# Conservation Integrals in Nonhomogeneous Materials with Flexoelectricity
## 3 Conservation Integrals Relevant to the Electric Gibbs Function
### Application of Translation and J-Integral
\begin{array}{ll}\tau_{ijm}\left[\delta\upsilon_{ijm}\right]=\tau_{ijm} \left[\partial_{k}w_{ijm}\right]\left(\delta x_{k}\right)\\ =\left[\left(\tau_{ijm}u_{i,k}\right)_{,m}-\tau_{ijm,m}u_{i,jk}\right]\left( \delta x_{k}\right)\\ =\left\{\left(\tau_{ijm}u_{i,jk}\right)_{,m}-\left[\left(\tau_{ijm,m}u_{i,k} \right)_{,j}-\tau_{ijm,mj}u_{i,k}\right]\right\}\left(\delta x_{k}\right)\\ =\left[\left(\tau_{ijm}u_{i,jk}\right)_{,m}-\left(\tau_{ijm,m}u_{i,k}\right)_ {,j}+\tau_{ijm,mj}u_{i,k}\right]\left(\delta x_{k}\right)\\ \end{array}  \tag{18}\]  
\[ \begin{array}{ll}D_{i}\left(\delta E_{i}\right)=D_{i}\partial_{k}E_{i} \delta x_{k}=D_{i}\left(\partial_{i}E_{k}\right)\delta x_{k}\\ =\left[\partial_{i}(D_{i}E_{k})-\left(\partial_{i}D_{i}\right)E_{k}\right] \delta x_{k}=-\left[\partial_{i}(D_{i}\varphi_{k})-\left(\partial_{i}D_{i} \right)\varphi_{,k}\right]\delta x_{k}\\ \end{array}  \tag{19}\]  
\[ \begin{array}{ll}Q_{ij}\left[\delta V_{ij}\right]=Q_{ij}\left[\partial_{k} V_{ij}\right]\left(\delta x_{k}\right)\\ =-Q_{ij}\varphi_{,ijk}\left(\delta x_{k}\right)=-\left[\left(Q_{ij}\varphi_{, ik}\right)_{,j}-Q_{ij,ij}\varphi_{,jk}\right]\left(\delta x_{k}\right)\\ =-\left[\left(Q_{ij}\varphi_{,ik}\right)_{,j}-\left(Q_{ij,ij}\varphi_{,k} \right)_{,j}+Q_{ij,ji}\varphi_{,k}\right]\left(\delta x_{k}\right)\end{array}  \tag{20}\]  
Substituting Equations (14)-(20)and governing Equations (1) and (2) into Equation (12), one can get  
\[ \begin{array}{ll}\int_{V}\partial_{i}\Big{[}g^{E}\delta_{ik}-\sigma_{ij}u_{i,k }-\tau_{mj}u_{m,jk}+\tau_{ijm,m}u_{j,k}-D_{i}\varphi_{,k}-Q_{ji}\varphi_{,jk}+Q _{ij,j}\varphi_{,k}\Big{]}dV\\ =\int_{V}\left(A_{k}^{\rm G}+f_{i}u_{i,k}+\rho_{e}\varphi_{,k}\right)dV=J_{k}^{ \rm G}\end{array}  \tag{21}\]  
The integrand in the second integral in Equation (21) is the configurational force, which can be defined as  
\[F_{k}=A_{k}^{\rm G}+f_{i}u_{i,k}+\rho_{e}\varphi_{,k} \tag{22}\]  
There are three terms in the configurational force: the first term represents the inhomogeneity of the material, the second term is the configurational force on the body force which can be found in [PERSON]'s work [47], and the third term is the configurational force due to the free electric charge, which is the electrostatic part of the Lorentz force [48]. The integrand in the first integral in Equation (21) is the divergence of the generalized Eshelby stress tensor of nonhomogeneous flexoelectric solid, which is defined as  
\[P_{ik}=g^{E}\delta_{ik}-\sigma_{ij}u_{j,k}-\tau_{mj}u_{m,jk}+\tau_{jin,m}u_{j,k}- D_{i}\varphi_{j,k}-Q_{ji}\varphi_{j,k}+Q_{ij,j}\varphi_{j,k} \tag{23}\]  
The above generalized energy-momentum tensor can reduce to the conventional elastic Eshelby stress tensor [36], the electroelastic Eshelby stress tensor [49; 50]. Based on Equation (21), the relation between the Eshelby stress tensor and the configurational force is  
\[\partial_{i}P_{ik}=F_{k} \tag{24}\]  
Analogous to governing Equation (1) in classical mechanics in physical space, the above expression is the governing equation in configurational mechanics in material space [51]. Besides, using [PERSON] divergence theorem in Equation (21), one can find  
\[J_{k}^{G}=\int_{V}\partial_{i}P_{ik}dV=\int_{S}n_{i}P_{ik}dS \tag{25}\]  
Thus, \(J_{k}^{G}\) is the J-integral of the nonhomogeneous flexoelectric material, and its value does not equal zero which means this integral is not path-independent. In absence of body force and free electric charge, a path-independent integral \(G_{k}\) can be obtained by moving the inhomogeneity term to the left side in Equation (21),  
\[G_{k}^{l}=J_{k}^{G}-\int_{V}A_{k}^{G}dV \tag{26}\]  
Furthermore, if the flexoelectric material is homogeneous, the path-independent integral \(G_{k}^{l}\) will reduce to the J-integral derived from Noether theorem for homogeneous material in [30].