International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380
Damage in a rock core caused by induced tensile stress and its relation
to differential strain curve analysis
K. Sakaguchi*, W. Iino1, K. Matsuki
Department of Geoscience and Technology, Graduate School of Engineering, Tohoku University, 01 Aramaki-Aza-Aoba, Aoba-ku,
Sendai 980-8579, Japan
Accepted 8 April 2002
Abstract
A finite element analysis was carried out to analyze the distribution of tensile stress within and below a long HQ core stub for 77
in situ stress conditions. The maximum tensile stress experienced by the core along the axis during boring under in situ stress was
accumulated in an equal-area stereonet for a central part of the cross-section. The maximum tensile stress accumulated for a central
area of less than about 60% of the total cross-sectional area was concentrated in a certain direction, which was nearly the same
direction as the minimum principal stress for all of the stress conditions, except those in which the minimum principal stress (s3) was
equal to the intermediate principal stress (s2). When s2 ¼ s3; the direction of the cumulative maximum tensile stress lay
approximately in the plane of s2 ¼ s3; which is perpendicular to the maximum principal stress. Based on the assumption that a
penny-shaped crack is produced normal to the maximum tensile stress in proportion to the magnitude of such stress, the crack
density in the core was analyzed by calculating strain under hydrostatic pressure as in differential strain curve analysis (DSCA). The
maximum principal crack density in the central part of the core was much greater than the intermediate and minimum principal
crack densities, excluding special cases in which s2 ¼ s3: The direction of the maximum crack density was similar to that of the
accumulated maximum tensile stress. Thus, the direction of the maximum crack density obtained by DSCA predicts the direction of
the minimum principal stress rather than that of the maximum principal stress, if the distribution of pre-existing microcracks before
stress relief is isotropic and if additional microcracks are produced only by tensile stress during boring under in situ stress. To verify
this, crack parameters were measured by DSCA for two cores of quartz-diorite, which were taken by overcoring when the
hemispherical-ended borehole technique was used to measure in situ stress. The directions of the maximum crack parameters
measured by DSCA were nearly the same as that of the minimum principal stress for one of the cores. For the other core, for which
the magnitudes of the intermediate and minimum principal stresses were close to each other and, accordingly, the direction of the
minimum principal stress was uncertain, the direction of the maximum crack density estimated by damage analysis under the
assumption that s2 ¼ s3 coincided with the directions of the maximum crack parameters measured by DSCA. r 2002 Elsevier
Science Ltd. All rights reserved.
Keywords: Crack density; Tensile stress; DSCA; Damage of rock core; Boring under stress
1. Introduction
Differential strain curve analysis (DSCA) is a method
for estimating in situ stress by measuring the distribu-
tions of microcracks in an oriented core taken from
depth [1]. Since this method is simple and can be applied
to great depth, many researchers have used DSCA to
estimate three-dimensional directions of principal stres-
ses and the ratios of their magnitudes [2–6]. DSCA is
based on the assumption that the stress relief in the core
produces microcracks perpendicular to the principal
stresses in proportion to the magnitude of each principal
stress [1]. Carlson and Wang [7] showed that microcrack
porosity in a rock core increases with the mean in situ
stress, which may indirectly support the above assump-
tion. Charlez et al. [8] proposed a rheological relaxation
model to explain microfracturing induced by stress
relief. However, this assumption has not yet been
sufficiently verified. Furthermore, DSCA assumes that
*Corresponding author. Fax: +81-22-217-7381.
E-mail addresses: saka@rock.earth.tohoku.ac.jp (K. Sakaguchi),
matsuki@rock.earth.tohoku.ac.jp (K. Matsuki).
1Present address: Fuji Photo Film Co., Ltd.
1365-1609/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 1 3 6 5 - 1 6 0 9 ( 0 2 ) 0 0 0 3 9 - 4
rocks have no effective microcracks, which is not always
valid.
It is well known that core discing occurs when the
principal stresses perpendicular to the core axis are high
[9]. This phenomenon is believed to occur due to tensile
stress near the bottom of the core stub [10–12]. Li and
Schmitt [13] showed that tensile stress could produce
another type of crack, such as a petal fracture, under
different in situ stress conditions. Therefore, if a
macroscopic fracture is not created, the tensile stress
induced by boring may produce microfractures in a rock
core according to the in situ stress, and consequently the
core may be more or less damaged. Evaluating damage
in the rock core is also important for understanding how
the mechanical properties of rock samples from deep
underground differ from those in situ.
