Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models

As a first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler–Bernoulli beam bending models following a simplified version of Mindlin’s strain gradient elasticity theory of form II. For both models, this leads to sixth-order boundary value problems with new types of boundary conditions that are given additional attributes singly and doubly, referring to a physically relevant distinguishing feature between free and prescribed curvature, respectively. Second, the variational formulations are analyzed with rigorous mathematical tools: the existence and uniqueness of weak solutions are established by proving continuity and ellipticity of the associated symmetric bilinear forms. This guarantees optimal convergence for conforming Galerkin discretization methods. Third, the variational analysis is extended to cover two other generalized beam models: another modification of the strain gradient elasticity theory and a modified version of the couple stress theory. A model comparison reveals essential differences and similarities in the physicality of these four closely related beam models: they demonstrate essentially two different kinds of parameter-dependent stiffening behavior, where one of these kinds (possessed by three models out of four) provides results in a very good agreement with the size effects of experimental tests. Finally, numerical results for isogeometric Galerkin discretizations with B-splines confirm the theoretical stability and convergence results. Influences of the gradient and thickness parameters connected to size effects, boundary layers and dispersion relations are studied thoroughly with a series of benchmark problems for statics and free vibrations. The size-dependency of the effective Young’s modulus is demonstrated for an auxetic cellular metamaterial ruled by bending-dominated deformation of cell struts.

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

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

[3]  Mohammad Taghi Ahmadian,et al.  Vibrational analysis of single-walled carbon nanotubes using beam element , 2009 .

[4]  R. Rafiee,et al.  On the modeling of carbon nanotubes: A critical review , 2014 .

[5]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

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

[7]  E. Aifantis,et al.  On Some Aspects in the Special Theory of Gradient Elasticity , 1997 .

[8]  Yunan Prawoto,et al.  Seeing auxetic materials from the mechanics point of view: A structural review on the negative Poisson’s ratio , 2012 .

[9]  E. Aifantis,et al.  Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results , 2011 .

[10]  M. Pulvirenti,et al.  Macroscopic Description of Microscopically Strongly Inhomogenous Systems: A Mathematical Basis for the Synthesis of Higher Gradients Metamaterials , 2015, 1504.08015.

[11]  M. Lazar,et al.  Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity , 2005, cond-mat/0502023.

[12]  Elias C. Aifantis,et al.  Exploring the applicability of gradient elasticity to certain micro/nano reliability problems , 2008 .

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

[14]  Alessandro Reali,et al.  Isogeometric analysis for sixth-order boundary value problems of gradient-elastic Kirchhoff plates , 2017 .

[15]  Wolfgang H. Müller,et al.  Applications of Strain Gradient Theories to the Size Effect in Submicro-Structures incl. Experimental Analysis of Elastic Material Parameters , 2022 .

[16]  Alex J. Zelhofer,et al.  Auxeticity in truss networks and the role of bending versus stretching deformation , 2016 .

[17]  Pierre Seppecher,et al.  Linear elastic trusses leading to continua with exotic mechanical interactions , 2011 .

[18]  Meral Tuna,et al.  Exact solution of Eringen's nonlocal integral model for bending of Euler–Bernoulli and Timoshenko beams , 2016 .

[19]  J. N. Reddy,et al.  Bending of Euler–Bernoulli beams using Eringen’s integral formulation: A paradox resolved , 2016 .

[20]  K. A. Lazopoulos,et al.  Bending and buckling of thin strain gradient elastic beams , 2010 .

[21]  Jaehong Lee,et al.  Modal analysis of carbon nanotubes and nanocones using FEM , 2012 .

[22]  Francesco dell’Isola,et al.  Dynamics of 1D nonlinear pantographic continua , 2017 .

[23]  K. F. Wang,et al.  A review on the application of modified continuum models in modeling and simulation of nanostructures , 2016 .

[24]  Huu-Tai Thai,et al.  A review of continuum mechanics models for size-dependent analysis of beams and plates , 2017 .

[25]  Sergey Y. Yurish Sensors and Biosensors , MEMS Technologies and its Applications , 2013 .

[26]  D. E. Beskos,et al.  Finite element static and stability analysis of gradient elastic beam structures , 2015 .

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

[28]  Alireza Beheshti Large deformation analysis of strain-gradient elastic beams , 2016 .

