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<div class="articledetails article-header clearfix">
<p class="art-type">Research Article</p>
<p class="art-title">Applying Reactivation Tendency Analysis and
Mohr-space to Evaluate Shear Strength Decrease
and Anisotropy with Pre-existing Weakness(es) under Uniform Stress State</p>
<p class="art-author"><?php $authors="Hengmao Tong*"; echo (stristr($authors,$coauthor))?str_replace($coauthor,"<a href='".$extpath."authors/".$courl."' target='_blank'>".$coauthor."</a>",$authors):$authors; ?></p>
<p class="art-affl">State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China
</p>
<p class="art-aff"><b>*Corresponding author: <?php $corresponding_author="Hengmao Tong"; echo ($coauthor!="" && $coauthor==$corresponding_author)?"<a href='".$extpath."authors/".$courl."' target='_blank'>".$coauthor."</a>":$corresponding_author;?></b>, State Key Laboratory of Petroleum
Resources and Prospecting, China University of Petroleum, Beijing 102249, China, Tel: +86 10 89734607, Fax: +86 10 89734158, E-mail: <a href="mailto:tonghm@cup.edu.cn">tonghm@cup.edu.cn</a>&nbsp;,<a href="mailto:tong-hm@163.com">tong-hm@163.com</a>
</p>
<p class="art-aff"><b>Received:</b> February 22, 2017
<b>Accepted:</b> April 7, 2017
<b>Published:</b> April 13, 2017</p>
<p class="art-aff"><b>Citation:</b> Tong H. Applying
Reactivation Tendency Analysis and Mohrspace
to Evaluate Shear Strength Decrease and
Anisotropy with Pre-existing Weakness(es) under Uniform Stress State. <i>Int J Petrochem Res.</i> 2017; 1(1): 31-39. doi: <a href="https://doi.org/10.18689/ijpr-1000107">10.18689/ijpr-1000107</a></p>
<p class="art-aff"><b>Copyright:</b> &copy; 2017 The Author(s). This
work is licensed under a Creative Commons
Attribution 4.0 International License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the
original work is properly cited.</p>
<p><a href="<?php echo $extpath;?><?php echo $jres['journal_link'];?>/ijpr-1000107.pdf" class="btn btn-danger pull-right" target="_blank">Download PDF</a></p>
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<div class="articlecontent">
<p class="art-subhead">Abstract</p>
<p class="art-para">Understanding the mechanical controls on shear strength decrease due to
preexisting weakness is a fundamental problem in tectonic studies. In this study, by
applying reactivation tendency analysis theory, a theoretical framework and defined
Shear-strength Coefficient (f<sub>d</sub>) are developed for evaluating the shear-strength decrease
and anisotropies due to the presence of preexisting weakness(es). The proposed study
managed to overcome the restrictions of previous studies assumption that a pre-existing
weakness plane contains the intermediate stress (&#963;<sub>2</sub>
) and vertical or horizontal
orientations of principal stresses (Andersonian stress state). A new graphical technique (Mohr-space) was utilized to predict the shear-strength decrease and anisotropies
caused by preexisting weakness(es). The Mohr-space technique made easier to visualize
the state of stress and results of shear strength changes and able to build the quantitative
and intuitive relationship between Shear-strength Coefficient (f<sub>d</sub>) and weakness relativeorientation(&#952;&#700;,&#966;&#8217;), weakness mechanical properties (C<sub>w</sub> and &#181;<sub>w</sub>) and relative <img src="<?php echo $imgpath;?>images/IJPR-107-teq1.PNG"></a>
 in
any uniform tri-axial stress state.. In this study, Shear-strength decrease and anisotropies
of a rock sample are evaluated theoretically, and shear strength properties and
deformation characteristics of a geological body with multiple pre-existing weaknesses
are analyzed and predicted. </p>
<p class="art-para"><b>Key words:</b> Shear strength; preexisting weakness; reactivation tendency; Mohr-space; sandbox experiment. </p>
<p class="art-subhead">Introduction</p>
<p class="art-para">It is well documented that pre-existing weaknesses (fracture planes or faults, layering, fabrics etc.) can lead to decrease of shear strength and strength anisotropies
<a href='#1'>[1</a>-<a href='#11'>3]</a>, and the potential strength anisotropy created by pre-existing weakness is
considerable <a href='#4'>[4]</a>. Ranalli (1990) <a href='#5'>[5]</a> proposed a unified quantitative model to evaluate
strength anisotropies by the pre-existing weakness in terms of the three tectonic faulting
regimes when the weakness plane contains the intermediate stress (&#963;<sub>2</sub>
). However, to our
knowledge strength anisotropies evaluation with pre-existing weakness is limited to
two-dimensional cases (weakness plane containing &#963;<sub>2</sub>
) before slip-tendency theory was
proposed <a href='#6'>[6]</a>. However, strength anisotropies evaluation with slip-tendency <a href='#7'>[7]</a> is
confined to the qualitative analysis only. In addition, there is also some limitation in slip-tendency theory, i.e. the principal stresses are oriented either
vertically or horizontally, and the cohesive strength of all preexisting
weakness is neglected <a href='#6'>[6]</a>. However, principal stress
direction may depart significantly from vertical and horizontal
with depth in the upper crust <a href="#8">[8</a>-<a href="#9">9]</a>. Furthermore, some
weakness zones may possess cohesive strength, particularly in
cemented faults, and properties of weaknesses may vary <a href='#4'>[4]</a>. For a example, Sibson (1974) <a href='#10'>[10]</a> showed that cementation of
a fault zone can create 1.0-MPa cohesive strength or more. In
this study, Shear-strength Coefficient (f<sub>d</sub>) is defined, and
