## 2 Field Theory of Deformation And Fracture
### Field Equations
#### 2.2.1 Elasto-Plastic Dynamics  
A rigorous analysis that is based on the gauge theoretical approach identifies the Lagrangian density for the elasto-plastic dynamics, as follows. \[\mathcal{L}_{ep}=\frac{\rho v^{2}}{2}-\frac{G\omega^{2}}{2}+\frac{G}{c}j^{0}A _{0}+Gj^{i}A_{i} \tag{1}\]  
Here, \(G\) is the shear modulus, \(c=\sqrt{G/\rho}\) is the phase velocity of the wave characteristics, and \(j^{0}\) and \(j^{i}\) are the temporal and spatial components of the charge of symmetry, respectively. \(\mathbf{v}\) and \(\mathbf{\omega}\) are related to \(\mathbf{A}\), as follows. \[\mathbf{v} = \dot{\mathbf{A}} \tag{2}\] \[\omega^{k} = \frac{\partial A^{j}}{\partial x^{i}}-\frac{\partial A^{i}}{ \partial x^{j}}, \tag{3}\]  
Here, indexes \(i,j,k\) represent the spatial coordinate variables. Figure 1: Concept underlying present field theory. Potential associated with connection field logically connects local elastic deformations. The Euler-Lagrangian equation of motion in this case is given as follows. \[\partial_{t}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{t}A_{\mu})} \right)+\partial_{x}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{x}A_{ \mu})}\right)-\frac{\partial\mathcal{L}}{\partial A_{\mu}}=0 \tag{4}\]  
where \(\mu=t,x,y,z\). Substitution of Equation (1) into Equation (4) yields the following field equations. \[\
abla\cdot\mathbf{v} = -\mathbf{j}^{0} \tag{5}\] \[\
abla\times\mathbf{\omega} = -\frac{\rho}{G}\frac{\partial\mathbf{v}}{\partial t}-\mathbf{j} \tag{6}\]  
Multiplication of \(G\) on both-hand sides allows us to rewrite Equation (6) in the following form. \[\rho\frac{\partial\mathbf{v}}{\partial t}=-G(\
abla\times\mathbf{\omega})-G\mathbf{j} \tag{7}\]  
The left-hand side of Equation (7) is in the form of mass times acceleration. Thus, Equation (7) can be interpreted as an equation of motion. The first and second terms on the right-hand side of Equation (7) can be interpreted as the transverse (shear) force and longitudinal force acting on the unit volume that is represented by density \(\rho\) (see Chapter 5 of ref. [9]). Analysis indicates that the longitudinal force takes the following forms in the pure elastic and pure plastic dynamics, respectively. \[G\mathbf{j}_{e} = -(\lambda+2G)\
abla(\
abla\cdot\mathbf{\xi}) \tag{8}\] \[G\mathbf{j}_{p} = \sigma_{0}\rho(\
abla\cdot\mathbf{v})\mathbf{v} \tag{9}\]  
Here, \(\lambda\) is the Lame's first parameter, \(\sigma_{0}\) is a material constant (See Chapter 5 of ref. [9]), \(\mathbf{\xi}\) is the displacement vector. In Appendix B, conceptual images of the shear force that are represented by \(G\
abla\times\mathbf{\omega}\) and the longitudinal fore represented by \((\lambda+2G)\
abla(\
abla\cdot\mathbf{\xi})\) are illustrated. (\(\mathbf{\xi}\) appearing on the right-hand side of Equation (8) is the displacement vector associated with elastic force law and \(\mathbf{v}\) on the right-hand side of Equation (9) is the temporal derivative of vector potential \(\mathbf{A}\) whose dimension is displacement. Strictly speaking, the physical meaning of displacement \(\mathbf{A}\) is different from \(\mathbf{\xi}\). However, recent theoretical consideration indicates that, formulaically, we can replace \(\mathbf{A}\) with \(\mathbf{\xi}\) and interpret it as the total displacement vector (see Appendix C). Hereafter in this paper, \(\mathbf{\xi}\) is used to represent the displacement vector and \(\mathbf{v}=\mathbf{\xi}\)). Equations (8) and (9) indicate that the form of the longitudinal force term differentiates plastic deformation from elastic deformation. When the deformation is elastic, the longitudinal force is proportional to the differential displacement \(\
abla\cdot\mathbf{\xi}\), which is interpreted as the three-dimensional stretch or compression of a unit volume. When the deformation is plastic, the longitudinal force is proportional to the velocity through a variable coefficient \((\
abla\cdot\mathbf{v})\). This relation can be interpreted as the proportionality is not connected by a constant but a variable coefficient.