Timoshenko beam model for static bending of nanowires with surface effects

Abstract In this paper, surface effects on the elastic behavior of static bending nanowires (NWs) are studied by using a comprehensive Timoshenko beam model. For NWs with different boundary conditions, explicit solutions are derived to study the combined effects of residual surface stress, surface elasticity and shear deformation on the effective stiffness and Young's modulus of the NWs. The stiffness is found to be size-dependent and this dependence is more significant for slender NWs. Residual surface stress tends to increase or decrease the stiffness of NWs depending on their boundary conditions, and the shear deformation always makes the NWs softer compared with the results of the Euler–Bernoulli beam model. The solutions agree well with experimental measurement in predicting the Young's modulus, especially for NWs with small length-to-thickness ratio. This work might be helpful for characterizing the mechanical properties of NWs and the design of nanobeam-based devices.

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

[2]  Xiaodong Li,et al.  Young’s modulus of ZnO nanobelts measured using atomic force microscopy and nanoindentation techniques , 2006, Nanotechnology.

[3]  Min Zhou,et al.  Orientation and size dependence of the elastic properties of zinc oxide nanobelts , 2005 .

[4]  David N. McIlroy,et al.  On the importance of boundary conditions on nanomechanical bending behavior and elastic modulus determination of silver nanowires , 2006 .

[5]  Bernard Nysten,et al.  Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy , 2004 .

[6]  Michael L. Roukes,et al.  Very High Frequency Silicon Nanowire Electromechanical Resonators , 2007 .

[7]  Y. S. Zhang,et al.  Size dependence of Young's modulus in ZnO nanowires. , 2006, Physical review letters.

[8]  John E. Sader,et al.  Effect of surface stress on the stiffness of cantilever plates. , 2007 .

[9]  R. Rudd,et al.  Mechanics of silicon nanowires: size-dependent elasticity from first principles , 2008 .

[10]  Xi-Qiao Feng,et al.  Timoshenko beam model for buckling and vibration of nanowires with surface effects , 2009 .

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

[12]  Jian-Gang Guo,et al.  The size-dependent bending elastic properties of nanobeams with surface effects , 2007 .

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

[14]  Jianbin Xu,et al.  Surface effects on elastic properties of silver nanowires: Contact atomic-force microscopy , 2006 .

[15]  Xi-Qiao Feng,et al.  Effects of surface elasticity and residual surface tension on the natural frequency of microbeams , 2007 .

[16]  Harold S. Park,et al.  Quantifying the size-dependent effect of the residual surface stress on the resonant frequencies of silicon nanowires if finite deformation kinematics are considered , 2009, Nanotechnology.

[17]  Tungyang Chen,et al.  Derivation of the generalized Young-Laplace equation of curved interfaces in nanoscaled solids , 2006 .

[18]  M. E. Gurtin,et al.  A general theory of curved deformable interfaces in solids at equilibrium , 1998 .

[19]  Elisa Riedo,et al.  Elastic property of vertically aligned nanowires. , 2005, Nano letters.

[20]  J. R. Hutchinson.  Shear coefficients for Timoshenko beam theory , 2001 .

[21]  D. Srivastava,et al.  Silicon carbide nanowires under external loads: An atomistic simulation study , 2006 .

[22]  R. Williams,et al.  Mechanical properties of self-welded silicon nanobridges , 2005 .

[23]  Rashid Bashir,et al.  BioMEMS: state-of-the-art in detection, opportunities and prospects. , 2004, Advanced drug delivery reviews.

[24]  Zhong Lin Wang,et al.  Piezoelectric Nanogenerators Based on Zinc Oxide Nanowire Arrays , 2006, Science.

[25]  Khashayar Babaei Gavan,et al.  Size-dependent effective Young’s modulus of silicon nitride cantilevers , 2009 .

[26]  Charles M. Lieber,et al.  High Performance Silicon Nanowire Field Effect Transistors , 2003 .

[27]  A. Rafsanjani,et al.  Free vibration of microscaled Timoshenko beams , 2009 .

[28]  R. Cook,et al.  Diameter-Dependent Radial and Tangential Elastic Moduli of ZnO Nanowires , 2007 .

[29]  Tong-Yi Zhang,et al.  Size-dependent surface stress, surface stiffness, and Young’s modulus of hexagonal prism [111] β-SiC nanowires , 2008 .

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