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Dynamic stress around a cylindrical nano-inhomogeneity with an interface in a half-plane under anti-plane shear waves

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Abstract

Taking into account the size of the nanostructure, the effect of surface/interface stiffness on the dynamic stress around a cylindrical nano-inhomogeneity embedded in an elastic half-plane subjected to anti-plane shear waves is investigated. The boundary condition at the straight edge of the half-plane is traction free, which is satisfied by the image method. The analytical solutions of displacement fields are expressed by employing a wave function expansion method. The addition theorem for a cylindrical wave function is applied to accomplish the superposition of wave fields in the two half-planes. Analyses show that the effect of the interface properties on the dynamic stress is significantly related to the nano-scale distance between the straight edge and the center of the cylindrical nano-inhomogeneity. The frequency and incident angle of incident waves and the shear modulus ratio of the nano-inhomogeneity to matrix also show different effect on the dynamic stress distribution when the inhomogeneity shrinks to nano-scale. Comparison with the existing results is also given.

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Abbreviations

a :

radius of the nano-inhomogeneity

b :

nano-scale distance between the straight edge and the center of the nano-inhomogeneity

ω :

wave frequency

θ 0 :

incident angle of waves

μ I,ρ I :

shear modulus and mass density of nano-inhomogeneity

μ M,ρ M :

shear modulus and mass density of matrix

μ S,ρ S :

shear modulus and mass density of interface

W :

displacement component in the z direction

τ zx ,τ zy :

shear stresses in the matrix

k :

wave number of waves in the matrix

k 0 :

wave number inside the nano-inhomogeneity

c :

wave speed of shear waves

J n (⋅):

nth Bessel function of the first kind

\(H_{n}^{(1)}(\cdot)\) :

nth Hankel function of the first kind

A n ,B n :

scattering mode coefficients around the nano-inhomogeneities

C n :

refracted mode coefficients around the nano-inhomogeneities

\(\tau_{\theta z}^{S}\) :

circumferential interfacial stress at the interface

\(\varepsilon_{\theta z}^{S}\) :

interfacial strain component

τ 0 :

residual surface stress

W 0 :

amplitude of the incident waves

τ in :

maximum dynamic stress resulting from the incident waves

DSCF :

dynamic stress concentration factor

(in):

incident waves in the matrix

(sc):

scattered waves

(t):

total wave field in the matrix

(r):

refracted waves

I :

nano-inhomogeneity

M :

matrix

S :

surface/interface

1:

actual nano-inhomogeneity

2:

image nano-inhomogeneity

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Authors and Affiliations

  1. Department of Engineering Mechanics, Shijiazhuang Railway Institute, Shijiazhuang, 050043, P.R. China

    Xue-Qian Fang, Le-Le Zhang & Jin-Xi Liu

  2. Institutes of Engineering Mechanics, Beijing Jiaotong University, Beijing, 100044, China

    Le-Le Zhang

Authors
  1. Xue-Qian Fang
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  2. Le-Le Zhang
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  3. Jin-Xi Liu
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Correspondence to Xue-Qian Fang.

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Fang, XQ., Zhang, LL. & Liu, JX. Dynamic stress around a cylindrical nano-inhomogeneity with an interface in a half-plane under anti-plane shear waves. Appl. Phys. A 106, 625–633 (2012). https://doi.org/10.1007/s00339-011-6633-4

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  • DOI: https://doi.org/10.1007/s00339-011-6633-4

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