Linear eigenvalue analysis of laminated thin plates including the strain gradient effect by means of conforming and nonconforming rectangular finite elements

Abstract The paper presents a finite element method to investigate the critical buckling loads and the natural frequencies of laminated Kirchhoff plates including the nonlocal strain gradient effect, which could have considerably consequences at the nanoscale. With respect to the existing literature, the proposed numerical methodology is developed to deal with general stacking sequences of orthotropic layers with arbitrary orientations and various boundary conditions. The resulting membrane-bending coupling is emphasized in the formulation, which requires to study the whole set of partial differential equations. The membrane and bending degrees of freedom are all approximated by means of Hermite interpolating functions with higher-order continuity requirements. To this aim, regular rectangular finite elements based on conforming (C) and nonconforming (NC) approaches are used. A wide validation procedure is carried out to prove the effectiveness of the proposed formulation. A set of new results is presented for general mechanical configurations with arbitrary restraints.

[1]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[2]  Jeremy Bleyer,et al.  Automated formulation and resolution of limit analysis problems , 2020, Computers & Structures.

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

[4]  Moshe Eisenberger,et al.  Shear Buckling of Thin Plates with Constant In-Plane Stresses , 2007 .

[5]  Shenjie Zhou,et al.  A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory , 2014 .

[6]  Elias C. Aifantis,et al.  Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua , 2010 .

[7]  D. W. Scharpf,et al.  The TUBA Family of Plate Elements for the Matrix Displacement Method , 1968, The Aeronautical Journal (1968).

[8]  Pradeep Sharma,et al.  Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities , 2003 .

[9]  A. Giannakopoulos,et al.  Variational analysis of gradient elastic flexural plates under static loading , 2010 .

[10]  P. Hu,et al.  A 4-node quasi-conforming quadrilateral element for couple stress theory immune to distorted mesh , 2016 .

[11]  H. Altenbach,et al.  On the Direct Approach in the Theory of Second Gradient Plates , 2015 .

[12]  R. Luciano,et al.  Hygro-thermal vibrations and buckling of laminated nanoplates via nonlocal strain gradient theory , 2020 .

[13]  K. A. Lazopoulos,et al.  On the gradient strain elasticity theory of plates , 2004 .

[14]  Zhang Ran,et al.  NONLINEAR TRANSIENT ANALYSIS FOR LARGE-SCALE DYNAMICS OF MICROELECTROMECHANICAL SYSTEMS WITH THE MOR-PIM METHOD , 2010 .

[15]  R. Luciano,et al.  Buckling loads of nano-beams in stress-driven nonlocal elasticity , 2020, Mechanics of Advanced Materials and Structures.

[16]  R. Luciano,et al.  Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams , 2019, Composites Part B: Engineering.

[17]  L S D Morley,et al.  The constant-moment plate-bending element , 1971 .

[18]  A new general‐purpose least‐squares finite element model for steady incompressible low‐viscosity laminar flow using isoparametric C1‐continuous Hermite elements , 1994 .

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

[20]  Junfeng Zhao,et al.  A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory , 2011 .

[21]  K. Bell.  A refined triangular plate bending finite element , 1969 .

[22]  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 .

[23]  Marco Amabili,et al.  Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments , 2004 .

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

[25]  F. Fraternali,et al.  Size effect and dynamic properties of 2D lattice materials , 2017 .

[26]  R. Luciano,et al.  Critical Temperatures for Vibrations and Buckling of Magneto-Electro-Elastic Nonlocal Strain Gradient Plates , 2021, Nanomaterials.

[27]  P. Trovalusci,et al.  Constitutive Relations for Elastic Microcracked Bodies: From a Lattice Model to a Multifield Continuum Description , 1999 .

[28]  P. Trovalusci,et al.  Stress distribution around an elliptic hole in a plate with ‘implicit’ and ‘explicit’ non-local models , 2021, Composite Structures.

