A method to evaluate the three-dimensional roughness of fracture surfaces in brittle

geomaterials
Bryan S. A. Tatone and Giovanni Grasselli

Citation: Review of Scientific Instruments 80, 125110 (2009); doi: 10.1063/1.3266964

View online: http://dx.doi.org/10.1063/1.3266964
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 REVIEW OF SCIENTIFIC INSTRUMENTS 80, 125110 共2009兲

A method to evaluate the three-dimensional roughness of fracture surfaces

in brittle geomaterials
Bryan S. A. Tatone and Giovanni Grassellia兲
Department of Civil Engineering, Geomechanics Research Group, Lassonde Institute, University of Toronto,
35 St. George St., Toronto, Ontario M5S 1A4, Canada
共Received 11 June 2009; accepted 2 November 2009; published online 30 December 2009兲
Conventionally, the evaluation of fracture surface roughness in brittle geomaterials, such as concrete
and rock, has been based on the measurement and analysis of two-dimensional profiles rather than
three-dimensional 共3D兲 surfaces. The primary reason for doing so was the lack of tools capable of
making 3D measurements. However, in recent years, several optical and mechanical measurement
tools have become available, which are capable of quickly and accurately producing high resolution
point clouds defining 3D surfaces. This paper provides a methodology for evaluating the surface
roughness and roughness anisotropy using these 3D surface measurements. The methodology is
presented step-by-step to allow others to easily adopt and implement the process to analyze their
own surface measurement data. The methodology is demonstrated by digitizing a series of concrete
fracture surfaces and comparing the estimated 3D roughness parameters with qualitative
observations and estimates of the well-known roughness coefficient, Rs. © 2009 American Institute
of Physics. 关doi:10.1063/1.3266964兴

I. INTRODUCTION rameters are limited in terms of their ability to describe the

anisotropy in roughness that is often observed on fracture
For many years, the quantification of fracture surface surfaces.10,17,18
roughness in geomaterials, such as rock and concrete, was
restricted to the analysis of two-dimensional 共2D兲 profiles. II. RATIONALE
This restriction stemmed from a lack of tools with sufficient
Observation of sheared rock joint surfaces in the labora-
accuracy, resolution, and ease of use to characterize three-
tory often reveals that only a fraction of the total surface area
dimensional 共3D兲 surface topography did not exist. In recent
of a joint is damaged during testing.7,19–22 The shape, extent
years, however, several optical instruments have become
and distribution of these damaged zones are controlled by
commercially available to digitize fracture surfaces in the
many factors including: the roughness of the surface, which
laboratory and in situ with high resolution and accuracy.
accounts for the size, and the shape of the asperities; the
These systems include: laser scanners,1,2 close-range photo- shear direction; the applied normal stress; the total displace-
grammetric systems,3–5 and stereotopometric scanners.6–8 ment; and the mechanical properties the intact asperities.
Considering the Advanced TOpometric Scanner 共ATOS兲 II Nevertheless, the damaged areas are strictly related to the
employed in the current study, up to 1.4 million points on a specific surface topography and are typically restricted to
surface can be obtained in a matter of seconds.9 The resulting those asperity faces that have a local dip direction opposite
collection of 3D points, referred to as a point cloud, forms a the shear direction7,23 and preferentially develop in areas
highly detailed digital model of the surface of interest. Nev- comprised of the steepest faces.7,23–26 Thus, it follows that a
ertheless, the point cloud alone is only a collection of spatial roughness parameter describing the topography of the sur-
points on the surface and does not provide a “measure” of face should be based on the distribution of asperity angles
surface roughness. An objective parameterization of the with respect to the shear direction to fully capture its influ-
roughness requires further processing and analysis of the ence on shear strength. Extending the same approach, it
point cloud. would be possible to measure the roughness with respect to
Despite the increased availability and use of 3D surface several potential shear directions, and, by including it into
measurement equipment, many investigations continue to existing strength criteria,27–30 one can also account for the
evaluate roughness based on 2D profiles extracted from the influence of roughness on the anisotropic shear strength of
3D surface data. This approach can lead to incomplete and discontinuities, which has received limited attention over the
potentially biased representation of surface roughness.10–12 last two decades.10,17,31–33
Although some 3D roughness parameters have been previ- The sole objective of this paper is to propose a standard
ously developed to characterize joint surfaces in geomateri- methodology for quantifying the 3D roughness and aniso-
als to overcome the drawbacks of 2D profiles,13–16 these pa- tropy of fracture surfaces based on 3D measurements of sur-
face topography. The method is based on the angular thresh-
a兲
Author to whom correspondence should be addressed. Electronic mail: old concept, which was initially proposed as a means of
giovanni.grasselli@utoronto.ca. identifying potential damaged areas during direct shear test-

0034-6748/2009/80共12兲/125110/10/$25.00 80, 125110-1 © 2009 American Institute of Physics

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 125110-2 B. S. A. Tatone and G. Grasselli Rev. Sci. Instrum. 80, 125110 共2009兲

FIG. 1. 共Color online兲 Schematic illustrating triangulation of the measured

point cloud: 共a兲 3D view of final triangulated point cloud; 共b兲 zoomed in
view of original point cloud; 共c兲 connection of points to their natural neigh-
bors to form triangular elements; and 共d兲 the rendered triangular mesh that is
exported in STL format for subsequent analysis.