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

[30]  K. Harris,et al.  Flexible electronics under strain: a review of mechanical characterization and durability enhancement strategies , 2016, Journal of Materials Science.

[31]  Demosthenes Polyzos,et al.  Bending and stability analysis of gradient elastic beams , 2003 .

[32]  Gérard A. Maugin,et al.  A Historical Perspective of Generalized Continuum Mechanics , 2010 .

[33]  Xiangyang Zhang,et al.  A Review of Graphene on NEMS. , 2016, Recent patents on nanotechnology.

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

[35]  Robert Bogue,et al.  Recent developments in MEMS sensors: a review of applications, markets and technologies , 2013 .

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

[37]  Hwan-Sik Yoon,et al.  A Review on Electromechanical Devices Fabricated by Additive Manufacturing , 2017 .

[38]  Majid Minary-Jolandan,et al.  A Review of Mechanical and Electromechanical Properties of Piezoelectric Nanowires , 2012, Advanced materials.

[39]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[40]  Antti H. Niemi,et al.  Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems , 2016 .

[41]  Viacheslav Balobanov,et al.  Variational formulations and isogeometric analysis for the dynamics of anisotropic gradient-elastic Euler-Bernoulli and shear-deformable beams , 2018 .

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

[43]  Andrei V. Metrikine,et al.  Mechanics of generalized continua : one hundred years after the Cosserats , 2010 .

[44]  Pierre Seppecher,et al.  Truss Modular Beams with Deformation Energy Depending on Higher Displacement Gradients , 2003 .

[45]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[46]  S. K. Park,et al.  Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem , 2007 .

[47]  L. Tjeng,et al.  Orbitally driven spin-singlet dimerization in S=1 La4Ru2O10. , 2006, Physical review letters.

[48]  Sergei Khakalo,et al.  Gradient-elastic stress analysis near cylindrical holes in a plane under bi-axial tension fields , 2017 .

[49]  J. Niiranen,et al.  Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates , 2017 .

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

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

[52]  F. Auricchio,et al.  Single-variable formulations and isogeometric discretizations for shear deformable beams , 2015 .

[53]  R. D. Mindlin Second gradient of strain and surface-tension in linear elasticity , 1965 .

[54]  B. Akgöz,et al.  Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory , 2012 .

[55]  Robert Bogue Towards the trillion sensors market , 2014 .

[56]  Hendrik Speleers,et al.  A locking-free model for Reissner-Mindlin plates: Analysis and isogeometric implementation via NURBS and triangular NURPS , 2015 .

[57]  Gérard A. Maugin,et al.  Generalized Continuum Mechanics: What Do We Mean by That? , 2010 .

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

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

[60]  Gary R. Consolazio,et al.  Finite Elements , 2007, Handbook of Dynamic System Modeling.

[61]  D. Beskos,et al.  Wave dispersion in gradient elastic solids and structures: A unified treatment , 2009 .

[62]  C. Polizzotto A gradient elasticity theory for second-grade materials and higher order inertia , 2012 .

[63]  J. Paavola,et al.  Size effects on centrosymmetric anisotropic shear deformable beam structures , 2017 .

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

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

[66]  M. Nikkhah-bahrami,et al.  A discussion on incorporating the Poisson effect in microbeam models based on modified couple stress theory , 2015 .

[67]  I. Vardoulakis,et al.  Bifurcation Analysis in Geomechanics , 1995 .

[68]  Katja Bachmeier,et al.  Finite Elements Theory Fast Solvers And Applications In Solid Mechanics , 2017 .

[69]  J. Bleustein A note on the boundary conditions of toupin's strain-gradient theory , 1967 .

[70]  M E Khater,et al.  A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams , 2016 .

[71]  S. M. Mousavi,et al.  Strain and velocity gradient theory for higher-order shear deformable beams , 2015 .

[72]  Sergei Khakalo,et al.  Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software , 2017, Comput. Aided Des..

[73]  K. Novoselov,et al.  The mechanics of graphene nanocomposites: A review , 2012 .

[74]  S. Shen,et al.  A new Bernoulli-Euler beam model based on a simplified strain gradient elasticity theory and its applications , 2014 .

[75]  Leopoldo Greco,et al.  B-Spline interpolation of Kirchhoff-Love space rods , 2013 .