Cantilever Nanobeam Research Articles

Actively tunable laser action in GeSn nanomechanical oscillators.

Mechanical forces induced by high-speed oscillations provide an elegant way to dynamically alter the fundamental properties of materials such as refractive index, absorption coefficient and gain dynamics. Although the precise control of mechanical oscillation has been well developed in the past decades, the notion of dynamic mechanical forces has not been harnessed for developing tunable lasers. Here we demonstrate actively tunable mid-infrared laser action in group-IV nanomechanical oscillators with a compact form factor. A suspended GeSn cantilever nanobeam on a Si substrate is resonantly driven by radio-frequency waves. Electrically controlled mechanical oscillation induces elastic strain that periodically varies with time in the GeSn nanobeam, enabling actively tunable lasing emission at >2 μm wavelengths. By utilizing mechanical resonances in the radio frequency as a driving mechanism, this work presents wide-range mid-infrared tunable lasers with ultralow tuning power consumption.

In-plane bending vibration of L-shaped cantilever nanobeams carrying a tip nanoparticle by nonlocal elasticity

In-plane bending vibration of L-shaped cantilever nanobeams carrying a tip nanoparticle by nonlocal elasticity

Application of Surface Stress-Driven Model for Higher Vibration Modes of Functionally Graded Nanobeams.

This paper employs a surface stress-driven nonlocal theory to investigate the synergistic impact of long-range interaction and surface energy on higher vibration modes of Bernoulli-Euler nanobeams made of functionally graded material. It takes into account surface effects such as the surface modulus of elasticity, residual surface stresses, surface density, and rotary inertia. The governing equation is derived through the application of Hamilton's principle. The novelty of this work lies in its pioneering approach to studying higher-order vibrations, carefully considering the combination of long-range interactions and surface energy in nanobeams of functionally graded materials through a well-posed mathematical model of nonlocal elasticity. This study conducts a parametric investigation, examining the effects of the nonlocal parameter and the material gradient index for four static schemes: Cantilever, Simply-Supported, Clamped-Pinned and Clamped-Clamped nanobeams. The outcomes are presented and discussed, highlighting the normalized nonlocal natural frequencies for the second through fifth modes of vibration in each case under study. In particular, this study illustrates the central role of surface effects in the dynamic response of nanobeams, emphasizing the importance of considering them. Furthermore, the parametric analysis reveals that the dynamic response is influenced by the combined effects of the nonlocal parameter, the material gradient index, the shapes of the cross-sections considered, as well as the static scheme analyzed.

Nonlocal boundaries and paradoxes in thermoelastic vibrations of functionally graded Non-Uniform cantilever nanobeams and annular nanoplates

The paradoxical behaviour of free vibrations of cantilever nanostructures using Eringen's nonlocal elasticity model is always a debatable issue for researchers in the past few years. According to this, the frequency of cantilever nanobeams increases with the increase in the value of the nonlocal parameter while it decreases for other boundary conditions. In thermoelastic vibrations, the frequency of nanobeams increases and decreases with the increase in temperature at the surfaces for clamped-free and other boundary conditions, respectively. Therefore, investigation of these structures is of critical importance. In view of this, the authors have proposed the modified boundary conditions for the well-established differential model of Eringen’s nonlocal elasticity theory to deal with the free edges of the structural components. Using these conditions, the vibration analysis of nonuniform functionally graded nanobeams and axisymmetric annular nanoplates is presented for clamped-free boundary conditions on the basis of first-order shear deformation theory and physical neutral plane. The thickness of these structural elements (nanobeams and annular nanoplates) is assumed to be varying linearly and parabolically in the axial direction. The temperature-dependent mechanical properties of the material are obtained by the power law. The nonlocal governing equations and modified boundary conditions for the structural elements are discretized using the generalized differential quadrature method and solved by MATLAB to obtain the fundamental frequency. For the present modifications, the behaviour of frequency parameter for different values of nonlocal parameter, volume fraction index, temperature difference, thickness parameter, and taper parameter is discussed.