Reactivation Tendency analysis theory <a href='#11'>[11]</a>, and Mohr-space
<a href='#12'>[12]</a> are applied to evaluate strength decrease and anisotropies
due to the presence of pre-existing weakness(es) with arbitrary
azimuth in any uniform tri-axial stress state. Strength decrease
and anisotropies caused by weakness may be intuitively
simplified and quantitatively analyzed and evaluated with
Mohr space.</p>
<p class="art-para"><b>Mohr-space, pole (&#963;<sub>n</sub>, &#964;<sub>n</sub>) of any oriented plane in tri-axial
stress state</b></p>
<p class="art-para">Mohr diagrams, which was introduced by Otto Mohr (1882), is one of the most used and useful tools in structural
geology <a href='#13'>[13]</a>, and has been used extensively in mechanical
problems, such as stress analysis, failure envelopes, fractures
opening and reactivation <a href='#14'>[14]</a> <a href='#15'>[15]</a> <a href='#16'>[16]</a> <a href='#17'>[17]</a>. Although real
three-dimensional Mohr diagrams do exist for any tri-axial
stress state <a href='#13'>[13]</a>, Mohr-diagram is usually considered to be
two-dimensional <a href='#13'>[13]</a>, which is well known as Mohr-circles. Mohr-cyclides, which can be used to represent any second
rank tensor (including stress tensor) was introduced by Coelho
and Passchier (2008) <a href='#13'>[13]</a>. However, the stress components (&#963;<sub>n</sub>
, &#964;<sub>n</sub>
) of any given plane are the most important, and the
general diagrams of Mohr-cyclides are not so convenient to
be used and prepared. In contrast, the Mohr space, which was
proposed by Tong and Yin (2011) <a href='#11'>[11]</a> can be used to express
the normal stress and shear stress of a plane with an arbitrary
azimuth in an arbitrary three-dimensional stress state.</p>
<p class="art-para">In any given stress state, the pole (&#963;<sub>n</sub>
, &#964;<sub>n</sub>
) of any plane (i.e. pre-existing weakness plane, defined by dip direction &#952; and
dip angle &#966;) is either located on the three Mohr-circles (i.e. P<sub>1</sub>
on large Mohr circle &#963;<sub>1</sub>
-&#963;<sub>3</sub>
, P<sub>2</sub>
 and P<sub>3</sub>
 on small Mohr circles &#963;<sub>1</sub>
-
&#963;<sub>2</sub>
 and &#963;<sub>2</sub>
-&#963;<sub>3</sub> respectively, Figure 1) or in the area (grey area in
Figure 1) between large Mohr-circle and two small Mohrcircles(i.e. P<sub>4</sub>
 in Figure 2) in &#963;<sub>n</sub>
-&#964;<sub>n</sub>
 coordinate system <a href='#2'>[2</a>,<a href='#18'>18]</a>. There is one to one correspondence relationship between any
plane and its pole (&#963;<sub>n</sub>
, &#964;<sub>n</sub>
) in &#963;<sub>n</sub>
-&#964;<sub>n</sub>
 coordinate system. This
space (i.e. the three Mohr-circles and the area between them) is called Mohr-space <a href='#11'>[11]</a>. With the contour lines (pink and
green dotted lines in Figure 2) of plane angles (pseudo-dip
direction &#952;&#8217; and pseudo-dip angle &#966;&#8217;, &#952;&gt;= &#945;&gt; +90 &#176;, where &#945;&gt; is the angle between the plane and the intersection line of the
plane and &#963;<sub>2</sub>
 -&#963;<sub>3</sub> plane; &#966;&gt; the angle between the plane and &#963;<sub>2</sub>
-&#963;<sub>3</sub>
 plane), the pole of the plane can be found in the Mohrspace. The relationship between &#952;&gt;, &#966;&gt;, &#952;, and &#966; can be
determined with transformation of coordinates <a href='#11'>[11]</a>, and the
contour lines of angles can be compiled with equation (1) <a href='#11'>[11]</a>. </p>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-eq1.PNG" class="img-responsive center-block"/></div>
<p class="art-para">Coordination definition and transformation can be seen
in Tong and Yin (2011) <a href='#11'>[11]</a>.</p>
<p class="art-para">Thus, by applying Mohr-space, normal stress (&#963;<sub>n</sub>
) and
shear stress (&#964;<sub>n</sub>
) of any plane can be conveniently and
intuitively determined, and the changes of normal and shear
stresses with plane relative-orientation (&#952;&acute;,&#966;&gt;) can be easily
analyzed. </p>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-figure1.PNG" class="img-responsive center-block"/></div>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-figure2.PNG" class="img-responsive center-block"/></div>
<p class="art-para"><b>Reactivation Tendency Factor and its expressions in
Mohr-space</b></p>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-f2.PNG" class="img-responsive center-block"/></div>
<p class="art-para">(where &#964;<sub>n</sub>
, &#963;<sub>n</sub>
 are shear and normal stresses on the weakness
plane, respectively; &#964;<sub>n</sub><sup>w</sup> is the corresponding rupture shear
strength; C<sub>w,</sub> and &#181;<sub>w</sub> are cohesiveness and frictional coefficient
of weakness plane, respectively), which is extended from Sliptendency
<a href='#6'>[6]</a> and is used to evaluate the reactivation likelihood
of pre-existing weakness, is proposed by Tong and Yin<a href='#11'>[11]</a> (2011). Reactivation Tendency Factor (f<sub>a</sub>
) is determined by its
relative-orientation (&#952;&acute;,&#966;&acute;), mechanical properties (C<sub>w</sub>, &#181;<sub>w</sub> ) of
the weak plane, and the stress tensor. f<sub>a</sub>
=1.0 shows that the
pre-existing weakness is in critical state of reactivation; when
f<sub>a</sub>
&lt;1, it has reactivated and when f<sub>a</sub>
&gt;1, it is in a stable state. The weaknesses, which f<sub>a</sub>
&#8805;1.0 in the critical stress state of
Coulomb rupture, will reactivate one by one in progressive
deformation according to their f<sub>a</sub>
 value order.</p>
<p class="art-para">Applying Mohr-space, f<sub>a</sub>
 can be intuitively expressed
