Energy jump during bond breaking

In current fracture theory, the fracture stress is related to the surface energy on the basis of linear elastic theory. However, the fracture stress does not necessarily exceed the stress required to break atomic bonds. Here, we show that a jump in the inelastic separation energy is generated by fracture, where the inelastic separation energy is the energy between the separation planes measured by excluding the contribution of elastic relaxation, and the stress at the onset of the energy jump is the fracture stress. Analysis of the electronic states of $\beta -\mathrm{SiC}$ (cubic SiC), Ge, and Cu by first-principles tensile tests shows that the electrons redistribute during surface formation in the transition from the onset to the end of the energy jump. Therefore, it is suggested that the inelastic separation energy at the end of the energy jump can be identified with the fracture energy. Also, first-principles shear tests show that an energy jump occurs during shearing for $\beta -\mathrm{SiC}$, but not for Ge and Cu. Thus, an energy jump is a sign of fracture (bond breaking), and an energy jump during shearing is a good indicator estimating the ductile and brittle character. These principles can hold for any solid and will therefore be beneficial for the fundamental understanding of the mechanical properties of solids and for their industrial applications.

Figure 1

Variation in the total energies of the (a) $\beta -\mathrm{SiC}$, (b) Ge, and (c) Cu cells as a function of strain. The variation in the total energy was determined by relaxed-type and unrelaxed-type first-principles tensile tests. In the unrelaxed-type simulations, the separation planes move along the tensile direction with all of the atoms fixed. In the relaxed-type simulations, only the atoms located in the outermost two layers of the cell are fixed for tensile straining, while the other atoms can move freely. The total energy is found to continuously vary in the unrelaxed-type simulations. In the relaxed-type simulations, after the total energy continuously varies, it suddenly decreases in the final stage.

Figure 2

Variation in the inelastic separation energy of the separation planes as a function of the displacement between them: (a) $\beta -\mathrm{SiC}$, (b) Ge, and (c) Cu. The variation in the inelastic separation energy was determined by relaxed-type and the unrelaxed-type first-principles tensile tests. The inelastic separation energy continuously is found to vary in the unrelaxed-type simulations. In the relaxed-type simulations, the inelastic separation energy continuously varies until a certain displacement, and it then discontinuously jumps to its final value. The stress at the onset of the energy jump can be defined with the fracture stress. Although it appears that the number of data for the relaxed-type test is less than that for the unrelaxed-type test, some data for the relaxed-type test overlap, and the number of data for the relaxed-type test is the same as that for the unrelaxed-type test.

Figure 3

Charge density distributions before deformation, at the OPB, and at the EPB: (a) $\beta -\mathrm{SiC}$, (b) Ge, and (c) Cu. The charge density around the separation plane slightly decreases at the OPB compared with the charge density before deformation, but fracture is not completed at the OPB. On the other hand, the charge density around the separation plane decreases to zero at the EPB, which indicates that fracture is completed at the EPB.

Figure 4

Partial DOS before deformation (initial), at the OPB and at the EPB for $\beta -\mathrm{SiC}$ and Cu. (a) $p$ orbitals of a Si atom, (b) $p$ orbitals of a C atom, and (c) $s$ orbitals of a Cu atom at the separation plane before deformation, at the OPB, and at the EPB. The partial DOS at the OPB is similar to that before deformation. The electrons are localized near the Fermi level at the EPB.

Figure 5

Variation in the inelastic separation energy as a function of displacement for pure alias shear: (a) $\beta -\mathrm{SiC}$, (b) Ge, and (c) Cu. The variation in the inelastic separation energy was measured by first-principles pure alias shear tests. A jump in the inelastic separation energy occurs for $\beta -\mathrm{SiC}$ before the inelastic separation energy reaches the USFE. No jump in the inelastic separation energy occurs for Ge and Cu.