Abstract
By using the complex variable function theory and the conformal mapping method, the scattering of plane shear wave (SH-wave) around an arbitrary shaped nano-cavity is studied. Surface effects at the nanoscale are explained based on the surface elasticity theory. According to the generalized Yong–Laplace equations, the boundary conditions are given, and the infinite algebraic equations for solving the unknown coefficients of the scattered wave solutions are established. The numerical solutions of the stress field can be obtained by using the orthogonality of trigonometric functions. Lastly, the numerical results of dynamic stress concentration factor around a circular hole, an elliptic hole and a square hole as the special cases are discussed. The numerical results show that the surface effect and wave number have a significant effect on the dynamic stress concentration, and prove that our results from theoretical derivation are correct.
1. Introduction
Since the scattering problem plays an important role in understanding the propagation phenomena of various waves in engineering materials and structures, the elastic wave scattering by the cavity embedded in the elastic matrix have always been a hot topic in wave motion theory. The application and development of mineral exploration, petroleum acquisition, quantitative nondestructive exploration, radar, underwater sonar, blasting and other technologies are all based on understanding the relationship between the scattering of elastic waves and the geometrical dimensions and physical parameters of defective bodies in elastic matrix. Pao and Mow [1] discussed in detail the diffraction of elastic waves in cavity/inclusion in an elastic medium by the methods of wave functions expansion, integral equation, and integral transform method. Using the complex variable function theory, Liu [2] discussed the dynamic stress concentration around a circular hole caused by SH-wave in an anisotropic medium. Subsequently, Liu et al. [3, 4] studied the scattering of SH-wave around an elliptical hole and cracks. In recent years, the scattering of circular cavity in circular domain to SH waves was analyzed by Wang and Jiang [5]. Liuet al. [6] discussed the scattering of plane SH-waves by an arbitrary shaped cavity embedded in a wedge-shaped domain by the method of IBEM. Then Yang et al. [7, 8] studied the scattering of shear waves by a circular or an elliptical cavity in inhomogeneous medium. Further, Ghafarollahi and Shodja [9] proposed an analytical treatment for the scattering of SH-waves by an elliptical cavity/crack which is embedded near the interface between exponentially graded and homogeneous half-spaces.
However, the above studies were carried out at the macroscale, so the influence of the surface stresses was not considered. With the rapid development of nanotechnology, it is necessary to understand the mechanical behavior of nanomaterials and nanostructures. At the nano-scale, the surface effect becomes significant due to the increasing ratio of surface to bulk volume [10]. The surface elasticity theory that considers the surface/interface influences were proposed by Gurtin et al. [11], and Miller and Shenoy’s [12, 13] results agreed with the surface elasticity theory. Subsequently, a geometric illustration to prove the generalized Yong–Laplace equation was presented by Chen et al. [14]. Therefore, the surface elasticity theory has been widely applied to study the various mechanical behaviors of materials and constructers at nano-scale [15–17].
By using the surface elasticity theory, Wang et al. [18, 19] considered the diffractions of -wave by a nano-cavity. Further, Ou and Lee [20] discussed the effects of interface energy on the scattering of plane elastic wave by a nano-sized coated fiber. Ru et al. [21–23] studied the diffraction of the elastic waves around a cylindrical nano-inclusion and then studied the surface effects of the scatterings of the vertical shear wave by a cluster of nanosized cylindrical holes. Wu and Ou [24, 25] studied the interface effects of SH-waves’ scattering around a cylindrical nano-inclusion by wave functions expansion method and complex variable function theory respectively. However, thanks to the complex boundary conditions of the arbitrary shaped cavity, most of those studies are confined to the circular hole or the spherical cavity/inclusion at the nano-size. Up to now, very few studies thought about the scattering of SH-wave by an arbitrary shaped nano-cavity. Therefore, it is necessary for us to study the relevant contents of the issue.
In this work, based on the theory of surface elasticity, the scattering of SH-wave around an arbitrary shaped nano-cavity embedded in an infinite elastic medium is studied, in which the numerical solutions of displacement fields are expressed by employing the complex variable function theory and the conformal mapping. The numerical results of dynamic stress concentration factor about a circular hole, an elliptic hole are illustrated graphically. The effects of surface energy on the dynamic stress concentration factor in the matrix material are analyzed. The paper is organized as follows. Section 2 briefly introduces the theory of surface elasticity. The solution for SH wave field around an arbitrary shaped nano-cavity is obtained using the method of conformal mapping. The governing equation and the corresponding boundary condition for the problem under consideration are presented in Section 3. In Section 4 discusses the surface effects on the dynamical stress concentration induced by incident SH-wave with a circular hole and an elliptic hole. The conclusions are presented in Section 5.
