Anisotropy-Based Failure Criterion for Interphase Systems
Jianfeng Wang1; Joseph E. Dove, P.E., M.ASCE2; and Marte S. Gutierrez3
Abstract: This paper presents a methodology for estimating the shear strength of interphase systems composed of granular materials and
planar inclusions having various degrees of roughness. Existing empirical and semiempirical relationships between strength and surface
roughness do not appear to be general and are unable to account for surface-particle interactions at the appropriate scales. The proposed
method is based on the contact force anisotropy of those particles that touch the inclusion surface. It was developed using two-dimensional
discrete element method simulations of interphase systems constructed within a direct interface shear test device. Particles consist of
polydisperse and monodisperse spheres of constant median grain diameter. Surface roughness was varied by using profiles with regular
and random asperities, and profiles of manufactured surfaces. Results indicate that the magnitude and direction of average contact total
force at the interface controls strength. A bilinear relationship, independent of particle to surface friction coefficient, exists between the
principal direction of contact total force anisotropy and strength. Results using the proposed criterion are in good agreement with
laboratory results using spheres and subrounded sand.
DOI: 10.1061/共ASCE兲1090-0241共2007兲133:5共599兲
CE Database subject headings: Anisotropy; Interfaces; Failures; Discrete elements; Interfaces.
Introduction
An interphase within a geotechnical composite system is a region
that consists of an inclusion surface, with its asperities, and a
variable thickness of granular material directly adjacent to the
surface. The interface is the boundary between the surface, with
its asperities, and the particles touching the surface. For the purpose of this paper, the inclusion is sufficiently rough to engage the
particles at the interface but it is planar at greater length scales
due to small surface waviness 共ISO 4287 1997兲. The thickness of
the granular portion of the interphase is the thickness of the shear
band, which is found through experimental results 共Westgate and
DeJong 2006兲 and numerical simulations 共Wang et al. 2005a兲 to
extend approximately 6–14 median particle diameters from the
interface, depending on the relative grain to surface geometry.
Shear displacement of the continuous material with respect to the
particles is resisted by the shear strength developed by the granular material within the interphase. The interphase governs the
overall behavior of a composite system and the ability to quantify
this behavior is of considerable significance to civil engineering.
1
Postdoctoral Associate, Dept. of Civil and Environmental
Engineering, 200 Patton Hall, Virginia Tech, Blacksburg, VA 24061.
E-mail: [email protected]
2
Research Assistant Professor, Dept. of Civil and Environmental
Engineering, 200 Patton Hall, Virginia Tech, Blacksburg, VA 24061
共corresponding author兲. E-mail: [email protected]
3
Associate Professor, Dept. of Civil and Environmental Engineering,
200 Patton Hall, Virginia Tech, Blacksburg, VA 24061. E-mail:
[email protected]
Note. Discussion open until October 1, 2007. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and possible publication on January 11, 2006; approved on November 6, 2006.
This paper is part of the Journal of Geotechnical and Geoenvironmental
Engineering, Vol. 133, No. 5, May 1, 2007. ©ASCE, ISSN 1090-0241/
2007/5-599–608/$25.00.
The goal of this paper is to present a new interphase strength
failure criterion that is based on anisotropy of contact forces acting at the interface.
Background
A substantial body of research is available in geotechnical engineering whereby the “roughness” of surfaces is related to the
strength behavior of natural and engineered interphase systems
共Patton 1966; Ladanyi and Archambault 1969; Barton and
Choubey 1977; Ladanyi and Archambault 1980; Desai 1981;
Yoshimi and Kishida 1982; Uesugi and Kishida 1986a,b; Kishida
and Uesugi 1987; Uesugi et al. 1988; Paikowsky 1989; Irsyam
and Hryciw 1991; Hryciw and Irsyam 1993; Tejchman and Wu
1995; Dove et al. 1997; Ebeling et al. 1997; Day and Potts 1998;
Dove and Frost 1999; Gomez 2000; Esterhuizen et al. 2001; Dove
and Jarrett 2002兲. Despite these important contributions, general
predictive relationships that account for granular material and inclusion surface properties remain elusive. This is less of a problem for fully dilative interface systems 共Dove and Jarrett 2002兲
that result in a high degree of granular material strength mobilization. However, it is a considerable problem for the majority of
interphase systems that mobilize the strength of the soil somewhere in between a nondilative and a fully dilative system. This is
because mechanisms responsible for interphase strength develop
at the particle to surface contacts. Statistical parameters, singlevalue parameters, fractals, and Fourier series approximations used
in surface characterization result in homogenization of the profile.
Normalized parameters that account for the relative grain to surface geometry 共Yoshimi and Kishida 1982; Kishida and Uesugi
1987兲 are useful for some surfaces. However, these and other
parameters are insufficiently robust to capture the full complexity
of relative geometry.
Discrete simulations and experimental micromechanics provide realistic insight into fundamental mechanisms of interphase
behavior but the number of studies is limited. Paikowsky and Xi
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2007 / 599
With strength data from shear box simulations it is possible to
express the results in terms of interface efficiency, as described
later.
Granular Material
Granular materials included both monodisperse 共uniform兲 and
polydisperse 共well-graded兲 particle size distributions. Particles
were modeled as spheres for calculation of contact stress. The
maximum and minimum particle diameters 共Dmax, Dmin兲 were
0.735 and 0.665 mm, respectively, for the monodisperse material,
and 1.05 and 0.35 mm, respectively, for the polydisperse material.
