Abstract
In this paper, at first the attenuation of Lamb waves in threelayer adhesive joints, including two elastic plates bonded together by a viscoelastic adhesive layer, is investigated using Global matrix method and then suitable incidence angle is theoretically calculated to generate lowattenuation Lamb waves using angle beam transducer. Theoretical boundary value problem in threelayer adhesive joints with perfect bond and tractionfree boundary conditions on their outer surfaces is solved to find a combination of frequencies and modes with lowest attenuation. Characteristic equation is derived by applying continuity and boundary conditions in threelayer joints using Global matrix method. Phase velocity dispersion curves and attenuation intensity plot in high and low frequencies are obtained with numerical solution of this equation by a computer code for a threelayer joint, including an aluminum repair patch bonded to the aircraft aluminum skin by a layer of viscoelastic epoxy adhesive. To validate the numerical solution results of characteristic equation, wave structure curves are plotted for a special mode in two different frequencies in the adhesive joint. Also, transducer incidence angle is calculated in terms of frequency for different modes using theoretical method to generate Lamb wave modes with low attenuation level by angle beam transducer. These modes are recognizable by transducers in inspections with Lamb waves because of low attenuation level.
Keywords:
Threelayer adhesive joints; viscoelastic; lamb wave generation; attenuation; transducer incidence angle
1 INTRODUCTION
The applications of adhesive joints in plates and pipes to increase life, improve stiffness, protect against corrosion, and electrical insulation have been caused the widespread use of viscoelastic polymer material like epoxy with plates and pipes as adhesive and coating. An example of these joints' application is a threelayer adhesive joint including an aluminum patch bonded to a surface, like aircraft aluminum skin, by a viscoelastic epoxy adhesive layer. Repair patches are used to extend the life of the aircraft. Ultrasonic guided waves are used to inspect these adhesive joints. Lamb waves have applications in nondestructive inspection of elasticviscoelastic multilayer joints and plates. Some modes of these waves have frequencies with minimum attenuation and are recognizable in inspection by transducer, and they can also detect the defects in the structures. Lowattenuation Lamb waves can be produced in multilayer structures using angle beam transducers for inspection purposes.
Different studies have been carried out to obtain propagated modes and frequencies in multilayer structures (or dispersion curves). Thomson (1950)Thomson, W.T., (1950). Transmission of elastic waves through a stratified solid medium. J. Appl. Phys. 21: 8993.and Haskell (1953)Haskell, N.A., (1953). Dispersion of surface waves on multilayered media. Bull. Seism. Soc. Am. 43: 1734. first investigated the equations of elastic waves propagation in planar multilayers with arbitrary number of layers using transfer matrix method or ThomsonHaskell method. They introduced a transfer matrix that shows the relationship between displacement and stress in bottom of a layer in comparison to their values in top of the same layer. One of the difficulties when using transfer matrix method is the instability of the solution whenever the product of frequency in thickness increases. Dunkin (1965)Dunkin, J.W., (1965). Computation of modal solutions in layered elastic media at high frequencies. Bull. Seism. Soc. Am. 55: 335358.introduced delta operator technique to solve this difficulty. Knopoff (1964)Knopoff, L., (1964). A matrix method for elastic wave problems. Bull. Seism. Soc. Am. 54: 431438.was the first to use Global matrix method to investigate the propagation of elastic waves in multilayers. In this method a global matrix is used which is derived from putting together the equations of continuity and boundary conditions in all the layers. In investigated studies, the effect of wave energy attenuation in materials is not taken into consideration.
