Some studies on evaluation of degradation in composite adhesive joints using ultrasonic techniques

Ultrasonics 53 (2013) 1150–1162

Contents lists available at SciVerse ScienceDirect

Ultrasonics journal homepage: www.elsevier.com/locate/ultras

Some studies on evaluation of degradation in composite adhesive joints using ultrasonic techniques R.L. Vijaya Kumar ⇑, M.R. Bhat, C.R.L. Murthy Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India

a r t i c l e

i n f o

Article history: Received 16 July 2012 Received in revised form 26 January 2013 Accepted 27 January 2013 Available online 7 March 2013 Keywords: Adhesive joints Ultrasonics Adhesive degradation Interfacial stiffness

a b s t r a c t Experimental and theoretical studies on degradation of composite-epoxy adhesive joints were carried out on samples having different interfacial and cohesive properties. Oblique incidence ultrasonic inspection of bonded joints revealed that degradation in the adhesive can be measured by significant variation in reflection amplitude as also by a shift in the minima of reflection spectrum. It was observed that severe degradation of the adhesive leads to failure dominated by interfacial mode. Through this investigation it is demonstrated that a correlation exists between the bond strength and a frequency shift in reflection minimum. The experimental data was validated using analytical models. Though both bulk adhesive degradation and interfacial degradation influences the shift in spectrum minimum, the contribution of the latter was found to be significant. An inversion algorithm was used to determine the interfacial transverse stiffness using the experimental oblique reflection spectrum. The spectrum shift was found to depend on the value of interfacial transverse stiffness using which a qualitative assessment can be made on the integrity of the joint. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Adhesively bonded joints are being increasingly used in the assembly of composite structures. While there are obvious advantages of using adhesive joints instead of bolted or riveted joints, they also pose challenges when it comes to evaluating them for their mechanical properties owing to associated issues and variables such as, degradation in service, variation in bond line thickness, and surface roughness of the adherends. Many aerospace and automobile structures are assembled using adhesive joints. The strength and life expectancy of such a structure depends critically on the bond integrity between components. Despite great attention given to maintain quality in manufacturing processes, imperfections such as cracks, porosity, and inclusions, do creep in the bond line and can significantly degrade the performance of the joint. Often the imperfections are confined to a very thin interface separating the adhesive and the substrate. Many researchers have done extensive work to identify different types of defects in adhesive joints and have suggested suitable non destructive test methods to evaluate them [1–5]. Significant effort has also been made to develop non destructive testing methods to test adhesive bond quality and to evaluate its integrity [6–14].

⇑ Corresponding author. Tel.: +91 8023640797. E-mail address: [email protected] (R.L. Vijaya Kumar). 0041-624X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2013.01.014

Failure of bonded joints can be either adhesive or cohesive in nature. Environmental conditions and stresses degrade the interface as well as bulk of the adhesive in the bond line. Ultrasonic methods have been widely used in the non destructive testing and inspection of adhesive joints. The cohesive properties of an adhesive layer in the joint were studied using ultrasonic spectroscopy by Cawley et al. [15,16]. Efforts to evaluate the interface characteristics of adhesive joints using normal and oblique incidence ultrasonic methods are also reported [17–25]. The properties of the adhesive joints viz., the interfacial stiffness, density, attenuation of the ultrasonic longitudinal and shear waves in the adhesive layer, etc. have been computed using different algorithms [26–28]. Bulk of previously reported work on NDT of adhesively bonded joints using ultrasonic techniques were carried out on metal to metal joints in which the impedance mismatch between the adhesive and the substrate is quite high. Few attempts have also been made to inspect composite to composite adhesive joints [29–34]. The work presented in this paper relates to the non destructive evaluation of carbon fiber reinforced polymer (CFRP) epoxy adhesive joints. Single lap shear joints were prepared using CFRP substrates and a two part epoxy adhesive. Different degrees of degradation in the adhesive layer were achieved by using different amounts poly vinyl alcohol (PVA) mixed during execution of the joint. The joints were subjected to oblique incidence ultrasonic inspection and then subsequently subjected to mechanical loading in a testing machine till failure to determine its strength. Efforts were made to correlate the parameters of ultrasonic examination to bond strength and to measure the interfacial properties of the joints.

