for the force on a dielectric in an electrical field E and bounded by a surface S with unit normal n. He used an operational \"definition\" of stress and concluded that the compensating mechanical forces which must be introduced operationally are not derivable from a tensor. It is suggested here that Slepian's analysis is essentially correct and that the difficulty arises because of the choice of an operationally \"defined\" stress. This choice is inconsistent with the existence of an electrical surface stress-which is familiar, in the magnetic analogue, in studies of the form effect-and it is argued here that the Euler-Cauchy definition of stress is the appropriate one. The Definition of Stress.-In authoritative works on continuum mechanics stress is introduced by means of the stress hypothesis of Euler and Cauchy,2 that is, by asserting that, acting upon any imagined closed geometrical surface a within the body, there exists a field of stress vectors t which has an equivalent effect to the (interparticle) forces exerted by the material outside aupon the material within. For a dielectric material the interaprticle (i.e., intermolecular) forces are partly long-range in character and they may therefore contribute not only to t but also to f, the body force per unit volume. For the present purpose, however, the important point to note is that ais an imagined geometrical surface and not a physical surface of separation within the material. An alternative procedure is to use the operational definition of stress in which it is imagined that a physical cut is made in the material along an internal element of surface dd = nda. If means are then introduced for keeping the strains in the material on both sides of the cut the same as they were before the cut was made, then the force introduced by these means is t'do-, where t' is the operationally defined stress vector. In adopting this operational definition, Slepian commented: \"It is not assumed that the cut and the introduced means do not disturb the microstructure and micromechanics of the material. For example, in the case of a fluid the cut and means would cause molecules to be reflected which would otherwise pass through the geometric element of surface dS. It is assumed, however, that in spite of the change in the micromechanics, there is no change in the observable macromechanics.\"' It may also be noted there is a further element of idealization involved in that the cut is imagined to be of finite extent: in practice, as discussed later in this paper, it is only possible to measure the force on an element of volume when the element is completely separated from the rest of the body. For an ordinary elastic material the stress acting at a physical surface of separa-

The elastic dielectric.

The fundamental motivation of this research is to investigate the effect of flexoelectricity on a piezoelectric nanobeam for the first time involving internal viscoelasticity. To date, the effect of flexoelectricity on the mechanical behavior of nanobeams has been investigated extensively under various physical and environmental conditions. However, this effect as an internal property of materials has not been studied when the nanobeams include an internal damping feature. To this end, a closed-circuit condition is considered taking converse piezo–flexoelectric behavior. The kinematic displacement of the classical beam using Lagrangian strains, also applying Hamilton’s principle, creates the needed frequency equation. The natural frequencies are measured in nanoscale by the available nonlocal strain gradient elasticity model. The linear Kelvin–Voigt viscoelastic model here defines the inner viscoelastic coupling. An analytical solution technique determines the values of the numerical frequencies. The best findings show that the viscoelastic coupling can directly affect the flexoelectricity property of the material.

https://www.mdpi.com/2073-8994/12/4/643/pdf

On the Dynamics of a Visco─Piezo─Flexoelectric Nanobeam

On the piezoelectric effect on stability of symmetric FGM porous nanobeams

Nonlinear eigenfrequency characteristics of multi-directional functionally graded porous panels

Based on the Timoshenko beam theory, von Karman geometric nonlinearity assumption and the modified couple stress theory, this paper presents linear and nonlinear free vibration analysis of rotating two-dimensional functionally graded micro-beam with even and uneven porous distributions. The material properties vary along the axial and thickness directions. The NURBS function is employed as the basic one to overcome the inherent difficulty of constructing higher order continuous displacement function for modified couple stress theory based finite element formulation. The validity of the present simulation is validated by comparing numerical results with those available literatures for rotating functionally graded micro-beams. Finally, the effects of several parameters such as porosity, power law index, material length scale parameter, rotational speed and hub radius on dimensionless frequencies and nonlinear frequency ratios are discussed.

Nonlinear free vibration analysis of a rotating two-dimensional functionally graded porous micro-beam using isogeometric analysis

In this paper, based on the strain gradient theory, a size-dependent porous axially functional gradient (AFG) flexoelectric Euler-Bernoulli nanobeam model is developed. A modified power-law formula incorporating porosity volume fraction is presented to describe material properties of porous AFG nanobeam, and two typical porosity distributions patterns are considered. The generalized differential quadrature method is employed to discretize governing equations into a series of linear algebraic equations, so that the static bending deflections and free vibration frequencies are calculated conveniently for various boundary conditions. The credibility of the present mathematical formulation and the associated numerical solution are validated by comparing the results in previous literatures. Parametric studies are performed to illustrate the effects of the flexoelectricity, material length scale parameters, functionally graded index, porosity volume fraction, porosity distribution patterns on the static bending and free vibration behaviors of porous AFG flexoelectric nanobeam. The numerical results may provide support for the application of porous AFG flexoelectric nanobeam in Nano-electro-mechanical system.

Effects of porosity and flexoelectricity on static bending and free vibration of AFG piezoelectric nanobeams

This paper investigates the effect of flexoelectricity on the vibration responses of functionally graded porous piezoelectric sandwich nanobeam reinforced by graphene platelets (GPLs). The Euler–Bernoulli beam theory and the general modified strain gradient theory are employed to formulate the constitutive equations. Different distributions of porosity and GPLs dispersion patterns are considered and the Halpin–Tsai model is used to predict Young’s modulus and density of the nanobeam. The governing equations and boundary conditions are derived based on the general strain gradient theory and solved by differential quadrature method. A parametric study is accomplished to investigate the effects of flexoelectricity, size-dependence, porosity coefficient, GPLs weight fraction, porosity distributions as well as GPLs dispersion patterns on the fundamental frequency of composite nanobeam. Numerical results indicate that the porosity, GPLs and flexoelectricity can effectively influence the vibration behavior of nanobeam.

Size-dependent free vibration analysis of functionally graded porous piezoelectric sandwich nanobeam reinforced with graphene platelets with consideration of flexoelectric effect

Size-dependent constituent equations of piezoelectric bimorphs

https://iopscience.iop.org/article/10.1088/1361-665X/ab1cfc/pdf

Zongjun Li

Papers

Nonlinear free vibration analysis of a rotating two-dimensional functionally graded porous micro-beam using isogeometric analysis

Effects of porosity and flexoelectricity on static bending and free vibration of AFG piezoelectric nanobeams

Size-dependent free vibration analysis of functionally graded porous piezoelectric sandwich nanobeam reinforced with graphene platelets with consideration of flexoelectric effect

Size-dependent constituent equations of piezoelectric bimorphs

Size-dependent nonlinear bending and vibration of flexoelectric nanobeam based on strain gradient theory