Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams

This paper is proposed to study the coupled effects of surface properties and nonlocal elasticity on vibration characteristics of nanobeams by using a finite element method. Nonlocal differential elasticity of Eringen is exploited to reveal the long-range interactions of a nanoscale beam. To incorporate surface effects, Gurtin-Murdoch model is proposed to satisfy the surface balance equations of the continuum surface elasticity. Euler-Bernoulli hypothesis is used to model the bulk deformation kinematics. The surface layer and bulk of the beam are assumed elastically isotropic. Galerkin finite element technique is employed for the discretization of the nonlocal mathematical model with surface properties. An efficiently finite element model is developed to descretize the beam domain and solves the equation of motion numerically. The output results are compared favorably with those published works. The effects of nonlocal parameter and surface elastic constants on the vibration characteristics are presented. Also, the effectiveness of finite element method to handle a complex geometry is illustrated. The present model can be used for free vibration analysis of single-walled carbon nanotubes with essential, natural and nonlinear boundary conditions.

[1]  Vijay B. Shenoy.  Size-dependent rigidities of nanosized torsional elements , 2001 .

[2]  J. N. Reddy,et al.  Nonlocal theories for bending, buckling and vibration of beams , 2007 .

[3]  I. Elishakoff,et al.  Surface Elasticity Effects Can Apparently Be Explained Via Their Nonconservativeness , 2011 .

[4]  Reza Ansari,et al.  BENDING BEHAVIOR AND BUCKLING OF NANOBEAMS INCLUDING SURFACE STRESS EFFECTS CORRESPONDING TO DIFFERENT BEAM THEORIES , 2011 .

[5]  Y. Mirzaei,et al.  Free transverse vibrations of cracked nanobeams with surface effects , 2011 .

[6]  Antonina Pirrotta,et al.  Mechanically-based approach to non-local elasticity: Variational principles , 2010 .

[7]  J. N. Reddy,et al.  Non-local elastic plate theories , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[8]  A. Eringen,et al.  On nonlocal elasticity , 1972 .

[9]  Dominic G.B. Edelen,et al.  Nonlocal continuum mechanics , 1971 .

[10]  Harold S. Park,et al.  A continuum model for the mechanical behavior of nanowires including surface and surface-induced initial stresses , 2011 .

[11]  Win-Jin Chang,et al.  Surface effects on frequency analysis of nanotubes using nonlocal Timoshenko beam theory , 2010 .

[12]  F. F. Mahmoud,et al.  Vibration analysis of Euler–Bernoulli nanobeams by using finite element method , 2013 .

[13]  F. Daneshmand,et al.  Stress and strain-inertia gradient elasticity in free vibration analysis of single walled carbon nanotubes with first order shear deformation shell theory , 2013 .

[14]  H. Sheng,et al.  Free vibration analysis for micro-structures used in MEMS considering surface effects , 2010 .

[15]  Jin He,et al.  Surface effect on the elastic behavior of static bending nanowires. , 2008, Nano letters.

[16]  F. F. Mahmoud,et al.  Nonlocal finite element modeling of the tribological behavior of nano-structured materials , 2010 .

[17]  M. A. Eltaher,et al.  Free vibration analysis of functionally graded size-dependent nanobeams , 2012, Appl. Math. Comput..

[18]  E. Kröner,et al.  Elasticity theory of materials with long range cohesive forces , 1967 .

[19]  Ö. Civalek,et al.  Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory , 2011 .

[20]  Quan Wang,et al.  A Review on the Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphenes , 2012 .

[21]  Yiming Fu,et al.  Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies , 2011 .

[22]  C. W. Lim,et al.  Size-dependent nonlinear response of thin elastic films with nano-scale thickness , 2004 .

[23]  F. F. Mahmoud,et al.  Static analysis of nanobeams including surface effects by nonlocal finite element , 2012 .

[24]  J. Reddy.  Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates , 2010 .

[25]  Pradeep Sharma,et al.  A novel atomistic approach to determine strain-gradient elasticity constants: Tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies , 2007 .