The median particle diameter 共D50兲 was held constant at 0.7 mm
so that the influence of gradation could be isolated.
Parametric Study
Fig. 1. Surface profiles used in numerical simulations: 共a兲 Groups
1–3; 共b兲 Group 4; and 共c兲 example of measured profile from real
surface of Group 5
共1997兲 performed an extensive laboratory study using photoelastic methods coupled with discrete modeling of micro- and macromechanical interface behavior. Their study contains valuable
fundamental experimental and numerical simulation results. Numerical simulations by Jensen et al. 共1999兲, Claquin and Emeriault 共2001兲, and Frost et al. 共2002兲 provide insight into surface
geometry and hardness effects on strength and stiffness.
This paper presents a failure criterion that relates the principal
direction of surface normal distribution, ␪a, to interphase strength.
The criterion is rooted in the fundamental source of particulatesolid interphase strength; the grains in direct contact with the
inclusion surface. It implicitly accounts for the inclusion surface
roughness.
Experimental Methods
A parametric numerical study was conducted using the twodimensional discrete element program PFC2D, Version 3.0 共Itasca
Consultants, Inc.兲. A direct interface shear device model was developed 共Wang et al. 2005a,b兲 and validated against laboratory
data 共Dove and Jarrett 2002; Johnson 2000兲.
A direct shear box model was also developed to determine the
stress-displacement behavior and peak strength of the same
granular material 共Wang 2006兲 used in the interphase simulations.
Five groups of numerical simulations were conducted at 100 kPa
normal stress. Two-dimensional surface profiles were created analytically and directly measured on construction materials using a
stylus profilometer 共Johnson 2000兲. Profiles with triangular asperities of equal size were used in Groups 1, 2, and 3 with asperity height 共Rt兲 asperity width 共Sw兲, and spacing between asperities
共Sr兲 varied within each group, respectively 关Fig. 1共a兲兴. Table 1
provides the ranges of variables and constant values in these experiments. All asperity sizes are normalized with respect to D50.
Irregular asperities were randomly generated in Group 4 to
investigate how profile complexity influences interphase behavior.
These profiles are described by the parameters asperity unit 共Au兲,
asperity ratio 共Ar兲, and overall maximum asperity height 共Rtmax兲,
as shown on Fig. 1共b兲. Au = unit length of the projection of any
linear asperity segment in the horizontal direction. Ar = ratio of the
unit length of a projection of any linear asperity segment in the
vertical direction to that in the horizontal direction. Rtmax
= prescribed maximum elevation difference between the peak and
the valley over the whole surface. Au and Rtmax are normalized
with respect to D50. A random number multiplier is applied to the
unit segment length in both horizontal and vertical directions
using the values of Au, and Ar as starting values. Control of profile
geometry is through assigning different values to the abovementioned three parameters. Parameter values used in the seven
numerical experiments of Group 4 are provided in Table 2.
A stylus profilometer was used to record surface profiles of
natural and manufactured surfaces 共Group 5兲 that included three
coextruded geomembranes, wood, stone, and concrete. The profile horizontal data point spacing was 1 ␮m. These profiles have a
richer frequency content than those in Groups 1–4 due to the
more complex texture patterns 关Fig. 1共c兲兴.
Global Model Parameters
The Hertz-Mindlin contact model was implemented in all simulations. A particle rolling resistance model 共Wang et al. 2004兲 was
Table 1. Simulation Parametric Study for Groups 1–3
Constant valuesa
Group
Variable
1
Rt / D50
2
Sr / D50
3
Sw / D50
a
Normal stress= 100 kPa in all simulations.
Value of variable
Rt / D50
Sr / D50
Sw / D50
0.1, 0.2, 0.5, 1, 1.2, 1.5, 2, 3, 5
0.2, 0.5, 1, 2, 3, 5, 8, 10
0.2, 0.5, 1, 2, 3, 4, 5, 10
Varied
1
1
0
Varied
0
2
2
Varied
600 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2007
Table 2. Values of Au, Ar, and Rt max Used in Experimental Group 4
Numerical experiment
Variable
A
B
C
D
E
F
G
Au / D50
Ar
Rt max / D50
1
1
2
2
1
2
2
0.5
2
4
2
4
2
0.25
1
4
0.25
2
4
4
4
also applied at both particle-particle contacts and particleboundary contacts. Physical constants used in the simulations include: Particle density 2,650 kg/ m3, shear modulus 29 GPa, and
Poisson’s ratio of the particles 0.3. Critical normal and shear viscous damping coefficients were set equal to 1.0. A time step 5.0
⫻ 10−5 s was used.
The shear modulus used in this study was determined experimentally using results of microindentation tests on soda lime
glass beads 共Dove et al. 2006兲. Both particles and surfaces were
assumed to be rigid in this study. PFC2D simulates particle deformation by overlapping particles and surfaces. Therefore
changes in particle shear modulus will not alter the basic mechanism and failure criterion described herein.
The interparticle friction coefficient was 0.5. Three particle to
lower surface friction coefficients 共tan ␾␮兲 of 0.05, 0.2, and 0.5
were used. These represent the friction coefficients of a single
particle sliding against solid surfaces. The above-mentioned values span the range experimentally determined from interface
shear tests using a pin-on-disk tribometer 共Dove et al. 2006兲.
