E.E., Gens, A. (Eds.), Unsaturated Soils. Taylor & Francis, pp. 91–112. doi: http://dx.doi.org/10. 1201/b10526-8. Sheng, D., 2011b. Review of fundamental principles in modelling unsaturated soil behaviour. Comput. Geotech. 38, 757–776. https:// doi.org/10.1016/j.compgeo.2011.05.002.
Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
6
D. Tang et al. / Soils and Foundations xxx (2017) xxx–xxx
Sun, D., Sheng, D., Xu, Y., 2007. Collapse behaviour of unsaturated compacted soil with different initial densities. Can. Geotech. J. 44, 673– 686. https://doi.org/10.1139/t07-023. Tang, D., Li, D.Q., Cao, Z.J., 2017. Slope stability analysis in the Three Gorges Reservoir area considering effects of antecedent rainfall. Georisk 11 (2), 161–172. Terzaghi, K., 1943. Theoretical soil mechanics, 1st ed. John Wiley & Sons, New York. doi: http://dx.doi.org/10.1680/geot.1964.14.1.1. Vanapalli, S., Nicotera, M., Sharma, R., 2008. Axis translation and negative water column techniques for suction control. Geotech. Geol. Eng. 26, 645–660. Wheeler, S.J., Sivakumar, V., 1995. An elasto-plastic critical state framework for unsaturated soil. Ge´otechnique 45, 35–53. https://doi. org/10.1680/geot.1995.45.1.35.
Wroth, C.P., Wood, D.M., 1978. The correlation of index properties with some basic engineering properties of soils. Can. Geotech. J. 15, 137– 145. https://doi.org/10.1139/t78-014. Zhang, L.M., Li, X., 2010. Micro-porosity structure of coarse granular soils. J. Geotech. Geoenviron. Eng. 136 (10), 1425–1436. Zhou, A., Sheng, D., Sloan, S.W., Gens, A., 2012a. Interpretation of unsaturated soil behaviour in the stress – saturation space, I: Volume change and water retention behaviour. Comput. Geotech. 43, 178–187. https://doi.org/10.1016/j.compgeo.2012.04.010. Zhou, A., Sheng, D., Sloan, S.W., Gens, A., 2012b. Interpretation of unsaturated soil behaviour in the stress-saturation space. II: Constitutive relationships and validations. Comput. Geotech. 43, 111–123. https://doi.org/10.1016/j.compgeo.2012.02.009.
Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
Available online at www.sciencedirect.com
ScienceDirect Soils and Foundations xxx (2017) xxx–xxx www.elsevier.com/locate/sandf
On modelling the compressibility of unsaturated fine-grained soils Dong Tang a,b, Eshan Ganju c, Shahedur Rahman c, Dianqing Li a,⇑, Monica Prezzi c, Rodrigo Salgado c a
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan 430072, PR China b Hunan Lishui Hydro & Power Co., Ltd., Changsha 410004, PR China c Lyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907-1284, USA Received 9 October 2016; received in revised form 12 June 2017; accepted 4 July 2017
Abstract An empirical expression linking the slope k of the normal compression line (NCL) for unsaturated soils to the effective degree of saturation Se is studied in this technical note. The parameters in this empirical expression are intrinsic soil parameter a, the slope of the normal compression line (NCL) at full saturation, k100%, and the slope of the normal compression line at the driest state achievable by a soil, kd (a state at which only adsorbed water remains in the pores). The determination of k at any other level of saturation usually requires a considerable amount of test data. This technical note explores the relationship between regression parameter a and the particle size, quantified by the D10 of the soil, based on test data for five different soils taken from the literature. A regression equation is developed that relates a to D10. Based on the proposed regression equation, the slope of the normal compression line for unsaturated finegrained soils at any degree of saturation can be estimated when the test data are limited. Ó 2017 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Unsaturated soil; Effective stress; Volume change behavior; Normal compression line (NCL); Particle size
1. Introduction The principles of soil mechanics are well understood and widely accepted for saturated soils (Salgado, 2008; Terzaghi, 1943). However, those of unsaturated soils are not understood to the same degree. This technical note focuses on the response of the normal and isotropic compression of unsaturated soils. This response is expressed in saturated soil mechanics using the normal compression line (NCL), which is defined as a straight line in the plane of specific volume t versus the natural logarithm of mean effective stress p0 .