Watson (1972)Watson, T.H., (1972). A real frequency, complex wavenumber analysis of leaking modes. Bull. Seism. Soc. Am. 62: 369541. obtained the complex roots of the characteristic equation in earth layers and showed that imaginary part of the wave number is the same as the attenuation in multilayered. Hosten and Castaings (1993)Hosten, B., Castaings, M., (1993). Transfer matrix of multilayered absorbing and anisotropic media: Measurements and simulations of ultrasonic wave propagation through composite materials. J. Acoust. Soc. Am. 94: 14881495. applied the transfer matrix method in multilayered anisotropic and damping media. The use of this method in high frequencies is accompanied with numerical instability. Castaings and Hosten (1994)Castaings, M., Hosten, B., (1994). Delta operator technique to improve the ThomsonHaskell method stability for propagation in multilayered anisotropic absorbing plates. J. Acoust. Soc. Am. 95: 19311941. applied delta operator technique to improve the stability of transfer matrix method in multilayered anisotropic damping plates. Lowe (1995)Lowe, M.J.S., (1995). Matrix techniques for modeling ultrasonic waves in multilayered media. IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 42: 525542. presented a summary of the matrix methods for modeling the propagation of ultrasonic waves in multilayered media. Both global matrix method and transfer matrix method are used in this study. These techniques can be used to obtain attenuation and phase velocity dispersion curves in viscoelastic materials. Pan et al. (1999)Pan, E., Rogers, J., Datta, S.K., Shah, A.H., (1999). Mode selection of guided waves for ultrasonic inspection of gas pipelines with thick coating. Mech. Mater. 31: 165174.investigated the propagation of ultrasonic guided waves in gas pipelines with a thick coating to choose the suitable mode for inspection. Both the effect of coating thickness and the effect of coating damping on dispersion curves and mode shapes were investigated and the modes being the least affected by coating thickness and coating damping were identified. To model the viscoelastic behavior, the coating is assumed a linear standard solid. Seifried et al. (2002)Seifried, R., Jacobs, L.J., Qu, J., (2002). Propagation of guided waves in adhesive bonded components. NDT & E Int. 35: 317328. investigated the propagation of guided waves in multilayered adhesive structures by taking into consideration the low stiffness and viscoelastic behavior of adhesive layer. To better understand the guided waves behavior and to obtain dispersion curves, they used analytical, experimental and transient FEM simulation methods.
Simonetti (2004)Simonetti, F., (2004). Lamb wave propagation in elastic plate coated with viscoelastic materials. J. Acoust. Soc. Am. 115: 20412053.investigated the propagation of Lamb wave in elastic plates coated with viscoelastic materials, and considered the viscoelastic coatings effect on dispersion properties of Lamb wave propagation in elastic plates. To do this, he used Superposition Partial Bulk Waves (SPBW) method to model the wave. Simonetti and Cawely (2004)Simonetti, F., Cawley, P., (2004). On the nature of shear horizontal wave propagation in elastic plates coated with viscoelastic materials. Proc. R. Soc. Lond. 460: 21972221. investigated the propagation of shear horizontal (SH) waves in an elastic plate coated with viscoelastic material. Material damping causes an excessive reduction of applied signal in ultrasonic test. In this research, SH wave dispersion curves for metal plates coated with viscoelastic layers are obtained using SPBW method. Barshinger and Rose (2004)Barshinger, J.N., Rose, J.L., (2004). Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material. IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 51: 15471556. investigated the propagation of guided waves in elastic hollow cylinders with viscoelastic coating using experimental and analytical methods. Wave equation is solved using theoretical boundary value problem and the best modes are specified. In this research, Global matrix method is used to obtain the roots of characteristic equation. It should be noted that in this paper, the viscoelastic characteristics of the coating are obtained using the transient wave propagation method. Shorter (2004)Shorter, P.J., (2004). Wave propagation and damping in linear viscoelastic laminates. J. Acoust. Soc. Am. 115: 19171925.investigated the wave propagation in linear viscoelastic laminates using spectral finite element method or semianalytical finite element method (SAFE). In this reference, damping loss factor is estimated for waves in low frequencies, and also stiffness matrix is assumed to be real. Birgersson et al. (2005)Birgersson, F., Finnveden, S., Nilsson, C.M., (2005). A spectral super element for modelling of plate vibrationpart 1: general theory. J. Sound Vib. 287: 297314. investigated damping loss factor using SAFE method and taking into consideration the complex stiffness matrix. Bartoli et al. (2006)Bartoli, I., Marzani, A., Lanza di Scalea, F., Viola, E., (2006). Modeling wave propagation in damped waveguides of arbitrary crosssection. J. Sound Vib. 295: 685707. investigated the wave propagation in viscoelastic waveguides with an arbitrary crosssection. To model ultrasonic wave propagation in different waveguides, SAFE method is used. The results of group velocity and phase velocity dispersion curves (for undamped media), attenuation and energy velocity (for damped media), and crosssection mode shapes are obtained which are used in nondestructive inspection. The results accuracy is validated compared to the SPBW method. Marzani et al. (2008)Marzani, A., Viola, E., Bartoli, I., Lanza di Scalea, F., Rizzo, P., (2008). A semianalytical finite element formulation for modeling stress wave propagation in axisymmetric damped waveguides. J. Sound Vib. 318: 488505.used SAFE method to analyze wave propagation in viscoelastic axisymmetric waveguides. The results accuracy of the dispersion curves is validated compared to the SPBW method. Puthillath and Rose (2010)Puthillath, P., Rose, J.L., (2010). Ultrasonic guided wave inspection of a titanium repair patch bonded to an aluminum aircraft skin. Int. J. Adhes. 30: 566573. inspected the titanium repair patches bonded to the aircraft aluminum skin using ultrasonic guided waves. They plotted the wave structures using a theoretical method and selected the mode shape with maximum inplane displacement for inspection, although they didn't take into consideration the effect of material damping.
