Deformation theory and energy mechanism of cyclic dynamic mechanical damage for granite in the diversion tunnel under cyclic loading and unloading

Abstract

The diversion tunnel is frequently subjected to cyclic dynamic loads during blasting and mechanical excavation. To explore the theory and mechanism of cyclic dynamic mechanical damage for granite in a diversion tunnel under cyclic loading-unloading, the incremental cyclic loading-unloading test and numerical simulation were conducted on granite samples from the diversion tunnel. According to the mechanical and deformation characteristics of rock samples in the process of cyclic loading-unloading, the stress-strain normalization theory evolution model based on viscoelastoplasticity was established, and the cyclic dynamic damage evolution mechanism of rock samples was revealed. The evolution characteristics of stress-strain curves of rock samples under cyclic loading-unloading were effectively described, and the loading-unloading two-stage constitutive equation of rock samples under cyclic loading-unloading was established. The energy and damage evolution mechanism of the final full stress-strain curve and the influence mechanism of shrinkage and expansion of rock particles on the mechanical properties of granite were discussed.

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Article
iScience
Deformation theory and energy mechanism of cyclic
dynamic mechanical damage for granite in the diversion
tunnel under cyclic loading-unloading
Graphical abstract
Highlights
dTheoretical model of stress-strain based on
viscoelastoplasticity was established
dDynamic cyclic damage mechanism and the Poisson’s ratio
evolution law were revealed
dLoading-unloading two-stage constitutive equation was
established
dMechanism of energy and damage evolution in the final
stress-strain was discussed
Authors
Rongzhou Yang, Ying Xu
Correspondence
rongzhouy@outlook.com
In brief
Applied sciences; Engineering; Materials
science
Yang & Xu, 2025, iScience 28, 111583
January 17, 2025 ª2024 The Author(s). Published by Elsevier Inc.
https://doi.org/10.1016/j.isci.2024.111583 ll

iScience
Article
Deformation theory and energy mechanism of cyclic
dynamic mechanical damage for granite in the diversion
tunnel under cyclic loading-unloading
Rongzhou Yang
1,2,3,4,
*and Ying Xu
1,2,3
1
Anhui Key Laboratory of Mining Construction Engineering, Anhui University of Science and Technology, Huainan 232001, China
2
School of Civil Engineering and Architecture, Anhui University of Science and Technology, Huainan 232001, China
3
State Key Laboratory of Mining Response and Disaster Prevention and Control in Deep Coal Mines, Anhui University of Science and
Technology, Huainan 232001, China
4
Lead contact
*Correspondence: rongzhouy@outlook.com
https://doi.org/10.1016/j.isci.2024.111583
SUMMARY
The diversion tunnel is frequently subjected to cyclic dynamic loads during blasting and mechanical excava-
tion. To explore the theory and mechanism of cyclic dynamic mechanical damage for granite in a diversion
tunnel under cyclic loading-unloading, the incremental cyclic loading-unloading test and numerical simula-
tion were conducted on granite samples from the diversion tunnel. According to the mechanical and defor-
mation characteristics of rock samples in the process of cyclic loading-unloading, the stress-strain normal-
ization theory evolution model based on viscoelastoplasticity was established, and the cyclic dynamic
damage evolution mechanism of rock samples was revealed. The evolution characteristics of stress-strain
curves of rock samples under cyclic loading-unloading were effectively described, and the loading-unloading
two-stage constitutive equation of rock samples under cyclic loading-unloading was established. The energy
and damage evolution mechanism of the final full stress-strain curve and the influence mechanism of
shrinkage and expansion of rock particles on the mechanical properties of granite were discussed.
INTRODUCTION
The diversion tunnel project is an important ecological project
and livelihood project. The construction of the diversion tunnel
project can well realize urban emergency standby water supply,
flood control and drainage, farmland irrigation, and ecological
water replenishment of the river network. In the process of diver-
sion tunnel construction, the drilling-blasting and mechanical
excavation (such as TBM) is the main method of tunnel excava-
tion.
1–3
However, blasting and mechanical rock breaking inevi-
tably exert a frequent dynamic load on the surrounding rock of
the tunnel, especially the surrounding rock mass under complex
geological conditions is likely to affect its structural stability due
to the cyclic cumulative damage effect. The field of engineering
has demonstrated
4,5
that the construction of deep-buried tun-
nels often encounters significant challenges such as the large
deformation of surrounding rock and rockburst, which are
considered to be engineering geological disasters. Currently,
the research on rock mechanics is focused on the dynamic dam-
age mechanism and the evolution law of Poisson’s ratio for rock
materials under dynamic cyclic loading, which remains a com-
plex and highly debated topic in this field.
6,7
Therefore, it is
essential to conduct comprehensive research on the dynamic
behavior and evolution characteristics of rock samples from
diversion tunnels under cyclic loading and unloading. It is crucial
to deeply understand the cyclic damage mechanism and the
evolution law of Poisson’s ratio in rock materials. This will further
establish a theory for the cyclic damage and deformation of rock
materials, which is highly significant for developing safe and effi-
cient blasting and mechanical construction technology for multi-
section diversion tunnels.
In recent years, numerous scholars have conducted cyclic
loading-unloading tests on rocks subjected to frequent
dynamic disturbances, yielding significant research findings
regarding the evolution characteristics of dynamic mechanical
behavior.
8–11
In terms of the mechanical properties, Gautam
et al.
12,13
studied the evolution characteristics of the strength,
deformation, failure, and acoustic emission (AE) of granite and
shale at different cyclic loading-unloading rates (0.5, 1.0, 1.5,
2.0, and 2.5 kN/s) under the damage-controlled cyclic loading-
unloading tests. Quantitative damage based on the Felicity effect
at different cyclic loading-unloading rates was analyzed, Kaiser
and Felicity effects based on AE parameters were verified, and
the peak stress and Kaiser effect stress of each cyclic loading-un-
loading of rock specimens were discussed. The results showed
that the Kaiser effect occurs in the linear elastic phase, while
the Felicity effect occursin the crack initiation and crack propaga-
tion phase. Yang et al.
14
conducted multi-stage cyclic compres-
sion tests on red sandstone and granite under various confining
pressures. They proposed an improved Hardin hyperbolic model
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to describe the backbone curve, analyzed the relationship be-
tween dynamic shear modulus, damping ratio, and cyclic shear
strain, and established an empirical formula for estimating dy-
namic shear modulus and damping ratio. Zhang et al.
15
conduct-
ed multi-level cyclic loading tests on artificial columnar jointed
rock mass specimens with varying joint dip angles. They qualita-
tively and quantitatively discussed the energy evolution charac-
teristics and failure modes of these specimens under multi-level
cyclic loading and elucidated the mechanism of fatigue effect
on the mechanical properties of anisotropic artificial columnar
jointed rock mass specimens. Zhou et al.
16
conducted a numer-
ical study on the mechanical properties of rock materials under
single-level and multi-level cyclic loading using the dynamic
constitutive model of rock materials. They analyzed the mechan-
ical evolution characteristics of rock samples, focusing on stress-
strain curves, elastic modulus, damage, and strength. In terms of
the damage characteristics, Moghaddam and Golshani
17
con-
ducted a study on the fatigue damage mechanical behavior of
rock under cyclic loading using the discrete element method.