Experimental values ranged from 0.10 for glass beads on polyvinylchloride sheet to 0.54 for subrounded sand on untextured high
density polyethylene 共HDPE兲 geomembrane. A particle to boundary friction of 0.9 was applied on the top wall and two lateral
walls, which is different from the particle to lower surface friction
coefficient above. This prevented sliding of the particles against
the boundary walls and proved essential for obtaining the correct
interface stiffness behavior 共Wang 2006兲.
f̄ s共␪兲 = corresponding density distribution functions; f kn and f sk
= contact normal force and shear force, respectively;
n = 共cos ␪ , sin ␪兲 = unit contact normal vector; t = 共−sin ␪ , cos ␪兲
= vector perpendicular to n; and Nc = total number of contacts in
the volume. f̄ 0⫽average contact normal force calculated by
1
f̄ 0 =
2␲
冕
2␲
0
Nc
1
f̄ n共␪兲d␪ =
fk
Nc k=1 n
兺
共3兲
In addition, Wang et al. 共2007兲 defined the average contact
total force tensor Rij as
Representation of Discrete Data
The analysis consists of using discrete data to compute continuum
quantities such as fabric and contact force anisotropy within the
assemblage 共Wang 2006兲. Anisotropy refers to the deviation of
contact and contact force orientation from the isotropic state. This
methodology is briefly reviewed in the following sections.
Second-Order Tensors of Contact Force Anisotropies
The two-dimensional, second-order density distribution tensors
共Rothenburg 1980兲 were used to describe the anisotropy of contact forces in both the granular assemblage and for the particles
touching the surface. These tensors were computed from the discrete simulation data and are defined as follows:
Nij =
1
2␲
冕
1
Sij =
2␲
2␲
f̄ 0
0
冕
f̄ n共␪兲
2␲
0
f̄ s共␪兲
f̄ 0
N
nin jd␪ =
c
f kn k k
1
nn
Nc k=1 f̄ i j
0
兺
共1兲
N
c
f sk k k
1
tin jd␪ =
tn
Nc k=1 f̄ i j
0
兺
共2兲
where Nij and Sij = average contact normal force and average
contact shear force tensors, respectively; f̄ n共␪兲 and
Fig. 2. Contact force anisotropies and their principal directions at the
particle–surface boundary: 共a兲 contact normal force; 共b兲 contact total
force. Shaded region—computed distribution, unshaded region—
Fourier series approximation.
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2007 / 601
Fig. 3. Determination of ␪a from median particle centroid trace profile: 共a兲 original surface profile and median particle centroid trace
profile; 共b兲 contact normal distribution. Shaded region—computed
distribution, unshaded region—Fourier series approximation.
1
Rij =
2␲
冕
2␲
f̄ r共␪兲
0
f̄ r0
N
c
f rk k k
1
mim jd␪ =
mi m j
Nc k=1 f̄
r0
兺
共4兲
where f̄ r共␪兲 = average contact total force density distribution function; f r = contact total force; and m = 共cos ␣ , sin ␣兲 = unit vector in
the direction of contact total force. f̄ r0 = average contact total force
calculated from
1
f̄ r0 =
2␲
冕
2␲
0
Fig. 4. Contact force network inside the simulated interface shear
device: 共a兲 Group 1, ⌬ = 45°, Rt / D50 = 1.0; 共b兲 coextruded geomembrane
Nc
1
f̄ r共␪兲d␪ =
fk
Nc k=1 r
兺
共5兲
Fourier Series Approximations
Second-order Fourier series expressions for f̄ n共␪兲, f̄ s共␪兲, and f̄ r共␪兲
were used to approximate the density distributions and compute
the principal directions of anisotropy. The following were proposed by Bathurst and Rothenburg 共1990兲 and Wang et al.
共2007兲:
f̄ n共␪兲 = f̄ 0关1 + an cos 2共␪ − ␪n兲兴
共6兲
f̄ s共␪兲 = f̄ 0关− as sin 2共␪ − ␪s兲兴
共7兲
f̄ r共␪兲 = f̄ r0关1 + ar cos 2共␪ − ␪r兲兴
共8兲
In Eqs. 共6兲–共8兲 an, as, and ar = coefficients of contact normal
force, contact shear force and contact total force anisotropies,
respectively; and ␪n, ␪s, and ␪r = corresponding principal directions of these contact force anisotropies. These anisotropy parameters are invariant quantities of the second-order tensors defined
in Eqs. 共1兲, 共2兲, and 共4兲. Particularly, an, as, and ar are related to
the eigenvalues of tensor Nij, Sij, and Rij, respectively; and the
cosine and sine of ␪n, ␪s, and ␪r⫽eigenvectors of these tensors.
An example of typical particle contact force anisotropies and their
principal directions are shown in Fig. 2.
Physically, ␪n reflects the statistics of the relative particle to
surface geometry and ␪r accounts for both surface roughness and
friction. They are expressed as angular deviations of the principal
directions from vertical 共Wang et al. 2007兲. ␪r and ␪n are not
direct measures of the inclusion surface roughness but result from
actual mechanical interaction between the surface asperities and
granular media.