In the present study, the propagation of Lamb waves in elasticviscoelastic threelayer joints, including two elastic plates bonded together with a layer of viscoelastic adhesive, is investigated using Global matrix method and considering viscoelastic layer damping effect. Then, the suitable incidence angle is theoretically calculated to generate Lamb wave mode with low attenuation using angle beam transducer. Also, wave structure is plotted for a specific mode in two different frequencies to verify that continuity and boundary conditions are satisfied and also to explain the attenuation behavior of waves in joints. Adhesive damping causes the excessive reduction of sending signal amplitude in ultrasonic test; so, modes and frequencies with minimum attenuation should be specified. Because these waves travel the maximum possible distance in joints and can detect the different defects namely interfacial defects.
2 THEORETICAL MODELING OF LAMB WAVES PROPAGATION IN THREELAYER ADHESIVE JOINTS
Lamb waves are propagated in thin platelike mediums in which planar dimensions are far greater than the thickness of plate and wavelength of the same order with plate thickness (Su and Ye, 2009Su, Z., Ye, L., (2009). Identification of damage using lamb waves: from fundamentals to applications, Berlim Heidelberg: SpringerVerlag Press.). Free upper and lower surfaces in plate guide movement of these waves. Lamb waves have infinite modes and their propagation properties depend on wave entry angle, frequency, and structure geometry. Figure 1 shows Lamb wave propagation in an adhesive joint which is comprised of three layers. The first and the third layers, which are elastic and isotropic, are bonded together by the second layer which is an isotropic viscoelastic layer. The layers are perfectly bonded together and the free surfaces at the top and the bottom of the three layers are tractionfree. Layers thickness is shown by h_{1}, h_{2} , and h_{3} . A local Cartesian coordinate system is used to investigate the propagation of Lamb waves in the three layers. Because of the propagation of the Lamb waves in threelayer joint, the problem is investigated as plain strain, and also the wave propagation is considered harmonic. In each layer, Lamb wave are comprised of shear and longitudinal waves superposition (Rose, 2004Rose, J.L., (2004). Ultrasonic waves in solid media, 1st ed., Cambridge: Cambridge University Press.). L+ and L show the propagation of longitudinal waves downwards and upwards the plate, and S+ and S show the propagation of shear waves downwards and upwards the plate, respectively.
Assuming that the wave propagation in threelayer adhesive joints in terms of time is harmonic, stressstrain equations of viscoelastic layer are similar to those in elastic layer, except that material properties of viscoelastic layer are complex numbers and a function of frequency (Christensen, 2010Christensen, R.M., (2010). Theory of viscoelasticity, 2nd ed., New York: Dover Publications.). This dependency between elastic and viscoelastic material in harmonic state is called Alfrey's Correspondence Principle (Flugge, 1975Flugge, W., (1975). Viscoelasticity, 2nd ed., New York: SpringerVerlag. and Ferry, 1980Ferry, J.D., (1980). Viscoelastic properties of polymers, 3nd ed., New York: John Wiley & Sons.). Also, Navier's equation of motion in viscoelastic layer is similar to the elastic layer and is expressed by Eq. (1):
In Eq. (1), λ, μ, and ρ are Lame constants and density, respectively. Lame constants in viscoelastic material are complex numbers and a function of frequency which are measured using experimental methods such as ultrasonic tests (Barshinger and Rose, 2004Barshinger, J.N., Rose, J.L., (2004). Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material. IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 51: 15471556.). In Eq. (1), the displacement field, u, can be decomposed as a combination of the gradient of a scalar potential field, Φ, and the curl of a vector potential field, H (Helmholtz decomposition) (Rose , 2004Rose, J.L., (2004). Ultrasonic waves in solid media, 1st ed., Cambridge: Cambridge University Press. and Graff, 1991Graff, K.F., (1991). Wave motion in elastic solids, 1st ed., New York: Dover Publications.):
Substituting Eq. (2) in Eq. (1), scalar and vector equations are obtained respectively:
Eq. (3) shows the propagation of longitudinal wave, and Eq. (4) shows the propagation of transverse wave in structures, and C _{1} and C _{2} quantities are longitudinal and shear wave velocities in medium, respectively. Since Lame constants in viscoelastic material are complex numbers and a function of frequency, wave velocities are also complex numbers and a function of frequency.