FIG. 2. 共Color online兲 Schematic diagram illustrating the geometric defini-

ing of rock joints.7,29,30 This article presents a step-by-step tion of azimuth, dip and apparent dip in relation to the selected analysis
description of the methodology and demonstrative examples. direction 共Refs. 7, 8, and 29兲.
Although the method described herein was developed in
response to the increased availability of 3D surface measure- kind was applied in the current study to remove measure-
ment data obtained with optical instruments, it is indepen- ment noise. Further discussion on the influence of such mea-
dent on the measurement system as it only requires a 3D surement noise is provided in Sec. V.
mesh describing the surface as input. Therefore, it can be Following triangulation, the surface mesh is exported
utilized to characterize the roughness of surfaces digitized from the measuring system in binary stereolithography
with mechanical instruments as well 共e.g. Develi34兲. 共STL兲 format; a ubiquitous 3D file format in the rapid pro-
totyping industry. The STL file is then converted to a format
III. METHODOLOGY FOR QUANTIFICATION OF 3D called 3s,35 which is preferred over the STL format as it
ROUGHNESS maintains exactly the same topology, but is more efficient in
The overall methodology for estimating the 3D rough- terms of file size.
ness can be divided into four smaller steps. The procedure
begins with the acquisition of the 3D surface topography. B. Alignment of triangulated surface
Next, the 3D measurements are preprocessed, aligned, and
Following data acquisition and the creation of a triangu-
then analyzed. Lastly, the roughness of the surface is calcu-
lated surface file in the desired format 共3s兲, a reference co-
lated. A detailed description of these above steps is provided
ordinate system must be established to allow the orientation
in the following subsections of this paper.
of each triangle comprising the surface to be measured. To
A. Acquisition and preprocessing of surface
do so, a best-fit plane is established through the surface to be
topography measurements analyzed and the coordinate system is transformed such that
the best-fit plane defines the xy plane 共i.e., z = zero兲. Follow-
The first step in analyzing surface roughness involves ing the definition of the xy plane, the x and y axes can be
digitizing the surface of interest using an optical or mechani- rotated about the z-axis to any desired direction of interest
cal measuring system. Typically these systems provide a 共e.g., along the dip direction of a rock joint, or parallel to the
dense point cloud comprised of 3D coordinates defining the axes of a sample兲.
surface of interest. A point cloud, however, is not well suited
for the analysis of surface roughness; therefore, it must be
converted into a triangular irregular network 共TIN兲 before C. Analysis of triangulated surface
analysis can proceed. To begin analyzing the triangulated surface, a specific
Several triangulation algorithms are available to create a analysis direction 共t兲 must be selected. Afterwards, the ori-
TIN from point cloud data, each with advantages and disad- entation of each individual triangle forming the rough sur-
vantages in terms of computational efficiency, repeatability, face can be uniquely identified by its dip 共␪兲 and azimuth
and accuracy. Delaunay triangulation was employed in the 共␣兲; where dip is defined as the maximum angle between the
current study. This technique connects each point in the point best-fit plane through the entire rough surface and the indi-
cloud to its natural neighbors to form a unique surface de- vidual triangles and azimuth is defined as the angle measured
fined by contiguous triangles, as outlined in Fig. 1. It is noted clockwise between the selected analysis direction and the
that this triangulation of the point cloud was based on the projection of the true dip vector 共d兲 onto the best-fit plane
original data points. In other words, no smoothing of any 共Fig. 2兲.
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 125110-3 B. S. A. Tatone and G. Grasselli Rev. Sci. Instrum. 80, 125110 共2009兲

