Study on thermal damage mechanism and energy evolution characteristics of granite after high temperature based on discrete element method

doi:10.21203/rs.3.rs-3154898/v1

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Study on thermal damage mechanism and energy evolution characteristics of granite after high temperature based on discrete element method

https://doi.org/10.21203/rs.3.rs-3154898/v1

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In order to explore the thermal cracking behavior and energy conversion mechanism of granite at different temperatures from a mesoscopic perspective. Based on the laboratory experimental, PFC^2D was used to construct different particle cluster model for research. The mechanical properties of granite under uniaxial compression under laboratory experimental and numerical simulation were compared and analyzed. The evolution laws of microcracks and particle displacement during high temperature treatment were explored. The relationship between energy conversion mechanism and crack evolution of granite after exposure to different temperatures was emphatically analyzed. The results show that: the stress-strain curve of laboratory experimental and the stress-strain curve under numerical simulation have similar evolution laws, the relative error between the numerical simulation results of peak strength and laboratory experimental is less than 5%, and the relative error of elastic modulus is less than 10%. The higher the temperature, the more thermal induced cracks are produced and the larger the particle displacement is. The cracks and particle displacement during heating are larger than those during cooling. Thermal induced cracks are mainly intra-granular tension, and when T ≥ 600 ℃, the shear cracks appear. The storage of elastic energy and the slow dissipation of energy are the main factors before the granite peak, and the dissipation energy increases abruptly after the peak, and the elastic strain energy is released rapidly. The higher the temperature, the more the number of microcracks before the peak of granite, and the greater the damage degree before the peak. Therefore, the weaker the energy storage property, the stronger the energy release property, and the easier the energy driven destruction.

Granite

Numerical simulation

Thermal induced cracks

Particle displacement

Energy evolution

Building different particle cluster models based on the Grain flow principle can well characterize the mechanical behavior characteristics of granite after high temperature.
There are significant differences in the evolution characteristics of microcracks in granite during heating and cooling processes, and the distribution of different types of thermal induced cracks has a significant temperature effect.
The particle cluster model constructed can effectively simulate the energy evolution characteristics after different high temperatures; The mechanism of pre peak energy conversion is significantly influenced by temperature.

The problem of rock thermal damage caused by high temperature is an important subject explored by many scholars in rock mechanics in recent years(Miao et al.,2020). For example, deep geological disposal of nuclear waste (Meng et al.,2019; Vidana Pathiranagei et al.,2021), Drilling process of ultra-deep well in metal mine(Zhang et al.,2018), And the development of geothermal resources (Liu et al.,2021).The surrounding rock in geotechnical engineering is in a high temperature environment. The physical and mechanical properties will change significantly after being affected by temperature, resulting in extremely high safety hazards in the process of engineering advancement(Jiang et al.,2022). In underground engineering operations, rock cracks develop, expand, and connect with each other, accompanied by energy absorption and dissipation (Gao et al.,2022; Li and Cai,2021) Therefore, exploring the influence of temperature on the physical and mechanical properties and energy conversion characteristics of rock can provide new ideas for exploring the mechanism of rock thermal damage, and it is of great significance to guide the construction of underground high temperature environment engineering and disaster prevention.

