The Effect of Specimen Height on the
Uniaxial Compressive Experiment of
Rocks
Li Ximeng*, Liu Changyou, Guo Weibin, Qiu Jiaojian
School of Mines，Key Laboratory of Deep Coal Resource Mining, Ministry of
Education of China, State Key Laboratory of Coal Resources and Safe Mining，
China University of Mining & Technology，Xuzhou Jiangsu 221116，China
*Corresponding author, e-mail:[email protected]
ABSTRACT
In order to study the effect of specimen height on the uniaxial compressive test of rocks, an uniaxial
compression test was conducted on five groups of sandstone samples which diameters were 2 inches
and heights were 3.0 inches, 3.5 inches, 4.0 inches, 4.5 inches, 5.0 inches respectively by using MTS
electro-hydraulic servo testing machine. Based on this experiment, the relationship between the
calculated elastic modulus and samples’ height was analyzed theoretically and a revised formula on the
calculation of elastic modulus of the rock was developed. The results showed that there was no
significant effect on the rock uniaxial compressive strength when the height-diameter ratio range was
between 1.5~2.5; however, along with the increase of sample height, the elastic modulus calculated
with displacement data obtained from the MTS software changed accordingly with sample's height; in
the occasions when strain gages are not accessible in the experiments due to restricted experimental
condition, the modifier formula can revise the experimental results to calculate the elasticity modulus
of material itself more precisely. The experimental results have significant referential meaning both for
the correction of uniaxial compression strength values of other rocks and rock-like materials and for
the selection of parameters in numerical simulation experiments.
KEYWORDS: Specimen; height; uniaxial: compressive strength; rock mechanics; height
effect; MTS electro-hydraulic servo testing machine
INTRODUCTION
How to control rock stratum is an eternal issue in mining. In order to reasonably control the
movement of rock stratum in mining, it is necessary to test the rock and obtain some important
parameters like the strength of rocks. Uniaxial compression test is the most popular way to study the
rock performances in the laboratory. Researchers had recognized that the size of rock is strongly
associated with its strength several centuries ago. As early as 15th century, Italian scientist Leonardo
da Vinci and French scholar Edmé Mariotte had pointed out that the longer an object was, the less its
strength was[1]. The study that how and to what extent the rock's length affect its strength in
mechanism has great significance even in nowadays both in theory and practice. That’s why there are
always so many scholars interested in this topic. CAE member Liu Baocheng [2] studied the results of
experiment in mechanics on seven rocks, and calculated an empirical formula on the compressive
strength of rock. Haftani [3] and other scholars have studied the effect of size on the evaluation of the
strength test of detritus indentation. The results showed that detritus area parallel to the loading
direction had little effect on the indentation index while the thickness of samples played a major role.
The indentation index was also standardized by the thickness relevant function in the results. Weiss
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[2] and other scholars have studied the size effect on brittle material like rock from the perspective of
statistical physics and established progressive damage model, regarding the compression failure as a
critical transitional phase related to the rock length during the compression. Yang Shengqi [3] and
other scholars, through a compression experiment on the marble specimens, studied the effects of the
sample’s length on characteristics of deformation and concluded that the length of rock had no
significant influence on the deformation characteristics below stress peek value, but deformation
characteristics were significantly changed after the peek value: the greater the length of specimen
was, the more brittle the specimen was. Ni Hongmei[2] and other scholars used numerical simulation
software RFPA2D, conducted an uniaxial compression numerical simulation comparative research on
sample rocks of the same diameter but different lengths under the conditions of with-friction and
without-friction respectively and believed that the reason why rock’s length affected the rock’s
strength was because of the friction existed on the ends of rock samples, rather than its heterogeneity
of the material itself. Yazici and other scholars[3] have studied the compressive strength of cylindrical
cement samples with two different sizes of 150/300mm and the 120/200mm, conducted linear and
nonlinear regression analysis, and obtained the compressive strength ratio of the two samples was
103%. You Mingqing[4], together with other scholars studied the size effect of rocks and pillar
supporting performances under uniaxial compression, discussed the effect of specimen’s shapes and
sizes on post-stress-peak softening process, illustrated that bigger pillars had a better performance and