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

2. Specimen preparation Single lap adhesive joints (Fig. 1) were prepared as per ASTM D 5868 standard using carbon fiber reinforced plastic (CFRP) as substrates and a two part epoxy adhesive; Araldite AV138M/Hardener HV 998. Substrates were obtained from a unidirectional CFRP laminate fabricated using CP150 ns carbon prepreg, cured in an autoclave according to manufacturer’s recommendation. The resulting CFRP plate was subjected to ultrasonic C-scan evaluation to ensure that it is free from gross defects and anomalies. It was then cut to size (102 mm  25 mm); the thickness of each substrate remained constant at 2.55 ± 0.05 mm. Surface preparation was carried out according to ASTM D 2093 standard for surface preparation of plastics. An area of 25.4 mm  25.4 mm was bonded by taking 100 parts of adhesive and 40 parts of hardener by weight as recommended by the manufacturer. Joint was executed using a specially designed mold (Fig. 1c) which helps in keeping the alignment of the substrates during the process of curing and also in maintaining a uniform bond line thickness of 0.7 ± 0.06.

1151

A total of 30 samples were prepared and were divided into five different sets having six samples each, depending on the percentage of poly vinyl alcohol (PVA) added to the epoxy resin to achieve varied quality of adhesive. The samples which did not have any PVA were treated as healthy samples (H), while others were denoted as P10, P20, P30, and P40 depending on the amount of PVA added. Details of the samples prepared are presented in Table 1. PVA is water soluble compound which does not mix with epoxy; it is used as a mold releasing agent in composite structure fabrication. PVA was dissolved in water and mixed with the adhesive. When liquid PVA is exposed to open atmosphere the water gradually evaporates. Similar phenomenon occurs when an adhesive mixed with PVA is cured; the water in liquid PVA spreads in bulk adhesive and reaches the interface and creates small pores both at the interface as well as in the bulk adhesive (refer Fig. 11). The more the percentage of PVA we add more will be the amount of pores and hence more severe will be the degradation. This is analogous to the environmental degradation encountered by adhesive joints in real situations due to moisture ingress in the bondline area.

Fig. 1. (a) Single lap joint dimensions, (b) CFRP single lap shear joints and (c) mold designed to accomplish the joint.

1152

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

Table 1 Details of adhesive joint samples prepared. Adhesive joint set

Sample designation

Percentage of PVA by weight of epoxy (%)

1 2 3 4 5

H P10 P20 P30 P40

0 10 20 30 40

3. Experiments Fig. 2. Oblique incidence ultrasonic inspection.

Ultrasonic methods have been widely used in the evaluation and characterization of adhesive joints [1]; these methods have generally been regarded as potentially the most useful techniques for NDE of multilayered composite structures and adhesive joints. The joints were subjected to different ultrasonic testing techniques viz., pulse echo and through transmission using both immersion as well as contact transducers. The details of the Normal incidence ultrasonic investigations carried out and the results are reported in reference [35]. To start with the properties of the adhesive and the CFRP substrate were determined experimentally; bulk adhesive samples with different percentages of PVA were prepared and were subjected to ultrasonic inspection after 24 h of cure. Properties like density, longitudinal and shear wave velocities, Elastic and shear modulus, and longitudinal and shear wave attenuations were determined. A similar procedure was followed to determine the above said parameters for the unidirectional CFRP substrates. Table 2 gives a summary of material property values determined.