As the most common igneous rock, granite is widely used in deep geotechnical engineering due to its low permeability and high strength(Miao et al.,2020; Zhang et al.,2018). Therefore, granite is the main research object in high temperature rock mechanics. After high temperature, the granite shows the change of physical and mechanical properties in the macro, and the change of crystal and material composition in the mining area in the micro structure. Firstly, the mass, density, longitudinal wave velocity, uniaxial compressive strength and elastic modulus of granite decrease with the increase of temperature (Gautam et al.,2018; Liu et al.,2021; Meng et al.,2019; Vidana Pathiranagei et al.,2021; Yin et al.,2016); The volume and porosity increase with the increase of temperature.(You-Liang Chen 2017). In terms of microstructure, granite mainly occurs the passage of absorbed water and the expansion of mineral particles before 200℃((Wu et al.), Mineral crystal water and structural water are evaporated at 200℃~400℃(Wu et al.,2019);at 400℃~600℃, some minerals are decomposed and volatilized, and the oxidation-reduction reaction of iron ore, as well as the phase transformation of quartz (Wang et al.,2022b)༛When the temperature exceeds 600 ℃, some minerals will melt and decompose, such as chlorite, illite, etc (Fang et al.,2019). At the same time, scholars have used other methods to study the properties of granite after high temperature. Li et al. (2023) conducted cyclic thermal shock tests on hollow cylindrical granite, and established a breakable numerical model based on discrete element method, which can capture intergranular fracture and crack initiation and propagation. It is found that after cyclic thermal shock, the longitudinal wave velocity and tensile strength of granite are significantly reduced, and the porosity is significantly increased, and the higher the temperature, the more obvious. Wu et al. (2021) obtained by means of acoustic emission that when the stress reached the crack initiation stress, the cumulative acoustic emission count showed a sharp increase. Kahraman (2022) studied the influence of water cooling and air cooling on the physical and mechanical properties of granite after high temperature. It was found that the S-P wave velocity, hardness and uniaxial compressive strength of water-cooled granite samples decreased more than those of air-cooled samples. In addition, the porosity of water-cooled granite samples also increased. Xi et al. (2023) conducted SHPB test on granite under different cooling methods, and analyzed the tensile and failure mechanism under different heat treatment and cooling methods combined with DIC technology. It was concluded that the higher the temperature, the longer the central crack time after impact, the larger the area of shear failure zone and tensile failure Zone, and the natural cooling < water cooling < liquid nitrogen cooling. You-Liang Chen (2017) obtained by SEM that the connection between minerals in granite became weaker with the increase of temperature. Intergranular cracking was the main failure mode for samples below 800 ℃, while intergranular failure was the main failure mode for samples above 1000 ℃.

The gradual damage and failure of rock is accompanied by energy exchange with the external environment, and its failure is caused by internal energy driven by external force (Wang et al.,2022a). Therefore, the damage evolution process of rock is inseparable from the energy conversion mechanism. Huang et al. (2023) conducted CDTTs on granite specimens, and discussed the influence of confining pressure and loading rate on energy conversion mechanism. It was found that the greater the loading rate, the less the proportion of elastic energy, and the greater the confining pressure, the greater the elastic energy and dissipation energy. Qiao et al. (2022) found that the ability of sandstone to store elastic strain energy at high temperature has nothing to do with its mechanical properties, and established an energy self suppression model (EII) based on the energy dissipation and release theory. Zhang et al. (2023) used digital speckle technique to find that the energy accumulation and release law of coal and rock mass is related to the evolution of localization zone, and the strain energy is different in different regions outside of localization. Xu et al. (2021) carried out uniaxial compression and cyclic loading and unloading tests on five kinds of prefabricated red sandstone specimens with different angle fractures. It was found that the input energy density, elastic energy density and dissipated energy density of five kinds of red sandstone with different prefabricated fractures increased significantly with the actual unloading stress. According to the linear energy storage law, the peak strength energy density of red sandstone with different fracture angles was determined quantitatively. Zhao et al. (2020) studied the energy evolution characteristics of granite under different loading rates, and found that the loading rate increased from 0.001mm/s to 0.05mm/s, the total strain energy increased by 91%, the elastic energy increased by 48%, and the dissipation energy increased by 184%.

With the rapid development of computer technology, numerical simulation has become one of the important means to study the meso damage mechanism of rock. In the study of rock mechanics, common numerical simulation methods include Abaqus(Yang et al.,2023) and COMSOL(Wang et al.,2023b) based on finite element method; FLAC based on finite difference method,(Wang and Konietzky,2019); PFC based on particle flow program, etc.(Zhao et al.,2021)。Among them, PFC can well reproduce the meso damage evolution process of rock, and its two-dimensional model (PFC^2D) is widely used because of its short calculation time and simple boundary and loading conditions(Ao et al.,2023). For example, Ding et al. (2023) established 15 PFC numerical models and found that the energy of pre splitting coal in sub unstable state accounted for more than 49% of the whole process under graded loading, and the slip energy and damping energy began to separate at this stage. Zhu and Dong (2022) used PFC to establish a fluid structure coupling hydraulic pressure simulation model considering fatigue damage, and combined with the proposed parallel corrosion model to achieve fatigue degradation. Tian et al. (2020) used PFC to establish a cluster particle flow model, and studied the influence of granite particle size on mechanical properties and crack evolution after high temperature. It was found that coarse-grained granite samples were prone to macro cracks after high temperature treatment, but the number of cracks in granite decreased with the increase of experienced size.

