Abstract
The flexoelectricity, which is a new electromechanical coupling phenomenon between strain gradients and electric polarization, has a great influence on the fracture analysis of flexoelectric solids due to the large gradients near the cracks. On the other hand, although the flexoelectricity has been extensively investigated in recent decades, the study on flexoelectricity in nonhomogeneous materials is still rare, especially the fracture problems. Therefore, in this manuscript, the conservation integrals for nonhomogeneous flexoelectric materials are obtained to solve the fracture problem. Application of operators such as grad, div, and curl to electric Gibbs free energy and internal energy, the energy-momentum tensor, angular momentum tensor, and dilatation flux can also be derived. We examine the correctness of the conservation integrals by comparing with the previous work and discuss the operator method here and Noether theorem in the previous work. Finally, considering the flexoelectric effect, a nonhomogeneous beam problem with crack is solved to show the application of the conservation integrals.
Highlights
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
If the piezoelectric effect is omitted, the conservation integral will reduce to the classical elasticity: M02 h3
Summary
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]. For systems without a Lagrangian, Honein et al [32] developed a novel methodology called Neutral Action method to obtain the conservation laws directly from the partial differential equations. Their method was used to obtain conservation integrals for nonhomogeneous plane problems [33] and nonhomogeneous. Kirchner [38] derived the energy-momentum tensor for nonhomogeneous anisotropic linearly elastic three-dimensional solids Following this procedure, Lazar and Kirchner [39], Kirchner and Lazar [40], obtained the Eshelby stress tensor and conservation laws for gradient elasticity of nonhomogeneous, incompatible, linear, anisotropic media, and bone growth and remodeling, respectively.
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