Nonlinear bandgap properties in a nonlocal piezoelectric phononic crystal nanobeam

Applying nonlocal elasticity theory, von Kármán type nonlinear strain-displacement relation and plane wave expansion (PWE) method to Euler-Bernoulli beam, the calculation method of band structure of a nonlinear nonlocal piezoelectric phononic crystal (PC) nanobeam is proposed and formulized. In order to investigate the properties of wave propagating in the nanobeam in detail, band gaps of first four orders are picked, and the corresponding influence rules of electro-mechanical coupling fields, nonlocal effect and geometric parameters on band gaps are studied. During the researches, external electrical voltage and axial force are chosen as the influencing parameters related to electro-mechanical coupling fields. Scale coefficient is chosen as the influencing parameter corresponding to nonlocal effect. Length ratio between materials PZT-4 and epoxy and height-width ratio are chosen as the influencing parameters of geometric parameters. Moreover, all the influence rules are compared to those in linear nanobeam. The results are expected to be of help for the design of micro and nano devices based on piezoelectric periodic nanobeam.     

Longitudinal wave propagation in a piezoelectric nanoplate considering surface effects and nonlocal elasticity theory

The propagation characteristics of the longitudinal wave in a piezoelectric nanoplate were investigated in this study. The nonlocal elasticity theory was used and the surface effects were taken into account. In addition, the group velocity and phase velocity were derived and investigated, respectively. The dispersion relation was analyzed with different scale coefficients, wavenumbers, and voltages. The results showed that the dispersion degree can be strengthened by increasing the wavenumber and scale coefficient.     

Dynamic analysis of Bernoulli-Euler piezoelectric nanobeam with electrostatic force

The effect of electrostatic force on the dynamic response of a Bernoulli-Euler piezoelectric nanobeam is analyzed in this paper.The governing equations with the electrostatic stress are derived based on a variational principle.Static bending problem of simply supported and cantilever beam is considered.The influence of the electrostatic force on the first four natural frequencies is discussed.It is shown that when the beam thickness is small,the effect of the electrostatic force is significant.When the beam thickness is large,the electrostatic force is insignificant and can be neglected.The results also indicate that one can adjust the natural frequency of a nanobeam by applying appropriate voltage.     

Excitation of surface elasticity waves in piezoelectric media by ion beams∗

Kinetically it has been shown that by considering a semi-bounded piezoelectric medium with hexagonal symmetry with an ion beam flowing on its free surface, the surface elasticity waves can be excited on their surface-vacuum interface as a result of the piezoelectric effect.     

On correct nonlocal generalized theories of elasticity

The paper discusses nonlocal elasticity theories among which are models of media with defect fields, gradient elasticity theories, and hybrid nonlocal elasticity theories. Gradient theories are analyzed, and their correctness properties are examined. Applied theories that satisfy the correctness conditions are developed, and known applied gradient theories are verified for the correctness properties. A new nonlocal generalized theory has been developed for which the operator of balance equations is represented as the product of the equilibrium operator of classical elasticity theory and the Helmholtz operator. It is shown that this theory is one-parameter and is the only representative of hybrid models constructed by a complete system of equations for forces and moments. Unlike classical elasticity that is free from scale parameters characterizing the internal material structure, nonlocal elasticity theories naturally incorporate these parameters. That is why they are suitable for the modeling of scale effects and find application in the solution of numerous applied problems for heterogeneous structures with developed phase interfaces where the degree of influence of scale effects depends on the density of phase boundaries. Nonlocal continuum models are especially attractive for modeling the properties of various micro/nanostructures, elastic properties of composites and structured materials with submicron- and nanosized internal structures in which effective properties are to a great extent defined by the scale effects (short-range interaction effects of cohesion and adhesion). Generalized elasticity theories even for isotropic materials contain many additional physical constants that are difficult or impossible to determine experimentally. Applied models with a small number of additional physical parameters are therefore of great interest. However, the reduction of nonlocal theories aimed at reducing the number of additional parameters is a nontrivial task and may lead to incorrect theories. The goal of this paper is to study the symmetry properties in gradient theories, to analyze the correctness of gradient theories, and to develop applied one-parameter elasticity theories.     

A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration

In the present study, a generalized nonlocal beam theory is proposed to study bending, buckling and free vibration of nanobeams. Nonlocal constitutive equations of Eringen are used in the formulations. After deriving governing equations, different beam theories including those of Euler–Bernoulli, Timoshenko, Reddy, Levinson and Aydogdu [Compos. Struct., 89 (2009) 94] are used as a special case in the present compact formulation without repeating derivation of governing equations each time. Effect of nonlocality and length of beams are investigated in detail for each considered problem. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes.     

Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory

The aim of this paper is to study the longitudinal frequency of a cracked nanobeam. The frequency equation of the nanobeam with clamped–clamped and clamped–free boundary conditions is derived based on the nonlocal elasticity theory. According to the equation, it can be found that the effects of the crack parameter, crack location, and nonlocal parameter on the longitudinal frequency of the cracked nanobeam are significant. The frequency decreases with an increase of the crack parameter. However, the increasing nonlocal parameter results in a decrease of the crack effect on the frequency. In addition, when the crack location is near the support, a larger decrease in the frequency can be observed.     