analysis direction 共t兲 共Fig. 2兲. Mathematically, the apparent

dip is related to the true dip as follows:
tan ␪ⴱ = − tan ␪ · cos ␣ . 共1兲
Based on the apparent dip angle of each triangle making
up the surface, it is possible to distinguish the fraction of the
surface that is more steeply inclined than progressively
greater threshold values of ␪ⴱ. This fractional area is referred
to as the normalized area, A␪ⴱ, in that it is defined by the area
of the surface with an apparent dip greater than a selected
threshold value normalized with respect to the total area of
the surface, At.
This concept is further illustrated in Fig. 3. Figure 3共a兲
displays a TIN model of a tensile fracture surface in lime-
stone. The individual triangles with an apparent dip greater
FIG. 3. 共Color online兲 Schematic diagram illustrating the use of the angular than thresholds of 0°, 5°, 10°, 20°, and 30° are identified in
threshold concept to characterize surface roughness in one analysis direc- black in Figs. 3共b兲–3共f兲, respectively. In the same respective
tion: 共a兲 example of a TIN surface; and 关共b兲–共f兲兴 triangular elements that are
steeper than angular thresholds, ␪ⴱ, of 0°, 5°, 10°, 20°, and 30°, respectively. order, the corresponding normalized areas, A␪ⴱ, defined by
The fraction of the total surface area, Athetaⴱ, corresponding to normalized these triangles are 0.540, 0.399, 0.271, 0.092, and 0.022. If
areas, A␪ⴱ, corresponding to each threshold in 共b兲–共f兲 are 0.540, 0.399, the value of A␪ⴱ were determined for several additional
0.271, 0.092, and, 0.022, respectively. threshold values between 0° and 90° 共the upper and lower
bound values兲, the cumulative distribution of A␪ⴱ as a func-
Given the dip and azimuth, it is possible to define the tion of the various threshold values of ␪ⴱ can be established
apparent inclination of each triangle facing the specified 关Fig. 4共a兲兴. This process can be considered analogous to the
analysis direction. This apparent inclination is termed the sieve analysis that is commonly used to determine the cumu-
apparent dip angle 共␪ⴱ兲 and can be obtained by projecting the lative grain size distribution of a soil in geotechnical engi-
true dip vector 共d兲 onto a vertical plane oriented along the neering.

FIG. 4. 共Color online兲 Example of the distribution of normalized area, A␪ⴱ, as a function of different threshold, value of ␪ⴱ for the analysis directions indicated
by the arrows in the top right corners of 共a兲, 共b兲, 共c兲, and 共d兲.
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 125110-4 B. S. A. Tatone and G. Grasselli Rev. Sci. Instrum. 80, 125110 共2009兲

0.6 9
A0 = 0.540

0.5 8
C = 3.35

C=0
0.4
∗ d Area, Aθ

A0 [
max/(C+1)]]
=
6

0.
25
C
0.3

=
Normalized

0.
5
C
= 5

*
Limestone
1
0.2
C

granite
=2

4 gneiss
C=
C=

0.1 marble
5
10

3 sandstone
* o
θ max = 52.1 serpentinite
0.0
0 10 20 30 40 50 60 70 80 90 2
∗ 5 10 15 20 25
Apparent Dip, θ , Threshold (degrees)

*max/C
FIG. 5. 共Color online兲 Lines defined by Eq. 共2兲 compared to the measured
ⴱ
distribution presented in Fig. 4共a兲. FIG. 6. Plot of the area under the curve given by Eq. 共2兲 vs ␪max / C for 37
tensile fracture surfaces in 6 rock types.

Recalling that the cumulative distribution of A␪ⴱ, such as

that shown 关Fig. 4共a兲兴, represents only one analysis direction,
the above process must be repeated to characterize the areal parameter C, upon initial inspection, appears to provide an
distributions in other directions 关e.g., Figures 4共a兲–4共c兲兴. objective measure of surface roughness in the selected analy-
sis direction. However, as described in the following para-
graphs, the value of C alone is not adequate to parameterize
D. Roughness metric
roughness.
Under the assumption that a surface with a larger pro- In an attempt to validate C as a measure of surface
portion of steeply dipping asperities is rougher than a surface roughness, a previous study by Grasselli et al.7 considered 39
dominated by lower angle asperities, the relative roughness tensile rock fracture specimens comprised of 6 different rock
ⴱ
of surfaces can be objectively evaluated by comparing the types and determined A␪ⴱ, ␪max , and C along with the corre-
cumulative distributions of A␪ⴱ. 共i.e., Figure 4兲. As described sponding shear strength via direct shear tests. The results
by Grasselli29 and Grasselli et al.,7 the relationship between indicated that the values of A0 for all surfaces remained
the A␪ⴱ and ␪ⴱ can be expressed by the following equation: roughly constant at 0.5 while, ␪max ⴱ
typically ranged between

