Noël Challamel

Bending, Buckling, and Vibration of Micro/Nanobeams by Hybrid Nonlocal Beam Model

The hybrid nonlocal Euler-Bernoulli beam model is applied for the bending, buckling, and vibration analyzes of micro/nanobeams. In the hybrid nonlocal model, the strain energy functional combines the local and nonlocal curvatures so as to ensure the presence of small length-scale parameters in the deflection expressions. Unlike Eringens nonlocal beam model that has only one small length-scale parameter, the hybrid nonlocal model has two independent small length-scale parameters, thereby allowing for a more flexible and accurate modeling of micro/nanobeamlike structures. The equations of motion of the hybrid nonlocal beam and the boundary conditions are derived using the principle of virtual work. These beam equations are solved analytically for the bending, buckling, and vibration responses. It will be shown herein that the hybrid nonlocal beam theory could overcome the paradoxes produced by Eringens nonlocal beam theory such as vanishing of the small length-scale effect in the deflection expression or the surprisingly stiffening effect against deflection for some classes of beam bending problems.

Calibration of Eringen's small length scale coefficient for initially stressed vibrating nonlocal Euler beams based on microstructured beam model

In this paper, we calibrate Eringens small length scale coefficient e0 for an initially stressed vibrating nonlocal Euler beam via a microstructured beam modelled by some repetitive cells comprising finite rigid segments and elastic rotational springs. By adopting the pseudo-differential operator and Pades approximation, an analytical solution for the vibration frequency in terms of initial stress may be developed for the microstructured beam model. When comparing this analytical solution with the established exact vibration solution from the nonlocal beam theory, one finds that the calibrated Eringens small length scale coefficient e0 is given by where σ0 is the initial stress and is the mth mode buckling stress of the corresponding local Euler beam. It is shown that e0 varies with respect to the initial axial stress, from at the buckling compressive stress to when the axial stress is zero and it monotonically increases with increasing initial tensile stress. The small length scale coefficient e0, however, does not depend on the vibration/buckling mode considered.

Carbon nanotubes and nanosensors: vibration, buckling and ballistic impact

The main properties that make carbon nanotubes (CNTs) a promising technology for many future applications are: extremely high strength, low mass density, linear elastic behavior, almost perfect geometrical structure, and nanometer scale structure. Also, CNTs can conduct electricity better than copper and transmit heat better than diamonds. Therefore, they are bound to find a wide, and possibly revolutionary use in all fields of engineering. The interest in CNTs and their potential use in a wide range of commercial applications; such as nanoelectronics, quantum wire interconnects, field emission devices, composites, chemical sensors, biosensors, detectors, etc.; have rapidly increased in the last two decades. However, the performance of any CNT-based nanostructure is dependent on the mechanical properties of constituent CNTs. Therefore, it is crucial to know the mechanical behavior of individual CNTs such as their vibration frequencies, buckling loads, and deformations under different loadings. This title is dedicated to the vibration, buckling and impact behavior of CNTs, along with theory for carbon nanosensors, like the Bubnov-Galerkin and the Petrov-Galerkin methods, the Bresse-Timoshenko and the Donnell shell theory.

Nonlocal Equivalent Continua for Buckling and Vibration Analyses of Microstructured Beams

This paper is focused on the buckling and the vibration analyses of microstructured structural elements, i.e., elements composed of repetitive structural cells. The relationship between the discrete and the equivalent nonlocal continuum is specifically addressed from a numerical and a theoretical point of view. The microstructured beam considered herein is modeled by some repetitive cells composed of finite rigid segments and elastic rotational springs. The microstructure may come from the discreteness of the matter for small-scale structures (such as for nanotechnology applications), but it can also be related to some larger scales as for civil engineering applications. The buckling and vibration results of the discrete system are numerically obtained from a discrete-element code, whereas the nonlocal-based results for the equivalent continuum can be analytically performed. It is shown that Eringens nonlocal elasticity coupled to the Euler-Bernoulli beam theory is relevant to capture the main-scale phenomena of such a microstructured continuum. The small-scale coefficient of the equivalent nonlocal continuum is identified from the specific microstructure features, namely, the length of each cell. However, the length scale calibration depends on the type of analysis, namely, static versus dynamic analysis. A perfect agreement is found for the microstructured beam with simply supported boundary conditions. The specific identification of the equivalent stiffness for modeling the equivalent clamped continuum is also discussed. The equivalent stiffness of the discrete system appears to be dependent on the static-dynamic analyses, but also on the boundary conditions applied to the overall system. Satisfactory results are also obtained for the comparison between the discrete and the equivalent continuum for other type of boundary conditions.

Hencky bar-chain model for buckling and vibration of beams with elastic end restraints

This paper presents the Hencky bar-chain model (HBM) for buckling and vibration analyses of Euler–Bernoulli beams with elastic end restraints. The Hencky bar-chain comprises rigid beam segments (of length a = L/n where L is the total length of beam and n the number of beam segments) connected by frictionless hinges with elastic rotational springs of stiffness EI/a where EI is the flexural rigidity of the beam. The elasticity and the mass of the beam are concentrated at the hinges with rotational springs. The key contribution of this paper lies in the modeling of the elastic end restraints of the Hencky bar-chain that will simulate the same buckling and vibration results as that furnished by the first-order central finite difference beam model (FDM) which was earlier shown to be analogous to the HBM. The establishment of such a physical discrete beam model allows one to obtain solutions for beam-like structure with repetitive cells (or elements) as well as to calibrate the Eringens coefficient e0 in the nonlocal beam theory that captures the small length scale effect.