accordingto the following steps. (1) The lines &#964;<sub>n</sub><sup>w</sup>=&#177; (C<sub>w</sub> + &#181;<sub>w</sub>&#963;<sub>n</sub>
) are two symmetrical straight lines (when &#181;<sub>w</sub> is a constant)or
curved lines(b and b&acute; in Figure 2, when &#181;<sub>w</sub> is variable) in &#963;<sub>n</sub>
-&#964;<sub>n</sub>
coordinate system, which were called &#8220;weakness reactivation
lines&#8221; <a href='#19'>[19]</a> or &#8220;shear strength line of weakness &#8221; <a href='#11'>[11]</a>, where (0, C<sub>w</sub> ) is the starting point, &#181;<sub>w</sub> is the slope of the line. If C<sub>w</sub> = 0 (weakness without cohesion), the line cross the point of origin. (2) The pole (&#963;<sub>n</sub>
, &#964;<sub>n</sub>
) of a weakness plane (P in Figure 2) can be
easily plotted in Mohr-space according to its relative-orientation (&#952;&acute;and &#966;&acute;) and its &#963;<sub>n</sub>
, &#964;<sub>n</sub>
w can be intuitively determined (Figure
2). As such, f<sub>a</sub>
 (&#964;<sub>n</sub>
/&#964;<sub>n</sub>
w) of the weakness plane can be easily and
intuitively determined (PP<sub>0</sub>
/PP<sub>R</sub> in Figure 2). (3) When the pole
of the weakness plane is on its &#8220;weakness reactivation line&#8221; (P<sub>1</sub>
in Figure 2), f<sub>a</sub>
=1.0 and the pre-existing weakness is in the
critical state of reactivation. However, when the pole is located
outside the two &#8220;weakness reactivation lines&#8221; (striped yellow
area in Figure 2, i.e. P<sub>2</sub>
), f<sub>a</sub>
>1.0 and the weakness plane has
reactivated. When the pole is located inside the two weakness
reactivation lines (yellow area without stripes in Figure 2, i.e. P<sub>3</sub>
), f<sub>a</sub>&lt;1 indicating that the plane is in a stable state.</p>
<p class="art-para"><b>The definition of Shear-strength coefficient and its
relationship to Reactivation Tendency Factor of
pre-existing weakness</b></p>
<p class="art-para">As a weakness plane may be reactivated when stress is
below Coulomb rupturing stress state, a pre-existing weakness
will lead to decrease in shear strength <a href='#1'>[1]</a> <a href='#2'>[2]</a> <a href='#7'>[7]</a>. However, the
weakness with f<sub>a</sub>
&lt;1.0 will not reactivate at critical stress of
Coulomb rupture <a href='#11'>[11]</a> according to Reactivation Tendency
analysis theory. This means that not all weakness planes will lead
to decrease in shear strength. &#65293;Therefore, in the following
shear strength analysis, we only consider weaknesses with f<sub>a</sub>&#8805;1.0.</p>
<p class="art-para">Consider a weakness plane (P<sub>w</sub>) with a normal n and its
reactivation tendency factor f<sub>a</sub>
&#8805;1.0 at critical uniform stress
state (&#963;<sub>1</sub>
, &#963;<sub>2</sub>
 and &#963;<sub>3</sub>
) of Coulomb rupture (Figure 3). P<sub>w</sub> has
been reactivated at critical stress state of Coulomb rupture
<a href='#11'>[11]</a>, and its pole in Mohr-space is (&#963;<sub>n</sub>
, &#964;<sub>n</sub>
) (Point A in Figure 3). With the same normal stress &#963;<sub>n</sub>
, P<sub>w</sub> will be at critical state of
reactivation when its shear stress is &#964;<sub>n</sub><sup>w</sup> (Point B in Figure 3, &#963;&acute;1
, &#963;&acute;2
 and &#963;&acute;3
 are the three principal stresses) and &#964;<sub>n</sub><sup>w</sup><&#964;<sub>n</sub>
. The
large Mohr circle (&#963;&acute;1
-&#963;&acute;3
 circle) of critical stress state of
weakness reactivation is smaller than the &#963;<sub>1</sub>
-&#963;<sub>3</sub>
 Mohr circle of critical stress state of Coulomb rupture (Figure 3), that means
&#963;&acute;1 - &#963;&acute;3
 < &#963;<sub>1</sub> - &#963;<sub>3</sub>
. Thus, in Mohr-space (Figure 3), it is very easy
to understand that a weaknesswith f<sub>a</sub>
>1.0will lead to decrease
in shear strength. </p>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-figure3.PNG" class="img-responsive center-block"/></div>
<p class="art-para">Because shear strength is related to normal stress and
increases with increasing of normal stress, relative instead of
absolute shear strength is more useful to consider.In order to
quantitatively evaluate the shear strength decrease due to the
existence of a weakness, we define the parameter f<sub>d</sub> called
Shear-strength coefficient (f<sub>d</sub> = &#964;<sub>n</sub><sup>w</sup> /&#964;<sub>n</sub>
, where &#964;<sub>n</sub><sup>w</sup> is the critical
shear stress of weakness reactivation, &#964;<sub>n</sub>
 is the shear stress on
the same plane with the same normal stress &#963;<sub>n</sub>
 at critical stress
state of Coulomb rupture (&#963;<sub>1</sub>
, &#963;<sub>2</sub>
 and &#963;<sub>3</sub> stress state in Figure
3)). It is easy to understand that the smaller Shear-strength
coefficient (f<sub>d</sub> ) is the larger decrease will be in shear strength.</p>
<p class="art-para"> When f<sub>a</sub>
&#8805;1.0, &#964;<sub>n</sub><sup>w</sup> can never be greater than &#964;<sub>n</sub>
 (as stress will
drop down when weakness reactivates according to
Reactivation Tendency analysis theory <a href='#20'>[20]</a> <a href='#21'>[21]</a>. Furthermore, when f<sub>a</sub>
&#8804; 1.0, which means that weakness will not be reactivated
at critical stress state of Coulomb rupture and will not lead to
shear strength decrease(i.e., f<sub>d</sub>=1.0). Therefore, shear-strength
coefficient can never be greater than 1.0 (f<sub>d</sub> &#8804; 1.0).</p>
<p class="art-para">It is easy to find the following relationship between f<sub>d</sub> and
Reactivation Tendency Factor (f<sub>a</sub>
) at critical stress state of
Coulomb rupturing according to the definition of shearstrength
