Vibration of quadrilateral embedded multilayered graphene sheets based on nonlocal continuum models using the Galerkin method

Free vibration analysis of quadrilateral multilayered graphene sheets(MLGS) embedded in polymer matrix is carried out employing nonlocal continuum mechanics.The principle of virtual work is employed to derive the equations of motion.The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis.The dependence of small scale effect on thickness,elastic modulus,polymer matrix stiffness and interaction coefficient between two adjacent sheets is illustrated.The non-dimensional natural frequencies of skew,rhombic,trapezoidal and rectangular MLGS are obtained with various geometrical parameters and mode numbers taken into account,and for each case the effects of the small length scale are investigated.     

Buckling analysis of pristine and defected graphene

The buckling behavior of monolayer graphene (pristine and vacancy-defected) and bilayer graphene (pristine) loaded in the armchair direction was simulated for different boundary conditions using a truss FE model, representing the exact atomic lattice of graphene, and a FE model of an equivalent 2D plate. The critical buckling stress of pristine monolayer graphene was derived as a function of aspect ratio. The results from the two FE models coincide and are in very good agreement with established analytical solutions. With increasing the aspect ratio, the critical buckling stress of monolayer graphene decreases until the value of 2 from which the effect starts to diminish. Using the truss FE model, the effect of randomly dispersed vacancies on the critical buckling stress and buckling mode of monolayer graphene was studied. It was found that the critical buckling stress decreases dramatically with increasing the defect density: for a defect density of 10%, the critical buckling stress decreases by almost 50%. Moreover, the presence of defects was found to affect the highest buckling modes (above 3) even at low densities. Bilayer graphene has a totally different critical buckling stress than monolayer graphene due to the effect of van der Waals forces which depends on the applied boundary conditions.     

Nanoscale mass sensing based on vibration of single-layered graphene sheet in thermal environments简

Based on vibration analysis, single-layered graphene sheet （SLGS） with multiple attached nanoparticles is developed as nanoscale mass sensor in thermal environments. Graphene sensors are assumed to be in simplysupported configuration. Based on the nonlocal plate the- ory which incorporates size effects into the classical theory, closed-form expressions lot the frequencies and relative fre- quency shills of SLGS-based mass sensor are derived using the Galerkin method. The suggested model is justified by a good agreement between the results given by the present model and available data in literature. The effects of tem- perature difference, nonlocal parameter, the location of the nanoparticle and the number of nanoparticles on the relative frequency shift of the mass sensor are also elucidated. The obtained results show that the sensitivity of the SLGS- based mass sensor increases with increasing temperature difference.     

Damped vibration of a graphene sheet using a higher-order nonlocal strain-gradient Kirchhoff plate model

A higher-order nonlocal strain-gradient model is presented for the damped vibration analysis of single-layer graphene sheets (SLGSs) in hygrothermal environment. Based on Kirchhoff plate theory in conjunction with a higher-order (bi-Helmholtz) nonlocal strain gradient theory, the equations of motion are obtained using Hamilton's principle. The higher-order nonlocal strain gradient theory has lower- and higher-order nonlocal parameters and a material characteristic parameter. The presented model can reasonably interpret the softening effects of the SLGS, and indicates a reasonably good match with the experimental flexural frequencies. Finally, the roles of viscous and structural damping coefficients, small-scale parameters, hygrothermal environment and elastic foundation on the vibrational responses of SLGSs are studied in detail.     

基于新修正偶应力理论的Mindlin层合板稳定性分析

基于新的各向异性修正偶应力理论提出一个Mindlin复合材料层合板稳定性模型。该理论包含纤维和基体两个不同的材料长度尺度参数。不同于忽略横向剪切应力的修正偶应力Kirchhoff薄板理论,Mindlin层合板考虑横向剪切变形引入两个转角变量。进一步建立了只含一个材料细观参数的偶应力Mindlin层合板工程理论的稳定性模型。计算了正交铺设简支方板Mindlin层合板的临界载荷。计算结果表明该模型可以用于分析细观尺度层合板稳定性的尺寸效应。     

Torsional buckling of a double-walled carbon nanotube embedded in an elastic medium

The torsional buckling of a double-walled carbon nanotube embedded in an elastic medium is studied in this paper. The effects of surrounding elastic medium and van der Waals forces between the inner and outer nanotubes are taken into account. Using continuum mechanics, an elastic double-shell model is presented for the torsional buckling of a double-walled carbon nanotube. Based on the model, a condition is derived in terms of the buckling modes of the shell and the parameters describing the effect of van der Waals interaction and surrounding elastic medium. A simplified analysis is also carried out estimate the critical torque for torsional buckling of the double-walled carbon nanotube.     