Effect of surface elasticity on scattering of elastic P-waves from a nanofiber including an inhomogeneous interphase

Scattering of elastic P-waves from a nanofiber in a matrix is studied analytically throughout this paper. An inhomogeneous interphase region is considered between the nanofiber and the matrix. Dividing the interphase into homogeneous sublayers, surface elasticity effects are studied in the layers adjacent to matrix and nanofiber. Wave function expansion method is used to solve the corresponding equations in all three phases including fiber, interphase, and matrix. Dynamic stress concentration factors around the nanofiber are calculated and utilizing a parametric study, effects of different parameters, such as nanoscale interface, interphase thickness, and interphase rigidity are investigated. The results indicate that considering the effects of surface elasticity in wave scattering problems from inhomogeneous interphases show a major impact on the results. The dimensionless equations presented in this paper provide the possibility of further numerical studies.     

Vibration analysis of defective graphene sheets using nonlocal elasticity theory

Many papers have studied the free vibration of graphene sheets. However, all this papers assumed their atomic structure free of any defects. Nonetheless, they actually contain some defects including single vacancy, double vacancy and Stone-Wales defects. This paper, therefore, investigates the free vibration of defective graphene sheets, rather than pristine graphene sheets, via nonlocal elasticity theory. Governing equations are derived using nonlocal elasticity and the first-order shear deformation theory (FSDT). The influence of structural defects on the vibration of graphene sheets is considered by applying the mechanical properties of defective graphene sheets. Afterwards, these equations solved using generalized differential quadrature method (GDQ). The small-scale effect is applied in the governing equations of motion by nonlocal parameter. The effects of different defect types are inspected for graphene sheets with clamped or simply-supported boundary conditions on all sides. It is shown that the natural frequencies of graphene sheets decrease by introducing defects to the atomic structure. Furthermore, it is found that the number of missing atoms, shapes and distributions of structural defects play a significant role in the vibrational behavior of graphene. The effect of vacancy defect reconstruction is also discussed in this paper.     

Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions

This paper investigates the thermo-electro-mechanical vibration of the rectangular piezoelectric nanoplate under various boundary conditions based on the nonlocal theory and the Mindlin plate theory. It is assumed that the piezoelectric nanoplate is subjected to a biaxial force, an external electric voltage and a uniform temperature rise. The Hamilton's principle is employed to derive the governing equations and boundary conditions, which are then discretized by using the differential quadrature (DQ) method to determine the natural frequencies and mode shapes. The detailed parametric study is conducted to examine the effect of the nonlocal parameter, thermo-electro-mechanical loadings, boundary conditions, aspect ratio and side-to-thickness ratio on the vibration behaviors.     

Flexural sensitivity and resonance of cantilever micro-sensors based on nonlocal elasticity theory

Sensors based on microcantilevers, especially ones with uniform structure, have ultrahigh sensitivities. The normalized natural frequencies and the sensitivity of lateral vibration of an elastic microcantilever sensor in contact with a surface are derived analytically based on the Euler–Bernoulli beam theory by taking into account the small scale effect. The interaction of the sensor with the surface is modeled by linear springs, which restricts the results to experiments involving low-amplitude excitations. The results show that the normalized natural frequencies of nonlocal microcantilever are smaller than those for its local counterpart, especially for higher values of small scale parameters. Also, each mode has a different sensitivity to variations in surface stiffness. Moreover, the most sensitivity is observed at the first mode of vibration. When the nonlocal effect is not taken into account, the natural frequencies and the sensitivity of the microcantilever in contact with the surface are compared with those obtained in previous study without considering the nonlocal effect.     

Bending analysis of a functionally graded piezoelectric cantilever beam

A new analysis based on Airy stress function method is presented for a functionally graded piezoelectric material cantilever beam. Assuming that the mechanical and electric properties of the material have the same variations along the thickness direction, a two-dimensional plane elasticity solution is obtained for the coupling electroelastic fields of the beam under different loadings. This solution will be useful in analyzing FGPM beam with arbitrary variations of material properties. The influences of the functionally graded material properties on the structural response of the beam subjected to different loads are also studied through numerical examples.     

Hyperelasticity, viscoelasticity, and nonlocal elasticity govern dynamic fracture in rubber

Dynamic cracks in rubber can spontaneously oscillate under certain biaxial strain conditions [R. D. Deegan et al., Phys. Rev. Lett. 88, 014304 (2002)]. We have found that this unusual phenomenon can be understood from the unique mechanical properties of rubber: hyperelasticity, viscoelasticity, and nonlocal elasticity. While all these are important, the decisive role of nonlocality needs to be particularly emphasized. Through numerical simulations with a lattice model, we have quantitatively reproduced the experimental results.     

Gradient expansion of the kinetic energy including nonlocal exchange contributions

We discuss the gradient expansion of the kinetic energy density, explicitly including nonlocal exchange contributions to ordere ². Restricting the expansion to orderh ²2 one finds that the expression obtained in the standard TFDW model is unchanged.