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| Exact Solution of Free Vibration and Buckling of One-Dimensional Hexagonal Simple-Supported and Layered Quasicrystal Beams |
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Abstract The pseudo-Stroh formula can transform the governing equations of multi-field coupling materials such as quasicrystals into a linear eigen-system and obtain the exact solution of multilayered structures with simple supported boundary conditions, which provides an important reference for various numerical and experimental methods of quasicrystal beams in engineering practice. This paper focuses on the exact analysis of free vibration and buckling problem of one-dimensional hexagonal simply-supported and layered quasicrystal beams. The relation of the extended displacement (phonon displacement and phason displacement) and the extended stress (phonon stress and phason stress) is established firstly, and the governing equations of one-dimensional hexagonal quasicrystal beams are then deduced by using the pseudo-Stroh formula. By solving the linear eigensystem, the general solution of one-dimensional hexagonal quasicrystal homogeneous beam is obtained. Furthermore, the exact solutions of the natural frequencies of free vibration and the critical buckling load of layered quasicrystal beams are derived by propagating matrix method. The correctness and effectiveness of the pseudo-Stroh formula presented in this paper are verified by comparing with the previous results of the shear deformation theory of beams. The numerical examples are illustrated to show the effect of stacking sequence, high-span ratio, layer-thickness ratio and the number of layers on the natural frequency, critical buckling load and mode shape of two kinds of sandwich quasicrystal beams. The results indicated that the stacking sequence, the high-span ratio and the layer-thickness of beam had a great effect on the natural frequency of free vibration and the critical buckling load of quasicrystal beams. When the quasicrystals with high stiffness were used as the surface layers of the layered beam, the natural frequency and critical buckling load of the quasicrystal beams could be enhanced by increasing the high-span ratio and the layer-thickness. Thus, the optimal natural frequency and the critical buckling load of quasicrystal beams could be obtained by adjusting the geometrical size and the stacking sequence of the beams. Besides, the presented results would be a foundation on the further study of layered quasicrystal beams at micro-/nano-scale.
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Received: 22 August 2022
Published: 17 February 2023
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2023, Vol. 44 
