Premium
Strain gradient differential quadrature finite element for moderately thick micro‐plates
Author(s) -
Zhang Bo,
Li Heng,
Kong Liulin,
Zhang Xu,
Feng Zhipeng
Publication year - 2020
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6513
Subject(s) - finite element method , gaussian quadrature , mathematics , quadrature (astronomy) , discretization , stiffness matrix , mass matrix , mathematical analysis , vibration , geometry , quadrilateral , structural engineering , nyström method , boundary value problem , engineering , physics , electrical engineering , quantum mechanics , neutrino , nuclear physics
Summary In this study, we integrate the advantages of differential quadrature method (DQM) and finite element method (FEM) to construct a C 1 ‐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 C 1 ‐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.