Abstract
Based on the surface elasticity theory, the scattering of shear wave (SH-wave) by a cylindrical nano-inclusion with an interface in a right-angle plane is studied using the method of complex variable function. The dynamic stress concentration factor along the interface of inclusion by the SH-wave and scattering cross section are derived and numerically evaluated. The surface effect, the incident wave’s frequency, the shear modulus, and the distances from the center of nano-inclusion to the right-angle boundaries show the different degrees effects on the DSCF. Our results can aid in analyzing the mechanical properties of nonuniform nanocomposites. The proposed method can better solve the scattering problem of the holes/inclusions on noninfinite elastic substrates.
1. Introduction
It is a classical problem to study the geometric and physical properties of discontinuities in the structures. Elastic waves can be used to simulate various types of loads in practical engineering and technology. By analyzing the propagation and scattering of elastic wave in solids, the dynamic and static responses of structures under various loads can be determined. Cylindrical holes or inclusions are commonly used as models for discontinuities in solids. Since the groundbreaking work of Pao and Mow [1], because of its important application in materials science and engineering, the study of elastic wave scattering around holes/inclusions brought about the widespread attention from scholars.
Nanocomposites are a kind of novel materials. The surface effect becomes significant due to the increasing ratio of surface to bulk volume [2]. In recent years, with the rapid development of nanotechnology, the problem of elastic nanocavities has aroused the interest of scholars. Gurtin et al. [3] presented the theory of surface elasticity that considers the surface/interface elasticity. Miller and Shenoy’s [4, 5] results agreed well with direct atomic simulations. Therefore, the surface elasticity theory has widely been applied to study various mechanical behaviors at the nanoscale [6–9]. In recent years, surface effects have received more attention from scholars in various fields. Fang et al. discussed the surface energy effect on nonlinear free vibration of piezoelectric nanoshells [10, 11]. Karami et al. [12] developed mechanical analysis of anisotropix nanoparticles by the three-dimensional (3D) elasticity theory in conjunction with nonlocal strain gradient theory. At the same time, in nanotechnology, extensive research has been conducted on functionally graded materials. Boutaleb et al. [13] studied the dynamic analysis of the functionally graded rectangular nanoplates. Based on a nonlocal strain gradient refined plate modeled, Karami et al. [14] dealt with the size-dependent wave propagation analysis of functionally graded anisotropic nanoplates. The static and dynamic behavior of functionally graded nanotube-reinforced porous sandwich polymer plate is considered by Medani et al. [15].
Researchers have been relatively late in the study of the propagation of the elastic wave in nanocomposites. Using Gurtin’s surface elasticity theory, Wang et al. [16, 17] studied the scattering of the plane wave in the infinite nanocomposites with surface effect by the wave functions expansion method. Ou and Lee [18] considered interface energy’s effects on plane elastic wave’s of by a nanosized coated fiber. Furthermore, Ru et al. [19–21] studied elastic wave’s diffraction by a cluster of nanosized cylindrical holes. Using the complex variable function theory, Wu [22] discussed the interface effects of SH-waves’ scattering around a cylindrical nano-inclusion.
It can be found that these studies assume that elastic wave propagates in the infinite nanocomposites, which simplifies the process of solving the wave function. However, in most practical problems, nanocomposites are finite, and the reflection elastic waves at the edge of the material greatly affect the wave fields. In some cases, it may play an important role. The stress field is very different when it is near the free surface compared to the infinite space. Using the mirror method, it is possible to solve the wave field generated by the circular scattering in the free surface. Fang et al. [23, 24] considered a cylindrical nano-inhomogeneity in a half-plane. So far, there has been little discussion about the nano-inclusions in the right-angle plane. Therefore, it is imperative to investigate related issues.
In this work, dynamic stress around a cylindrical nano-inclusion with an interface in a right-angle plane under SH-wave is studied within the framework of surface elasticity. Consider the boundary conditions of traction free on two straight edges. The scattering wave field satisfying the boundary condition is constructed by using the mirror method. The analytical solutions of displacement fields are expressed by employing the method of wave function expansion and the complex variable function theory. The numerical results of dynamic stress concentration factors about nano-inclusion are illustrated graphically. The effects of surface elasticity on the dynamic stress concentration factor in the matrix material are analyzed. The effects of the frequency of the incident wave and the shear modulus ratio of the nano-inclusion to the matrix and the distances from the center of nano-inclusion to the right-angle boundaries on the DSCF are also analyzed.
2. Model and Analysis of the Problem
A right-angle elastic medium containing a cylindrical nano-inclusion is considered, as depicted in Figure 1. An SH-wave with frequency impinges on the elastic medium, and its incident angle is . It is assumed that the two edges of right-angle elastic medium are traction free. The shear modulus and mass density of the nano-inclusion are denoted by and , which are different from those ( and ) of the matrix in general. The radius of the nano-inclusion is . The distance from the center of nano-inclusion to the horizontal and vertical boundaries are and , respectively. , , and are the inclusion, horizontal, and vertical interfaces, respectively.
