**4. Discussion**

#### *4.1. Degree, Complexity, and Fractal Dimension of Fracture*

In order to characterize the fracture morphology characteristics and to link them with mechanical behaviors, quantitative digital characterization techniques are required. In this paper, a characterizing method was defined and used to analyze the experiment results, including fracture degree and complexity, and fractal dimension of the cracks. Through this method, different types of fracture morphology can be distinguished quantitatively, and the results were also proved by SEM detection.

Fracture degree *D* is defined as the percentage of the volume of the cracks (i.e., the void volume) in the specimen; while the fracture complexity *C* is defined as the ratio of the fracture surface area to the fracture volume. In order to non-dimensionalize this fracture complexity, the fracture surface area should be multiplied with a voxel width. As shown in Formulas (2) and (3),

$$D = \frac{V\_{factor}}{V\_{specium}}\tag{2}$$

$$\mathbf{C} = \frac{\mathbf{S}\_{\text{fracture}} \cdot \mathbf{W}\_{\text{voxel}}}{V\_{\text{fracture}}} \tag{3}$$

where *Vf racture* is the volume of fractures, *Vspecimen* is the volume of specimen, *Sf racture* is the area of fracture surface, and *Wvoxel* is the voxel width.

Fracture degree is used to characterize the level of the fractures. A larger fracture degree describes a more serious fracture; while, the fracture complexity is used to characterize the tortuosity character of the fractures. A larger fracture complexity describes a more tortuous fracture. In porous media, tortuosity is commonly used to describe diffusion [26], which is defined as the arc-chord ratio: the ratio of the length of the curve to the distance between the ends of it. However, the real fractures are in three-dimensional, the whole volume and surface area of the fractures should be considered. Therefore, the fracture complexity is defined in this way.

The fracture degree and complexity of cracks at different temperatures are shown in Figure 6. The fracture degree shows a slight fluctuation around 1% at different temperatures, while the fracture complexity shows an increasing trend with the rise of temperature. Because, when temperature rises, the brittleness of the specimen will be weakened, while the viscosity will be enhanced. Therefore, the cracks at lower temperature will be more straight, while at higher temperature, it will be more tortuous, which can be also seen from the 3D crack morphology image directly (shown in Figure 4).

**Figure 6.** Fracture degree and complexity at different temperatures.

Fractal dimension of fracture is used to describe the roughness of the crack surface. Fractal geometry was firstly formed and defined in order to describe fractal features in 1983 [27]. Fractal feature is a self similarity of geometric characteristics, which means that the object always has a self-similar structure at a smaller scale. Obviously, if the fracture has a fractal feature, then the crack surface is not smooth in a micro-scale. Therefore the fractal dimension of fracture is commonly used to

describe the roughness of the crack surface [28]. Previous research in brittle materials found that the fractal dimension of fracture is related to fracture toughness [29–33]. This can be explained as that a larger fracture fractal dimension implies a larger ratio of transgranular or trans-particle fracture, while a smaller fractal dimension implies a larger ratio of intergranular or inter-particle fracture.

In this paper, a box-counting method was used to calculate the fractal dimension of cracks at different temperatures. The box-counting dimension is defined, as following:

$$d = \lim\_{\varepsilon \to 0} \left( \frac{\log N(\varepsilon)}{\log(1/\varepsilon)} \right) \tag{4}$$

where *d* is the fractal dimension, *ε* is the chosen scale, *<sup>N</sup>*(*ε*) is the volume of the crack under the scale *ε*. The algorithm based on box-counting dimension goes as following:


C. fit {log(*N*(*<sup>ε</sup>i*))} and {log(1/*<sup>ε</sup>i*)}.

If there is a strong linear correlation, then the fractal characteristic is typical and obvious, and the slope is the fractal dimension of the fracture. In this paper, all of the *R<sup>2</sup>* in fractal dimension fit is above 0.99, with the *p*-value being far smaller than 0.01, which means that the cracks have a typical and obvious fractal characteristic. The fracture fractal dimensions at different temperatures are shown in Figure 7.