[29]  B. P. Patel,et al.  Analytical solution for strain gradient elastic Kirchhoff rectangular plates under transverse static loading , 2019, European Journal of Mechanics - A/Solids.

[30]  D. E. Beskos,et al.  Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates , 2008 .

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

[32]  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 .

[33]  A. Ibrahimbegovic,et al.  Variational formulation of micropolar elasticity using 3D hexahedral finite-element interpolation with incompatible modes , 2018, Computers & Structures.

[34]  P. Trovalusci,et al.  Deformation of atomic models and their equivalent continuum counterparts using Eringen’s two-phase local/nonlocal model , 2019, Mechanics Research Communications.

[35]  Ian M. Smith.  A finite element analysis for “moderately thick” rectangular plates in bending , 1968 .

[36]  Xiaopeng Zhang,et al.  Vibration and thermal buckling analysis of rotating nonlocal functionally graded nanobeams in thermal environment , 2020 .

[37]  Nathan Ida,et al.  Introduction to the Finite Element Method , 1997 .

[38]  A. Tessler,et al.  A robust four-node quadrilateral element for laminated composite and sandwich plates based on Refined Zigzag Theory , 2021 .

[39]  R. Luciano,et al.  Semi-analytical static analysis of nonlocal strain gradient laminated composite nanoplates in hygrothermal environment , 2021, Journal of the Brazilian Society of Mechanical Sciences and Engineering.

[40]  P. Trovalusci,et al.  A continuum model with microstructure for materials with flaws and inclusions , 1998 .

[41]  Chong-Jun Li,et al.  A quadrilateral spline element for couple stress/strain gradient elasticity , 2014 .

[42]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[43]  S. H. Lo,et al.  A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity , 2011 .

[44]  E. Kosko,et al.  A HIGH PRECISION TRIANGULAR PLATE-BENDING ELEMENT, , 1968 .

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

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

[47]  B. P. Patel,et al.  A new computationally efficient finite element formulation for nanoplates using second-order strain gradient Kirchhoff's plate theory , 2019, Composites Part B: Engineering.

[48]  Paul Steinmann,et al.  Reduced-Order Modelling and Homogenisation in Magneto-Mechanics: A Numerical Comparison of Established Hyper-Reduction Methods , 2019, Mathematical and Computational Applications.

[49]  Finite Anticlastic Bending of Hyperelastic Solids and Beams , 2018 .

[50]  Rui Xu,et al.  Data-driven multiscale simulation of FRP based on material twins , 2021 .

[51]  A. Ferreira,et al.  Conforming and nonconforming laminated finite element Kirchhoff nanoplates in bending using strain gradient theory , 2020 .

[52]  N. Fantuzzi,et al.  Static finite element analysis of thin laminated strain gradient nanoplates in hygro-thermal environment , 2020, Continuum Mechanics and Thermodynamics.

[53]  R. Luciano,et al.  Analytical solution of cross- and angle-ply nano plates with strain gradient theory for linear vibrations and buckling , 2019, Mechanics of Advanced Materials and Structures.

[54]  Baolin Wang,et al.  Vibration of nanoscale plates with surface energy via nonlocal elasticity , 2011 .

[55]  L. Morley.  The Triangular Equilibrium Element in the Solution of Plate Bending Problems , 1968 .

[56]  M. Amabili,et al.  A comprehensive electro-magneto-elastic buckling and bending analyses of three-layered doubly curved nanoshell, based on nonlocal three-dimensional theory , 2020 .

[57]  Le Van Lich,et al.  Size and surface effects on mechanical behavior of thin nanoplates incorporating microstructures using isogeometric analysis , 2019, Computers & Structures.

[58]  E. Aifantis,et al.  Finite element analysis with staggered gradient elasticity , 2008 .

[59]  E. Aifantis,et al.  A simple approach to solve boundary-value problems in gradient elasticity , 1993 .

[60]  R. Luciano,et al.  Trigonometric Solution for the Bending Analysis of Magneto-Electro-Elastic Strain Gradient Nonlocal Nanoplates in Hygro-Thermal Environment , 2021 .

