Continuum Beam Model Research Articles

Shear vibration modes of granular structures: Continuous and discrete approaches

AbstractThis work is dedicated to the study of shear vibration modes of one‐dimensional granular microstructured beams using a discrete Cosserat model. The dynamic response of the one‐dimensional granular beam (discrete beam) is investigated for various boundary conditions, through the analytical resolution of an exact discrete eigenvalue problem. For an infinite number of grains, the dynamic behavior of the discrete problem converges towards a Bresse–Timoshenko continuum beam model. First, the effective modeling parameters, the constitutive equations and the governing equations of motion are derived in a difference form. For simply supported granular beams, the appearance of pure shear modes of vibration is highlighted (vibration modes without deflection), a phenomenon which is already known in the free vibration of a continuous Bresse–Timoshenko beam problem. The eigenfrequencies of these pure shear modes for the discrete granular beam coincide with the critical frequencies of the discrete system. These pure shear modes may appear for granular beams composed of few grains, which give a physical support of a phenomenon well studied in the literature from a continuum beam model. The eigenfrequencies of the discrete granular beam are also calculated for various boundary conditions, including clamped, hinge and free end boundary conditions. Based on the discrete element method (DEM), several numerical simulations of dynamic tests in some representative examples have been performed. The responses of the numerical approach are accurately compared against the exact results of the analytical solutions based on the difference eigenvalue problem, which illustrates the relevance of the proposed approach.

Multiquadric radial basis function approximation method and asymptotic numerical method for nonlinear and linear static analysis of single-walled carbon nanotubes

In this paper, a numerical algorithm combining the meshless collocation technique based on the globally supported Multiquadric Radial Basis Function (MQRBF) approximation method and the Asymptotic Numerical Method (ANM) is developed to investigate the nonlinear and linear static behaviors of Single-Walled Carbone Nanotubes (SWCNTs) based on a Nonlocal Continuum Beam Model (NCBM). The NCBM which accounts for the effects of small scale, shear deformation and geometric nonlinearity of von-Kármán type is constructed by implementing the Nonlocal Elasticity Theory of Eringen (NET) into the First-Order Shear Deformation Elastic Beam Theory (FOSDEBT) and used to model the SWCNTs as a Continuous Elastic Nanobeam (CEN). Based on the Total Lagrangian Formulation (TLF) the nonlocal nonlinear governing equations of SWCNT are elaborated in a quadratic matrix strong form. These equations are then solved numerically by applying the proposed algorithm. This algorithm permits to transform the quadratic nonlocal nonlinear governing equations into a sequence of linear ones shared the same tangent stiffness matrix which are then solved numerically by the MQRBF approximation method, to get the whole branch of solution a continuation procedure is required. In order to highlight the reliability and efficiency of the present numerical algorithm, comparative studies are conducted.

Asymptotic Modeling of Nonlinear Bending and Buckling Behavior of Carbon Nanotubes

The present work investigates the nonlinear bending and buckling behavior of carbon nanotubes (CNTs) using variational asymptotic method. Considering a CNT as a slender beam structure, an asymptotically-correct nonlinear continuum beam model is presented. Through the resulting nonlinear moment-curvature relationship, the model captures the phenomenon of ovalization of the cross sections and local buckling of the CNT, which arises due to their geometrical nature. Further studies are performed in order to explore the effect of CNT wall thickness on the nonlinear bending behavior of the CNT structure. It is shown that the continuum modeling approach can capture the ovalization and further localization of the CNT deformation under bending. The study aims to provide a reduced-order modeling framework analyzing the inherent nonlinearities associated with the geometrical nature of CNTs.

