Elsevier

Wave Motion

Volume 101, March 2021, 102686
Wave Motion

Influence of contact interface morphology on the nonlinear interaction between a longitudinal wave and a contact interface with friction : A numerical study

https://doi.org/10.1016/j.wavemoti.2020.102686Get rights and content

Highlights

  • Numerical and experimental investigation of generation of second harmonic for contact nonlinearity.

  • Experimental evidence of different nonlinear signatures of real cracks contact.

  • Numerical investigation of influence of morphologies of contact on nonlinear signature.

  • Local analysis of the second harmonic generation.

Abstract

The detection and evaluation of closed cracks are of prime interest in industry. Whereas conventional ultrasonic methods fail to detect these defects, nonlinear methods based on activation of the nonlinear behavior of closed cracks constitute an interesting alternative. The aim of this article is to give a better understanding of interactions between cracks and a longitudinal elastic wave for a quantitative investigation into nonlinear signatures. Using a 1D approach based on the literature, the nonlinear signature of the contact interface is analyzed in two cases. In the first, the interface is initially open and in the second, it is initially closed before interaction with an elastic wave. These signatures were qualitatively observed experimentally in real cracks. Next, in order to investigate the influence of the coexistence of open and closed zones within the interface, a numerical 2D-study is proposed. Two configurations are considered involving two steel blocks in contact, with different contact interface morphologies. The first configuration is a perfectly plane contact interface, while the second one involves an interface between a concave surface and a plane surface. A non-plane wave is also considered. This study attempts to establish a link between local second harmonic generation and interface parameters (pre-stress, gap) that can be exploited for the nondestructive quantitative evaluation of interfaces or cracks.

Introduction

The evaluation of damage at an early stage of fracture is relevant in many industrial applications such as aeronautics or nuclear plants. Ultrasonic methods based on linear wave scattering are efficient for detecting defects and for characterizing material elasticity, but are less sensitive to micro-cracks or closed cracks. In this context, nonlinear acoustics constitutes a good alternative for detection and evaluation of these defects, taking advantages of the nonlinear behavior of contact dynamics induced by an acoustic wave if its amplitude activates sufficiently nonlinear contact behavior.

When elastic waves and contact interfaces interact, specific nonlinear acoustic phenomena can be observed [1]: DC-effect, subharmonics generation or hysteresis/storage effects, etc. These non-classical effects are due to complex contact behavior: an asymmetrical normal stiffness, the presence of asperities and multiphysical interactions on contact surfaces that result in specific contact acoustic nonlinearity (CAN). In order to exploit these nonlinear effects it is essential to understand the complex interactions between waves and contact interfaces, and this can be achieved using analytical and numerical models. As it is not possible to include the whole complexity of contact interfaces in a model, the contact effects and study methodology have to be carefully chosen according to the objectives of the study.

Previous studies have investigated the effect of contact nonlinearity on nonlinear acoustic signatures by considering phenomenological contact models [2], [3], [4]. They lead to a mean stress/strain relation that is able to describe the observed nonlinear behavior. The advantage of these approaches is that observed behavior can be described, however they do not make it possible to understand the physical behavior of interfaces. Other approaches, called the “physical models”, based on physical contact behavior have been used to investigate the nonlinear interaction of waves and contact interfaces [5], [6]. Aleshin et al. included the influence of asperities on nonlinear acoustics [7], [8], in a model based on memory-diagrams which has been implemented in a Finite Element (FE) code.

Richardson [9] provides an analytical study of the interaction between a longitudinal plane wave (therefore infinite) and an infinite contact interface between two identical media considering a unilateral contact law. It corresponds to an infinitely rigid contact that cannot support tension and hence opens up in the tension phase [9]. In the case of an incident sinusoidal wave, second harmonic generation efficiency (which corresponds to the ratio of the second harmonic to the incident wave amplitude) can be determined, and shows a specific nonlinear signature. Based on this work, other studies aimed at explaining the relation between contact nonlinearity and harmonic evolution considering an interface [10], [11], [12], [13] or a crack [6]. They show that nonlinear signatures provide some information on interface parameters (contact stresses, friction induced energy dissipation, crack orientation, etc.) if they are measurable. These studies take into account both the “clapping” effect and sliding with friction on a perfectly plane interface. They give a good understanding of interactions between wave and contact in a model case, but do not take into account asperities and non-conforming profiles that actually exists in the case of real cracks and that can play an important role during dynamic interactions with waves.

