Elsevier

Ultrasonics

Volume 49, Issue 8, December 2009, Pages 682-695
Ultrasonics

Acoustic resonance scattering from a multilayered cylindrical shell with imperfect bonding

https://doi.org/10.1016/j.ultras.2009.05.007Get rights and content

Abstract

The method of wave function expansion is adopted to study the three dimensional scattering of a time-harmonic plane progressive sound field obliquely incident upon a multi-layered hollow cylinder with interlaminar bonding imperfection. For the generality of solution, each layer is assumed to be cylindrically orthotropic. An approximate laminate model in the context of the modal state equations with variable coefficients along with the classical T-matrix solution technique is set up for each layer to solve for the unknown modal scattering and transmission coefficients. A linear spring model is used to describe the interlaminar adhesive bonding whose effects are incorporated into the global transfer matrix by introduction of proper interfacial transfer matrices. Following the classic acoustic resonance scattering theory (RST), the scattered field and response to surface waves are determined by constructing the partial waves and obtaining the non-resonance (backgrounds) and resonance components. The solution is first used to investigate the effect of interlayer imperfection of an air-filled and water submerged bilaminate aluminium cylindrical shell on the resonances associated with various modes of wave propagation (i.e., symmetric/asymmetric Lamb waves, fluid-borne A-type waves, Rayleigh and Whispering Gallery waves) appearing in the backscattered spectrum, according to their polarization and state of stress. An illustrative numerical example is also given for a multi-layered (five-layered) cylindrical shell for which the stiffness of the adhesive interlayers is artificially varied. The sensitivity of resonance frequencies associated with higher mode numbers to the stiffness coefficients is demonstrated to be a good measure of the bonding strength. Limiting cases are considered and fair agreements with solutions available in the literature are established.

Introduction

The high diversity and severity of demands as well as of operating conditions imposed on structural elements by today’s advanced technologies have resulted in the need for new types of structures providing performance unattainable by the classical structures built up by traditional methods. In particular, sandwich cylindrical shells have been used increasingly as high performance structural components in numerous applications in various industrious fields (e.g., aerospace, marine, petrochemical, and power generation technologies) due to the versatility of such structures in providing enhanced mechanical properties.

The resonances of an elastic object are considered as its fingerprints. They are intrinsic characteristics of the object which are completely independent of the source of excitation and depend only on its bulk physical properties (e.g., bulk density and elastic constants) and geometry. The resonance effects may be caused by the excitation of eigenvibrations of an elastic component by an incident acoustic wave. When an elastic target is insonified by an acoustic wave, a geometric reflection is returned from the target and various types of surface waves are generated inside the target as well as the surrounding fluid medium. The resonance modes in the back-scattered spectrum are primarily linked to the standing surface waves which are formed around the cylindrical shell. For an obliquely incident wave upon the cylindrical structure, λnt=λn/cosβ is known as the projection of the total helical wavelength (λn) in the transverse rθ-plane, where β is the wave refraction angle. Knowing that a resonance indicates a standing wave pattern, an integral number (n) of the transversal component of wavelengths may be fitted around the circumference of the resonating cylindrical structure such that λnt=2πa/n, where a indicates the outer radius of the shell. These standing waves demonstrate themselves in the back-scattered signal as a spectra superimposed on a smooth background (geometric reflection). To extract resonance scattering spectra from the global scattering spectra, this non-resonance contribution (background), has to be known. The non-resonance contributions can be in general approximated with a relevant impenetrable scatterer (e.g., hard and soft scatterer for thick and very thin shells). The rigid background is suitable for isolating the resonances of a very dense (heavy) cylinder, while the soft background has proved useful in extracting the resonances of a low density cylinder. Clearly, when densities of the solid and fluid mediums are of the same order of magnitude, neither rigid nor soft backgrounds are applicable. In cases where the impedance ratio is close to unity, the proper background behaves intermediately between the rigid and soft backgrounds. Several models have been proposed especially for the shells that can neither be considered thick nor thin and that the above backgrounds are not applicable [1], [2], [3], [4], [5], [6]. Recently exact backgrounds of cylindrical shells have been found based on the absorbing scatterer, in which the elastic waves fade out quickly without forming resonances while the inertial interaction of the shell with the surrounding fluid is taken into account [1], [2]. Pure resonances in the scattering spectra can be isolated by subtracting these inherent backgrounds from the global scattered spectra. Based on this approach, the background signal complicates the interpretation of scattered sound signal and proper identification of resonances.

In recent years, many researches have tried on this research subject. Veksler and Izbicki [7] proposed a procedure for modal resonance isolation in the scattering problems of a plane acoustic wave by cylindrical and spherical shells. Choi et al. [1], [8], [9] considered resonance scattering of acoustic waves from submerged penetrable targets of canonical geometry (e.g., an empty cylindrical or spherical elastic shell in a fluid) and proposed exact expressions, named the inherent background coefficients, which is obtained from the zero-frequency limit of an equivalent fluid target, in order to properly describe the acoustical background over the entire frequency range. Joo et al. [2] subsequently extended the concept of the inherent background to multilayered elastic cylindrical structures by solving the problem of acoustic wave scattering by an analogous liquid structure. Just recently, Hasheminejad and Rajabi [10] employed an exact treatment based on the inherent background coefficients to investigate the resonance scattering of time-harmonic plane acoustic waves by isotropic functionally graded cylindrical shells.