Effective Beam Stiffness Properties of n-Strut Cylindrical Tensegrity Towers

In this paper, a method to obtain effective continuum beam stiffness properties of tensegrity towers with struts in each bay is developed. Long tensegrity towers for space applications can be modeled as beam-like structures, and axial, bending, shear, and torsional stiffness properties can be used to make comparisons between alternatives to determine the optimum design for a mission. Continuum beam models and effective stiffness properties of Class-2 tensegrity towers are obtained using a modified energy equivalency approach. Self-stress levels (i.e., cable tension and strut compression) are varied to observe their influence on the effective stiffness properties. Axial and torsional rigidities are found to be highly affected by the self-stress level, while bending and shear rigidities are not affected as much. The obtained effective stiffness properties are validated with nonlinear finite element analyses. With these properties, alternative tower designs for a specific application can be readily evaluated without the need for numerous computationally expensive structural analyses.

Free vibration analysis of spatial Bernoulli–Euler and Rayleigh curved beams using isogeometric approach

This paper deals with the linear free vibration analysis of Bernoulli–Euler and Rayleigh curved beams using isogeometric approach. The geometry of the beam as well as the displacement field are defined using the NURBS basis functions which present the basic concept of the isogeometric analysis. A novel approach based on the fundamental relations of the differential geometry and Cauchy continuum beam model is presented and applied to derive the stiffness and consistent mass matrices of the corresponding spatial curved beam element. In the Bernoulli–Euler beam element only translational and torsional inertia are taken into account, while the Rayleigh beam element takes all inertial terms into consideration. Due to their formulation, isogeometric beam elements can be used for the dynamic analysis of spatial curved beams. Several illustrative examples have been chosen in order to check the convergence and accuracy of the proposed method. The results have been compared with the available data from the literature as well as with the finite element solutions.

Mechanics of polymer brush based soft active materials– theory and experiments

A brush-like structure emerges from stretching of long polymer chains, densely grafted on to the surface of an impermeable substrate. The structure arises due to the competition between conformational entropic elasticity of polymer chains and excluded volume interactions leading to intra and interchain monomer repulsions. Recently, soft materials based on stimuli responsive polymer brushes have been developed to produce controllable and reversible large bending deformation of the host substrates. To understand such systems and improve their functional properties, we study the stress distribution in a brush, and develop surface stress-curvature relation for an elastic beam of a soft material grafted with a neutral polymer brush. In the strongly stretched brush regime, we combine mean field theory from polymer physics with a continuum mechanics model and show that the residual stress variation in a brush is a quartic function of distance from the grafting surface, with a maximum occurring at the grafting surface. By idealizing a brush as a continuum elastic surface layer with residual stress, we derive a closed form expression for surface stress and surface elasticity of the layer as a function of brush parameters, such as graft density and molecular weight. A generalized continuum beam model accounts for the Young–Laplace and Steigmann–Ogden curvature elasticity correction terms, and yields a surface stress-curvature relation, that contains existing relations in the literature as special cases. Further, we report experiments on a thermoresponsive random copolymer brush, Poly(N- isopropylacrylamide)-co-Poly(N,N-Dimethylacrylamide) (PNIPAm-co-PDMA) brush, grafted on one side of a plasticized poly(vinyl chloride) (pPVC) thin film. Estimated surface stress from measured curvature is on the order of −10N/m, and its magnitude decreases gradually, and reversibly, on increasing ambient temperature from 15 °C to 55 °C.

Size-dependent behavior of slacked carbon nanotube actuator based on the higher-order strain gradient theory

This paper proposes to investigate the nonlinear size dependent behavior of electrically actuated carbon nanotube (CNT) based nano-actuator while including the higher-order strain gradient deformation, the geometric nonlinearity due to the von Karman nonlinear strain as well as the slack effect, and the temperature gradient effects. The assumed non-classical beam model adopts some internal material size scale parameters related to the material nanostructures and is capable of interpreting the size effect that the classical continuum beam model is unable to pronounce. The higher-order governing equations of motion and boundary conditions are derived using the so-called extended Hamilton principle. A Galerkin based reduced-order model (ROM) modal decomposition is developed to prescribe the non-classical nanotube mode shape as well as its static behavior under any applied DC actuation load. Results of the static analysis is compared with those obtained by both classical elasticity continuum and strain gradient theories. A Jacobian method is utilized to determine the variation of the natural frequencies of the nanobeam with the DC load as well as the slack level. A thorough parametric study is conducted to study the influences of the size scale dependent parameters, the geometric nonlinearity, the initial curvature, the gate voltage, and the temperature gradient effect on structural behavior of the CNT-based nano-actuator. It is found that the size effect based on the strain gradient deformation has significant influence on the fundamental nanotube natural frequency dispersion. Also, varying this size effect have revealed the offering of numerous possibilities of modes veering and crossing, all shown to be dependent of the strain gradient parameters as well as the CNT slack level.