could prevent the occurrence of pillar rockburst from aspects of roof supporting load and settlement
displacement. Khair and other scholars[5] studied the influence of size effect on compression strength
of cube shaped coal samples with the length of 5cm, 10cm, 15cm, and 20cm respectively. The
research results showed that the size effect had an upper limit value. Below the upper limit value,
there’s a correlation between the compression strength and the size of the samples, whereas, when
exceed this limit value, the variation of sizes have no significant effect on compression strength of the
samples. Based on Grey Theory, Zhu Zhende together with other scholars [10] studied the size effect
on the compressive strength of brittle rock and verified the rules that compressive strength of brittle
rock decreases along with the increase of brittle rocks’ height-diameter ratio in a certain range. In
addition, some other scholars [11 12 13 14 15,16, 17, 18, 19, 20, 21, 22] both from home and abroad
also did some research works from the perspectives of energy, micro- and macro- structures,
theoretical analysis, numerical simulation, etc.The research achievements of predecessors show that the effect of rock sizes has always been a
hot issue in the study of rock mechanics. Although lot of researches have been done on it, in actually
sampling, there are a number of nonstandard specimen height, how the sample’s height affects the
result of the experimental has not been studied in detail, in addition, there is no
practical and theoretical standards for the reliability of experimental results, which all make the
further studies on this issue is necessary. Based on the previous researches, this research further
studied the law of influence of rock specimen height on its strength, and by controlling variables
method, an uniaxial compressive experiment was carried out on the rock samples of
the same diameters with different heights, by using MTS electro-hydraulic servo testing machine to
study the effect of sample heights on the uniaxial compressive strength of rocks and elastic modulus.
The results of the experiment have been verified through theoretical derivation and a revised elastic
modulus formula was developed to provide experimental and theoretical reference for
compression experiment on rock materials and rock-like materials.
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UNIAXIAL COMPRESSION TESTS ON DIFFERENT
HEIGHTS OF ROCK SPECIMENS
Preparation of the specimens
The specimens used in the test were from Princeton, a city in West Virginia of the United States,
which belongs to the Appalachian Mountains. The cores were drilled and packaged by a geological
drilling team and shipped to the laboratory, as shown in Figure 1. Then they were cut, polished, and
tested in the laboratory. Equipment for cutting, grinding, and testing are as shown in Figure 2. The
testing machine is electro-hydraulic servo testing machine which is on the right most of Figure 2.
Figure 1: Original rock core
a. Cutter
b. automatic grinder
c. MTS electrohydraulic servo tester
Figure 2: Experimental equipment
Previous studies [Error! Bookmark not defined.,Error! Bookmark not defined.,Error! Bookmark not defined.] showed that, the
strength parameters of rock were different, revealing some discreteness. Besides the variable nature of
the sample, experimental condition and degree of the processing precision are also closely related to
the accuracy of the experiment result. Therefore, in order to obtain reliable results, the rock samples
in this research were selected in the same location and processed strictly in accordance with the
standards in literature [6]. The specimen surface non-parallelism of error at both ends was less than
0.05mm, the sample surface was perpendicular to the axis with less than 0.25° error.
Experimental program
The purpose of the experiment was to monitor the state of stress-strain during the uniaxial
compression and study the features that specimens’ heights affected the result of uniaxial
compression. The loading equipment was shown in Figure 3. The testing machine was loaded and
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controlled by the way of axial displacement, the rate was 0.5mm / min, the strength in entrance was
10N, the sample diameters were 5cm (2 inches), the heights of each group were 75mm (3.0in),
87.5mm (3.5in), 100cmm (4.0in), 112.5mm (4.5in), 125mm (5.0in) respectively, and height-diameter
ratios were 1.5,1.75,2.0,2.25,2.5 respectively. In order to exclude the contingency in the results and
make it more reliable, five groups of experiments were done, with the strain gages were posted both
in axial and circumferential directions on the samples in the fifth experiment. The test samples were
prepared as shown in Figure 4.
Figure 3: Test system of uniaxial compression test
Figure 4: Test samples
Before the experiment, the diameter, circumference, weight, and other parameters of each sample
were measured and the sample density was calculated accurately. To ensure the accuracy of the
results, the specimens were measured multiple times and then calculated their average values.
Specific sample group numbers and the measurement results were shown in Table 1.
