## 2 Field Theory of Deformation And Fracture
### Field Equations
#### 2.2.2 Fracture Dynamics  
The Lagrangian for the fracture dynamics is as follows.  
\[\mathcal{L}_{ep}=\frac{\rho v^{2}}{2}-\frac{G\omega^{2}}{2}+\frac{G}{c}\mathbf{j} ^{0}A_{0}+G\mathbf{j}^{i}A_{i}+\frac{E}{2}\left(\partial_{x_{s}}^{2}\mathbf{\xi} \right)^{2}\left(\delta x_{s}\Delta x_{s}\right) \tag{10}\]As compared with Lagrangian density (1), expression (10) has the extra term at the end of the right-hand side. This term represents the potential energy that is associated with the secondary spatial derivatives of displacement. Because the first-order derivative of displacement is strain, this term represents the potential energy associated with the spatial gradient of the strain. In the pre-fracturing stage, deformation becomes so nonuniform that the differential strain generates potential energy. In other words, this term represents how nonuniform the deformation field becomes.  
In this case, since the Lagrangian density has a term depending on the second-order spatial differential term, the Euler-Lagrangian equation of motion has the extra term.  
\[\partial_{t}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{t}A_{\mu})} \right)+\partial_{x}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{x}A_ {\mu})}\right)-\partial_{xx}\left(\frac{\partial\mathcal{L}}{\partial( \partial_{xx}A_{\mu})}\right)-\frac{\partial\mathcal{L}}{\partial A_{\mu}}=0 \tag{11}\]  
Tensile experiments [14; 15; 16; 17] indicate that under the pre-fracture condition, \((\
abla\cdot\mathbf{v})\) is one-dimensional and oriented at approximately \(45^{\circ}\) to the tensile axis. Under the one-dimensional condition, the substitution of Equation (10) into Equation (11) yields the following field equation corresponding to (7).  
\[\rho\frac{\partial\mathbf{v}}{\partial t}=-G(\
abla\times\mathbf{\omega})\frac{ \lambda+2G}{\rho}\frac{d\xi_{s}}{dx_{s}}-\sigma_{0}\rho\frac{dv_{s}}{dx_{s}}v _{s}-\left(E\delta x_{s}\Delta x_{s}\right)\left(\partial_{x_{s}}^{4}\xi\right) \tag{12}\]  
Here, \(x_{s}\) is the axis set along the one-dimensional \((\
abla\cdot\mathbf{v})\). Accordingly, \(d\xi_{s}/dx_{s}\) and \(dv_{s}/dx_{s}\) in the first and second terms on the right-hand side are the one-dimensional form of \((\
abla\cdot\xi)\) and \((\
abla\cdot\mathbf{v})\) appearing in Equations (8) and (9), respectively. It is reasonable to assume that a fracturing unit volume does not exert elastic resistant force proportional to the volume expansion. When the volume is about to fracture, the resistant force does not increase as the volume stretches. Instead, the surrounding unit volumes that border the fracturing unit volume at \(x_{s}\) and \(x_{s}+\Delta x_{s}\) exert elastic force proportional to the local stretch occurring within the small region expressed by \(\delta x_{s}\) (in other words, the strain energy is associated with the differential strain, not strain. See Figure 2 and the associated text). The third term on the right-hand side of Equation (12) represents this force. Thus, we can set the first term on the right-hand side of Equation (12) to null and rewrite the equation, as follows.  
\[\rho\frac{\partial\mathbf{v}}{\partial t}=-\sigma_{0}\rho\frac{dv_{s}}{dx_{s}}v_{s }-\left(E\delta x_{s}\Delta x_{s}\right)\left(\partial_{x_{s}}^{4}\xi\right) \tag{13}\]  
As will be discussed later in this paper this assumption is consistent with the experiment.