Peer review under responsibility of The Japanese Geotechnical Society. ⇑ Corresponding author. E-mail address: [email protected] (D. Li).
t ¼ N k ln
p0 pr
ð1Þ
where pr is a reference stress used for normalization (usually set to 1 kPa); N is the specific volume at the chosen reference stress; and k is the slope of the normal compression line, observed to be a constant for a specific soil. Ideally, in formulating an effective stress-based formulation for unsaturated soil mechanics, the NCL should be unique in the t vs ln p0 space, with p0 defined in a way that takes into account the degree of saturation of the soil and the resulting suction (Sheng, 2011a). Lu and Likos (2006) have, in essence, separated the sources of the interparticle forces that can be determined (total stress, pore water pressure, and air pressure) from those that cannot be measured or separately determined (surface tension, double layer forces, and van der Waals forces). These forces should all satisfy the governing equations, starting
https://doi.org/10.1016/j.sandf.2017.08.020 0038-0806/Ó 2017 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
2
D. Tang et al. / Soils and Foundations xxx (2017) xxx–xxx
with equilibrium in a static problem. They are interrelated, depending on the same intrinsic soil variables, and are enabled by each other. For example, surface tension only exists because of the coexistence of air and water; and thus, the difference between air and water pressures is directly related to it. Since surface tension, double-layer, van der Waals, and any other electro-chemical forces cannot be measured, a definition of effective stress accounting for suction (the difference between air and water pressures enabled by these forces) that recovers the principle of effective stress for fully saturated and practically dry soil is desirable. This definition requires a parameter in order to reflect the complex interaction between these forces. To the degree that these interactions cannot be accurately captured, nonuniqueness in certain constitutive relationships may occur, including on the normal compression line (NCL) in space. The main objectives of this technical note are to study the problem of volume change in unsaturated soils due to 1-D or isotropic compression and to assess how the slope k of the normal compression line changes with the degree of saturation. Firstly, the proper selection of a stress variable (effective stress) is discussed as the use of ad-hoc stress variables can result in non-uniqueness for the normal compression line (NCL) in the t – ln p0 space. Then, the expression proposed by Zhou et al. (2012a), used to model the k of unsaturated soil as a function of the degree of saturation (in the form of the effective degree of saturation (Alonso et al., 2010)), is introduced. Subsequently, data available from the literature on the effect of the degree of saturation on k are provided. Lastly, the effect of the particle size on k is considered and a connection is made between an intrinsic parameter a (used to model the variation in k with the degree of saturation) and the soil particle size distribution. 2. Stresses between soil particles Considering a representative elementary volume (REV) of a soil–water–air mixture, as done by Lu and Godt (2013), and working with the assumption made by Houlsby (1997) that the areas on which pore-air and pore-water pressures act are proportional to their volume fraction, the following effective stress expression can be derived using the macroscopic stresses: r0 ¼ ðrtotal ua Þ þ S e ðua uw Þ
ð2Þ
where r0 is the ratio of the summation of the soil-particle contact forces to the gross cross-sectional area of the soil; rtotal is the total external applied stress; ua is the pore-air pressure; uw is the pore-water pressure; and Se is the effective degree of saturation, defined by Alonso et al. (2010) as Se ¼
S r S res r 1 S res r
ð3Þ Sres r
where Sr is the degree of saturation and is the residual degree of saturation at which the pore water in the soil only exists in micro-pores or in the adsorbed state.