Table 1: Parameters for different rocks tested
Group
A
B
No.
A1
A2
A3
A4
A5
B1
B2
B3
B4
Depth/m
318.9
318.8
318.7
318.6
318.5
319.3
319.2
319.1
319.6
Diameter/mm (in
47.498 (1.870)
47.447 (1.868)
47.473 (1.869)
47.396 (1.866)
47.396 (1.866)
47.549 (1.872)
47.473 (1.869)
47.396 (1.866)
47.371 (1.865)
Height/mm (in)
126.898 (4.996)
114.275 (4.499)
101.625 (4.001)
88.697 (3.492)
76.022 (2.993)
126.873 (4.995)
113.894 (4.484)
101.270 (3.987)
88.722 (3.493)
Weight/g
597.7
539.4
477.4
417.2
356.5
596.2
534.8
475.1
416.4
Density（Average）/g/cm3
2.661
2.656
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C
D
S
B5
C1
C2
C3
C4
C5
D1
D2
D3
D4
D5
S1
S2
S3
S4
S5
319.5
320.7
320.2
320.5
320.4
320.3
321.4
321.3
321.2
321.1
321.0
323.7
323.4
323.5
324.6
324.5
1477
47.447 (1.868)
47.498 (1.870)
47.498 (1.870)
47.447 (1.868)
47.422 (1.867)
47.498 (1.870)
47.498 (1.870)
47.498 (1.870)
47.549 (1.872)
47.549 (1.872)
47.473 (1.869)
47.447 (1.868)
47.473 (1.869)
47.523 (1.871)
47.523 (1.871)
47.473 (1.869)
76.352 (3.006)
126.619 (4.985)
113.944 (4.486)
101.397 (3.992)
88.824 (3.497)
76.454 (3.010)
126.721 (4.989)
114.122 (4.493)
101.981 (4.015)
88.925 (3.501)
76.632 (3.017)
130.785 (5.149)
118.516 (4.666)
105.639 (4.159)
93.548 (3.683)
80.391 (3.165)
358.9
594.7
546.0
476.3
420.3
364.0
593.5
535.1
474.8
415.5
358.6
611.7
556.6
496.4
440.3
376.5
2.676
2.637
2.649
ANALYSIS OF EXPERIMENTAL RESULTS
Effect of sample height on the
uniaxial compressive strength
Five groups of rock samples with different heights were tested in the uniaxial compressive
experiment and the experimental results were calculated as shown in Table 2, where the standard
deviations and the means were calculated by removing the minimum values, and the minimum values
in stress-strain curve in the uniaxial compressive experiment were also removed to reduce error.
Table 2: The results of UCS of the samples
Group\No.
A
B
C
D
S
Average
1
152.5
149.4
148.0
#
130.7
161.3
152.8
2
*116.4
158.0
175.0
#
124.9
155.1
151.1
3
148.4
*116.5
144.1
#
128.6
*150.0
139.7
4
136.4
139.2
164.8
#
*107.8
169.7
152.5
5
Standard Deviation
126.1
12.0
134.5
10.5
#
*129.7
14.5
150.3
11.4
159.9
6.1
142.7
*Minimum UCS within the same group；#Minimum UCS within the same height
The curves shown in Figure 5, which were the relationship between uniaxial compressive
strengths and the sample heights in five groups, can be divided into four types: the first type was
fluctuation type, such as group C. In this group, the data had a large discreteness, the standard
deviation is 14.5 which is the largest in this 5 groups. The main reason was that the physical
characteristic parameters of rocks were hard to keep consistency in all aspects and that sample’s
discreteness of physical characteristics was variable, so the result showed that there was no
correlation between sample height and the strength; the second type was declined gentlely, such as
group D, in which uniaxial compressive strength gradually decreased along with the increase of
sample height. This trend was in line with the previous intuitive understanding that the uniaxial
compressive strength of rock declined with the increase of size; the third type rose slowly, such as
group A and group B, in which uniaxial compressive strength of rock increased with the rise of
sample's height; the fourth was smooth type, such as group S. Along with the increase of sample
height, the uniaxial compressive strength was relatively stable. According to the experimental result,
the standard deviation of uniaxial compressive strength of the group was 6.1, which was the smallest
among the five groups. The result had a high-level consistency and the data was reliable.