Buckling analysis of Euler columns embedded in an elastic medium with general elastic boundary conditions

In this article, an efficient analytical method for elastically restrained Euler columns embedded in an elastic medium has been proposed to calculate buckling loads. The lateral deflection function under compression is represented by a Fourier sine series. Stokes’ transformation is employed to develop the legitimized stability equations. Explicit analytical expressions are derived, which can be used for any type of boundary conditions. The efficiency of present formulation is demonstrated by comparing the results to those obtained by imposing three well-known boundary conditions available in the literature. A very good agreement has been obtained. The present formulation permits to have more efficient coefficient matrix for calculating the buckling loads of Euler columns with any desired boundary conditions.     

Buckling of embedded microtubules in elastic medium

Motivated by the application of Winkler-like models for the buckling analysis of embedded carbon nanotubes, an orthotropic Winkler-like model is developed to study the buckling behavior of embedded cytoskeletal microtubules within the cytoplasm. Experimental observations of the buckling of embedded cytoskeletal microtubules reveal that embedded microtubules bear a large compressive force as compared with free microtubules. The present theoretical model predicts that embedded microtubules in an elastic medium bear large compressive forces than free microtubules. The estimated critical pressure is in good agreement with the experimental values of the pressure-induced buckling of microtubules. Moreover, due to the mechanical coupling of microtubules with the surrounding elastic medium, the critical buckling force is increased considerably, which well explains the theory that the mechanical coupling of microtubules with an elastic medium increases compressive forces that microtubules can sustain. The model presented in the paper is a good approximation for the buckling analysis of embedded microtubules.     

Axial vibration analysis of nanocones based on nonlocal elasticity theory

Carbon nanocones have quite fascinating electronic and structural properties,whose axial vibration is seldom investigated in previous studies.In this paper,based ona nonlocal elasticity theory,a nonuniform rod model is applied to investigate the small-scale effect and the nonuniformeffect on axial vibration of nanocones.Using the modifiedWentzel-Brillouin-Kramers(WBK) method,an asymptoticsolution is obtained for the axial vibration of general nonuniform nanorods.Then,using similar procedure,the axial vibration of nanocones is analyzed for nonuniform parameters,mode number and nonlocal parameters.Explicit expressionsare derived for mode frequencies of clamped-clamped andclamped-free boundary conditions.It is found that axial vibration frequencies are highly overestimated by the classicalrod model because of ignorance of the effect of small lengthscale.     

Nonlinear buckling of a spherical shell embedded in an elastic medium with imperfect interface

This work analyzes nonlinear buckling of a single spherical shell imperfectly bonded to an infinite elastic matrix under a compressive remote load. The inclusion is modeled using a nonlinear shell formulation and the matrix is treated as a linear elastic body. Imperfect bonding conditions are realized through a linear spring interface model. A variational method is used to derive the governing differential equations, which are cast into a tractable set of nonlinear algebraic equations using the Galerkin method. An incremental iterative technique based on the modified Newton–Raphson method is employed to find the critical load of the system. The accuracy and convergence properties of the proposed method are validated through finite element analysis. The study is relevant to the analysis of compressive failure of syntactic foams used in marine and aerospace applications. Results are specialized to glass particle-vinyl ester matrix syntactic foams to test the hypothesis as to whether microballoons’ buckling is a dominant failure mechanism in such composites under compression. Parametric studies are conducted to understand the effect of interfacial properties and inclusion wall thickness on the overall mechanical behavior of the composite. Comparisons between analytical findings and experimental results on compressive response of syntactic foams and isolated microballoons indicate that inclusion buckling is unlikely a determinant of compressive failure in vinyl ester-glass systems. In particular, the matrix is found to exert a beneficial stabilizing effect on the inclusions, which fail under brittle fracture before the onset of buckling.     

Nonlinear softening and hardening nonlocal bending stiffness of an initially curved monolayer graphene

In the present article, the governing nonlinear nonlocal elastic equations are obtained for a monolayer graphene with an initial curvature and the related softening and hardening bending stiffness is analytically calculated. The effects of large deformation, initial curvature, discreteness and direction of chiral vector on the bending stiffness of the monolayer graphene are discussed in detail. A behavior more complex than previously reported in the literature emerges. It is found that the bending stiffness of graphene strongly depends on the initial configuration, showing not obvious maxima and minima, and suggesting the possibility of a smart tuning.     

Free axial vibration of nanorods with elastic medium interaction based on nonlocal elasticity and Rayleigh model

In this paper, the free axial vibration of single walled carbon nanorod embedded in an elastic medium is investigated by the use of Rayleigh model. The stress gradient model introduced by Eringen is used to formulate the governing equations. Explicit expressions are derived for eigenfrequencies of fixed-fixed and fixed-free boundary conditions.     

Unbonded contact of a Mindlin plate on an elastic half-space

Summary In this paper a penalty formulation of the frictionless unilateral contact problem between an elastic rectangular plate and an elastic half-space is presented. In order to take into account the effects of the shear stress, the Mindlin plate model is analyzed. Some numerical results, obtained via finite elements, are given.

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Sommario In questo lavoro viene presentata una formulazione penalty del problema di contatto unilaterale senza attrito tra una piastra rettangolare elastica ed un semispazio elastico. Per la piastra si utilizza il modello di Mindlin, che consente di tener conto dell'effetto delle tensioni da taglio. Si forniscono alcuni risultati numerici ottenuti mediante discretizzazione agli elementi finiti.