The matrix and inclusion are both assumed to be isotropic and elastic. For the antiplane problem, the shear wave is defined by [1]
In the absence of body force, the equation of motion in the matrix is written as [1]where and are the shear stresses in the matrix, respectively.
The relation between stress components and displacement are [1]
Substituting equation (3) into equation (2), we obtain the following equation:
Equation (4) is the equation of motion in terms of the displacement vector for a homogeneous elastic body and applies to both matrix and inclusion. and are the shear modulus and mass density of the material, respectively. For the steady-state response, the dependence on time may be separated as , and then equation (4) can be written as follows:where is the displacement function and , in which is the media’s shear velocity.
Based on the complex variable function theory, we introduce complex variables and . Equations (3) and (5) are
In the cylindrical coordinate system , equation (6) can be expressed as
As shown in Figure 1, the incident angle is . According to equation (7), the general solution of the incident plane SH-wave function in the coordinate system is expressed as [1]where , in which is the shear velocity of the nano-inclusion.
Due to the action of the incident wave, the two right-angle boundaries produce the following reflected waves [1]:
For convenient calculation, using multipolar coordinate transformation and introducing complex variables and , equations (9)–(12) are expressed as follows:wherewhere is the incident wave’s amplitude and .
The incident and reflected waves meet the stress-free conditions of two right-angle boundaries. So, by the mirror method as shown in Figure 2, the scattering wave function in the coordinate system is expressed aswherewhere is the th order Hankel function of the first kind and are unknown coefficients to be determined by the boundary conditions.
Also, due to the action of the incident wave, refracted wave function generated in nano-inclusion in the coordinate system is expressed aswhere is the th order Bessel function of the first kind, are unknown coefficients to be determined from the boundary conditions, and , in which is the shear velocity of the nano-inclusion.
The total wave function in the matrix and the nano-inclusion are determined by
According to equation (8), we can obtainwherewhereand thus,
3. Basic Equations of Surface Elasticity Theory
According to the surface elasticity theory, a surface is regarded as negligibly thin membranes that adheres to the matrix without slipping. The classical theory of elasticity is still applicable in the matrix, but the presence of surface stress leads to nonclassical boundary conditions. The equilibrium equations and the isotropic constitutive relations in the matrix are the same as those in the classical theory of elasticity:where is the time, is the material’s mass density, and and are the shear modulus and Poisson’s ratio, respectively. and are the stress and strain tensors in the bulk material, respectively. The strain tensor is related to the displacement vector by
The surface stress tensor is related to the surface energy density by [3]where is the second-rand tensor of surface strain and is the Kronecker delta. Einstein’s summation convention is adopted for all repeated Latin indices and Greek indices throughout the letter.
Assume that the surface adheres perfectly to the bulk material without slipping. By the generalized Young–Laplace equation [25], the equilibrium equations and the constitutive relations on the surface arewhere , and are the matrix, inclusion, and interface stress, respectively. is the interface divergence, and is the residual surface tension under the unstrained condition. denotes the normal vector of the surface, and and are the two surface constants, respectively.
Surface elasticity theory has been used to study surface/interface effect on the solids with nano-inhomogeneities [26–29]. In what follows, we derive the solutions for elastic fields near the cylindrical nano-inclusion in a right-angle plane, in the framework of surface elasticity theory.
4. Boundary Conditions at the Interface of the Nano-Inclusion
According to the continuity of displacements, on the interface , we have
On the nano-inclusion’s interface, the strain component can be obtained from equation (25):
The interface stress can be obtained from equation (29):and thus,
According to equation (28), we can obtain the boundary condition on the interface
Substituting equation (33) into equation (34), we have the following boundary condition aswhere , which is a dimensionless parameter that reflects the effect of the interface at the nanoscale. For a macroscopic inclusion, the value of is big enough , and thus, the surface effect can be ignored. However, when the radius of the inclusion shrinks to the nanoscale, becomes noticeable and the surface effect should be considered [15–24].
5. Determination of Mode Coefficients
Substituting equations (18), (22), and (23) into equations (30) and (35), then multiply on both sides of the equations, and then apply integral from to , we havewhere
Using equations (36)–(41), the scattering and refracting coefficients are determined. When the surface effect is neglected , the results are consistent with the results of Shi et al. [30]. Thus, it can be determined that our results are correct.
6. Numerical Results and Analysis
In the presence of nano-inclusion, the stress field exhibits a significant difference due to the scattering and refraction of the wave. To study the effect of surface effects on dynamic stress concentration factors (DSCF), we define DSCF to be [1]where is the bulk stress in the medium along the surface and is the stress intensity in the SH-wave’s propagation direction. From equations (36)–(41), it can be seen that surface elasticity parameter has a great impact on DSCF.