Determination of ␪a Based on Surface Normal
Distributions
The principal direction of the surface normal distribution, ␪a, is
determined by the relative grain to surface geometry and is
closely related to ␪r and ␪n. In this study, the relative geometry
between grains and surface was incorporated into the calculation
of ␪a by using a particle centroid trace surface profile 共DeJong
et al. 2002兲 obtained using a particle with median diameter of
0.7 mm 关Fig. 3共a兲兴. An example of a surface normal distribution
and its principal direction is shown in Fig. 3共b兲 for a sawtooth
surface 关Fig. 1共a兲兴 with Sr = 0. The distribution is populated only
in the upper hemisphere.
Relative motion between the grains and the surface during
shear displacement results in contacts along the interface located
predominately on portions of the asperities that oppose shear. Accordingly, the portion of the surface normal distribution located in
the first quadrant with respect to the surface is neglected 关Fig.
3共b兲兴. The principal direction, ␪a, is based on the Fourier series
approximation to the normals occurring in the second quadrant of
the surface only.
Results
Mechanism for Shear Strength Mobilized
inside the Interphase
An understanding of the mechanism by which contact forces are
transmitted from the interface into the soil mass is required before
a failure criterion can be developed. Fig. 4 illustrates examples of
contact force networks recorded at peak stress state from a surface with repeating triangular asperities and one coextruded
HDPE geomembrane surface. The individual figures have slightly
different scales therefore thicknesses of force chains are not com-
602 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2007
Fig. 6. Physical meaning of ␪r: 共a兲 ␪r measured at the interface is the
resultant angle of the sum of the normal and shear forces acting on
the asperities and is approximately equal to ␾ds at the interface;
共b兲 when measured within the interphase, ␪r corresponds to the
orientation of the major principal stress. ␪r is not equal to ␾ds in the
interphase.
Fig. 5. Relation between peak stress ratio, ␪n and ␪r: 共a兲, 共b兲 measurements made at the surface; and 共c兲, 共d兲 measurements made in
sampling region
parable. From these plots it is possible to visualize how the orientation of major force chains changes as they extend into the
matrix soil from the surface. Fig. 4共a兲 is a profile with triangular
asperities of 45° slope. The contact force chains are perpendicular
to the asperity surface and then curve toward horizontal only
slightly as they extend into the interphase region. These extended
force chains engage a large number of adjacent particles that force
shear failure to occur inside the interphase soil. In a continuum
sense, the surface is able to develop anisotropy of fabric and
contact forces inside the interphase region that are similar to those
occurring inside the granular media subject to direct shear 共Wang
2006兲. The shear band is fairly thick and the stress ratio is high
for this combination of relative particle to surface roughness.
These observations agree with the experimental results of Majmudar and Behringer 共2005兲.
From Fig. 4共b兲 it is seen that force chains preferentially
develop on the larger asperities, or asperities that are spaced
wider than the grain diameter. Many of the contact orientations
are vertical, or nearly so. Force chains curve and become closer to
horizontal as they extend into the interphase region. This is because the force chains at the surface are too steep to involve a
large number of particles above the surface in intense shearing;
therefore horizontal forces applied during shear influence their
direction. Smaller degrees of fabric and contact force anisotropies
are developed in the interphase region. Particle sliding also occurs
along the surface, leading to a very thin shear zone just above the
surface.
The above-described mechanism is illustrated by sampling two
different regions within the simulated interface shear box and
computing the corresponding stress ratio. Fig. 5 shows the variation in average stress ratio measured inside the interphase region
from all simulations using polydisperse material with ␪n and ␪r
measured at the surface asperities at peak stress state. The “average stress ratio” is computed using the average stress tensor
共Rothenburg and Selvadurai 1981; Christoffersen et al. 1981兲
within a rectangle that extends into the interphase a distance of
14 mm from the interface, with length corresponding to the length
of the rough surface. In contrast, “stress ratio” is the ratio of the
sum of horizontal and vertical components of contact forces that
arise from the first row of particles touching the asperities. This
stress ratio is equal to the tangent of the direct shear friction
angle, ␾ds, at the surface 关Fig. 6共a兲兴. Moreover, ␪r measured at the
surface is the principal angle of the distribution of all contact total
forces acting on the asperities. The magnitude of ␪r is approximately equal to ␾ds at the surface. However, within the interphase, ␪r corresponds to the orientation of the major principal
stress and is not equal to ␾ds 关Fig. 6共b兲兴.
The bilinear relationships 共Fig. 5兲 developed are observed to
be similar in form between the two sampling locations. The magnitude of the average stress ratio measured in the 14 mm high
sampling region near the interface is slightly lower than the stress
ratio computed from force summation at the surface asperities.
This is because stresses developed at the surface diminish in the
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2007 / 603
Fig. 7. Density distributions and principal directions of normals to
the inclusion surface based on particle centroid trace profile: 共a兲
Group 1, Rt / D50 = 1.0; 共b兲 Group 5, geomembrane. Quadrant locations are given in the inset. Shaded region—computed distribution,
unshaded region—Fourier series approximation.
interphase region away from the surface. It must be noted that
inside the interphase, stress ratio is determined by the combined
effects of fabric and contact force anisotropies 共Wang et al.
2007兲. It is also affected by the horizontal stress ␴xx, which is not
applicable when summing forces at the asperities. Within the
granular material, both ␪n and ␪r coincide with the principal stress
direction throughout the shear process 关Fig. 6共b兲兴.