They proposed a unique perspective on the damage fracture
and geometric shape evolution of rock samples under cyclic
loading and revealed the distribution characteristics and rules
of rock fracture samples throughout the entire sample volume.
To investigate the mixed mode fracture damage characteristics
of the rock-concrete interface under cyclic loading, Guo et al.
18
conducted a study to quantify the influence of rock type and
modal angle on interface damage mechanisms using a damage
identification method. They also determined the temporal and
spatial evolution of tensile and shear damage under cyclic
loading, and further elucidated the influence mechanism of dam-
age evolution law on crack propagation path. To investigate the
macroscopic and mesoscopic cumulative damage mechanism
of the structural plane in soft and hard sandwiched rock mass,
Xu et al.
19
conducted laboratory pre-peak cyclic shear tests
and PFC2D meso-numerical calculations under constant normal
load. The study took into consideration the effects of moisture
Figure 2. Stress-strain and damage evolution characteristics during cyclic loading-unloading
(A) Stress-strain curves and macro-crushing characteristics.
(B) Deformation and rupture and AE damage evolution.
Figure 1. Evolution characteristics of load and deformation during cyclic loading-unloading
(A) Load time history curves.
(B) Deformation time history curves.
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content, normal stress, shear rate, shear amplitude, first-order
micro-convex angle, and cyclic shear times. In the aspect of
the constitutive theory, to fully simulate the cyclic behavior of
rock, Hu et al.
20
conducted a comprehensive analysis of the me-
chanical characteristics relevant to fully simulating the cyclic
behavior of rock. This analysis included considerations of the
massing effect, ratcheting effect, Kaiser effect, and Felicity effect.
Based on an understanding of the deformation mechanism, they
proposed a constitutive model realized through the strain incre-
ment method. Liu et al.
21
proposed a novel damage constitutive
model based on energy dissipation to characterize the behavior
of rock under cyclic loading. They introduced a damage variable
based on energy dissipation, calculated the damage evolution
equations for two typical rock types based on the results of uni-
axial cyclic loading tests, and introduced the concept of compac-
tion coefficient to describe the degree of compaction. This was
used to modify the damage constitutive model obtained by the
Lemaitre strain equivalent hypothesis. Ren et al.
22
derived the
compaction damage variable from the nonlinear stress-strain
relationship in the initial stage of compaction. They also proposed
the crack damage variable based on statistical damage theory
and established the damage constitutive equation of rock under
cyclic compression.
To sum up, the current research still lacks research on the me-
chanical characteristics, damage mechanism, and Poisson’s ra-
tio evolution law of granite in diversion tunnels under cyclic
Figure 3. Shear stress nephograms during cyclic unloading
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compression. Based on this, in conjunction with the actual con-
struction project of the diversion tunnel, an incremental cyclic
loading-unloading test was conducted on typical hard granite
within the diversion tunnel. The dynamic damage mechanism
and evolution law of Poisson’s ratio for diversion tunnel granite
under cyclic loading-unloading were thoroughly investigated.
Additionally, a two-stage constitutive equation for rock samples
under cyclic loading-unloading was established, and the energy
Figure 4. AE damage nephograms during cyclic unloading (Note: The white circles represent compression-shear damage, the red circles
represent tensile damage, and the larger the density and distribution range of the circles, the more serious the damage.)
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and damage evolution mechanism of the final full stress-strain
curve for granite under cyclic loading-unloading was discussed.
The findings of this study hold significant importance in promot-
ing efficient construction and ensuring safety during the blasting
and mechanical excavation of hard rock in diversion tunnels.
RESULTS
Mechanical properties and damage evolution
characteristics
Load/deformation time history curves
The load/deformation time history curves of the granite sample
under cyclic loading-unloading are shown in Figure 1. It can be
seen from Figure 1 that the cyclic load showed the perfect linear
evolution with time, and the loading force curves and unloading
force curves of each cycle had a vertical axisymmetric relation-
ship with each other, which ensured the accuracy of the test
data of granite samples in the process of cyclic loading-unload-
ing by loading method of control load fore. The difference was
that the cyclic deformation showed a regular nonlinear evolution
with time, and the loading deformation curves and unloading
deformation curves of each cycle were approximately vertical
axisymmetric to each other. It can be found that the loading
deformation curves and unloading deformation curves showed
the characteristics of deformation advance and deformation
hysteresis with time respectively, which further led to the evolu-
tion characteristics of residual deformation accumulating over
time. The above results showed that the complex mechanical
behavior evolution mechanism of granite samples in the process
of cyclic loading-unloading needs to be further revealed.
Stress-strain curves and the characteristics of damage-
fracture
The damage-fracture characteristic mechanism of rock samples
is the key mechanical response contained in the evolution char-
acteristics of stress-strain curves of rock samples.
23–26
The
stress-strain curves and macroscopic crushing characteristics
and deformation and rupture and AE damage evolution of granite
sample under cyclic loading-unloading are shown in Figure 2.As
can be seen from Figure 2A, with the increase of the number of
cycles, both the loading stress-strain curve and the unloading
stress-strain curve gradually moved to the right, reflecting the re-
sidual strain caused by the compaction effect and damage effect
under the loading-unloading action of each cycle. It can be seen
from the local magnification region of the stress-strain curves
that the stress-strain curves showed the evolution characteristic
of "from dense to sparse" with the increase of the number of cy-
cles, which further reflected that the damage of rock samples
showed the evolution characteristic of "from non-structural
Figure 5. Meso-fracture morphology characteristics of rock samples
(A) Compression-tension fracture surfaces.
(B) Compression-shear fracture surfaces.
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damage to structural damage" with the increase of the number of
cycles. In particular, the obvious elastic-brittle failure of the rock
sample occurred during the 15th cyclic loading, and the frag-
ments of the rock sample ejected violently around, showing
the characteristics of a violent "rockburst," and the structural
damage was maximized. As can be seen from Figure 2B, During
the evolution process of cyclic loading-unloading deformation,
AE damage counts mainly occurred during cyclic loading defor-
mation, and almost no AE damage counts occurred during cyclic
unloading deformation. With the evolution of cyclic loading
deformation, AE damage counts gradually increased, especially
during the 10th-15th cyclic loading. At the same time, it can be
found that the degree of damage and rupture of the rock samples
gradually increased, showing a relatively complex comprehen-
sive failure mode.
Combining the stress and AE damage nephograms obtained
from numerical simulation (Figures 3 and 4), it can be seen that
the fragmentation degree of the rock sample was serious, and
the failure modes were more complex, which were mainly char-
acterized by compression-tension failure and compression-
shear failure. Through the analysis of the meso-morphological
characteristics of the compression-tension fracture surfaces
and the compression-shear fracture surfaces, it is known that
both the compression-tension fracture surfaces and the
compression-shear fracture surfaces showed clearly colored
rock matrix and mineral composition (Figure 5A). The difference
was that the compression-tension fracture surfaces were rough
and showed exfoliation fracture characteristics, while the
compression-shear fracture surfaces were relatively flat and
showed obvious shear-slip fracture characteristics (Figure 5B).
