Flexoelectricity and apparent piezoelectricity of a pantographic micro-bar

We discuss a homogenized model of a pantographic bar considering flexoelectricity. A pantographic bar consists of relatively stiff small bars connected by small soft flexoelectric pivots. As a result, an elongation of the bar relates almost to the torsion of pivots. Taking into account their flexoelectric properties we find the corresponding electric polarization. As a results, the homogenized pantographic bar demonstrates piezoelectric properties inherited from the flexoelectric properties of pivots. The effective stiffness properties of the homogenized bars are determined by the geometry of the structural elements and shear stiffness whereas the piezoelectric properties follow from the flexoelectric moduli of the pivots.