In summary, the research on high temperature damage of granite mostly focuses on the changes of mechanical properties and microstructure, but the research on the evolution law of thermal damage of granite in the heating and cooling stages during high temperature treatment and the energy evolution mechanism in the process of progressive failure of granite under high temperature are rarely seen. In this study, based on the characteristics of granite mineral crystal composition, considering the thermal mechanical coupling process of granite, the discrete element model of different particle cluster composition which can characterize the mechanical properties of granite after high temperature is constructed by using PFC^2D particle flow software. The consistency between the uniaxial compression test results of granite samples treated at different temperatures and the stress-strain curves of numerical simulation, as well as the rationality of the peak strength and elastic modulus of numerical simulation are analyzed. In addition, the variation of crack number and displacement with temperature during heating and cooling and the development characteristics of different types of cracks with temperature were explored. The relationship between energy and crack evolution of granite at different temperatures is studied from the perspective of meso mechanism, and the energy conversion mechanism of granite after high temperature is revealed. In this study, the damage failure mechanism and energy evolution mechanism of granite after high temperature are explored through indoor experiments and numerical simulation. The research results can provide reference for the design and early warning of related rock engineering.

2.1 Sample preparation

Granite samples were obtained from Luotian, Hubei in China. According to the requirements of ISRM, the granite samples were made into a standard samples with a diameter of 50mm and a height of 100mm. The parallelism deviation between the upper and lower end faces is less than 0.05mm.The XRD and microscope analyses showed that the graine was comprised of approximately 37% quartz(Q), 38.8% albite(A), 22.4% potassium feldspar(P) and 1.8% mica(M). The microstructure and main mineral composition of granite samples at normal temperature are shown in Fig. 1.

2.2 Test equipment and procedure

Firstly, Standard granite samples were dried in the oven for 24h. Then non-metal wave velocity detector was used to conduct P-wave detection on the granite samples. The granite samples with the same or similar wave velocity were selected and divided into 6 groups according to room temperature, 200℃, 400℃, 600℃, 800℃ and 1000℃, the quality and volume of each group of samples were measured. Subsequently, each group of granite samples were placed in the box Muffle furnace according to the set target temperature for high temperature treatment (excluding room temperature), in which the heating rate was set to 5℃/min and the constant temperature time was set to 4h. After the constant temperature time is over, the Muffle furnace is powered off, so that the samples were naturally cooled to room temperature in the furnace. The wave velocity, mass and volume of the sample after high temperature treatment were measured. Then the RMT-150C rock mechanic test system was used to carry out uniaxial compression test on the granite samples after exposure to different temperature, and the displacement loading rate was set at 0.002mm/s. The test procedure and loading diagram are shown in Fig. 2.

2.3 Numerical model setup and Micro-parameters calibration

Granite is composed of different types and spatial geometric crystals bonded with each other. The size and shape of mineral particles, mechanical properties and bonding strength between particles will affect the macro mechanical properties of granite. In order to better reflect the mechanical behavior of granite, The particle cluster model was constructed to study the characteristics of granite after high temperature. The specific steps are: (1) First, an initial specimen with a height of 100mm and a width of 50mm is generated; (2) Rhino is used to randomly generate polygon geometry and import it into PFC^2D; (3) According to the actual mineral composition ratio, the particle clusters of mineral crystals with corresponding proportions are generated; (4) Add a contact model to the particles between and within the particle clusters, as shown in Fig. 3. A total of 13005 particles were generated this time, including 4812 quartz particles, 5046 albite particles, 2913 potassium feldspar particles and 234 mica particles. In order to better reflect the actual mechanical properties of granite through PFC^2D, the PB model contact between particles was calibrated by "trial and error method" (Tian et al.,2020). The particle stiffness and parallel bond stiffness between particle clusters are about twice as large as those between particle clusters. In order to improve the efficiency, firstly, the elastic modulus of simulated granite samples in PFC^2D was calibrated according to the elastic modulus of granite at 25℃; When the calibrated elastic modulus conforms to the actual test value, the uniaxial compressive strength is calibrated by adjusting the PB strength parameter. A set of the most reasonable calibration values can be obtained by repeating this for 2–3 times, as shown in Table 1.