A ␪ⴱ = A 0 冉 ⴱ
␪max
ⴱ
− ␪ⴱ
␪max
冊 C
, 共2兲
20° and 90° 共with nominal point spacing⫽250 ␮m. The re-
sulting values of C for each specimen did not correlate well
with the observed degree of roughness, indicating that C,
where A0 is the normalized area of the surface corresponding alone, was an unsuitable metric for roughness. Interestingly,
ⴱ
to an angular threshold of 0° in the chosen analysis direction however, when the ratio of ␪max / C for each discontinuity
共i.e., the surface area defined by an apparent dip greater than specimen was compared to the corresponding shear strength,
ⴱ ⴱ
0° normalized with respect to the total surface area兲; ␪max is a strong correlation was found.7 As a result, the ratio ␪max /C
36
the maximum apparent dip angle of the surface in the chosen was adopted as a measure of the surface roughness the
ⴱ
analysis direction; and C is a dimensionless fitting parameter, acceptance of ␪max / C as a suitable measure of surface rough-
calculated via a nonlinear least-squares regression analysis ness was based on purely empirical observations. Therefore,
共see Sec. II兲, which characterizes the shape of the the physical meaning of the parameter and why it had a
distribution.7 positive correlation with discontinuity shear strengths re-
Using the measured distribution of A␪ⴱ previously plot- mained unknown.
ted in Fig. 4共a兲 as an example, the measured data points and As part of the current study, 37 of the 39 tensile fracture
the best-fit line defined by Eq. 共2兲 with C = 3.35 are plotted in surfaces of Grasselli et al.7 were reanalyzed and the values of
ⴱ
Fig. 5. Also plotted, are the lines defined by different values A␪ⴱ, ␪max , and C re-evaluated to seek a physical basis for the
ⴱ ⴱ
of C. It is observed that with constant A0 and ␪max , the value use of ␪max / C as a roughness metric. In doing so, it was
ⴱ
of C controls the concavity of the curve defined by Eq. 共2兲. discovered that the values of ␪max / C for a surface had a
Theoretically, C can range from 0 to infinity. A value of C positive correlation with the area under the corresponding
equals 0 characterizes a saw tooth profile in which all of the best-fit curves given by Eq. 共2兲 共Fig. 6兲. This correlation was
asperity faces have the same dip angle, while a value ap- deemed significant as the Pearson product-moment correla-
proaching infinity is indicative of a perfectly smooth surface. tion coefficient, r, was 0.97, which is greater than the critical
Since a surface with a higher proportion of steeply dipping directional value of 0.418 at the 0.5% significance level.
asperities is defined by a lower C value 共less concave兲 and a Large areas under the curve indicate that the surface contains
surface with high proportion of shallowly dipping asperities of a larger proportion of steeply dipping asperities and, thus,
is defined by higher C values 共more concave兲, the fitting greater relative roughness. In contrast, smaller areas under
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 125110-5 B. S. A. Tatone and G. Grasselli Rev. Sci. Instrum. 80, 125110 共2009兲

the curve are characteristic of comparatively smoother sur-

faces. By evaluating the definite integral of Eq. 共2兲 between
(a)
ⴱ
0 and ␪max , the area under the curve is given as follows:

A0 冕 冉
ⴱ
␪max

0
ⴱ
␪max
ⴱ
− ␪ⴱ
␪max
冊 冏 冉 冊
C
d␪ⴱ = − A0
ⴱ
␪max
C+1

冉 冊 冏
ⴱ
C+1 ␪max
␪ⴱ
⫻ 1− ⴱ
␪max 0

冉 冊 = A0
ⴱ
␪max
C+1
, 共3兲
Y

where A0 is the normalized surface area steeper than 0° in the

ⴱ
analysis direction; and the term, ␪max / 共C + 1兲, characterizes
the mean apparent dip angle of the surface in the analysis
direction and is equivalent to the apparent dip of a single saw
tooth asperity with the same area under the curve as the
original surface.
Seeing as the parameter ␪max ⴱ
/ 共C + 1兲 is very similar to Z
ⴱ X
the empirical parameter ␪max / C, which was proven to be a
reasonable measure of roughness,7,29,30 it is proposed that
ⴱ ⴱ
(b) 90
o

␪max / 共C + 1兲 could be adopted in place of ␪max / C as the met- 15

ric of surface roughness in the future. Considering the poten-
ⴱ
tial range of C 共0 − ⬁兲, the value of ␪max / 共C + 1兲 can range
ⴱ 10
from 0 for a perfectly smooth surface to ␪max for a saw tooth
Roughness ( max / [C+1])

surface with a consistent inclination anywhere between 0°

and 90°. 5
It should be noted that, although A0 was experimentally
shown to be nearly constant at 0.5 for the 39 tensile fracture


o o
0 180 0
surfaces, this is not true for all fracture surfaces. In some
cases A0 could vary significantly in different directions 共e.g.,
ripple marks兲, such that it may be necessary to include the 5
ⴱ
value of A0 along with ␪max / 共C + 1兲 to fully describe the
roughness anisotropy. In these cases, it is proposed that
ⴱ 10
2A0关␪max / 共C + 1兲兴 be used as the new metric. This expression
ⴱ
simplifies to ␪max / 共C + 1兲 when A0 is 0.5, yet maintains the
same magnitude for the roughness parameter in cases where 15
o
A0 varies in different directions. 270

