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This research investigates the free vibration of a rotating annular microplate under the flexoelectric effect. Initially, the Kirchhoff plate theory assumptions are used to express the displacement fields. After considering the displacement field, strains and their gradients are derived and substituted into the electric enthalpy and kinetic energy expressions. Subsequently, by applying Hamilton’s principle to the aforementioned equations, the electric and mechanical equations are computed. To derive the equations of motion, initially, the polarization vectors and their gradients are derived from the electric equations and associated boundary conditions. Subsequently, these are incorporated into the mechanical equations, which also encompass electric components. It is notable that by removing the time-dependent terms from the in-plane equations of motion, the static displacement due to rotation at each speed is obtained. After deriving the equations of motion and boundary conditions, these equations are non-dimensionalized using non-dimensionalizing relations. In the next step, Hamilton’s principle is used to discretize the equations and boundary conditions. Consequently, by applying the generalized differential quadrature method and extracting the stiffness and mass matrices resulting from the transverse equation of motion and boundary conditions, the natural transverse frequency of the rotating annular microplate under the flexoelectric effect is calculated. The results of this research are useful for promoting the use of rotating annular microplants under flexoelectric effect for microelectromechanical systems designers with high efficiency.

Flexoelectricity describes the coupling between polarization and strain gradients and presents a strong size dependence at nanoscale. In the current work, based on the extended linear piezoelectricity theory with flexoelectricity, we study bending of piezoelectric beams with consideration of flexoelectric effect. When a concentrated force at any position and electric voltage is exerted, the expression for bending deflection of simply-supported and clamped beams is derived. The obtained results show that flexoelectric effect can cause a softer elastic behavior of simply-supported and clamped beams. Sensitivity analysis of the transverse deflection and bending moment is made for two typical boundary conditions. Flexoelectric effect has a more significant effect on the bending response of a piezoelectric beam with smaller thickness.

This paper presents the snap-through and bifurcation elastic stability analysis of nano-arch type structures with the Winkler foundation under transverse loadings by the strain gradient and stress gradient (nonlocal) theories. The equations of equilibrium are derived by using the variational method and virtual displacement theorem of minimum total potential energy. In the elastic stability analysis, von Karman's nonlinear strain component is included, with the deformation represented by a series solution. It is concluded that in general, the strain gradient theory pushes the system away from instability as compared to the classical theory. However, the nonlocal theory does the reverse and causes the system to experience instability earlier than that of the classical theory. Moreover, theories with different small-size considerations change the mechanism of instability in different ways. For example, in similar conditions, the strain gradient theory causes the system to reach a snap-through point, while the nonlocal theory causes the system to stop at a bifurcation critical point.

Dynamic analysis of functionally graded size-dependent annular nano-plate is the main concern in this study. To obtain the vibrational behavior of this plate, the stress-driven nonlocal integral elasticity, as well the strain gradient theory were used in conjunction with the classical plate theory. The resulting equilibrium equations were solved using the generalized differential quadrature rule (GDQR) and the influences of various parameters such as; size-effect parameter, material heterogeneity index, the aspect ratio of the inner to outer radii, and the effects of different boundary conditions were investigated on the vibrational behavior of the nano-plate, based on different types of boundary conditions. Results indicate that the natural frequencies increase with an increase in the heterogeneity index $n$ and the increase in size-effect parameter shows a similar effect in both models. Additionally, for the simply supported and free-edge boundary conditions (for both edges), as well as the free and knife-edges, and simply supported-free edges, the strain gradient theory predicts higher values of frequency ratios as $L_c$ was increased. Similar results were obtained for the remaining types of boundary conditions, with a higher sensitivity to $L_c$, provided the stress-driven model is used. This behavior can be interpreted as the sensitivity of the nano-plate to $L_c$ that is manifested by the use of the stress-driven model for the prediction of vibrational behavior of the nano-plate.