[61]  R. Melosh.  BASIS FOR DERIVATION OF MATRICES FOR THE DIRECT STIFFNESS METHOD , 1963 .

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

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

[64]  H. Askes,et al.  Finite element implementation of a multi-scale dynamic piezomagnetic continuum model , 2020 .

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

[66]  M. Mahmoodi,et al.  A nonlinear thick plate formulation based on the modified strain gradient theory , 2018 .

[67]  S. C. Pradhan,et al.  Nonlocal elasticity theory for vibration of nanoplates , 2009 .

[68]  E. Aifantis,et al.  On the structure of the mode III crack-tip in gradient elasticity , 1992 .

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

[70]  Gerardo Carpentieri,et al.  Three-dimensional modeling of the wave dynamics of tensegrity lattices , 2017 .

[71]  F. Bogner,et al.  The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulae , 1965 .

[72]  Xu Yin,et al.  Generalized conforming plate bending elements using point and line compatibility conditions , 1995 .

[73]  Alberto Milazzo,et al.  A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels , 2018 .

[74]  R. Luciano,et al.  Solution for cross- and angle-ply laminated Kirchhoff nano plates in bending using strain gradient theory , 2019, Composites Part B: Engineering.

[75]  Richard H. Gallagher,et al.  Finite Element Analysis: Fundamentals , 1975 .

[76]  William Duncan,et al.  The effectiveness of excessive nodal continuities in the finite element analysis of thin rectangular and skew plates in bending , 1970 .

[77]  Jae-Hoon Choi,et al.  A 10-node tetrahedral element with condensed Lagrange multipliers for the modified couple stress theory , 2021 .

[78]  P. Trovalusci,et al.  Mechanical Behavior of Anisotropic Composite Materials as Micropolar Continua , 2019, Front. Mater..

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

[80]  Fulvia Taddei.  Introduzione al metodo degli elementi finiti , 2012 .

[81]  Robert E. Fulton,et al.  Accuracy and Convergence of Finite Element Approximations , 1968 .

[82]  J. N. Reddy,et al.  Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates , 2009 .

[83]  Holm Altenbach.  On the determination of transverse shear stiffnesses of orthotropic plates , 2000 .

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

[85]  The Bending of Beams in Finite Elasticity , 2020, Journal of Elasticity.

[86]  E. Aifantis.  Update on a class of gradient theories , 2003 .

[87]  J. N. Reddy,et al.  A general third-order theory of functionally graded plates with modified couple stress effect and the von Kármán nonlinearity: theory and finite element analysis , 2015 .

[88]  S. Hosseini-Hashemi,et al.  On the local/nonlocal piezoelectric nanobeams: Vibration, buckling, and energy harvesting , 2021 .

[89]  C. Medaglia,et al.  Stress-driven two-phase integral elasticity for torsion of nano-beams , 2018, Composites Part B: Engineering.

[90]  T. Vo,et al.  Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis , 2017 .

[91]  P. Trovalusci.  Molecular Approaches for Multifield Continua: origins and current developments , 2014 .

[92]  K. K. Żur,et al.  Free vibration and buckling analyses of magneto-electro-elastic FGM nanoplates based on nonlocal modified higher-order sinusoidal shear deformation theory , 2020 .

[93]  R. Luciano,et al.  A Fully Gradient Model for Euler-Bernoulli Nanobeams , 2015 .

[94]  O. C. Zienkiewicz,et al.  The finite element method, fourth edition; volume 2: solid and fluid mechanics, dynamics and non-linearity , 1991 .

[95]  M. Roukes.  Nanoelectromechanical Systems , 2000, cond-mat/0008187.

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

[97]  B. M. Fraeijs de Veubeke,et al.  A conforming finite element for plate bending , 1968 .

[98]  P. C. Dunne.  Complete Polynomial Displacement Fields for Finite Element Method , 1968, The Aeronautical Journal (1968).