Mathematical modelling of phononic nanoplate and its size-dependent dispersion and topological properties |
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| Affiliation: | 1. Department of Architecture and Civil Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong SAR, P.R. China;2. Department of Engineering Mechanics, Zhejiang University, Yuquan Campus, Hangzhou 310027, China;3. Mechanical Engineering Department, Texas A&M University, College Station, Texas 77843-3123, USA;1. Computational Mechanics and Scientific Computing Group, Technical University of Cartagena, Campus Muralla del Mar, Cartagena 30202, Murcia, Spain;2. INEI, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain;1. Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin 150001, China;2. College of Mathematics, Sichuan University, Chengdu 610043, China;1. Institue of Solid Mechanics (IFKM), Technische Universität Dresden, Dresden 01062, Germany;2. Institute of Textile Machinery and High Performance Material Technology (ITM), Technische Universität Dresden, Dresden, Germany;1. School of Science, Henan Institute of Technology, Xinxiang 453003, PR China;2. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China;3. School of Ocean Engineering, Guangdong Ocean University, Zhanjiang 524088, PR China;4. School of Naval Architecture, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, PR China;5. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, PR China;1. Laboratory of Harbour Works, School of Civil Engineering, National Technical University of Athens, 5 Heroon Polytechniou Str., 15780 Zografou, Greece;2. Scientia Maris, 10 Agias Elenis Str., 15772 Zografou, Greece |
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| Abstract: | A new model with analysis for the propagation of flexural waves in a phononic plate at nanoscale is developed. The Gurtin-Murdoch theory for surface elasticity is adopted to model the surface heterogeneity. The Mindlin (or first-order) plate theory is further generalized to establish the governing equations for flexural waves in a phononic plate with surface effect, for which the plane wave expansion method is applied to derive the dispersion relation. A numerical model is developed using the finite element method and very good consistency between theory and numerical solution is observed. It is found that the surface density and the surface residual stress play the main role that affects the band structures. The surface effect can be approximately regarded as the competition between frequency decrease due to surface density and frequency increase caused by surface residual stress, which effectively increases the low-frequency bands but decreases the high-frequency bands. The quantum spin Hall effect is observed in the phononic plate at nanoscale, and the surface effect is studied numerically. By applying the k.p perturbation method, a theoretical framework is established to calculate the spin Chern number, which is an important topological invariant that determines the quantum spin Hall effect. Based on the topological analysis, an efficient waveguide with a zig-zag path is designed, in which a topologically protected wave in the interface state can robustly propagate along the path against disorders. The theory and numerical study developed in this paper will help better understand the size-dependent quantum spin Hall effect in nanostructures and it may also provide guidance for the design of topological wave devices at nanoscale. |
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