# Conservation Integrals in Nonhomogeneous Materials with Flexoelectricity
## 1 Introduction  
Unlike the piezoelectric effect only existing in noncentrosymmetric dielectrics, flexoelectricity is an important electromechanical coupling phenomenon theoretically existing in materials with all possible symmetries [1; 2]. The main characteristic of the flexoelectricity is that the solids deform due to the electric field gradient and vice versa, they produce electric polarization due to the strain gradient. In recent decades, researchers have researched flexoelectric effects from theories [3; 4; 5], atomic simulation and molecular dynamics simulation [6; 7; 8], experimental measurement [9; 10; 11; 12], and device application [13; 14; 15; 16]. For example, from the phenomenological and microscopic perspectives, [PERSON] [17; 18] systematically analyzed the flexoelectric effects and piezoelectric effects and provided a method to calculate the values of the flexoelectric coefficients. For nanosized dielectrics, Hu and Shen [3; 19], Shen and Hu [4] established the dynamic variational principle for flexoelectric dielectrics with surface effects and obtained generalized governing equations and Young-Laplace equations. However, there have been few studies on the flexoelectric effect introduced by the inhomogeneity of the solids. Besides, flexoelectricity may change the strength and fracture toughness of the material [20] due to the large gradients near the defects. Thus, it is necessary to accurately understand the fracture problems for the flexoelectric solids, further for nonhomogeneous flexoelectric solids. For classical mechanical or piezoelectric fracture problems, the governing equations and the corresponding boundary conditions are solved directly using the complex potential method [21; 22]. However, it is difficult to obtain the exact solutions for some crack problemsdue to the mathematical complexities. These difficulties urged researchers to obtain the conservation laws or conservation integrals such as J-, M-, and L- integrals to bypass the complexities around the crack tip. J-, M-, and L-integrals are related to the energy release rates for crack translation, crack rotation, and self-similar expansion [23]. Then, the stress intensity factor (SIF) can be obtained naturally from the J-, M-, and L-integrals, which is helpful to study fracture problems. Several methods of constructing conservation integrals can be classified into the following categories:  
1. The procedure used most widely is Noether theorem. Since [PERSON] demonstrated the corresponding relation between the conservation integrals and the symmetries in 1918 [24], conservation integrals derived from the Noether theorem have been extensively developed by constructing different symmetries. Usually, according to [PERSON] theorem, J-, M-, and L-integrals can be derived from the corresponding coordinate translation symmetry, scaling symmetry, and coordinate rotation symmetry, respectively. For instance, beginning with these symmetries, [PERSON] and [PERSON] [25], and [PERSON] [26], developed conservation laws in elastostatics and linear elastodynamics, respectively. [PERSON] and [PERSON] [27] obtained the conservation laws for linear piezoelectric materials. Other applications of [PERSON] theorem can be found in second gradient electroelasticity [28], flexoelectric solids [29], dissipative electrochemical coupling processes [30; 31]. 2. The Neutral Action method for dissipative systems or nonhomogeneous solids. It is well known that the classical [PERSON]'s theorem is only applicable to the Lagrangian systems. For systems without a Lagrangian, [PERSON] et al. [32] developed a novel methodology called Neutral Action method to obtain the conservation laws directly from the partial differential equations. Then, their method was used to obtain conservation integrals for nonhomogeneous plane problems [33] and nonhomogeneous Bernoulli-Euler beams [34]. [PERSON] and [PERSON] [35] applied the Neutral Action method to Schrodinger equation and found a new conservation law. 3.