The scattered pressure field from the target contains valuable information about the characteristics of the target and the surrounding medium. Appropriate exploitation of this information and proper identification of the resonance frequencies of the elastic object can serve as a powerful tool in many applications such as material characterization and non-destructive testing/evaluation of materials [11], [12], remote classification of submerged targets [13], [14], and on-line monitoring of elastic components [15]. The Comprehensive reviews of the sound scattering problems from cylindrical components and extensive bibliographies can be found in the works of Gaunaurd [16], Uberall [17], and Veksler [18].

The assumption of perfect bonding between adjacent layers of a multilayered structure is an ideal assumption. In practice, the bonding adhesives may either be initially imperfect because of the introduction of small micro defects in the process of manufacture or become deteriorated during its long period of service. During the two recent decades, many efforts have been devoted to study the usefulness of ultrasonic inspection methods to evaluate the bonding condition in composite laminated elastic structures. For example, Aboudi [19] presented a continuum theory for elastic wave propagation in three-dimensional composite materials with imperfect bonding between the phases. Rose et al. [20] proposed the oblique incidence technique by utilizing shear waves, obliquely incident upon an interface, applicable for the non-destructive evaluation of the adhesive bonds. Nagy [21] suggested a special high-frequency, high-inspection angle ultrasonic technique to improve the detectability of kissing bonds. It is shown that this technique is essential to interrogate the adhesive–adhered interface from the adhered side without sending the ultrasound through the inherently inhomogeneous and highly attenuating adhesive layer. Nagy [22] proposed a simple technique to distinguish and classify the types of imperfections, such as kissing, partial, and slip bonds, by using the ratio between the normal and transverse interfacial stiffness measured by comparing the longitudinal and shear reflection coefficients at normal incidence. Ko and Adler [23] used the lowest order mode of the generalized Lamb waves to examine the interfacial properties of imperfect layered substrates. Their results showed that the position of the turning point in the dispersion curve in addition to its shape was found to be sensitive to bond quality variation. Chiroiu et al. [24] used the Stoneley waves, propagating along the boundary between dissimilar layered media, to examine and analyze the behavior of the adhesive joint in the interface region. Singher [25] proposed a quasi-static model for ultrasonic guided waves interaction with imperfect interfaces in order to assess the bond strength. This work presented the results of a series of experiments with bonds of varying quality and thickness. Kosbi et al. [26] used the quantitative scanning acoustic microscopy (SAM) surface acoustic wave (SAW), to measure the dispersion curves of a glass substrate coated by gold, in order to evaluate the bonding condition. They found the significant effect of the interfacial stiffness constants and thickness of silicone oil adhesive film on the dispersion curves. Seifried et al. [27] used the analytical and numerical models in combination with experimental measurements to develop quantitative understanding of the propagation of guided Lamb waves in multilayered, adhesive bonded components. Their studies showed the sensitivity of this type of waves to the low stiffness of adhesive bond and its viscoelastic behaviour. Just recently, Leiderman and Braga [28] present an analytical–numerical method to simulate the scattering of ultrasonic waves by submerged anisotropic laminated plates with non-homogeneous adhesive defects. They combine the quasi-static approximation with a very high-order regular perturbation series to allow modelling of non-uniform interfacial flaws. They discussed how the results of their simulations can be used to indicate the frequencies and angles of incidence where scattering from potential defects is strongest to offer the best potential for flaw characterization.

In this paper, we employ a laminate approximate model along with the so-called state space formulation in conjunction with the transfer matrix (T-matrix) approach and novel features of Resonance Scattering Theory (RST) to carry out an analysis for scattering of acoustic waves by a multilayered cylindrical shell with imperfect bonding which is submerged in and filled with compressible ideal fluids. Primary attention is focused on studying the effect of the interfacial imperfection on the sensitivity of resonances in the scattered field components. Finally, an illustrative numerical example is given for a multi-layered air filled cylindrical shell to examine the effect of imperfection on the modal response of the structure. The proposed model is of high practical value in ultrasonic characterization or testing of sandwich cylindrical vessels, storage tanks, and pipelines which are of important application in oil, gas, water transport, power generation, and chemical processing industries [11], [12]. It is also of interest in other closely related technical applications such as remote classification of submerged targets (shells) [13], [14] or on-line monitoring of cylindrical structures [15].

Section snippets

Basic acoustic field equations

Consider a time harmonic infinite plane acoustic wave, with the circular frequency ω, obliquely incident at an angle α on a submerged and fluid-filled multilayered cylindrical shell of infinite length, inner radius a1 and outer radius aq+1, fabricated of q individual layers. The thickness of k’th layer denoted by hk=ak+1-ak, for k=1,2,,q. The problem geometry is depicted in Fig. 1, where (x,y,z) is the Cartesian coordinate system with origin at O, the z direction is coincident with the axis of

Numerical results

In order to illustrate the nature and general behaviour of the solution, we consider some numerical examples. Realizing the large number of parameters involved here while keeping in view our computing hardware limitations, we confine our attention to some particular model. The surrounding and filling fluids are respectively assumed to be water (ρ1=1000kg/m3,c1=1480m/s) and air (ρ2=1.2kg/m3,c2=344m/s) at atmospheric pressure and ambient temperature. Since that an increase in the incidence angle

Conclusions

The present work is concerned with acoustic wave interaction with a thick-walled orthotropic multilayered cylindrical shell with interlaminar bonding imperfections. An approximate laminate model in the context of the so-called state space formulation along with the classical T-matrix solution technique involving a system global transfer matrix as the product of the individual transfer matrices is employed to solve for the unknown modal scattering and transmission coefficients. A linear spring

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