Based on the four types above, the types of fluctuation, gently decline, and slowly rising were all
because of the natural otherness of rocks. Excluding the effect of rock’s the heterogeneous and other
random external factors, the author of this research thought that within the scope of this experiment
(sample heights were between 75mm to 125mm, height-diameter ratios were between 1.5 to 2.5), the
effect of sample height on the compressive strength of the rock was small. This result also could be
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seen from the average value curve in Figure 5. The reason that rock strengths increased or decreased
with sample heights was mainly caused by the experimental error or heterogeneity of rocks.
Generally, when a rock mechanics experiment is carried out, in order to obtain accurate rock
parameters to improve the accuracy and reliability of the experimental structure and make the results
less discrete, a high accuracy in the processing of the sample should be maintained at first; In
addition, the testing machine should be operated in according with standards and the samples should
contact with the test machine close enough to reduce the shock failure of the sample cause by the
point contact.
Figure 5: The relationship between the UCS and height of samples
Effect of sample height on elasticity modulus of rock
Table 3 showed the statistical elasticity modulus of the five group samples.
The data of axial displacements in groups of A, B, C and D was monitored and recorded constantly
by MTS software. Then the strain data of samples was calculated based on the displacements and
specimen heights. The axial loading value was recorded by MTS in real time, and the data of sample
stress was calculated based on this loading value and the cross sectional area of specimen. But the
data of axial strain in group C was obtain directly from strain gauges. Obviously, when compared
with the data in group S, the four groups of A, B, C ,D had bigger discreteness on elastic modulus,
and their standard deviations were greater than 2GPa-about 11% of average value of the elastic
modulus in each group. However, the standard deviations of elastic modulus in group S is only
0.5GPa- about 1% of average value of the elastic modulus in this group.
Table 3: The elastic modulus of the samples
Group/No.
1
2
3
4
5
A
B
C
D
*S
23.3
22.6
22.0
22.1
37.5
20.9
22.2
28.9
21.1
38.6
21.9
19.0
21.9
20.2
38.3
19.8
19.9
22.2
17.6
38.2
17.2
17.8
20.4
17.9
38.8
*Strain data was recorded through strain gage in group S
Standard
Deviation
2.3
2.1
3.3
2.0
0.5
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Take the group A as an example, stress-strain graph of each sample were drown as shown in
Figure 6, in which we found out the point where the half of compressive stresses was, drew another
300 points both above and below that point, made straight lines by the method of linear fitting (see
the black line in Figure 6), and obtained the slopes of the straight lines where the line segment
located, which were also the modulus of elasticity of the sample. To compare the relationship of
elasticity modulus of samples in four groups, we combined the starting points of four elasticity
segments of A1, A2, A3 and A4 in Figure 6 and got four radials, A1-E, A2-E, A3 -E and A4-E. The
slope of each straight line where the radial located represented the elasticity modulus of each sample
respectively. It can be seen that from Figure 6 that the slope of straight line increase correspondingly
with the sample height rising from A5 to A1, showing that the sample’s elastic modulus increased
correspondingly. To further understand the relationship between the sample height and elastic
modulus, based on the experimental results, we took the averages of the elastic modulus and sample
heights in group A, B,C and D, and made Figure 7, from which we can see that the elastic modulus
has a rising trend along with the increase of sample length.