Table 1

Micro-parameters of cluster model in PFC^2D
Micro-parameters	Values
Minimum radius(mm)	0.035
Maximum radius(mm)	0.05
Effective modulus of the particle(GPa)	10
Ratio of normal to shear stiffness of the particle, kn/ks	0.8
Particle friction coefficient, u	0.5
Parallel-bone radius multiplier	1.0
Effective modulus of the parallel bond(GPa)	10
Ratio of normal to shear stiffness of the parallel bond(MPa)	0.8
Parallel bond normal strength of intra-cluster(MPa)	100
Parallel bond shear strength of intra-cluster(MPa)	120
Parallel bond normal strength of inter-cluster(MPa)	68
Parallel bond shear strength of inter-cluster(MPa)	90

2.4 Principles of thermodynamic calculation

The built-in thermodynamic module of PFC can simulate the transient heat conduction and storage problems of materials composed of particles, and can simulate the deformation and force changes of particles caused by thermal effects. In PFC, each particle is considered a container for storing heat; Heat conduction is carried out between each heat exchanger through a heat pipe connection. The essence of its thermal coupling is that the change in particle temperature leads to a change in its own volume, which in turn affects the stress state of the model. This process is mainly achieved by defining the thermal expansion coefficient of particles. Based on the grouping of mineral particles mentioned above, the thermal expansion coefficients of particles belonging to different mineral components are defined as follows: Quartz:24.3×10^− 6k^− 1, Albite:14.1×10^− 6k^− 1, Potassium feldspar:8.7×10^− 6k^− 1, Mica:3×10^− 6k^− 1 (Zhao,2016)。According to the defined coefficient of thermal expansion, the thermal expansion of particles can be achieved by Eq. (1)

$$\Delta R=\alpha R\Delta T$$

Where $\alpha$ is the thermal expansion coefficient of the particle, is the particle radius,$\Delta T$ is the temperature increment,$\Delta R$ is the particle radius increment. The normal force caused by thermal expansion is (Huang et al.,2022)

$$\Delta \overline {{{F_n}}} = - \overline {{{k_n}}} A\Delta {U_n}= - \overline {{{k_n}}} A(\overline {\alpha } \overline {L} \Delta T)$$

where $\overline {{{k_n}}}$ is the bond normal stiffness, A is the area of the bond cross-section, $\overline {\alpha }$ is the expansion coefficient of the bond (average value of the expansion coefficient of two particles), $\overline {L}$ is the distance between the two centers of contacting particle, and $\Delta T$ is the temperature increment. Based on the above principles, conduct high-temperature simulation according to the following process. Firstly, in order to simulate the uniform heating conditions around the muffle furnace, a target temperature is applied to the outermost particles of the numerical model and the temperature value is fixed. Subsequently, both thermodynamic and mechanical calculations were initiated. At this point, the outermost particles began to transfer heat to the center of the sample through a heat pipe, and the heated particles experienced thermal expansion accompanied by microcracks; Then, when the sample is uniformly heated, thermodynamic calculations are stopped, which is a complete heating process. Finally, change the temperature of the outermost particles to 25℃ and repeat the heating process for cooling. When the sample cools to room temperature and the thermodynamic calculation module is closed, it represents the completion of high-temperature simulation and allows for uniaxial loading. The schematic diagram and failure mechanism of temperature rise and fall are shown in Fig. 4.