In the following numerical analysis, we choose as the characteristic length, where is the nano-inclusion’s radius. Define dimensionless variables and constants as follows: the incident wave number is , the distance between the horizontal straight edge and the center of nano-inclusion is , the distance between the vertical straight edge and the center of nano-inclusion is , and the shear modulus ratio is .
In order to verify the correctness of our results, we compare our numerical results with those of results of Shi et al. as shown in Figures 3 and 4. It can be seen that whether it is hard or soft inclusion, and whether it is low or high frequency, the present results for DSCF around the nano-inclusion with is consistent with the results of Shi et al. [30], i.e., . On the contrary, we can see that the DSCF decreases as the inclusion’s hardness increases.
6.1. Effects of a Low-Frequency Incident Wave on the DSCF
For a low-frequency harmonic incident wave with , DSCF’s distributions on the interface for various values of are shown in Figure 5. For soft inclusion, , which indicates clearly the interface’s noticeable effect on the DSCF near the nano-inclusion. As increases, the DSCF decreases. The maximum stress concentration occurred at and , and the minimum stress concentration occurred at , , and , which are quite similar to the results of previous research [15–24].
To examine the effect of horizontal and vertical boundaries, Figure 6 displays the DSCF increase as increases. From Figure 7, we can see that the DSCF decreases as increases. When , the effect of on DSCF is very small. In this case, the influence of the horizontal boundary on the nano-inclusion can be ignored.
To examine the bulk properties’ effect, Figure 8 shows the DSCF near the inclusion for various ratios of modulus . For a soft nano-inclusion (), the DSCF increased as increased; the maximum stress concentration occurred at and , and the minimum stress concentration occurred at , , and . For a hard nano-inclusion (), the results are just the opposite. That is, the DSCF decreased as increased; the maximum stress concentration occurred at , , and , and the minimum stress concentration occurred at and .
6.2. Effects of a High-Frequency Incident Wave on the DSCF
For a high-frequency harmonic incident wave with , Figure 9 shows the DSCF’s distributions on the interface for various values of . It can be seen that the distributions of the DSCF is no longer symmetric about . As increases, the DSCF decreases. The results are similar to those at low frequency. It indicates that the interface effects tend to suppress the perturbation of the DSCF. Multiple peaks of the DSCF occurred along the surface due to the interference between the incident and reflected waves.
It can be seen from Figures 10 and 11 that the influence of the boundaries on the inclusion is no longer as significant as the low frequency due to the interference of the high frequency.
Figure 12 shows the DSCF’s distributions for different ratios of shear modulus . For a soft nano-inclusion , the DSCF increases as increases. For a hard nano-inclusion , the results are just the opposite. That is, the DSCF decreases as increases. These results are similar to that at a low frequency. However the values of DSCF are no longer symmetrical.
7. Conclusions
In this paper, the surface/interface elastic theory has been adopted to study the SH-wave’s scattering by a cylindrical nano-inclusion embedded in a right-angle plane. The wave function expansion method and the complex variable function theory are used to obtain the free field in the right-angled plane. The scattering and refraction fields in the right-angled plane are established by the mirror method. The total wave field in the same coordinate system is written by the Graf addition formula. The stress boundary condition and the displacement continuous condition are obtained using the generalized Young–Laplace equation, and the infinite algebraic equations are obtained to solve the unknown coefficients in the scattered and refracted waves. For example, various factors’ influence on the DSCF around the nano-inclusion is analyzed to obtain numerical results. When the radius of the inclusion shrinks to nanoscale, the interface effect becomes obvious and must to take into account. The combination of the mirror method and multipolar coordinates can better solve the scattering problem of the holes/inclusions on noninfinite elastic substrates. This study is helpful to analyze the mechanical properties of nonuniform nanocomposites, such as the vibration of defective body models in micro-nano-mechanical systems and the propagation of semiconductor nanodevices.
The main results of this paper are as follows:(1)Whether it is high-frequency or low-frequency incident wave, the interface effect tend to suppress the perturbation of the DSCF. As increases, the values of the DSCF decrease continuously across almost the entire range.(2)Whether it is high-frequency or low-frequency incident wave, for a soft nano-inclusion, the DSCF increases as increases. For a hard nano-inclusion, the DSCF decreases as increases.(3)For a low-frequency incident wave, the DSCF’s distributions are symmetrical. But for a high frequency, multiple peaks of the DSCF occurred along the surface due to the interference between the incident and reflected waves. The distributions of the DSCF are no longer symmetrical.(4)For a low-frequency incident wave, the influence of boundary on the inclusion has some certain regularity. But for a high frequency, the influence of the boundary on the inclusion is no longer as significant as the low frequency due to the interference of the high frequency.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 11362009 and 11862014).