     

An analytical symplectic approach to the vibration analysis of orthotropic graphene sheets

A nonlocal continuum orthotropic plate model is proposed to study the vibration behavior of single-layer graphene sheets (SLGSs) using an analytical symplectic approach.A Hamiltonian system is established by introduc-ing a total unknown vector consisting of the displacement amplitude,rotation angle,shear force,and bending moment. The high-order governing differential equation of the vibra-tion of SLGSs is transformed into a set of ordinary differential equations in symplectic space.Exact solutions for free vibra-tion are obtianed by the method of separation of variables without any trial shape functions and can be expanded in series of symplectic eigenfunctions. Analytical frequency equations are derived for all six possible boundary con-ditions. Vibration modes are expressed in terms of the symplectic eigenfunctions.In the numerical examples,com-parison is presented to verify the accuracy of the proposed method. Comprehensive numerical examples for graphene sheets with Levy-type boundary conditions are given.A para-metric study of the natural frequency is also included.     

Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity

The axial vibration of single walled carbon nanotube embedded in an elastic medium is studied using nonlocal elasticity theory. The nonlocal constitutive equations of Eringen are used in the formulations. The effect of various parameters like stiffness of elastic medium, boundary conditions and nonlocal parameters on the axial vibration of nanorods is discussed. It is obtained that, the axial vibration frequencies of the embedded nanorods are highly over estimated by the classical continuum rod model which ignores the effect of small length scale.     

Buckling modes of a compressed plate on an elastic substrate

The buckling modes of a homogeneously compressed elastic plate on a soft elastic substrate are studied. The critical compression is uniquely determined by the bifurcation equation, but this compression is associated with a wide set of buckling modes. It was proved that any solution of the Helmholtz equation satisfies the bifurcation equation. At the same time, in microelectronics, it is required to know which buckling mode is realized. Experimental and theoretical investigations show that the chessboard-like buckling mode should be expected. In what follows, this problem is discussed theoretically. The expected buckling mode can be found by analyzing the energy of the initial postcritical deformation, and the desired mode is determined from the condition of its minimum. The analytic expression of this energy is obtained. Its minimization results in the chessboard-like buckling mode.     

A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales

In this work, the question of homogenizing linear elastic, heterogeneous materials with periodic microstructures in the case of non-separated scales is addressed. A framework if proposed, where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low-pass filters on the fine scale fluctuations. The present theory extends the classical linear homogenization by substituting averaging operators by integral operators, and localization tensors by nonlocal operators involving appropriate Green functions. As a result, the obtained constitutive relationship at the mesoscale appears to be nonlocal. Compared to nonlocal elastic models introduced from a phenomenological point of view, the nonlocal behavior has been fully derived from the study of the microstructure. A discrete version of the theory is presented, where the mesoscopic strain field is approximated as a linear combination of basis functions. It allows computing the mesoscopic nonlocal operator by means of a finite number of transformation tensors, which can be computed numerically on the unit cell.     

Motion of a thin plate in an elastic medium

A solution to the problem of motion of a thin rigid plate in an elastic medium is obtained using the Smirnov-Sobolev method for solving a two-dimensional wave equation.     

Nonlocal vibration analysis of circular double-layered graphene sheets resting on an elastic foundation subjected to thermal loading

Based on the nonlocal elasticity theory, the vibra-tion behavior of circular double-layered graphene sheets (DLGSs) resting on the Winkler- and Pasternak-type elas-tic foundations in a thermal environment is investigated. The governing equation is derived on the basis of Eringen’s nonlocal elasticity and the classical plate theory (CLPT). The initial thermal loading is assumed to be due to a uniform temperature rise throughout the thickness direc-tion. Using the generalized differential quadrature (GDQ) method and periodic differential operators in radial and cir-cumferential directions, respectively, the governing equation is discretized. DLGSs with clamped and simply-supported boundary conditions are studied and the influence of van der Waals (vdW) interaction forces is taken into account. In the numerical results, the effects of various parameters such as elastic medium coefficients, radius-to-thickness ratio, thermal loading and nonlocal parameter are examined on both in-phase and anti-phase natural frequencies. The results show that the thermal load and elastic foundation respec-tively decreases and increases the fundamental frequencies of DLGSs.     

Buckling load and critical length of nanowires on an elastic substrate

This paper considers the stability of nanowires on an elastic substrate. The problem is converted to a generalized Euler problem containing rotational spring restraint. When distributed loading and tip forces are simultaneously applied, the buckling problem of a heavy nanocolumn with rotational spring junction is reduced to an integral equation. An approximate buckling load equation is derived explicitly. The critical length of nanocantilevers is given in closed form. Results indicate that spring stiffness increases the critical length of nanowires. The effect of self-weight on the critical length is pronounced for small tip forces, and becomes weaker for larger tip forces.