TY - JOUR
T1 - Strain gradient differential quadrature finite element for moderately thick micro-plates
AU - Zhang, Bo
AU - Li, Heng
AU - Kong, Liulin
AU - Zhang, Xu
AU - Feng, Zhipeng
N1 - Funding Information:
National Natural Science Foundation of China, 11602204; 11672251; Research Grants Council, University Grants Committee, 15204719; 15209918; General Research Fund of Hong Kong SAR Government (GRF), 152099/18E; Hong Kong Polytechnic University, 152047/19E Funding information
Funding Information:
information National Natural Science Foundation of China, 11602204; 11672251; Research Grants Council, University Grants Committee, 15204719; 15209918; General Research Fund of Hong Kong SAR Government (GRF), 152099/18E; Hong Kong Polytechnic University, 152047/19EThis work is financially supported by National Natural Science Foundation of China (Nos.11602204 and 11672251), the project of the Hong Kong Polytechnic University (PolyU 152047/19E), the General Research Fund (GRF) Grant (BRE/PolyU 152099/18E).
Publisher Copyright:
© 2020 John Wiley & Sons Ltd
PY - 2020/12/30
Y1 - 2020/12/30
N2 - In this study, we integrate the advantages of differential quadrature method (DQM) and finite element method (FEM) to construct a C1-type four-node quadrilateral element with 48 degrees of freedom (DOF) for strain gradient Mindlin micro-plates. This element is free of shape functions and shear locking. The C1-continuity requirements of deflection and rotation functions are accomplished by a fourth-order differential quadrature (DQ)-based geometric mapping scheme, which facilitates the conversion of the displacement parameters at Gauss-Lobatto quadrature (GLQ) points into those at element nodes. The appropriate application of DQ rule to non-rectangular domains is proceeded by the natural-to-Cartesian geometric mapping technique. Using GLQ and DQ rules, we discretize the total potential energy functional of a generic micro-plate element into a function of nodal displacement parameters. Then, we adopt the principle of minimum potential energy to determine element stiffness matrix, mass matrix, and load vector. The efficacy of the present element is validated through several examples associated with the static bending and free vibration problems of rectangular, annular sectorial, and elliptical micro-plates. Finally, the developed element is applied to study the behavior of freely vibrating moderately thick micro-plates with irregular shapes. It is shown that our element has better convergence and adaptability than that of Bogner-Fox-Schmit (BFS) one, and strain gradient effects can cause a significant increase in vibration frequencies and a certain change in vibration mode shapes.
AB - In this study, we integrate the advantages of differential quadrature method (DQM) and finite element method (FEM) to construct a C1-type four-node quadrilateral element with 48 degrees of freedom (DOF) for strain gradient Mindlin micro-plates. This element is free of shape functions and shear locking. The C1-continuity requirements of deflection and rotation functions are accomplished by a fourth-order differential quadrature (DQ)-based geometric mapping scheme, which facilitates the conversion of the displacement parameters at Gauss-Lobatto quadrature (GLQ) points into those at element nodes. The appropriate application of DQ rule to non-rectangular domains is proceeded by the natural-to-Cartesian geometric mapping technique. Using GLQ and DQ rules, we discretize the total potential energy functional of a generic micro-plate element into a function of nodal displacement parameters. Then, we adopt the principle of minimum potential energy to determine element stiffness matrix, mass matrix, and load vector. The efficacy of the present element is validated through several examples associated with the static bending and free vibration problems of rectangular, annular sectorial, and elliptical micro-plates. Finally, the developed element is applied to study the behavior of freely vibrating moderately thick micro-plates with irregular shapes. It is shown that our element has better convergence and adaptability than that of Bogner-Fox-Schmit (BFS) one, and strain gradient effects can cause a significant increase in vibration frequencies and a certain change in vibration mode shapes.
KW - C-type four-node quadrilateral element
KW - differential quadrature method
KW - finite element method
KW - geometric mapping technique
KW - strain gradient Mindlin micro-plates
UR - http://www.scopus.com/inward/record.url?scp=85090168153&partnerID=8YFLogxK
U2 - 10.1002/nme.6513
DO - 10.1002/nme.6513
M3 - Journal article
AN - SCOPUS:85090168153
SN - 0029-5981
VL - 121
SP - 5600
EP - 5646
JO - International Journal for Numerical Methods in Engineering
JF - International Journal for Numerical Methods in Engineering
IS - 24
ER -