coefficient and Reactivation Tendency analysis
theory <a href='#11'>[11]</a></p>
 
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-eq2.PNG" class="img-responsive center-block"/></div>
<p class="art-para">The relationship between Shear-strength Coefficient (f<sub>d</sub>
 ) and
the ratio of Differential stress can be derived at Coulomb rupture ((&#963;<sub>1</sub>
- &#963;<sub>3</sub>
) C =&#963;<sub>1</sub>
- &#963;<sub>3</sub>
 in Figure 3) and at critical state of weakness
reactivation ((&#963;<sub>1</sub>
- &#963;<sub>3</sub>
) W =&#963;<sub>1</sub>
&acute;- &#963;<sub>3</sub>
&acute; in Figure 3) according to the
definition of Shear-strength Coefficient and using Mohr-space, and it means that f<sub>d</sub>
 = (&#963;<sub>1</sub>
- &#963;<sub>3</sub>
)<sub>W</sub> / (&#963;<sub>1</sub>
- &#963;<sub>3</sub>
)<sub>C</sub>. As the relative position
of the weakness pole in (&#963;<sub>1</sub>
, &#963;<sub>2</sub>
, &#963;<sub>3</sub>
)-Mohr-space (point A) and in (&#963;<sub>1</sub>
&acute;, &#963;<sub>2</sub>
&acute;, &#963;<sub>3</sub>
&acute;)-Mohr-space (point B) is the same, OA/OB is equal to
the ratio of radius of the two big Mohr circle ( 2
&#963;<sub>1</sub> &#8722;&#963; 3 and
2
&#963;<sub>1</sub>
&acute;&#8722;&#963; 3&acute;), which means that OA/OB = (&#963;<sub>1</sub>
- &#963;<sub>3</sub>
)/(&#963;<sub>1</sub>
&acute;- &#963;<sub>3</sub>
&acute;) = (&#963;<sub>1</sub>
- &#963;<sub>3</sub>
) C/ (&#963;<sub>1</sub>
- &#963;<sub>3</sub>
) W. While two triangles AOC and BO&acute;C are similar, as a result, BC /AC
=OB /OA and f<sub>d</sub>
 =&#964;<sub>n</sub>
w /&#964;<sub>n</sub>
 = BC /AC = OB / OA = (&#963;<sub>1</sub>
- &#963;<sub>3</sub>
) C/ (&#963;<sub>1</sub>
- &#963;<sub>3</sub>
) W, which means that f<sub>d</sub>
 = (&#963;<sub>1</sub>
- &#963;<sub>3</sub>
) C/ (&#963;<sub>1</sub>
- &#963;<sub>3</sub>
) W when f<sub>a</sub>
&#8805;1.0. So, f<sub>d</sub>
 (1/
f
a
) is related to weakness orientation (&#952;&acute;and &#966;&acute;), mechanical
properties (C<sub>w</sub>, &#181;<sub>w</sub> ) and the stress tensor according to
Reactivation Tendency analysis theory <a href='#11'>[11]</a>, and it can be
quantitatively calculated using equations 1 and 2. Since f<sub>d</sub> is
related to weakness relative-orientation, weakness can lead to
shear-strength anisotropies.</p>
<p class="art-subhead">Analysis of shear strength affection
factors due to pre-existing weakness(es)</p>
<p class="art-para">Shear-strength coefficient (f<sub>d</sub>) is used to discuss how the
related factors (mechanical properties and orientation of
weakness, and <img src="<?php echo $imgpath;?>images/IJPR-107-teq2.PNG">) affect shear strength due to preexisting
weakness in triaxial stress state. As f<sub>d</sub> = 1/f<sub>a</sub>
, and fa
 can be
intuitively and conveniently expressed in Mohr-space, Mohrspace
is applied to do these analysis.</p>
<p class="art-para"><b>The relationship between f<sub>d</sub> and weakness relativeorientation (&#952;&acute;,&#966;&acute;)</b></p>
<p class="art-para">It is easy to see that when &#952;&acute;=90&#176;, and &#966;&acute; =&#177;&#966;&acute;0 = &#177; (45&#176;+0.5arctg (&#181;<sub>w</sub>) (&#966;&acute;0
 is usually about 60&#176;) (points A and A&acute; in
Figure 4), f<sub>a</sub>
 reaches the highest value, whereas f<sub>d</sub>
 reaches the
smallest value (Figure 4). Therefore, (90&#176;, &#177; &#966;&acute;0
) (two points) are
the two weakest relative-orientation of the weakness plane(s). As
&#952;&acute; decreases from 90&#176;, and &#966;&acute; deviates from &#966;&acute;0
, f<sub>d</sub>
 will increase. In
Mohr-space (Figure 4), the points of intersection between
&#8220;weakness reactivation lines&#8221; and Mohr circles are demarcation
points of weakness reactivation (B1
, B2
 and B&acute;1
, B&acute;2
 on &#963;<sub>1</sub>
-&#963;<sub>3</sub>
 large
Mohr circle, C1
, C2
 and C&acute;1
, C&acute;2
 on &#963;<sub>2</sub>
-&#963;<sub>3</sub>
 small Mohr circle, Figure
4). The corresponding &#966;&acute; of B1
, B2
, B&acute;1
 and B&acute;2
 is &#966;&acute;1
, &#966;&acute;2
, &#966;&acute;1
, &#966;&acute;2
, and the corresponding &#952;&acute; of C1
, C2
 and C&acute;1
, C&acute;2
 is &#952;&acute;1
, &#952;&acute;2
, (180&#176; - &#952;&acute;1
) and (180&#176; -&#952;&acute;2
) respectively. It is easy to find that when &#966;&acute;<&#966;&acute;1
 or
&#966;&acute;> &#966;&acute;2
, or &#952;&acute;<&#952;&acute;1
 or &#952;&acute;2
<&#952;&acute;&lt;180&#176;-&#952;&acute;2
 or &#952;&acute;>180&#176;-&#952;&acute;1
, the weakness
will all locate inside the two weakness reactivation lines (grey
area in Figure 4) and f<sub>a</sub> will always &lt;1.0, which means there exist
critical angle &#966;&acute;1
, &#966;&acute;2
 for &#966;&acute; and &#952;&acute;1
,&#952;&acute;2
 and (180&#176; - &#952;&acute;1
),(180&#176; -&#952;&acute;2
) for &#952;&acute;, when &#966;&acute;<&#966;&acute;1
 or &#966;&acute;> &#966;&acute;2
, or &#952;&acute;<&#952;&acute;1
or &#952;&acute;2
<&#952;&acute;&lt;180&#176;-&#952;&acute;2
 or
&#952;&acute;>180&#176;-&#952;&acute;1
, fa will always &lt;1.0 and the weakness cannot
reactivate and will not lead shear strength decrease (f<sub>d</sub>
 = 1.0).</p>
<p class="art-para">Point A and A&acute; (Figure 4) are two Coulomb rupture points, and
f<sub>d</sub>
 of the same oriented weakness plane is the lowest. Weakness
plane with low f<sub>d</sub>
 value concentrates around point A (or A&acute;) in the
yellow area of Figure 4(&#952;&acute;=90&#176;~75&#176; and &#966;&acute;=&#966;&acute;0
&#177; 15&#176;). Change of f<sub>d</sub>
