Modeling of tensile strength on moist granular earth material at low water content

Modeling of tensile strength on moist granular earth

material at low water content

Tae-Hyung Kima,*, Changsoo Hwangb

aDepartment of Civil and Environmental Engineering, Lehigh University, 13 East Packer Avenue, Bethlehem, PA 18015, USAbDepartment of Civil, Environmental and Architectural Engineering, University of Colorado at Boulder, Boulder, CO 80309, USA

Received 7 August 2002; accepted 13 December 2002

Abstract

While most research has mainly focused on the volume change, flow, and shear strength of unsaturated earth materials,

investigations of tensile strength of unsaturated earth materials especially granular materials have not received much attention

except for cemented and clayey materials. Thus, direct tension experiments were carried out to quantify the actual magnitude of

tensile strength induced by water in moist granular soil at especially low water contents (w< 4%). The magnitudes of the

measured tensile strength are significantly different from zero. A simple experimental tensile strength model is proposed.

Practicing engineers can use this model for approximate estimation of the tensile strength of unsaturated granular soils without

experiments and for precise design or analysis of most engineered facilities relying on the unsaturated granular soils in the

vadose zone. The experimental data are also compared with a theoretical model developed for monosized spheres at low water

contents, and its application for a real granular earth material having a variety of particles is discussed. The nonlinear behavior

of the tensile strength for moist granular soil is appropriately simulated with a model.

D 2003 Elsevier Science B.V. All rights reserved.

Keywords: Unsaturated; Tensile strength model; Low water contents; Granular soil

1. Introduction

In the field, unsaturated earth materials in vadose

zone are found in many construction environments,

such as utility trenches, foundations, slopes, embank-

ments, and roadways. Experimental and theoretical

investigations for unsaturated earth materials have

been widely conducted over the past couple of decades

(Fredlund et al., 1978; Rahardjo and Fredlund, 1995;

Rohm and Vilar, 1995; Rao, 1996). Most researchers

have mainly focused on the volume change, flow, and

especially shear strength of the unsaturated earth

materials. Several models devised for predicting the

permeability function and the shear strength have been

also investigated (Fredlund et al., 1994, 1996). How-

ever, investigations of tensile strength of unsaturated

earth materials especially granular soils have not

received much attention except for cemented and

clayey materials (Al-Hussaini and Townsend, 1973;

Yong and Townsend, 1981; Biarez et al., 1997).

The reason is that, in engineering practice, it is

generally assumed that granular earth materials

exhibit only shear strength and insignificant or no

tensile strength and cohesion. Even in the case where

more complex analysis methods can consider both

friction and tension (sometimes as cohesion), most

designers will assign a value of zero to the tensile

0013-7952/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0013-7952(02)00284-3

* Corresponding author. Fax: +1-610-758-6405.

E-mail address: [email protected] (T.-H. Kim).

www.elsevier.com/locate/enggeo

Engineering Geology 69 (2003) 233–244

strength of the granular material because of a lack of

data or due to the perception that cohesion or tensile

strength must be insignificant in granular materials.

However, experience shows that, within the unsatu-

rated earth zone near subsurface, a steep excavated

surface in fine-grained granular soils can be sustained

for some unknown but non-zero period of time and

some slopes may remain stable for extended time

periods due to capillary forces. Although the unsup-

ported excavation surfaces or slopes will naturally

collapse over time due to dissipation of negative pore

pressures or due to dynamic forces, both observations

clearly indicate small but non-zero values for tensile

strength and cohesion in unsaturated granular soils.

At low water contents in an assembly of granular

particles, small amounts of water form water bridges at

contact points and these bridges result in capillary

bonding between particles, giving rise to both cohesion

and tensile strength. Fig. 1 shows the states of satu-

ration in unsaturated granular materials (Schubert et al.,

1975). At low water content level (in the pendular

state), which is defined as the state where the water is

disconnected excepted for the very thin films of water

surrounding the solids, the capillary forces are devel-

oped from water bridges between the particles.

In this paper, the tensile strength of granular soil at

low water contents is examined experimentally and

theoretically especially focused on modeling based on

the results of direct tension experiments. Within the

scope of this work, purely mechanical water-induced

strength issues on granular materials are investigated.

2. Capillary bonding forces at low water contents

Capillary bonding in bridge systems generally

leads to two force components at low water content

levels: (1) the surface tension force, Fs, acting along

the water–particle contact line and (2) the force, Fc,

due to the difference in the pressures outside and

inside the bridge acting on the cross-sectional area.

