Reflection and transmission coefficients of the SH0 mode in the adhesive structures with imperfect interface

Accepted Manuscript Reflection and transmission coefficients of the SH0 mode in the adhesive structures with imperfect interface Juncai Ding, Bin Wu, Cunfu He PII: DOI: Reference:

S0041-624X(16)30060-9 http://dx.doi.org/10.1016/j.ultras.2016.05.010 ULTRAS 5284

To appear in:

Ultrasonics

Received Date: Revised Date: Accepted Date:

5 January 2016 18 April 2016 15 May 2016

Please cite this article as: J. Ding, B. Wu, C. He, Reflection and transmission coefficients of the SH0 mode in the adhesive structures with imperfect interface, Ultrasonics (2016), doi: http://dx.doi.org/10.1016/j.ultras.2016.05.010

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Reflection and transmission coefficients of the SH0 mode in the adhesive structures with imperfect interface Juncai Ding, Bin Wu, Cunfu He College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, PR China

Abstract: Compared with body waves, ultrasonic guided waves can provide more local characteristic information about the interface in the defect detection of adhesive structures. In the paper, the expressions of the reflection and transmission coefficients of the lowest SH mode (SH0) in multilayered plate-like adhesive structure were deduced on the basis of wave propagation controlling equations and tangential stiffness coefficient KT was contained in the expressions. Then, the expressions were compared with the previous results to verify their applicability and correctness. Then, aluminum/epoxy resin/aluminum adhesive structures were used to explore the effects of the changes in incident angle, frequency-thickness product and tangential stiffness coefficient on SH wave propagation characteristics in adhesive structures with different interface quality (perfect, weak bonding, and slip/debonding interfaces). The results showed that the propagation mode of SH wave in adhesive structures was mainly determined by the incident angle, frequency, adhesive layer thickness and tangential stiffness coefficient. With the increase in the frequency-thickness product, multi-order resonance is generated in the reflection and transmission coefficient curves of SH wave under the perfect and weak bonding interfaces. If proper values of the incident angle of acoustic waves and frequency-thickness product are selected, the perfect, weak bonding, and slip/debonding interfaces can be differentiated from each other, but the slip and debonding interfaces cannot be distinguished from each other. The study provides theoretical contribution to the detection of multilayered plate-like adhesive structure by SH wave. Key words: Adhesive structure; Lowest SH mode; Tangential stiffness coefficient; Interface bonding conditions; Multi-order resonance

1. Introduction An adhesive structure has high specific strength, high specific modulus, superior sealing, and

vibration attenuation properties and is extensively applied in machinery, construction, electronics, aerospace, and other fields [1-3]. Adhesive quality is mainly determined by the adhesive operation process. Influenced by adhesive quality and adhesive process design, holes, weak bonding, shear slip, partial debonding and other defects in the implementation process are prone to emerge at the interfaces and seriously affects mechanical properties of adhesive structures [4-6]. Thus, it is necessary to perform nondestructive testing and evaluation of interface quality of adhesive structure [7]. At present, ultrasonic testing has become one of the most extensively applied technologies in nondestructive testing of adhesive structures [8]. In general, the contact interface between two solids has special and complex physical properties, which are different from but depend on the properties of the two solids. Since the propagation mode of ultrasonic waves will be changed when the waves pass different contact forms of interfaces, the interface states may be analyzed through investigating the characteristics of reflection or transmission spectrum of ultrasonic waves passing through the interfaces [9]. The studies on ultrasonic propagation mechanism in adhesive structures with different interface quality and nondestructive testing and evaluation of bonding interface have been widely concerned [10]. Brekhovskikh et al. [11] adopted impedance and sound pressure continuation to deduce the reflection and transmission coefficient expressions of body waves in multilayered homogeneous medium and studied the interface layer of adhesive structure. Baik et al. [12] put forward a quasi-static model, adopted the vertical incidence method to study the reflection and transmission characteristics of body waves in the adhesive structure which was not connected perfectly, and greatly improved computational accuracy of reflection and transmission coefficients by introducing inertia mass. Based on the transfer matrix method, Thomson [13] theoretically analyzed the characteristics of adhesive layer with the reflection and transmission coefficients of oblique incident acoustic waves. Rokhlin et al. [14-16] regarded the bonding interface as a viscoelastic thin layer, studied the propagation characteristics of body waves in multilayered adhesive structure when the adhesive layer with finite thickness was between two semi-infinite isotropic media, and effectively identified the perfect and debonding interfaces. Similarly, Wang et al. [17] explored the reflection and transmission characteristics of sound in layered adhesive structure with perfect and slip interfaces with the transfer matrix method and derived the analytical expressions of reflection and transmission coefficients for body waves from unilateral substrate. Based on the spring model, Qiu