[26]  F. F. Mahmoud,et al.  Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams , 2013 .

[27]  Samir A. Emam,et al.  Static and stability analysis of nonlocal functionally graded nanobeams , 2013 .

[28]  Yiming Fu,et al.  INFLUENCES OF THE SURFACE ENERGIES ON THE NONLINEAR STATIC AND DYNAMIC BEHAVIORS OF NANOBEAMS , 2010 .

[29]  Xi-Qiao Feng,et al.  Surface effects on buckling of nanowires under uniaxial compression , 2009 .

[30]  H. P. Lee,et al.  Thin plate theory including surface effects , 2006 .

[31]  Paolo Fuschi,et al.  Nonlocal integral elasticity: 2D finite element based solutions , 2009 .

[32]  Azim Eskandarian,et al.  Atomistic viewpoint of the applicability of microcontinuum theories , 2004 .

[33]  M. Şi̇mşek,et al.  Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle , 2011 .

[34]  Metin Aydogdu,et al.  A GENERAL NONLOCAL BEAM THEORY: ITS APPLICATION TO NANOBEAM BENDING, BUCKLING AND VIBRATION , 2009 .

[35]  C. Ricciardi,et al.  A new Finite Element approach for studying the effect of surface stress on microstructures , 2010 .

[36]  Castrenze Polizzotto,et al.  Nonlocal elasticity and related variational principles , 2001 .

[37]  K. M. Liew,et al.  Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures , 2007 .

[38]  Seyyed M. Hasheminejad,et al.  Surface effects on nonlinear free vibration of nanobeams , 2011 .

[39]  Abdelouahed Tounsi,et al.  The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory , 2012 .

[40]  Reza Ansari,et al.  A sixth-order compact finite difference method for vibrational analysis of nanobeams embedded in an elastic medium based on nonlocal beam theory , 2011, Math. Comput. Model..

[41]  R. Ansari,et al.  Surface stress effects on the free vibration behavior of nanoplates , 2011 .

[42]  D. Huang.  Size-dependent response of ultra-thin films with surface effects , 2008 .

[43]  Morton E. Gurtin,et al.  A continuum theory of elastic material surfaces , 1975 .

[44]  Chien Ming Wang,et al.  Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model , 2010 .

[45]  Chunyu Li,et al.  A STRUCTURAL MECHANICS APPROACH FOR THE ANALYSIS OF CARBON NANOTUBES , 2003 .

[46]  J. Reddy,et al.  Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method , 2011 .

[47]  Lin Wang,et al.  Vibration analysis of fluid-conveying nanotubes with consideration of surface effects , 2010 .

[48]  John Peddieson,et al.  Application of nonlocal continuum models to nanotechnology , 2003 .

[49]  A. C. Eringen,et al.  Nonlocal polar elastic continua , 1972 .

[50]  A. Eringen,et al.  Nonlocal Continuum Field Theories , 2002 .

[51]  S. C. Pradhan,et al.  Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates , 2010 .

[52]  Vijay B. Shenoy,et al.  Size-dependent elastic properties of nanosized structural elements , 2000 .

[53]  A. Eringen.  On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves , 1983 .

[54]  Morton E. Gurtin,et al.  Surface stress in solids , 1978 .

[55]  Jin-Xing Shi,et al.  Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model , 2012 .

[56]  Saeed Reza Mohebpour,et al.  Small-scale effect on the vibration of non-uniform carbon nanotubes conveying fluid and embedded in viscoelastic medium , 2012 .

[57]  Ömer Civalek,et al.  Free Vibration and Bending Analyses of Cantilever Microtubules Based on Nonlocal Continuum Model , 2010 .

[58]  Win-Jin Chang,et al.  Surface effects on axial buckling of nonuniform nanowires using non-local elasticity theory , 2011 .

[59]  C. Lim,et al.  Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory , 2009 .

[60]  Gang Wang,et al.  Effects of surface stresses on contact problems at nanoscale , 2007 .

[61]  B. Farshi,et al.  Size dependent vibration of curved nanobeams and rings including surface energies , 2011 .