shows that weakness relative-orientation is the predominant factor which leads to shear strength anisotropies. It is noteworthy
tounderline that (&#952;&acute;, &#966;&acute;) of weakness is determined jointly by
orientations of weakness and three axes of principal stresses.</p>
 
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-figure4.PNG" class="img-responsive center-block"/></div>
<p class="art-para"><b>The relationship between f<sub>d</sub> and weakness mechanical
properties (C<sub>w</sub> and &#181;<sub>w</sub>)</b></p>
<p class="art-para">In Mohr-space (Figure 5), it is easy to find that the change
of mechanical property of a weakness plane (C<sub>w</sub> and &#181;<sub>w</sub>) will
cause the change of the position (determined by C<sub>w</sub>) and the
slope (determined by &#181;<sub>w</sub>) of weakness reactivation lines. It is easy
to understand that f<sub>d</sub> will decrease with decreasing of C<sub>w</sub> and &#181;<sub>w</sub>.</p>
 
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-figure5.PNG" class="img-responsive center-block"><img src="<?php echo $imgpath;?>images/IJPR-107-figure5a.PNG" class="img-responsive center-block"/></div>
<p class="art-para">Weakness mechanical properties are the predominant
factors which lead to shear strength decrease. For example, low friction coefficient (&#181;<sub>w</sub>< 0.2 <a href='#22'>[22]</a>) and no cohesion along
the weakness may lead to more than 90% shear strength
decrease (f<sub>d</sub>< 0.1).</p>
<p class="art-para"><b>The relationship between f<sub>d</sub>
 and relative value of &#963;<sub>2</sub></b></p>
<p class="art-para">Shear strength is also related to relative value of <img src="<?php echo $imgpath;?>images/IJPR-107-teq1.PNG"> <a href='#7'>[7]</a>. However, our analysis in Mohr-space provided
intuitive results which are easy to follow. It is easy to see that
relative <img src="<?php echo $imgpath;?>images/IJPR-107-teq1.PNG"> determines the shape of Mohr-space (Fig. 6). The relationship between <img src="<?php echo $imgpath;?>images/IJPR-107-teq2.PNG"> and f<sub>d</sub> is complicated and
depends on &#952;&acute; and &#966;&acute;, particularly &#966;&acute;(Fig. 6). In general, &#966;&acute; can
be divided into 3 intervals: &#966;&acute;&#8805;70&#176;, 40&#176;<&#966;&acute;&lt;70&#176;, and &#966;&acute;&#8804;40&#176;. When &#966;&acute;&#8805;70&#176;f<sub>d</sub> will increase (P<sub>w1</sub> in Figure 6). When 40&#176;<&#966;&acute;&lt;70&#176;, f
d decreases a little (P<sub>w3</sub>, P<sub>w4</sub> and P<sub>w5</sub> in Figure 6), and when
&#966;&acute;&#8804;40&#176;,f<sub>d</sub> changes little (P<sub>w6</sub> and P<sub>w7</sub> in Figure 6). These three
cases are valid when&#963;<sub>2</sub>
 decrease(or <img src="<?php echo $imgpath;?>images/IJPR-107-teq2.PNG"> increases, Fig. 6). As such, the change of &#963;<sub>2</sub>
 should not be ignored in shear
strength analysis in the presence of pre-existing weakness. However, when &#966;&acute;&#8805;70&#176; or &#966;&acute;&#8804;40&#176; (particularly when &#966;&acute;&#8804;40&#176;), most of the weaknesses usually cannot be reactivated at
critical stress state of Coulomb rupturing and thus doesnot
lead shear-stress decrease (f<sub>d</sub> = 1.0). As a result, the effect of
relative &#963;<sub>2</sub>
is most prominant only in the interval 40&#176;<&#966;&acute;&lt;70&#176;; under normal circumstances, f<sub>d</sub> will decrease a little with
decreasing &#963;<sub>2</sub>
 (Fig. 6).</p>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-figure6.PNG" class="img-responsive center-block" ><img src="<?php echo $imgpath;?>images/IJPR-107-figure5a.PNG" class="img-responsive center-block"/></div>
<p class="art-para">Mechanical properties and relative-orientation, which are
the governing factors of shear strength decrease and strength
anisotropies, respectively, are internal factors, while <img src="<?php echo $imgpath;?>images/IJPR-107-teq2.PNG"> is
an external factor and not so important according to the
above analysis,. When the relative-orientation (&#952;&acute;, &#966;&acute;) and
mechanical properties (C<sub>w</sub>, &#181;<sub>w</sub> ) of pre-existing weakness, and
<img src="<?php echo $imgpath;?>images/IJPR-107-teq2.PNG"> are given, the Shear-strength Coefficient can be
quantitatively evaluated in Mohr-space.</p>
<p class="art-para"><b>Shear strength evaluation for rock samples and geological
bodies: theoretical analysis and verification</b></p>
<p class="art-para">There are many kinds of pre-existing weakness, such as
faults, geologic contacts, bedding and foliation that affect
shear strength <a href='#23'>[23]</a> <a href='#24'>[24]</a> <a href='#25'>[25]</a>. Based on their structure, Morley (2002) <a href='#26'>[26]</a> divided weaknesses into two types: &#8220;discrete&#8221; and
&#8220;pervasive&#8221;. Based on the value of cohesive strength, Tong
and Yin (2011) <a href='#11'>[11]</a> divide weaknesses into &#8220;strong weakness&#8221; (with relatively large cohesive strength, such as bedding, foliation etc.) and &#8220;weak weakness&#8221; (with relatively small or
zero cohesive strength (i.e. ignorable), such as faults, fracture
planes etc.). &#8220;Discrete&#8221; weakness is usually a &#8220;weak weakness&#8221;, and &#8220;pervasive&#8221; weakness probably is a &#8220;strong weakness&#8221;. We will evaluate the effect of &#8220;strong weakness&#8221; and &#8220;weak
weakness&#8221; on shear strength in rock samples, and analyze
deformation sequences with multiple pre-existing weaknesses
in geological bodies.</p>
<p class="art-para"><b>Rock samples</b></p>
<p class="art-para">&#8220;Pervasive&#8221; weaknesses (or &#8220;strong weakness&#8221;, bedding in