Effective stress, as defined for use in Eq. (2), is the interparticle force averaged over the gross surface area of the REV. The efficacy of the effective stress definition using Se is affected by the type of soil considered, particularly the particle size. Surface tension and electrochemical forces are all affected by saturation and the particle size. Important for reflection in the present work, Lu and Godt (2013) showed that the electric double-layer repulsive forces, van der Waals forces, and surface tension forces all decrease with the increasing particle size and are weak in sand and gravel. As these forces are very low in sand, regardless of the degree of saturation, the compressibility of sand is not significantly influenced by the degree of saturation, an observation supported by Cho and Santamarina (2001) for spherical glass beads. 3. Normal compression line (NCL) of unsaturated soils To extend Eq. (1) to the unsaturated state, two alternative assumptions have been proposed in the literature (Zhou et al., 2012a): (1) parameters N and/or k are assumed to be functions of suction s and (2) parameters N and/or k are assumed to be functions of the degree of saturation, as expressed by either Sr or Se. In most geotechnical tests, it is possible to control and measure suction s using the axis-translation technique (Hilf, 1956; Vanapalli et al., 2008; Zhang and Li, 2010; Tang et al., 2017). Therefore, researchers initially attempted to link N and/or k to s. This approach was widely used in early work on the constitutive modelling of unsaturated soils. Sheng (2011a,b) and Zhou et al. (2012a) pointed out that there are some limitations to this approach. Further investigation into the limitations of this approach is beyond the scope of this technical note. To overcome these limitations, Al-Badran and Schanz (2010, 2014) proposed the concept of ‘‘Sr – lines”, according to which N and k both increase with a decrease in Sr. Other researchers suggested linking only k to Sr (Sheng, 2011a) or Se (Zhou et al., 2012a,b), as, conceptually, parameter N should be a constant because it corresponds to the state at which the soil has experienced no consolidation at all (since it is equal to the value of the specific volume at a very small effective stress). The NCL in the t vs ln p0 space can be written with its slope as a function of the degree of saturation, as suggested by Zhou et al. (2012a). t ¼ N kðS e Þ ln
p0 pr
ð4Þ
The value of k varies from that at full saturation (k100%) to that at the driest state (kd). Zhou et al. (2012a) suggested the following form for the expression of k: kðS e Þ ¼ k100% ð1 S e Þa ðk100% kd Þ
ð5Þ
where a is a fitting parameter that controls the variation in k with Se. Zhou et al. (2012a) suggested that kd may be taken as the slope of the elastic compression line in the
Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
D. Tang et al. / Soils and Foundations xxx (2017) xxx–xxx
ð6Þ
MH CH ML ML ML Clay and silt division is 5 lm; sand and silt division is 75 lm. a
Clay (%)
90 47 44 32 22 10 48 56 48 39
Silt (%) Sand (%)
0 5 0 20 39 2.61 2.69 2.71 2.64 – 32 33 13 11 16 32 29 20 18 16 0.16 0.25 0.54 0.41 0.86 Speswhite Kaolin Fujinomori Clay Catalpo Clay Soil CL-1 Soil EDO-1
64 62 33 29 32
Percentage by weighta Gs PI PL LL D10 (lm)
Table 1 Summary of soil properties.
To quantify the influence of the degree of saturation and the particle size on k, test data were collected from the literature. To ensure the quality of the test data, some requirements were necessary: (1) detailed soil characterization, especially the grain size distribution; (2) onedimensional compression data for both saturated and unsaturated states; (3) void ratio e, the net stress, the matric suction, and the degree of saturation data for each point on the NCL; and (4) sufficient amount of data so as to ensure reliability. Data satisfying these requirements are not widely available in the literature. Data for five soils – Speswhite Kaolin (Wheeler and Sivakumar, 1995), Fujinomori Clay (Kikumoto et al., 2010; Li et al., 2009), Catalpo Clay (Honda, 2000), Soil CL-1 (Burland et al., 2003; Jotisankasa, 2005), and Soil EDO-1 (Barrera, 2002) – satisfied these requirements adequately and were used in this study. The properties of these soils are given in Table 1 and the grain size distributions are presented in Fig. 1. A regression analysis of the test data using Eq. (5) is presented graphically in Fig. 2 and in tabular form in Table 2. For the regression analysis, the values for k100% and kd of each soil were obtained from experimental data available in the literature (1-D or isotropic compression). A reasonably good fit of Eq. (5) to the experimental data indicates a strong correlation of k with Se. The shape of the regression curves is governed by three parameters: (1) a, (2) k100%, and (3) kd. For example, Catalpo Clay has a k100% of 0.108 and a kd of 0.008, while EDO-1 has a k100% of 0.073 and a kd of 0.011. Therefore, Catalpo Clay has a wider compressibility range, which may be attributed to the fact that the average particle size of Catalpo Clay is smaller than that of EDO-1. While k100% and kd control the range of k, the slope of the k vs. Se curves is controlled by parameter a. An a value of 1 leads to a linear relationship between k and Se. The regression analysis of the soils presented in this technical note yielded a values in the range of 1.49 (for Speswhite Kaolin) to 2.7 (for EDO-1). For soils with limited compressibility
USCS
4. Effect of saturation on slope of normal compression line
Isotropic 1-D 1-D Isotropic 1-D
Type of consolidation test
Source
where PI is the plasticity index and Gs is the specific gravity of the soil. The effects of saturation on the slope of the normal compression line and the determination of fitting parameter a are discussed next. It should be noted here that while k100% may be obtained from an empirical expression similar to Eq. (6), the value of k100% was obtained in this technical note from 1-D and isotropic compression test data. Eq. (6) is provided only as an alternative method for obtaining k100% when sufficient experimental data are not available.