The surface tension tends to force the particles

together, whereas the force due to the pressure differ-

ence can only contribute to particle adhesion if there is

a net pressure deficiency within the bridge. Due to the

presence of water bridges between the particles, these

two forces act together as a total bonding force, Ft.

Schubert (1984) and Pierrat and Caram (1997)

determined theoretically the magnitude of the capillary

bonding forces, Ft, Fs, and Fc in the pendular state. For

instance, if a water bridge exists between two particles

of diameter, d, and separated by a distance, a, as shown

in Fig. 2, the surface tension for the liquid, Fs, is given

in a dimensionless form as

Fs

ad¼ psinhsinðh þ dÞ ð1Þ

where h is the filling angle, and a and d are the surface

tension and contact angle, respectively. The force, Fc, is

due to the difference of the pressure within the bridge

acting on the cross-sectional area and is also given in a

dimensionless form as

Fs

ad¼ p

sinh2

� �21

r*� 1

h*

� �ð2Þ

h* ¼ h

d¼ sinh

2þ r

d½sinðh þ dÞ � 1�

r* ¼ r

d¼ ð1� coshÞ þ a=d

2cosðh þ dÞwhere h* and r* are the two dimensionless radii of

curvature of the water bridge, when taken as arcs of a

circle. The total dimensionless bonding force, Ft, is the

Fig. 1. States of saturation in unsaturated granular materials: (a) pendular state, (b) funicular state, and (c) capillary state (from Schubert et al.,

1975).

T.-H. Kim, C. Hwang / Engineering Geology 69 (2003) 233–244234

sum of the two components as a function of the contact

and filling angles (d and h) and the dimensionless

separation distance (a/d).

Fs

ad¼ psinh sinðh þ dÞ þ sinh

4

1

r*� 1

h*

� �� ð3Þ

3. Direct tension experiments

All specimens were prepared using F-75 Ottawa

silica granular material obtained from the Ottawa

Silica. The material is a fine-grained natural quartz

material of uniform gradation with a mean particle

Fig. 2. (a) Water bridge bonding two spherical particles (from Pierrat and Caram, 1997) and (b) one-dimensional free-body diagram of bonding

forces.

Fig. 3. Grain size distribution curve for F-75 Ottawa granular soil.

T.-H. Kim, C. Hwang / Engineering Geology 69 (2003) 233–244 235

Fig. 4. Direct tension apparatus.


size of 0.22 mm (Fig. 3). Two categories of F-75

Ottawa material mixtures were prepared for the

experiments. The first version of the material (F-

75-C) was prepared by washing through a No. 200

sieve to remove particles smaller than 0.075 mm. A

second version (F-75-F) was prepared by adding 2%

fines by weight.

The direct tension experiments were conducted on

specimens having three different relative densities,

Dr (30%, 50%, and 70%), and four different water

contents in the range of 0.5–4% with a newly

developed tension apparatus shown in Fig. 4. This

apparatus was modified from the original version

developed by Perkins (1991). The specimen box

(178� 178� 178 mm) was split in two equal halves.

Inside the box, four triangular wedges were attached

in order to facilitate contact between the specimen

and container as tension is developed across the

plane of separation. Sandpaper was attached to the

sides of the wedges to provide a no-slip condition

between the sides of the wedges and the soil. The

direct tension device rests on a table equipped with

two pulleys, which connect thin wires to two grav-

ity-loading containers.To set up a homogeneous specimen in box (see

Fig. 4), soil and water were first thoroughly mixed.

Distilled water was used to avoid introducing other

physicochemical factors during mixing. Individual

specimens were prepared in four lifts within the

specimen container box by compaction with a drop

hammer furnished with an angular foot, which facil-

itates compaction in corner regions.

The tensile load on the sample was slowly and

steadily applied by introducing water into the front

loading container at a rate of about 170 g/min or

about 0.03 N/s until failure occurred. The error in

load-mass measurement was F 0.01 g. The tensile

strength was directly calculated by dividing the

failure load by the separated area. Immediately after

completion of each test, the density and moisture

content of the entire specimen were measured.

Brittle failure occurred at very small displacements

( < 0.1 mm) in all experiments. The details of this

experimental work included apparatus, material, and

procedure can be found in Kim and Stein (2002).

The results of the experiments performed on F-75

sand are shown in Table 1. The data clearly shows

that the magnitudes of the tensile strengths of moist

granular soil are not equal to zero, as is widely as-

sumed.