3.1. Oblique incidence ultrasonic inspection Inspection of an adhesive joint using normal incidence ultrasound involves the bulk longitudinal waves which are not very sensitive to the degradation at the adhesive–adherend interface [24]. Often the degradation is due to the diffusion of water (moisture) into the interface which renders the normal incidence ultrasonic inspection ineffective. However, it is found that oblique incidence methods, which generate shear waves, are sensitive to interface properties [36–38]. Thus, the samples prepared were subjected to oblique incidence ultrasonic inspection using a Perspex wedge and a pair of contact transducers (Fig. 2). The angle of incidence was computed using Snell’s law (between Perspex and CFRP) and was set at 70°; more than the first critical angle (65°) so that only the shear wave component existed in the bond line. Fig. 3 shows the experimental waveform obtained from oblique incidence ultrasonic inspection of adhesive joint samples using 5 MHz contact type transducers. It can be seen that the amplitude of reflection from the interface of a healthy joint is much less compared to that from a degraded joint. The waveform for a healthy adhesive joint also shows the reflection from the second interface which is not seen in case of degraded samples.

Multiple reflection echoes with increased amplitude can also be seen in the wave form for a healthy joint, this can be attributed to constructive interference taking place in the bond line [36]. The variation of amplitude of reflection of shear waves with different percentages of PVA is shown in Fig. 4, it can be seen that the amplitude of reflection increases with increase in PVA percentage. The experimental wave forms were digitized using Yokogawa DLM 2022, 200 MHz digital oscilloscope and then processed in the frequency domain using an FFT program and deconvolved from a reference signal taken from the CFRP adherend. It was found that the degradation of adhesive causes changes in the spectra of reflected ultrasonic signal from the bondline [39]. Reflection from an adherend can be separated in the time domain and the signal reflected from the adhesive layer can easily be gated out and analyzed in the frequency domain. Fig. 5 shows the reflection spectra for two samples, the solid line is for a healthy sample without any degradation, while the dashed line is for a degraded sample with 40% PVA. The minima observed in the spectra are due to interference of the ultrasonic signals within the adhesive layer. An average shift of 0.7 MHz towards lower frequency was observed in the minima for 40% PVA samples near the test frequency of 5 MHz, this can be attributed to distributed damage at the interface, which causes pulse widening and scattering [39]. The negative frequency shift observed experimentally is very sensitive and can change significantly due to variation in parameters like bondline thickness, surface roughness, and change in service conditions due to moisture and temperature. The samples were subsequently loaded till failure to determine their strength and the fractured surfaces were inspected. While the healthy samples had an interfacial failure area of less than 50%, the degraded samples had higher percentage of interfacial failure area. Fig. 6 shows the failure surface of three different samples a healthy sample, a sample with 20% PVA and 40% PVA. The images were taken with 10 magnification using a high resolution camera. It can be seen that fractured surface of a healthy sample reveal traces of fibers on the adhesive indicating a strong bond. The percentage of interfacial failure area increased with increasing percentages of PVA. A complete interfacial failure was observed in a couple of highly degraded samples i.e., P40 samples. A closer inspection of fractured surface revealed distributed pores

Table 2 Material property values determined using ultrasonic inspection. Sample

Density ‘q’ (kg/mm3)

Longitudinal velocity ‘Vl’ (m/s)

Shear wave velocity ‘Vt’ (m/s)

Modulus of elasticity ‘E’ (GPa)

Shear modulus ‘G’ (GPa)

Longitudinal wave attenuation ‘al’ (Nepers/m)

Shear wave attenuation ‘at’ (Nepers/m)

CFRP P0 P10 P20 P30 P40

1700 1548 1429 1322 1213 1107

3100 2380 2342 2223 2065 1914

1800 1400 1370 1300 1210 1150

12 (E3) 4.7 4.2 3.5 2.8 2.2

5.5 3.0 2.7 2.3 1.8 1.5

15 20 26 34 39 48

45 53 63 75 89 104

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

1153

Fig. 3. Ultrasonic waveforms from oblique incidence ultrasonic inspection.