Influence of the nonlocal parameter on the transverse vibration of double-walled carbon nanotubes

AbstractA high-order nonlocal continuum beam model is proposed, which can be applied to study the transverse vibrations of double-walled carbon nanotubes (DWCNTs), including those that could have initial deformations due to defects or external actions. A beam element is developed adopting Hermite cubic polynomials as shape functions, and mass and elastic stiffness matrix are presented. The influence of the nonlocal parameter on the vibrational properties of DWCNTs is studied. Using the proposed model, it was found that the nonlocal parameter has a strong influence on the natural frequencies.

Controlled Manipulation of Flexible Carbon Nanotubes through Shape-Dependent Pushing by Atomic Force Microscopy

A systematic approach to manipulating flexible carbon nanotubes (CNTs) has been developed on the basis of atomic force microscope (AFM) based pushing. Pushing CNTs enables efficient transport and precise location of individual CNTs. A key issue for pushing CNTs is preventing defective distortion in repetitive bending and unbending deformation. The approach presented here controls lateral movement of an AFM tip to bend CNTs without permanent distortion. The approach investigates possible defects caused by tensile strain of the outer tube under uniform bending and radial distortion by kinking. Using the continuum beam model and experimental bending tests, dependency of maximum bending strain on the length of bent CNTs and radial distortion on bending angles at a bent point have been demonstrated. Individual CNTs are manipulated by limiting the length of bent CNTs and the bending angle. In our approach, multiwalled CNTs with 5-15 nm diameter subjected to bending deformation produce no outer tube breakage under uniform bending and reversible radial deformation with bending angles less than 110°. The lateral tip movement is determined by a simple geometric model that relies on the shape of multiwalled CNTs. The model effectively controls deforming CNT length and bending angle for given CNT shape. Experimental results demonstrate successful manipulation of randomly dispersed CNTs without visual defects. This approach to pushing can be extended to develop a wide range of CNT based nanodevice applications.

Interior and Edge Elastic Waves in Graphene

Elastic waves propagating in graphene nanoribbons were studied using both continuum modeling and molecular dynamics simulations. The Mindlin's plate model was employed to model the propagation of interior waves of graphene, and a continuum beam model was proposed to model the propagation of edge waves in graphene. The molecular dynamics results demonstrated that the interior longitudinal and transverse wave speeds of graphene are about 18,450 m/s and 5640 m/s, respectively, in good agreement with the Mindlin's plate model. The molecular dynamics simulations also revealed the existence of elastic edge waves, which may be described by the proposed continuum beam model.

The anisotropic hyperelastic biomechanical response of the vocal ligament and implications for frequency regulation: A case study

One of the primary mechanisms to vary one's vocal frequency is through vocal fold length changes. As stress and deformation are linked to each other, it is hypothesized that the anisotropy in the biomechanical properties of the vocal fold tissue would affect the phonation characteristics. A biomechanical model of vibrational frequency rise during vocal fold elongation is developed which combines an advanced biomechanical characterization protocol of the vocal fold tissue with continuum beam models. Biomechanical response of the tissue is related to a microstructurally informed, anisotropic, nonlinear hyperelastic constitutive model. A microstructural characteristic (the dispersion of collagen) was represented through a statistical orientation function acquired from a second harmonic generation image of the vocal ligament. Continuum models of vibration were constructed based upon Euler-Bernoulli and Timoshenko beam theories, and applied to the study of the vibration of a vocal ligament specimen. From the natural frequency predictions in dependence of elongation, two competing processes in frequency control emerged, i.e., the applied tension raises the frequency while simultaneously shear deformation lowers the frequency. Shear becomes much more substantial at higher modes of vibration and for highly anisotropic tissues. The analysis was developed as a case study based on a human vocal ligament specimen.