**Figure 7.** The relation of fracture fractal dimension and temperature.

The fracture fractal dimension showed a decreasing trend with the rise of temperature, which is rational because at a lower temperature, especially lower than the glass transition temperature of the binder, the brittleness of binder is enhanced, so more transgranular and trans-particle fracture will occur; while at a higher temperature, especially higher than the glass transition temperature of the binder, the viscosity of the binder is enhanced so more intergranular and inter-particle fracture will occur. This result agrees with the previous research on TATB-PBX by atomic force microscopy (AFM) [34]. This phenomenon can also be seen directly through SEM images, which was discussed in the previous section.

#### *4.2. Interior Displacement Field of PBX*

Through calculating the morphological connectivity of the binary images that were obtained from CT slices, the "center of pixel mass" of each particle can be obtained by using the coordinate distribution of the particles' pixels, which is the position coordinate of the center of the particle, as shown below.

$$\left(\left(x, y\right)\_{particle} = \frac{particlc\left(x\_i, y\_i\right)\_{particle} dS\_i}{S\_{particle}}\right) \tag{5}$$

In the Formula (5), (*<sup>x</sup>*, *<sup>y</sup>*)*particle* is the position coordinate of the particle; (*xi*, *yi*)*particle* is the *i*th pixel's coordinate in the particle; *dSi* is the area of the *i*th pixel, which equals 1 here; and, *Sparticle* is the area of the whole particle, which equals the amount of the pixels that are included in the particle.

Because of the little loading displacement and the advantage of in-situ experiment, the thickness and the position of the specimen barely changed after loading. As a result, the particular middle slice can be accurately identified and located, and the particles barely moved in the thickness position of the specimen. When comparing the particle distribution morphology before and after loading, the interior displacement field of PBX at different temperatures can be obtained, as shown in Figure 8. The red ones and red circle markers show the original positions of particles, while the blue ones and blue star markers show the positions of particles after loading.

Through the above analysis, it can be concluded that: 1. at −20 ◦C to room temperature, which are all under the glass transition temperature of the binders, there are slipping and shear among the particles. Dispersion of particles can also be seen, which is proved as particle break by SEM detection; 2. at 0 ◦C, it can be observed that left half of particles move top-left, while right half of particles move top-right, which is exactly same as the test loading method, the top loading head is stable while the bottom loading head move upwards. This implies that the micro-structure evolution is the same as the macro-mechanical behavior; 3. at room temperature, shear along the loading direction is observed, which is because the limitation and the slight unbalance of the loading heads; 4. particles in the middle part of the specimen tend to disperse, which agrees with the fracture behavior under Brazilian test loading method; 5. when the temperature is up to 55 ◦C, which is above the glass transition temperature of binders, the binders begin to transform from glass state to high elastic state, so the binders soften. Therefore, the binder's module reduces while the viscosity enhances, and it has better liquidity. So, the displacement of most particles is smaller, while the particles tend to disperse rather than displacing. The volume of binders tends to expending, accordingly, it can be seen that the particle size tends to reduce; 6. when temperature is higher and up to 70 ◦C, the binders are totally on high elastic state, the viscosity and the liquidity are more significant, resulting in a greater tendency of binder expending and particle reorganization. Particles tend to reshape rather than displacing, and there is complex structure evolution inside the specimen; and, 7. when comparing with the SEM images, it can be concluded that at lower temperatures, the dispersion of particles is mainly the break of particles, while at higher temperatures, is mainly debondings and break of binders.

**Figure 8.** *Cont*.

**Figure 8.** Displacement field of particles at (**a**) −20 ◦C; (**b**) 0 ◦C; (**c**) 22 ◦C; (**d**) 55 ◦C; and (**e**) 70 ◦C, respectively. The red ones and red circle markers show the original positions of particles, while the blue ones and blue star markers show the positions of particles after loading.