sedimentary rocks, foliation in metamorphic rocks) do exist in
rocks. There may also exist&#8220;pervasive&#8221; weaknesses in magmatic
rocks with ductile deformation or flow foliation. In
homogeneous-looking rock samples there may exist
&#8220;pervasive&#8221; (particularly in sedimentary or metamorphic rocks) and /or &#8220;discrete&#8221; weaknesses (&#8220;weak weakness&#8221;, internal
small or micro-fracture) leading to shear strength anisotropies. In order to quantitatively evaluate shear strength decrease
and anisotropies of rock samples with &#8220;pervasive&#8221; or &#8220;discrete&#8221; weakness, the situations of &#9312;C<sub>w</sub> =0.5C, 0.33C and 0.2C for
&#8220;pervasive&#8221;weakness, and C<sub>w</sub> =0 for &#8220;discrete&#8221; weakness, and
&#9313; <img src="<?php echo $imgpath;?>images/IJPR-107-teq2.PNG"> = 0.5, 1.0 and 2.0 are considered (let &#181;<sub>w</sub> = &#956; = 0.6, C = 20MPa). Applying Mohr-space, the calculated values of
Shear-strength Coefficient can be seen in Table 1. For other C
values, the result is completely the same, while the result will
change a little bit as &#956; ( &#181;<sub>w</sub> ) changes (if &#181;<sub>w</sub> = &#956; ). However, if &#181;<sub>w</sub> changes while &#956; isconstant, f<sub>d</sub> will change proportionally
with &#181;<sub>w</sub> .The results show thata &#8220;discrete&#8221; weakness may lead
to more than 80% maximum drop of shear strength, while
&#8220;pervasive&#8221; weakness usually leads to 20-60% maximum shear
strength decrease for rock samples.</p>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-table1.PNG" class="img-responsive center-block"/></div>
<p class="text-center"><b>Table 1. </b> Shear-strength Coefficient (f<sub>d</sub>
) value in different value of C<sub>W,</sub>relative &#963;<sub>2</sub>
, &#952;&acute; and &#966;&acute;</p>
<p class="art-para">Haimson and Rudnicki (2010) <a href='#27'>[27]</a> conducted true tri-axial
compression tests on siltstone samples, and showed that shear
strength is related to &#963;<sub>2</sub>
. Although the orientation of siltstone
bedding is not mentioned in the paper, we speculate that the
shear strength change is caused by &#8220;pervasive&#8221; weakness (siltstone bedding). If bedding does not lead to mechanical
anisotropy (i.e. when it is mechanically homogeneous), shear
strength will not change with &#963;<sub>2</sub>
. The tri-axial compression tests
of Haimson and Rudnicki (2010) <a href='#27'>[28]</a> may be the verification for
the above theoretical analysis of dependece of shearstrength on
&#963;<sub>2</sub>
. Naturally, availability of more experiment data is needed.</p>
<p class="art-para">In the presence of multiple pre-existing weaknesses in a rock
sample, each weakness has its own Shear-strength Coefficient. The
overall Shear-strength Coefficient is determined by and is equal to
that of the weakest weakness, where the Shear-strength Coefficient
is the smallest, if interactions of weaknesses are ignored.</p>
<p class="art-para"><b>Geological bodies and deformation sequence</b></p>
<p class="art-para">There are the two main differences between geological
bodies and rock samples when the size and locationof
weakness are concerned. It is possible to prepare relatively
homogeneous small rock samples (it is necessary for regular
rock mechanics test). However, it is inevitable that there are
pre-existing weaknesses more or less in the geological bodies, which will lead to shear strength decrease (shear strength is
much smaller than that of rock samples <a href='#1'>[1]</a> <a href='#2'>[2]</a> <a href='#7'>[7]</a>.</p>
<p class="art-para">On a larger scale, there may or probably exist multiple
weaknesses <a href='#13'>[13]</a> in geological bodies (i.e. rift basins, orogenic
belts, or a part of them). Different relative-orientation of the
weaknesses and/or their mechanical properties will lead to
different f<sub>d</sub> of weakness. The relative-orientation and/or
mechanical properties may also vary greatly along large scale
pre-existing weakness (i.e. big pre-existing faults) and will lead
to different f<sub>d</sub> in different segments along the same weakness.</p>
<p class="art-para">Unlike rock samples, in the presence of multiple pre-existing
weaknesses in geological bodies, one of the weaknesses
reactivation will lead to stress drop and form local stress field
only in the area along and near the weakness <a href='#20'>[20]</a> <a href='#21'>[21]</a> instead of
the whole geological body. Differential stress within other areas
of the body away from the weakness can increase with progressive
deformation until another weakness reactivates or Coulomb
rupturing occurs. In otherwords, a weakness can lead to shear
strength decrease only within a part of geological bodies (along
and near a &#8220;prefered&#8221; weakness). As a result, in the condition of
homogeneous regional stress field, it is easy to understand that
the weaknesses, which are away from each other and f<sub>d</sub>
< 1.0, can
reactivate according to their f<sub>d</sub>
 values (from small to large) in the
progressive deformation: the weakest weakness with the smallest
f<sub>d</sub>
 value reactivates first, then the second weakest, and so on. Coulomb rupturing will occur at last in the area away from the
weaknesses.</p>
<p class="art-para">In order to verify the above statements, a sandbox model
is run, where multiple pre-existing weaknesses are built away
from each other. The f<sub>d</sub> values of these weaknesses can be
quantitatively determined, and the regional stress field can be
regarded as homogeneous.</p>