3.1 Stress-strain response

Based on the laboratory experiments and numerical simulation methods mentioned above, the stress-strain curves, peak strength, and elastic modulus of granite after exposure to different under uniaxial compression are shown in Fig. 5. It’s illustrated that the stress-strain curves of different types of granite samples treated at different temperatures in numerical simulations and laboratory tests behave relatively similarly. From Fig. 5 (a) and (b), it can be seen that as the temperature increases, the peak strength of granite gradually decreases, and under laboratory experiments conditions, the increase in temperature will lead to an enhancement of the nonlinear characteristics of the stress-strain curve of granite under initial loading. This is due to the increase in temperature leading to an increase in micro pores and cracks inside the granite; However, due to the simulation characteristics of PFC^2D, this characteristic cannot be clearly reflected. Figure 5 (c) and (d) show the variation of peak strength and elastic modulus with temperature through laboratory experiments and numerical simulations. From the figure, it can be seen that the peak strength and elastic modulus obtained by the two show good consistency. Under numerical simulation, the relative errors between the uniaxial compressive strength at each temperature and the experimental values are 3.18%, -2.93%, -1.80%, 2.68%, 3.19%, and 4.78%, respectively; The relative errors between the elastic modulus at each temperature and the experimental values are 1.37%, 6.44%, 9.73%, 8.47%, 7.72%, and 9.84%. Therefore, in conclusion, the Stress–strain curve of granite numerical simulation after exposure to different temperatures is highly similar to the laboratory test results, and the relative error of peak strength and elastic modulus is more than 10%. This reflects the credibility of the above calibrated microscopic parameters and the feasibility of using numerical simulation to study the changes in granite properties after exposure to different temperatures.

3.2 The influence of temperature on granite

3.2.1 Development law of heat-induced crack

In order to investigate the effect of temperature treatment on granite, Fig. 6 shows the crack evolution patterns of granite specimens during heating and cooling under different temperature conditions. As shown in Fig. 6, when granite is subjected to heat transfer from the surrounding environment, heat transfer occurs from the surface of the sample to the center of the sample. Therefore, the granite sample is more prone to thermal induced cracks near the outside. At 200℃~400℃, the generation of thermal induced cracks is concentrated between the outermost particles. When the temperature reaches above 400℃, thermal induced cracks begin to appear inside the sample, and the number of cracks significantly increases, The particles inside the sample are subject to more constraints from the surrounding environment. Due to the constraints of the surrounding particles, the contact between the particles inside the sample is more difficult to fracture than the contact near the surface. In addition, it is not difficult to find from Fig. 6 that the number of cracks generated during the heating process is smaller than the number of cracks generated during the cooling process. In order to further understand the variation pattern of cracks in the sample during heating and cooling, statistical analysis was conducted on the microcracks generated during heating and cooling at different temperatures, as shown in Fig. 7. As shown in the figure, as the temperature increases, the number of cracks in both the heating and cooling processes increases. The number of cracks generated during the heating process from 200℃ to 1000℃ is 19, 82, 1169, 1551, and 3132, respectively. The number of cracks generated during the cooling process is 2, 5, 348, 490, and 1073, respectively. The number of cracks generated during the heating process is more than three times that of the cooling process. In the process of numerical simulation, particles expand when heated, and when the temperature is high enough, thermal stress is generated between local particles. When the thermal stress exceeds the bonding strength between particles, tensile or shear failure occurs; During the cooling process, due to the concentration of stress released by local mineral particles, relatively few cracks are generated(Wang and Konietzky,2019; Wang et al.,2023a). This also indicates that the main cause of thermal induced cracks is the interaction between mineral particles due to phase transformation, expansion, and other factors during the heating process, while fewer cracks are generated during the cooling process.

Granite is a polycrystalline bonded geometry. There are four types of microcracks: Intra-granular tensile fracture, Intra-granular shear fracture, Inter-granular tensile fracture and Inter-granular shear fracture. By monitoring the thermal crack of different mineral particles, the influence of temperature change on the thermal crack is analyzed. First, it can be seen from Fig. 8 that the number of cracks always increases with the increase of temperature, which also reflects the reason why the temperature increases and the strength of the sample decreases. Among all the cracks, tensile cracks are the main ones, and tensile fracture is the most likely to occur in the mineral crystal. Secondly, the occurrence of shear cracks requires a high level of temperature. After treatment at 200℃ and 400℃, the specimen only produces tensile cracks. When the temperature reaches and exceeds 600℃, the shear cracks begin to appear. It shows that the particles with the same thermal expansion coefficient are relatively easy to expand and extrude after heating and stretch after cooling and shrinkage, while the inter-granular cracks are less due to the difference of thermal expansion coefficient between crystals, and the difference of expansion and shrinkage between particles with relatively small thermal expansion coefficient and particles with relatively large thermal expansion coefficient. In the actual heating process of granite, when the temperature exceeds 573℃, when quartz transforms from$\alpha$ phase to $\beta$ phase, the volume of granite expands obviously. At this time, the thermal stress generated between mineral particles is large enough to enhance the interaction between different mineral particles, resulting in local compression and extension and shear movement at the level of mineral particle size, which makes the shear crack increase significantly after 400℃~600℃. This also shows that the granite sample was seriously thermal damaged at this time.