et al. [18, 19] and Wu et al. [20] adopted the methods of vertical and oblique incidence to study the propagation characteristics of body waves in weak adhesive structure, distinguished the weak bonding interface from debonding interfaces well, and conducted a numerical simulation with COMSOL. Wu et al. [21] studied the longitudinal wave transmission characteristics of adhesive structures under the condition of water immersion and oblique incidence, and gained a complete longitudinal wave transmission coefficient expression. However, the incident ultrasonic guided waves were not studied. Based on ultrasonic body waves, Zhou et al. [22] investigated the relationship between the inherent frequency and adhesive layer thickness of adhesive structures with different substrate materials, and obtained the consistent theoretical and experimental results. Tohmyoh et al. [23] used the resonant frequency of high-frequency ultrasound to test steel/coating/air structure and measured the coating thickness through analyzing the changes of interface reflection coefficient. Deng et al. [24] used the semi-analytical finite element method to analyze the propagation characteristics of guided waves in the adhesive structure with tapered adhesive layer. Cerniglia et al. [25] studied the propagation of guided waves in multilayered adhesive structure through 3D simulation and experiments. He et al. [26] derived the wave equations of coupled Lamb waves in multilayered arbitrary anisotropic composite laminates with the Legendre orthogonal polynomial approach, and solved the coupled equations of wave motion. Based on the robustness algorithm of global matrix method, Demčenko et al. [27] calculated the dispersion curves of Lamb waves in aluminum/steel two-layered structure and aluminum/fibre/glass five-layered structure, and indicated the feasibility of the analysis of isotropic plane structures with global matrix method. Wang et al. [28] simplified the weak contact interface between two solids as a spring model, deduced the dispersion equations of Lamb waves when aluminum/copper interface was in perfect and slip connection, and analyzed the influence of tangential stiffness coefficient on lower-order Lamb modes. On the basis of the results by Wang et al. [28], Ning et al. [29] derived the characteristic equation for Lamb waves in three-layered solid composites with arbitrary layer thickness and explored the particle displacement distributions along the thickness direction under the two lowest modes A0 and S0. Pant et al. [30] obtained the dispersion equation of Lamb waves in n-layered composite laminated plates on the basis of linear 3D elastic theory and experimentally verified the equation. Crom et al. [31] got the dispersion curve of SH waves in aluminum/composite plate adhesive structure by numerical

simulation, and analyzed the effects of the thickness of aluminum and composite plates and shear modulus on phase velocity of SH waves. Banerjee et al. [32] theoretically analyzed the propagation mode of guided waves in the adhesive structure and found that the response signal was controlled by the first-order anti-symmetric mode during vertical simulation and that the high-order symmetric and anti-symmetric controlling signals during horizontal stimulation was multi-mode signals. Chaudhary et al. [33] derived the reflection and transmission coefficient expressions of plane SH waves in “sandwich” structure where the self-reinforced elastic layer was between two homogeneous semi-infinite solids. Yew et al. [34] used SH waves to evaluate the adhesive layer quality of adhesive structure and indicated that the cut-off frequency of the second-order mode of SH waves in the adhesive structure mainly depended on the thickness of adhesive layer and its mechanical properties. Dravinski et al. [35] and Sheikhhassani et al. [36] utilized the boundary integral method to investigate the dispersion characteristics of SH waves in multilayered structure. Castaings [37] adopted the finite element and experiment methods to explore the SH wave propagation in aluminum/plexiglass/aluminum overlapping structure with different interface quality and indicated that the mode SH0 was sensitive to the change of interface quality. Predoi et al. [38] investigated the change rules of adhesive layer thickness through analyzing the reduction of bonding interface strength. In conclusion, with different methods, simplified interface models, and the combination of ultrasonic body or guided waves, many scholars studied the adhesive structure. In fact, previous studies on adhesive structures mainly focused on interface characteristics. Compared with ultrasonic body waves, guided waves can provide more local feature information about the interface in the detection of adhesive structures. Previous studies were mainly performed through simulations or experiments without necessary theoretical supports, especially theoretical studies on identification of interface forms of adhesive structures by SH waves. Weak bonding interface is a special interface form, but the contact interfaces in the above literatures are mostly in perfect or debonding state. The SH wave propagation characteristics in the bonding structure with weak interface were not reported. In the paper, we introduced tangential stiffness KT into wave propagation controlling equations under the condition of plane harmonic SH wave incidence and investigated the changes in the refection and transmission characteristics of SH wave propagating in the plate-like adhesive structures with the perfect, weak bonding, and slip/debonding interfaces. Firstly, the analytical

expressions of reflection and transmission coefficients of the lowest SH mode (SH0) containing tangential stiffness coefficient were derived theoretically. Then, we obtained the reflection and transmission coefficient curves of SH waves in the adhesive structure with different interface bonding conditions through numerical simulation. Finally, we explored the effects of the changes in the incident angle of acoustic waves, frequency-thickness product or tangential stiffness coefficient on the reflection and transmission characteristics of SH waves.