Strain behavior characteristics
The strain behavior characteristics of granite samples under cy-
clic loading-unloading are shown in Figure 6. As can be seen
from Figure 6A, with the increase of the number of cycles, the
Figure 6. Characteristics of strain behavior
(A) Strain.
(B) Strain difference.
Figure 7. Characteristics of modulus behavior
(A) Deformation modulus.
(B) Deformation modulus difference.
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loading strain, cumulative residual strain, and elastic strain
all increased linearly. The magnitude relation of strain under
the same number of cycles was: loading strain > elastic
strain > cumulative residual strain, and there was little difference
between loading strain and elastic strain, but both loading strain
and elastic strain were significantly greater than cumulative re-
sidual strain, indicating that the loading strain generated during
cyclic loading was mainly converted into elastic strain during cy-
clic unloading. In other words, for the granite material with
elastic-plastic, the elastic deformation of the rock samples was
dominant, which highlighted the elastic-brittle characteristics
of the rock samples. As can be seen from Figure 6B, as a whole,
with the increase of the number of cycles, the loading strain dif-
ference, cumulative residual strain difference, elastic strain dif-
ference, and unclosed degree all showed the evolution charac-
teristics of "nonlinear decreasing /horizontal fluctuation." On
the whole, the magnitude relation of strain difference under the
same number of cycles was: loading strain difference > elastic
strain difference > unclosed degree > cumulative residual strain
difference, and there was little difference between loading strain
difference and elastic strain difference, and there was little differ-
ence between unclosed degree and cumulative residual strain
difference, but loading strain difference and elastic strain differ-
ence were significantly larger than unclosed degree and cumula-
tive residual strain difference. It can be found that the strain dif-
ference decreased obviously in the 1st 3rd cycle, the strain
difference was basically stable in the 4th 10th cycle and fluc-
tuated obviously in the 11th 14th cycle. This phenomenon
showed that the plastic deformation of rock samples in the initial
cycle was mainly compaction deformation, and the plastic defor-
mation of rock samples tended to fluctuate and stabilize due to
the weakening of the compaction deformation effect and the
enhancement of the elastic deformation effect in the subsequent
cycle. However, with the continuous increase of the number of
cycles, the cumulative effect of plastic damage increased, which
led to the obvious fluctuation of the plastic deformation of rock
samples in the later cycle.
Modulus behavior characteristics
The modulus behavior characteristics of granite samples under
cyclic loading-unloading are shown in Figure 7. From Figure 7A,
it can be seen that the evolution characteristics of loading defor-
mation modulus and unloading deformation modulus with the
number of cycles were basically the same, and the deformation
modulus showed a "convex" nonlinear growth characteristic due
to the stiffness strengthening effect, but the stiffness strength-
ening effect will gradually weaken with the increase of the num-
ber of cycles due to the cumulative effect of plastic damage. At
the same number of cycles, the loading deformation modulus
was less than the unloading deformation modulus, which was
mainly due to the inevitable plastic deformation of rock samples
in each cycle. According to the fitting curves, the difference in
unloading-loading deformation modulus was larger in the 1st 
3rd and 11th 14th cycles, and smaller in the 4th 10th cycle.
As can be seen from Figure 7B, as a whole, with the increase of
the number of cycles, both the loading deformation modulus dif-
ference and the unloading deformation modulus difference
showed the evolution characteristic of nonlinear decrease, while
the unloading-loading deformation modulus difference showed
the evolution characteristic of "nonlinear reduction /horizontal
fluctuation." The deformation modulus difference decreased
obviously in the 1st 3rd cycle, the deformation modulus differ-
ence was basically stable in the 4th 10th cycle, and fluctuated
obviously in the 11th 14th cycle, which reflected that the rock
samples mainly went through three evolution stages in the whole
cycle: initial cycle compaction (non-structural damage), middle
cycle stability (non-structural damage), and later cycle damage
(structural damage).
Energy behavior characteristics
The energy density-strain curves of granite samples under cyclic
loading-unloading are shown in Figure 8. As can be seen from
Figure 8A, the input energy density and elastic energy density
of rock samples showed completely different evolution charac-
teristics with the increase of strain. In the process of loading,
the energy density input to the rock sample showed a "concave"
Figure 8. Energy density-strain curves
(A) Input energy density.
(B) Elastic energy density.
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non-linear growth trend with the increase of strain, the energy
input rate of the rock sample increased with the increase of
strain, and the input energy density curve gradually moved to
the right with the increase of the number of cycles. It can be
seen from Figure 8B that during unloading, the elastic energy
density released from rock samples showed a "convex"
nonlinear growth trend with the decrease of strain, the energy
release rate of rock samples slowed down with the decrease
of strain, and the elastic density curve gradually moved to the
right with the increase of the number of cycles.
The energy behavior characteristics of granite samples under
cyclic loading-unloading are shown in Figure 9. It can be seen
from Figure 9A that the input energy density, elastic energy den-
sity, dissipated energy density, and cumulative dissipation
energy density of rock samples all showed a "concave" non-
linear growth trend with the increase of the number of cycles,
and well obeyed the characteristics of power function distribu-
tion. Under the same number of cycles, the magnitude relation
of energy density was: input energy density > elastic energy
density > cumulative dissipation energy density > dissipated en-
ergy density, and there was little difference between input energy
density and elastic energy density, which reflected that the en-
ergy input into the rock sample during cyclic loading was mainly
accumulated as elastic strain energy. However, due to the exis-
tence of the compaction effect and cumulative damage effect, a
small part of the input energy was inevitably transformed into
dissipated energy due to plastic deformation.
As can be seen from Figure 9B, on the whole, the evolution
characteristics of the input energy density difference and the
elastic energy density difference with the number of cycles
were basically the same, and were obviously larger than the
dissipated energy density difference. In the process of the 1st
10th cycle, the input energy density difference curve and
elastic energy density difference curve basically coincided, and
showed the characteristic of linear growth with the increase of
the number of cycles. With the increase of the number of cycles,
the cumulative dissipation energy density difference showed the
evolution characteristic of horizontal stability with near zero. Dur-
ing the 11th 14th cycle, the input energy density difference,
elastic energy density difference, and cumulative dissipation en-
ergy density difference fluctuated obviously with the increase of
the number of cycles. The above phenomena showed that the
rock sample was basically in a state of dynamic equilibrium in
which the input energy density was transformed into elastic
Figure 9. Characteristics of energy behavior
(A) Energy density.
(B) Energy density difference.
(C) Energy ratio.
(D) Linear energy storage and energy dissipation law.
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energy density and dissipated energy density during the 1st 
10th cycle, but with the continuous increase of the number of cy-
cles, this dynamic equilibrium state tended to fluctuate due to
the enhancement of cumulative damage effect, and was finally
broken because the elastic energy broke the energy storage limit
in the loading process of the 15th cycle.
