Shock and Vibration

Volume 2018, Article ID 7516402, 14 pages

https://doi.org/10.1155/2018/7516402

## Study on the Single Scattering of Elastic Waves by a Cylindrical Fiber with a Partially Imperfect Bonding Using the Collocation Point Method

^{1}College of Aerospace Engineering, Chongqing University, Chongqing 400044, China^{2}Chongqing Key Laboratory of Heterogeneous Material Mechanics, Chongqing University, Chongqing 400044, China^{3}State Key Laboratory of Silicate Materials for Architecture, Wuhan University of Technology, Wuhan 430070, China^{4}Institute of Microstructure Technology (IMT), Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany

Correspondence should be addressed to Jun Zhang; nc.ude.uqc@gnahzjem and Chuanlin Hu; nc.ude.tuhw@nilnauhc

Received 9 January 2018; Revised 17 February 2018; Accepted 8 March 2018; Published 23 April 2018

Academic Editor: Nerio Tullini

Copyright © 2018 Jun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The single scattering of P- and SV-waves by a cylindrical fiber with a partially imperfect bonding to the surrounding matrix is investigated, which benefits the characterization of the behavior of elastic waves in composite materials. The imperfect interface is modelled by the spring model. To solve the corresponding single scattering problem, a collocation point (CP) method is introduced. Based on this method, influence of various aspects of the imperfect interface on the scattering of P- and SV-waves is studied. Results indicate that (i) the total scattering cross section (SCS) is almost symmetric about the axis with respect to the location () of the imperfect interface, (ii) imperfect interfaces located at and highly reduce the total SCS under a P-wave incidence and imperfect interfaces located at reduce the total SCS most significantly under SV-incidence, and (iii) under a P-wave incidence the SCS has a high sensitivity to the bonding level of imperfect interfaces when is small, while it becomes more sensitive to the bonding level when is larger under SV-wave incidence.

#### 1. Introduction

A large number of interfaces exist in composite materials and play a very important role in the performance of composite materials, for example, the transmission of the load between the matrix and fibers. To simplify the analysis and calculation, the fibers/particulates are generally assumed to be perfectly bonded to the surrounding matrix [1–4]. However, this is not always the case, which has consequently led to the increased research effort to take the bonding conditions into account [5–16]. Among them, the debonding case [9–13, 16–18], which is assumed to be a crack with noncontacting faces, has been well investigated. Specifically, Norris and Yang [11–13] investigated the single scattering of antiplane shear and P- and SV-waves by a partially debonded fiber through the expansion of the crack opening displacement (COD) or the stress at the neck in terms of Chebyshev polynomials. The results indicated that a resonance happens at low frequencies when the neck joining the fiber to the matrix is thin. Recently, Fang et al. [16] extended this method to the case of pipelines. Based on the same method, Kim [9] studied the effects of interface debonding on the attenuation and speed of antiplane shear wave propagating in fiber-reinforced composites. The corresponding static effective mechanical properties of fiber-reinforced composites were obtained from the asymptotic solutions of dynamic harmonic analysis by Liu and Kriz [10]. Besides the technique proposed by Norris and Yang [11–13], Y. S. Wang and D. Wang [17, 18] proposed a different method, where the dislocation density function was introduced and the interface conditions were transformed to standard Cauchy singular integral equations. Solving these equations gives rise to the COD at each debonding/crack. Compared with the method of Norris and Yang, the one proposed by Y. S. Wang and D. Wang has a high convergence rate especially when the size of cracks is small. Besides these theoretical investigations, numerical simulations using the boundary element methods were also conducted, for example, Biwa and Shibata [19, 20] and Sato and Shindo [21].

The perfect bonding and the debonding are the two extreme cases of the bonding conditions of interfaces. Actually, there are other conditions which extensively exist between the perfect bonding and the debonding, that is, imperfect condition. For this reason, several theoretical models were proposed during the past few years [22–24]. Among them, the spring model [22] attracted the most attention. In the spring model, the stresses are assumed to be continuous across the interface, while the displacements jump. Furthermore, the displacement jump is assumed to be linearly proportional to the stress at the interface. The stress to the displacement jump ratio is defined as the stiffness of springs. Thus, the stiffness of the spring reflects the bonding level of the imperfect interface. It is clear that when the stiffness of the spring tends towards infinity, the imperfect bonding becomes the usual perfect bonding, while it approaches the debonding when the stiffness approaches zero.