Analysis of wave propagation in micro/nanobeam-like structures: A size-dependent model

By incorporating the strain gradient elasticity into the classical Bernoulli-Euler beam and Timoshenko beam models, the size-dependent characteristics of wave propagation in micro/nanobeams is studied. The formulations of dispersion relation are explicitly derived for both strain gradient beam models, and presented for different material length scale parameters (MLSPs). For both phenomenological sizedependent beam models, the angular frequency, phase velocity and group velocity increase with increasing wave number. However, the velocity ratios approach different values for different beam models, indicating an interesting behavior of the asymptotic velocity ratio. The present theory is also compared with the nonlocal continuum beam models.

On the influence of van der Waals coefficient on the transverse vibration of double walled carbon nanotubes

The influence of van der Waals interaction coefficient on the vibrational properties of double-walled carbon nanotubes (DWCNTs) is studied. A high order continuum beam model is used, which can be applied to study the transverse vibrations of MWCNTs, including those that could have initial deformations due to defects or external actions. A numerical study was carried out and it was found that the non-coaxial intertube frequencies are strongly influenced by the coefficient used. The importance of this result can be valued considering that the concentric structure of DWCNTs is a crucial geometrical characteristic and a non-coaxial vibration would affect their electronic and optical properties.

Molecular investigation of the mechanical properties of single actin filaments based on vibration analyses

The mechanical vibration properties of single actin filaments from 50 to 288 nm are investigated by the molecular dynamics simulation in this study. The natural frequencies obtained from the molecular simulations agree with those obtained from the analytical solution of the equivalent Euler–Bernoulli beam model. Through the convergence study of the mechanical properties with respect to the filament length, it was found that the Euler–Bernoulli beam model can only be reliably used when the single actin filament is of the order of hundreds of nanometre scale. This molecular investigation not only provides the evidence for the use of the continuum beam model in characterising the mechanical properties of single actin filaments, but also clarifies the criteria for the effective use of the Euler–Bernoulli beam model.

Evaluation of the in-plane effective elastic moduli of single-layered graphene sheet

In this paper, an equivalent continuum-structural mechanics approach is used to characterize the mechanical behaviour of nanostructured graphene. The in-plane elastic deformation of armchair graphene sheets is simulated by using finite element modelling. The model is based on the assumption that force interaction among carbon atoms can be modelled by load-carrying beams in a representative two-dimensional honeycomb lattice structure. The elastic properties of beam elements are determined by equating the energies of the molecular structure and the continuum beam model subjected to small strain deformation. Then an equivalent continuum technique is adopted to estimate effective elastic moduli from which elastic constants are extracted. A comparison of elastic constants obtained from current modelling concur with results reported in literature. With the multifunctional properties of graphene sheets as manifested in a broad range of industrial applications, determination of their elastic moduli will facilitate a better design of the corresponding materials at macroscopic level.

On the influence of the shear deformation and boundary conditions on the transverse vibration of multi-walled carbon nanotubes

The vibrational properties of multi-walled carbon nanotubes (MWCNTs) are studied, especially the influence of the shear deformation and the boundary conditions on the non-coaxial frequencies. A high order continuum beam model is proposed, which can be applied to study the transverse vibrations of MWCNTs, including those that could have initial deformations due to defects or external actions. Using the model it was found that the non-coaxial intertube frequencies are independent of the shear deformation and the boundary conditions. This result is important considering that the concentric structure of MWCNTs is a crucial geometrical characteristic and a non-coaxial vibration would affect their electronic and optical properties.