2. Mathematics and formulation of the adhesive structure The schematic diagram of SH wave propagation is shown in Fig. 1. The y axis of Cartesian coordinate system is set on the upper interface (interface 1) of the three-layered plate-like adhesive structure and the plate thickness is set along the x direction. Semi-infinite solid media (1 and 3) are respectively the upper and lower substrates of the adhesive structure. Medium 2 is the adhesive layer with the thickness of h and media 1, 2, and 3 are isotropic elastic solid media. In Fig. 1, SHi1 is the incident SH wave; SHt 2 and SHt 3 are transmitted SH waves in the adhesive layer and substrate 3, respectively; SH r1 and SH r 2 are reflected SH waves in substrate 1 and the adhesive layer, respectively;  refers to the propagation angle of SH waves in the adhesive layer;  and  are respectively the incident angle (or the reflected angle) of SH waves in substrate 1 and the transmitted angle of SH waves in substrate 3. Since the wave conversion will not happen when SH waves propagate in the isotropic media, there are only SH waves in media 1, 2, and 3. If plane harmonic SH waves with an angular frequency of  enter the multilayered adhesive structure from substrate 1, the wave vector components of incident waves and all the reflected or transmitted waves are equal along the y direction according to Snell’s Law. If the nonlinear effect of wave propagation is ignored, the incident and reflected waves in the same medium can be superposed linearly. In addition, it should be noted that the propagation attenuation of SH waves in the structure is not considered in this paper. For isotropic elastic solid media, the Navier controlling equation can be expressed as [39]: (   )u j ,ij  ui , jj   fi  ui

(i, j  x, y, z) .

(1)

If body force is ignored, Eq. (1) can be rewritten as the component forms of Cartesian coordinates as:

(   )

 2ux   ux u y uz  2       ux   2 , x  x y z  t

(2a)

(   )

 2u y   ux u y uz  2      u   ,   y y  x y z  t 2

(2b)

(   )

 2uz   ux u y uz  2      u   ,   z z  x y z  t 2

(2c)

where  and  represent Lame constants;  is density; u x , u y , and u z are displacement components; t refers to time;  2 

2 2 2  2  2 is Laplace operator. 2 x y z

Since the transverse waves belong to equivoluminal waves,

ux u y uz    0 . Then Eq. (2) x y z

can be rewritten as:  2ux  C 2T  2ux , t 2  2u y

(3a)

 C 2T  2u y ,

(3b)

 2uz  C 2T  2 u z , t 2

(3c)

t 2

where CT   /  represents transverse wave velocity. For SH wave, its displacement component is nonzero only in the z direction, so ux  u y  0 . Then Eq. (3) can be simplified as:  2uz  C 2T  2 u z . t 2

(4)

As shown in Fig. 1, it is assumed that the displacement components along the z direction in substrate 1, adhesive layer 2 and substrate 3 (marked as u z1 , u z 2 and u z 3 , respectively) can be expressed as:  u z1  g1 ( x)ei ( ky t )  i ( ky  t ) u z 2  g 2 ( x)e  u  g ( x)ei ( ky t ) 3  z3

x0 0  x  h, xh

(5)

where k and  respectively represent wave number and angular frequency. The reason for choosing such form of solution is that it represents wave motion along the y direction (expressed as the exponential term), and there is certain distribution in the x direction (given by g1 ( x) , g 2 ( x) and g3 ( x) ). Generally, practical physical displacement field is the real part of the right term in Eq.

(5). Combining Eq. (4) with Eq. (5), we can know that g1 ( x) , g 2 ( x) and g3 ( x) are the solutions to the following differential equations actually:

where Q12 

c 2  C 2T (1) C 2T (1)

; Q2 2 

g1 ( x)  k 2Q12 g1 ( x)  0 ,

(6a)

g2 ( x)  k 2Q2 2 g 2 ( x)  0 ,

(6b)

g3 ( x)  k 2Q32 g3 ( x)  0 ,

(6c)

c 2  C 2T (2) C 2T (2)

; Q32 

c 2  C 2T (3) C 2T (3)

; c   / k refers to the phase velocity;

CT (1) , CT (2) , and CT (3) are respectively the transverse wave velocities in media 1, 2 and 3. As

mentioned above, the wave vector components of incident, reflected and transmitted waves along the interface are equal. Therefore, at interfaces 1 and 2, c

c

CT (1) sin 

CT (2) sin 





CT (2) sin 

,

(7a)

.