Eringen's small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model

This paper presents the determination of Eringens small length scale coefficient e0 for buckling of nonlocal Timoshenko beam from a microstructured beam model. The microstructured beam model is composed of discrete rigid elements (of equal length), which are connected by rotational and shear springs that model the bending and shearing behaviors in a beam. The exact solution of e0 is given for nonlocal Timoshenko beam with small length scale term appearing in the normal stress-strain relation only. It is shown that e0 approaches 1/12≈0.289 which coincides with the one calibrated for nonlocal Euler beams.

Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

The present study takes an analytical approach for solving the free vibration problem of a microstructured beam model, in which transverse displacement springs are added to allow for the transverse shear deformation effect in addition to the rotational springs. The exact vibration frequencies for the discrete microstructured beam model with simply supported ends are obtained via matrix decomposition. In addition, a general solution technique involving the use of Pade approximants for the continualization procedure is proposed in order to obtain the continuous equivalent system for the discrete microstructured beam model. The analytical vibration solutions of the equivalent continuous system are obtained and their accuracy is assessed by using the exact solutions. It is found that the solutions of the equivalent continuous system have a first order accuracy when compared with the exact solutions of their discrete counterpart. The length scale coefficient in the nonlocal Timoshenko beam model is calibrated by using the analytical solutions. Two nonlocal Timoshenko beam models, i.e., the Wang model (without the length scale effect in the shear stress strain relation) and the Reddy model, are evaluated based on their ability to capture the nonlocal effect.

Eringen's stress gradient model for bending of nonlocal beams

This paper is concerned with the bending response of nonlocal elastic beams under transverse loads, where the nonlocal elastic model of Eringen, also called the stress gradient model, is used. This model is known to exhibit some paradoxical responses when applied to beams with certain types of boundary conditions. In particular, for clamped-free boundary condition, this nonlocal model is not able to predict scale effects in the presence of concentrated loads, or it leads to an apparent stiffening effect for distributed loads in contrast to other boundary conditions for which softening effect is observed. In the literature, these paradoxes have been resolved by changing the kernel of the nonlocal model or by modifying the standard boundary conditions. In this paper, the paradox is solved from the nonlocal differential model itself via some related discontinuous nonlocal kinematics. It is shown that the kinematics related to the nonlocal constitutive law lead to the use of moment or shear discontinuities. With such a nonlocal differential model coupled with the nonlocal discontinuity requirements, the beam effectively shows a softening response irrespective of the boundary conditions studied, including the clamped-free boundary conditions, and thereby resolves the paradox. The model is also compared to lattice-based solutions where an excellent agreement between the present nonlocal model and the lattice one is obtained. Finally, the stress gradient model is shown to be cast in a stress-based variational framework, which coincides with a Timoshenko-type model where the shear effect is shown to play the nonlocal role.

Eringen’s Length-Scale Coefficients for Vibration and Buckling of Nonlocal Rectangular Plates with Simply Supported Edges

For the nonlocal theory of structures, Eringens small length-scale coefficient e0 may be identified from atomistic modeling or experimental tests. In this study, Eringens small length-scale coefficients are presented for the vibration and buckling of nonlocal rectangular plates with simply supported edges. The coefficients are calibrated by comparing the vibration frequency and buckling loads obtained from a nonlocal plate and a microstructured beam-grid model with the same characteristic length. The beam-grid model is composed of rigid beams connected by rotational and torsional springs. It is found that the small length-scale coefficient e0 varies with respect to the initial stress, rotary inertia, mode shape, and aspect ratio of the rectangular plate.

Boundary-Layer Effect in Composite Beams with Interlayer Slip

An apparent analytical peculiarity or paradox in the bending behavior of elastic-composite beams with interlayer slip, sandwich beams, or other similar problems subjected to boundary moments exists. For a fully composite beam subjected to such end moments, the partial composite model will render a nonvanishing uniform value for the normal force in the individual subelement. This is from a formal mathematical point of view in apparent contradiction with the boundary conditions, in which the normal force in the individual subelement usually is assumed to vanish at the extremity of the beam. This mathematical paradox can be explained with the concept of boundary layer. The bending of the partially composite beam expressed in dimensionless form depends only on one structural parameter related to the stiffness of the connection between the two subelements. An asymptotic method is used to characterize the normal force and the bending moment in the individual subelement to this dimensionless connection parameter. The outer expansion that is valid away from the boundary and the inner expansion valid within the layer adjacent to the boundary (beam extremity) are analytically given. The inner and outer expansions are matched by using Prandtl’s matching condition over a region located at the edge of the boundary layer. The thickness of the boundary layer is the inverse of the dimensionless connection parameter. Finite-element results confirm the analytical results and the sensitivity of the bending solution to the mesh density, especially in the edge zone with stress gradient. Finally, composite beams with interlayer slip can be treated in the same manner as nonlocal elastic beams. The fundamental differential equation appearing in the constitutive law associated with the partial-composite action in a nonlocal elasticity framework is discussed. Such an integral formulation of the constitutive equation encompassing the behavior of the whole of the beam allows the investigation of the mechanical problem with the boundary-element method.