Soil
k100% ¼ 0:217 PI Gs
Wheeler and Sivakumar (1995) Kikumoto et al. (2010) and Li et al. (2009) Honda (2000) Burland et al. (2003) and Jotisankasa (2005) Barrera (2002)
absence of experimental data. The slope k100% of the NCL at full saturation must be obtained experimentally, but may also be estimated from correlations with index properties available in the literature (Wroth and Wood, 1978).
3
Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
4
D. Tang et al. / Soils and Foundations xxx (2017) xxx–xxx
note, D10 was used as the representative particle size for two main reasons: (1) D10 has routinely been used in geotechnical research to develop practical correlations between the grain size distribution and the hydromechanical properties of soil and (2) D10 is conceptually a representative particle size because fines tend to control the hydro-mechanical response of the soil. Parameters k100% and kd are unique for each soil and can be acquired from consolidation test data. The fitting parameter a appearing in Eq. (5) should also be unique for each soil, and a simple regression analysis of a vs D10 yields the following expression:
100
60
40
Speswhite Kaolin Fujinomori Clay Catalpo Clay Soil CL-1 Soil EDO-1
20 Sand
Silt
Clay 0 0.1
1
10
100
1000
10000
Particle size (μm)
Fig. 1. Grain size distribution of soils.
0.14 0.12 0.10
λ
0.08 0.06 Speswhite Kaolin Fujinomori Clay Catalpo Clay Soil CL-1 Soil EDO-1
0.04 0.02 0.00
a ¼ 2:77ðD10 Þ
0:35
ð7Þ
Fig. 3 shows how a varies with D10 and the trend line that best describes the relation between these two variables. As can be seen in this figure, a increases with an increase in D10. In addition to the trend line, the lines corresponding to the 95% confidence intervals are also presented in Fig. 3. While more data are required to further refine the relationship between a and D10, the use of Eqs. (5) and (7) makes it possible to estimate the value of k of the unsaturated soils for which there are limited or no test data. To verify the effectiveness of Eqs. (5) and (7) in the estimation of the value of k, the isotropic consolidation test data for Pearl Clay (Sun et al., 2007) are used. The D10 of Pearl Clay is 0.9 lm, the plasticity index is 22%, the 3.0 2.8
0
10
20
30
40
50
60
70
80
90
100
Se (%) Fig. 2. Regression analysis of experimental data.
data, such as EDO-1, the assessment of the a values becomes subject to some degree of uncertainty outside the available data range. A more complete dataset would yield a more accurate estimate of the a value. 5. Effects of particle size on slope of normal compression line Since k is influenced by the saturation and the particle size of the soil, at a specific saturation it can be viewed primarily as a function of the particle size. In this technical
Fitting parameter α
Passing (%)
80
2.6
α =2.77 ( D10 )
2.4
R 2 = 0.97
0.35
2.2 2.0 1.8
α from test data
1.6
Fitting curve for α 95% confidence interval
1.4 1.2 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Particle size parameter - D10 (μm) Fig. 3. Fitting parameter a versus D10.