(7b)

CT (3) sin 

Its general solution of displacement can be gained easily according to Eq. (6):  uz1  A1eik ( y  Q1 x  ct )  B1eik ( y Q1x  ct )  ik ( y  Q2 x  ct )  B2 eik ( y Q2 x ct ) u z 2  A2 e  u  A eik ( y  Q3 x  ct ) 3  z3

x0 0 x  h, xh

(8)

where A1 and B1 respectively refer to the amplitudes of incident and reflected SH waves in substrate 1; A2 and B2 respectively represent the amplitudes of transmitted and reflected SH waves in the adhesive layer; A3 is the amplitude of transmitted SH waves in substrate 3. The spring model is applicable to analyze solid structures with small displacement and small deformation, so the bonding interface of adhesive structure may be studied with the spring model. Interface stiffness is one of the critical factors to evaluate the interface bonding quality. For the longitudinal wave or vertical shear wave (SV transverse wave) incidence, according to boundary conditions of the spring model, if normal stiffness coefficient K N and tangential stiffness coefficient KT of the interface are combined together and different values are attributed to the two parameters, the mechanical state of bonding interface may be indirectly characterized

mathematically. Different from longitudinal wave and SV transverse wave, plane harmonic SH wave belongs to horizontally polarized wave, so the values of particle displacement and stress caused by SH waves propagating in a medium are only related to tangential stiffness coefficient KT and do not involve normal stiffness coefficient K N [37]. Thus, for SH wave incidence, we may use KT   , let 0  KT   , and KT  0 to represent the perfect interface, the weak bonding

interface, and the slip or debonding interface, respectively. The symbol “  ” means the maximum tangential stiffness value for the adhesive structure with the perfectly connected interface. Through the assumptions above, we can find that due to the particularity of SH wave vibration form, the slip and debonding interfaces cannot be differentiated from each other when SH waves are used to detect the adhesive structure. Since substrates 1 and 3 are semi-infinite isotropic solid structures, the stress or displacement of the upper interface of substrate 1 or the lower interface of substrate 3 may not be taken into account. For interfaces 1 and 2, tangential stiffness coefficients KT (1) and KT (2) may be used to describe the mechanical properties of the bonding interface between substrate 1 and the adhesive layer and the bonding interface between the adhesive layer and substrate 3, respectively. KT (1) and KT (2) respectively refer to tangential stiffness coefficients of interfaces 1 and 2. The interface connection conditions can be expressed as follows:  xz1   xz 2   (1) ( xz1   xz 2 ) / 2  KT (uz1  uz 2 )  xz 2   xz 3   (2) ( xz 2   xz 3 ) / 2  KT (uz 2  uz 3 )

where  xz1  1

x=0 ,

(9a)

x=h ,

(9b)

u uz1 u ,  xz 2  2 z 2 and  xz 3  3 z 3 represent tangential stress components in x x x

substrate 1, adhesive layer and substrate 3, respectively. Combing Eq. (9) with Eq. (8), four linear equations containing tangential stiffness coefficients KT (1) and KT (2) as well as five unknown numbers A1, B1, A2, B2, and A3 can be acquired as follows:

1Q1 ( A1  B1 )  2 Q2 ( A2  B2 )   ik 1Q1 ( A1  B1 )  KT (1) ( A1  B1  A2  B2 )  .  2 Q2 ( A2 eikQ2 h  B2 e ikQ2 h )  3Q3 A3eikQ3h  ikQ2 h   B2 e ikQ2 h )  KT (2) ( A2 eikQ2 h  B2 e ikQ2 h  A3eikQ3 h ) ik 2 Q2 ( A2 e

(10)

Generally, the amplitude A1 of incident SH waves is a known quantity, so Eq. (10) is a system of linear equations containing four unknown numbers (B1, A2, B2 and A3). The four unknown

numbers are expressed by A1, KT (1) , and KT (2) and Mathematica is applied to solve the system of equations. Finally, with tangential stiffness coefficients KT (1) and KT (2) , the analytical expressions of reflection and transmission coefficients of SH waves in three-layered plate-like adhesive structure can be gained as follows: B1 1Q1 F  2 KT (1) Q2 ( H  1) ,  A1 1Q1 F  2 KT (1) Q2 ( H  1)

(11a)

A3 21 KT (1) KT (2) Q1 ( H  e 2ikQ2 h )e ik (Q3 Q2 ) h ,  A1 [ KT (2)  ik 3Q3 ][ 2 KT (1) Q2 ( H  1)  1Q1 F ]

(11b)

R T

where R and T respectively represent the reflection and transmission coefficients and H

[( KT (2)  ik 3Q3 ) 2Q2  3Q3 KT (2) ] 2ikQ2 h e [( KT (2)  ik 3Q3 ) 2Q2  3Q3 KT (2) ]

F  KT (1) ( H  1)  ik 2Q2 ( H  1)

It should be noted that the reflection and transmission coefficients of the lowest SH mode (SH0) are solved in this paper.