As can be seen from Figure 9C, as a whole, the elastic energy
ratio showed the evolution characteristic of "first increasing, then
decreasing" with the increase of cycle times, and the dissipated
energy ratio shows the evolution characteristic of "first
decreasing, then increasing" with the increase of the number
of cycles, and the relationship between elastic energy ratio curve
and dissipated energy ratio curve showed a horizontal axis sym-
metry. At the same number of cycles, the elastic energy ratio
(0.66–0.88) was significantly higher than that of dissipated en-
ergy (0.12–0.34), which also reflected that the input energy of
rock samples was more transformed to elastic energy in the pro-
cess of cyclic loading. As can be seen from Figure 9D, with the
increase of input energy density, both elastic energy density
and dissipated energy density showed obvious linear growth
characteristics, which highlights the linear energy storage law
of elastic energy and the linear energy dissipation law of dissi-
pated energy.
Damage behavior characteristics
The damage behavior of granite samples under cyclic loading-
unloading is shown in Figure 10. It can be seen from Figure 10
that in the process of 0–14th cycle, strain damage, energy dam-
age, and cumulative AE counts showed the evolution character-
istics of nonlinear and slowly increasing with the increase of the
number of cycles, and well obeyed the power function distribu-
tion. During the 14th 15th cycle, both strain damage and en-
ergy damage showed the evolution characteristics of linear
and rapidly increasing with the increase of the number of cycles.
On the whole, the damage curves can be divided into three
Figure 11. Evolution characteristics of axial and lateral strain during cyclic loading-unloading
(A) Axial strain time history curves.
(B) Lateral strain time history curves.
Figure 10. Characteristics of damage behavior
(A) Strain damage and energy damage.
(B) AE damage counts.
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evolution stages: early non-structural damage stage (0–10th cy-
cle), later structural damage stage (10th 14th cycle), and final
structural damage stage (14th 15th cycle).
(1) In the early non-structural damage stage (0–10th cycle):
At this stage, the rock sample mainly occurred compac-
tion deformation and elastic deformation, the skeleton
structure of the rock sample was intact, almost no struc-
tural damage occurred in the rock sample, and the dam-
age of the rock sample was at a low level (0–0.08). It is
worth noting that the strain damage and energy damage
of rock samples under the 0th and 10th cycles were equal,
in addition, the strain damage was larger than the energy
damage. The reason that the strain damage was larger
than the energy damage was mainly caused by the
compaction effect, which led to the difference between
the evolution path of strain damage and energy damage.
(2) In the later structural damage stage (10th 14th cycle): At
this stage, the slight structural damage of rock samples
mainly occurred due to the enhancement of the cumula-
tive damage effect, and the damage level of rock samples
was improved (D
S
: 0.08–0.11, D
E
: 0.08–0.25). It is worth
noting that the strain damage increased slowly, while
the energy damage increased faster, which indicated
that the energy damage was more sensitive to the in-
crease of dissipated energy and can better characterize
the damage evolution characteristics of rock samples.
(3) In the final structural damage stage (14th 15th cycle): At
this stage, the elastic energy accumulated in the rock
sample broke through the energy storage limit, and the
complete loss of the bearing capacity of the skeleton
structure of the rock sample led to the sudden release
of the large number of elastic energy, which was mainly
transformed into the fracture dissipation energy of the
rock sample and the ejection kinetic energy of the rock
sample fragments. The strain damage and energy dam-
age increased rapidly to 1 and showed the shock failure
characteristics of strain rockburst and energy rockburst
respectively.
Based on the above analysis results, the mechanism of differ-
ences in damage evolution characteristics between the non-
structural damage stage (0–10th cycle) and the structural dam-
age stage (10th 15th cycle) can be further analyzed. The strain
damage of rock samples is mainly caused by the cumulative ef-
fect of residual strain during each cycle, and the energy damage
Figure 12. Evolution law of Poisson’s ratio
(A) Poisson’s ratio time history curves.
(B) Poisson’s ratio at the unloading point.
Figure 13. Elastic deformation-linear
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of rock samples is mainly caused by the cumulative effect of
damage energy consumption during each cycle.
Before structural damage occurs (0–10th cycle), since the rock
sample does not experience obvious structural damage during
this process, the main source of residual strain is caused by
the compaction deformation effect of the rock sample during
each cycle, and the main source of damage energy consumption
is the non-structural damage effect of the rock sample during
each cycle. However, the non-structural damage effect of rock
samples during this process is negligible. Therefore, before
structural damage occurs (0–10th cycle), strain damage is
greater than energy damage.
During the process of structural damage (10th 15th cycle),
Since the rock samples suffer obvious structural damage during
this process, the main source of residual strain is caused by the
damage and deformation effect of the rock sample during each
cycle, and the main source of damage energy consumption is the
structural damage effect of the rock samples during each cycle.
However, the structural damage effect of rock samples is basi-
cally fully reflected in this process. Therefore, during the occur-
rence of structural damage (10th 15th cycle), energy damage
is greater than strain damage.
Axial/lateral strain time history curves and Poisson’s
ratio evolution law
Axial/lateral strain time history curves
The axial/lateral strain time history curves in the middle of granite
samples under cyclic loading-unloading are shown in Figure 11.
As can be seen from Figure 11A, the evolution characteristics of
Figure 14. Elastic deformation-nonlinear
Figure 15. Viscoelastic deformation
the axial strain time history curve in the
middle of the rock sample were basically
consistent with the axial deformation time
history curve of the rock sample as a
whole, which ensured the accuracy of
the data collected by the strain gauge.
The difference was that the axial
loading-unloading strain in the middle of
the rock sample showed a linear charac-
teristic with time, and there was no
obvious residual strain, which indicated that the axial strain in
the middle of the rock sample was almost completely elastic
strain. It can be seen from Figure 11B that the lateral loading-un-
loading strain in the middle of the rock sample also showed a
linear characteristic with time. The transverse expansion defor-
mation of the rock sample under axial load was mainly elastic
deformation, but it had obvious residual strain, and the cumula-
tive residual strain increased gradually with the number of
cycles.
Poisson’s ratio evolution law
The variation characteristics of Poisson’s ratio of rock is an
important reflection of its loading and deformation, which con-
tains the mechanical evolution law that needs to be further re-
vealed.
27
The evolution characteristic curves of Poisson’s ratio
in the middle of granite samples under cyclic loading-unloading
are shown in Figure 12. It can be seen from Figure 12A that the
effective value range of Poisson’s ratio in the middle of the
rock sample was mainly between 0 and 1 during the whole cyclic
loading-unloading process. It can be found that in the process of
cyclic loading, Poisson’s ratio attenuated rapidly with time, and
in the process of cyclic unloading, Poisson’s ratio increased
rapidly with time. The main reason for this phenomenon is that
the axial deformation rate of rock samples is always larger than
that of transverse deformation. It can be seen from Figure 12B
that the Poisson’s ratio at the unloading point in the middle of
the rock sample increased with the increase of the number of cy-
cles because the degree of lateral cumulative residual strain is
larger than that of axial cumulative residual strain. According to
the evolution characteristics of deformation modulus, the almost
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complete elastic deformation stage of rock samples mainly
occurred in the process of the 6th 8th cycle, so the Poisson’s
ratio range of rock samples in the elastic deformation stage was
0.19–0.23.
Theory and mechanism of cyclic dynamic deformation
and damage
Further research on the theory and mechanism of dynamic dam-
age of rock under cyclic load is of great significance to solving
practical engineering problems and promoting the development
of rock mechanics.
