Differential quadrature element for second strain gradient beam theory

In this paper, first we present the variational formulation for a second strain gradient Euler-Bernoulli beam theory for the first time. The governing equation and associated classical and non-classical boundary conditions are obtained. Later, we propose a novel and efficient differential quadrature element based on Lagrange interpolation to solve the eight order partial differential equation associated with the second strain gradient Euler-Bernoulli beam theory. The second strain gradient theory has displacement, slope, curvature and triple displacement derivative as degrees of freedom. A generalize scheme is proposed herein to implement these multi-degrees of freedom in a simplified and efficient way. The proposed element is based on the strong form of governing equation and has displacement as the only degree of freedom in the domain, whereas, at the boundaries it has displacement, slope, curvature and triple derivative of displacement. A novel DQ framework is presented to incorporate the classical and non-classical boundary conditions by modifying the conventional weighting coefficients. The accuracy and efficiency of the proposed element is demonstrated through numerical examples on static, free vibration and stability analysis of second strain gradient elastic beams for different boundary conditions and intrinsic length scale values.

[1]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE AND LONG-TERM INTEGRATION , 1971 .

[2]  R. D. Mindlin,et al.  On first strain-gradient theories in linear elasticity , 1968 .

[3]  W. T. Koiter.  Couple-stresses in the theory of elasticity , 1963 .

[4]  Alfred G. Striz,et al.  High-accuracy plane stress and plate elements in the quadrature element method , 1995 .

[5]  Hejun Du,et al.  Application of generalized differential quadrature method to structural problems , 1994 .

[6]  Xinwei Wang,et al.  STATIC ANALYSIS OF FRAME STRUCTURES BY THE DIFFERENTIAL QUADRATURE ELEMENT METHOD , 1997 .

[7]  Charles W. Bert,et al.  The differential quadrature method for irregular domains and application to plate vibration , 1996 .

[8]  Bo Liu,et al.  High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain , 2009 .

[9]  G. Karami,et al.  Application of a new differential quadrature methodology for free vibration analysis of plates , 2003 .

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

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

[12]  Chien H. Wu.  Cohesive elasticity and surface phenomena , 1992 .

[13]  Xinwei Wang,et al.  Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method , 2004 .

[14]  Application of generalized differential quadrature to vibration analysis , 1995 .

[15]  Ö. Civalek.  Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns , 2004 .

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

[17]  D. Beskos,et al.  Static and Dynamic BEM Analysis of Strain Gradient Elastic Solids and Structures , 2012 .

[18]  Guo-Wei Wei,et al.  A note on the numerical solution of high-order differential equations , 2003 .

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

[20]  D. Beskos,et al.  Dynamic analysis of gradient elastic flexural beams , 2003 .

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

[22]  Ö. Civalek,et al.  Harmonic differential quadrature (HDQ) for axisymmetric bending analysis of thin isotropic circular plates , 2004 .

[23]  S. Timoshenko.  Theory of Elastic Stability , 1936 .

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

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

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

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

[28]  N.,et al.  A PHENOMENOLOGICAL THEORY FOR STRAIN GRADIENT EFFECTS IN PLASTICITY , 2002 .

[29]  Alfred G. Striz,et al.  Static analysis of structures by the quadrature element method , 1994 .

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

[31]  Gui-Rong Liu,et al.  Differential quadrature solutions of eighth-order boundary-value differential equations , 2002 .

[32]  Xinwei Wang,et al.  Free vibration analysis of multiple-stepped beams by the differential quadrature element method , 2013, Appl. Math. Comput..

[33]  Ghodrat Karami,et al.  A new differential quadrature methodology for beam analysis and the associated differential quadrature element method , 2002 .

[34]  Chunhua Jin,et al.  Novel weak form quadrature element method with expanded Chebyshev nodes , 2014, Appl. Math. Lett..

[35]  Guirong Liu,et al.  Application of generalized differential quadrature rule to sixth‐order differential equations , 2000 .

[36]  H. Zhong,et al.  Analysis of thin plates by the weak form quadrature element method , 2012 .

[37]  C. Bert,et al.  Differential Quadrature Method in Computational Mechanics: A Review , 1996 .

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

[39]  A. K. Lazopoulos,et al.  Dynamic response of thin strain gradient elastic beams , 2012 .

[40]  S. Timoshenko,et al.  Vibration problem in engineering / S. Timoshenko , 1955 .

[41]  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 .