Unified theory of beam bending within flexoelectricity with including piezoelectricity

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 1 order shear deformation theory (Timoshenko theory) and 3 order shear deformation theory. The unified formulation allows switching between theories with various bending assumptions by a proper selection of two key factors.


Introduction
In non-centre-symmetric dielectric crystals, the polarization vector is related to the 2 nd order strain tensor through the 3 rd order piezoelectric tensor which must vanish for all dielectrics with inversion-centre symmetry. Therefore piezoelectricity is not observed in centresymmetric dielectric crystals [1,2]. However a net electrical dipole moment is generated also upon application of non-uniform strain, i.e. strain gradients, even in originally centresymmetric dielectric crystals. 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]. Thus the flexoelectric effect can be incorporated into macroscopic phenomenological theory by consideration of higher-grade continuum theory involving also the 2 nd 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 1D formulation will be derived in a unified form with including the deformation assumptions of three theories for bending of elastic beams. Making use such a unified formulation, one can switch between three various theories by a proper selection of two key factors. The derivation of the governing equations and the boundary conditions is performed in a consistent way with using variational principle.

Stationary electro-elasticity with including piezo-and flexoelectric effects
In contrast to higher-order theories, the number of degrees of freedom is not changed as compared with classical theory, i.e. the independent field variables are the same as in classical theory, but some additional field-gradient measures appear in higher-grade theories. Therefore also the number of governing equations is not changed, while the order of the differential equations as well as and the number of boundary conditions are increased. Assuming small derivatives of field variables, the general linear constitutive laws can be derived from the quadratic energetic functional of the derivatives of field variables. In case of dielectric solids, the electric enthalpy can play the role of the energetic functional. Assuming the higher-grade theory of dielectric solids with including the 2 nd order derivatives of field variables, the electric enthalpy density is considered as quadratic functional , , , Note that the index following a comma denotes the partial derivative with respect to the corresponding Cartesian coordinate. In Eq. (1), ijkl c is the tensor of elastic coefficients, flexoelectric and converse flexoelectric coefficients, respectively. The third-rank piezoelectric tensor vanishes in crystalline centrosymmetric dielectrics. In the above formulation, the contribution to the piezoelectric as well as flexoelectric polarization is considered as a response to an applied macroscopic strains and its gradients. Bearing mind the bulk contribution to the polarization, the direct and converse flexoelectricity terms in (1) can be expressed in only one term [5,6] as The symmetry properties of tensors of material coefficients depend on symmetry of elastic dielectric crystals. For crystals of cubic symmetry [7] these tensors are given as In non-centrosymmetric crystals exhibiting mm2 class of symmetry with 3 x being the poling axis, the piezoelectric coefficients are given as ijk  , and i D being the stress tensor, higher order stress tensor, and the electric displacements, respectively.

Derivation of the formulations for beam bending
Let us consider a beam of thickness b ( Assuming the translational symmetry along 2 x , we may write the displacement field distribution as () wx being axial displacement, rotation of the beam cross-section and deflection, respectively, and From (6), one can obtain displacement gradients, strains and  x -coordinate. In order to get a pure 1D formulation for considered electroelastic problems in thin beam structures, it is meaningful to adopt the assumption for distribution of electric potential as 2 33 ) are three new field variables, with two of them being determined by the boundary conditions on the bottom and top surface of the beam as in which A g and B g coefficients are specified in Table 1 according to considered either Dirichlet b.c.
If we consider a beam without free bulk electric charge and external body forces, the 1D formulation (governing equations and boundary conditions) can be derived from the variational principle where the semi-integral fields are defined as Now the governing equations (17) can be rewritten as Thus, the governing equations are given by the system of the 6 th order ordinary differential equations. Similarly, one can rewrite also the Neumann boundary conditions resulting from the boundary restrictions (18) with using the expressions given by (19). From this general formulation, one can obtain the formulations corresponding to deformation assumptions of particular beam bending theories by proper selection of two key-factors 1 c and 2 c .

Conclusions
In this paper, we presented the consistent derivation of 1D formulation for behaviour of dielectric elastic beams subject to stationary electro-mechanical loading. The derivation starts from the higher-grade continuum theory for elastic dielectrics with including flexoelectric and piezoelectric effects. The deformation assumptions of three beam bending theories are incorporated in the derived unified formulation and switching among these three theories is allowed by proper selection of two key-factors.
The financial support by the Slovak Research and Development Agency through grant SK-CN-RD-18-0005 as well as APVV-18-004 and VEGA 2/0061/20 is gratefully acknowledged.