# Conservation Integrals in Nonhomogeneous Materials with Flexoelectricity
## 3 Conservation Integrals Relevant to the Electric Gibbs Function
### Application of Translation and J-Integral  
Using Equation (10), the total electric Gibbs free energy is  
\[G^{E}=\int_{V}g^{E}dV=\frac{1}{2}\int_{V}\left(\sigma:\epsilon+\dot{\mathbf{\tau}} \dot{\mathbf{w}}-\mathbf{D}\cdot\mathbf{E}-\mathbf{Q}:\mathbf{V}\right)dV \tag{11}\]  
Now let us take an arbitrary infinitesimal functional derivative \(\delta G^{E}\) of the energy density. Substituting the constitutive Equation (9) into Equation (11), one can obtain  
\[ \begin{array}{l}\delta G^{E}=\int_{V}\delta g^{E}dV\\ =\int_{V}\delta\begin{pmatrix}\frac{1}{2}C_{ijkl}\epsilon_{ij}\epsilon_{kl}+r _{ijklm}\epsilon_{ij}w_{klm}-d_{ijk}\epsilon_{ij}E_{k}-e_{ijkl}\epsilon_{ij}V _{kl}+\frac{1}{2}g_{ijklmn}w_{ijk}w_{lmn}\\ -f_{ijkl}E_{i}w_{jkl}-\eta_{ijkm}V_{ij}w_{kmn}-\frac{1}{2}\delta_{kl}E_{k}E_{l} -h_{ijk}E_{i}V_{jk}-\frac{1}{2}b_{ijkl}V_{ij}V_{kl}\end{pmatrix}dV\end{array}  \tag{12}\]  
In the following, the functional derivative will be converted to be translational  
\[\delta=(\delta x_{k})\frac{\partial}{\partial x_{k}}=(\delta x_{k})\partial_{k} \tag{13}\]  
Using Equation (13), one can get from the left-hand side of Equation (12)  
\[\delta G^{E}=\int_{V}\delta g^{E}dV=\int_{V}\left(\partial_{k}g^{E}\right)( \delta x_{k})dV=\int_{V}\partial_{i}\Big{(}g^{E}\delta_{ik}\Big{)}(\delta x_{ k})dV \tag{14}\]Considering that the material coefficients are also the function of the space coordinates and using the symmetry of these material coefficients, the integrand in the second integral in Equation (12) can be derived as follows  
\[ \begin{array}{ll}\delta\big{(}g^{E}\big{)}&=\frac{1}{2}\left[\partial_{k}C_{ijkl }\right]\varepsilon_{ij}\varepsilon_{kl}\delta x_{k}+\left[\partial_{k}r_{ijkl }\right]\varepsilon_{ij}w_{klm}\delta x_{k}\\ &-\left[\partial_{k}d_{ijk}\right]\varepsilon_{ij}E_{k}\delta x_{k}-\left[ \partial_{k}\epsilon_{ijkl}\right]\varepsilon_{ij}V_{kl}\delta x_{k}\ +\ \frac{1}{2}\left[\partial_{k}S_{ijklmn}\right]w_{ijk}w_{ lmn}\delta x_{k}\\ &-\left[\partial_{k}f_{ijkl}\right]E_{i}w_{jkl}\delta x_{k}-\left[\partial_{k} \eta_{ijkm}\right]V_{ij}w_{kmn}\delta x_{k}\\ &-\frac{1}{2}[\partial_{k}a_{kl}]E_{k}E_{l}\delta x_{k}-\left[\partial_{k}h_{ ijk}\right]E_{i}V_{jk}\delta x_{k}-\frac{1}{2}\left[\partial_{k}b_{ijkl} \right]V_{ij}V_{kl}\delta x_{k}\\ &+\sigma_{ij}\left[\delta\varepsilon_{ij}\right]+\tau_{ijm}\left[\delta \upsilon_{ijm}\right]-D_{i}[\delta E_{i}]-Q_{ij}\left[\delta V_{ij}\right]\\ &=A_{k}^{\rm G}\delta x_{k}+\sigma_{ij}\left[\delta\varepsilon_{ij}\right]+ \tau_{ijm}\left[\delta\upsilon_{ijm}\right]-D_{i}[\delta E_{i}]-Q_{ij}\left[ \delta V_{ij}\right]\end{array}  \tag{15}\]  
For convenience, mark the first four lines in Equation (15) as \(A_{k}^{\rm G}\delta x_{k}\) with  
\[ \begin{array}{ll}A_{k}^{\rm G}=\frac{1}{2}\left[\partial_{k}C_{ijkl}\right] \varepsilon_{ij}\varepsilon_{kl}+\left[\partial_{k}r_{ijklm}\right] \varepsilon_{ij}w_{klm}\ -\left[\partial_{k}d_{ijk}\right]\varepsilon_{ij}E_{k}\\ \ -\left[\partial_{k}\varepsilon_{ijkl}\right]\varepsilon_{ij}V_{kl}\ +\ \frac{1}{2}\left[ \partial_{k}S_{ijklmn}\right]w_{ijk}w_{lmn}\ -\left[\partial_{k}f_{ijkl}\right]E_{i}w_{ jkl}\\ \ -\left[\partial_{k}\eta_{ijkm}\right]V_{ij}w_{kmn}\ -\frac{1}{2}\left[ \partial_{k}a_{kl}\right]E_{k}E_{l}-\left[\partial_{k}h_{ijk}\right]E_{i}V_{jk }-\frac{1}{2}\left[\partial_{k}b_{ijkl}\right]V_{ij}V_{kl}\end{array}  \tag{16}\]  
Thus, \(A_{k}^{\rm G}\) is the so-called material inhomogeneity force which represents the nonhomogeneous property of the material. The following derivation can be obtained for the terms in the last line in Equation (15)  
\[\sigma_{ij}\left[\delta\varepsilon_{ij}\right]=\sigma_{ij}\left[\partial_{k} \varepsilon_{ij}\right]\left(\delta x_{k}\right)=\left[\left(\sigma_{ij}u_{i,k }\right)_{,j}-\sigma_{ij,ij}u_{i,k}\right]\left(\delta x_{k}\right) \tag{17}\]  
\[