28–31
Aiming at the phenomenon of loading
strain advance effect and unloading strain hysteresis effect of
granite samples in the process of incremental cyclic load, based
on the author’s previous research
32
and combined with the
physical mechanics and deformation characteristics of mate-
rials, in the process that the cyclic load uniformly increases
and decreases with time, the deformation characteristics of the
materials can be divided into five cases to further analyze the cy-
clic dynamic damage theory and mechanism, the five normalized
theoretical evolution models about stress and strain were estab-
lished accordingly.
Case 1: Elastic deformation-linear
As shown in Figure 13, in the process of incremental cyclic load
in which the cyclic load increases and decreases uniformly with
time, the material has no loading strain advance effect and un-
loading strain hysteresis effect, and the stress and strain always
change synchronously. The loading deformation can be elasti-
cally restored in the original path during unloading, which makes
Figure 16. Elastoplastic deformation
Figure 17. Viscoelastoplastic deformation
the material free of residual deformation
and no damage and energy dissipation
during cyclic loading-unloading.
Case 2: Elastic deformation-
nonlinear
As shown in Figure 14, in the process of
incremental cyclic load in which the cy-
clic load increases and decreases uni-
formly with time, the material has the
loading strain advance effect and
unloading strain hysteresis effect, and the stress and strain
can not change simultaneously. However, loading deformation
can be elastically restored in the original path during
unloading (caused by the loading strain advance effect equal
to the unloading strain hysteresis effect), which makes the
material also has no residual deformation in the cyclic
loading-unloading process, and there is no any damage and
energy dissipation.
Case 3: Viscoelastic deformation
As shown in Figure 15, in the process of incremental cyclic load
in which the cyclic load increases and decreases uniformly with
time, the material has the loading strain advance effect and un-
loading strain hysteresis effect, and the stress and strain can
not change synchronously. The loading deformation can not
be elastically restored in the original path during unloading
(caused by the loading strain advance effect being less than
the unloading strain hysteresis effect). When the material does
not have residual deformation in the cyclic loading-unloading
process, the material does not have any damage but has damp-
ing energy dissipation.
Case 4: Elastoplastic deformation
As shown in Figure 16, in the process of incremental cyclic load
in which the cyclic load increases and decreases uniformly with
time, the material has no loading strain advance effect and un-
loading strain hysteresis effect, and the stress and strain can
change synchronously. However, when the loading deformation
can not be elastically restored in the original path during unload-
ing (caused by plastic deformation), the residual deformation
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occurs in the cyclic loading-unloading process. The material has
damage (there may be no structural damage), and there is plastic
energy dissipation.
Case 5: Viscoelastoplastic deformation
As shown in Figure 17, in the process of incremental cyclic load
in which the cyclic load increases and decreases uniformly with
time, the material has the loading strain advance effect and un-
loading strain hysteresis effect, and the stress and strain can
not change synchronously. The loading deformation can not
be elastically restored in the original path during unloading
(caused by the loading strain advance effect being less than
the unloading strain hysteresis effect and plastic deformation).
In the process of cyclic loading and unloading, the material has
residual deformation and damage (there may be no structural
damage), and there is damping energy dissipation and plastic
energy dissipation.
Combined with the above experimental and theoretical re-
sults, it can be seen that in the process of incremental cyclic
load in which the cyclic load increases and decreases uniformly
with time, the mechanical deformation characteristics of granite
samples were consistent with the viscoelasticplastic deforma-
tion characteristics of the material in case 5. The cyclic dynamic
damage theory and mechanism of granite samples were deeply
analyzed and revealed.
Constitutive theory analysis
Two-stage constitutive equation of loading-unloading
stress-strain curves
The constitutive relation of material is the essential reflection of
its mechanical behavior in theory.
33
Based on the author’s
previous research,
32
the cyclic loading-unloading stress-strain
curves of granite samples were fitted by an exponential function
Figure 18. Theoretical fitting curves of stress-strain curves under cyclic loading-unloading
(A) 3rd cycle.
(B) 4th cycle.
Table 1. Theoretical fitting parameters of stress-strain curves
Load type Loading Unloading
Equations sL=aLexpðhLεLÞ+bLsU=aUexpðhUεUÞ+bU
Number of cycles (n)aLhLbLR2aUhUbUR2
1 3.03214 850.77714 2.32952 0.99801 1.15187 1332.37705 1.72848 0.99784
2 4.95888 685.53194 5.64426 0.99926 3.01106 853.69266 4.73117 0.99933
3 6.36646 610.90123 8.03733 0.99905 4.64235 700.95818 7.54072 0.99953
4 8.91915 522.36426 11.90334 0.9984 7.69681 560.36788 12.35724 0.99904
5 12.93256 437.61472 17.53553 0.99785 11.93012 457.29049 18.64562 0.99866
6 18.00125 372.10374 24.48107 0.99749 17.19937 382.81149 26.03941 0.9984
7 24.04808 321.33828 32.40736 0.99733 23.52909 327.05966 34.54408 0.99818
8 31.43235 278.9856 41.70269 0.99717 30.84897 283.58423 44.00644 0.99812
9 40.98214 241.13002 53.34777 0.99702 39.32372 247.74341 54.58712 0.99816
10 51.19294 212.23738 65.56372 0.9971 48.44415 219.16375 65.79378 0.99817
11 62.12509 189.11199 78.58492 0.99714 56.2825 190.556 75.54474 0.99689
12 86.3596 147.11291 106.3724 0.99739 77.38104 157.03059 99.99618 0.99848
13 102.70647 131.4475 124.65374 0.99752 90.09634 142.24286 114.46891 0.99851
14 137.97689 108.11731 163.29513 0.99669 105.7925 121.46932 131.62504 0.99874
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(s=aexpðhεÞ+b)(Figure 18). The theoretical fitting parameters
of the stress-strain curves are shown in Table 1. As can be seen
from Figure 18 and Table 1, all fitting correlation coefficients R
2
were larger than 0.996, and the fitting effect was accurate, which
can fully reflect the evolution characteristics of stress-strain
curves of granite samples under cyclic loading-unloading.
Based on this, the two-stage constitutive equation (Equation
1) of granite samples under cyclic loading-unloading can be
proposed:
sL=aLexpðhLεLÞ+bLðLoadingÞ
sU=aUexpðhUεUÞ+bUðUnloadingÞ(Equation 1)
where: aL,hL,bLare the constitutive mechanical parameters of
the stress-strain curves during loading; aU,hU,bUare the consti-
tutive mechanical parameters of the stress-strain curves during
unloading.
To further determine the mechanical parameters of the
loading-unloading two-stage constitutive equation, the expo-
nential function (s=aexpðhεÞ+b) was also used to fit and
analyze the variation of the mechanical parameters of the consti-
tutive equation with the number of cycles n(Figure 19). All fitting
correlation coefficients R
2
were larger than 0.996, the fitting ef-
fect was accurate, and the mechanical parameter equation of
the constitutive equation can be determined accurately. As a
result, the loading-unloading two-stage constitutive equation
of granite samples under cyclic loading-unloading can be
established.