and voids (refer Fig. 11), these defects contribute significantly to the variation of spectra. Table 3 summarizes the results obtained from experimental investigations and mechanical testing of adhesive joint samples. It can be seen that the average shear strength of the samples decreases with an increase of interfacial failure area. The variation of negative frequency shift in reflection minimum with interfacial failure area and the average shear strength is shown in Figs. 7 and 8 respectively. It can be observed that the shift of reflection minimum increases with the percentage of interfacial failure area. This shift can be due to degradation of the interface or due to degradation of the bulk adhesive or both. It is necessary to stress here that the spectrum characteristics of the signal reflected from the adhesive joint bondline depends on several joint parameters, including adhesive layer thickness, bulk adhesive properties, and interfacial bond quality. Thickness of the adhesive layer was maintained constant and its effect on the frequency minimum shift is considered negligible. The mechanism of joint degradation, including that of the interfacial region, is not completely clear and different mechanisms of interface degradation are possible. Theoretical modeling [24,39] has shown that the experimentally observed negative frequency shift of spectrum minimum can be explained by the degradation

of the interface. Nguyen et al. [40] studied the diffusion of water through an epoxy layer to the interface with a transparent crystal substrate using infrared spectroscopy and found that water forms a thin interfacial layer of thickness less than 40 nm. Angel and Achenbach [41] developed an elasto dynamic theory to explain the interaction of normally incident ultrasonic waves from a planar array of cracks. Thompson and Fiedler [42] replaced an array of cracks by an equivalent layer of springs. Baik and Thompson [43] developed a quasi static model for ultrasonic transmission and reflection at imperfect interfaces. Achenbach and Li [44] studied stiffness deterioration in single-lap adhesive joint due to distributed damage, disbonds were modeled by normal and equivalent shear springs with masses. Interface is also modeled as a weak boundary layer by Rokhlin et al., [14,24] where defects like microporosity and microcracks are formed at the interface due to diffusion of water molecules to the primer layer through bulk of the adhesive, this will lead to reduction in interfacial stiffness and strength. As the amount of degradation increases the fraction of the interfacial region damaged increases by either enlargement of the defects or by increase of their number leading to its failure. Though this model explains the frequency shift of the reflection minimum, the frequency shift is described as a function of primer layer thickness and not the interfacial degradation alone. In the current work two different models are considered to explain the phenomenon of interfacial degradation and the associated frequency shift in the reflection minimum. The first one is known as transfer matrix model where degradation in bulk adhesive properties and its effect on the reflection and transmission coefficients are considered. The second model describes the interfacial degradation as an array of liquid filled disbonds having very small thickness. Spring boundary conditions are used and interfacial degradation is taken into account by decreasing the interfacial spring stiffness value. 4. The transfer matrix model

Fig. 4. Variation of reflection amplitude of shear wave from the interface of a joint.

Analytical investigations were carried out using transfer matrix model to verify experimental results. A transfer matrix relates the displacements and stresses at the bottom of a layer to that at the top of the layer. These transfer matrices for a number of layers can be coupled to yield a single matrix for a complete system, the reflection and transmission characteristics of a multilayered

1154

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

Fig. 5. Experimental spectra of reflected wave from the adhesive layer for a healthy sample and a P40 sample.

Fig. 6. Failure surface (10) of (a) healthy sample, (b) P20 sample and (c) P40 sample.

structure can then be computed by applying appropriate boundary conditions [45]. Adhesive joint was modeled as a structure having three layers (Fig. 9), the two CFRP adherends were treated as semi infinite half spaces and the adhesive layer was modeled as a viscoelastic mate0 Prime rial with complex bulk modulus k ¼ k  ik and shear modulus l ¼ l0  ilPrime : The adhesive was assumed to be a Maxwell material with a single relaxation time ‘s’. The Maxwell model is represented by a series connection of viscosity and stiffness elements [46]. In this case the bulk and Shear moduli are expressed as shown below

  x2 s 2 xs k ¼ ko þ ðk1  ko Þ  i 1 þ x2 s 2 1 þ x2 s2

ð1Þ

Table 3 Summary of experimental results. Sample

Average shear strength (MPa)

Percentage interfacial failure area (Average) (%)

Average shift in reflection minima (MHz)