3. Numerical calculation and analysis Based on the model mentioned above, a computer program was developed by Matlab to calculate the reflection and transmission coefficients of SH waves and then the reflection and transmission characteristics of acoustic waves passing through different interfaces were studied. In this paper, aluminum was chosen as the substrate material of the adhesive structure and the adhesive was epoxy resin. The characteristic parameters of the two materials are listed in Table 1. Table 1 Physical properties of common materials Density

L-wave velocity

T-wave velocity

 (kg/m3)

CL (m/s)

CT (m/s)

Aluminum

2700

6320

3080

Epoxy resin

1300

2800

1100

Materials

In theory, if specific values are attributed to KT (1) and KT (2) to describe corresponding interfaces, for any interface form, the suitable values of KT (1) and KT (2) are unlimited. However, the selected values are limited in the practical calculation. Thus, aiming at different interface quality, a group of appropriate data are chosen to characterize corresponding interfaces. For aluminum/epoxy

resin/aluminum adhesive structure, the maximum tangential stiffness value is about 3 1015 N/ m3 under the condition of perfect interface. Hence, to get the more precise calculation, KT (1) (or KT (2) )  3 1016 N/ m3 is used to represent the perfect interface; KT (1) (or KT (2) )  7 1012 N/ m3 is used to

represent the weak bonding interface; KT (1) (or KT (2) )  0 is used to represent the slip or debonding interface. Although only a group of representative data are listed in this paper, other values may be attributed to tangential stiffness coefficient as long as they meet the conditions. 3.1. Comparison with existing data To verify the correctness of the proposed method, we compared the results in this paper with previous results. Under the condition of perfectly connected aluminum/epoxy resin/aluminum adhesive interfaces, the influence of the changes in the adhesive layer thickness on the reflection and transmission coefficients of SH waves is shown in Fig. 2. The frequency of SH waves is 2 MHz and the angle of incidence is 45°. In Fig. 2, the solid and dashed lines represent the previous calculation results [40], while the square and circular dotted lines represent the calculation results obtained with Eq. (11) in the paper. In the matrix method [40], it is assumed that the tangential stress and displacement are the same on the upper and lower sides of the interface in perfect connection. Different from the above assumption, the same value of KT (1) and KT (2) of 3 1016 N/ m3 in this paper indicates the interface in perfect connection. Compared to the matrix method, the simplified spring model of the interface can be easily used to characterize the interface of different qualities, especially for the weak bonding interface. Nevertheless, if the matrix method is used to deal with the adhesive structure, different boundary conditions should be added according to diverse interface quality, thus leading to the complex formula derivation. More importantly, it is difficult to deal with the weak interface. In Fig. 2, if the frequency and incident angle are fixed, multi-order resonance occurs on the reflection and transmission coefficient curves of SH waves with the increase in the adhesive layer thickness. The change trend is in accordance with the acoustic propagation theory in multilayered structure. According to the comparison results shown in Fig. 2, the results in this paper are consistent with the previous results [40], verifying the effectiveness of the method proposed in this paper. 3.2. Influence of incident angle on the reflection and transmission characteristics

The changes in the reflection and transmission coefficients of SH waves in aluminum/epoxy resin/aluminum adhesive structure with different interface quality during the acoustic waves incident at different angles were explored in this section. Here was studied the case of a perfect connection at interface 2 ( KT (2)   ) but interface 1 was in non-perfect connection. The case of the non-perfect connection at the interface 2 was not studied in this paper. For the actual adhesive structure, the thickness of the adhesive layer is generally tens to hundreds of microns, so epoxy resin with the thickness of 0.1 mm is chosen as the adhesive layer. In addition, the frequency of SH wave is set to be 1 MHz. As shown in Fig. 3, when interface contact forms are

