Shock and Vibration

Volume 2018 (2018), Article ID 2341797, 11 pages

https://doi.org/10.1155/2018/2341797

## Scattering Attenuation of SH-Waves in Fiber-Reinforced Composites with Partial Imperfectly Bonded Interfaces

^{1}College of Aerospace Engineering, Chongqing University, Chongqing 400044, China^{2}Chongqing Key Laboratory of Heterogeneous Material Mechanics, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Jun Zhang

Received 28 October 2017; Accepted 1 January 2018; Published 20 February 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 influence of partial imperfectly bonded interfaces between each fiber and the matrix on the scattering attenuation of coherent SH-waves in fiber-reinforced composites is investigated. The imperfection of interfaces is modelled using the spring model, in which the level of imperfection is characterized by a parameter , which is called the stiffness of the imperfect interfaces. First, the single scattering of SH-waves by a cylindrical fiber with such a partial imperfectly bonded interface to the matrix is formulated and subsequently solved using the collocation point (CP) method. Later on, based on the analysis of the corresponding single scattering problem, effects of several parameters (i.e., frequency of the incident wave, level of imperfection of interfaces, and width of the imperfect interfaces) on the far-field scattering magnitude, scattering cross section, and scattering attenuation coefficients of coherent SH-waves are shown graphically. The potential application of the current results to nondestructive evaluation of interfaces in composites is also discussed.

#### 1. Introduction

Various interfaces exist in the nature, such as interfaces between different constituents of composite materials and those between distinct phases in a polycrystalline material. For composite materials, the interfaces play very important roles, such as transferring loads from the matrix to fibers. Accordingly, evaluation of the interfaces of composite materials has been the focus of many researchers [1–6].

Currently, the ultrasonic technique has become a major method for the nondestructive evaluation of interfaces, due to the fact that the behavior (i.e., attenuation and wave speed) of elastic waves in composite materials is highly related to the interfaces [1, 2, 7]. It is well known that, before using this ultrasonic technique to evaluate composite materials, the relationship between characteristics of elastic waves in composites and their properties should be established. It is in this spirit that, to date, numerous theoretical models [8–20] for the evaluation of attenuation and wave speed of coherent elastic waves in composite materials have been proposed. It is worthwhile to note that, among these models, there is a simple one used for cases with dilute concentration of inclusions, which is called the independent scattering model [10, 17–19, 21]. The model indicates that the scattering attenuation coefficient of coherent elastic waves in composites can be calculated as , where is the scattering cross section [22] and is the number of fibers/particulates inside a unit area/volume. Several experiments [1, 9] have shown that this independent scattering model has a good performance.

To date, in most of published relevant works, the fibers and the surrounding matrix are usually assumed to be perfectly bonded. However, this is not always true in practice. Therefore, during the past few years, several theoretical models have been developed to model the imperfect situations of interfaces. These models include, but are not limited to, the spring model [23] and the spring-mass model [24]. Specifically, in the spring model, the stress is assumed to be continuous across the interfaces, while allowing the displacement has a jump across the interfaces. This jump is assumed to be proportional to the stresses at the interfaces and has the relation , where denotes the jump of the displacements across the interface and is the parameter to characterize the imperfection level of the interfaces. It is clear that, when , the spring model becomes the familiar perfect bonding and it is degenerated to the debonding case as . Other intermediate values of represent the* “imperfect bonding”* between the* “perfect bonding”* and the* “debonding.”* The validity of this spring model has been verified in several works [1, 7, 25]. Compared with the spring model, the additional inertia effect of the interphases has been considered in the spring-mass model. Detailed information on these models has been provided in [26]. This spring model can be also used to model the imperfect interfaces in other physical fields, that is, electric and magnetic [27].