Using Cartesian coordinate system, the potential vector, H, can be defined as the Eq. (5):
Since this problem is assumed as a plain strain, then the equation u_{z} = ∂/∂z = 0 should be satisfied. This happens when H_{x} and H_{y} components equal zero and only H_{z} remains. The scalar potential function, Φ, should also be a function of x and y.
The solutions of Eqs. (3) and (4) for a harmonic wave propagates along the positive x direction, are assumed as Eqs. (6) and (7):
In Eqs. (6) and (7), k and w are wave number and angular frequency, respectively.
Substituting Eqs. (6)and (7) in Eqs. (3) and (4) and taking into consideration that the two components of vector potential function are equal to zero, and after solving the differential equations, the solutions are obtained as:
The solutions of Eqs. (8) and (9) are known as the partial waves solution. The four terms obtained from Eqs. (8) and (9) show the longitudinal waves propagation, L, and transverse waves propagation, S, upwards and downwards the layer. Constant values show the amplitude of propagated waves; for instance, A_{(L+)} shows the longitudinal wave amplitude propagates towards the bottom of the layer.
Substituting vector and scalar potential functions from Eqs. (8) and (9) in Eq. (2), the displacement field in adhesive joint is obtained in terms of unknown constants of the shear and longitudinal wave amplitudes:
Eqs. (10) and (11) can also be expressed as Eqs. (13) and (14):
In Eqs. (13) and (14), U_{x} and U_{y} are unattenuated displacement amplitudes.
Using Hooke and straindisplacement relations, stresses in the adhesive joint can be obtained in terms of the unknown constants of shear and longitudinal wave amplitudes:
In order to obtain Lamb waves dispersion curves for elasticviscoelastic threelayer adhesive joint, continuity and boundary conditions should be applied.
3 FORMULATION OF CONTINUITY AND BOUNDARY CONDITIOND USING GLOBAL MATRIX METHOD
Global matrix method is a suitable method for formulation of problems concerning multilayers. Continuity and boundary conditions are needed for this formulation. Using this method, continuity and boundary conditions can be shown as matrices and vectors. This method can simultaneously consider effects of material damping and wave leakage to the environment. In this method a global matrix is used to describe all the continuity and boundary conditions, and when it comes to numerical stability, it is better than other matrix methods (Lowe, 1995Lowe, M.J.S., (1995). Matrix techniques for modeling ultrasonic waves in multilayered media. IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 42: 525542.).
Figure 2 shows the boundary conditions of a threelayer adhesive joint including stress and displacement continuity in layers interfaces and tractionfree conditions in up and bottom surfaces of the elasticviscoelastic threelayer adhesive joint.
The bond between layers is perfect, and there's no shear and normal stress on freesurfaces at the top and bottom of the threelayer. This condition is shown in vector Eq. (20). Continuity of interfaces conditions include continuity of displacement components, and shear and normal stresses components. As an example, continuity between m and m+1 layers are shown by vector Eq. (21).
Before applying continuity and boundary conditions, a vector relation for displacement and stress in each layer is necessary which is obtained using Eqs. (10), (11), (16), and (18) and is shown by vector Eq. (22):
In which D is the layer matrix and is expressed as Eq. (23):
Before applying continuity and boundary conditions using Eq. (22), layer matrix in interfaces of each layer is calculated. This is achieved from Eq. (23), by substituting y = h/2 for layer top interface, and y = h/2 for layer bottom interface. These two new layer matrices are shown by Dt and Db, respectively, in which the subscripts t and b show the top and bottom interfaces of layer, respectively. Local coordinate system is used to derive these matrices, which are shown in Figure 2, and therefore can be derived for all layers by substituting material properties and thickness.