FIG. 7. 共Color online兲 共a兲 Example of a triangulated surface and 共b兲 the
ⴱ
corresponding distribution of ␪max / 共C + 1兲 displayed on a polar plot.
1. Quantifying anisotropy in surface roughness
Recall that to fully characterize the 3D surface rough- the polar plot. For an isotropic surface this value approaches
ⴱ
ness, the parameters A0, C, and ␪max must be calculated in 1 while, anisotropic surfaces would display values ⬎1. In the
several different directions and the resulting values of case of Fig. 7共b兲, the polar plot displays an elliptical shape
ⴱ
␪max / 共C + 1兲 obtained. Adopting a counter-clockwise angular with maximum and minimum values of ␪max ⴱ
/ 共C + 1兲 equal to
convention, in which the positive x-direction is considered 12.86 共at 165°兲 and 11.06 共at 275°兲, respectively, resulting in
ⴱ
0°, the values of ␪max / 共C + 1兲 corresponding to each analysis an anisotropy value of 1.16.
direction between 0° and 360° in 5° increments can be dis-
played on a polar plot to visualize anisotropy in surface
roughness 共Fig. 7兲. 2. Estimation of C
Qualitatively, the anisotropy can be described by the Equation 共2兲 has the form of a power law, y = axb, where
ⴱ ⴱ
shape of the polar plot. Roughness values that are approxi- a = A0; x = 共␪max − ␪ⴱ兲 / ␪max ; and b = C. Therefore, it would be
mately the same in all directions 共i.e., isotropic兲 produce a expected that the value of the roughness parameter C could
nearly circular polar plot, while roughness values that dis- be determined by performing a least square linear regression
play a distinct difference with direction can result in ellipti- on the logarithmic form of Eq. 共2兲. However, when plotting
cal or sinusoidal-shaped plots. the logarithmic form of the equation the relationship often
ⴱ ⴱ
A simple quantitative description of the anisotropy in becomes nonlinear for the lowest values of 共␪max − ␪ⴱ兲 / ␪max ,
surface roughness can be obtained by considering the ratio skewing the best-fit line 关Fig. 8共a兲兴. As a result, the estimate
between the maximum and minimum roughness values on of C 共the slope of the best-fit line兲 does not produce a good
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 125110-6 B. S. A. Tatone and G. Grasselli Rev. Sci. Instrum. 80, 125110 共2009兲

(a) 0
(a) 0.6

Measured Measured
-2 Linear fit of ln-ln relationship 0.5 Non-linear regression fit

-4
0.4
y = 4.845x - 0.679
2
R = 0.935
-6
03
0.3
A0 = 0.507
ln[A


C = 6.30


6.30
y = 0.507x

A

o
-8
*max = 89.7 2
0.2 R = 0.999

-10
0.1

-12
0.0
-14
-3.5
35 -3.0
30 -2.5
25 -2.0
20 -1.5
15 -1.0
10 -0.5
05 0
0.0
0 00
0.0 02
0.2 04
0.4 06
0.6 08
0.8 10
1.0
     
ln[(
max
)/
max] (
max
)/
max
(b) 0.6 (b) 0

Measured Measured
Resulting fit of Equation (1) -2 ln(non-linear regression fit)
0.5

-4
0.4

A0 = 0.507 -6
0.3
C = 4.85
ln[A
]

A


o

*max = 89.7 -8
0.2

-10
0.1
-12
0.0
-14
0.0 0.2 0.4 0.6 0.8 1.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
  
 
(
max
)/
max

ln[(
max
)/
max]

FIG. 8. 共Color online兲 Typical plots showing normalized area, A␪ⴱ, vs nor- FIG. 9. 共Color online兲 Typical plots showing normalized area, A␪ⴱ, vs nor-
ⴱ ⴱ
malized angular threshold 共␪max ⴱ ⴱ
− ␪ⴱ兲 / ␪max: 共a兲 In transformed data with malized angular threshold 共␪max − ␪ⴱ兲 / ␪max: 共a兲 data with best-fit nonlinear
best-fit linear regression line; 共b兲 resulting fit of Eq. 共2兲 to the measured regression line; 共b兲 resulting linear fit of In transformed data.
data.

