# The *J*-integral in flexoelectric solids

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## Abstract

The flexoelectric effect is a significant electromechanical coupling phenomenon between strain gradients and electric polarization. Since the design of materials with high flexoelectricity should be accompanied with stress concentration/intensity, the strength and fracture analysis of flexoelectric materials with large strain gradients is desired. The famous *J*-integral can be used to characterize the singularity at crack tips and predict the fracture behavior of flexoelectric solids. However, the definition of *J*-integral in flexoelectric solids is lacked or incomplete in the open literature. In this study, an explicit expression of *J*-integral associated with material configurational forces is derived from the gradient operation of electric enthalpy density function for centrosymmetric flexoelectric solids, where the electric enthalpy density depends not only on the strain and strain gradient, but also on the polarization and polarization gradient. The path-independence of *J*-integral in flexoelectric solids is also examined through the Gauss–Green’s theorem. Then the derived *J*-integral is applied to study a cylindrical cavity and a mode III crack problem in flexoelectric solids. The results indicate that, in flexoelectric solids, there is a conservation law of the *J*-integral. That is, the *J*-integral defined in a global coordinate system vanishes when the integration contour chosen to calculate the *J*-integral encloses whole cavity. The present complete expression of *J*-integral in flexoelectric solids is addressed from the self-consistent theory of flexoelectricity. It corrects the inaccurate definition of *J*-integral in the previous literature. The *J*-integral obtained in this paper will provide a useful way to study fracture problems in flexoelectric solids.

## Keywords

Flexoelectricity Strain gradient Configurational force*J*-integral Fracture

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 11772245, 11472205, 11672222), the 111 Project (B18040), and the Fundamental Research Funds for the Central Universities in China.

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