Finite Element Analyses of the Modified Strain Gradient Theory Based Kirchhoff Microplates

Kandaz M., DAL H.

SURFACES, cilt.4, sa.2, ss.115-156, 2021 (ESCI) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 4 Sayı: 2
  • Basım Tarihi: 2021
  • Doi Numarası: 10.3390/surfaces4020014
  • Dergi Adı: SURFACES
  • Derginin Tarandığı İndeksler: Emerging Sources Citation Index (ESCI), Scopus, INSPEC, Directory of Open Access Journals
  • Sayfa Sayıları: ss.115-156
  • Anahtar Kelimeler: microplates, modified strain gradient theory, higher-order elasticity, finite element method, Kirchhoff plate theory, SHEAR DEFORMATION-THEORY, FREE-VIBRATION, FORMULATION, MODEL
  • Orta Doğu Teknik Üniversitesi Adresli: Evet

Özet

In this contribution, the variational problem for the Kirchhoff plate based on the modified strain gradient theory (MSGT) is derived, and the Euler-Lagrange equations governing the equation of motion are obtained. The Galerkin-type weak form, upon which the finite element method is constructed, is derived from the variational problem. The shape functions which satisfy the governing homogeneous partial differential equation are derived as extensions of Adini-Clough-Melosh (ACM) and Bogner-Fox-Schmit (BFS) plate element formulations by introducing additional curvature degrees of freedom (DOF) on each node. Based on the proposed set of shape functions, 20-, 24-, 28- and 32- DOF modified strain gradient theory-based higher-order Kirchhoff microplate element are proposed. The performance of the elements are demonstrated in terms of various tests and representative boundary value problems. Length scale parameters for gold are also proposed based on experiments reported in literature.