H P10 P20 P30 P40

9.97 7.46 6.66 6.47 5.27

42 48 59 73 82

0 0.1 0.3 0.6 0.7

l ¼ l1



x2 s 2 xs i 1 þ x2 s 2 1 þ x2 s2

 ð2Þ

where k1 ; l1 are the bulk and shear modulus of a fully cured adhesive respectively, ko is the bulk modulus of an adhesive in a degraded state, x is the circular frequency, s ¼ g=l1 is the relaxation time and g is the static viscosity of an adhesive. s is the time for stress relaxation towards equilibrium to 1/e of its value after a sudden unit deformation. The parameters required in Eqs. (1) and (2) were obtained from manufacturers data for AV138M epoxy adhesive, where k1 = 5.5 GPa, l1 = 3 GPa, ko = 2.8 GPa. Changing the value of xs transforms the material properties of an adhesive, which is convenient for numerical simulation of the ultrasonic wave interaction with an interface layer having variable properties. A higher value of xs P 10; implies a fully cured adhesive while the degradation in its properties are represented by a decrease in its value. The ultrasonic reflection coefficients can be computed using the transfer matrix model. The incoming shear and longitudinal waves are designated as ‘S+’ and ‘L+’ respectively while the outgoing waves are designated as ‘S’ and ‘L’. It is convenient to refer the layers and interfaces in terms of their vertical positions in a stack and with respect to the top and bottom surfaces, as shown in Fig. 9. Accordingly the X2 direction is defined downwards from the top to the bottom of the plate. Each layer has its own X2 origin, defined at its top interface, except for the first layer which has its origin at its interface with the second layer in order to avoid having

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

1155

Fig. 7. Frequency shift of reflection minimum with average percentage interfacial failure area.

Fig. 8. Frequency shift of reflection minimum with average shear strength of adhesive joint.

an origin at 1. When the two semi infinite half spaces are solids, the transfer matrix formulation can be expressed as shown by the following equation:

wave speed ‘Cs’), the frequency used (x), and the plate wave number (K1). [S] is known as the system matrix consisting of the product of the layer matrices.

8 8 9 9 AðLþÞ > AðLþÞ > > > > > > > > > > > > > A A ðSþÞ > ðSþÞ > > > > > > > > : : ; ; AðSÞ l3 AðSÞ l1

½S ¼ ½Ll2 ½Ll3    ½Llðn1Þ ð3Þ

where AðLþÞ; AðLÞ, etc., are the wave amplitudes. [D] is known as a field matrix; it describes the relationship between wave amplitudes and displacements, stresses at any location in a layer. Its coefficients depend on the position of the layer in the plate, the material properties (density ‘q’, Longitudinal wave speed ‘CL’, and transverse

Fig. 9. Adhesive joint modeled as a three layered plate.

ð4Þ

1156

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

where ‘n’ is the number of layers. The [L] matrix for a layer can be obtained by multiplying the field matrices at the bottom and top of that layer. i.e.,

½Ll2 ¼ ½Dl2;bottom ½D1 l2;top

ð5Þ

The reflection coefficients can be computed if any four of the eight amplitude components in Eq. (3) are known; it is a practice to assume the input wave amplitude AðSþÞ = 1, while amplitudes AðLþÞ for layer one and, AðLÞ , AðSÞ for layer three are treated as zero. So that Eq. (3) becomes

8 9 8 9 Tl > 0> > > > > > > > > > <0= = l 1 ¼ ½Dl3;top ½S½Dl1;ðx2¼0Þ > Ts > > > 1> > > > > > : > : > ; ; Rs l1 0 l3

ð6Þ

where T i ’s and Ri ’s are the transmission and reflection coefficients respectively. A detailed illustration of transfer matrix method and the elements associated with different matrices, etc. has been given by Lowe [45]. Some modifications have to be done to account for the attenuation present in layers, in the current work the attenuation in CFRP adherend semi space is neglected and it is taken into account for the adhesive layer by treating it as a viscoelastic material. This makes the wave number complex, the real part of which describes the propagation of the wave and imaginary part describes attenuation. The reflection coefficients for a healthy adhesive joint was obtained using Eq. (6) by assuming xs ¼ 10 which is the relaxation parameter for a completely cured solid adhesive [46], while the degradation for a P40 sample was represented using a lower value of relaxation parameter xs ¼ 5. A very low value of relaxation parameter (xs ¼ 1) implies the adhesive is a liquid. The adhesive used in the present work is a thixotropic paste, hence a value of greater than ‘1’ was assigned to the relaxation parameter and the corresponding shift in the reflection coefficient was observed in each case, it was found on a trial and error basis that the value of ‘5’ gave closer correlation to the experimental results. The angle of refraction for the shear wave transmitted from CFRP adherend to the adhesive layer was computed using Snell’s law and found to be 50°. The thickness of the adhesive layer was assumed constant at 0.7 mm and the amplitude of reflection coefficient obtained for different frequencies were plotted as shown in Fig. 10. It can be seen

