A new meshfree method for modeling strain gradient microbeams

The incapability of classical elasticity theory of accurately modeling the deformation behavior of structures at micro- and nanoscales has necessitated the development of more advanced theories. The strain gradient theory, being one of such theories, involves higher-order spatial derivatives of the field variables. However, except for few cases, there exists no analytical solution based on the strain gradient theory. This paper proposes a novel meshfree method with modified point interpolation functions possessing the Kronecker delta property; it is proposed to incorporate the strain gradient formulation into the Euler–Bernoulli beam theory. In the present method, the continuity of the shape function and its higher derivatives, appearing in the general form of the strain gradient theory, can be much more conveniently accommodated compared to the FEM. In addition, the present approach is based on the global weak form which is computationally less costly as compared to meshfree methods based on local weak form. The validity of the method is demonstrated by comparing the results with both analytical and experimental results for beams at macro- and microscales for static and dynamic loadings.

[1]  J. Oden,et al.  H‐p clouds—an h‐p meshless method , 1996 .

[2]  E. Aifantis.  On the role of gradients in the localization of deformation and fracture , 1992 .

[3]  René Chambon,et al.  Large strain finite element analysis of a local second gradient model: application to localization , 2002 .

[4]  Hyun Gyu Kim,et al.  Analysis of thin beams, using the meshless local Petrov–Galerkin method, with generalized moving least squares interpolations , 1999 .

[5]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[6]  Satya N. Atluri,et al.  The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics , 2000 .

[7]  C. Lim.  On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection , 2010 .

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

[9]  Mohammad Taghi Ahmadian,et al.  Strain gradient beam element , 2013 .

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

[11]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[12]  R. D. Mindlin.  Micro-structure in linear elasticity , 1964 .

[13]  J. F. T. Pittman,et al.  Isoparametric Hermite elements , 1994 .

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

[15]  Satya N. Atluri,et al.  Analysis of Materials with Strain-Gradient Effects: A Meshless Local Petrov-Galerkin(MLPG) Approach, with Nodal Displacements only , 2003 .

[16]  Ioannis Vardoulakis,et al.  A finite element displacement formulation for gradient elastoplasticity , 2001, Bifurcation and Localisation Theory in Geomechanics.

[17]  Chen Wanji,et al.  Finite element formulations of strain gradient theory for microstructures and the C0–1 patch test , 2004 .

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

[19]  J. Monaghan.  Why Particle Methods Work , 1982 .

[20]  R. Batra,et al.  Free vibrations of a strain gradient beam by the method of initial values , 2012 .

[21]  V. Kouznetsova,et al.  Multi-scale second-order computational homogenization of multi-phase materials : a nested finite element solution strategy , 2004 .

[22]  K. Liew,et al.  Effect of higher-order deformation gradients on buckling of single-walled carbon nanotubes , 2014 .

[23]  Guirong Liu.  Mesh Free Methods: Moving Beyond the Finite Element Method , 2002 .

[24]  V. Kouznetsova,et al.  Size of a representative volume element in a second-order computational homogenization framework , 2004 .

[25]  G. E. Exadaktylos,et al.  Gradient elasticity with surface energy: Mode-I crack problem , 1998 .

[26]  N. Fleck,et al.  FINITE ELEMENTS FOR MATERIALS WITH STRAIN GRADIENT EFFECTS , 1999 .

[27]  R. D. Mindlin.  MICROSTRUCTURE IN LINEAR ELASTICITY , 1999 .

[28]  Ellen Kuhl,et al.  Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity , 2010 .

[29]  Shenjie Zhou,et al.  A micro scale Timoshenko beam model based on strain gradient elasticity theory , 2010 .

[30]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[31]  E. Aifantis,et al.  Numerical modeling of size effects with gradient elasticity - Formulation, meshless discretization and examples , 2002 .

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

[33]  N. Aravas,et al.  Mixed finite element formulations of strain-gradient elasticity problems , 2002 .

[34]  C. Sansour,et al.  A strain gradient generalized continuum approach for modelling elastic scale effects , 2009 .

[35]  S. Chaterjee,et al.  A large deflection model for the pull-in analysis of electrostatically actuated microcantilever beams , 2009 .

[36]  YuanTong Gu,et al.  A local point interpolation method (LPIM) for static and dynamic analysis of thin beams , 2001 .

[37]  Farhang Daneshmand,et al.  Combined strain-inertia gradient elasticity in free vibration shell analysis of single walled carbon nanotubes using shell theory , 2014, Appl. Math. Comput..

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

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

[40]  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 .

[41]  M. Porfiri,et al.  Vibrations of narrow microbeams predeformed by an electric field , 2008 .

[42]  E. Aifantis,et al.  Cracks in gradient elastic bodies with surface energy , 1996 .

[43]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

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