Loading stage (Equation 2):
sL=5:10757e0:23616n3:36814e½959:67298e0:166311n+22:98663ε
+9:41486e0:20645n+8:72014(Equation 2)
Unloading stage (Equation 3):
sU=11:82541e0:16671n14:30766
3e½1572:19985e0:33771n+149:82042ε
+23:37356e0:13772n+26:8508
(Equation 3)
Figure 19. Parameter fitting curves of loading-unloading two-stage damage constitutive equation
(A) a.
(B) h.
(C) b.
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Figure 20. Verification of loading-unloading two-stage constitutive equation
(A) n=1.
(B) n=2.
(C) n=5.
(D) n=6.
(E) n=7.
(F) n=8.
(G) n=9.
(H) n= 10.
(I) n= 11.
(J) n= 12.
(K) n= 13.
(L) n= 14.
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Verification of loading-unloading two-stage constitutive
equation
The cyclic loading-unloading stress-strain curves of rock sam-
ples were theoretically fitted by using the loading-unloading
two-stage constitutive equation, and the accuracy of the
loading-unloading two-stage constitutive equation was verified
by comparing the experimental curves with the theoretical
curves. The data presented in Figure 20 demonstrates the effi-
cacy of the loading-unloading two-stage constitutive equation
proposed in this study, which accurately characterizes the evo-
lution characteristics of stress-strain curves exhibited by rock
samples under cyclic loading-unloading conditions. This finding
underscores the precision and reliability of our approach in char-
acterizing the mechanical behavior of rocks subjected to
repeated loading and unloading cycles.
DISCUSSIONS AND LIMITATIONS
Energy dissipation principle and energy damage theory
From the point of view of energy evolution, the process of rock
compression deformation and failure is essentially a series of
complex evolution processes of external energy input, releas-
able elastic energy accumulation, energy dissipation, and
release, that is, the essence of rock sample deformation, dam-
age, and failure under external load is the process of internal en-
ergy exchange.
34,35
From the meso-scale of the rock damage
process, the deformation and failure process is the process of
compression and closure of internal primary micro-defects, initi-
ation, and development of new cracks, and finally converging to
form macro-cracks. The evolution process of dissipated energy
just represents and reflects the evolution law of internal micro-
defects and damage degree of rock.
35,36
For the rock after
many times of cyclic compression, the degree of internal primary
micro-defects must be increased due to the cumulative damage.
Therefore, a deep discussion and analysis of the evolution mech-
anism of the stress-strain curve and energy curve of rock sam-
ples after repeated cyclic compression is of great significance
to further reveal the mechanism of dynamic damage and insta-
bility of surrounding rock under cyclic dynamic disturbance.
Assuming that there is no heat exchange between the internal
energy conversion of the rock sample and the external environ-
ment during the final uniaxial cyclic loading, it can be obtained
Figure 21. Energy evolution characteristics of the final full stress-strain curve
(A) Final stress-strain curve.
(B) Final energy-strain curves.
Figure 22. Energy damage evolution characteristics of the final full stress-strain curve
(A) Energy damage-strain curve.
(B) Structural damage-strain curve.
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according to the first law of thermodynamics (Equations 4,5,6,
and 7):
34,36–38
Ufi =Ufe +Ufd (Equation 4)
Ufi =Z
εfc
εfo
sdε(Equation 5)
Ufe =1
2sεfe =1
2Efu
s2(Equation 6)
Ufd =Z
εfc
εfo
sdε1
2Efu
s2(Equation 7)
where: Ufi,Ufe , and Ufd are the input energy density, elastic en-
ergy density, and dissipated energy density of the rock sample
during the final uniaxial cyclic loading, respectively; εfo ,εfc, and
εfe are the initial strain, the final strain, and the elastic strain of
the rock sample during the final uniaxial cyclic loading, respec-
tively; Efu is the unloading elastic modulus of the rock sample
during the final uniaxial cyclic loading.
The effective prediction of rock mass damage and fracture by
theoretical method is very important to accurately predict the
cataclysm of deep rock mass mechanical behavior.
39
Based
on the energy damage dissipation mechanism of rock, the ratio
of dissipated energy density to total dissipated energy density
UfdT was defined as energy damage, and then the damage var-
iable Dfwith dissipated energy as the characteristic parameter
was obtained (Equation 8):
Df=Ufd
UfdT
(Equation 8)
Energy and damage evolution mechanism of the final full
stress-strain curve
Calculated from Equations 4,5,6,7, and 8, the evolution curves
of input energy, elastic energy, dissipated energy, and energy
damage with strain of granite sample under uniaxial cyclic
compression can be obtained as shown in Figures 21 and 22.
It can be seen from Figure 21A that the evolution characteristics
of the final full stress-strain curve can be divided into three
stages: pre-peak plastic compaction deformation stage (oa),
pre-peak linear elastic deformation stage (ab), and post-peak
fracture deformation stage (bc), and the evolution mechanisms
of input energy, elastic energy, dissipated energy, and energy
damage with strain in different stages of the final full stress-strain
curve of granite samples under uniaxial cyclic compression were
deeply analyzed.
(1) In the pre-peak plastic compaction deformation stage
(oa): The final full stress-strain curve showed the non-
linear growth characteristic with a "concave" shape,
which mainly reflected the secondary compaction effect
under cyclic cumulative damage. The dissipated energy
curve increased slowly, which was due to both the dissi-
pative energy of the primary micro-defects inside the rock
and the secondary micro-defects caused by the cumula-
tive damage of the first 14 cycles of compression. The
input energy, elastic energy, and dissipated energy were
all at a very low level, the growth of energy damage was
extremely slow, and there was no substantial structural
damage in rock samples at this stage.
(2) In the pre-peak linear elastic deformation stage (ab): The
final full stress-strain curve showed the characteristic of
linear growth, which mainly reflected the effect of second-
ary energy accumulation under cyclic cumulative dam-
age. The input energy and elastic energy increased
rapidly, while the dissipated energy increased extremely
slowly and tended to stabilize gradually, which showed
that almost all the energy input from the outside was
stored in the rock sample in the form of elastic energy.
The dissipated energy was still at a very low level, and
the energy damage increased extremely slowly and
tended to stabilize gradually, and there was no substantial
structural damage in the rock samples at this stage.
(3) In the post-peak fracture deformation stage (bc): The final
full stress-strain curve showed the characteristic of
elastic-brittle impact failure, which mainly reflected the
"rockburst" effect under cyclic cumulative damage. The
elastic energy decreased rapidly at an approximate verti-
cal trend, while the dissipated energy increased rapidly at
an approximate vertical trend, which indicated that partial
elastic energy accumulated in the rock sample was con-
verted into the energy dissipation with the fracture and
fragmentation of the rock sample, and the remaining
part of the elastic energy was mainly converted into the
energy dissipation with the dynamic ejection of the rock
sample fragments. The energy damage also increased
rapidly at an approximate vertical trend, and substantial
structural damage occurred in the rock samples at this
stage.
Influence mechanism of shrinkage and expansion of
rock particles on mechanical properties during cyclic
loading-unloading
Influence on the internal structure of the rock
Initiation and propagation of microcracks: During the loading
process, the grains inside the granite are squeezed, and the con-
tact stress between the grains increases, causing the initiation of
microcracks. As the number of cycles increases and the stress
changes continuously, the microcracks will gradually expand.
