Size-dependent analysis of homogeneous and functionally graded microplates using IGA and a non-classical Kirchhoff plate theory

Abstract The paper presents an effective thin plate formulation based on isogeometric analysis (IGA) and a non-classical Kirchhoff plate theory to study static bending, free vibration, and buckling behaviors of homogeneous and functionally graded microplates. The small scale effects are captured using a non-classical Kirchhoff plate theory which is developed based on a modified couple stress theory. The requirement for C 1 -continuity in terms of the non-classical Kirchhoff plate theory is straightforwardly possessed with the aid of inherent high-order continuity of non-uniform rational B-spline (NURBS). Studies on convergence and comparison with reference solutions are demonstrated in order to show the effectiveness and accuracy of the proposed method. Numerical examples are presented to illustrate the effects of small scale on the mechanical response of homogeneous and functionally graded microplates. The results reveal that the small scale effects lead to a reduction of deflection and an increase in frequency and buckling loads because of an increase in plate stiffness, and more importantly the small scale effects are significant for the thin plates.

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

[2]  Sohichi Hirose,et al.  NURBS-based isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method , 2016 .

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

[4]  J. N. Reddy,et al.  Energy principles and variational methods in applied mechanics , 2002 .

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

[6]  Tinh Quoc Bui,et al.  On the thermal buckling analysis of functionally graded plates with internal defects using extended isogeometric analysis , 2016 .

[7]  Bert Jüttler,et al.  IETI – Isogeometric Tearing and Interconnecting , 2012, Computer methods in applied mechanics and engineering.

[8]  T. Q. Bui,et al.  A simple FSDT-based isogeometric analysis for geometrically nonlinear analysis of functionally graded plates , 2015 .

[9]  Reza Ansari,et al.  Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates: An isogeometric analysis , 2016 .

[10]  Sohichi Hirose,et al.  A cutout isogeometric analysis for thin laminated composite plates using level sets , 2015 .

[11]  Huu-Tai Thai,et al.  SIZE-DEPENDENT FUNCTIONALLY GRADED KIRCHHOFF AND MINDLIN PLATE MODELS BASED ON A MODIFIED COUPLE STRESS THEORY , 2013 .

[12]  Hung Nguyen-Xuan,et al.  Isogeometric analysis for functionally graded microplates based on modified couple stress theory , 2016, ArXiv.

[13]  Xin-Lin Gao,et al.  A non-classical model for circular Kirchhoff plates incorporating microstructure and surface energy effects , 2015 .

[14]  H. Noori,et al.  The size-dependent vibration analysis of micro-plates based on a modified couple stress theory , 2011 .

[15]  Mohammad Taghi Ahmadian,et al.  A nonlinear Timoshenko beam formulation based on the modified couple stress theory , 2010 .

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

[17]  Ernst Rank,et al.  Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries , 2014 .

[18]  Nuttawit Wattanasakulpong,et al.  Buckling isogeometric analysis of functionally graded plates under combined thermal and mechanical loads , 2017 .

[19]  Wanji Chen,et al.  A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation , 2011 .

[20]  J. N. Reddy,et al.  A Nonclassical Reddy-Levinson Beam Model Based on a Modified Couple Stress Theory , 2010 .

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

[22]  F. F. Mahmoud,et al.  A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects , 2014 .

[23]  T. Rabczuk,et al.  Extended isogeometric analysis for dynamic fracture in multiphase piezoelectric/piezomagnetic composites , 2016 .

[24]  B. Zhang,et al.  Size-dependent functionally graded beam model based on an improved third-order shear deformation theory , 2014 .

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

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

[27]  B. K. Mishra,et al.  Numerical simulation of functionally graded cracked plates using NURBS based XIGA under different loads and boundary conditions , 2015 .

[28]  Chao-feng Lu,et al.  Nonlinear Responses of Nanoscale FGM Films Including the Effects of Surface Energies , 2011, IEEE Transactions on Nanotechnology.

[29]  Roland Wüchner,et al.  Analysis in computer aided design: Nonlinear isogeometric B-Rep analysis of shell structures , 2015 .

[30]  L. Ke,et al.  Flow-induced vibration and instability of embedded double-walled carbon nanotubes based on a modified couple stress theory , 2011 .

[31]  Huu-Tai Thai,et al.  A size-dependent functionally graded Reddy plate model based on a modified couple stress theory , 2013 .

[32]  Liao-Liang Ke,et al.  Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory , 2011 .

[33]  Sven Klinkel,et al.  The weak substitution method – an application of the mortar method for patch coupling in NURBS‐based isogeometric analysis , 2015 .

[34]  Xin-Lin Gao,et al.  A Nonclassical Model for Circular Mindlin Plates Based on a Modified Couple Stress Theory , 2014 .

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

[36]  Qin Qian,et al.  Vibration Analysis of Microscale Plates Based on Modified Couple Stress Theory , 2010 .

[37]  Yuri Bazilevs,et al.  The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches , 2010 .

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

[39]  T. Q. Bui,et al.  Buckling and vibration extended isogeometric analysis of imperfect graded Reissner-Mindlin plates with internal defects using NURBS and level sets , 2016 .

[40]  Onur Atak,et al.  A robust patch coupling method for NURBS-based isogeometric analysis of non-conforming multipatch surfaces , 2017 .

[41]  Jie Yang,et al.  Nonlinear free vibration of size-dependent functionally graded microbeams , 2012 .

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

[43]  F. F. Mahmoud,et al.  Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects , 2014 .

[44]  H. Nguyen-Xuan,et al.  Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory , 2015 .

[45]  X.-L. Gao.  A new Timoshenko beam model incorporating microstructure and surface energy effects , 2015 .

[46]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[47]  Hung Nguyen-Xuan,et al.  An efficient computational approach for size-dependent analysis of functionally graded nanoplates , 2015 .

[48]  A. Rinzler,et al.  Carbon nanotube actuators , 1999, Science.

[49]  T. Q. Bui.  Extended isogeometric dynamic and static fracture analysis for cracks in piezoelectric materials using NURBS , 2015 .

[50]  Alireza Nateghi,et al.  Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory , 2012 .

[51]  G. Y. Zhang,et al.  A microstructure- and surface energy-dependent third-order shear deformation beam model , 2015 .

[52]  J. Reddy,et al.  A non-classical third-order shear deformation plate model based on a modified couple stress theory , 2013 .

[53]  Yuri Bazilevs,et al.  Isogeometric fluid–structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines , 2012 .

[54]  Xin-Lin Gao,et al.  A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects , 2016 .

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

[56]  Sritawat Kitipornchai,et al.  Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory , 2012 .

[57]  Norman A. Fleck,et al.  A phenomenological theory for strain gradient effects in plasticity , 1993 .

[58]  J. N. Reddy,et al.  A non-classical Mindlin plate model based on a modified couple stress theory , 2011 .

[59]  Stéphane Bordas,et al.  Isogeometric locking-free plate element: A simple first order shear deformation theory for functionally graded plates , 2014 .

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