The Mode-I full field solution in flexoelectric materials

The flexoelectric effect refers to strain-gradient-induced polarization and has significant applications in nanoelectromechanical systems. Various defects will inevitably exist in these nanoelectromechanical devices, thus significantly affecting their electromechanical coupling properties. In particular, Mode-I cracks, as a classical type of defect, have high theoretical value for research. Motivated by this, this paper derives the full-field solution for Mode-I cracks in flexoelectric materials for the first time. Using the Fourier transform, we transform the Mode-I crack problem in flexoelectric materials into a problem of solving a set of hypersingular integral equations. Then, we numerically solve these hypersingular integral equations by employing the collocation approach and the Chebyshev polynomial expansion technique. Based on the derived full-field solution, we comprehensively investigate the role of the strain-gradient length scale and flexoelectric coefficients on the displacement, stress, potential, electric fields, and energy release rate due to cracking. Furthermore, a mixed finite-element study of the Mode-I crack problem in flexoelectric materials is performed to compare it with the hypersingular integral equation solutions. The high degree of consistency between the results of the two methods further validates the rigor of this study. This study contributes to a deeper understanding of the multi-physics field coupling mechanism near cracks in flexoelectric materials, thus guiding the design of flexoelectric devices.