Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory

Abstract Dynamic stability of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and Timoshenko beam theory. This non-classical Timoshenko beam model contains a material length scale parameter and can interpret the size effect. The material properties of FGM microbeams are assumed to vary in the thickness direction and are estimated though Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using the Hamilton’s principle. The differential quadrature (DQ) method is employed to convert the governing differential equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions are determined by Bolotin’s method. Free vibration and static buckling are also discussed as subset problems. A parametric study is conducted to investigate the influences of the length scale parameter, gradient index and length-to-thickness ratio on the dynamic stability characteristics of FGM microbeams with hinged–hinged and clamped–clamped end supports. Results show that the size effect on the dynamic stability characteristics is significant only when the thickness of beam has a similar value to the material length scale parameter.

[1]  D. Clarke,et al.  Size dependent hardness of silver single crystals , 1995 .

[2]  Shenjie Zhou,et al.  Static and dynamic analysis of micro beams based on strain gradient elasticity theory , 2009 .

[3]  T. Kocatürk,et al.  Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load , 2009 .

[4]  Shenjie Zhou,et al.  The size-dependent natural frequency of Bernoulli–Euler micro-beams , 2008 .

[5]  Yong Qing Fu,et al.  TiNi-based thin films in MEMS applications: a review , 2004 .

[6]  Wang Daobin,et al.  Dynamic stability analysis of FGM plates by the moving least squares differential quadrature method , 2007 .

[7]  M. Ashby,et al.  Strain gradient plasticity: Theory and experiment , 1994 .

[8]  P. Tong,et al.  Couple stress based strain gradient theory for elasticity , 2002 .

[9]  A. Witvrouw,et al.  The Use of Functionally Graded Poly-SiGe Layers for MEMS Applications , 2005 .

[10]  M. Şi̇mşek,et al.  Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories , 2010 .

[11]  S. Kitipornchai,et al.  Flexural Vibration and Elastic Buckling of a Cracked Timoshenko Beam Made of Functionally Graded Materials , 2009 .

[12]  M. Şi̇mşek.  NON-LINEAR VIBRATION ANALYSIS OF A FUNCTIONALLY GRADED TIMOSHENKO BEAM UNDER ACTION OF A MOVING HARMONIC LOAD , 2010 .

[13]  M. Asghari,et al.  On the size-dependent behavior of functionally graded micro-beams , 2010 .

[14]  Lin Wang,et al.  Size-dependent vibration characteristics of fluid-conveying microtubes , 2010 .

[15]  Fan Yang,et al.  Torsion and bending of micron-scaled structures , 2001 .

[16]  Renato Natal Jorge,et al.  Natural frequencies of functionally graded plates by a meshless method , 2006 .

[17]  Kyung-Su Na,et al.  Three-dimensional thermomechanical buckling analysis for functionally graded composite plates , 2006 .

[18]  George C. Tsiatas,et al.  A new Kirchhoff plate model based on a modified couple stress theory , 2009 .

[19]  M. Şi̇mşek,et al.  VIBRATION ANALYSIS OF A FUNCTIONALLY GRADED BEAM UNDER A MOVING MASS BY USING DIFFERENT BEAM THEORIES , 2010 .

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

[21]  K. M. Liew,et al.  Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading , 2001 .

[22]  V. Radmilović,et al.  Metallic NEMS components fabricated from nanocomposite Al–Mo films , 2006 .

[23]  H. F. Tiersten,et al.  Effects of couple-stresses in linear elasticity , 1962 .

[24]  R. Jorge,et al.  A radial basis function approach for the free vibration analysis of functionally graded plates using a refined theory , 2007 .

[25]  A. Tounsi,et al.  An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions , 2010 .

[26]  Andrew W. Mcfarland,et al.  Role of material microstructure in plate stiffness with relevance to microcantilever sensors , 2005 .

[27]  M.Kh. Elmarghany,et al.  An efficient and simple refined theory for bending and vibration of functionally graded plates , 2009 .

[28]  Santosh Kapuria,et al.  Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation , 2008 .

[29]  R. Toupin.  Elastic materials with couple-stresses , 1962 .

[30]  S.M.R. Khalili,et al.  A MIXED RITZ-DQ METHOD FOR FORCED VIBRATION OF FUNCTIONALLY GRADED BEAMS CARRYING MOVING LOADS , 2010 .

[31]  M. Şi̇mşek,et al.  VIBRATION ANALYSIS OF A SINGLE-WALLED CARBON NANOTUBE UNDER ACTION OF A MOVING HARMONIC LOAD BASED ON NONLOCAL ELASTICITY THEORY , 2010 .

[32]  Jin H. Huang,et al.  Post-buckling analysis of functionally graded rectangular plates , 2007 .

[33]  S. K. Park,et al.  Bernoulli–Euler beam model based on a modified couple stress theory , 2006 .

[34]  Fan Yang,et al.  Experiments and theory in strain gradient elasticity , 2003 .

[35]  Hiroyuki Matsunaga,et al.  Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory , 2009 .

[36]  Mansour Darvizeh,et al.  Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading , 2008 .

[37]  E. Aifantis.  Strain gradient interpretation of size effects , 1999 .

[38]  J. N. Reddy,et al.  A microstructure-dependent Timoshenko beam model based on a modified couple stress theory , 2008 .

[39]  David C C Lam,et al.  Strain gradient plasticity effect in indentation hardness of polymers , 1999 .

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

[41]  K. Lazopoulos.  On bending of strain gradient elastic micro-plates , 2009 .

[42]  Jie Yang,et al.  Postbuckling analysis of edge cracked functionally graded Timoshenko beams under end shortening , 2009 .

[43]  V. V. Bolotin,et al.  Dynamic Stability of Elastic Systems , 1965 .

[44]  C. Shu.  Differential Quadrature and Its Application in Engineering , 2000 .

[45]  Anthony G. Evans,et al.  A microbend test method for measuring the plasticity length scale , 1998 .

[46]  Mohammad Taghi Ahmadian,et al.  Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory , 2010 .

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

[48]  M. R. Eslami,et al.  Buckling of thick functionally graded plates under mechanical and thermal loads , 2007 .

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

[50]  Romesh C. Batra,et al.  Pull-In Instabilities in Functionally Graded Microthermoelectromechanical Systems , 2008 .

[51]  Jun Zhu,et al.  DYNAMIC STABILITY OF FUNCTIONALLY GRADED PIEZOELECTRIC CIRCULAR CYLINDRICAL SHELLS , 2005 .

[52]  M. Ganapathi,et al.  Dynamic stability characteristics of functionally graded materials shallow spherical shells , 2007 .

[53]  Lin Wang,et al.  Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration , 2010 .

[54]  Jie Yang,et al.  Nonlinear vibration of edge cracked functionally graded Timoshenko beams , 2009 .