linear portion of the logarithmically transformed data. As a

fit to the measured data 关Fig. 8共b兲兴. To overcome this prob- result, it provides an improved estimate of the “fitting param-
lem, estimation of C requires the use of either a maximum eter,” C.
likelihood estimation of the exponent or nonlinear least- It should be noted that when applying an iterative least-
squares regression. squares regression technique, a reasonable initial estimate of
For the current study, nonlinear regression analysis was the value of C is required to ensure the Solver add-in not
performed by implementing an iterative least-squares ap- only finds a solution more quickly but finds the absolute
proach using the Solver add-in for Microsoft Excel.37,38 By minimum and not a local minimum of Eq. 共4兲. In this study,
altering the value of C in Eq. 共2兲, the sum of squares of the initial value of C was taken as the slope of the best-fit
residuals, as given by Eq. 共4兲: straight line through the logarithmic form of Eq. 共2兲 关Fig.
8共a兲兴. This approach has been successful for all roughness
SSresiduals = 兺 共A␪ⴱcalc. − A␪ⴱmeasured兲2 , 共4兲 analyses completed to date 共over 300 surfaces兲 including:
fractures in different rock types, concrete, cement, and plas-
was minimized. The value of C corresponding to the abso-
tic materials.
lute minimum of Eq. 共4兲 defines the “roughness parameter”
for the surface under consideration. Using the same data set
shown in Fig. 8, the best-fit of Eq. 共2兲 to the measured data IV. DEMONSTRATIVE EXAMPLE
as determined by nonlinear regression is shown in Fig. 9共a兲
and the corresponding linear fit of the logarithmically trans- A. Specimen description
formed data is shown in Fig. 9共b兲. It is evident, both visually To demonstrate the application of the roughness evalua-
and by the improved R2 value, that this approach provides a tion methodology described in the preceding section, fracture
better fit of Eq. 共2兲 to the measured data and honors the surfaces from four failed concrete beams of varying com-
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 125110-7 B. S. A. Tatone and G. Grasselli Rev. Sci. Instrum. 80, 125110 共2009兲

(a) (b)

Sample ID: AT-1

Sample ID: DB 180
(c) (d)

Sample ID: DB 0.530 Sample ID: DB 120

0 10 20 30 50 mm

FIG. 10. 共Color online兲 Photographs of fracture surfaces from concrete

members failed in 3-point bending with the 150⫻ 150 mm2 area selected
for digitization and roughness analysis outlined in black.

pressive strength were digitized and analyzed. Failure of

these beams in the laboratory, via three-point bending, com-
prised part of the ongoing investigations of the shear strength
of deep concrete members at the University of Toronto. Fig-
ure 10 provides photographs of the fracture surfaces from
each beam and outlines the 150⫻ 150 mm2 area selected for FIG. 11. 共Color online兲 The 3D stereotopometric measurement system uti-
digitization and roughness analysis. A full description of the lized for this study, the Advanced TOpometric Sensor 共ATOS兲 II manufac-
tured by GOM mbH.
casting and testing procedure of the DB and AT-1 series
beams is given in Angelakos et al.39 and Lubell et al.,40
respectively.
recognized by the ATOS software based on their relative po-
B. Selected measurement technique: Equipment and sition, enable the system to automatically transform mea-
methodology surements into a common coordinate system. Following ac-
quisition, the individual measurements were aligned such
For this study, a 3D stereotopometric measurement sys-
that the average deviation between redundant data 共i.e., over-
tem, the ATOS II manufactured by GOM mbH, was used to
lapping measurements兲 was minimized. This minimized
digitize the fracture surfaces of the four concrete beam speci-
value, termed the mesh deviation, serves as an estimate of
mens. The ATOS II system consists of a sensor head contain-
the intensity of the average measurement noise of the
ing a central projector unit and two charge-coupled device
system.9 With the configuration used in this study, the mesh
共CCD兲 cameras, along with a high-performance Linux PC to
deviation was less than 10 ␮m for all specimens, which is
pilot the system 共Fig. 11兲. To measure an object, the system
less than 5% of the average point spacing. After minimiza-
projects various structured white-light fringe patterns onto
tion of the mesh deviation, the point clouds for each speci-
the object’s surface. Images of these patterns, which become
men were polygonized and analyzed according the method-
distorted due to the relief of the surface, are automatically
ology described in Sec. III.
captured by the two CCD cameras. From these image pairs,
the software automatically computes precise 3D coordinates
for each pixel based on the principle of triangulation. Given
the resolution of the CCD cameras used in the current study (a) (b)