180
160
A3
Axis Stresss, σa/MPa
140
A1
A4
120
A5
100
80
A1-E
60
A3-E
40
A4-E
A5-E
20
0
0.0
0.5
1.0
1.5
Axial Strain, εa/%
Figure 6: The stress-strain curves of samples in Group A
24
Eastic modulus, E/GPa
23
22
21
20
19
18
17
16
7.0
9.0
11.0
13.0
Sample height, h/cm
Figure 7: The relationship between elastic modulus and samples height
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THEORETICAL ANALYSIS
Elastic modulus, an intrinsic property of a rock, does not change with the height of the sample. As
mentioned before, the displacement in four groups were monitored by MTS software. The axial
displacement recorded in the experiment was considered as the displacement of sample, which was
based on the assumption that the stiffness of testing machine was much larger than that of the rock
and that the deformation only occurred in the specimens. In fact, the effect of stiffness of the testing
machine does exist, and can’t be ignored sometimes. The displacements that recorded by software
were numbers by adding samples’ displacements and the deformation of test machine. Now we
assume that the integral rigidity of the testing machine is 𝑘𝑘, the original length of sample is 𝐿𝐿0 , its
diameter is 𝐷𝐷𝑠𝑠 , cross-sectional area is 𝐴𝐴0 , the intrinsic elasticity modulus of rock is 𝐸𝐸𝑟𝑟 (𝐸𝐸𝑟𝑟 is a
constant value), the axial displacement monitored by MTS is 𝜂𝜂, the amount of shorten of rock sample
in axial direction is △ 𝑙𝑙, the amount of shorten of testing machine is δ, then under the conditions that
axial load is 𝑃𝑃, the calculating formula for the elasticity modulus is as following:
𝐸𝐸𝑟𝑟 =
𝑃𝑃⁄𝐴𝐴0
（1）
△𝑙𝑙⁄𝐿𝐿0
In the equation: 𝐸𝐸𝑟𝑟 is a constant elasticity modulus of the rock. For a specific rock type, its value
is constant; △ 𝑙𝑙 is the amount of shorten of rock sample in axial direction; but the MTS software
monitored the whole system of displacement 𝜂𝜂 rather than the axial displacement of rock samples，
𝜂𝜂 =△ 𝑙𝑙 + 𝛿𝛿. Under the condition of ignoring MTS testing machine system deformation δ, and
assuming that only rock samples deformed, equation (1) therefore can be rewritten as:
Er ≈
P⁄A0
η⁄L0
（2）
In equation (2), rock’s intrinsic elasticity modulus 𝐸𝐸𝑟𝑟 can be calculated from the right part of
approximate formula, let
𝐸𝐸𝑐𝑐 =
𝑃𝑃⁄𝐴𝐴0
𝜂𝜂 ⁄𝐿𝐿0
（3）
In this formula: 𝐸𝐸𝑐𝑐 represents rock’s elasticity modulus that calculated with the displacement
amount obtained from monitoring system, when 𝜂𝜂 ≈△ 𝑙𝑙, 𝐸𝐸𝑐𝑐 is approximately equal to the intrinsic
elasticity modulus of the rock sample, and when MTS testing machine system deformation amount is
δ, equation (3) becomes:
𝑃𝑃⁄𝐴𝐴
𝐸𝐸𝑐𝑐 = (△𝑙𝑙+𝛿𝛿)0⁄
𝐿𝐿0
（4）
The total deformation amount of the system consists two parts, the axial displacement amount of
sample △ 𝑙𝑙 and deformation amount of test systems δ, because that the total amount was obtained by
automatic monitoring MTS software, there were no specific amounts for △ 𝑙𝑙 and δ respectively, and
because we previously assumed that the system stiffness was k, so with the given loading 𝑃𝑃, we can
get the equation:
From the equation (1) , we can be told:
𝛿𝛿 =
△ 𝑙𝑙 =
𝑃𝑃
（5）
𝑃𝑃⁄𝐴𝐴0
（6）
𝑘𝑘
𝐸𝐸𝑟𝑟 ⁄𝐿𝐿0
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Substituting Equation (5) and (6) in equation (4), we obtain the following equation:
𝐸𝐸𝑐𝑐 =
1
（7）
1
𝐴𝐴
+ 0
𝐸𝐸𝑟𝑟 𝑘𝑘∙𝐿𝐿0
In the equation,𝐸𝐸𝑟𝑟 a constant, is the inherent elasticity modulus of sample rock, and 𝐴𝐴0 is the