Now, we express threelayer joint continuity and boundary conditions in the form of a global matrix which is shown in Eq. (24). Am and 0 vectors in this matrix are shown by Eq. (25):
In Eq. (24) the subscript 34 shows the rows 3 and 4 of the layer matrix.
Global matrix method is a 4n × 4nsystem of equations, in which n is number of layers, and the global matrix for an elasticviscoelastic threelayer is 12 × 12. In order to the nontrivial solution to exist, the determinant of global matrix should become zero. This is shown by Eq. (26), which is called characteristic or dispersion equation of Lamb waves. With the aid of the roots of this equation, attenuation and phase velocity dispersion curves are plotted in terms of frequency.
4 NUMERICAL SOLUTION METHOD OF CHARACTERISTIC EQUATION
Characteristic equation roots in the threelayer adhesive joint are obtained using numerical solution method. In characteristic equation, frequency, w, is the independent variable, and wave number, k, is the dependent variable. The wave number in a desired frequency is obtained by solving this equation. To find characteristic equation roots, computer code is written in Matlab software. These roots are shown by curves called dispersion curves.
Finding complex roots of a characteristic equation concerning a threelayer adhesive joint of which at least one layer is viscoelastic, is a difficult task. In linear viscoelasticity, if harmonic wave propagation is desired, transverse and longitudinal velocities, and Lame constants of viscoelastic layer, are complex and a function of frequency. The transverse and longitudinal velocities are calculated from Eqs. (27) and (28) (Christensen, 2010Christensen, R.M., (2010). Theory of viscoelasticity, 2nd ed., New York: Dover Publications.):
In Eqs. (27) and (28), c _{1} and c _{2} are bulk velocities of longitudinal and transverse waves, and α_{1} and α_{2} are bulk attenuations of longitudinal and transverse waves of viscoelastic layer. Bulk attenuation and velocity values for viscoelastic material can be calculated in terms of frequency, using experimental test such as ultrasonic test (Barshinger and Rose, 2004Barshinger, J.N., Rose, J.L., (2004). Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material. IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 51: 15471556.).
Before introducing a method for finding the attenuation and phase velocity numerical results, wave number should be defined in terms of imaginary and real parts. Eq. (29) shows the wave number as complex (Blanc, 1993Blanc, R.H., (1993). Transient wave propagation methods for determining the viscoelastic properties of solids. J. Appl. Mech. 60: 763768.):
Eq. (29) enables us to solve the viscoelastic characteristic equation in terms of attenuation, k_{I} , and phase velocity, c_{ph} , instead of wave number, k. In this case, the attenuation and phase velocity dispersion curves are obtained directly.
One solution method for finding the viscoelastic characteristic equation roots is taking into consideration the minimum of characteristic equation absolute value. In this case, the problem becomes three dimensions in which the characteristic equation absolute value is a function in terms of the attenuation and phase velocity. In this method we seek to find minimum value of this function. The main issue in this method is finding all the roots.
Figure 3 shows a minimization process of characteristic equation absolute value in order to find characteristic equation complex roots.
Using the process shown in Figure 3, a computer code can be written to find characteristic equation roots. This process can be applied for all desired frequencies, and attenuation and phase velocity can be obtained in terms of frequency. Attenuation constant can be converted to attenuation in decibel per length unit, using Eq. (30). This conversion magnifies the attenuation values.
5 THEORETICAL MODELING OF LAMB WAVE MODE GENERATION
Different methods exist to generate and receive ultrasonic guided waves. Viktorov (1967)Viktorov, I.A., (1967). Rayleigh and lamb waves, New York: Plenum Press.was the first to evaluate the dispersive properties of Lamb wave he also investigated generation methods of Lamb waves. He investigated four methods to generate Lamb waves in plates. These methods can also be used in other structures.
One method to generate Lamb wave is using a longitudinal wave transducer on a plexiglass wedge, which is also called angle beam transducer. Figure 4shows the Lamb wave generation method in an elasticviscoelastic threelayer adhesive joint using an angle beam transducer. In this method, according to Snell's law transducer incidence angle depend on wedge velocity and Lamb wave phase velocity. Eq. (31) shows Snell's law, in which θi is the plexiglass wedge angle, c_{plexi} is the longitudinal wave velocity of the wedge, and c_{ph} is the Lamb wave phase velocity in adhesive joint.