In this study, we analyzed the history of the tensile
principal stress experienced by an HQ core during
boring under in situ stress by using finite element
method (FEM) to calculate the tensile stress distribu-
tions within and below a long HQ core under a general
state of in situ stress. First, the maximum tensile stress
experienced by the core along the axis was accumulated
in an equal-area stereonet for a central part of the cross-
section. Next, based on the assumption that the
maximum tensile stress produces a penny-shaped crack
in proportion to the magnitude of such stress at each
position in the cross-section, the crack density in the
core was evaluated. Our analysis showed that the
direction of the maximum crack density is nearly
the same as that of the minimum principal stress
rather than that of the maximum principal stress,
contrary to the assumption in DSCA. Finally, by using
two rock cores taken by overcoring when in situ stress
was measured by the hemispherical-ended borehole
technique, we measured the crack parameters in the
specimens by DSCA and compared the principal
directions with those of in situ stress. For both cores,
the directions of the maximum crack parameters
measured by DSCA were consistent with the results
obtained by the damage analysis.
2. Analysis of tensile principal stress
2.1. Finite element method
A finite element code for an axisymmetric body
subjected to non-axisymmetric boundary conditions [14]
was used to calculate stresses in the model shown in
Fig. 1. Fig. 1(b) is an enlarged diagram of part (b) in
Fig. 1(a). In this stress analysis, boundary conditions
were expressed with a Fourier series and the results
obtained for each component were superimposed to
obtain the stress distributions in an arbitrary vertical
cross-section (Fig. 1). The core was an HQ (core radius/
borehole radius=0.6612) with a length of 4 times the
core radius (R). The tip of the core bit was assumed to
be a semi-torus. The z coordinate (core axis=Z-axis)
was set to be zero at the beginning of the semi-circular
cross-section of the bit (Fig. 1(b)). Since the core was
sufficiently long, no appreciable stress was produced in
the upper part of the core. The model was composed of
957 6-point and 8-point elements with 2688 nodes.
Poisson’s ratio was assumed to be 0.25. A tensile
principal stress analysis was carried out for a region
within and below the core stub, shown by thick lines in
Fig. 1(b), which is referred to the analyzed region below.
As shown in Fig. 2, the entire horizontal cross-section of
the core was divided into 168 equal-area elements for the
analyzed region (z=R ¼ �4 to 4); 7 for the radial
direction (from r1 to r7; see Fig. 1(b) and Fig. 2) and
24 for the circumferential direction (from y1 to y24; see
Fig. 2), and the stresses at the center of gravity for each
element were analyzed. Thus, the stress in each element
is equivalent in integrating the effects on the horizontal
cross-section of the core. The circumferential coordinate
(y) was taken from the X -axis to the Y -axis, as shown in
Fig. 2. The maximum principal stress lies within the
ZX -plane, as will be described later.
If all of the principal in situ stresses are in compres-
sion, the maximum (s1) and intermediate (s2) principal
stresses normalized by the mean stress (sm) lie within the
triangular area shown in Fig. 3. To consider general in
situ stress conditions, 11 sets of normalized principal
stress magnitudes were adopted for the analysis, as
shown with open circles in Fig. 3. The magnitudes of the
normalized principal stresses are summarized in Table 1.
The direction of s1 was limited to within the ZX -plane
and the inclination angle of s1 from the core axis (f1)
was varied from 01 to 901 in steps of 301 (Fig. 4). The
direction of s2 was varied in the plane perpendicular to
s1 by increasing the angle (g2) from the Y -axis towards
the X -axis from 01 to 901 in steps of 301. As a result, the
analysis was performed for 77 in situ stress conditions.
2.2. Analysis of maximum tensile stress
Fig. 5 shows examples of the tensile principal stress
distribution along the core axis at certain positions in
the horizontal cross-section (s1=sm ¼ 1:5; s2=sm ¼ 1;
s3=sm ¼ 0:5; f1 ¼ 601; g2 ¼ 301). The tensile principal
stress (st1) and the z coordinate are normalized by the
mean in situ stress (sm) and the core radius (R),
respectively. As a horizontal cross-section approaches
the bottom of the core, the tensile principal stress
increases to a maximum near the bottom of the core and
decreases to be negligible at z=R greater than 3. This
figure also shows an example of the history of the tensile
principal stress experienced by the core at an arbitrary
position in the horizontal cross-section until the boring
is complete, if no macroscopic fractures are created. The
K. Sakaguchi et al. / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380368
same tendency in the tensile stress distribution was
observed for all of the in situ stress conditions used in
this study. We examined the maximum tensile stress
experienced by the core along the core axis at each
position in the horizontal cross-section, since the
damage due to tensile stress may be determined by
the maximum tensile stress (st max) and, accordingly, the
Fig. 1. Mesh diagram used in FEM analysis for (a) the whole model and (b) an enlarged diagram for part (b) in (a).