that there is shift of reflection minimum towards lower frequencies due to degradation in the adhesive, a similar trend was observed in experimental investigations. 5. Interface as an array of liquid filled disbonds The mechanism of joint degradation is not completely clear and different mechanisms are possible [40–44]. This model describes the interface as an array of very thin interfacial disbonds filled with water. The degradation becomes severe either by widening of existing disbonds or by appearance of new disbond regions or both. The fractured surface of a P40 sample can be seen in Fig. 11 which shows small pores and voids distributed throughout. A solid–solid interface can have either a rigid or slip boundary condition; these boundary conditions are mathematically described as

ry ¼ r0 y; uy ¼ u0 y; syz ¼ s0 yz; uz ¼ u0 z

ð7Þ

ry ¼ r0 y; uy ¼ u0 y; syz ¼ s0 yz ¼ 0

ð8Þ

where ‘ry’ is the stress normal to the interface, ‘syz’ is the shear stress, ‘uz’ and ‘uy’ are the displacements. Primed and non primed variables correspond to the two sides of the interface. It can be seen from Eqs. (7) and (8) that the normal stresses and displacements at the interface are continuous. The effect of a thin water film at the interface on the reflected ultrasonic wave can be modeled using slip boundary condition (8) which results in zero resistivity to the motion of adjacent phases transverse to each other while ensuring the continuity of stresses and displacements [11,14]. The degraded interface can be represented by an array of circular disbonds with thickness approaching zero (Fig. 12). Each disbond can be characterized by slip boundary conditions (Kn ! 1; Kt ! 0), while the non-damaged area corresponds to welded boundary condition. The properties of the interface as a whole can be modeled by spring boundary conditions (9) with complex transverse spring constant Kt, satisfying 0 < jKtj < 1

s0 yz ¼ Ktðuz  u0 zÞ; syz ¼ s0 yz; uy ¼ u0 y; ry ¼ r0 y

ð9Þ

This model describes an interface using a homogenized distribution of springs. When the geometry of a disbond is specified, the spring stiffness constants can be obtained by calculating the

Fig. 10. Amplitude of Reflection coefficient for shear waves obtained using transfer matrix model.

1157

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

failure area ‘Ad ! 0’ indicating a healthy joint, on the other hand ‘Kt ! 0’ for a complete interfacial failure. Fig. 14 shows the adhesive joint model with a distinct interface of thickness ‘h’ separating the adhesive and the adherend. Different boundary conditions can be applied to this interface. Slip boundaries between layers may be modeled by uncoupling the shear stresses between the layer matrices [47]. One convenient approach is to include a boundary condition matrix in the assembly of the layers. Boundary condition matrices may also be used to define specific values of stiffness at an interface [46]. The particle displacements (ui) and stresses (rik) on the upper and lower interfaces can be related to each other for arbitrary layer thickness by a transfer matrix.

Fig. 11. Magnified (10) image of Fractured surface of an adhesive joint with 40% PVA.

deformation of the bonded structure under static loading. For a simple disbond pattern as shown in Fig. 12 the stiffness constant ‘Kt’ is given by [24].