In order to theoretical modeling of the Lamb wave generation, at first attenuation and phase velocity in the joint are obtained, then a combination of modes and frequencies which have low attenuation are selected, and finally the suitable incidence angle for generation is calculated using Snell's law and phase velocity of these modes. These angles are used in inspections.
6 DISCUSSION OF RESULTS FOR A SPECIFIC APPLICATION
Solving the characteristic equation by a computer code for a threelayer adhesive joint, including an aluminum repair patch bonded to the aircraft aluminum skin with a viscoelastic epoxy adhesive layer, the attenuation intensity plot and phase velocity dispersion curves in high and low frequencies for this specific application are generated. Also, acceptable attenuation level is calculated for ultrasonic inspection using a single transducer of the adhesive joint with 200 mm length and suitable modes are selected. Geometric and acoustic properties of elasticviscoelastic threelayer adhesive joint can be seen in Table 1. Aluminum and Mereco Epoxy 303 acoustic properties are picked up from (Barshinger and Rose, 2004Barshinger, J.N., Rose, J.L., (2004). Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material. IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 51: 15471556.).
Wave structure for a mode in two different frequencies is plotted to validate numerical solution results. Finally, transducer incidence angle is plotted in terms of frequency for different modes, and suitable wedge angles are selected to generate lowattenuation Lamb wave modes in the adhesive joint with 200 mm length.
6.1 Phase velocity dispersion curves and attenuation intensity plot
Figure 5 shows the phase velocity dispersion curves in terms of frequency for different modes in the elasticviscoelastic threelayer adhesive joint, the properties of which are shown in Table 1. The lamb wave modes are identified with M and numbers in Figure 5. Investigating this curve it can be seen that in the frequency intervals of 150 kHz to 250 kHz only three modes of M1, M2, and M3 propagate, and other modes don't propagate in these low frequencies. 250 kHz frequency is cutoff frequency of M4 mode; because, M4 mode doesn't propagate in frequencies lower than this frequency. Also, in the frequency intervals of 275 kHz to 575 kHz only four modes of M1, M2, M3, and M4 propagate, and 575 kHz is the cutoff frequency of M5 mode.
Phase velocity dispersion curves in terms of frequency in the threelayer adhesive joint: aluminumepoxyaluminum.
The attenuation is shown superimposed on the Lamb wave dispersion curves with the intensity plot in Figure 6. In this paper, acceptable range of attenuation is calculated for ultrasonic inspection of the adhesive joint with 200 mm length using a single transducer. The suitable range of attenuation depends on wave propagation distance in a wave roundtrip to transducer and on the signal to noise ratio (SNR). In inspection with guided waves, SNR is a measure for detecting small defects and is the ratio of reflected signal from defects to return signal from grains (as noise) to transducer. Minimum identifiable SNR in guided wave test is 6 dB (Barshinger and Rose, 2004Barshinger, J.N., Rose, J.L., (2004). Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material. IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 51: 15471556.). In guided wave test, defect signal is usually 20 dB higher than the noise signal; therefore, a 14 dB signal can be lost because of the guided wave mode attenuation, and if attenuation is more than which, defects are not detectable. The distance that a wave travels in a roundtrip to transducer is twice the length of the plate and equal to 400 mm, and maximum attenuation that the wave can have in a roundtrip equals 14 dB / 0.4 m or 35 dB m^{1}; therefore, the suitable range of attenuation is from 0 to 35 dB m^{1}.
Variation of the attenuation superimposed over the Lamb wave dispersion curves for the threelayer adhesive joint: aluminumepoxyaluminum.
The modes with acceptable attenuation level (0 to 35 dB m^{1)} are selected for inspection in high and low frequencies. Generating these modes, adhesive joint inspection can be carried out to find the defects. From curves in Figure 6, it can be seen that M1 mode in frequency range of 150 kHz to 500 kHz has a suitable attenuation level for inspection, and in frequencies higher than 500 kHz, a sudden and excessive increase can be seen in attenuation. This mode in high frequencies is not suitable for inspection. Attenuation in M2 mode in low frequencies, in the range of 150 kHz to 250 kHz increases extremely, and has a sudden and excessive increase in frequencies higher than 800 kHz. M2 Mode in frequency range of 325 kHz to 800 kHz and M3 mode in frequency ranges of 150 kHz to 675 kHz and 1.2 MHz to 1.775 MHz have suitable attenuation levels for inspection of the adhesive joint with 200 mm length. M3 mode in 500 kHz has an attenuation equal to 5.1 dB m^{1} which is the lowest attenuation level in frequency range of 0 to 3 MHz. M5 mode has a suitable attenuation level in high frequencies and is suitable for inspection in 1.9 MHz to 3 MHz frequency range, and it also has negligible attenuation about 0.27 dB m^{1} in frequencies near 3 MHz.