2. Surface Elasticity
According to the surface elasticity theory, a surface is considered to be a negligibly film that adheres to the matrix without slipping. The equilibrium equations and the isotropic constitutive relations in the matrix are the same as those in the classical theory of elasticity, but the presence of surface stress leads to nonclassical boundary conditions. To learn more about it, refer to Gurtin [11].
The surface stress tensor and the surface energy density satisfy the following formula
where is the second-rank tensor of surface strain, is the Kronecker delta. All repeated Latin indices and Greek indices in the entire paper use Einstein’s summation convention.
Based on the generalized Yong–Laplace equation [14], the equilibrium equations and the constitutive relations on the surface are
where , , are the stress tensors of the matrix, the inclusion, and the surface, respectively, denotes the normal vector of the surface, and is the interface divergence. and are two surface constants. The residual surface tension is not considered in our dynamic analysis, i.e. .
Since the classical elastic theory is established in the matrix, the equilibrium equations and isotropic constitutive relations are written as
where is the time, is the mass density of the material, and are the shear modulus and the Poisson’s ratio, respectively, and , are stress tensor and strain tensor in the matrix material, respectively. is the component of displacement, and the strain tensor is related to the displacement vector by
3. The Scattering of SH-Wave Around an Arbitrary Shaped Nano-Cavity
Based on surface elasticity theory, we discuss the scattering of SH-wave by an arbitrary shaped nano-cavity embedded in an infinite elastic medium, as shown in Figure 1.
The anti-plane governing equation in the matrix is
where , are the shear stresses in the bulk. is the displacement along the axis.
The relations between stresses and displacement are
By substituting Equation (8) into Equation (7), we obtain
For the steady-state response, the time can be separated as , then Equation (9) can be expressed as
where is the displacement function, is the wave number, is the shear velocity of the media, is the circular frequency, and is the mass density.
Based on the complex variable function theory, we introduce complex variables , . Equations (10) and (8) are
In order to solve the scattering of the noncircular cavity, the mapping function maps the arbitrary shaped nano-cavity boundary of the - plane to a conformal circle of radius with the boundary of the - plane, as shown in Figure 2.
Introduce the conformal mapping function
Substituting the mapping function into Equation (11), the following equation is obtained
Introducing the cylindrical coordinates , in -plane, we have
Assume that a harmonically plane SH-wave propagates in the positive -direction, as shown in Figure 2. According to Equation (11), the general solution of incident plane SH-wave function in the cylindrical coordinates is expressed as [1]
where is the amplitude of the incident wave, is the order Bessel function of the first kind, and the coefficient for and for .
The scattered wave function in the cylindrical coordinates is expressed as [1]
where is the order Hankel function of the first kind, and are unknown coefficients to be determined by the boundary conditions.
The total wave function in the matrix is given by the addition of and
According to the Equations (15)–(17), we can obtain
where
The corresponding stresses are
According to the Equations (1)–(6), we can obtain the boundary condition on the surface
where , which is a dimensionless parameter that reflects the effect of surface at nano-scale. For a macroscopic cavity, the radius is large enough (). Therefore, there is no need to consider surface effects. However, when the radius of the hole is reduced to the nano-scale, becomes apparent and the surface effect should be considered [26–28].
Substituting Equations (19)–(27) into Equation (28), we have
where , , , are presented in Appendix.
It is evident that Equation (29) is an equation related to though there are still some unknown coefficients. Multiplying both sides of Equation (29) by , then integrating respect to between the limits and , we obtain
From Equation (30), an infinite algebraic equation about unknown constants can be obtained. By solving the Equation (30), the unknown constants can be calculated. It should be noted that it is mathematically convenient to truncate the infinite algebraic equation in Equation (30) with during numerical calculation.
4. Numerical Results and Discussion
In order to study the effect of surface effects on dynamic stress concentration factors (DSCF), we define DSCF to be
where is the bulk stress in the medium along the surface, and is the stress intensity in the propagation direction of SH-wave. From the Equation (30), It is easy to see that surface elasticity parameter has a great impact on DSCF.