For example, when the stress reaches a certain level, the adhe-
sion between grains is destroyed, and microcracks will continue
to develop along grain boundaries or weak areas. The existence
of these microcracks not only changes the internal structure of
the rock, but also provides a potential path for subsequent defor-
mation and failure. Changes in pore structure: The shrinkage and
expansion of grains will change the pore structure inside the
rock. During loading, the pores between particles may be com-
pressed and the pore volume may decrease; during unloading,
due to incomplete elastic recovery of the particles, the pores
may not be fully restored to their original state, resulting in a
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more complex pore structure. This change in pore structure will
affect the mechanical properties such as permeability and
strength of rock.
Influence on rock strength
Strength reduction: Repeated shrinkage and expansion of parti-
cles during cyclic loading will cause damage to the interior of the
rock, thereby reducing the strength of the rock. On the one hand,
the continuous expansion of micro-cracks will destroy the integ-
rity of the rock, resulting in a decrease in its ability to resist
external forces; on the other hand, the weakening of the bonding
force between particles will also reduce the strength of the rock.
For example, after multiple cyclic loading, the uniaxial compres-
sive strength, tensile strength, and so forth of granite will be
significantly lower than the strength under monotonic loading.
Fatigue failure: The shrinkage and expansion of particles accel-
erate the fatigue failure of rock. Under cyclic loading, the stress
distribution inside the rock continues to change, and the parti-
cles are constantly stretched and compressed, which will grad-
ually accumulate damage inside the rock. When damage accu-
mulates to a certain extent, fatigue failure occurs in the rock,
even if the applied stress is much lower than its static strength.
Influence on rock deformation characteristics
Increased plastic deformation: The shrinkage and expansion of
grains cause more plastic deformation of the rock. During cyclic
loading, grains in the rock will undergo irreversible deformation
behaviors such as sliding and rotation, and these deformations
will gradually accumulate, increasing the plastic deformation of
the rock. For example, in the uniaxial cyclic loading test, it can
be observed that the axial strain and lateral strain of granite in-
crease with the increase of cycles. Change in deformation
modulus: The deformation modulus is an indicator of the ability
of a rock to resist deformation. In the early stage of cyclic
loading, the deformation modulus may increase due to the
adjustment of the internal structure of the rock and the rear-
rangement of grains; but as the number of cycles increases
and damage accumulates, the deformation modulus will gradu-
ally decrease. This is because changes in micro-cracks and pore
structure within the rock reduce its ability to resist deformation.
Influence on rock energy dissipation
Increased energy dissipation: Energy is consumed during the
shrinkage and expansion of particles and cannot fully recover
when unloaded, resulting in increased energy dissipation. This
energy dissipation can be expressed by a hysteresis loop. The
larger the area of the hysteresis loop, the greater the energy
dissipation of the rock during cyclic loading. Change in damping
characteristics: Changes in the damping characteristics of rock
lead to changes in energy dissipation characteristics. During cy-
clic loading, the damping characteristics of rock change due to
the shrinkage and expansion of particles, which affects the
response of rock under dynamic loading. It should be noted
that due to the small proportion of damping energy consump-
tion, no in-depth research on damping energy has been carried
out in this study in terms of energy.
Conclusions
In this study, the incremental cyclic loading-unloading test of
typical hard granite in the diversion tunnel was conducted in
conjunction with the actual construction project. The theory and
mechanism of cyclic dynamic mechanical damage for granite in
the diversion tunnel under cyclic loading-unloading were thor-
oughly investigated. The following conclusions are drawn.
(1) The cyclic stress-strain curves showed the evolution
characteristic of "from dense to sparse" with the increase
of the number of cycles, which reflected the evolution
characteristic of rock sample damage "from non-struc-
tural damage to structural damage." The plastic deforma-
tion of rock samples in the initial cycle was mainly
compaction deformation, and the plastic deformation of
rock samples tended to fluctuate and stabilize due to
the weakening of the compaction deformation effect
and the enhancement of the elastic deformation effect in
the subsequent cycle, but with the continuous increase
of the number of cycles, the cumulative effect of plastic
damage increased, resulting in the obvious fluctuation
of plastic deformation of rock samples in the later cycle.
(2) During the evolution process of cyclic loading-unloading
deformation, AE damage counts mainly occurred during
cyclic loading deformation, and almost no AE damage
counts occurred during cyclic unloading deformation.
With the evolution of cyclic loading deformation, AE dam-
age counts gradually increased, especially during the
10th-15th cyclic loading. The degree of damage and
rupture of the rock samples gradually increased, showing
a relatively complex comprehensive failure mode with
both the compression-tension fracture and the compres-
sion-shear fracture.
(3) In the process of cyclic loading, the energy input into the
rock sample was mainly accumulated as elastic energy,
but due to the existence of the compaction effect and cu-
mulative damage effect, a small part of the input energy
was inevitably converted into dissipated energy due to
plastic deformation. The evolution curves of strain dam-
age and energy damage with the number of cycles can
be divided into three evolution stages: early non-struc-
tural damage stage (0–10th cycle), later structural dam-
age stage (10th 14th cycle), and final structural damage
stage (14th 15th cycle).
(4) In the process of cyclic loading, Poisson’s ratio
decreased rapidly with time, while in the process of cyclic
unloading, Poisson’s ratio increased rapidly with time.
The almost complete elastic deformation stage of the
rock sample mainly occurred in the process of the 6th 
8th cycle, so the Poisson’s ratio range of the rock sample
in the elastic deformation stage was 0.19–0.23.
(5) In the process of incremental cyclic loading-unloading,
when the cyclic load uniformly increases and decreases
with time, the rock sample has the loading strain leading
effect and the unloading strain hysteresis effect, and the
stress and strain can not change synchronously. The
loading deformation can not elastically recover to the
original path during the unloading process, the residual
deformation occurs in the cyclic loading-unloading pro-
cess, the rock sample occurs damaged, and there is
damping energy dissipation and plastic energy dissipa-
tion.
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(6) The stress-strain curves of granite samples under cyclic
loading-unloading were fitted by an exponential function
(s=aexpðhεÞ+b), which fully reflected the evolution
characteristics of stress-strain curves of granite samples
under the cyclic loading-unloading, and the loading-un-
loading two-stage constitutive equation of granite sam-
ples under cyclic loading-unloading was established.
(7) The evolution characteristics of the final full stress-strain
curve can be divided into three stages: pre-peak plastic
compaction deformation stage (oa), pre-peak linear
elastic deformation stage (ab), and post-peak fracture
deformation stage (bc), and the evolution mechanisms
of input energy, elastic energy, dissipated energy, and en-
ergy damage with strain in different stages of the final full
stress-strain curve of granite samples under uniaxial cy-
clic compression were deeply analyzed.
Limitations of the study
Based on the actual construction environment and conditions,
this study only carried out a relatively simple amplitude-type
low-order cyclic loading-unloading test on the diversion tunnel
rock samples, reflecting the limitations of the research method
Figures 23,24,25,26,27, and 28 and content. In the later stage,
we should continue to conduct in-depth high-order cyclic
loading tests on rock samples from diversion tunnels under
different initial conditions (high and low temperatures, different
initial damage types and degrees, and so forth). In addition,
the accuracy of corresponding numerical simulations under
complex cyclic loading-unloading modes should be further
improved.