共1392⫻ 1040 pixels兲 a point cloud of up to 1.4 million sur-

face points can be obtained in one measurement. The average
spacing between these points can be varied by changing the
lenses of the CCD cameras and projector. With the configu-
ration selected for this study, points were obtained on a
(c) (d)
roughly 250 ␮m xy grid over a nominal area of 150
⫻ 150 mm2.
To fully digitize a 150⫻ 150 mm2 area on each speci-
men with the ATOS system, multiple individual measure-
ments from different angles and positions were required.
Transformation of these individual measurements into a
common coordinate system was achieved by affixing several
FIG. 12. 共Color online兲 Images of the 150⫻ 150 mm2 digitized area of the
3 mm reference points around the perimeter of the area to be concrete fracture specimens: 共a兲 DB 180; 共b兲 AT-1; 共c兲 DB 0.530; and 共d兲
digitized. These reference points, which are automatically DB 120.
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 125110-8 B. S. A. Tatone and G. Grasselli Rev. Sci. Instrum. 80, 125110 共2009兲

o
90 24
30 DB 180
AT-1
25 DB 0.530 22
DB 120
20

15
Roughness (
mmax / [C+1])

20 Range of measured
*

max / (C+1)
10

max / (C+1)
5 18


0 o
180 0
o

*
5 16

10
14 *
15 Average
max / (C+1)

20
12
25 1.08 1.12 1.16 1.20 1.24

30 Rs
o
270
ⴱ
ⴱ FIG. 14. Average ␪max / 共C + 1兲 vs the roughness coefficient, Rs, for the con-
FIG. 13. 共Color online兲 Polar plot of the 3D roughness values, ␪max / 共C crete beam fracture surfaces.
+ 1兲, of the four concrete beam fracture surfaces.

C. Results of 3D roughness evaluation At

Rs = , 共5兲
The digitized areas of each fracture surface are illus- An
trated in Fig. 12. Qualitative observation of the four surfaces where At is the true area of the surface defined by the sum-
reveals an increase in roughness from Figs. 12共a兲–12共d兲. Re- mation of the area of each triangular element, Ai, comprising
sults of the 3D roughness analysis performed according to the 3D surface:14
the methodology described in Sec. III are presented in the
form of a polar plot 共Fig. 13兲. The plot quantitatively illus- At = 兺
surface
Ai , 共6兲
trates relative differences in roughness that are in agreement
with the qualitative observations. Furthermore, the polar plot and An is the nominal area defined by the projection of the
shows that there is a small degree of roughness anisotropy as actual area onto a best-fit plane through the surface. Using
the plot has a slight elliptical shape with the major axis ori- this method it follows that given two surfaces, the rougher
ented in the 90°–270° direction for the DB series beams and surface will display a greater ratio of the true surface area to
in the 0°–180° direction for the AT-1 fracture surface. A tabu- the nominal surface area due to the increased undulation and
lated summary of the minimum, maximum, and average unevenness of the surface. For a perfectly flat surface, Rs
ⴱ
value of ␪max / 共C + 1兲 for each fracture specimen is provided equals a minimum value of 1 while, most fractures in brittle
in Table I. materials have an Rs values between 1 and 2.13
Table I summarizes the values of Rs obtained for the four
concrete fracture surfaces. The values of Rs plotted against
1. Comparison with 3D roughness coefficient, Rs ⴱ
the arithmetic average of the ␪max / 共C + 1兲 values obtained in
The surface roughness coefficient, Rs, was adopted to all directions demonstrate a positive correlation 共Fig. 14兲.
characterize the roughness of the 4 concrete fracture surfaces This correlation was deemed significant as the Pearson
to facilitate comparison with the results obtained with the product-moment correlation coefficient, r, was 0.99, which is
proposed methodology. Rs is a simplistic 3D roughness pa- greater than the critical directional value of 0.917 at the 0.5%
rameter that has been used to quantify the roughness of frac- significance level. This relationship indicates that average
ⴱ
ture surfaces in a variety of materials.6,13,14,41–45 The value of ␪max / 共C + 1兲 provides an estimate of relative roughness that
Rs is given by El-Soudani43 as follows: is consistent with existing methods. However, the proposed

TABLE I. Tabulated results of 3D roughness analyses on the 150⫻ 150 mm2 areas on the four concrete fracture
surfaces shown in Fig. 10.

ⴱ
Roughness, ␪max / 共C + 1兲
Compressive strength
Sample ID 共MPa兲 Max Min Average Anisotropy Roughness coefficient, Rs

DB 180 80 14.73 13.51 14.09 1.09 1.09

AT-1 64 15.16 12.55 13.89 1.21 1.10
DB 0.530 32 20.58 18.06 19.65 1.14 1.18
DB 120 21 23.79 21.63 22.78 1.10 1.24

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 125110-9 B. S. A. Tatone and G. Grasselli Rev. Sci. Instrum. 80, 125110 共2009兲

