2.4. Calculation of the Strain and Stress

In the SHPB test, the three-wave method [27,28] was used to analyze the dynamic characteristics of granite specimens.

The axial stress σs, strain εs and strain rate ε˙s can be calculated as Equation (1):

where A₀ and E₀ are the cross-sectional areas and elastic modulus of the bar. A_s, C₀ and A_s are the cross-sectional area, P-wave velocity, and length of the specimen. εI, εR and εT are the incident, reflected, and transmitted strain of the test, respectively. ls is the length of the granite specimen.

In the SHPB test, the air pressure makes the bullet move, and the kinetic energy of the bullet is converted into the incident energy. Ignoring the kinetic energy of rock fragments and the heat exchange between the rock and outside, the energy consists of three parts: the first part of the energy return incident bar is carried by a reflected wave, the second arrives at the transmitted bar by a transmitted wave, and the last is absorbed by the granite specimen. The strain energy in bars [29] can be calculated as Equation (2):

where, Wi, Wr and Wt are then carried by the incident, reflected, and transmitted waves, respectively.

The calculation method of the absorbed energy of specimen Wd [30] is shown in Equation (3):

The index of energy consumption density w_s is introduced to evaluate the absorption of energy in rock, and the calculation is shown in Equation (4):

where V is the volume of the granite specimen.

In the SHPB, the incident energy affects the dynamic characteristics of the specimen, and the energy magnitude has received widespread attention in previous studies. However, the action time of the energy also affects the dynamic characteristics of the specimen. This energy structure with time is determined as energy time density w_td, and the calculation is shown in Equation (5):

where Tw is the duration of the stress wave.

The functions of energy time density are shown in Equation (6):