These results provide the basis for understanding the mechanism of force transmission resulting from particle contacts at the
interface and within the interphase region and serve as the basis
for the failure criterion. Fig. 5 indicates that this force transmission mechanism, and its effect on mobilized shear strength, can be
quantified using the principal directions of the contact force distributions computed over the profile.
Failure Criterion for Interphase Systems
A failure criterion relating the interphase strength to easily measured properties of granular material and inclusion surfaces can
be established using Fig. 5 and the principle of contact force
anisotropy. The key to developing the failure criteria was to find a
relationship between the anisotropy parameters and some measure
of interphase strength. Unfortunately, ␪n and ␪r can only be derived from DEM simulations; however ␪a can be determined from
a surface profile and a standard grain size distribution analysis. ␪a
can then be correlated to ␪r as described below.
Polar distributions of surface normals and their principal directions for two selected surfaces are shown as the shaded regions
in Fig. 7. Fig. 7共a兲 is from an idealized surface 关Fig. 1共a兲兴 with
Sr = 0 and Fig. 7共b兲 is from a manufactured surface with a high
percentage of horizontal segments which yield vertical normals.
The principal directions were determined using the Fourier approximation to the surface normal distribution in the second quadrant 共unshaded regions兲.
Fig. 8 presents the relationship between peak stress ratio measured at the interface and ␪a for all five simulation groups. Values
of ␪a are referenced counterclockwise from the vertical. In Fig.
8共a兲, ␪a is calculated using all contact normals, including the horizontal segments. Significant deviation of data from Groups 2 and
5 is observed. The cause of the scatter at stress ratios less than the
maximum is the high percentage of vertical normals, as shown in
Fig. 7共b兲. These contacts do not contribute in a significant way to
Fig. 8. Peak stress ratio ␪a relation for well-graded material: 共a兲, 共c兲,
共e兲 all surface normals included; and 共b兲, 共d兲, 共f兲 surface normals less
than 92° filtered from the average
the total strength measured along the surface because particles are
either “trapped” within the asperities or undergo sliding along the
smooth surface.
Exclusion of these horizontal segments of the profile leads to
the correlation shown in Fig. 8共b兲, where ␪a is calculated using
surface normals with normal angle greater than 92° from horizontal. Distinct bilinear relationships are observed with interface
strength ratio in the lower interface friction cases 共tan ␾␮ = 0.05
and 0.2兲. Greatest variation is observed in the high interface friction case 共tan ␾␮ = 0.5兲. From Figs. 8共b, d and f兲, it is observed
that profiles with horizontal spacing between asperities 共Group 2兲
have ␪a of approximately 42° even though the stress ratio varies
slightly. This indicates that surfaces with regular discontinuous
asperity spacing require additional consideration. Such surfaces
can result from machining of metals and forming of concrete. The
threshold value of 92° was selected empirically and was found to
give the most distinct bilinear correlations in Fig. 8共b兲.
Fig. 9 presents correlations between ␪a and ␪r for various particle to inclusion friction cases. Fig. 9共a兲 shows simulation data
over all groups for values of tan ␾␮ of 0.05, 0.2, and 0.5. Except
for Group 2 profiles, a bilinear correlation exists between ␪a and
␪r. It can be seen that when ␪a 艋 40°, the slope of the linear
portion decreases as interface friction coefficient increases. The
same linear equation applies for all interface friction cases when
␪a ⬎ 40° 关Fig. 9共b兲兴. The vertical axis intercept is approximately
equal to ␾␮. This indicates that for a given surface with a known
interface friction coefficient, ␪r can be determined from ␪a.
Fig. 9共a兲 shows that the correlation for Group 2 profiles with
the high proportion of vertical normals is independent of ␪a. Fig.
10 provides correlations based on additional parametric studies
between ␪a and ␪r / ␪a for these surfaces over asperity slopes, ⌬
关Fig. 1共a兲兴, of 22, 33, and 45° 共corresponding to Sw / D50 = 5, 3, and
2, respectively兲. For each given asperity slope, spacing between
asperities was increased to give the variation of peak strength
ratio. The use of these figures is discussed in the next section.
604 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2007
Fig. 9. Charts to determine ␪r from ␪a at peak stress state for wellgraded material: 共a兲 influence of tan ␾␮; and 共b兲 combined chart for
irregular profiles and profiles of construction material surfaces
Fig. 10. Correlations between ␪r from ␪a for surfaces with high
proportion of horizontal segments. Note: ␪a is referenced counterclockwise from the horizontal.
Criterion Based on Efficiency Parameter
Use of the Failure Criterion
Peak efficiency or strength ratio for interphase systems composed
of materials similar to those used herein can be predicted with the
proposed failure criterion. First, ␪a is determined from the profile
data using the centroid trace methodology based on median grain
diameter. A computer program is available from the writers that
determines ␪a using a surface profile and median grain diameter
data as inputs. The program computes the centroid trace profile,
the distribution of surface normals, arithmetic average asperity
slope of asperities, and the value of ␪a for all profiles used in this
study. It is recommended at this time to use unfiltered, centerline
corrected profiles 40 mm in length 共Johnson 2000兲. Filtered profiles can be used depending the surface and project requirements.