3.2.3 Changes in particle displacement after different temperatures

In PFC^2D, the essence of the failure of the simulated sample is that after the particles are heated, expanded and cooled, the displacement generated is large enough to cause the parallel bond fracture between particles and produce microcracks. Therefore, the influence of temperature on rock is studied by analyzing the displacement change of particles after heating. As shown in Fig. 9, the displacement change law of mineral particles after treatment at different temperatures is shown. Firstly, it can be seen from the figure that with the increase of temperature, the displacement after heating and cooling increases significantly. Secondly, the displacement of mineral particles after heating is significantly greater than that of particles after cooling. At the same time, the above law of the increase of crack number with the increase of temperature and the law of the increase of heating cracks more than cooling cracks are verified. And when the temperature increases from 400℃ to 600℃, the displacement of mineral particles increases significantly, indicating that the mineral particles expand greatly in the temperature range, resulting in large displacement changes in the process of heating and cooling. The main change in the temperature range from 400℃ to 600℃ is quartz phase transition, which is the main reason for the deterioration of granite strength, the significant increase of particle displacement and the number of cracks after this temperature range. Comparing the particle displacement distribution after heating and cooling at all temperatures, it is found that the particle displacement near the center is always at the minimum value, or even does not change, while there is always a significant displacement change after heating and cooling near the outside particles, which also proves that the constraint degree of particles near the center of the sample is greater than that near the outside particles. Therefore, the displacement change after heating is small, indicating that the minerals in the outer layer of the rock always change most after the temperature, and the microcracks are also generated from the outside first and then from the inside.

4.1 Energy mechanism in PFC

PFC^2D can monitor the changes of absorption, storage and release of energy during the simulation load process. Therefore, PFC can be used to monitor the energy evolution of granite samples after different temperatures, and explore the influence of temperature on the energy failure mechanism. According to the laws of thermodynamics, the essence of the change process of physical properties during energy conversion (Li and Cai,2021)。The sample absorbs energy under external load. Part of the energy is stored in the rock in the form of elastic strain energy, and the other part is used for the initiation and propagation of rock cracks, which is called dissipative energy. Therefore, the essence of rock failure under load is the result of the interaction and transformation of energy storage and dissipation energy release. Assuming that there is no exchange heat between the system and the external environment, the relationship among the input energy, stored elastic energy and released dissipation energy of the external environment during the loading process is shown in Eq. (3):

$$U={U^e}+{U^d}$$

Where: is the external input energy, that is, the work done by the loading plate on the sample; ${U^e}$ is the stored elastic energy; ${U^d}$ is the dissipated energy for plastic deformation and crack initiation and propagation. In PFC, the input energy under uniaxial compression is the sum of the work done by the upper and lower loading plates on the specimen; Elastic strain energy is particle strain energy and parallel bonding strain energy; Dissipative energy is composed of particle sliding friction energy, damping energy and kinetic energy. The above energies are directly monitored by PFC command statement, and the total energy, elastic strain energy and dissipative energy strain are calculated.