The electro-mechanical coupling responses of functionally graded piezoelectric nanobeams with flexoelectric effect

The aim of this paper is to study the electro-mechanical coupling responses of functionally graded (FG) piezoelectric cantilever nanobeam under the concentrated load, and the material properties follow an exponential distribution in the thickness direction. The constitutive equations are derived from the electric Gibbs free energy density function. The governing equations and boundary conditions of the Euler–Bernoulli FG beam can be obtained by variational principle, and then, the analytical expressions for its deflection, polarization intensity, and induced potential are obtained. The results show that the flexoelectric effect has a significant impact on the deflection, polarization intensity, and induced potential of the FG cantilever nanobeam.

Interaction of the fundamental frequencies of a torsional cantilever nanobeam and spring mass system single degree of freedom (SDOF) under axial load, including buckling

In this study we present the interactions of the fundamental frequencies of a nanomanufacturing coupled system by exploring the natural frequencies of the subsystems. These nanomanufacturing subsystems function in concert, e.g., a cantilever beam with spring-mass. The individual subsystems are studied under free vibration to generate the natural and buckling frequencies. These subsystems, when under free vibration, generate unique local natural frequencies that interact to form a unique global natural frequency. This allows for greater control and improved sensitivity for scanning and shaping nano surfaces, by allowing selective variation of the local frequency of one system to influence the global system frequencies. In this investigation, a nanobeam with arbitrary boundary conditions is used to model the system and the effects on the parameters of interest are studied. Euler–Bernoulli theory is applied in conjunction with Eringen’s theory of nonlocal continuum theory to model the small-scale effects due to the size of the beam under consideration. The coupled equations are solved using separation of variables for the local and global frequencies. The nanobeam is restrained with an adjustable torsional spring and pin at one end. The boundary condition at the free end is a spring-mass system with axial load. Altering the torsional, transverse spring stiffness and mass increases or decreases the natural frequencies. The motions of the beam and the tip-mass generates a frequency response during contact interactions. The tip response frequency is used to determine the maximum displacements (penetration depth) and accelerations (contact forces) in a sample during nanomanufacturing.

Nonlocal large deflection analysis of a cantilever nanobeam on a nonlinear Winkler-Pasternak elastic foundation and under uniformly distributed lateral load

Nonlocal large deflection analysis of a cantilever nanobeam on a nonlinear Winkler-Pasternak elastic foundation and under uniformly distributed lateral load

Nonlocal analysis of edge‐cracked nanobeams under Mode I and Mixed‐Mode (I + II) static loading

AbstractIn the present paper, the behavior of an edge‐cracked nanobeam under both Mode I and Mixed‐Mode (I + II) static bending loading is analyzed by using the stress‐driven nonlocal model (SDM), being the SDM applied for the first time in the case of Mixed‐Mode loading. More precisely, the cracked nanobeam is modeled by using a modification of the classical cracked‐beam theory, consisting in dividing the beam into two beam segments connected through a massless elastic rotational spring located at the cracked cross section. The bending stiffness of the cracked cross section is computed by exploiting: the Griffith energy criterion and the conventional linear elastic fracture mechanics. Cracks under both Mode I and Mixed‐Mode (I + II) loading are considered. Finally, an edge‐cracked cantilever nanobeam subjected to a transversal point load at the free end is analyzed. The static deflection is calculated and discussed for different values of relative crack depth, crack location, and crack orientation.