For the majority of construction material profiles, ␪a is computed using normals greater than 92°. Then, ␪r is estimated from
the correlation shown in Fig. 9共b兲. However, for those surfaces
that have regular horizontal spacing between asperities 关constant
Sr in Fig. 1共a兲兴, the following procedures are used to determine ␪r.
First, ␪a is computed using all surface normals 共no filtering兲. Then
Fig. 10 is used to determine ␪r using the principal direction ratio
共␪r / ␪a兲. Values of ␪a in Fig. 10 are referenced counterclockwise
from the horizontal. The value of asperity slope used to select a
particular curve in Fig. 10 is the arithmetic average slope of the
individual asperities without the horizontal segments included.
For slope values other than 22, 33, and 45°, interpolation is
needed to calculate the value of ␪r / ␪a.
Peak efficiency 共E p兲 is defined as E p = tan ␦peak / tan ␾peak, where
tan ␦peak = peak effective stress friction coefficient from a direct
interface shear test and tan ␾peak = peak effective stress friction
coefficient of the granular media alone. Fig. 11 shows the bilinear
relationship between peak efficiency and ␪r for all numerical experiments grouped by particle to surface friction coefficient.
Some variation in efficiency values is observed in the region
where ␪r is greater than about 30° from vertical. This variation is
mainly for irregular surfaces, which have higher contact forces
lying in the third quadrant of the particle, opposing shear and
reducing the total shear strength. However these data points represent the maximum efficiency that highly irregular surfaces typically can achieve. Within each group of surfaces, however, the
Fig. 11. Failure criterion in terms of peak interphase efficiency, E p
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2007 / 605
Fig. 12. Comparisons among simulations, failure criterion predictions, and laboratory results: 共a兲 uniform glass beads/aluminum
surface; 共b兲 failure criterion, glass beads/aluminum surface; 共c兲
failure criterion, Ottawa 20/ 30 sand/aluminum surface; and 共d兲 failure criterion, Ottawa 20/ 30 sand/HDPE geomembranes, 300 kPa
normal stress
variation in this region is relatively small. For guidance, it is
observed that most surfaces give efficiencies of approximately
0.9–1.0, or within the mid to upper region of the data band.
Verification of Failure Criterion
Verification of the failure criterion based on ␪a was performed
using data from laboratory direct shear and direct interface shear
tests conducted at 100 kPa normal stress with 0.7 mm glass
beads, 0.5 mm glass beads, and Ottawa 20/ 30 sand. The granular
materials were placed at a relative density of 80%. Idealized aluminum surfaces with triangular asperities and Sr = 0 关Fig. 1共a兲兴
were used in the direct interface shear tests by Dove and Jarrett
共2002兲. Fig. 12共a兲 shows the comparison between simulation and
laboratory data for 0.7 mm glass beads. In Fig. 12共a兲, the measured peak interface strength ratios are plotted against ␪a computed from profiles of the individual surfaces. It can be seen that
the peak strength ratios from laboratory data and simulation data
are in good agreement. The simulation data cover a much broader
range of relative particle to surface geometry than can be accommodated in laboratory equipment, therefore there are no laboratory data for ␪a greater than 40°.
Figs. 12共b and c兲 compare the peak interface efficiency data
for glass beads and Ottawa sand measured in the laboratory with
those estimated using the failure criterion. The surfaces used in
the laboratory tests and simulations have the same relative particle to surface geometry. Figs. 12共b and c兲 show the maximum
difference between the estimated and measured interface friction
angles are less than 4° up to ␪a of about 30°. For 0.7 mm glass
beads and Ottawa 20/ 30 sand, the differences are less than 1°,
and 1 to 2° for 0.5 mm glass beads. The measured interphase
efficiency values in Fig. 12共c兲 for ␪a greater than 30° fall in the
middle of the upper bound range given in Fig. 11.
Test results for glass beads and Ottawa sand on smooth aluminum surfaces 共␪a = 0兲 are also shown in Figs. 12共b and c兲. Smooth
surfaces have only particle to surface friction. It can be seen that
efficiencies based on particle to surface friction coefficients of 0.1
for glass beads and 0.2 for Ottawa sand in the failure criterion
are, in general, only slightly lower than the laboratory peak
efficiencies.
Fig. 12共d兲 presents laboratory and failure criterion results for
Ottawa 20/ 30 sand in contact with HDPE smooth and textured
geomembranes. The laboratory tests were conducted at a normal
stress of 300 kPa and relative density of 80%. Estimated interphase efficiency values for the textured materials are based on a
particle to surface friction coefficient of about 0.5. This value is
realistic for manufactured textured surfaces because the richer
frequency content of the roughness will result in engagement of
the particles over a wide range of length scales. A particle to
surface friction coefficient of 0.3 was used for the smooth surface
共␪a = 0兲. Strength estimates for these irregular surfaces are in good
agreement with the laboratory data.
For smooth surfaces and nonspherical particles, it is recommended that a particle to surface friction coefficient in the range
of 0.2–0.3 be used for estimating peak efficiency. These values
are approximately equal to the particle to surface friction coefficients just after initiation of particle sliding as determined by
Dove et al. 共2006兲. In general, a higher value is representative of
rougher particles against softer surfaces and a lower value would
be expected for smoother, rounder particles against harder surfaces. For smooth surfaces and spherical particles, a value of 0.1
is suggested.