different, the reflection or transmission coefficient curves of SH waves show significant differences, which indicates that the interactions between SH waves and diverse interfaces are different. Within the same angle scope, as the interface bonding quality declines, the amplitude of SH wave reflection coefficient shows a rising trend, but the amplitude of transmission coefficient gradually decreases because more acoustic waves will be reflected back to substrate 1 when the interfacial adhesiveness is worse. When the angle is close to 80°, the amplitude difference of reflection or transmission coefficients under different quality of interfaces is the largest, indicating that if aluminum/epoxy resin/aluminum adhesive structure with 0.1 mm-thick bonding layer is tested by SH waves with the frequency of 1 MHz, the incident angle of acoustic waves should be about 80°. When the interface is in the slip/debonding state, the amplitudes of reflection and transmission coefficients of SH waves are respectively 1 and 0 because the stress generated by SH waves is parallel to the interface and the slip/debonding interface cannot transmit the stress of incident waves. According to the results shown in Fig. 3, we can draw the conclusion that the critical angle does not exist when the adhesive structure is tested by SH waves. 3.3. Influence of frequency-thickness product on the reflection or transmission characteristics For adhesive structure testing, frequency-thickness product is a basic parameter. For an excitation with a central frequency in the order of megahertz which is commonly used in ultrasonic testing, its relative bandwidth is usually between 50% and 80% and the frequency-thickness product for detection is no more than 5 MHz mm. Hence, the range of frequency-thickness product is set to be from 0 to 5 MHz mm in this section. Fig. 4 and Fig. 5 respectively show the effects of the changes in the frequency-thickness product on the reflection or transmission characteristics of acoustic waves

under the condition of SH wave incidence at specific angles (0° or 50°) in aluminum/epoxy resin/aluminum adhesive structure. Irrespective of the angle, when the interface is in the perfect or weak bonding state, the reflection and transmission coefficients of SH waves present the obvious resonance with the increase in the frequency-thickness product. At the resonance point shown in Fig. 4(b), the reflection coefficient shows the minimal value, while the transmission coefficient presents the maximum value. Under the condition of vertical incidence of acoustic waves (Fig. 4), the resonant period of the reflection and transmission coefficient curves in perfect interface is always 0.55 MHz mm, and the amplitudes of maximum and minimal value points of reflection and transmission coefficient curves are constant. The amplitude of reflection coefficient curve at the minimal value point rises gradually in weak bonding interface, while the amplitude of transmission coefficient curve at maximum value point decreases gradually. No matter whether the reflection coefficient or transmission coefficient method of SH waves is applied, if the proper value of frequency-thickness product is selected, the perfect, weak bonding, and slip/debonding interfaces could be distinguished from each other. For the reflection coefficient or transmission coefficient method of SH waves, the frequency-thickness product of 0.67 MHz mm or 1.1 MHz mm could be used to identify the forms of interfaces, respectively. Unlike the results in Fig. 4, only the incident angle of SH waves is changed in Fig. 5. When the incident angle of SH waves increases to 50°, the position of the corresponding-order resonance point of the reflection or transmission coefficient curves shown in Fig. 5(b) shifts to the right by about 3.64% under perfect and weak bonding interfaces, compared with that in Fig. 4. When the interface is in the perfect state, the amplitudes at maximum value point of reflection coefficient curve and minimal value point of transmission coefficient curve of SH waves are respectively decreased by 7.45% and increased by 45.45%. Similarly, when the angle of incidence is 50°, the reflection coefficient or transmission coefficient method of SH waves can be used to identify the interface forms easily under the condition of the frequency-thickness product meet 0.69 MHz mm or 1.12 MHz mm, respectively. Furthermore, if the interface is in the slip/debonding state, the amplitudes of reflection and transmission coefficients of SH waves are respectively always 1 and 0, which indicate that the slip/debonding interface cannot transmit the stress generated by SH waves irrespective of the incident angle of acoustic waves or frequency-thickness product. 3.4. Three-dimensional relationship among the frequency, adhesive layer thickness, and the