Including asperities in the model leads to difficulties of interpretation if too many parameters are involved. For example, in [7], the contact model introduces a nonlinear normal contact stiffness in compression, and consequently other effects are introduced during the interactions between waves and contact interfaces that, as far as we know, are not fully analyzed and understood. The present paper considered non-conforming profile interface between two deformable solids. A unilateral contact law with Coulomb’s friction is applied locally in order to investigate the effect of the coexistence of open and closed zones in the interaction zone. This is physical approaches to contact modeling. If it does not take into account a large number of contact effects (adhesion, asperities, contact stiffness, wear), but it describes interface closing/opening (“clapping” effect) and sliding with friction (“sliding” effect) with only one parameter: the friction coefficient. In the first part of the paper, a 1D model of the nonlinear interaction between a plane wave and an infinite interface is used to analyze second harmonic evolution as a function of pre-stress and incident wave magnitude in the cases of closed and open contact interfaces. Next, the experimental results of second harmonic evolution as a function of applied stress on a real crack are presented. In the third part, the effect of the coexistence of closed and open parts within the interaction zone on second harmonic generation is investigated numerically using a 2D-FE model. Investigation of local contact behavior in relation to local harmonic generation is a way to go further in the evaluation of cracks and contact interfaces.

Section snippets

1D-Numerical study of a longitudinal wave reflected from a unilateral contact interface

In this part, a homogeneous, isotropic elastic half-space defined as Ω is assumed to be in perfect contact with a rigid wall on a line Γc at x=0. The half-space and the rigid plane are brought into contact under a given compression normal stress σ0 (σ0<0) or maintained open with a normal initial displacement gap u0 (u0<0) (Fig. 1). An incident plane longitudinal wave is generated at x=L and propagates linearly in Ω in the positive x-direction with a velocity cL (approximately 6000 m.s−1 in

Experimental evidence of different evolutions of the second harmonic for a real fatigue crack

In this section, experimental results are presented for the second harmonic generation of a real fatigue crack. The experimental set-up and cracked sample are shown in Fig. 4(a) and (b) respectively. Compression force loading is applied to the sample using a threaded rod and nuts. This force is monitored through a load-cell located between the screw thread and the sample. In practice the system is mounted horizontally to allow the application of loads lower than the sample weight. A 32 mm

2D-Numerical study of a longitudinal wave interacting with contact interfaces of different morphologies

Two configurations (Fig. 7) are considered both including two steel blocks in contact. They differ in their contact interface morphologies. The first configuration (Fig. 7(a)) consists in a perfectly plane contact interface, while the second involves an interface between a concave surface and a plane surface (Fig. 7(b)). For the plane interface, the normal applied force will result in a quasi-uniform pre-stress. In this case, the interface will remain closed whatever the applied force.

Local analysis of contact acoustic nonlinearity and harmonic generation

In this part, a local analysis of CAN is proposed through local dimensionless parameters ξL(x) and ψL(x) along a contact interface defined as: ξL(x)=|σ0(x)|σincmax(x)ψL(x)=|u0(x)|uincmax(x) where σincmax(x) and uincmax(x) correspond to maximal values of incident stress and displacement respectively. These parameters are dependent on x as the incident wave is non-plane (Fig. 9), and the pre-stress and initial normal gap are non-uniform in the concave case (Fig. 8). In the case of a plane

Conclusion

In this paper, the interaction of a longitudinal wave and a unilateral contact interface has been analyzed. First, a 1D analysis was carried out to retrieve evolutions of the second harmonic as a function of the dimensionless parameters ξL and ψL already described in the literature. These evolutions were compared to some experimental results in a sample with a real crack for different transmitter and receiver positions. These results motivated a 2-D FE analysis considering two blocks in contact

CRediT authorship contribution statement

Abdelkrim Saidoun: Conception and design of study, Acquisition of data, Analysis and/or interpretation of data, Writing - review & editing. Anissa Meziane: Conception and design of study, Acquisition of data, Analysis and/or interpretation of data, Writing - original draft, Writing - review & editing. Mathieu Renier: Conception and design of study, Acquisition of data, Analysis and/or interpretation of data, Writing - review & editing. Fan Zhang: Analysis and/or interpretation of data, Writing

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors would like to thank Fondation CETIM for their support. Approval of the version of the manuscript to be published.

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