Shear-horizontal waves in periodic layered nanostructure with both nonlocal and interface effects
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APPLIED MATHEMATICS AND MECHANICS (ENGLISH EDITION) https://doi.org/10.1007/s10483-020-2660-8
Shear-horizontal waves in periodic layered nanostructure with both nonlocal and interface effects∗ Ru TIAN1,2 , Jinxi LIU3,4,† ,
E. N. PAN2 , Yuesheng WANG1
1. Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, China; 2. Department of Civil Engineering, University of Akron, Akron, OH 44325-3905, U. S. A.; 3. Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China; 4. Hebei Key Laboratory of Mechanics of Intelligent Materials and Structures, Shijiazhuang Tiedao University, Shijiazhuang 050043, China (Received May 24, 2020 / Revised Jul. 10, 2020) Abstract The propagation of shear-horizontal (SH) waves in the periodic layered nanocomposite is investigated by using both the nonlocal integral model and the nonlocal differential model with the interface effect. Based on the transfer matrix method and the Bloch theory, the band structures for SH waves with both vertical and oblique incidences to the structure are obtained. It is found that by choosing appropriate interface parameters, the dispersion curves predicted by the nonlocal differential model with the interface effect can be tuned to be the same as those based on the nonlocal integral model. Thus, by propagating the SH waves vertically and obliquely to the periodic layered nanostructure, we could invert, respectively, the interface mass density and the interface shear modulus, by matching the dispersion curves. Examples are further shown on how to determine the interface mass density and the interface shear modulus in theory. Key words shear-horizontal (SH) wave, nonlocal theory, interface effect, nanostructure, integral model, differential model Chinese Library Classification O347 2010 Mathematics Subject Classification
74J05
List of symbols csh , ekl , fl , h, hj ,
bulk shear wave speed, m/s; strain components; body force density in the l-direction (l = x, y, z), m/s2 ; thickness of the unit cell, m; thickness of the jth layer (j = 1, 2),
m; h (= hj /h), dimensionless thickness; k, wavenumber, m−1 ; Bloch wavenumber in kx , x-direction, m−1 ; kh/π, dimensionless wavenumber;
the
∗ Citation: TIAN, R., LIU, J. X., PAN, E. N., and WANG, Y. S. Shear-horizontal waves in periodic layered nanostructure with both nonlocal and interface effects. Applied Mathematics and Mechanics (English Edition) (2020) https://doi.org/10.1007/s10483-020-2660-8 † Corresponding author, E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos. 11472182 and 11272222) and the China Scholarship Council (No. 201907090051) c The Author(s) 2020
2
Ru TIAN, Jinxi LIU, E. N. PAN, and Yuesheng WANG
l1 ,
material intrinsic length which represents the size of the interface mass density, m; l2 , material intrinsic length which represents the size of the interface shear modulus, m; l1 /h, dimensionless material intrinsic length which represents the size of the interface mass density; l2 /h,
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