4. Experimental modeling of tensile strength

Based on test results, an experimental model that

can predict tensile strength was developed as a function

of water content, relative density, and fines. The goal of

this work is to devise a simple model that can be used

by practicing engineers at especially low water con-

tents (w < 4%).

Table 1

Summary of direct tension test results

Test w (%) S (%) Dr (%) rt (Pa)

(a) F-75-C granular soil

Loose 1 0.46 1.73 32 409.68

Loose 2 1.01 3.77 30 580.67

Loose 3 1.07 3.96 28 586.11

Loose 4 2.13 7.85 27 704.93

Loose 5 4.04 14.83 26 873.03

Loose 6 4.02 14.89 28 850.64

Medium 1 0.46 1.91 52 473.35

Medium 2 1.01 4.17 51 623.86

Medium 3 2.05 8.37 49 886.48

Medium 4 2.08 8.46 48 856.53

Medium 5 4.11 17.04 52 1073.41

Dense 1 0.47 2.15 71 498.52

Dense 2 1.02 4.70 72 730.45

Dense 3 1.04 4.74 70 732.94

Dense 4 2.05 9.24 68 981.97

Dense 5 3.89 17.53 68 1164.45

Dense 6 4.06 18.00 65 1150.84

(b) F-75-F granular soil

Loose 1 0.47 1.76 30 425.51

Loose 2 1.02 3.79 29 608.71

Loose 3 2.05 7.56 27 811.37

Loose 4 2.00 7.41 28 744.02

Loose 5 4.03 14.99 29 951.11

Loose 6 4.06 15.03 28 914.59

Medium 1 0.46 1.91 52 460.17

Medium 2 0.99 4.08 51 681.96

Medium 3 1.00 4.15 52 697.69

Medium 4 2.06 8.54 52 994.31

Medium 5 4.02 16.50 50 1169.23

Dense 1 0.41 1.89 72 524.13

Dense 2 1.01 4.58 69 823.28

Dense 3 2.03 9.16 70 1065.36

Dense 4 2.03 9.25 70 1050.99

Dense 5 4.00 18.12 69 1346.73


To develop a model, first, the data were plotted in

terms of tensile strength along the ordinate and relative

density along the abscissa for different water contents

and were analyzed via a regression analysis by per-

forming a least-squares fit using the natural logarithm

function, which fitted the data well (Fig. 5). Regression

equations were obtained for different water contents as

shown in Table 2. According to these regression

equations, the value of the tensile strength was normal-

ized by dividing it by Prln(Dr), wherePr is the reference

pressure ( = 1 Pa) expressed in the same unit as the

stress and is included to obtain a dimensionless format

of the equation. The normalized tensile strengths were

then recorded versus the water content (Fig. 6). These

data were regressed again by performing a least-

squares fit using the natural logarithm function. The

regressed curves fitted the test data fairly well and the

regression equations were also presented in Fig. 6.

By summing up these results, the tensile strength

equation for F-75 Ottawa granular soil can be

derived as a function of water content, w, relative

density, Dr, and presence of fines.

rt

PrlnðDrÞ¼ 69 lnðwÞ þ 170

ðfor F� 75� CÞ

rt

PrlnðDrÞ¼ 77 lnðwÞ þ 186

ðfor F� 75� F granular soilÞð4Þ

This relationship is important for the prediction of

the tensile strength of unsaturated granular soils

having low moisture content levels in the field.

The measurements of tensile strength using this

relationship are simply made through the use of

physical soil parameters, relative density, and mois-

ture content. As expected, the predicted tensile

strength in Eq. (4) shows good agreement with the

measured data of F-75 granular soils (Fig. 7).

From Figs. 5–7 and Eq. (4), the tensile strength

tends to increase as the water content increases. This

can be explained by considering the capillary bonding

forces between the particles. At low water contents,

Table 2

Regression equations of tensile strength as a function of relative

density

Material Water content,

w (%)

Regression equation

(Pa)

R2

F-75-C 0.5 rt = 113ln(Dr) + 19 0.991

1.0 rt = 159ln(Dr) + 14 0.896

2.0 rt = 298ln(Dr)� 281 0.986

4.0 rt = 355ln(Dr)� 332 0.997

F-75-F 0.5 rt = 107ln(Dr) + 57 0.900

1.0 rt = 226ln(Dr)� 107 0.849

2.0 rt = 286ln(Dr)� 153 0.979

4.0 rt = 436ln(Dr)� 526 0.984

Fig. 5. Results of direct tension testing: tensile strength versus relative density for different water content.


water bridges form at the particle–particle contact

points. As the water content increases, the water

bridges become more developed and expanded in the

contact geometries, and the tensile strength increases.