In what follows, as special cases, we discuss the influence of the surface effects on the DSCF around a circular hole and an elliptic hole.
4.1. Circular Hole
For a circular hole, the expression of can be written as
Substituting Equation (32) into Equation (30), the coefficients can be determined as
For a macroscopic circular hole, the surface effect is ignored , our results for are consistent with results of [1].
Thus, it is evident to see that our methods are correct, and the elastic stress fields the SH-wave induces can be obtained.
In order to consider the effects of surface on DSCF, the distributions of DSCF on the surface for different values of with low frequency incident wave are shown in Figure 3, which indicates obviously the surface effect on the DSCF near the hole. As increases, the DSCF decreases continuously, the maximum DSCF decreases from to . The maximum DSCF appears at about , and there are two peak values in the interval . Similarly, the distributions of DSCF on the surface for different values of with high frequency incident wave are shown in Figure 4, which also indicates obviously the surface effect on the DSCF near the hole. Again, as increases, the DSCF decreases continuously too, the maximum DSCF decreases from to . However, due to the high frequency, the distributions of DSCF changes frequently. The maximum value of DSCF occurs at angles of about .
4.2. Elliptic Hole
For an elliptic hole, the expression of is written as
where , .
The distributions of DSCF on the surface for different values of with , and low frequency incident wave are shown in Figure 5. It is remarkable that the surface effect on the DSCF near the elliptic hole. Compare with Figure 3, the results are similar to those for the circular hole in and , as increases, the DSCF decreases continuously, while in and , as increases, the DSCF increases. However, the maximum DSCF appear at angles of about , and the maximum DSCF decreases from to . Similarly, the distributions of DSCF on the surface for different values of with high frequency incident wave are shown in Figure 6, and the results are similar to those for in most regions. However, for different , the location of the maximum DSCF is different, the maximum DSCF decreases from to . The change of DSCF at high frequency is larger than that at low frequency.
4.3. Square Hole
For a square hole, the expression of is written as
The distributions of DSCF on the surface for different values of with are shown in Figure 7. It is remarkable that the surface effect on the DSCF near the square hole. Compare with Figures 3 and 5, the results are different from the circular hole and the elliptic hole, as increases, the DSCF increases in and , and in other intervals, the trend is not obvious. For different , the location of the maximum DSCF is different. When and , the changes are very significant. Similarly, the distributions of DSCF on the surface for different values of with high frequency incident wave are shown in Figure 8, and the results are different from . For different , the location of the maximum DSCF is approximately the same, the maximum DSCF appear at angles of about and . However, due to the frequent disturbance of the incident wave and the complexity of the expression of , with the increases of , there are no certain rule in the changes of DSCF.
It is evident from Figures 9 to 11 that the influence of the wave number is great on the DSCF. The DSCF decreases as increases in most regions. Stress concentration around the square hole is more significant, the maximum value of DSCF is . However, the maximum value of DSCF is about around the circular hole and the elliptical hole. In three cases, the symmetry of the DSCF with respect to .
In summary, the numerical results show that the surface energy and the wave numbers have significant effects on the DSCF in the different shapes such as circular, ellipse or square.
5. Conclusion
In the present paper, we investigate the scattering of SH-wave by an arbitrary shaped nano-cavity embedded in an infinite elastic matrix based on the surface elasticity. By using the complex variable function theory and conformal mapping method, the numerical solutions of the stress field are obtained around the cavity. Simultaneously, the analytical solutions of the stress field around a circular hole are obtained. Finally, the obtained numerical results are used to analyze the influence of the surface energy and the wave numbers on the DSCF around a circular hole, an ellipse hole and a square hole. When the radius of hole shrinks to nanometers, the surface effects become significant and should be taken into account. Regardless of the frequency of incident wave, the surface effect weakens the DSCF around the circular and ellipse hole in most regions, but it is different for the square hole. At the same time, the DSCF depends strongly on the wave numbers, with the wave numbers increases, the DSCF gradually decreases in most regions. This study is helpful to analyze the mechanical properties of nonuniform nano-composites, such as the vibration of defective body models in micro-nano mechanical systems, the propagation of semiconductor nano-devices, and so on.
Appendix
Appendix A
Data Availability
The data used to support the findings of this study are included within Figures 3–11 of the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (Grant No. 11362009 and No. 11862014).