A nonlocal Kirchhoff plate model with the van der Waals (vdW) interactions taken into consideration is developed to study the vibration of double-layered graphene sheets (DLGS). The dynamic equations of multi-layered Kirchhoff plate are derived based on strain gradient elasticity. An explicit formula is derived to predict the natural frequency of the DLGS with all edges simply supported. Then a 4-node 24-degree of freedom (DOF) Kirchhoff plate element is developed to discretize the higher order partial differential equations with the small scale effect taken into consideration by the theory of virtual work. It can be directly used to predict the scale effect on the vibrational DLGS with different boundary conditions. A good agreement between finite element method (FEM) results and theoretical natural frequencies of the vibration simply supported double-layered graphene sheet (DLGS) validates the reliability of the FEM. Finally, this new FEM is used to investigate the effect of vdW coefficients, sizes, nonlocal parameters, vibration mode and boundary conditions on the vibration behaviors of DLGS.

A size-dependent nonclassical Bernoulli–Euler beam model based on the strain gradient elasticity is proposed for piezoelectric nanowires. The governing equations and the corresponding boundary conditions are naturally derived from the variational principle. Different from the classical piezoelectric beam theory, the electric field–strain gradient coupling and the strain gradient elasticity are both taken into account. Static bending problem of a cantilever piezoelectric nanobeam is solved to illustrate the effect of strain gradient. The present model contains material length scale parameters and can capture the size dependent piezoelectricity and elasticity for nanoscale piezoelectric structures. The numerical results reveal that the deflections predicted by the present model are smaller than that by the classical beam theory and the effective electromechanical coupling coefficient is dramatic enhanced by the electric field–strain gradient coupling effect. However, the differences in both the deflections and effective EMC coefficient between the two models are very significant when the beam thickness is very small; they are diminishing with the increase of the beam thickness. This model is helpful for understanding the electromechanically coupling mechanism and in designing piezoelectric nanowires based devices.

Different mechanisms such as glide of lattice dislocation and grain boundary mediated processes may be active during the plastic deformation of polycrystals with small grain size. A continuum model for polycrystal plasticity has been developed to capture such a feature. Specifically, the strain gradient effect due to dislocation pile-up, dislocation emission/absorption at surface and grain boundary sliding have been taken into account from the perspective of energy storage. As an application of the model, a bicrystal under plane constrained shear has been considered. The dependence of yield strength on the thickness of the bicrystal has been investigated. It has been demonstrated the present model predicts the yield strength of the bicrystal first increases and then decrease with decrease in the thickness, which is similar to the inverse Hall–Petch behavior but with a different scaling. Such behavior is attributed to the transition in different dominant deformation mechanisms at different values of the thickness.

The elastic-plastic deformation of rotating functionally graded (FG) cylinders is investigated based on strain gradient theory. The governing equations are obtained based on the modified von Mises yield criterion, linear work hardening and plane strain assumptions. An analytical solution for the obtained equations is presented by which the deformation, strain and stress components for any point of the cylinder can be obtained. After verification of the formulation by comparing the obtained results with the reported results in the literature, some studies are presented to investigate the effects of cylinder size on the stress distribution and elastic-plastic interface radius of the rotating FG cylinder under internal and external pressure. The effects of the strain gradient coefficient, angular velocity, and the heterogeneity constant of the material are investigated. The results show that increasing the heterogeneity constant of the material and decreasing the cylinder radius lead to increasing the strength of material and decreasing the elastic-plastic interface radius. Moreover, classical theory is compared with this study and the range of the sizes in which both the theories leading to the same results, are defined.

A new functionally graded non-classical Timoshenko microbeam model is developed by using a variational formulation. The new model incorporates strain gradient, couple stress (rotation gradient) and velocity gradient effects and explains a power-law variation of two-phase materials through the thickness direction. The new model contains three additional material constants to account for the relative three gradient effects. The static bending, buckling and free vibration problems of the newly developed functionally graded material (FGM) beam are then analytically solved. Numerical results show that the differences between the prediction results of the current model and the classic model are significant when FGM beam thickness is very small, but they are diminishing as the thickness increases.

The ingredient devices tend to be designed and fabricated with microscale, high accuracy, and complex geometry and are subjected to complex loading. Many experiments have proved that the size effect plays a significant role in designing and manufacturing an engineering component. This size effect is largely attributed to the grain size and the strain gradient. While the grain size effect was omitted in conventional strain gradient theories, an extended model that considers both effects of grain size and strain gradient has been proposed in the current work (denoted as GMSG). The proposed GMSG model has been implemented into the finite element method (denoted as GMSG-FEM) to investigate the size effect of copper wires under complex working conditions. The simulation results show that the hollow structure can improve the bearing capacity of the micro-wires under torsion. Among micro-wires with different cross-sections, the bearing capacity of the micro-wire with a circular section is the largest, followed by those with square and triangle sections. Complex loading also has an important influence on stress distribution. Based on the current study, it can be envisaged that the GMSG-FEM could be a useful tool in engineering applications where the size effect has to be considered.