On the Static Flexural Response of Shear Deformable Isotropic Rectangular Nanobeams under the Parabolic Loading

It is well known that size-dependent effects, which are insignificant for macroscopic structures, become significant for small-scale structures like nanobeams. In addition, effects of the beam transverse shear deformation, which are insignificant for slender beams, become significant for shear deformable beams. Hence the need arises for a beam theory which is simple-to-use as well as which accounts for just-mentioned features of shear deformable nanobeams. Recently, the authors have developed a single variable new first-order shear deformation nonlocal beam theory (NFSDNBT, doi: 10.1007/s40430-019-2128-6). The NFSDNBT is applicable for the flexure of linear isotropic nanobeams undergoing small deformations. Displacement functions of the NFSDNBT give rise to the constant transverse shear strain through the beam thickness. Hence similar to the nonlocal Timoshenko beam theory (NTBT), the NFSDNBT also requires a shear correction factor. In the NFSDNBT, nonlocal differential stress-strain constitutive relations of Eringen are utilized, through which size-dependent effects have been taken into account. These relations relate not only the beam axial stress with the beam axial strain but also the beam transverse shear stress with the beam transverse shear strain. The governing equation of the NFSDNBT is obtained by utilizing beam gross equilibrium equations and it has a strong resemblance with the governing equation of the nonlocal Bernoulli-Euler beam theory (NBEBT). In this paper, the NFSDNBT is utilized for finding the static flexural response of shear deformable isotropic rectangular nanobeams under the action of parabolically distributed transverse loading. For the simply-supported, clamped-clamped and cantilever nanobeams, effects of variations in values of the nonlocal parameter of Eringen and beam thickness-to-length ratio on the maximum non-dimensional beam transverse displacement are presented. Obtained results are compared with corresponding results obtained by utilizing the NTBT and NBEBT so as to demonstrate the efficacy of the NFSDNBT. Along-the-length profiles of the non-dimensional beam transverse displacement for the just-mentioned cases of nanobeams, for various values of the nonlocal parameter and beam thickness-to-length ratio are also presented.

The Sub and Super-Tangential Nonconservative Load in Stability Problem of Nanobeams with Sprung Masses

In this study, the critical load and natural vibration frequency of Euler–Bernoulli single nanobeams based on Eringen’s nonlocal elasticity theory are investigated. Cantilever nanobeams with attached sprung masses were subjected to compressed concentrated and distributed follower forces. The parameter that determines the direction of nonconservative follower forces was given the positive and negative values, therefore, sub-tangential and super-tangential load were analyzed. The stability analysis is based on dynamical stability criterion and was carried out using a numerical algorithm for solving segmental nanobeams with many boundary conditions. The presented algorithm is based on the exact solutions of motion equations which are derived from equilibrium conditions for each separated segment of the nanobeam. Two comparison studies are conducted to ensure the validity and accuracy of the presented algorithm. The excellent agreement of critical load for Beck’s nano-column on Winkler foundation observed was confirmed as reported by other researchers. The effect of different values of the nonlocality parameter, tangency coefficient, spring stiffness coefficient, location of sprung mass and the greater number of attached sprung masses on a critical load of nanobeams compressed by nonconservative load are discussed. One of the presented results shows that significant differences between local and nonlocal theory appear when the beam subjected to follower forces loses its stability by flutter.

Vibration analysis of the rotating multilayer piezoelectric Timoshenko nanobeam

The vibration behavior of multilayers piezoelectric nanobeam is investigated in this study. The governing equations of rotating piezoelectric beams are obtained by utilizing the Timoshenko beam theory. The nonlocal continuum theory along with surface elasticity theory are handled to derive the differential governing equation of motion. The differential governing equation of motions is solved by employing the DQM. The results show a great correlation between the numerical results and the presented results in the literature. The results are obtained for four different boundary conditions such as cantilever nanobeam (clamped-free or CF), simply-simply (SS), clamped-simply (CS), and clamped-clamped (CC). The reaction of vibration frequencies to the change of different parameters such as the external voltage, the thermal stress, the angular velocity, etc. is investigated. The results show that the boundary condition type, thermal stress, and external voltage have a vital consequence on the vibration frequency. The results of this study can be appropriated to design and fabricated the nanodevices such as nanomotors, nano turbines, biosensors, nano-electromechanical systems, and micro-electromechanical systems (NEMS/MEMS) devices.

