Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5023127, 7 pages

http://dx.doi.org/10.1155/2016/5023127

## Ultrasonic Nonlinearity Evaluation of the Cracked Interface

^{1}School of Civil Engineering, Southwest Jiaotong University, Chengdu 610031, China^{2}State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China^{3}Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China^{4}College of Aerospace Engineering, Chongqing University, Chongqing 400044, China

Received 14 December 2015; Revised 5 May 2016; Accepted 15 May 2016

Academic Editor: Roman Lewandowski

Copyright © 2016 Yanjun Qiu 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

This paper derives a novel analytical solution for acoustic nonlinearity evaluation of the cracked interface. When microcracks exist at the interface, the tensile and compressive effective moduli of the cracked interface are considered to be different. It is clearly shown that the tension and compression elastic asymmetry can result in acoustic nonlinearity. In addition, numerical simulations using the finite element method are conducted to validate the theoretical solution. It is shown that numerical results agree well with the analytical solution. Finally, two factors affecting acoustic nonlinearity are studied based on the analytical solution. One is the tension and compression elastic asymmetry and another is the frequency of incident wave. Different from acoustic nonlinearity parameter of the general second harmonics, it is found that acoustic nonlinearity parameter is a function of two factors.

#### 1. Introduction

Adhesion bonding technology for fiber reinforced composite has been widely applied in aerospace industry. Generally, the cohesive interface is very thin but can provide the powerful strength and resiliency. However, the capability of the cohesive interface could be greatly affected by the imperfect condition, such as debond, fatigue damage, and microcracks. Thus, there is a great need for development of nondestructive testing methods to detect the imperfect condition.

Nonlinear ultrasonic methods have the powerful ability to characterize the material nonlinearity change caused by plasticity [1], imperfect interface [2–4], microcracks [5, 6], and fatigue damage [7, 8]. When a time-harmonic longitudinal wave propagates through a cracked solid or interface, it will cause the tension and compression elastic asymmetry; then the waveform will be distorted and higher-order harmonic waves are generated [9–11]. And considerable experimental evidence [12, 13] has shown that ultrasonic waves do interact with microcracks in a nonlinear fashion, but those researches may fail when the crack size is only tens of microns, which is one of the initial factors leading to interface degradation.

For the researches on acoustic nonlinearity induced by microcracks [14–16], majority are concentrated either on the scattering of elastic waves by a single crack or an array of cracks [17–20] or on the propagation of elastic waves in a cracked medium [21–23]. The first study on such contact-induced acoustic nonlinearity is probably the paper by Richardson [24] who considered the contact interface between two semi-infinite half-spaces. He has analyzed one-dimensional nonlinear wave propagation in a system composed of an unbounded planer interface separating two semi-infinite linear elastic media. The nonlinearity is caused by the opening and closing of the interface. However, in Richardson’s analysis, the interface stiffness varying continuously is not accounted for. Improving Richardson’s theory, Biwa et al. [25] have analyzed a nonlinear interface stiffness model, where the stiffness property of the contact interface is described as a function of the nominal contact pressure. Furthermore, by assuming the interface of the adhesive as a nonlinear spring, Achenbach and Parikh [26] have investigated theoretically to obtain information on the adhesive bond strength from ultrasonic test results, and it is shown that the nonlinear adhesive bond behavior could cause the generation of higher harmonics.

In this paper, based on the researches of Richardson and Biwa, one theoretical solution for one-dimensional nonlinear wave propagation in a system composed of a multicrack-included interface separating two semi-infinite elastic media is derived, from which the acoustic nonlinearity caused by the tension and compression elastic asymmetry is clearly shown. To validate the analytical solution, numerical results from the FEM simulations are presented. Comparison between the analytical predictions and the FEM simulation results shows good agreement. Finally, we also study two factors affecting acoustic nonlinearity. One is the tension and compression elastic asymmetry of the cracked interface and another is the frequency of incident wave.

#### 2. Solution for Harmonic Wave Incidence

We consider a system composed of a cracked interface separating two semi-infinite elastic media. In the presence of microcracks, the interface will respond to the tensile and compressive loadings differently, which is the tension and compression elastic asymmetry. According to [11], the elastic constants of the cracked medium can be considered to be dependent with crack density, the friction of the crack faces, and frequency of incident wave:where and are Young’s modulus and Poisson’s ratio of the uncracked solid, respectively. And and are, respectively, the corresponding effective Young’s modulus and Poisson’s ratio of the cracked solid under tension () and under compression (). Meanwhile, and for plane stress, and and for plane strain. The crack density is defined as , where is the average half-length of the cracks; is the area of the solid containing randomly distributed and randomly oriented two-dimensional microcracks.

We haveIn the above, is the Euler-Mascheroni constant, is a dimensionless wavenumber, is a dimensionless parameter, is the coefficient of friction, and

Therefore, in this paper, the conclusion of [11] will be adopted, and the interface can be regarded as an equivalent medium consisting of the random distribution cracks. Actually this assumption has been proven in [25] for investigating the void inclusion medium subjected to the ultrasonic wave loading conditions.

##### 2.1. The General Solution

Firstly, we assume that the elasticity of the cracked interface is a function of time, and the stress in the interface is uniform. In Figure 1 we illustrate a one-dimensional system schematically to consider elastic longitudinal wave propagation along -axis, where and represent the incident and transmitted wave functions, respectively. The blue area represents the cracked interface.