4.1 Energy calculation
The laws of thermodynamics show that the deformation and destruction of materials is the intrinsic essence of the internal energy conversion process, and the destruction of rocks can be better analysed from the perspective of energy. Failure (in rock) is a progressive damage and rupture process. The energy is mainly present as elastic energy, plastic energy, thermal energy, radiation energy, kinetic energy, etc. (Xie et al. 2005). Limited by the existing monitoring technology, it is difficult to realise real-time monitoring of various forms of energy. Considering the proportion of various energy in the process of rock failure, we focus here on the total mechanical input energy, elastic deformation energy, and dissipation of energy (via plasticity and friction).
Assuming the deformation and failure of the rock under compression is a closed system, all mechanical work done by the external force is transformed and absorbed: the first law of thermodynamics gives:
where U0 is the work done by the loading test machine on the rock sample, Ue denotes the elastic energy and strain energy accumulated in the rock specimen, Ud is the energy dissipated by the deformation of the rock, which is mainly used for fracture initiation, expansion, and mineral particle deformation.
Figure 9 shows the stress-strain curve of an element in a rock mass: E0 is the modulus of elasticity in the elastic stage, Ei is the unloading modulus of elasticity, Uie represents the internal releasable deformation energy, and Uid represents the energy dissipated under deformation.
The total input work, elastic deformation energy, and dissipation of energy under uniaxial compression are given by:
where σi is the stress, and εi is the strain. It is worth mentioning that the elastic modulus is taken as the elastic modulus E0 when calculating the energy in the present research.
4.2 Microwave effects on energy evolution
According to formulae (1) to (3), the energy evolution curve during the deformation process of the rock sample under different microwave irradiations is as shown in Figs. 10 and 11: under different microwave heating paths, the energy evolution during uniaxial compression deformation is similar. Before the peak, it shows energy accumulation and dissipation: post-peak, it shows energy release.
The main characteristics of energy evolution in each stage of rock compression deformation are: the crack closure and compaction stage (I): part of the work done by the testing machine on the rock sample is converted into elastic deformation energy and stored inside the rock sample. This part of elastic energy accounts for more than 53.6% of the total energy. The other part is dissipated, and contributes to closure of the original micro-cracks and the deformation of mineral particles.
Linear elastic deformation stage (II): the elastic energy change curve is close to the mechanical total input energy curve, and the dissipated energy curve is approximately parallel to the strain. At this stage, when the irradiation power is 1 kW, the elastic deformation energy accounts for 94.8–96.6% of the total energy. When the irradiation power is 3 kW, the elastic deformation energy accounts for 80.6–92.6% thereof.
Stable crack-development stage (III) and unsteady crack-development stage (IV): with the development and expansion of internal cracks in the rock sample, the amount of energy dissipated tends to increase and the rate of change thereof increases. At an irradiation power of 1 kW and 3 kW at this stage, the elastic energy accounted for 86–94.8% and 89.2–70.9% of the total mechanical input energy, respectively.
Post-peak phase (V): the elastic energy stored inside the rock sample is released instantly, the elastic energy increases rapidly, and that dissipated increases sharply. At this stage, most of the work done by the testing machine is dissipated. The internal fissures of the rock penetrated rapidly, and significant macroscopic cracks appeared.
The energy storage limit, residual elastic density, and maximum energy dissipation in each specimen were extracted to further quantify the influence of microwave irradiation on rock samples (Fig. 12).
At 1 kW and 3 kW, the energy storage limits are 0.35, 0.37, 0.24, 0.25, 0.24, and 0.17 with the increasing irradiation times, respectively. The evolution of residual elastic energy is akin to the deformation of stored elastic energy, and both decrease with the increase of irradiation power and time, while the maximum energy dissipation shows the opposite trend. Therefore, in engineering practice, by increasing the microwave irradiation power and irradiation time, the energy storage limit and residual elastic energy of granite can be reduced. This will transfer the accumulated elastic energy to greater depth within the rock mass, reducing the threat of potential dynamic disasters.
4.3 Microwave effects on the energy dissipation ratio
Rock deformation and failure as an asymptotic damage process can usually be reflected by energy dissipation. To reduce the influence of the discreteness of the rock sample, the ratio of the dissipated energy of the specimen to the total input energy is defined as the energy dissipation ratio (Fig. 13). It can be seen from this that the evolution of the capacity dissipation ratio of each specimen is similar, and it generally shows a trend of “rapid rise-rapid decline-slow decline-steep rise” as the strain increases, however, there are certain differences in the influence of microwave irradiation. Comparing rock samples, with the increase of microwave irradiation power and time in the pre-peak stage, the proportion of energy dissipated is found to increase. In the post-peak phase, the energy dissipation ratio rises at a different rate; for example, the 5 kW 3 min and 5 kW 5 min specimens rise in multiple steps after the peak stress is reached.
4.4 Microwave effects on the energy dissipation conversion rate
After the rock reaches its peak strength, the internal fissures develop and penetrate rapidly. The energy release rate after the peak is closely related to the conversion of the dissipated energy at the peak stress. Dividing the increment of dissipated energy at the peak point by the corresponding time increment gives the conversion rate of dissipated energy at the peak point. According to data statistics, the relationship between the conversion rate of the energy dissipated in the specimen and the irradiation time is shown in Fig. 14.
It can be seen from the figure that under the same microwave irradiation time, the conversion rate of high-power energy dissipation energy is lower than that at low-power: as the irradiation time increases, the energy dissipation conversion rate under the action of high power decreases 3.15 times more than that at low power. This reflects the fact that with the increase of microwave irradiation power and time, the lower the conversion rate of dissipated energy at the peak stress on the granite specimens, the slower the rate of energy release after the peak. That is, microwave irradiation reduces the brittleness of the rock, and renders it more ductile, therefore, in the excavation of hard rock, the weakening of the rock mass by microwave irradiation can reduce the risk of dynamic disasters.