6.2 Validation of numerical solution results
One method to validate the numerical solution results of characteristic equation, which are the same as attenuation and phase velocity, is the investigation of the interfacial continuity equations and boundary conditions in the adhesive joint. The wave structure of propagated modes in the threelayer adhesive joint is plotted to validate whether interfacial continuity equations and boundary conditions are satisfied. Wave structure curves are the same as stress and displacement amplitudes across threelayer adhesive joint thickness.
The curve in Figure 7 is the M2 mode wave structure in 500 kHz frequency with an attenuation of 13.5 dB m^{1}. As the Figure 7 shows, shear and normal stresses don't exist in free surfaces at the top and bottom of the three layers; also, interfacial continuity conditions including continuity of shear and normal stresses and displacement components, are satisfied. M2 mode wave structure in 1.25 MHz frequency with the attenuation of 222.8 dB m^{1} is also plotted in the curves of Figure 8, in which continuity and boundary conditions are also satisfied. Because attenuation level is high in wave structure curve of Figure 8, most of the displacement exists in the viscoelastic layer.
M2 mode wave structure in 500 kHz frequency with the attenuation of 13.5 dB m1 in a threelayer adhesive joint (a) normalized displacement wave structure (b) normalized stress wave structure.
M2 mode wave structure in 1.25 MHz frequency with the attenuation of 222.8 dB m1in a threelayer adhesive joint (a) normalized displacement wave structure (b) normalized stress wave structure.
6.3 Transducer incidence angles to generate lowattenuation Lamb wave modes
In this section, at first the transducer incidence angle curves in terms of frequency for generating lamb wave modes in the threelayer adhesive joint is plotted using Eq. (31) and phase velocity values, these curves are shown in Figure 9. Then the suitable incidence angles are specified to generate lowattenuation modes by transducer. Investigating Figure 9 it can be seen that the incidence angles for generating some modes such as M1 mode don't exist, the generation of which is impossible by the transducer for all the frequency ranges. M2 mode has a low attenuation level and a suitable incidence angle for inspection in 150 kHz to 200 kHz and 650 kHz to 800 kHz frequency ranges. To generate this mode in 200 kHz frequency with low attenuation level of 7.34 dB m^{1}, the transducer incidence angle should be 32.4 degree. Simultaneously investigating the attenuation values and transducer incidence angle it can be seen that M3 mode in 150 kHz to 675 kHz frequency range, M4 mode in 300 kHz to 725 kHz frequency range, and M5 mode in 875 kHz to 1.075 MHz and 1.9 MHz to 3 MHz frequency ranges have low attenuation level and suitable incidence angle for the generation of Lamb wave. Transducer incidence angle should be 30.6 degree to generate M3 mode in 500 kHz frequency and low attenuation level of 5.1 dB m^{1}, and 66 degree to generate M5 mode in 3 MHz frequency and negligible attenuation level of 0.27 dB m^{1}.
Transducer incidence angle curves in terms of frequency for generating Lamb wave modes in the threelayer adhesive joint: aluminumepoxyaluminum.
7 CONCLUSIONS
The obtained results of Lamb wave theoretical modeling in threelayer adhesive joint with a viscoelastic adhesive can be used in inspections using ultrasonic guided waves in threelayer structures. The results of the present paper can be summarized as:

1. Lamb waves have many modes, some of which don't propagate in frequencies lower than the cutoff frequency.

2. Some modes, such as M1, M2, and M3, have an acceptable attenuation level for inspection with guided waves in low frequencies and some others, such as M5, in high frequencies.

3. Investigating the wave structure curves it can be seen that the interfacial continuity and boundary conditions is satisfied in adhesive joint. This result validates the numerical solution results of characteristic equation.

4. Transducer incidence angle obtained from theoretical modeling of Lamb wave mode generation can be used to inspect adhesive joints and generate Lamb wave with low attenuation level in joints.
References
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Publication Dates

Publication in this collection
Mar 2015
History

Received
22 Dec 2013 
Accepted
12 Jan 2015