RESOURCE AVAILABILITY
Lead contact
Further information and requests for resources should be directed to and will
be fulfilled by the lead contact, Rongzhou Yang (Rongzhouy@outlook.com).
Materials availability
The present study did not generate new unique reagents.
Data and code availability
dData reported in this article will be shared by the lead contact upon
request.
dThis article does not report original code.
dAny additional information required to reanalyze the data reported in this
article is available from the lead contact upon request.
Figure 24. Experimental equipment and rock sample loading-unloading
Figure 23. Rock sample preparation process
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ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support from the Open Fund
for Anhui Key Laboratory of Mining Construction Engineering, Anhui University
of Science and Technology (GXZDSYS2023103), Natural Science Research
Project of Anhui Educational Committee (2023AH051167), Scientific Research
Foundation for High-level Talents of Anhui University of Science and Technol-
ogy (2022yjrc84), and the National Natural Science Foundation of China
(52074009).
AUTHOR CONTRIBUTIONS
R.Y.: Conceptualization, methodology, data curation, formal analysis, re-
sources, funding acquisition, investigation, project administration, writing-
original draft, visualization, validation, and writing-review and editing. Y.X.:
Conceptualization, methodology, resources, funding acquisition, supervision,
and writing-review and editing.
DECLARATION OF INTERESTS
The authors declare no competing interests.
STAR+METHODS
Detailed methods are provided in the online version of this paper and include
the following:
dKEY RESOURCES TABLE
dMETHOD DETAILS
BCyclic loading-unloading test
BEstablishment of numerical models
BAnalysis principle and variable definition
dQUANTIFICATION AND STATISTICAL ANALYSIS
SUPPLEMENTAL INFORMATION
Supplemental information can be found online at https://doi.org/10.1016/j.isci.
2024.111583.
Received: July 26, 2024
Revised: October 22, 2024
Accepted: December 9, 2024
Published: December 12, 2024
Figure 27. Principle of modulus analysisFigure 26. Principle of strain analysis
Figure 25. Cyclic loading-unloading
scheme
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STAR+METHODS
KEY RESOURCES TABLE
METHOD DETAILS
Cyclic loading-unloading test
Rock sample preparation
The rock samples used in the test were taken from the surrounding rock of a diversion tunnel located in Zhejiang Province, and the
rock cores were obtained by in-situ coring. According to the specifications GB/T 23561-2009 and GB/T 50266-2013, the rock core
was further processed into the standard size range of rock samples (F55 mm 3h110 mm) by cutting. The process for preparing the
rock sample is illustrated in Figure 23.
Test methods and results
The incremental cyclic loading-unloading test of granite samples was carried out by using the RMT-150B rock testing machine, and
the axial and transverse strains in the middle of rock samples were collected by ST-3C strain testing system, to further analyze the
damage evolution characteristics and Poissons ratio law of rock samples in the process of cyclic loading-unloading test. The physical
pictures of the test equipment and the cyclic loading-unloading diagram of rock sample are shown in Figure 24. Test scheme: The
test loading mode was slope, the increased amplitude of per cyclic load was 30 kN, and the cyclic loading-unloading rate was 5 kN/s.
To effectively consider the possible initial irregularity of the rock samples and realize the complete unloading process accurately, the
test preload was carried out before the formal loading-unloading. The relevant parameters of the cyclic loading-unloading methods
are shown in Figure 25 and Table S1. The test results showed that the rock sample was completely destroyed in the form of the "rock-
burst" during the 15th cyclic loading, that was, the rock sample had undergone 14 complete cyclic loading-unloading processes. In
addition, to further confirm the consistency between the test results and the numerical simulation results, the RFPA cyclic load-un-
loading version of the numerical calculation software was used to carry out numerical simulation research corresponding to the cyclic
load-unloading physical test.
It should be noted that in the actual drilling and blasting construction process of the Zhejiang diversion tunnel, taking into account
all the factors that can be taken into account (geological environment, rock properties, blasting network, construction process, con-
struction period, cost control, etc.), the optimal number of periodic blasting is determined: The blasting was carried out twice a day,
with an average excavation extension of about 3.5 m per blasting. Each blasting process was regarded as a cyclic loading-unloading
process, and one week was a surrounding rock damage inspection cycle (the damage caused by blasting disturbance to the sur-
rounding rock was checked once a week), that was, there were 14 cyclic loading-unloading processes in one surrounding rock dam-
age inspection cycle. It can be seen that in terms of the number of cycles, the design of the test scheme is more consistent with the
actual situation of the project.
Establishment of numerical models
Using the cyclic loading version of RFPA
2D
numerical calculation software and combining a series of mesoscopic physical and me-
chanical parameters of the diversion tunnel granite in Table S2, the two-dimensional (2D) numerical calculation model of rock sample
under cyclic loading-unloading was established (Table S2). The meso-element sizes of the model grids were l0.55 mm 3w0.55 mm.
According to the conversion relationship Equations 9 and 10
40–42
between the macro-mechanical parameters and the meso-me-
chanical parameters in the RFPA
2D
numerical calculation, based on the macro-mechanical parameters of granite in the physical
test, the meso-mechanical parameters of granite in Table S2 can be obtained.
smes =smac
0:2602 ln m+0:0233 ð1:2%m%50Þ(Equation 9)
Emes =Emac
0:1412 ln m+0:6476 ð1:2%m%50Þ(Equation 10)
REAGENT or RESOURCE SOURCE IDENTIFIER
Software and algorithms
Microsoft 365 (Office) Microsoft https://www.office.com
Origin 2024 OriginLab https://www.originlab.com
RFPA-cyclic load Mechsoft http://www.mechsoft.cn
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where: smac and smes are the physical static load compressive strength and numerical meso-element compressive strength of the
rock sample, respectively; Emac and Emes are the physical static load elastic modulus and numerical meso-element elastic modulus
of the rock sample, respectively; mis the homogeneity coefficient of the material, which characterizes the homogeneity of materials
such as rocks. The greater the value of m, the higher the homogeneity of the material.
Analysis principle and variable definition
The mechanical parameters such as stress, strain, modulus, energy, and damage of rock samples during cyclic loading-unloading
were important reflections and characterizations of their mechanical behavior and damage evolution mechanism.
43–46
To deeply
study the dynamic damage mechanism and Poisson’s ratio evolution law of granite under cyclic loading-unloading, combined
with the analysis principles involved in related studies,
47–50
the mechanical variables of stress, strain, modulus, energy, and damage
were comprehensively defined from many angles.
Stress
The definition of stress variables is shown in Table S3.
Strain
The definition of strain variables is shown in Table S4. The principle of strain analysis is shown in Figure 26.
Modulus
The definition of modulus variables is shown in Table S5. The principle of modulus analysis is shown in Figure 27.
Energy
The definition of energy variables is shown in Table S6. The principle of energy analysis is shown in Figure 28.
Damage
The definition of damage variables is shown in Table S7.
QUANTIFICATION AND STATISTICAL ANALYSIS
There are no quantification or statistical analyses to include in this study.
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