Mathematical model of a three-layer micro- and nano-beams based on the hypotheses of the Grigolyuk–Chulkov and the modified couple stress theory

J. Awrejcewicz, V. A. Krysko, M. V. Zhigalov, A. V. Krysko

Research output: Contribution to journalArticle

17 Citations (Scopus)


The mathematical model of three-layered beams developed based on the hypothesis of the Grigolyuk–Chulkov and the modified couple stress theory and the size depended equations governing the layers motions on the micro- and nano-scales is constructed. The Hamilton's principle yields the novel equations of motion as well as the boundary/initial conditions regarding beams displacement. The latter ones clearly exhibit the size dependent dynamics of the studied micro- and nano-beams, and the introduced theory overlaps with the classical beam equations for large enough layer thickness. In particular, a three-layer beam with the micro-layer thickness has been investigated with respect to the classical theory of Grigolyuk–Chulkov. The derived boundary problem is of sixth order and can be solved analytically in the case of statics. The carried out numerical experiments allowed to detect and explain size dependent effects exhibited by the micro-beams. The beam deflections and stress yielded by the employed couple stress model are less than those predicted by the classical Grigolyuk–Chulkov theory, while the estimated eigen frequencies are higher, respectively. It has been shown that the proposed model can be reduced to the classical three-layer Grigolyuk–Chulkov beam through increase of the layers thickness, which validates our approach.

Original languageEnglish
Pages (from-to)39-50
Number of pages12
JournalInternational Journal of Solids and Structures
Publication statusPublished - 15 Jun 2017



  • Analytical solution
  • Modified couple stress theory
  • Static/dynamic behavior
  • Three-layer beam

ASJC Scopus subject areas

  • Modelling and Simulation
  • Materials Science(all)
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics

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