Review of Kerogen’s Geomechanical Properties: Experiments and Molecular Simulation

Measuring the mechanical properties of kerogen, the predominant constituent of organic matter in shale is exceedingly difficult as it constitutes small-scale aggregates interspersed in rocks. Kerogen is characterized by significantly lower stiffness compared to inorganic minerals, thereby the kerogen regions are potential areas for study during, for example, drilling or macroscopic fracture propagation in the course of hydraulic fracturing. For instance, the elastic modulus of kerogen-rich spots is around 10 GPa, while it is about 70 GPa for quartz. Failure of the kerogen nanocantilever beam shows an elastic strain-hardening behavior, indicating a higher energy requirement to propagate a crack. Studies illustrated that the kerogen’s mechanical properties are controlled by maceral composition and are positively correlated to the maturity level. This paper provides a comprehensive review of how the mechanical properties of kerogen are elucidated experimentally and contrast the results with the properties delineated from molecular simulation. In addition, we relate kerogen innate attributes, such as maturity and type, to the physical qualities measured and substantiate why accurate knowledge of the mechanical characteristics is pivotal from a hydraulic fracturing perspective.

Transverse free vibration of nanobeams with intermediate support using nonlocal strain gradient theory

In the present study, the transverse free vibration of intermediately supported nanobeams is investigated in the framework of size-dependent nonlocal strain gradient theory. Unlike the classical elasticity theory, nonlocal strain gradient theory considers the micro/nano scale effects in mechanical analysis of size-dependent structures. The potential and kinetic energies have been derived for the nanobeam model and the Ritz method has been used to determine the natural frequencies of intermediately supported nanobeams. A cantilever nanobeam model with intermediate support is considered in the analysis. By changing the position of the intermediate support, dimensionless vibration frequencies of the nanobeam model have been obtained. Obtained results from the present nonlocal strain gradient theory showed that hardening or softening material responses have been observed according to the classical elasticity theory depending on the magnitude of the material length scale parameter and nonlocal scale parameter. This mechanical behavior can provide an advantage to designers while modeling the micro/nanostructures. Present results can be used for the design of nano-sensors, nanotube-based resonators, and oscillators.

Bending analysis of nanobeams based on the integral form of nonlocal elasticity using the numerical Rayleigh-Ritz technique

The nonlocal theory is commonly applied for nanomaterials due to its capability in considering size influences. Available studies have shown that the differential version of this theory is not suitable for some problems such as bending of cantilever nanobeams, and the integral version must be used to avoid obtaining inconsistent results. Therefore, an attempt is made in this paper to propose an efficient variational formulation based on the integral nonlocal model for the analysis of nanobeams. The formulation is developed in a general form so that it can be used for arbitrary kernel functions. The nanobeams are modeled using the Bernoulli-Euler beam theory, and their bending behavior is analyzed. Derivation of governing equations is performed according to an energy-based approach. Also, a numerical approach based on the Rayleigh-Ritz method is developed for the solution of problem. Moreover, the results of integral and differential models are compared. It is revealed that by the proposed numerical solution, the paradox in the behavior of nanocantilever is resolved.

Bending Analysis of Functionally Graded Nanobeam Using Chebyshev Pseudospectral Method

Static performance of functionally graded cantilever nanobeams exposed to lateral and axial loads from the end was examined by applying the Pseudospectral Chebyshev Method. A solution is given for bending analysis using Euler-Bernoulli beam theory. The nonlocal elasticity theory was first introduced by Eringen and is used to represent effect on a small scale. Using the aforementioned theory, the governing differential equations the phenomenon for functionally graded nanobeams are reproduced. It is supposed that the modulus of elasticity of the beam changes exponentially in the x-axis direction, except for the values taken as constant. The exponential change of material properties may not allow analytical problems to be solved with known methods. Therefore, numerical approach is inevitable for the solution of the problem.