Nonlocal beam models for buckling of nanobeams using state-space method regarding different boundary conditions

Buckling analysis of nanobeams is investigated using nonlocal continuum beam models of the different classical beam theories namely as Euler-Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Levinson beam theory (LBT). To this end, Eringen's equa- tions of nonlocal elasticity are incorporated into the classical beam theories for buckling of nanobeams with rectangular cross-section. In contrast to the classical theories, the nonlocal elastic beam models developed here have the capability to predict critical buckling loads that allowing for the inclusion of size effects. The values of critical buckling loads corresponding to four commonly used boundary con- ditions are obtained using state-space method. The results are presented for different geometric parameters, boundary conditions, and values of nonlocal parameter to show the effects of each of them in detail. Then the results are fitted with those of molecular dynamics simulations through a nonlinear least square fitting procedure to find the appropriate values of nonlocal parameter for the buckling analy- sis of nanobeams relevant to each type of nonlocal beam model and boundary conditions.analysis. Based on the above introduction, it seems that size-effects consideration in the analysis of nanobeams is necessary. In this work, different nonlocal beam models corresponding to the different classical beam theories (22-24) are presented on the basis of Eringen's equations of nonlocal elasticity (25) to predict the buckling behavior of nanobeams with four com- monly used boundary conditions. State-space method is used to solve the governing differential equations for each type of nonlocal beam model with different boundary conditions. Various numerical results are given to show the influences of boundary conditions, aspect ratio, and values of nonlocal con- stant, separately. Then the results are matched with those of molecular dynamics simulations which are available in the literature to extract the correct values of nonlocal parameter corresponding to each type of nonlocal beam model and boundary conditions.

Thermal effect on wave propagation in double-walled carbon nanotubes embedded in a polymer matrix using nonlocal elasticity

The thermal effect on wave propagation in double-walled carbon nanotubes (DWNTs) embedded in a polymer matrix is studied in this paper via nonlocal elasticity. The small-scale effects on vibration characteristics of carbon nanotubes are explicitly derived through a complete continuum beam model. In example calculations, the mechanical properties of carbon nanotubes and polymer matrix are treated as the functions of temperature change. The research work reveals the significance of the small-scale effect on wave propagation in DWNTs. It is also demonstrated that some properties of transverse vibrations of DWNTs are dependent on the change of temperature.

Dynamic Characteristic Analysis of Large Space Deployable Articulated Mast

Equivalent continuum beam model and finite element model of space deployable articulated mast are developed. Dynamic characteristics of the mast without and with tip load are analyzed based on two models respectively. Frequencies and mode shapes of the mast are calculated and simulated by Euler beam theory and finite element simulation, and theoretical solutions are compared with simulation results. Influence of mast structural parameters on mast natural frequencies is analyzed, the sensitivities of mast bending, torsional, axial frequencies to structural parameters are calculated. The measure for improving mast dynamic characteristic is proposed through sensitivity analysis.

Vehicle – structure interaction to analyse the primary dynamic aspects of railway bridges

A simplified approach to take into account the dynamic interaction between trains and bridge structures is presented in this paper. A system composed of a simple, supported, continuum beam model and a vehicle model allows the study of the dynamics of the bridge and of the suspended part of the train. A solution is obtained by using a modal decomposition method and a state-space formulation. The calibration of the beam for existing bridges is obtained through experimental analyses. Different vehicle models, with successively increasing levels of refinement, are considered in order to assess the influence of the bridge – vehicle interaction. The influence of the type of vehicle on the dynamics of the bridge is evaluated. The different solutions are compared to the ones obtained using a finite element model and the simplified methods proposed by the codes. This new approach seems promising in the evaluation of the dynamic magnification of the bridge response, which influences the resistance and durability of a bridge in general when subjected to different types of vehicles crossing at increasing velocities. This topic is important for old and new structures in order to assess their critical loading conditions in relation to changing train technology.