<p class="art-para">In the models, three pre-existing weakness planes (P<sub>w1</sub>, P<sub>w2</sub>
and P<sub>w3</sub>), which are oblique to the extension direction (&#963;<sub>3</sub>
), with (&#966;&acute;, &#952;&acute;) = (80&#176;, 55&#176;), (60&#176;, 70&#176;) and (30&#176;, 49&#176;) respectively, are set. The weakness planes are represented by a 80mm wide
slice of thin sheets of paper inserted into the base of a
homogeneous layer (8.0 cm thick) of dry quartz sand (Fig. 7). Using these sheets of paper allowed localizing fault initiation, and it was easy to determine friction angle between the paper
and loose sand. The friction angle of dry quartz sand is 31&#176;, while the friction angle of sand with paper is 20&#176; . So the &#181; and
&#181;<sub>w</sub> is 0.60 (= tg31&#176;) and 0.36 (= tg20&#176;). The cohesion of dry
sand is very small, so C and C<sub>w</sub> are both assumed to be equal
to 0. As a result, the relative Reactivation Tendency Factor of
the three weakness planes is f<sub>a1</sub> =1.59>f<sub>a2</sub> =1.42>1.0 >f<sub>a3</sub> =0.85
in the critical stress state of Coulomb rupture (Fig. 8), and
Shear-strength Coefficient f<sub>d1</sub> =0.63&lt;f<sub>d2</sub> = 0.70&lt;1.0= f<sub>a3.</sub></p>
<p class="art-para">With the above model (Figure 7), the weakness, with
Shear-strength Coefficient f<sub>d1</sub>&lt;1.0, reactivated to form several faults, one after another, in the progressive extension as
predicted, and Coulomb ruptures faults form at last (Figure 9). The deformation in the progressive extension is summarized
in the following:</p>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-figure7.PNG" class="img-responsive center-block"/></div>
<p class="text-center"><b>Figure 7. </b>  Experiment design of sandbox modeling. (a) Skeleton figure of experimental model. It is 40cm wide, 50cm long and 8cm thick. P<sub>w1</sub>, P<sub>w2</sub>, and P<sub>w3</sub> are weakness planes which is oriented oblique
to the extension direction, with (&#952;&acute;, &#966;&acute;) are (80&#176;, 55&#176;), (60&#176;, 70&#176;) and (30&#176;, 49&#176;) respectively, and represented by paper being inserted into
the homogeneous dry sand. Elastic rubber is under the sand layer
and connected to driving end. (b) Section of AA&acute; in figure (a).</p>
<p class="art-para">The weakness plane with the smallest Shear-strength
Coefficient (P<sub>w1</sub>) reactivated to form a fault at first (fault
number 1, Fig. 9b-1). Then, the second weakness plane (P<sub>w2</sub>) reactivated to form a fault (fault number 2, Fig. 9b-1). In the
middle stage (after the weakness reactivated and before the
fault near perpendicular to extension direction formed), weakness related faults which are parallel (or sub-parallel) to
and controlled by P<sub>w1</sub> or P<sub>w2</sub> (antithetic and synthetic faults, fault numbers 3-8in Figures 22b-2 and b-3) began to develop
near the weakness plane area. In the final stage (Fig. 9b-3), faults near perpendicular to the extension direction (small
faults in Figure 9b-3) began to develop. While the third
weakness plane, which Shear-strength Coefficientf<sub>d3</sub>=1.0 (f<sub>a3</sub>
=0.85&lt;1.0, Fig. 8), does not reactivate as predicted. Similar
experiments carried out by Tong and Yin (2011) <a href='#11'>[11]</a> and Tong
et al. (2014) <a href='#28'>[28]</a> showed the same results.</p>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-figure8.PNG" class="img-responsive center-block"/></div>
<p class="art-para">In general, a phase of deformation indicated by the
initiation of a new fault trend is attributed to a specific stress
regime. However, our analysis suggests that in a region where
the magnitude of differential stresses progressively increase
while their directions are kept constant, multiple phases of
fault initiation with different trends can occur. This could be
due to reactivation of pre-existing weaknesses. The sequence
of reactivation of any preexisting weakness is predictable
according to the relative-orientation and mechanical
properties of the weak planes.</p>
<p class="art-para">It is not easy to estimate the shear strength of pre-existing
weaknesses in the field. Even though, the relative-orientation
of a weakness may be determined accurately in some
circumstances (when high quality 3D seismic data is available, and fault systems in shallow level can be mapped
accurately),mechanical properties of the weakness (C<sub>w</sub>, &#181;<sub>w</sub> ) are not easily determined.However, if the sequence of faulting
in the geologic record is known , this information may be
reversely used for assessing the mechanical properties of the
preexisting weaknesses at the time of their reactivation.</p>
<div class="art-img">
<img src="<?php echo $imgpath;?>images/IJPR-107-figure9.PNG" class="img-responsive center-block"/></div>
<p class="art-subhead">Discussion</p>
<p class="art-para">Although this study is expanding the work of Ranalli (1990) <a href='#5'>[5]</a> and Morris et al <a href='#7'>[7]</a>, our newly defined Shearstrength
Coefficient, newly developed graphical techniqueMohr-space
and the underlying theoretically framework, which is based on Reactivation Tendency analysis theory (Tong and Yin, 2011), provide a much general and intuitive
treatment of the shear strength decrease and anisotropies
caused by pre-existing weakness(es). Specifically, the
assumption that the weakness plane containing the
intermediate stress (&#963;<sub>2</sub>
) in Ranalli&acute;s (1990) <a href='#5'>[5]</a> analysis can now
be neglected. Finally, we predicted that weaknesses will
reactivate sequentially according to the Shear- strength