Table 2 Regression results of a for five different soils. Soil
Speswhite Kaolin Fujinomori Clay Catalpo Clay Soil CL-1 Soil EDO-1
Soil properties
Regression results
D10 (lm)
k100%
kd
Fitting parameter a
R2
0.16 0.25 0.54 0.41 0.86
0.112 0.104 0.108 0.094 0.073
0.02 0.01 0.008 0.015 0.011
1.49 1.64 2.12 2.08 2.70
0.95 0.95 0.85 0.88 0.92
Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
D. Tang et al. / Soils and Foundations xxx (2017) xxx–xxx
5
0.14
Acknowledgements
0.12
This work was supported by the National Natural Science Foundation of China (Project Nos. 51329901, 51579190, 51528901) and the National Science Fund for Distinguished Young Scholars (Project No. 51225903). The authors wish to thank Prof. Adrian Russell, Prof. Tom Schanz, Prof. Zijun Cao, and Dr. Michinori Honda for their generous help in providing test data or valuable discussions.
0.10
λ
0.08 0.06 Data
λ100% from Eq. (6) and α from Eq. (7)
0.04
λ100% from test data and α from Eq. (7)
References
0.02 0.00 50
55
60
65
70
75
80
Se (%) Fig. 4. Comparison between values of k calculated from test data and from empirical equations for Pearl Clay (Sun et al., 2007).
specific gravity is 2.71, and k100% = 0.121. The values for k as a function of Se for Pearl Clay were calculated using Eq. (5) in two ways: (A) a was obtained from Eq. (7) and k100% was obtained from the saturated consolidation test data and (B) a was obtained from Eq. (7) and k100% was estimated using Eq. (6). Eq. (7) gave a value of 2.71 for a; using Eq. (6), k100% was estimated to be 0.130. For both cases, kd was assumed to be equal to the slope of the saturated recompression line, 0.012, as suggested by Zhou et al. (2012a). Fig. 4 compares the estimated k from the two cases against the test data from Sun et al. (2007). From Fig. 4, the estimated values for k in both cases (A) and (B) lie above, but close to, the values obtained from the test data, with the results from case (A) performing slightly better than those from case B. The comparison shows that the combined use of Eqs. (5) and (7) can produce estimates of the value of k that are reasonably accurate even when the test data are limited. 6. Conclusions The slope k of the normal compression line (NCL) for unsaturated soils was assumed to depend on the particle size and on the effective degree of saturation. This relationship was explored for five different soils. Effective particle size D10 was found to be the best variable for representing the dependence of a on the particle size. Regression analyses of the test data for these five soils led to an expression that allows for the estimation of the slope of the NCL k for unsaturated soils for any degree of saturation. The developed expression between a and D10 was compared against the data available in the literature, producing reasonable results. Further refinement of the expression proposed between a and D10 will become possible as more complete experimental data sets become available in the literature.
Al-Badran, Y., Schanz, T., 2014. Modelling the compaction curve of finegrained soils. Soils Found. 54, 426–438. https://doi.org/10.1016/j. sandf.2014.04.011. Al-Badran, Y., Schanz, T., 2010. Yielding surface model of volume change characteristics for unsaturated fine grained soils. In: Buzzi, O., Fityus, S., Sheng, D. (Eds.), Unsaturated Soils: Theoretical and Numerical Advances in Unsaturated Soil Mechanics – II. CRC Press. Alonso, E.E., Olivella, S., Vaunat, J., Pereira, J.M., 2010. A microstructurally based effective stress for unsaturated soils. Ge´otechnique 60, 913–925. https://doi.org/10.1680/geot.8.P.002. Barrera, M., 2002. Estudio experimental del comportamiento hidromeca´nico de suelos colapsables. Ph. D. thesis. Universitat Polite`cnica de Catalunya (In Spanish). Burland, J.B., Cunningham, M.R., Dineen, K., Ridley, A.M., Dineen, K., Burland, J.B., 2003. The mechanical behaviour of a reconstituted unsaturated silty clay. Ge´otechnique 53, 183–194. https://doi.org/ 