Using the above-introduced spring model, the effective mechanical properties of composite materials, such as the effective modulus [25, 26], thermal expansion coefficient [27], phase velocity, and attenuation [5, 8, 28], have been extensively investigated by taking the imperfect bonding into account. The spring model has also been used to study the influence of the imperfect bonding effect on the behaviors of the underground tunnels under various dynamic loadings [15, 24, 29, 30].

It is worthwhile to mention that, in most of the previous studies, the whole interface between each fiber/particulate and the matrix was usually assumed to simultaneously enter the imperfect bonding condition. However, this is not real case, assuming that the interface between each fiber and the surrounding matrix deteriorates as a process would be more reasonable. In this regard, less attention has been paid except for the partial debonding case. The limited works include those of Lopez-Realpozo et al. [31] and Guinovart-Díaz et al. [32], where the effect of the partially imperfect bonding on the effective mechanical properties of fiber-reinforced composites was investigated. To the best of our knowledge, investigation on the influence of this partially imperfect bonding of interfaces on the behavior of P- and SV-waves in composites is still scarce. It is well known that the study of the single scattering is crucial to understanding the overall behavior of elastic waves in composites. Towards this end, in this work, the single scattering of P- and SV-waves by a cylindrical fiber with a partially imperfect bonding to the surrounding matrix is studied. Compared with the single scattering with a fully perfect or imperfect interface, solving the single scattering with a partially imperfect bonding is much more complicated, since the conditions at the interface become discontinuous. In this work, the imperfect bonding is characterized by the spring model and a CP method is introduced to solve the corresponding single scattering problem.

In the following sections, firstly, the single scattering problem is formulated. Then, the CP method is introduced to solve the single scattering problem. After that, the influence of various aspects of the partially imperfect bonding on the single scattering is extensively studied numerically. Finally, a short conclusion is drawn.

#### 2. Single Scattering of P/SV-Waves by a Fiber with a Partially Imperfect Bonding

##### 2.1. Governing Equations for P- and SV-Waves in 2D Problems

The governing equations for P- and SV-waves in 2D problems with a homogeneous medium are decoupled based on the Helmholtz decomposition. In harmonic analysis, the two equations for P- and SV-waves can be expressed aswhere and denote the displacement potentials for P- and SV-waves, respectively; and is the Laplace operator. The symbols and are the wavenumbers of the P- and SV-waves, and the subscripts and represent the longitudinal and transverse wave. The wavenumbers are related to material properties aswhere is the circular frequency, is the Lame constant, is the shear modulus, and is the mass density. In the polar coordinate system, the displacements and stresses are expressed aswhere and represent the radial and circumferential displacements, and and are the radial and tangential stresses. The formula in ((1)–(3)) can be found in numerous classic books, of which we mention the monograph by Mow and Pao [33]. They are copied here to maintain the integrity of this work.

##### 2.2. Problem Formulation of the 2D Singe Scattering of P- and SV-Waves

Figure 1 shows a schematic of the single scattering of a plane P- or SV-wave by a cylindrical fiber partially imperfectly bonded to the surrounding matrix. The radius of the fiber is . The range of the imperfect interface is represented by angle , as indicated by the red solid curve shown in Figure 1. The symmetric axis of the imperfect interface is at angle with the horizontal axis. It is clear that when , the whole fiber is fully perfectly bonded to the matrix, and the whole interface becomes a fully imperfect bonding as . In this work, the imperfect bonding is modelled using the spring model introduced before, where it is characterized using two springs distributed along the radial and circumferential directions, with stiffness and stiffness , respectively. The incident wave is a plane P- or SV-wave propagating in the horizontal direction. In the polar coordinate system, general solutions to (1) are the products of Bessel functions and trigonometric functions. Hence, for harmonic analysis, the incident wave, the wave scattered by the fiber, and the wave fields inside the fiber can be expressed as series of cylindrical wave functions with the omission of :where are the polar coordinates with origins at the center of the fiber. The wavenumbers (), where is the wave speed of longitudinal () wave or transverse wave. The superscripts and indicate quantities associated with the matrix and the fiber, respectively. and are the Bessel and Hankel functions of the first kind; is the imaginary unit; and , , , and are the unknown coefficients, which are to be determined by the conditions at the interface. In the current case, the conditions of stresses at the interface are listed asAnd the conditions for the radial and circumferential displacements at the interface can be expressed as