4.2 Energy evolution characteristics

By monitoring the energy results of granite samples after exposure to different temperatures, the energy evolution mechanism of the samples in the loading process is analyzed. The energy evolution, stress-strain curve and crack curve in the loading process are shown in Fig.10. It can be seen from the figure that the evolution law of the energy evolution curve of the samples treated at different temperatures shows similarity, and the energy evolution curve is divided into four stages:

Initial compressive stage (OA): in this stage, the stress level is low, the particles in the model are compressed under the action of the loading plate. The input energy and elastic strain energy curves show nonlinear characteristics , the dissipation energy is nearly zero, so the input energy is almost converted into elastic energy storage.
Linear energy storage stage (AB): in this stage, the growth rate of curve of input energy and elastic strain energy gradually increases, showing strain hardening and energy accumulation. A small amount of strain energy is used for the friction between particles, so the dissipated energy increases slightly .In this stage, the main input energy still changes to elastic strain energy, which is still dominated by energy absorption and storage.
Energy consumption increase stage (BC): It can be seen that the dissipated energy has increased significantly and the increase rate of elastic strain energy curve has slowed down significantly, but the absorbed energy is still mainly stored in elastic strain energy. In addition, it can be seen from the figure that the microcracks began to increase at this stage, indicating that the dissipation of rock energy must correspond to the initiation and propagation of microcracks.
Post peak energy release stage (CD): in this stage, the elastic strain energy begins to show a downward trend after passing through the peak point, and the dissipation energy increases exponentially. It shows that after the sample reaches the energy storage limit, the crack propagation leads to a large amount of energy release. This can be well verified from the change law of the number of cracks.

In summary, there is no significant difference in the energy evolution of granite samples after high temperature treatment. In the pre peak stage, the input energy and elastic strain energy continue to increase, and the main input energy is stored in the rock as elastic strain energy, and only a small part of the energy is used for the initiation and propagation of microcracks. In the post peak stage, due to the formation of macro cracks, the energy accumulated before the peak is released rapidly, the elastic strain energy decreases, and the dissipated energy increases suddenly. Therefore, PFC can well simulate the energy evolution characteristics of rocks after high temperature.

4.2 Evolution mechanism of peak energy

After the rock is subjected to external force, the stress state of the rock will change from the current state to a new stress state. The change of the stress state corresponds to different deformation and failure of the rock, showing different energy release characteristics in the macro. Therefore, the study of the strain energy and strain energy ratio of granite at the peak point can fully explore the influence of temperature on the energy evolution mechanism of granite (Meng et al.,2022)

Table 2 shows the input energy, elastic strain energy, dissipation energy and corresponding energy ratio at the peak point of granite after exposure to different temperatures. It can be seen from the table that the peak input energy and peak elastic strain energy value are the largest at 25 ℃. With the further increase of temperature, the peak input energy and peak elastic strain energy are further reduced. The peak dissipated strain energy increases with the increase of temperature. In addition, the proportion of peak elastic strain energy is 0.67 ~ 0.954, and the proportion of peak dissipated strain energy is 0.046 ~ 0.331, indicating that the energy conversion at the pre peak stage is mainly based on the storage of elastic strain energy, and the proportion of dissipated strain energy is relatively small. Due to the influence of temperature, the proportion of elastic strain energy and dissipated strain energy is different, which reflects that the influence of temperature on the energy accumulation and dissipation of granite has obvious differences.

Figure 11 shows the variation of strain energy and strain energy ratio at the peak point of granite with temperature. It can be seen from Fig. 11 that the input energy and elastic strain energy show a decreasing trend as a whole, and the dissipated strain energy shows a gradually increasing trend with temperature; The input energy, elastic strain energy and dissipated strain energy all meet the linear relationship with temperature, and the goodness of fit is above 0.9. The input at the peak point can reflect the ability of granite to absorb energy before the peak, and the greater the value, the more energy it needs to absorb before the peak when granite is destroyed; The elastic strain energy represents its ability to store energy, and the greater its value, the greater the difficulty of granite failure driven by energy. Therefore, the variation law of the input energy and strain energy of granite at the peak point above shows that the energy absorption and energy storage properties of granite decrease linearly with the increase of temperature, and the granite at 1000 ℃ is most likely to be driven by energy to deform and destroy, that is, with the increase of temperature, the difficulty of deformation and destruction of granite driven by energy becomes easier. In addition, the energy dissipation in the pre peak stage is mainly used for the initiation and propagation of microcracks. In addition, the pre peak energy dissipation is mainly used for the initiation and propagation of microcracks. With the increase of temperature, the pre peak energy dissipation gradually increases, which means that the higher the temperature the specimen experiences, the greater the pre peak damage, and the more the number of microcracks in PFC. As Fig. 12 shows, through the statistics of the number of pre peak cracks in each temperature simulation sample, it is found that the number of microcracks before the peak increases with the increase of temperature. It shows that the dissipation of energy must be accompanied by the existence of damage development. The higher the temperature, the more serious the damage in the pre peak stage, and the higher the dissipated strain energy.