Higher relative densities also lead to more contacts

between soil particles, increasing the number of water

bridges, and thus causing higher measured tensile

strengths; this phenomenon becomes more pronounced

as the moisture content increases. At low water content

levels, the effects of density on the tensile strength are

not observed. For instance, the tensile strength at the

water content of 0.5% is almost the same value regard-

less of densities. The effect of fines content on tensile

strength is clearly shown in Fig. 5, which dictates that,

for a given water content and relative density, the ten-

sile strength of the sand containing fines is higher than

that of the clean sand, likely due to the increased num-

ber of contacts caused by the fines, as well as the coat-

ing of larger particles by finer particles. Furthermore,

capillary phenomena in a sand containing fines may

become more pronounced, because fines, with their

higher specific surface areas, can retain more water and

produce higher suction or capillary potential. Thus,

fines can increase the degree of bonding action within

the soil structure, resulting in higher tensile strengths.

Fig. 5 also shows that the fines effects are greatly

influenced by the water content. The variation of the

sand tensile strength with fines at the water content of

0.5% is almost identical to that observed for the clean

sand, i.e., no fines effects are observed at this level of

water content. As the water content increases, the effect

of fines on the tensile strength clearly appears and it is

more pronounced as the relative density increases.

5. Modeling of tensile strength utilizing Rumpf’s

theory

5.1. Tensile strength model for monosized sphere

Rumpf (1961) proposed the theory of tensile

strength (Eq. (5)) at the pendular state based on the

following assumptions. First, all the particles are

monosized spheres statistically (on the average uni-

formly) distributed in the agglomerate. Second, the

bonds are statistically distributed across the surface

and over the various spatial directions. Finally, the

effective bonding forces are distributed about a mean

value that can be used in the calculation.

rt ¼1� n

n

Ft

d2ð5Þ

where n is the porosity, d is the mean particle

diameter, and Ft is the bonding force between two

single particles. This theory can be applied for

bonding forces such as capillary, electromagnetic,

Fig. 6. Relationship between normalized tensile strength and water content (w< 4%).


and electrostatic forces. For capillary forces, Schu-

bert (1984) and Pierrat and Caram (1997) determined

theoretically the magnitude of the bonding forces, Ft,

in the pendular state. This total bonding force, Ft, is

the sum of the two components Fs and Fc as

explained in Section 2.

Pierrat and Caram (1997) used Rumpf’s model to

predict the tensile strength of glass beads and found

that the trend of predicted values simulates well the

experimental data with a sharp increase of the tensile

strength at low saturation and a plateau value for

higher saturation levels, though the model overpre-

dicts the experimental data. They also discussed the

reasons for the overprediction of Rumpf’s model. The

first reason is that, since Rumpf only considers

monosized particles (uniformly packed), the tensile

strength of the unevenly packed material is much

smaller than that expressed by the model. Therefore,

the first assumption reflected in Rumpf’s theory may

be an oversimplification and the error can increase for

Fig. 7. Comparison the measured and predicted tensile strength by experimental model (w < 4%).


particles having different diameters. Overprediction

by Rumpf’s model is also contributed by the second

assumption, which involves bonding forces distrib-

uted uniformly over all the directions, which corre-

sponds to an isotropic stress in the granular assembly.

However, a uniaxial stress condition is usually

encountered during tensile strength experiments. For

most granular particles, the uniaxial tensile stress has

been found to be 1.5 times smaller than the isotropic

tensile stress so that the values predicted by Rumpf’s

model overpredict the experimental data.

5.2. Application of Rumpf’s model for granular soil

5.2.1. Soil water characteristic curves (SWCC)

In order to apply the Rumpf’s model proposed to

predict the tensile strength for monosized spheres such

as beads at low water contents (in the pendular state),

first, the saturation state of F-75 Ottawa granular soil

used for the direct tension tests should be defined. A

suction-saturation test was thus conducted on F-75-C

with medium relative density (Dr = 50%) by using the

technique developed by Znidarcic and his students at

the University of Colorado at Boulder (Znidarcic et al.,

1991; Hwang, 2002). It is a faster and simpler way of

determining the soil water characteristic curves. The

equipment consists of a conventional triaxial cell, a

differential pressure transducer, the ceramic porous

plate, the flow pump, and the data acquisition system.