Stability of nanobeams and nanoplates with defects

The sensitivity of critical buckling load and critical stress concerning different geometrical and physical parameters of Euler-Bernoulli nanobeams with defects is studied. Eringen’s nonlocal theory of elasticity is used for the determination of critical buckling load for stepped nanobeams subjected to axial loads for different support conditions. An analytical approach to study the impact of discontinuities and boundary conditions on the critical buckling load and critical stress of nanobeams has been developed. Critical buckling loads of stepped nanobeams are defined under the condition that the nanoelements are weakened with stable crack-like defects. Simply supported, clamped and cantilever nanobeams with steps and cracks are investigated in this article. The presented results are compared with the other available results and are found to be in a close agreement.

Fracture behaviour of nanobeams through Two-Phase Local/Nonlocal Stress-Driven model

The aim of the present work is to employ the two-phase local/nonlocal Stress-Driven integral Model (SDM) to analyse the size-dependent Mode I fracture behaviour of Bernoulli-Euler cracked nanobeams, in terms of energy release rate, stress intensity factor and nonlocal stress field near the crack tip. Both edge- and centrally-cracked nanobeams, subjected to concentrated forces, are examined. The edge-cracked nanobeam is modelled as a pair of cantilever nanobeams, whereas the centrally-cracked nanobeam as a pair of double-clamped nanobeams with internal discontinuity due to concentrated loads. Moreover, a comparison with the results obtained from a gradient elasticity theory based model, available in the literature, is performed. From the present study, it is observed that the energy release rate decreases by increasing the nonlocality, showing the superior fracture performance of nanobeams with respect to large-scale beams.

Effect of memory dependent derivative and variable thermal conductivity in cantilever nano-Beam with forced transverse vibrations

This research work studies the effect of variable thermal conductivity on transversely isotropic thermoelastic (TIT) homogeneous nonlocal clamped-free beam. The beam undergoes forced vibrations due to two types of heat sources i.e. exponential time-varying and time-harmonic load. The mathematical model has been formed for a 2-D problem using a heat equation with two temperature (2T) involving memory-dependent derivatives (MDD) and Euler Bernoulli beam theory (EBBT). Laplace transforms are used to solve the problem. The dimensionless expressions for a thermal moment, axial stress, lateral deflection, and temperature change are calculated for these two types of forced vibrations. MATLAB software is used for programming and numerical data for cobalt nanobeam has been considered. The influence of variable thermal conductivity and kernel function of MDD has been depicted graphically on the physical quantities such as temperature change, lateral deflection, axial stress thermal moment, for the forced vibrations. Specific cases have also been mentioned.

Coupling Analysis of Flexoelectric Effect on Functionally Graded Piezoelectric Cantilever Nanobeams.

The flexoelectric effect has a significant influence on the electro-mechanical coupling of micro-nano devices. This paper studies the mechanical and electrical properties of functionally graded flexo-piezoelectric beams under different electrical boundary conditions. The generalized variational principle and Euler–Bernoulli beam theory are employed to deduce the governing equations and corresponding electro-mechanical boundary conditions of the beam model. The deflection and induced electric potential are given as analytical expressions for the functionally graded cantilever beam. The numerical results show that the flexoelectric effect, piezoelectric effect, and gradient distribution have considerable influences on the electro-mechanical performance of the functionally graded beams. Moreover, the nonuniform piezoelectricity and polarization direction will play a leading role in the induced electric potential at a large scale. The flexoelectric effect will dominate the induced electric potential as the beam thickness decreases. This work provides helpful guidance to resolve the application of flexoelectric and piezoelectric effects in functionally graded materials, especially on micro-nano devices.

Buckling analysis of cantilever nanobeams with defects

Cantilever nanobeams subjected to the axial pressure are considered under the assumption that the material behaviour can be modelled with the help of the nonlocal theory of elasticity. The nanobeams and nanocolumns under consideration have rectangular cross section with piecewise constant dimensions and are weakened by crack-like defects at the re-entrant corners of the steps. The critical buckling loads of stepped nanobeams are calculated and the sensitivity of critical loads with respect to crack parameters is studied. Results show the significant effect of crack parameters on the stability of stepped nanobeams. The numerical results are presented with the aid of MATLAB tools and are found to be in a close agreement with the available work in the literature.