Strain gradient and electric field gradient effects in piezoelectric cantilever beams

Abstract

A piezoelectric beam model with strain gradient and electric gradient effects is proposed. An energy variational principle with strain, strain gradient, electric field, and electric field gradient considered as independent variables is postulated to develop the governing equations and boundary conditions. Moreover, two strain gradient coefficients and one electric field gradient coefficient are introduced to account for higher order coupling effects of underlying microstructure. Then the bending problem of a cantilever beam is solved to illustrate the theory. It is found that the deflection of the cantilever may depend strongly on the so-introduced gradient effects. As a result of the electromechanical coupling, the generated electric field may also be affected by the strain gradient. However, electric field gradient has a limited effect on the electric field.

Gradient Extension of Classical Material Models: From Nuclear & Condensed Matter Scales to Earth & Cosmological Scales

“… ILG in Electrodynamics: The inclusion of higher-order gradients in deforming materials under the action of electromagnetic fields has also become very popular in recent years due to emerging applications and design of piezoelectric (induction of electricity due to applied pressure) and flexoelectric (induction of electricity due to strain gradients) components. The number of published articles is prohibitive to mention them here and we only refer to a couple of references by the author and coworkers [58][59][60], as well as the bibliography listed there for recent related literature on size effects.…”

A Concise Review of Gradient Models in Mechanics and Physics

“…The inclusion of higher order gradients in deforming materials under the action of electromagnetic fields has also become very popular in recent years due to emerging applications and novel design of piezoelectric (induction of electricity due to applied stress/strain) and flexoelectric (induction of electricity due to strain gradients) components. The number of written articles is prohibitive for mentioning them here, and we only refer to recent publications by the author and coworkers [35][36][37], as well as to the bibliography listed there for recent literature related to size effects in micro-/nanoelectromechanical systems.…”

Analytical solution of a plane strain pure bending problem in second gradient electroelasticity

“…Recently it was applied to the problems of fracture mechanics [46] and micromechanics [47,48]. Beam-type models in this theory were developed in [23] and with strain gradient effects only in [21,12]. Similar theory with distortion gradient and thermal effects were presented recently in [14].…”

“…Corresponding beam and plate theories were developed to explain non-classical dependence of apparent electromechanical properties of thin piezoelectric cantilevers and membranes on its size (see e.g. [2,20,21,22,23,24,25]). However, for the best of the authors knowledge, despite some experimental tests and molecular dynamics simulations [2], theoretical validation of these simplified models has not been presented yet.…”

“…cylindrical bending of a plate. For the same reason, simplified variant of constitutive equations with single additional length scale parameter is used following [47,23]. Thus, in the absence of electromechanical coupling, considered theory can be reduced to the widely known simplified strain gradient elasticity theory [49,50].…”