In the past few years, this imperfect bonding effect on the effective properties and the behavior of elastic waves in composite materials/structures have attracted attention of several researchers [1, 28–31]. Most of these works have accounted for the imperfect bonding effect through the spring model. Specifically, for example, the influence of the imperfect bonding between inclusions and the matrix on the effective elastic properties [28, 31, 32] and thermal [33] properties of composites has been investigated. For the behavior of elastic waves in composites, Rokhlin et al. [1] have investigated the imperfect bonding effect on the scattering of SH-, P-, and SV-waves in fiber-reinforced composites. Mal and Bose [30] have studied the phase velocity and attenuation of coherent P- and SV-waves in particulate composites taking the imperfect bonding effect into account. It should be pointed out that, in most of these published works, the whole interface between each fiber/particulate and the matrix is usually assumed to* enter the imperfect bonding situation simultaneously*. However, this is not always the case.* Assuming that the interface between each fiber and the matrix deteriorates as a process would be more reasonable*. In this regard, Lopez-Realpozo et al. [31] have investigated the nonuniform imperfect bonding effect on the effective shear stiffness of a periodic fibrous composite. The corresponding work on the effective thermal conductivity has been conducted in [33]. Several other authors [3, 6, 34–38] have investigated the influence of partial debonding of interfaces on elastic wave propagation in composite materials, which is an extreme case of the partial imperfect bonding. To the best of our knowledge, the investigation of the influence of the partial imperfect bonding of interfaces on the behavior of coherent elastic waves in composites is still scarce.

In this work, we aimed to investigate the influence of the partial imperfect bonding of interfaces on the scattering attenuation of coherent SH-waves in fiber-reinforced composite materials. The imperfection of interfaces is modelled using the spring model. The scattering attenuation is calculated using the previously introduced independent scattering model. The CP method is proposed to solve the corresponding single scattering problem with a partial imperfectly bonded interface. Compared with the existing techniques, the derivation process and implementation of this CP method are simple and it has a comparable accuracy. Based on this CP method, influence of various aspects of the imperfect interfaces between fibers and the matrix on the scattering attenuation of coherent SH-waves in composites is studied. Such investigation benefits the nondestructive evaluation of the interfaces in composites using the ultrasonic technique, as the results indicate that the working frequency of the scattering attenuation based ultrasonic technique to evaluate the interfaces should be carefully selected.

In the following sections, first, the single scattering of SH-waves by a cylindrical fiber with a partial imperfect bonding to the matrix is described in detail. The CP method is then introduced to solve such a single scattering problem. Based on the analysis of the single scattering problem, extensive parameter analysis on the far-field scattering magnitude, scattering cross section, and scattering attenuation is then conducted. At the end, a short conclusion is presented.

#### 2. Single Scattering of SH-Waves by a Fiber with a Partial Imperfect Bonding to the Matrix

##### 2.1. Formulation of the Problem

Figure 1 shows a schematic of the single scattering of a plane SH-wave by a cylindrical fiber with radius . As shown in this figure, only a portion of the fiber is perfectly bonded to the matrix while the remaining is not. The symmetric axis of the imperfect bonding is at an angle with the horizontal axis. The range of the imperfect bonding is represented by angle , as indicated by the red solid curve in Figure 1. It is clear that, when , the whole fiber is purely perfectly bonded to the matrix and the whole interface becomes a pure imperfect bonding as . The incident wave is a plane SH-wave with a unit magnitude () propagating in the positive horizontal direction. For harmonic analysis, the term is omitted for simplicity. In the polar coordinates, the incident wave, the wave scattered by the fiber, and the wave refracted into the fiber can be expressed as series [22] as follows: where is the polar coordinates with origin at the center of the fiber; and are the wave numbers in the matrix and the fiber, respectively; and are the Bessel and Hankel function of the first kind; is the imaginary unit; and are the unknown coefficients, which are to be determined by the conditions of displacements and stresses at the interface . In this case, the conditions at the interface can be listed as follows:where and are the axial shear stresses at the interface; the superscript “+” means values in the matrix and “−” denotes those in the fiber; variable is the stiffness of the interface adopted in the spring model. Substituting of the expression of displacements in (1) and the corresponding expression for the axial shear stress into (2), the conditions at the interface described in (2) can be further expressed in detail as follows:with . Here, and are the shear modulus of the matrix and the fiber. The symbol Prime means the first-order derivative. Unlike the pure perfect and imperfect bonding cases, the orthogonality of trigonometric functions has not been available to (3). Therefore, analytical solutions for and cannot be obtained as the pure perfect and imperfect bonding cases. In this work, (3) has been solved using the CP method [39].