### Abstract

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 language | English |
---|---|

Pages (from-to) | 39-50 |

Number of pages | 12 |

Journal | International Journal of Solids and Structures |

Volume | 117 |

DOIs | |

Publication status | Published - 15 Jun 2017 |

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### Keywords

- 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

### Cite this

*International Journal of Solids and Structures*,

*117*, 39-50. https://doi.org/10.1016/j.ijsolstr.2017.04.011

**Mathematical model of a three-layer micro- and nano-beams based on the hypotheses of the Grigolyuk–Chulkov and the modified couple stress theory.** / Awrejcewicz, J.; Krysko, V. A.; Zhigalov, M. V.; Krysko, A. V.

Research output: Contribution to journal › Article

*International Journal of Solids and Structures*, vol. 117, pp. 39-50. https://doi.org/10.1016/j.ijsolstr.2017.04.011

}

TY - JOUR

T1 - Mathematical model of a three-layer micro- and nano-beams based on the hypotheses of the Grigolyuk–Chulkov and the modified couple stress theory

AU - Awrejcewicz, J.

AU - Krysko, V. A.

AU - Zhigalov, M. V.

AU - Krysko, A. V.

PY - 2017/6/15

Y1 - 2017/6/15

N2 - 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.

AB - 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.

KW - Analytical solution

KW - Modified couple stress theory

KW - Static/dynamic behavior

KW - Three-layer beam

UR - http://www.scopus.com/inward/record.url?scp=85018581924&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018581924&partnerID=8YFLogxK

U2 - 10.1016/j.ijsolstr.2017.04.011

DO - 10.1016/j.ijsolstr.2017.04.011

M3 - Article

AN - SCOPUS:85018581924

VL - 117

SP - 39

EP - 50

JO - International Journal of Solids and Structures

JF - International Journal of Solids and Structures

SN - 0020-7683

ER -