The height of the sand specimen is 30 mm with a

diameter of 71 mm. Fig. 8 shows the test result, and the

pendular state is clearly defined and easily captured as

residual saturation, Sr, in the SWCC for both wetting

and drying curves. Sr corresponds to saturation levels at

about 18% as seen in Fig. 8. Since the 4%water content

actually corresponds to saturation levels of 15%, 17%,

and 18% for each of the relative densities, respective to

increasing, Dr, Rumpf’s model can be used for predict-

ing the tensile strength of F-75 granular soil.

5.2.2. Comparison between Rumpf’s model and

experimental results

The experimental data of F-75-C with medium

relative density were only compared for the sake of

convenience. To compute the model’s values, the

cubical-tetrahedral packing configuration (mean coor-

dination number, k= 8 and e = 0.65) was selected which

is a reasonable assumption because the void ratio tested

medium specimens in the experiment is at around 0.65.

The average particle size of d = 0.22 mm was used in

the grain size distribution curve (Fig. 3) and the contact

angle (d) was assumed to be zero. In practice, no

absolutely smooth particles are found so that the case

a/d = 0 seldom occurs in a real granular material. Pierrat

Fig. 8. Soil water characteristic curve of F-75-C medium granular soil.


and Caram (1997) suggested that values of a/d in the

range of 0.005–0.05 are more appropriate instead. This

assumption can be assured by an electron micro-

scope scan image of clean F-75 Ottawa granular soil

presented in Fig. 9, which shows that, even though

this granular material is clean, it has numerous

cracks, ridges, and fissures on its surfaces. Thus, it

does not make sense to use a/d = 0 in the computa-

tion.

The advantage of Rumpf’s model is that the tensile

strength can be predicted by simply changing fitting

parameters such as the dimensionless separation dis-

tance a/d (d is the diameter of particles and a is the

separation distance between particles in Fig. 2). The

several values of a/d were used for drawing curves

representing the predicted values calculated from Eq.

(5) as shown in Fig. 10.

The measured tensile strength steadily increases as

the saturation levels increases. However, the predicted

tensile strength for a/d = 0.005 sharply increases at low

saturation levels and a plateau value for saturation

greater than 2.5%. That is, the model overpredicts at

low saturation levels and does not simulate the pattern

of the data. For another separation distance, a/

d = 0.015, the curve does not really fit the data. The

curve still shows a sharp increase of the tensile strength

at low saturation levels and a plateau value for higher

saturation levels. This trend was observed for the

tensile strength experiments using glass beads (Pierrat

and Caram, 1997).

Thus, another attempt using a larger separation

distance a/d = 0.025 was conducted (Fig. 10). The

curve moves downward especially at low saturation

levels and makes it lie closer to the experimental trend

(nonlinear behavior); though at higher saturations, the

curve somewhat underpredicts the experimental data.

The nonlinear behavior is appropriate for granular soils

Fig. 9. Electron microscope scan image of F-75 Ottawa granular

soil.

Fig. 10. Comparison of tensile strength between measured and predicted by theoretical Rumpf’s model.


since it shows a steady increase of the tensile strength

as saturation level increases up to 17% (w = 4%). It

would be expected that the tensile strength would

eventually decay to near zero as the water content

approached saturation, but data were shown only to a

water content of 4% for this study. The tensile strength

of granular soil can be appropriately simulated with this

model by changing a fitting parameter, the dimension-

less separation distance a/d.

6. Summary and conclusions

This study was carried out to explore the tensile

strength models in quartz granular material at espe-

cially lowwater content levels (w < 4%). The following

conclusions can be drawn.

(1) The magnitudes of the tensile strengths of granular

soils are not equal to zero. The tensile strength

generally increases with increasing water content

and relative density. The presence of fines also

results in higher tensile strengths. The influences

of relative density and fines on the tensile strength

are substantially dependent on the water content.

(2) An experimental model was proposed for the

estimation of the tensile strength of an unsaturated

granular soil. The identification of tensile strength

is simply made through the use of physical soil

parameters, relative density and water content.

This simple model can be used by practicing

engineers on approximate estimation of the tensile

strength of an unsaturated granular material with-

out experiments.

(3) The experimental data were compared with the

values predicted by a theoretical model developed

for monosized spheres, such as glass beads at the

pendular state, and its application was discussed

for real granular soils. While the experimental

data for glass beads show a sharp increase of the

tensile strength at low saturation levels and a

plateau value for higher saturation levels, the

measured tensile strength of F-75 sand steadily

increases as the saturation levels increases (non-

linear behavior). The nonlinear behavior of gra-

nular soils is appropriately simulated with this

model by changing a fitting parameter, the

dimensionless separation distance a/d.