cross-sectional area of sample rock. So, for samples with same diameters in the same group, 𝐴𝐴0 is a
constant too, 𝑘𝑘 is the stiffness of the testing machine, for the same testing machine, the stiffness 𝑘𝑘 is
considered a constant. Therefore, in this experiment, only sample length 𝐿𝐿0 is a variable, from the
equation (7), there is the following equation when 𝑘𝑘 tends to infinity:
𝐸𝐸𝑐𝑐 = lim𝑘𝑘→∞
1
1
𝐴𝐴
+ 0
𝐸𝐸𝑟𝑟 𝑘𝑘∙𝐿𝐿0
= 𝐸𝐸𝑟𝑟
（8）
Therefore, by improving the stiffness of the system, the elasticity modulus calculated would be
closer to the real elasticity modulus of sample. We could be told from the equation (7) that for any
sample 𝑖𝑖:
𝐸𝐸𝑟𝑟𝑟𝑟 =
1
（9）
𝐴𝐴
1
− 0𝑖𝑖
𝐸𝐸𝑐𝑐𝑐𝑐 𝑘𝑘∙𝐿𝐿0𝑖𝑖
The equation (9) can solve the intrinsic elasticity modulus. Because 𝐸𝐸𝑟𝑟 , a constant, is the inherent
elasticity modulus of the rock, therefore, the denominator in equation 9 which should also be a
constant. To research the relationship between the sample’s elasticity modulus 𝐸𝐸𝑐𝑐 and sample’s length
𝐿𝐿0 , the formula (7) can be further transformed :
Let 𝑦𝑦 = 𝐸𝐸𝑐𝑐 ，𝑥𝑥 = 𝐿𝐿0 ，𝑎𝑎 = 𝑘𝑘，𝑐𝑐 =
𝑘𝑘
𝐸𝐸𝑐𝑐 =
𝐸𝐸𝑟𝑟
𝑘𝑘∙𝐿𝐿0
𝑘𝑘
∙𝐿𝐿0 +𝐴𝐴0
𝐸𝐸𝑟𝑟
（10）
，𝑑𝑑 = 𝐴𝐴0 ，the equation（10）can be transformed:
𝑦𝑦 = 𝑓𝑓(𝑥𝑥) =
𝑎𝑎𝑎𝑎
𝑐𝑐𝑐𝑐+𝑑𝑑
（11）
Based on this equation, we can be told that the elasticity modulus of the sample and the sample
𝑎𝑎
length is the fraction function relationship, asymptotic line on the y-axis of the function is y = =
𝑐𝑐
𝐸𝐸𝑟𝑟 . That is to say, along with the increasing of sample length, ignoring other factors, the calculated
elasticity modulus would be closer to the true value.
EXPERIMENTAL RESULTS VERIFICATION
Based on the theoretical analysis above and re-solving the elastic modulus of the samples, using
the equation (9), we can obtain each sample’s intrinsic elasticity modulus. As the sample used in this
experiment are the same kind of rock, so the elasticity modulus of each sample in theory should be
the same.
In equation (9), 𝐸𝐸𝑐𝑐 is a calculated modulus of elasticity of the sample, each sample’s elasticity
modulus 𝐸𝐸𝑐𝑐𝑐𝑐 was given in Table 3 and each sample’s length 𝐿𝐿0𝑖𝑖 and diameter 𝐷𝐷𝑠𝑠𝑠𝑠 were given in Table
1, so we can obtain each sample’s cross-sectional area 𝐴𝐴0𝑖𝑖 . The stiffness of MTS test machine used in
the test is k= 8.75 × 108 𝑁𝑁/𝑚𝑚. The recalculated values were shown in Table 4.
Vol. 20 [2015], Bund. 7
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Table 4: The corrected elastic modulus of samples
Group/No.
1
2
3
4
5
A
B
C
D
*S
37.1
35.4
34.0
34.3
37.5
33.2
36.8
#
59.5
33.8
38.6
38.8
30.7
38.9
33.8
38.3
36.2
36.3
44.9
29.5
38.2
31.6
33.8
44.6
33.9
38.8
Standard
Deviation
2.9
2.5
9.6
2.0
0.5
* Strain data was recorded through strain gage in group S；#Abnormal data
We can be told from Table 4 that the elastic modulus after revise increases significantly than
before revise. When removed those abnormal data due to experimental error and rock heterogeneity,
the discreteness of final values is diminished, the standard deviation of final value is around 7% of
average value (37.1GPa), and data stability increases by nearly 60%.
Meanwhile, from the stress-strain curves of samples in group S shown in Figure 8, we can be told
from the figure that the slope of straight section of each curve is almost the same, it meaning that the
data obtained by strain gauges is highly stable and perfectly avoid the effect caused by equipment
deformation and other unknown factors.
