Abstract
The behaviour of small size dielectric elastic beams is described within higher-grade theory with including electric polarization. The coupling between strain gradients and polarization is incorporated into the constitutive laws in the form of flexoelectricity, while piezoelectricity is involve in the classical form. Both the governing equations and boundary conditions are derived using variational formulation for electro-elastic continuous media and deformation assumptions employed in three various beam bending theories such as the classical theory (Euler-Bernoulli theory), the 1st order shear deformation theory (Timoshenko theory) and 3rd order shear deformation theory. The unified formulation allows switching between theories with various bending assumptions by a proper selection of two key factors.
Highlights
In non-centre-symmetric dielectric crystals, the polarization vector is related to the 2nd order strain tensor through the 3rd order piezoelectric tensor which must vanish for all dielectrics with inversion-centre symmetry
The existence of non-uniform strain due to relative displacements between the centres of oppositely charged ions is physically possible only provided that the centre-symmetry is broken and the contribution of macroscopic strain gradients to induced polarization is known as flexoelectric effect [3,4]
We presented the consistent derivation of 1D formulation for behaviour of dielectric elastic beams subject to stationary electro-mechanical loading
Summary
In non-centre-symmetric dielectric crystals, the polarization vector is related to the 2nd order strain tensor through the 3rd order piezoelectric tensor which must vanish for all dielectrics with inversion-centre symmetry. The existence of non-uniform strain due to relative displacements between the centres of oppositely charged ions is physically possible only provided that the centre-symmetry is broken and the contribution of macroscopic strain gradients to induced polarization is known as flexoelectric effect [3,4]. The flexoelectric effect can be incorporated into macroscopic phenomenological theory by consideration of higher-grade continuum theory involving the 2nd order derivatives of displacements besides the strains. Having used such a continuum model, we shall deal with behaviour of elastic dielectric beams under electro-mechanical loading [5,6]. The derivation of the governing equations and the boundary conditions is performed in a consistent way with using variational principle
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