Conclusions
This paper addressed the effects of relative particle to surface
geometry on the strength behavior of an interphase system composed of spherical particles in contact with natural and manufactured inclusion surfaces. Results indicate the magnitude and
direction of the average contact total force controls interface
strength behavior. A bilinear relationship independent of particlesurface friction coefficient exists between ␪r and the interphase
strength. A similar relationship holds for ␪n that is a function of
particle-surface friction coefficient. Full mobilization of the
granular material strength occurs when ␪r exceeds approximately
30°.
A failure criterion is presented based on ␪a determined from
median particle diameter centroid trace profiles. Good agreement
with the results of laboratory interface shear test data is observed
for rounded and subrounded particles. For the extreme case of
profiles with constant Sr, additional correlations are provided that
extend the range of surfaces that can be used.
606 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2007
Acknowledgments
This material is based upon work supported by the National
Science Foundation under Grant No. CMS-0200949. Any opinions, findings, conclusions or recommendations expressed in this
material are those of the writers and do not necessarily reflect the
views of the National Science Foundation.
Notation
The following symbols are used in this paper:
Ar ⫽ asperity ratio;
Au ⫽ asperity unit;
Dmax ⫽ maximum particle diameter 共mm兲;
Dmin ⫽ minimum particle diameter 共mm兲;
D50 ⫽ median particle diameter 共mm兲;
E p ⫽ interphase efficiency parameter at peak
strength;
f n ⫽ contact normal force 共N兲;
f r ⫽ contact total force 共N兲;
f s ⫽ contact shear force 共N兲;
Rt ⫽ asperity height 共␮m兲;
Rtmax ⫽ maximum accumulated asperity height 共mm兲;
Sr ⫽ spacing between asperities 共mm兲;
Sw ⫽ asperity width 共mm兲;
⌬ ⫽ asperity slope 共deg兲;
␦ ⫽ effective stress friction angle from direct
interface shear test 共deg兲;
␪a ⫽ principal direction of surface normal
distribution 共deg兲;
␪n ⫽ principal direction of average contact normal
force anisotropy 共deg兲;
␪r ⫽ principal direction of average contact total
force anisotropy 共deg兲;
␴xx , ␴yy ⫽ horizontal and vertical normal stresses,
respectively 共kPa兲;
␶yx,␶xy ⫽ shear stresses acting on element of particulate
media 共kPa兲;
␾ ⫽ effective stress friction angle of particulate
media alone 共deg兲;
␾ds ⫽ effective stress direct shear friction angle
共deg兲;
␾ps ⫽ effective stress plane strain friction angle
共deg兲; and
␾␮ ⫽ particle to surface friction angle 共deg兲.
References
Barton, N., and Choubey, V. 共1977兲. “The shear strength of rock joints in
theory and practice.” Rock Mech., 10, 1–54.
Bathurst, R. J., and Rothenburg, L. 共1990兲. “Observation on stress-forcefabric relationships in idealized granular materials.” Mech. Mater., 9,
65–80.
Christoffersen, J., Mehrabadi, M. M., and Nemat-Nasser, S. 共1981兲. “A
micro-mechanical description of granular material behavior.” J. Appl.
Mech., 48, 339–344.
Claquin, C., and Emeriault, F. 共2001兲. “Interface behavior of granular
materials: Discrete numerical simulation and statistical homogenization.” Powders and grains 2001, Kishino, ed., Swets & Zeitlinger,
Lisse, 323–326.
Day, R. A., and Potts, D. M. 共1998兲. “The effect of interface properties on
retaining wall behavior.” Int. J. Numer. Analyt. Meth. Geomech., 22,
1021–1033.
DeJong, J. T., Frost, J. D., and Saussus, D. R. 共2002兲. “Measurement of
relative surface roughness at particulate-continuum interfaces.”
J. Test. Eval., 共 30兲1, 8–19.
Desai, C. S. 共1981兲. “Behavior of interfaces between structural and geologic media.” Proc., Int. Conf. on Recent Advances in Geotechnical
Earthquake Engineering, Univ. of Missouri at Rolla, St. Louis,
619–638.
Dove, J. E., Bents, D. D., Wang, J., and Gao, B. 共2006兲. “Particle-scale
surface interactions of nondilative interface systems.” Geotext.
Geomembr., 24共3兲, 156–168.
Dove, J. E., and Frost, J. D. 共1999兲. “Peak friction behavior of smooth
geomembrane-particle interfaces.” J. Geotech. Geoenviron. Eng.,
125共7兲, 544–555.
Dove, J. E., Frost, J. D., Han, J., and Bachus, R. C. 共1997兲. “The influence of geomembrane surface roughness on interface strength.” Proc.,
Geosynthetics ‘97, North American Geosynthetics Society, Long
Beach, Calif., 863–876.
Dove, J. E., and Jarrett, J. B. 共2002兲. “Behavior of dilative sand interfaces
in a geotribology framework.” J. Geotech. Geoenviron. Eng., 12共1兲,
25–37.
Ebeling, R. M., Peters, J. F., and Mosher, R. L. 共1997兲. “The role of
nonlinear deformation analyses in the design of a reinforced soil berm
at Red River U-Frame Lock No. 1.” Int. J. Numer. Analyt. Meth.
Geomech., 21, 753–787.
Esterhuizen, J. J. B., Filz, G. M., and Duncan, J. M. 共2001兲. “Constitutive
behavior of geosynthetic interfaces.” J. Geotech. Geoenviron. Eng.,
127共10兲, 834–848.
