Electro-elastic dislocations in piezoelectric materials

ABSTRACT

In this paper, dislocations in piezoelectric materials are studied in the framework of linear incompatible theory of piezoelectricity with eigendistortion and eigenelectric field. We consider that both field variables, the displacement vector $\mathit{u}$ and the electrostatic potential ϕ, possess a jump discontinuity at the dislocation surface. This leads to the appearance of two eigenfields, the eigendistortion or plastic distortion tensor and the eigenelectric field vector providing the terminology of electro-elastic dislocations. These two eigenfields give rise to the well-known concept of dislocation density tensor and the new-introduced concept of the electric dislocation density vector, respectively, completing the basic framework of the mathematical modelling of electro-elastic dislocations in piezoelectric materials. Material balance (or broken conservation) laws which correspond to translation, scaling and rotation groups of transformations are derived for piezoelectric materials with electro-elastic dislocations considering additionally inhomogeneities, body forces and body charges. The terms breaking the translational, scaling and rotational symmetries, that is, the configurational or material forces (electro-elastic Peach–Koehler force, Cherepanov force, electrostatic part of the Lorentz force, piezoelectric inhomogeneity force or Eshelby force), the corresponding total configurational work and total configurational vector moment which give rise to the non-conserved $\mathit{J}$-, M- and $\mathit{L}$-integrals, respectively, are obtained.

KEYWORDS:

• Incompatible piezoelectricity
• eigenfields
• eigendistortion
• eigenelectric field
• dislocations
• $\mathit{J}$-, $M$- and $\mathit{L}$-integrals

Acknowledgments

The authors gratefully acknowledge a grant from the Deutsche Forschungsgemeinschaft (Grant No. La1974/4-1).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 See also [36] and [65] for a screw dislocation in incompatible elasticity.

Additional information

Funding

This work was supported by Deutsche Forschungsgemeinschaft [La1974/4-1].