**5. Plastic Work Analysis**

Since grain crushing is a process in which grains consume external work and result in energy loss, the evolution laws of volcanic soil grain crushing during the test can be studied in terms of plastic work (*Wp*) [37,43]. *Wp* is defined as the irrecoverable energy extracted by the sample during the test, and it is equal to the sum of plastic work done by shear stress (*W*1) and the plastic work done by mean effective stress (*W*2). Given that crushable coarse-grained soils undergo only slight elastic deformation during the test, dεd and dεv could be directly adopted to determine the *Wp* [44].

$$\mathcal{W}p = \mathcal{W}\_1 + \mathcal{W}\_2 = \int qd\varepsilon\_d^p + \int p d\varepsilon\_v^p \approx \int qd\varepsilon\_d + \int p d\varepsilon\_v \tag{4}$$

The results of plastic work for these two particle sizes group are shown in Table 4 (8th column); for the same particle size group, plastic work increases with the increase of confining pressure and relative density. For different particle size groups, the 2–5 mm particle size group needs to consume more plastic work, whereas the difference is not large. Under the same confining pressure and relative density conditions, for example, plastic work consumed by 2–5 and 0.3–100 kPa specimens and 0.5–2 to 0.3–100 kPa specimens are basically the same, but the difference between their relative breakage rate *Br* was larger, which indicates that the 0.5–2 mm particle group needs more plastic work than the 2–5 m particle group to reach the same relative breakage rate *Br*. The main reason is that the shape of the 2–5 mm particle group is more irregular, with larger internal flaws compared to the 0.5–2 mm particle group (Figure 4), which leads to a greater susceptibility to crushing for the same external load work conditions, causing a greater relative breakage rate. The relationship between the plastic work (*Wp*) and the relative breakage rate (*Br*) for these two particle sizes groups follows the power function (Figure 20a,b), which can be fitted using Equation (5), with *A*, *B* as the fitting parameters. For the same particle size groups, the relationship between plastic work and the relative breakage rate is not significantly influenced by the relative density and the confining pressure. Scholars' studies on calcareous sands also confirm this conclusion [37], however, in their research, *Br* = 1.17 × <sup>10</sup>−5*W p*1.42. By comparing the equations in this study with theirs, it can be found that the amount of crushing is greater under the same plastic work in this study, which is mainly because the main minerals strength of calcareous sand are stronger compared to volcanic soil particles, therefore, calcareous sand particles are less likely to crush under the external force, which also shows that the mineral composition has a great influence on particle breakage.

*Br* = *AW p<sup>B</sup>* (5)

**Figure 20.** Variations in relative breakage rate with plastic work: (**a**) 2–5 mm particle groups; (**b**) 0.5–2 mm particle groups.

#### **6. Conclusions**

A series of consolidation drainage shear tests were carried out to investigate the effects of particle size, confining pressure, and relative density on the mechanical properties and crushing characteristics of volcanic soils. The characteristics of soil stress–strain curve, shear strength index, critical state behavior, and quantification of particle fragmentation characteristics, using fractal dimension and relative breakage rate were systematically investigated. Finally, the quantitative relationship between external work and fragmentation amount was established through the energy perspective. The main findings are as follows:

Stress–strain curves characteristic of volcanic soil were affected by particle size; the large particle size groups (>0.5 mm) required larger axial strain to reach a stress steady state. The critical stress ratio was significantly influenced by particle size, with the largest critical stress ratio for the 2–5 mm particle size group and the smallest critical stress ratio for the 0.075–0.25 mm particle size group.

For crushable volcanic soils, there was a clear physical significance in using the peak internal friction angle index when considering the confining pressure effect; peak friction angle decreased significantly with the increase of confining pressure, and peak friction angle decreased as particle size increased and relative density decreased.

The particle crushing amount increased with increasing particle size, relative density, and confining pressure. For the same particle size, there was a good linear relationship between *Br* and *D*. Crushing patterns of large particle size groups could be classified as breakoff type. From an energy perspective, the power function relationship between *Wp* and *Br* of large particle size group was established, the relationship was only affected by particle size, and it was not significantly affected by the relative density and effective confining pressure. The results of comparison with similar materials [37] also confirm that the mineral composition has a large influence on the particle crushing amount.

In terms of engineering, based on the results of this study, it can be concluded that volcanic soil with fine particles used as roadbed filler can significantly reduce the deformation of the roadbed and improve the bearing capacity of the foundation, in addition to increasing the compaction degree of volcanic soil.

**Author Contributions:** Conceptualization, X.-Y.L.; methodology, X.-Y.L.; writing—original draft preparation, X.-Y.L.; investigation, H.-L.L. and D.W.; data curation, X.-Y.L., H.-L.L. and D.W.; writing—review and editing, C.-M.W., H.-L.L. and D.W.; project administration, C.-M.W.; funding acquisition, C.-M.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China (Grant Nos. 41972267).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**



#### **References**