reflection/transmission characteristics In this section, the relationship among the frequency, adhesive layer thickness, and the reflection or transmission coefficients of acoustic waves is studied under the incident angle of 50°. In Figs. 6 to 11, the shade of colors represents the amplitudes of reflection or transmission coefficients of SH waves. In addition, the similar curve part in Fig. 6(b) and Fig. 7(b) represent the minimal value points of reflection coefficient, while the similar curve in Fig. 9(b) and Fig. 10(b) represent the maximum value points of transmission coefficient. Here, the reflection coefficient (Figs. 6 to 8) is taken as an example in the study. According to the figures, when the interfaces are in the perfect (Fig. 6) and weak bonding (Fig. 7) states, multi-order resonance gradually appears in the reflection coefficient curve with the change in the acoustic wave frequency and adhesive layer thickness. As the interface bonding quality declines to the slip/debonding connection from the perfect connection, the amplitude of reflection coefficient at the minimal value point rises gradually to 1. In addition, another phenomenon can be gained under the perfect or weak bonding interfaces. When the frequency range of acoustic waves is fixed, the number of resonant frequency curves is increasing as the thickness of adhesive layer increases, and vice versa. Such phenomenon may be understood as follows: within the same range of adhesive layer thickness (or frequency), the resonant frequency curve drifts to the low frequency with the increase in acoustic wave frequency (or adhesive layer thickness). The analysis method in Figs. 9 to 11 is similar to that in Figs. 6 to 8, so no more description is given here. 3.5. Influence of changes in tangential stiffness coefficient on the reflection characteristics The weak bonding is a complex bonding condition. The distribution range of relevant tangential stiffness coefficients under the weak bonding is wide. The change rules of reflection or transmission coefficients of SH waves with the changes in the incident angle and frequency-thickness product were analyzed under the case of fixed KT (1) and KT (2) values above. Hence, it is necessary to study the influence of tangential stiffness coefficient on the propagation characteristics of acoustic waves. The changes in the reflection coefficient of SH waves under different interface stiffness are shown in Fig. 12. The incident angle of acoustic waves is 50° and the value of KT (2) is 3 1016 N/ m3 , indicating that aluminum/epoxy resin/aluminum bonding interface is in the perfect state. The values other than 3 1016 N/ m3 mean that the interface is in the weak bonding or slip/debonding state. It can be seen from Fig. 12 that when the values of tangential stiffness coefficients are 3 1016 N/ m3

and 3 1014 N/ m3 , respectively, the reflection coefficient curves of SH waves are almost superposed completely, indicating that if the tangential stiffness coefficient is greater than 3 1014 N/ m3 , the interface may be regarded as a perfect interface. As the tangential stiffness coefficient decreases, the amplitude of reflection coefficient of SH waves shows a rising trend. When the tangential stiffness coefficient decreases to 3 1010 N/ m3 , the amplitude curve of reflection coefficient of SH waves tends to be stable because under the weak bonding condition, when the tangential stiffness coefficient decreases to a smaller value, the reflection coefficient of SH waves is no longer sensitive to the changes in tangential stiffness. According to the results in Subsections 3.3 and 3.4, we note that the interface may be regarded as a slip/debonding interface when the tangential stiffness coefficient is less than 3 1010 N/ m3 . Furthermore, Fig. 12 also indicates that the value KT (1)  7 1012 N/ m3 may be used to characterize the weak bonding interface.

4. Discussion and conclusions A theoretical model is proposed to analyze the propagation characteristics of the lowest SH mode (SH0) in the non-perfect adhesive structure on the basis of wave propagation controlling equations in this paper. The following conclusions are obtained: The reflection and transmission coefficient expressions of the lowest SH mode containing tangential stiffness coefficient KT in the adhesive structure are derived. The relationship between the reflection and transmission coefficients of SH waves and the adhesive layer thickness in aluminum/epoxy resin/aluminum structure is analyzed in the case of the interfaces in the perfect connection. The correctness of the proposed equations and the feasibility of interface state characterization by KT assignment are verified through the comparison with existing data. Different from the body waves (longitudinal and SV transverse waves), SH waves passing through the bonding interface do not lead to the wave conversion. Different interface quality will lead to diverse sound reflection or transmission characteristics. For a specific adhesive structure, the propagation mode of SH waves in the structure mainly depends on the angle of incidence and frequency, etc. Even tiny changes in these parameters will produce large impacts on the reflection or transmission characteristics of acoustic waves. Therefore, if the adhesive structure is tested by SH waves, it is important to choose these parameters appropriately. No critical angle exists when the adhesive structure is tested by SH wave incidence. Under the

condition of the perfect or weak bonding interfaces, multi-order resonance will be generated on the reflection or transmission coefficient curves of SH waves as the frequency-thickness product increases. Within the same range of adhesive layer thickness (or frequency), the resonant frequency curve will drift to the low frequency with the rise in the frequency of acoustic waves (or adhesive layer thickness). If the values of incident angle of acoustic waves and frequency-thickness product are proper, the perfect, weak bonding and slip/debonding interfaces can be differentiated from each other, but it is unable to identify the slip and debonding interfaces by SH waves. Irrespective of the incident angle of acoustic waves or frequency-thickness product, when the interface is in the slip/debonding stage, the amplitude of reflection coefficient of SH waves is always 1 and the transmission coefficient is always 0. Besides, for the weak bonding interface, when the tangential stiffness coefficient is less than a certain value, the reflection coefficient of SH waves is no longer sensitive to the changes in the tangential stiffness coefficient. The study provides a theoretical basis for the ultrasonic evaluation and mechanism interpretation of adhesive interface characteristics. The proposed analysis method is applicable to detect the structure with the substrate and adhesive layer, which are isotropic elastic solid medium materials, by SH waves. The study results should be experimentally confirmed. However, due to the challenges of experimental conditions, we have not yet carried out related experiments. Nevertheless, the effectiveness of the method and the correctness of the formula have been verified by comparison with the relevant achievements.

Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 11132002, 11372016, 51235001, and 51475012).

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Figures

SH r1

SH i1

 

Substrate 1

o h

Adhesive layer 2

y Interface 1

 

SH t 2

SH r 2

Interface 2



Substrate 3

SH t 3

x

Fig. 1. Propagation mode of SH waves in an adhesive structure.

1.4 1.2

The results of this paper The results of this paper The results of paper The results of paper

Reflection coefficient Transmission coefficient

Amplitudes

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

The adhesive layer thickness (mm)

Fig. 2. Relationship between the adhesive layer thickness and the reflection/transmission coefficients of SH waves.

1.2

1.0

1.0

Transmission coefficient

Reflection coefficient

1.2

0.8 0.6 0.4 0.2 Perfect interface Weak bonding interface Slip/debonding interface

0.0 -0.2

0

10

20

30

40

50

60

70

80

0.8 0.6 0.4 0.2 0.0 -0.2

90

Perfect interface Weak bonding interface Slip/debonding interface

0

10

20

Angle of incidence (deg)

30

40

50

60

70

80

90

Angle of incidence (deg)

(a)

(b)

Fig. 3. Relationship between the incident angle and the reflection/transmission coefficient of SH waves, (a) reflection coefficient and (b) transmission coefficient.

1.4

1.4 0.94

Transmission coefficient

Reflection coefficient

1.2

Perfect interface Weak bonding interface Slip/debonding interface

1.0 0.8 0.6 0.4 0.2 0.0 0.55 MHz mm

-0.2

0

1

Perfect interface Weak bonding interface Slip/debonding interface

Resonance points

1.2 1.0 0.8 0.6 0.4 0.2

0.33

0.0

0.68 MHz mm

2

3

fh (MHz mm)

(a)

4

5

-0.2

0

1

2

3

4

5

fh (MHz mm)

(b)

Fig. 4. Relationship between the frequency-thickness product and the reflection/transmission coefficients of SH waves when the angle of incidence is 0°, (a) reflection coefficient and (b) transmission coefficient.

1.4

1.4

Transmission coefficient

Reflection coefficient

Perfect interface Weak bonding interface Slip/debonding interface

0.87

1.2 1.0 0.8 0.6 0.4 0.2 0.0

0.57 MHz mm

-0.2

0

1

Perfect interface Weak bonding interface Slip/debonding interface

First-order resonance Second-order resonance

1.2 1.0 0.8 0.6 0.4 0.2

0.48

0.0

0.7 MHz mm

2

3

4

5

-0.2

0

1

fh (MHz mm)

(a)

2

3

4

5

fh (MHz mm)

(b)

Fig. 5. Relationship between the frequency-thickness product and the reflection/transmission coefficients of SH waves when the angle of incidence is 50°, (a) reflection coefficient and (b) transmission coefficient.

(a)

(b)

Fig. 6. Relationship among the frequency, the adhesive layer thickness, and the reflection coefficient of SH waves under perfect interface, (a) stereogram and (b) top view.

(a)

(b)

Fig. 7. Relationship among the frequency, the adhesive layer thickness, and the reflection coefficient of SH waves under weak bonding interface, (a) stereogram and (b) top view.

(a)

(b)

Fig. 8. Relationship among the frequency, the adhesive layer thickness, and the reflection coefficient of SH waves under slip/debonding interface, (a) stereogram and (b) top view.

(a)

(b)

Fig. 9. Relationship among the frequency, the adhesive layer thickness, and the transmission coefficient of SH waves under perfect interface, (a) stereogram and (b) top view.

(a)

(b)

Fig. 10. Relationship among the frequency, the adhesive layer thickness, and the transmission coefficient of SH waves under weak bonding interface, (a) stereogram and (b) top view.

(a)

(b)

Fig. 11. Relationship among the frequency, the adhesive layer thickness, and the transmission coefficient of SH waves under slip/debonding interface, (a) stereogram and (b) top view.

1.6

31016 31014 31012 31010 3108

Reflection coefficient

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

0

1

2

3

4

5

fh (MHz mm)

Fig. 12. Reflection coefficients of SH waves under different

KT (1)

values.

Highlights We studied the SH0 mode in adhesive structures with different interface quality.