coefficient values order (from small to large) and new fractures (Coulomb rupture) form at last in the progressive deformation. It was verified by a simple sandbox experiment.</p>
<p class="art-para">Both in this study and that of Ranalli (1990) <a href='#5'>[5]</a>, it is
assumed that the preexisting weakness must be planar. This
assumption, however, does not prevent the application of the
analysis developed in this study to non-planar weaknesses, as
curved surfaces can always be divided into approximately
small planar segments. Another important assumption in our study is that fault formation, propagation and activity do not
affect the regional stress distribution. This assumption is
clearly an oversimplification, as both detailed analysis of faultzone
evolution and regional modeling show that frictional
sliding on faults is capable of creating local stress fields near
the faults that are different from the regional stress field <a href='#29'>[29]</a>
<a href='#30'>[30]</a>. However, the results of our sandbox experiment imply
that the effect of activating preexisting weakness does affect
local stress, but is minimal in changing regional stress. More
research is clearly needed to address this problem.</p>
<p class="art-para">Possible complex interactions between faults that lie near
one another were not considered in our model. As stress
concentrates at crack tips, it is expected that complex stress
fields can be induced by the presence of cracks or weaknesses
under uniform regional stress <a href='#30'>[30]</a> (e.g., Gudmundsson et al., 2009). Thus, our model should be considered as an idealized
conceptual guide, which can only be used in realistic situations
when the above factors are considered.</p>
<p class="art-para">Despite the complexities discussed above, our analysis
does provide a new insight into the temporal evolution of
multiple pre-existing weaknesses with different Shear&#8211; strength coefficient. Our analysis suggests that in a region
with a progressive increase in the magnitude of differential
stresses while the directions of the principal stresses are not
changed, multiple phases of fault initiation with different
trends can be generated. On the other hand, the results of
theoretical analysis will provide some information and clues
to understand actual shear strength decrease and anisotropies
due to the pre-existing weaknesses.</p>
<p class="art-subhead">Conclusion</p>
<p class="art-para">Shear-strength coefficient (f<sub>d</sub>), which is defined to evaluate
the shear strength decrease due to the presence of preexistingweaknesses, is determined by the orientation and
mechanical properties of weakness(the intrinsic factors) and
stress tensor (the external factors), and can be calculated
quantitatively. It also can be expressed intuitively/graphically
in Mohr-space. The results of theoretical analysis in Mohrspace
show that:</p>
<p class="art-para">(1)Weakness relative-orientation (&#952;&acute;, &#966;&gt;), which is
determined jointly by orientation of the weakness and three
principal stress axis, is the predominant factor leading to shear
strength anisotropies. (0&#176;, &#966;&acute;0
) and (180&#176;, &#966;&acute;0
) are two relativeorientations
with the lowest f<sub>d</sub>. As &#952;&acute; increases from 0&#176; or
decreases from 180&#176;, and &#966;&acute; deviates from &#966;&acute;0
, f<sub>d</sub> will increase. There are critical angles &#966;&acute;1
, &#966;&acute;2
 and &#952;&acute;1
, 180&#176;-&#952;&acute;1
, when &#966;&acute;1
<&#966;&acute;<
&#966;&acute;2
 or &#952;&acute;1
<&#952;&acute;&lt;180&#176;-&#952;&acute;1
, f<sub>d</sub> = 1.0. Low f<sub>d</sub> value of weakness plane
concentrates in the area around &#952;&acute;=0&#176;~15&#176; or 165&#176;~180&#176; and
&#966;&acute;=&#966;&acute;0
&#177; 15&#176;.</p>
<p class="art-para">(2)Weakness mechanical properties (C<sub>w</sub> and &#181;<sub>w</sub>) are the
predominant factorsthat lead to shear strength decrease. &#8220;Discrete&#8221; weaknesses may suffer more than 80% decrease in
maximum shear strength, while &#8220;pervasive&#8221; weaknesses
usually suffer 20-60% decrease in maximum shear strength.</p>
<p class="art-para">(3)The effect of relative <img src="<?php echo $imgpath;?>images/IJPR-107-teq1.PNG"> to shear strength is a
little complicated, and is related to &#952;&acute; and &#966;&acute;, particularly &#966;&acute;. The effect of relative &#963;<sub>2</sub> is most prominant only in the interval
40&#176;<&#966;&acute;&lt;70&#176;. Under normal circumstances, f<sub>d</sub> will decrease a
little with increasing of <img src="<?php echo $imgpath;?>images/IJPR-107-teq2.PNG">.</p>
<p class="art-para">In a region with a progressive increase in the magnitude
of differential stresses acting on multiple pre-existing
weaknesses, while the directions of the principal stresses
maintain the same, multiple phases of fault initiation with
different trends can be generated with predicted sequence
according to their shear&#8211;strength coefficient (from small to
large). This theoretical prediction wasverified by the results of
a sandbox model.</p>
<p class="art-subhead">Acknowledgement</p>
<p class="art-para">We would like to thank graduate students Mingyang
WANG and Hua wu HAO for drawing the figures of the Mohr
space and their help to complete the sandbox model.This
study was supported by China National Major Project of Oil
and Gas (2016ZX05024-005-004 & 2011ZX05023-004-
0122011zx05006-006-02-01) and China Natural Science
Foundation (Grant No. 41272160&40772086). HAK is
supported by the Swedish Research Council.</p>
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