We consider the multiple scattering of elastic waves (P-wave and SV-wave) by a cluster of nanosized cylindrical holes arranged as quadrate shape. When the radius of the holes shrinks to nanometers, the surface elasticity theory is adopted in analysis. Using the displacement potential method and wave functions expansion method, we obtain that the multiple scattering fields induced by incident P- and SV-waves around the holes are derived. The dynamic stress concentration around the holes is calculated to illustrate the effect of surface effects on the multiple scattering of P- and SV-waves.

The diffraction of elastic waves by a single inhomogeneity embedded in an elastic medium was discussed in detail by Pao and Mow [

Nanomaterials have different physical, optical, and mechanical properties distinct from their macroscopic counterparts [

At nanoscale, we consider the problem in the framework of surface elasticity theory because of the surface effect [

The surface stress tensor

Without residual surface tension, for an isotropic surface, the relationship between the surface stresses and the surface strains is [

Assume that the surface adheres perfectly to the bulk material without slipping, and then the equilibrium equations on the surface are [

In the bulk solid, the equilibrium and constitutive equations are the same as those in the classical theory of elasticity:

The strain tensor is related to the displacement vector

Based on surface elasticity theory, we derive the solutions for elastic fields near a cluster of cylindrical nanoholes arranged as quadrate shape induced by incident P-wave and SV-wave, respectively.

We consider the diffraction of elastic waves by a cluster of

The diffraction of elastic waves by a cluster of infinity nanosized cylindrical holes.

Dimensions and coordinate system for formulating the problem.

For the present plane strain problem, the surface strain component

In the bulk solid and the inclusion, the classical theory of elasticity still holds. Therefore, in each of them, the displacements can be expressed by two harmonic potential functions

Thereby, in terms of the displacement potentials, the stresses can be determined from (

Assuming a harmonically plane P-wave propagating in the positive

For an incident wave, SV-wave and P-wave are reflected from each hole. The displacement potentials of the diffracted waves due to the

Moreover, the total waves around the middle hole are determined by [

Using Graf addition theorem and then substituting (

Similarly, we can consider the diffraction of plane SV-wave by a cluster of cylindrical nanoholes arranged as quadrate shape. For an incident plane SV-wave propagating in the positive

The displacement potentials of the diffracted waves due to the

Substituting (

To examine surface effect on the multiple scattering of elastic waves, we consider the dynamics stress concentration around the middle hole.

Determine the dynamic stress concentration factor (DSCF) induced by P-wave as

It is seen that when the surface effect is taken into account, the dynamic stress depends not only on the wave number and Poisson’s ratio but also on the surface elasticity parameter

If

Comparisons of the present numerical results for the

For another case, when the distance

In what follows, we discuss the effects of the interface at an inclusion on the diffraction of elastic waves and on the dynamic stress concentration factors around the inclusion. We keep

In this case, wavelength (

Surface effect on the distribution of DSCF along the middle hole for

Distribution of DSCF along the middle hole for different separation of

For different separation of

In this case, the incident wavelength (

Surface effect on the distribution of DSCF along the middle hole for

Distribution of DSCF along the middle hole for different separation of

Figure

For a small separation of

Surface effect on the distribution of DSCF along the middle hole for

Distribution of DSCF along the middle hole for different separation of

For different value of

For

Surface effect on the distribution of DSCF along the middle hole for

Distribution of DSCF along the middle hole for different separation of

Figure

Based on the theory of surface elasticity, surface effect on the diffractions of plane elastic waves by a cluster of cylindrical nanoholes arranged as quadrate shape was theoretically investigated in this paper. Solutions for the elastic fields induced by P- and SV-waves near cylindrical nanoholes are obtained, respectively, and the effects of surface properties on the dynamic stress concentration near the nanoholes are discussed in detail.

It can be concluded that surface effect weakens the phenomenon of dynamic stress concentration. The DSCF depends not only on the surface effects but also on the separation between holes. For both low and high frequency, the interaction between holes is significant in a very small separation. With the increasing of separation between holes, the interaction can be ignored in a small separation for low frequency, but, for high frequency, the interaction can be neglected only for a much large separation.

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (11302166).