Table 2

There should be energy and strain energy ratio at the peak
Temperature/℃	U	U^e	U^d	U^e/ U	U^d/U
25	706.021	673.816	32.204	0.954	0.046
200	648.951	611.044	37.907	0.942	0.058
400	588.302	547.446	40.856	0.931	0.069
600	413.380	363.989	49.392	0.881	0.119
800	345.186	291.938	53.248	0.846	0.154
1000	176.240	117.939	58.301	0.670	0.331

Based on PFC^2D, a different cluster model was proposed. The mechanical properties, crack evolution characteristics and energy conversion mechanism of granite after high temperature were analyzed based on this model. Essential conclusions are as follows:

The accuracy of the particle cluster model is verified by the numerical simulation results. The stress-strain curve, peak strength and elastic modulus were compared and analyzed. The results show that the simulated values of stress-strain curves of granite samples exposure to different temperatures have similar variation rules with the experimental values in the laboratory, and the relative error of peak strength is less than 5%, and the relative error of elastic modulus is less than 10%.

In the process of simulating high temperature damage of granite, it is found that the number of microcracks induced by heat increases with the increase of temperature. Secondly, the number of thermal induced cracks generated during the heating process is more than three times that during the cooling process. In addition, the thermal induced cracks are mainly Intra-granular tension, and shear cracks only appear when the temperature reaches or exceeds 600℃.

With the increase of temperature, the displacement of particles increases gradually during heating and cooling. When the temperature increases from 400℃ to 600℃, the displacement of particles increases significantly. In addition, the displacement of the central particle changes little with temperature, and the displacement of the particle close to the external environment is larger.

Using PFC to monitor the energy evolution characteristics of granite after high temperature, it is found that the energy evolution characteristics of granite samples at different temperatures are the same. Before the peak, the storage of elastic energy is the main feature, accompanied by some energy dissipation applied to the initiation and propagation of microcracks; The elastic energy accumulated before the peak is released rapidly, and the dissipated energy increases suddenly.

The pre peak input energy and elastic strain energy decrease linearly with the increase of temperature, while the dissipation energy increases linearly with the change of temperature. The higher the temperature, the lower the energy absorption and storage properties, and the higher the energy release properties. In addition, the reason why the dissipated strain energy increases with the increase of temperature is that the higher the temperature, the greater the pre peak damage of the sample, and the more the number of microcracks, the greater the dissipated energy.

Ethics Approval Not applicable.

Conflict of interest The authors declare no competing interests.

Data Availability Some or all data, models, or codes generated or used during the study are available from the corresponding author by request.

Author Contribution Writing, original draft, and investigation, K.Z. and C.L.; conceptualization, writing, original draft, and data curation, L.X. and , C.G.；supervision and writing, review, P.Z. and Z.H., All the authors have read and agreed to the published version of the manuscript.

Funding This study was supported by the National Natural Science Foundation of China Youth Project (Grant No.52104086), National Natural Science Foundation of China(Grant No. 52164004) and Jiangxi province key research and development plan key projects(Grant No. 20212BBG71009)

Consent to publish All authors of this article consent to publish.

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No competing interests reported.

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Study on thermal damage mechanism and energy evolution characteristics of granite after high temperature based on discrete element method

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Abstract

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Article Highlights

1 Introduction

2 Sample preparation and experiment methodology

2.1 Sample preparation

2.2 Test equipment and procedure

2.3 Numerical model setup and Micro-parameters calibration

2.4 Principles of thermodynamic calculation

3 Results and analysis

3.1 Stress-strain response

3.2 The influence of temperature on granite

3.2.1 Development law of heat-induced crack

3.2.3 Changes in particle displacement after different temperatures

4 Simulation of energy evolution process of granite

4.1 Energy mechanism in PFC

4.2 Energy evolution characteristics

4.2 Evolution mechanism of peak energy

5 Conclusion

Declarations

References

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