180
160
140
Axis Stresss, δa/MPa
120
100
80
S1
S2
S4
S5
60
40
20
0
0.0
0.2
0.4
0.6
0.8
1.0
Axial Strain, εa/%
Figure 8: The stress-strain curves of samples of Group S
The average elastic modulus of samples after revise is around 37.1GPa and can approximately be
considered as the intrinsic modulus of elasticity of sample rocks. By the known tester stiffness
𝑘𝑘 = 8.75 × 108 𝑁𝑁/𝑚𝑚 and sample average cross-sectional area, we have the parameter values 𝑎𝑎, 𝑐𝑐, 𝑑𝑑
of equation (11 ), since the sample height range is 𝐿𝐿0 ∈ (0.07𝑚𝑚, 0.14𝑚𝑚), we can obtain equation
(12), then draw a changing curve that y value (𝐸𝐸𝑐𝑐 ) changes with x value (𝐿𝐿0 ) and compare it with the
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curve in figure 7. The two curves are shown in figure 9. Then we can see that the two curves are
similar, which can verify the correctness of the theoretical formula.
𝑦𝑦 = 𝑓𝑓(𝑥𝑥) =
8.75×108 ×𝑥𝑥
2.36×10−2 ×𝑥𝑥+1.77×10−3
, 𝑥𝑥 ∈ (0.07,0.14)
（12）
26
Elastic modulus, E/GPa
24
22
20
18
16
Theoretical curve
14
Experimental curve
12
10
7.0
8.0
9.0
10.0
11.0
12.0
13.0
Sample height, h/cm
Figure 9: The theoretical and experimental curve of elastic modulus
CONCLUSION
1) Within the scope of this experiment (sample heights are between 7.5cm ~ 12.5cm (3in ~ 5in),
height-diameter ratio is between 1.5 and 2.5), the effect of specimen height on the compressive
strength of rocks is small. The reason why the rock strength increased or decreased along with
sample’s height in the experiment is mainly due to experimental errors and the natural heterogeneity
of sample rocks. For the purpose of obtaining accurate parameters sample of rocks to improve
accuracy and reliability of experimental results and reduce discreteness when conducting a rock
mechanics experiment, we should follow these principles: first of all, we should maintain a high
accuracy during the processing of rock samples; second, we should follow the testing machine’s
standard operations, ensuring sample rocks contact with the testing machine tightly to reduce the
shock failure of the sample cause by the point contact.
2) In this experiment, when used MTS software to obtain displacement data and calculated the
elasticity modulus with it, the data showed some height effects. The calculated modulus increased
with the increasing of sample height. But by posting strain gauge to obtain displacement and
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calculated elasticity modulus, the modulus of elasticity showed high stability, and the height’s effect
was small on experimental results.
3) Theory proves that the reason why sample height affects elasticity modulus which is calculated
with the displacement data form MTS software is mainly due to the testing machine’s stiffness. For
the rock samples with higher hardness, the displacement of testing machine caused by its stiffness
cannot be ignored. The elasticity modulus formula deduced from theory that have considered the
stiffness of testing machine can revise experimental error caused by testing machine. In the places
where strain gages are not accessible in the experiments due to the restricted experimental condition,
this formula can revise the experimental results which can precisely represent the elasticity modulus
of material itself.
4) The relationship between the calculated elasticity modulus of the sample and sample’s height
is fraction function relationship, along with the increase of sample’s height, the effect of testing
machine on elasticity modulus decreased gradually. Ignoring other factors, the calculated elasticity
modulus data would gradually close to the true value, but there might be increased effects from other
factors of the samples. In actual sampling, under the special condition that height-diameter ration is
not 2.0, if rock sample’s length-diameter ratio is between 1.5-2.5, then the experimental results still
have a high reliability.
ACKNOWLEDGMENTS
Financial support for this work, provided by the National Nature Science Foundation of China
(No.51174192) is gratefully acknowledged. When author of this article studied in West Virginia
University, Syd S. Peng, a member of US National Academy of Engineering and a professor in West
Virginia University’s Mining Engineering Department, offered me countless selfless help in
providing experimental materials and facilities. I would like to express my deep gratitude to him here!
In addition, I would also like to express my appreciation to all the reviewers for reviewing my
manuscript.
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