Acknowledgements

The authors would like to acknowledge the

assistance of Dr. J. Ledlie Klosky, Associate Professor

at the United States Military Academy in performing

the direct tension experiments. The authors also wish to

acknowledge the assistance of Ms. Eleanor Nothelfer,

Departmental Library Coordinator at Lehigh Univer-

sity.

References

Al-Hussaini, M.M., Townsend, F.C., 1973. Tensile Testing of Soils

a Literature Review U.S. Army Engineer Waterways Experi-

ment Station, Vicksburg, Mississippi, USA.

Biarez, J., Saı̈m, R., Taı̈bi, S., Liu, H., Richane, B., Correia, A.G.,

1997. Analogy between the mechanical behavior of sand and

remoulded clays. Proceedings the XIVth International Confer-

ence on Soil Mechanics and Foundation Engineering, Hamburg,

Germany., 251–254.

Biarez, J., Saı̈m, R., Taı̈bi, S., Liu, H., Richane, B., Correia, A.G.,

1997. Analogy between the mechanical behavior of sand and

remoulded clays. Proceedings the XIVth International Confer-

ence on Soil Mechanics and Foundation Engineering, Hamburg,

Germany, pp. 251–254.

Fredlund, D.G., Morgenstern, N.R., Widger, D.A., 1978. The shear

strength of unsaturated soils. Canadian Geotechnical Journal 15,

261–276.

Fredlund, D.G., Xing, A., Huang, S., 1994. Predicting the perme-

ability function for unsaturated soils using the soil water char-

acteristic curve. Canadian Geotechnical Journal 31, 533–546.

Fredlund, D.G., Xing, A., Fredlund, M.D., BarBour, S.L., 1996.

The relationship of the unsaturated soil shear strength to the soil

water characteristic curve. Canadian Geotechnical Journal 32,

440–448.

Kim, T.-H., Stein, S., 2002. Moisture induced tensile strength in

sand. 6th International Symposium on Environmental Geotech-

nology and Global Sustainable Development, Seoul, Korea,

pp. 731–736.

Hwang, C., 2002. Determination of material functions for unsatu-

rated flow. Ph.D. dissertation, University of Colorado at Boulder.

Perkins, S.W., 1991. Modeling of regolith structure interaction in

extraterrestrial constructed facilities. Ph.D. dissertation, Univer-

sity of Colorado at Boulder.

Pierrat, P., Caram, H.S., 1997. Tensile strength of wet granular

materials. Powder Technology 91, 83–93.

Rahardjo, H., Fredlund, D.G., 1995. Pore pressure and volume

change behavior during undrained and drained loadings of an

unsaturated soil. In: Alonso, Delages (Eds.), Proceedings of the

1st International Conference on Unsaturated Soils, pp. 165–170.

Rao, S.M., 1996. Role of apparent cohesion in the stability of

dominican allophane soil slopes. Engineering Geology 43,

265–279.


Rohm, S.A., Vilar, O.M., 1995. Shear strength of unsaturated sandy

soil. In: Alonso, Delages (Eds.), Proceedings of the 1st Interna-

tional Conference on Unsaturated Soils, pp. 189–193.

Rumpf, H., 1961. The strength of granules and agglomerates. In:

Knepper, W.A. (Ed.), Agglomeration. Knepper, Interscience,

New York, pp. 379–418.

Schubert, H., 1984. Capillary forces-modeling and application in

particulate technology. Powder Technology 37, 105–116.

Schubert, H., Herrmann, W., Rumpf, H., 1975. Deformation behav-

ior of agglomerates under tensile stress. Powder Technology 11,

121–131.

Schubert, H., 1984. Capillary forces-modeling and application in

particulate technology. Powder Technology 37, 105–116.

Schubert, H., Herrmann, W., Rumpf, H., 1975. Deformation behav-

ior of agglomerates under tensile stress. Powder Technology 11,

121–131.

Yong, R.N., Townsend, F.C. (Eds.), 1981. Symp. Laboratory Shear

Strength of Soil. ASTM STP, vol. 740, p. 717.

Znidarcic, D., Illangasekare, T., Manna, M., 1991. Laboratory test-

ing and parameter estimation for two-phase flow problems.

ASCE Geotechnical Special Publications 27, 1089–1099.


Documents

Modeling of tensile strength on moist granular earth material at low water content