Fig. 2. Points at which tensile stress was analyzed in an arbitrary
horizontal cross-section of a core.
Fig. 3. Magnitudes of principal in situ stresses normalized by the mean
stress (s1=sm; s2=sm) used in FEM analysis.
K. Sakaguchi et al. / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380 369
damage in the core may be governed by the distribution
of st max accumulated for the cross-section of the core.
To obtain the cumulative distribution of st max; an
equal-area stereonet was divided into 20� 20 subdivi-
sions, and the magnitude of st max was accumulated in
one of these subdivisions according to the direction. The
results are shown as a contour diagram. The procedure
for determining a contour diagram is as follows:
(1) Prepare an equal-area stereonet with 20� 20
subdivisions (Xi; Yj) (upper hemisphere)(Fig. 6(a)), and
initialize the cumulative value Sij for each subdivision:
Sij ¼ 0:
(2) For r ¼ r1 to r2 and for y ¼ y1 to y24:
(i) Find the maximum tensile stress with respect to the
z coordinate. Let the direction of the maximum tensile
stress be (l, m, n) and the magnitude be st max (Fig. 6(b)).
(ii) Calculate the inclination angle relative to the core
axis (fe) and the azimuth (ye) of the direction (l; m; n):
tan fe ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l2 þ m2
p
n
and tan ye ¼
m
l
:
(iii) Calculate the position (xe; ye) of the direction
(l; m; n) in the equal-area stereonet:
xe ¼
ffiffiffiffi
2
p
sin
fe
2
cos ye and ye ¼
ffiffiffi
2
p
sin
fe
2
sin ye:
(iv) Find the subdivision (Xi; Yj) in the stereonet in
which (xe; ye) falls, and add the normalized magnitude
st max=sm to Sij:
Sij ¼ Sij þ st max=sm:
(3) Create contour lines of Sij by using software that is
commercially available.
Fig. 7 explains the meaning of symbols used in a
contour diagram that will be shown below. The smaller
circle in the diagram indicates an inclination angle of 601
Table 1
Magnitudes of principal in situ stresses normalized by the mean stress (s1=sm; s2=sm; s3=sm) used in this study
No. 1 2 3 4 5 6 7 8 9 10 11
s1=sm 1 1.5 2 2.5 3 1.25 1.5 1.5 2 2 2.5
s2=sm 1 0.75 0.5 0.25 0 1.25 1.5 1 1 0.75 0.5
s3=sm 1 0.75 0.5 0.25 0 0.5 0 0.5 0 0.25 0
Fig. 5. Examples of the distribution of normalized tensile principal
stress (st1=sm) along the core axis (z=R) at certain positions in the
cross-section (s1=sm ¼ 1:5; s2=sm ¼ 1; s3=sm ¼ 0:5; s1 ¼ 601;
s2 ¼ 301).
Fig. 4. Definitions for the directions of principal in situ stresses.
Fig. 6. (a) An equal-area stereonet with 20� 20 subdivisions (upper
hemisphere) and (b) the magnitude (st max) and direction (l; m; n) of the
maximum tensile stress determined with respect to the z coordinate for
a position in the cross-section.
K. Sakaguchi et al. / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380370
from the core axis. The rectangles indicate the directions
of in situ principal stresses (s1; s2; s3). The value of the
outermost contour is 0.1 and the numeral shown at the
bottom-right of the diagram indicates the maximum
value. Note that the contour lines exist in a small area.
The letter (c) indicates the classification of the cumula-
tive distribution, as will be described later. Fig. 8 shows
examples of the contour diagram for the cumulative
distribution of normalized maximum tensile stress
(st max=sm), which was accumulated for the central part
of the core (r ¼ r1 and r2; 28.6% of the horizontal cross-
sectional area) (s1=sm ¼ 1:5; s2=sm ¼ 1; s3=sm ¼ 0:5).
The cumulative distribution of st max is clearly concen-
trated nearly in the same direction as the minimum
principal stress (s3). This characteristic was found with
all of the in situ stress conditions used in this study,
except for special cases in which the direction of s3 is not
unique (s2 ¼ s3). The cumulative distribution of st max
can be classified into four types according to the
inclination of s3 from the core axis (f3). When f3 ¼
01; the direction of the maximum cumulative st max is
equal to that of s3 (type (a)). When 01of3o601; the
azimuth of the maximum cumulative st max is almost
equal to that of s3; but the inclination from the core axis
is smaller than that of s3 (type (b)). When 601of3o901;
the azimuth of the maximum cumulative st max is
deviated from that of s3 towards the Y -axis, and the
inclination from the core axis is greater than that of s3
(type (c)). When f3 ¼ 601; both types (b) and (c) appear.