approach has the added benefit of being able to describe the roughly 10 ␮m while the nominal point spacing of the mea-
anisotropy in roughness, as stated previously. surement was 250 ␮m. This low noise to point spacing ratio
make the surface measurements obtained with the ATOS sys-
V. MEASUREMENT CONSIDERATIONS tem ideal for roughness evaluations.
The quality of surface measurements is very important to
the estimation of roughness. The accuracy of the TIN model C. Curve fitting difficulties
of a surface is dependent on the density of measurement
points 共measurement resolution兲 and the precision with It should be noted that occasional difficulties may be
which these points can be located in space 共measurement encountered when fitting Eq. 共2兲 to the measured data. In
noise兲. The limitations of a measurement device, in terms of some cases the measured values of the normalized area, A␪ⴱ,
resolution and noise, must be appreciated before the data can corresponding to the steeper triangular facets 共i.e., beyond
be used to estimate roughness with the methodology pro- 45°兲 can significantly deviate from the power law relation-
posed in this article as well as for any other evaluation meth- ship given by Eq. 共2兲. These problems arise when the digi-
odology. tized surface area is very small. In these cases, there are a
limited number of the less frequent, steeply dipping triangu-
A. Measurement resolution
lar facets resulting in increased uncertainty in the corre-
sponding values of normalized area. An improved fit of Eq.
Considering a rough fracture surface, denser point 共2兲 can be obtained by increasing the size of the digitized
clouds are able to capture a greater amount of detail. As area, which effectively increases the number of steeply dip-
resolution decreases, surface features smaller than the nomi- ping discontinuities considered when calculating A␪ⴱ and in-
nal point spacing can no longer be digitized. This loss of creases the confidence in the calculated value.
detail essentially smoothes the resulting TIN surface model.
Hence, decreasing measurement resolution leads to de-
creased estimates of roughness. VI. SUMMARY AND CONCLUSIONS
Specifying an optimum resolution for evaluating the sur-
face roughness of fractures in brittle geomaterials is a chal- This paper has presented a detailed step-by-step method-
lenging task. Ultimately, the required resolution depends on ology to evaluate the 3D roughness and anisotropy of frac-
the purpose of the roughness estimates. For example, if the ture surfaces in brittle geomaterials, such as concrete and
goal is to simply compare the relative roughness of surfaces, rock. The proposed methodology makes use of 3D surface
using consistent resolution to digitize each surface is the measurements, which are becoming more widely available
most important factor. Conversely, if the roughness estimates with the increasing availability of commercial optical mea-
ⴱ
are to be used to estimate shear strength according to an suring devices. The proposed roughness metric, ␪max / 共C
empirical strength criterion, the resolution should be suffi- + 1兲, is based on the cumulative distribution of the apparent
ciently fine 共nominal point spacing ⬍0.5 mm兲 to capture dip of individual triangles of the TIN defining a surface. By
changes in the surface topography caused by the shearing considering several analysis directions, the anisotropy in
process 共i.e., the areas damaged during shearing兲. In general, roughness can be characterized and visualized using polar
the finest practical resolution should be used to digitize a plots, while avoiding potentially biased evaluations based on
surface for roughness evaluations. Most importantly, how- 2D profiles.
ever, the measurement resolution that is employed needs to Use of the proposed roughness evaluation methodology
be clearly documented to ensure roughness estimates are was demonstrated by digitizing and analyzing the fracture
comparable. surfaces of four failed concrete beam specimens. The pro-
posed method was shown to quantify the relative roughness
B. Measurement noise
of the surfaces consistent with qualitative observations. In
ⴱ
addition, the average value ␪max / 共C + 1兲 in all analysis direc-
Measurements of surface topography obtained with any tions was shown to be positively correlated with the well-
measurement device will contain some level of noise. Rec- known roughness coefficient, Rs, indicating the proposed
ognizing the magnitude of this noise and its influence on method provides estimates of roughness consistent with ex-
roughness estimates is essential regardless of the adopted isting methods but has the benefit of being able to character-
evaluation methodology. For a fixed measurement resolution, ize roughness anisotropy as well.
increased noise leads to increased roughness estimates as the
TIN surface models contain additional irregularities and un-
ACKNOWLEDGMENTS
dulations introduced by the measuring procedure. Although
some attempts have been made to remove such noise from This work has been supported by the Natural Science
the measurement by using various smoothing filters, this and Engineering Research Council of Canada in the form of
topic is beyond the scope of this article and will not be dis- Discovery Grant No. 341275 and RTI Grant No. 345516 held
cusses further. As a basic guideline if the magnitude of mea- by G. Grasselli and an Alexander Graham Bell Canada
surement noise exceeds the nominal point spacing of the Graduate Scholarship held by B.S.A. Tatone. The authors
surface measurements, the data cannot be used to evaluate would also like to thank Professor Evan Bentz at the Univer-
roughness. Considering the ATOS system demonstrated in sity of Toronto for making available the concrete fractures
Sec. IV, the estimated measurement noise had a magnitude of that were analyzed herein.
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 125110-10 B. S. A. Tatone and G. Grasselli Rev. Sci. Instrum. 80, 125110 共2009兲

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