10.1680/geot.2003.53.2.183. Cho, G.C., Santamarina, J.C., 2001. Unsaturated particulate materials— particle-level studies. J. Geotech. Geoenvironmental Eng. 127, 84–96. https://doi.org/10.1061/(ASCE)1090-0241(2001) 127:1(84). Hilf, J., 1956. An investigation of pore water pressure in compacted cohesive soils. Ph. D. thesis, University of Colorado. Honda, M., 2000. Research on the prediction methods for the mechanical behaviors of unsaturated soils. Ph. D. thesis, Kobe University (in Japanese). Houlsby, G.T., 1997. The work input to an unsaturated granular material. Ge´otechnique 47, 193–196. https://doi.org/10.1680/geot.1997.47.1.193. Jotisankasa, A., 2005. Collapse behaviour of a compacted silty clay. Ph. D. thesis. Imperial College London. Kikumoto, M., Kyokawa, H., Nakai, T., Shahin, M.H., 2010. A simple elasto-plastic model for unsaturated soils and interpretations of collapse and compaction behaviours. In: Alonso, E.E. (Ed.), Unsaturated Soils. CRC Press, pp. 849–855. https://doi.org/10.1201/b10526131. Li, J.-Z., Peng, F.-L., Xu, L., 2009. One-dimensional viscous behavior of clay and its constitutive modeling. Int. J. Geomech. 9, 43–51. https:// doi.org/10.1061/(ASCE)1532-3641(2009)9:2(43). Lu, N., Godt, J.W., 2013. Hillslope Hydrology and Stability. Cambridge University Press. Cambridge. http://dx.doi.org/10.1017/ CBO9781139108164. Lu, N., Likos, W.J.W., 2006. Suction stress characteristic curve for unsaturated soil. J. Geotech. Geoenvironmental Eng. 132, 131–142. https://doi.org/10.1061/(ASCE)1090-0241(2006)132:2(131). Salgado, R., 2008. The Engineering of Foundations, 1st ed. McGraw-Hill, New York. http://dx.doi.org/10.1017/CBO9781107415324.004. Sheng, D., 2011a. Constitutive modelling of unsaturated soils: Discussion of fundamental principles. In: Alonso, E.E., Gens, A. (Eds.), Unsaturated Soils. Taylor & Francis, pp. 91–112. doi: http://dx.doi.org/10. 1201/b10526-8. Sheng, D., 2011b. Review of fundamental principles in modelling unsaturated soil behaviour. Comput. Geotech. 38, 757–776. https:// doi.org/10.1016/j.compgeo.2011.05.002.
Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020
6
D. Tang et al. / Soils and Foundations xxx (2017) xxx–xxx
Sun, D., Sheng, D., Xu, Y., 2007. Collapse behaviour of unsaturated compacted soil with different initial densities. Can. Geotech. J. 44, 673– 686. https://doi.org/10.1139/t07-023. Tang, D., Li, D.Q., Cao, Z.J., 2017. Slope stability analysis in the Three Gorges Reservoir area considering effects of antecedent rainfall. Georisk 11 (2), 161–172. Terzaghi, K., 1943. Theoretical soil mechanics, 1st ed. John Wiley & Sons, New York. doi: http://dx.doi.org/10.1680/geot.1964.14.1.1. Vanapalli, S., Nicotera, M., Sharma, R., 2008. Axis translation and negative water column techniques for suction control. Geotech. Geol. Eng. 26, 645–660. Wheeler, S.J., Sivakumar, V., 1995. An elasto-plastic critical state framework for unsaturated soil. Ge´otechnique 45, 35–53. https://doi. org/10.1680/geot.1995.45.1.35.
Wroth, C.P., Wood, D.M., 1978. The correlation of index properties with some basic engineering properties of soils. Can. Geotech. J. 15, 137– 145. https://doi.org/10.1139/t78-014. Zhang, L.M., Li, X., 2010. Micro-porosity structure of coarse granular soils. J. Geotech. Geoenviron. Eng. 136 (10), 1425–1436. Zhou, A., Sheng, D., Sloan, S.W., Gens, A., 2012a. Interpretation of unsaturated soil behaviour in the stress – saturation space, I: Volume change and water retention behaviour. Comput. Geotech. 43, 178–187. https://doi.org/10.1016/j.compgeo.2012.04.010. Zhou, A., Sheng, D., Sloan, S.W., Gens, A., 2012b. Interpretation of unsaturated soil behaviour in the stress-saturation space. II: Constitutive relationships and validations. Comput. Geotech. 43, 111–123. https://doi.org/10.1016/j.compgeo.2012.02.009.
Please cite this article in press as: Tang, D. et al., On modelling the compressibility of unsaturated fine-grained soils, Soils Found. (2017), https://doi.org/ 10.1016/j.sandf.2017.08.020