Frost, J. D., DeJong, J. T., and Recalde, M. 共2002兲. “Shear failure behavior of granular-continuum interfaces.” Eng. Fract. Mech., 69,
2029–2048.
Gomez, J. E. 共2000兲. “Development of an extended hyperbolic model for
concrete to soil interfaces.” Ph.D. dissertation, Virginia Tech,
Blacksburg, Va.
Hryciw, R. D., and Irsyam, M. 共1993兲. “Behavior of sand particles around
rigid ribbed inclusions during shear.” Soils Found., 33共3兲, 1–13.
Irsyam, M., and Hryciw, R. D. 共1991兲. “Frictional and passive resistance
in soil reinforced by plane ribbed inclusions.” Geotechnique, 41共4兲,
485–498.
ISO. 共1997兲. “Geometrical product specifications 共GPS兲—Surface texture: Profile method—Terms, definitions, and surface texture parameters.” 4287:1997/Cor1:1998, Geneva, Switzerland.
Jensen, R. P., Bosscher, P. J., Plesha, M. E., and Edil, T. B. 共1999兲. “DEM
simulation of granular media-structure interface: Effects of structure
roughness and particle shape.” Int. J. Numer. Analyt. Meth. Geomech.,
23, 531–547.
Johnson, M. L. 共2000兲. “Characterization of geotechnical surfaces by
stylus profilometry.” MS thesis, Georgia Institute of Technology, Atlanta.
Kishida, H., and Uesugi, M. 共1987兲. “Tests of interface between sand and
steel in the simple shear apparatus.” Geotechnique, 37共1兲, 45–52.
Ladanyi, B., and Archambault, G. 共1969兲. “Simulation of shear behavior
of a jointed rock mass.” Proc., 11th Symp. on Rock Mechanics,
Berkeley, Calif., 105–125.
Ladanyi, B., and Archambault, G. 共1980兲. “Direct and indirect determination of shear strength of rock mass.” Society of Mining Engineering
Annual Meeting, Las Vegas.
Majmudar, T. S., and Behringer, R. P. 共2005兲. “Contact force measurements and stress-induced anisotropy in granular materials.” Nature
(London), 435, 1079–1082.
Paikowsky, S. G. 共1989兲. “A static evaluation of soil plug behavior with
applications to the pile plugging problem.” D.Sc. dissertation, MIT,
Cambridge, Mass.
Paikowsky, S. G., and Xi, F. 共1997兲. “Photoelastic quantitative study of
the behavior of discrete materials with application to the problem of
interfacial friction.” Research Rep., Air Force Office of Scientific Research, Research Grant F49620-93-0267.
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2007 / 607
Patton, F. D. 共1966兲. “Multiple modes of shear failure in rock.” Proc., 1st
Congress of the Int. Society on Rock Mechanics, Lisbon, 1, 509–513.
Rothenburg, L. 共1980兲. “Micromechanics of idealized granular system.”
Ph.D. dissertation, Carleton Univ., Ottawa.
Rothenburg, L., and Selvadurai, A. P. S. 共1981兲. “A micromechanical
definition of the Cauchy stress tensor for particulate media.” Proc.,
Int. Symp. on the Mechanical Behavior of Structured Media,
Selvadurai, ed., Ottawa, Part B, 469–486.
Tejchman, J., and Wu, W. 共1995兲. “Experimental and numerical study of
sand-steel interfaces.” Int. J. Numer. Analyt. Meth. Geomech., 19,
513–536.
Uesugi, M., and Kishida, H. 共1986a兲. “Frictional resistance at yield between dry sand and mild steel.” Soils Found., 26共4兲, 139–149.
Uesugi, M., and Kishida, H. 共1986b兲, “Influential factors of friction between steel and dry sands.” Soils Found., 26共2兲, 33–46.
Uesugi, M., Kishida, H., and Tsubakihara, Y. 共1988兲. “Behavior of sand
particles in sand-steel friction.” Soils Found., 28共1兲, 107–118.
Wang, J. 共2006兲. “Micromechanics of granular media: A fundamental
study of interphase systems.” Ph.D. dissertation, Virginia Tech,
Blacksburg, Va.
Wang, J., Dove, J. E., and Gutierrez, M. S. 共2007兲. “Determining
particulate—solid interphase strength using shear—Induced anisotropy.” Granular Matter 共in press兲.
Wang, J., Gutierrez, M. S., and Dove, J. E. 共2004兲. “Effects of particle
rolling resistance on interface shear behavior.” Proc., 17th Conf. on
Engineering Mechanics, ASCE, Newark, Del.
Wang, J., Dove, J. E., Gutierrez, M. S., and Corton, J. 共2005a兲. “Shear
deformation in the interphase region.” Powders and grains 2005,
Stuttgart, Germany, 747–750.
Wang, J., Gutierrez, M. S., and Dove, J. E. 共2005b兲. “Strain localization
during shear of an idealized interphase system.” Proc., 2005 Joint
ASME/ASCE/SES Conference on Mech, Mat., Baton Rouge, La, Paper
349.
Yoshimi, Y., and Kishida, T. 共1982兲. “A ring torsion apparatus for evaluating friction between soil and metal surfaces.” Geotech. Test. J., 4,
145–152.
608 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / MAY 2007