# Conservation Integrals in Nonhomogeneous Materials with Flexoelectricity
## 4 Conservation Integrals Derived from the Internal Energy  
In the previous section, several conservation integrals were obtained from the electric Gibbs free energy with the independent state variables \(\left(\varepsilon_{ij},w_{ijk},E_{i},V_{ij}\right)\). While in some situations, it is convenient to study the system through internal energy density function when the independent variables are \(\left(\varepsilon_{ij},w_{ijk},D_{i},Q_{ij}\right)\). Therefore, it is also essential to get the corresponding dual conservation integrals from the internal energy, which are presented as follows without the procedure of derivation.  
The total internal energy is  
\[U=\int_{V}udV=\frac{1}{2}\int_{V}\left(\mathbf{\sigma}:\mathbf{\varepsilon}+\dot{\mathbf{ \tau}}:\mathbf{w}+\mathbf{E}\cdot\mathbf{D}+\mathbf{V}:\mathbf{Q}\right)dV \tag{43}\]  
Specifying the functional derivative to be translational, dilatational, and rotational, respectively, the corresponding J-, M-, and L-integrals will be obtained.  
1. J-integral One can get \[\int_{V}\partial_{i}P_{ik}dV=\int_{V}\Big{(}B_{k}^{U}+f_{i}u_{i,k}+\varphi \partial_{k}\rho_{e}\Big{)}dV=J_{k}^{U}\] (44) with \[\begin{array}{l}B_{k}^{U}=\frac{1}{2}\Big{(}\partial_{k}C_{ijkl} \Big{)}\varepsilon_{ij}\varepsilon_{kl}+\Big{(}\partial_{k}\tau_{ijklm}\Big{)} \varepsilon_{ij}w_{klm}+\Big{(}\partial_{k}d_{ijk}\Big{)}\varepsilon_{ij}D_{ k}\\ +\Big{(}\partial_{k}\varepsilon_{ijkl}\Big{)}\varepsilon_{ij}Q_{kl}\ +\ \frac{1}{2}\Big{(}\partial_{k} \varepsilon_{ijklm}\Big{)}w_{ijk}w_{lmm}+\frac{1}{2}(\partial_{k}\alpha_{kl}) D_{k}D_{l}\\ +\Big{(}\partial_{k}f_{ijkl}\Big{)}D_{i}w_{jkl}+\Big{(}\partial_{k}\eta_{ijkm }\Big{)}Q_{ij}w_{kmn}+\Big{(}\partial_{k}h_{ijk}\Big{)}D_{i}Q_{jk}+\frac{1}{2 }\Big{(}\partial_{k}b_{ijkl}\Big{)}Q_{ij}Q_{kl}\end{array}\] (45) \[P_{ik}=u\delta_{ik}-\sigma_{ji}u_{j,k}-\tau_{mji}u_{m,jk}+\tau_{ jim,m}u_{j,k}+D_{i,k}\varphi+Q_{ji,k}\varphi_{,j}-Q_{ij,jk}\varphi\] (46)  
here, \(B_{k}^{U}\) represents the inhomogeneity of the material, \(P_{ik}\) is the generalized energy-momentum tensor. Obviously, due to the body force, free electric charge, gradient term, and inhomogeneity of material, the generalized J-integral is not path-independent. However, one can construct a conservation integral as  
\[U_{k}^{I}=J_{k}^{U}-\int_{V}\Big{(}B_{k}^{U}+f_{i}u_{i,k}+\varphi\partial_{k} \rho_{e}\Big{)}dV \tag{47}\]
2. M-integral Similarly, the M-integral derived from internal energy is \[M^{U}=\int_{V}\Big{[}x_{k}F_{k}+\frac{n-2}{2}f_{i}u_{i}-\tau_{mjk}u_{m,jk}- \frac{n+2}{2}\rho y_{e}\varphi-Q_{jk,k}\varphi_{,j}\Big{]}dV=\int_{V}\partial_ {i}Y_{i}dV\] (48)where \(Y_{i}\) is the dilatation flux and \[Y_{i}=x_{k}P_{ik}-\frac{n}{2}\big{(}\tau_{mji}u_{m,j}-Q_{ji}\varphi_{j}\big{)}- \frac{n-2}{2}\big{(}\sigma_{ji}-\tau_{jim,m}\big{)}u_{j}+\frac{n}{2}(D_{i}-Q_{ ik,k})\varphi\] (49) \(\quad\)L-integral \(\quad\)The L-integral can finally be obtained as follows \[L_{k}^{U}=\int_{S}\Gamma_{kj}n_{j}dS=\int_{V}\in_{kji}\big{[}x_{j}F_{i}+f_{i}u_ {j}+u_{m,j}(\partial_{n}\tau_{mni})-D_{i,j}\varphi\big{]}dV\] (50) with the angular momentum tensor \(\Gamma_{kn}\) defined as \[\Gamma_{kn}= \in_{kji}\big{[}x_{j}P_{in}+(\sigma_{in}-\tau_{nim,m})u_{i}+\tau_{ mni}u_{m,j}-Q_{in,j}\varphi\big{]}\] (51)