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.
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Abbreviations
- C sh :
-
bulk shear wave speed, m/s
- e kl :
-
strain components
- f l :
-
body force density in the l-direction (l = x, y, z), m/s2
- h :
-
thickness of the unit cell, m
- h j :
-
thickness of the jth layer (j = 1, 2), m
- \(\overline h (= {h_j}/h),\) (= hj/h):
-
dimensionless thickness
- k :
-
wavenumber, m−1
- k x :
-
Bloch wavenumber in the x-direction, m−1
- kh/π :
-
dimensionless wavenumber
- l 1 :
-
material intrinsic length which represents the size of the interface mass density, m
- l 2 :
-
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 :
-
dimensionless material intrinsic length which represents the size of the interface shear modulus
- R (=ε/h):
-
dimensionless internal characteristic length
- Rd (=εd/h):
-
dimensionless internal characteristic length in the nonlocal differential model
- Ri (=εi/h):
-
dimensionless internal characteristic length in the nonlocal integral model
- R :
-
transfer matrix of the sub-layer
- t :
-
time, s
- T :
-
transfer matrix between the two adjacent unit cells
- u l :
-
displacement in the l-direction, m
- V :
-
state vector
- x, x′ :
-
position vectors
- x, y, z :
-
variables in the rectangular coordinate system, m
- α(∣x′ − x∣):
-
influence function
- β :
-
dimensionless wavenumber in the y-direction
- ε :
-
internal characteristic length, m
- ξ, η :
-
dimensionless variables in the rectangular coordinate system
- θ :
-
incidence angle, °
- λ, β :
-
Lamé constants, N/m2
- ρ :
-
mass density, kg/m3
- ρ s :
-
interface mass density, kg/m2
- σ kl :
-
local stress components, N/m2
- τ kl :
-
nonlocal stress components, N/m2
- \(\tau _{zy}^{\rm{s}},\) :
-
interface stress components, N/m
- ω :
-
angular frequency, rad/s
- \(\varpi,\) :
-
dimensionless angular frequency
- Ω :
-
dimensionless frequency.
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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) 41(10), 1447—1460 (2020) https://doi.org/10.1007/s10483-020-2660-8
Project supported by the National Natural Science Foundation of China (Nos. 11472182 and 11272222) and the China Scholarship Council (No. 201907090051)
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Tian, R., Liu, J., Pan, E.N. et al. Shear-horizontal waves in periodic layered nanostructure with both nonlocal and interface effects. Appl. Math. Mech.-Engl. Ed. 41, 1447–1460 (2020). https://doi.org/10.1007/s10483-020-2660-8
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DOI: https://doi.org/10.1007/s10483-020-2660-8
Key words
- shear-horizontal (SH) wave
- nonlocal theory
- interface effect
- nanostructure
- integral model
- differential model