Size-dependent piezoelectricity

Abstract

In this paper, a consistent theory is developed for size-dependent piezoelectricity in dielectric solids. This theory shows that electric polarization can be generated as the result of coupling to the mean curvature tensor, unlike previous flexoelectric theories that postulate such couplings with other forms of curvature and more general strain gradient terms ignoring the possible couple-stresses. The present formulation represents an extension of recent work that establishes a consistent size-dependent theory for solid mechanics. Here by including scale-dependent measures in the energy equation, the general expressions for force-and couple-stresses, as well as electric displacement, are obtained.Next, the constitutive relations, displacement formulations, the uniqueness theorem and the reciprocal theorem for the corresponding linear small deformation size-dependent piezoelectricity are developed. As with existing flexoelectric formulations, one finds that the piezoelectric effect can also exist in isotropic materials, although in the present theory the coupling is strictly through the skew-symmetric mean curvature tensor.In the last portion of the paper, this isotropic case is considered in detail by developing the corresponding boundary value problem for two dimensional analyses and obtaining a closed form solution for an isotropic dielectric cylinder.

Boundary element formulation for plane problems in size-dependent piezoelectricity

“…In this section, we provide a brief overview of the two‐dimensional isotropic size‐dependent piezoelectricity theory. For a more detailed discussion on this consistent theory, the reader is referred to Reference .…”

“…This means that the flexoelectric coupling is between polarization and the curl of the rotation field. The detail of this theory shows that the size‐dependent piezoelectricity or flexoelectric effect may exist even in centrosymmetric dielectric materials . Interestingly, for isotropic and centrosymmetric cubic materials, there is only one material couple‐stress length scale and one flexoelectric parameter, which account for the size‐dependent piezoelectricity effect.…”

A Timoshenko dielectric beam model with flexoelectric effect

“…It should note that the firth and sixth order terms are neglected for simplicity as was done in Refs. [3,22], Hadjesfandiari [18] pointed out that if the effect of couple-stress is negligible, and the effect of flexoelectricity must be excluded. The effects of flexoelectricity on the electromechanical coupling behavior of Bernoulli-Euler piezoelectric beam were determined previously, it is shown that the effect of non-local elasticity can be neglected for piezoelectric materials with characteristic length scale at nanometer [11].…”

“…Hu and Shen [17] developed a theory for nano dielectrics with the effects of strain/electric field gradients as well as the surface, and Shen and Hu [7] established a theory for elastic dielectrics with the effects of flexoelectricity and surface. Very recently, Hadjesfandiari [18,19] developed a size-dependent piezoelectricity theory in which the symmetric part of force-stress and skew-symmetric part of the couple stress were introduced, and then a 2D finite element formulation was developed to consider the coupling between the electric field and the mean curvature [20]. A three-layer laminated piezoelectric beam model was also proposed based on the size-dependent piezoelectricity theory [21].…”

Size-dependent thermoelasticity

“…Therefore, this development can be extended quite naturally into many branches of continuum mechanics involving different multi-physics disciplines. Hadjesfandiari (2013) has already developed a size-dependent piezoelectricity for dielectric materials. Here we concentrate to develop the coupled sizedependent thermoelasticity for solids.…”