Abstract
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.
1. Introduction
The diffraction of elastic waves by a single inhomogeneity embedded in an elastic medium was discussed in detail by Pao and Mow [1]. With the development of composite materials, there is an increasing demand for understanding the dynamic behavior of composite materials, and much attention has been directed toward the multiple scattering of elastic waves. Fang et al. obtained the multiple diffraction fields by two-particle reinforced composite system [2]. Twersky investigated the scattering of acoustic or electromagnetic wave by an arbitrary configuration of parallel cylinders [3]. Lakhtakia et al. [4] observed the reflection and transmission of incident waves by an array of circular cylinders in an elastic slab. Wang and Sudak obtained the scattering field of elastic waves by a cluster of circular cylinders with imperfect interfaces [5].
Nanomaterials have different physical, optical, and mechanical properties distinct from their macroscopic counterparts [6]. At nanoscale, surface has significant effects on the physical and mechanical properties of solids, due to the increasing ratio of surface area to volume [7, 8]. To account for the surface effects, Gurtin et al. [9] developed a continuum model of surface elasticity. Based on the surface elasticity theory, Wang et al. [10] analyzed the diffraction of plane compressional wave (P-wave) by a nanosized circular hole. Ru et al. [11] obtained the scattering field of P- and SV-waves by a nanosized inhomogeneity. Wang [12] and Zhang et al. [13] analyze the diffraction of elastic waves by a pair and an array of nanosized inhomogeneities. Recently, Ru et al. discussed the surface effect on the scattering field induced by nanosized cylindrical holes [14]. Those investigations illustrated the importance of surface effects on the diffraction of elastic waves. In this paper, we discussed the multiple scattering of plane P- and SV-waves by a cluster of nanosized cylindrical holes arranged as quadrate shape.
2. Basic Equations
At nanoscale, we consider the problem in the framework of surface elasticity theory because of the surface effect [9]. According to the surface elasticity theory, a surface is regarded as a negligibly thin membrane adhered to the bulk without slipping and possesses material constants different from the bulk material. On the surface, the surface stress leads to a set of nonclassical boundary conditions. In the bulk, however, the classical theory of elasticity is still applicable.
The surface stress tensor is related to the surface energy density as [9]where is the Kronecker delta and is the second-rank tensor of surface strain. In this paper, Einstein’s summation convention is adopted for all repeated Latin indices (1, 2, 3) and Greek indices (1, 2).
Without residual surface tension, for an isotropic surface, the relationship between the surface stresses and the surface strains is [9]where and are two material constants on surface.
Assume that the surface adheres perfectly to the bulk material without slipping, and then the equilibrium equations on the surface are [15]where is the tangential component of the traction in the -direction, is the normal vector of the surface, is the curvature of the surface, and is stress tensor of the surface. Generally, the surface inertia force can be neglected for dynamic problems.
In the bulk solid, the equilibrium and constitutive equations are the same as those in the classical theory of elasticity:where is the mass density of the material, is the time, and are shear modulus and Poisson’s ratio, respectively, and and are stress tensor and strain tensor in the bulk material, respectively.
The strain tensor is related to the displacement vector by
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.
3. Diffraction of Elastic Waves by Cylindrical Nanoholes
We consider the diffraction of elastic waves by a cluster of ( approaching infinity) identical cylindrical holes with radius of in an infinite elastic matrix, as shown in Figure 1. The holes are arranged as quadrate shape and the distance between the centers and adjacent holes is , as shown in Figure 2. The global polar coordinate system is set up at the center of the middle hole. For convenience, at the center of th hole (), the local polar coordinate system is set up. The plane strain condition () is assumed; thus, .
For the present plane strain problem, the surface strain component on the surface of the hole is given byThe surface stress can be obtained from (5):On the surface, the equilibrium equations (3a) and (3b) with surface effects reduce toSubstituting (7) into (8) and then into (9a) and (9b), we havewherewith being a dimensionless parameter indicating the effect of surface at nanoscale. Equation (11) shows us that, for a macroscopic inclusion with a big value of , ; therefore, the surface effect can be neglected. However, when the radius of the inclusion shrinks to nanoscale, becomes noticeable and the surface effect should be considered in analysis [10–14].
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 and (see [1] for more details):Using the cylindrical coordinate system, the geometric relations becomeand the constitutive relations (5) can be rewritten as
Thereby, in terms of the displacement potentials, the stresses can be determined from (12a) to (14c). In what follows, we derive the solutions for the present problems. Since the wave diffractions around every hole are identical, for convenience, we consider only the wave diffractions around the middle hole.
3.1. Diffraction of P-Wave by a Cluster of Cylindrical Nanoholes
Assuming a harmonically plane P-wave propagating in the positive -direction, it can be described by using a displacement potential (see [1] for more details):where is the amplitude of the incident wave, , is the angle between -axis and the direction of incident wave, is time, is the circular frequency, and the normalized wave number of P-wave in the bulk is with the velocity of P-wave being
For an incident wave, SV-wave and P-wave are reflected from each hole. The displacement potentials of the diffracted waves due to the th (herein, ) cylindrical hole can be written aswhere is the th order Hankel function of the first kind, is the normalized wave number of SV-wave in the matrix, and is the velocity of SV-wave. and are two coefficients to be determined by the boundary conditions.