Flexoelectric effect and its influence on the application of multifunctional ferroelectrics have been investigated. Theory of flexoelectric coupling has indicated that mechanical strain gradient can impact polarization in a way analogous to electric field. Experimentally, magnitudes of the flexoelectric coefficients have been measured in ferroelectric, incipient ferroelectric and relaxor ferroelectric perovskites. Present data of flexoelectricity suggests that such unconventional electromechanical coupling could make unique contribution to properly engineered ferroelectric thin films and nanostructures. Flexoelectric effect is expected to intensify at small dimensions and get large enough at nanoscale to significantly impact phase transition and functional response in ferroelectrics.

It has been established that in plates made of hot-pressed ferroelectric ceramics PZT-19 ($T_c=300^{\circ }$C), after metal electrodes from Ag are burned into opposite surfaces with different depths of mechanical damage, counter-directional stationary deformation gradients and, accordingly, internal electric displacement fields create a stationary flexoelectric effect. As a result, unpolarized ceramics below the Curie temperature become a pyroelectric, and above it become a pyroelectric bolometer. After polishing the less damaged surface and replacing the Ag (valency $v=3$) electrode with Cr ($v=6$) by thermal evaporation in vacuum from the intact surface, a new thermally stable polarization is created, directed from the damaged surface in the interelectrode volume. Such plates have an asymmetric dielectric hysteresis loop, retain the values of the pyroelectric effect and piezoelectric effect in the ferroelectric phase, as well as the bolometric effect in the paraelectric phase after repeated heating cycles to $400^{\circ }$C and cooling to $T_{\mathrm{\text{norm}}}$.

Flexoelectric effect describes the electromechanical coupling between the strain gradient and its internal polarization in all dielectrics. Despite this universality, the resulting flexoelectric field remains small at the macroscopic level. However, in nanosystems, the size-dependent effect of flexoelectricity becomes increasingly significant, leading to a notable flexoelectric field that can strongly influence the material’s physical properties. This review aims to explore the flexoelectric effect specifically at the nanoscale. We achieve this by examining strain gradients generated through two distinct methods: internal inhomogeneous strain and external stimulation. In addition, advanced synthesis techniques are utilized to enhance the properties and functionalities associated with flexoelectricity. Furthermore, we delve into other coupled phenomena observed in thin films, including the coupling and utilization of flexomagnetic and flexophotovoltaic effects. This review presents the latest advancements in these areas and highlights their role in driving further breakthroughs in the field of flexoelectricity.

Hill's lemma for the Cauchy continuum has been playing an important role in micromechanics. An extended version of Hill's lemma for non-Cauchy continua is formulated using the simplified strain gradient elasticity theory (SSGET), which contains only one material length scale parameter and can account for the microstructure-dependent strain gradient effect. As a corollary of the extended Hill's lemma, the Hill–Mandel macro-homogeneity condition for non-Cauchy continua is obtained along with the general forms of kinematically and statically admissible boundary conditions that are required for constructing an energetically equivalent homogeneous comparison material. Based on these general forms, four sets of uniform boundary conditions are identified, which are implementable in material tests and can be directly used in homogenization analyses of heterogeneous materials. It is shown that when the strain gradient effect is suppressed, the extended Hill's lemma recovers the classical Hill's lemma for the Cauchy continuum and the extended Hill–Mandel condition reduces to its classical counterpart.

In this paper, a mechanism-based strain gradient dependent constitutive equation for two-phase particle reinforced metal matrix composites is presented. By using this strain gradient dependent constitutive equation and the linear perturbation analysis, the effect of strain gradient on adiabatic shear banding in particle reinforced metal matrix composites is investigated. The results have demonstrated that the onset of adiabatic shear banding in the composite with small particles is more prone to occur than in the composite with large particles. This result also means that high strain gradient is a strong driving force for adiabatic shear banding in metal matrix composites.