Fig. 8. Examples of contour diagrams in an equal-area stereonet of the cumulative maximum tensile stress normalized by the mean in situ stress
(st max=sm) accumulated for r ¼ r1 and r2 (28.6% of the horizontal cross-section) (upper hemisphere, s1=sm ¼ 1:5;s2=sm ¼ 1; s3=sm ¼ 0:5).
Fig. 7. Meaning of symbols in a contour diagram in an equal-area
stereonet of the cumulative maximum tensile stress normalized by the
mean in situ stress (st max=sm).
K. Sakaguchi et al. / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380 371
When f3 ¼ 901; the direction of the maximum cumula-
tive st max again agrees with that of s3 (type (d)).
Fig. 9 shows examples of the contour diagram of the
cumulative distribution of st max=sm accumulated for the
central part of the core (r ¼ r1 and r2) when s2 ¼ s3
(s1=sm ¼ 2; s2=sm ¼ s3=sm ¼ 0:5). The direction of the
maximum principal stress (s1) and the plane perpendi-
cular to s1 are shown in these diagrams. When the
inclination of s1 from the core axis (f1) is 01, the
direction of the maximum cumulative st max is dispersed
in the horizontal plane (XY-plane), since the tensile
stress was evaluated at every 151 in the circumferential
direction. When f1 ¼ 301 or 601, the direction of the
maximum cumulative st max is concentrated in nearly
the same direction as the Y -axis. When f1 ¼ 901;
the maximum cumulative st max is concentrated in the
direction of the core axis. For all cases in which s2 ¼ s3;
the direction of the maximum cumulative st max lies
approximately within the plane in which s3 (=s2) exists.
The vector of st max evaluated for a central part of the
core was summed to determine a unique direction for
the cumulative distribution of st max (denoted by sa) for
all stress conditions except cases in which s2 ¼ s3: The
results are shown in Fig. 10 for the principal axes of in
situ stress. Figs. 10(a) and (b) show the directions of
sa=sm accumulated for r1 and r2 (28.6% of the
horizontal cross-sectional area) and for r1 to r4 (57.1%
of the horizontal cross-sectional area), respectively. The
directions of sa=sm for these different areas are similar,
and are close to that of the minimum principal stress
(s3) regardless of the in situ stress conditions. Since the
direction of the tensile stress varies greatly at the outer
part of the core [12], the direction of sa=sm was more
dispersed and was less meaningful as the area considered
increased. The results shown in Fig. 10 are consistent
with core discing because core discing occurs due to
tensile stress in the direction of the core axis when s3 is
nearly in the same direction as the core axis (type (a))
[12]. We carried out a similar analysis for every tensile
principal stress, instead of only for the maximum tensile
stress, by accumulating the tensile stress along the core
axis. The results with respect to the direction are almost
Fig. 9. Examples of contour diagrams in an equal-area stereonet of the cumulative maximum tensile stress normalized by the mean in situ stress
(st max=sm) accumulated for r ¼ r1 and r2 (28.6% of the horizontal cross-section) (upper hemisphere, s1=sm ¼ 2;s2=sm ¼ s3=sm ¼ 0:5).
Fig. 10. Stereoplotted directions of the sums of the maximum tensile stress vectors in the principal axes of in situ stress, accumulated (a) for r ¼ r1
and r2 (28.6% of the horizontal cross-section) and (b) for r ¼ r1 to r4 (57.1%).
K. Sakaguchi et al. / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 367–380372
identical to those obtained for the maximum tensile
stress described above. This means that the directions of
the tensile principal stresses produced in the central part
(at least for 57.1% of the horizontal cross-sectional
area) of the core are fairly uniform.
3. Damage analysis of the core
3.1. Evaluation of crack parameters by DSCA
In DSCA, strain gauges are glued to a rock specimen,
which is usually cubic, on three surfaces perpendicular
to each other (see Fig. 11) and hydrostatic pressure is
applied both to the rock specimen and to fused silica as
a reference material. The use of fused silica is to
compensate errors due to drifts and temperature effects
in the measurement of strain by taking the difference
between strains of the rock specimen and those of fused
silica [15]. The difference between these strains is called
differential strain [15]. Fig. 11 shows a schematic of
normal strain of rock specimen in a certain direction
versus hydrostatic pressure. Note that the strain in this
figure is not a differential strain for convenience. At low
pressures, open cracks are closed, to cause a large
increase in strain. The initial gradient in the diagram
gives an effective linear compressibility (beff ). With an
increase in pressure, the gradient gradually decreases
due to the closure of cracks with smaller aspect ratios.
When all of the effective cracks are closed, the gradient
takes a constant value,