Moreover, the total waves around the middle hole are determined by [1]According to the Graf addition theorem [2], the transformation between two local cylindrical coordinate systems and isUsing this theorem, we can get the total waves around the arbitrary hole.
Using Graf addition theorem and then substituting (19a), (19b) and (12a), (12b) into (13a), (13b) and (13c) then into (14a), (14b) and (14c), one obtains the solutions for the stress fields in the bulk. From the surface conditions (10a) and (10b), we can determine the coefficients and in the solutions. Thus, the solutions for the elastic fields induced by incident P-wave near a cluster of cylindrical nanoholes are obtained.
3.2. Diffraction of SV-Wave by a Cylindrical Nanoinclusion
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 -direction, it can be described by using a displacement potential [1]:
The displacement potentials of the diffracted waves due to the th (herein, ) cylindrical hole can be written asSo, the total waves around the middle hole are
Substituting (23a), (23b) and (12a), (12b) into (13a), (13b) and (13c) then into (14a), (14b) and (14c), one obtains the solutions for the stress fields in the bulk and the inclusion, respectively. Again, using the surface conditions (10a) and (10b), the coefficients in the solutions can be determined. Thus, the solutions for the elastic fields induced by incident SV-wave near a cluster of cylindrical nanoholes are obtained. In what follows, we will discuss the surface effect on the elastic-wave-induced stress concentration near the cylindrical nanoholes.
4. Results and Discussions
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 asand the DSCF induced by SV-wave aswhere is the hoop stress along the middle hole; and are the stress intensity in the propagation direction of P-wave and SV-wave, respectively.
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 and the distance between holes.
If and keeping as a constant, then the incident P-wave represents a static biaxial loading with and [1]. Similarly, if and keeping as a constant, then the incident SV-wave represents a static loading with [1]. Moreover, let ; the present problem reduces to the classical problem without surface effect. For such a remote loading and , we calculate the stress field by ABAQUS. For a low frequency incident wave with for P-wave and , as shown in Figure 3(a), and for a low frequency incident wave with for SV-wave and , as shown in Figure 3(b), the present analytical result agrees well with the static finite element calculation.
(a)
(b)
For another case, when the distance between holes is much larger than the hole radius , the interaction between holes can be neglected and then the stress field around the hole approaches that for a single hole, as shown in Figures 5, 7, 9, and 11. These two particular cases have been used as a benchmark to test our numerical results.
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 the calculations.
4.1. Dynamic Stress Concentration Induced by P-Wave
4.1.1. Low Frequency Incident Wave with
In this case, wavelength () of the incident wave is much larger than the radius nanoholes. For , Figure 4 shows the distributions of DSCF around the surface of the middle hole for different values of . The maximum DSCF appears at about and . For the whole range, DSCF decreases with the increasing of .
For different separation of and keeping , the distributions of DSCF along the middle hole are shown in Figure 5. It is indicated that the larger the separation of is, the less the interaction between holes is. When the distance between holes is larger than , the interaction between holes can be negligible and the distribution of DSCF approaches that of a single hole. When the separation decreases, the DSCF decreases at and .
4.1.2. High Frequency Incident Wave with
In this case, the incident wavelength () is comparable with the hole radius. For a small separation of , the distributions of DSCF for various values of are shown in Figure 6. It is seen that multiple peak values are excited along the surface due to the interference between the incident and reflected waves. With the increasing of , the DSCF decreases almost in the whole range.
Figure 7 shows the variation of DSCF along the hole for different value of . The interference between incident and reflected waves is still obvious. When the separation between holes is large enough (e.g., which is much larger than that for a low frequency), the interaction can be neglected and the distributions of DSCF are approaching to that of a single hole.
4.2. Dynamic Stress Concentration Induced by SV-Wave
4.2.1. Low Frequency Incident Wave with
For a small separation of , the distributions of DSCF around the surface of the middle hole for different values of are indicated in Figure 8. The maximum DSCF appears at almost and . For the whole range, DSCF also decreases with the increasing of .
For different value of , let ; the distributions of DSCF along the middle hole are shown in Figure 9. For a larger value of , the interference between holes still holds. When the distance between holes is larger than , the distribution of DSCF approaches that of a single hole.
4.2.2. High Frequency Incident Wave with
For , Figure 10 shows the distributions of DSCF for various values of . Multiple peak values are excited along the surface due to the interference between the incident and reflected waves. With the increasing of , the DSCF decreases almost in and .
Figure 11 shows the variation of DSCF along the hole for different values of . The interference between incident and reflected waves is obvious. When the separation between holes is large enough (), the interaction can be neglected and the distributions of DSCF are approaching to that of a single hole. With the increasing of , the DSCF decreases almost in the whole range.
5. Conclusion
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.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (11302166).