Kt 

  2t p E00 1 ð1:299723Ad1=2  0:9952365Ad 2 8 ð1  t2 Þ a þ 0:6672023Ad3=2  0:42308925A2d þ 0:1406982A5=2 d  0:0295401A3d þ

0:149058Ad1=2 1 þ Ad1=2

ð10Þ

where ‘Ad’ is the fraction of disbond area, ‘a’ is the distance between centers of disbonds and E00 is the effective young’s modulus obtained using the following equation:

EPrime ¼

2E1 E2 E2 ð1  t21 Þ þ E1 ð1  t22 Þ

The elements of transfer matrix [B] depend on the layer properties and are illustrated in detail by Rokhlin and Wang [46]. Eq. (12) may be regarded as the boundary condition which relates the stress and particle displacements across a layered interface. If the wavelength ‘k’ in the interface layer is much greater than its thickness (h), the matrix B may be asymptotically expanded and can be further reduced making some assumptions [46]. It finally takes the form.

1 0 0 6 0 1 ix=k n 6 ½B ¼ 6 40 0 1

1=2

ð11Þ

where El, E2 and t1, t2 are Young’s moduli and Poisson’s ratio for the two materials in contact, (i.e. substrate and adhesive). Poisson’s ratio t is taken as an average of t1 and t2. Table 4 shows the material property values considered in the present work. The mechanism of disbond area growth is still not clear, interfacial damage may increase either by the enlargement of individual disbonds or by the appearance of new disbonds. However it is suggested that the growth is due to intermediate mechanism i.e., the sizes of disbonded regions increase along with the appearance of small new disbonds [25]. Fig. 13 shows the variation of ‘Kt’ with the percentage interfacial failure area ‘Ad’ plotted by taking the value of ‘a’ as unity. The interfacial stiffness ‘Kt ! 1’ as interfacial

ð12Þ

syz

syz

2

 1:86868 lnð1 þ Ad Þ

 0:419904 lnð1  Ad1=2 ÞÞ1

8 8 0 9 9 uz > uz > > > > > > > > > > = y y ¼ ½B 0 > ryy > > > ryy > > > > > > > : : 0 > ; ;

0

0

0

ix=kt 0 0

3 7 7 7 5

ð13Þ

1

The associated boundary condition is given by Eq. (14). The limiting case of the welded boundary condition between two solids, which consists of continuity of the stresses and displacements on the interfaces, follows from Eq. (14) when kn ! 1 and kt ! 1. The B matrix in this case is a unit matrix.

r0yy ¼ K n ðuy  u0y Þ; s0yz ¼ K t ðuz  u0z Þ; ryy ¼ r0yy ; syz ¼ s0yz

ð14Þ

Similarly one can obtain slip boundary condition when kn ! 1 and kt ! 0, the shear stress syz in this case will be equal to zero and the transverse component (uz) of the displacement is discontinuous at the interface. In some cases reflection coefficient from an imperfect interface is computed by considering only the transverse spring stiffness Kt and assuming normal stiffness as infinity. Numerical calculations show that this approach may give significant error even for thin layers; however this approximation can be used when the interfacial layer has properties close to that of the viscous liquid. When both the normal and transverse stiffness are considered satisfactory results for reflection coefficients can be obtained [46]. Reflection coefficients from the interface between a CFRP adherend and the adhesive were computed using the model described above. Bulk adhesive properties were kept constant and only the transverse interfacial spring stiffness (Kt) was varied depending on the interfacial failure area according to Eq. (10). The normal

Table 4 Material property values used in the calculations.

Fig. 12. Interface as an array of circular disbonds.

Material

Modulus of elasticity (GPa)

Poisson’s ratio (t)

Adherend (CFRP UD + Epoxy) Adhesive (AV 138 M + HV 998)

12.0 (E3) 4.70

0.31 0.38

1158

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

Fig. 13. Reduction of transverse interfacial stiffness (Kt) with interfacial failure area.

Fig. 14. Adhesive joint model with interfaces.

spring stiffness ‘Kn’ was assumed to be twice that of transverse stiffness ‘Kt’. Fig. 15 shows the variation of the reflection coefficient with frequency, a negative shift in the reflection minima can be observed for a degraded adhesive joint sample. However the shift is less compared to experimentally observed shift, this discrepancy can be attributed to change in the bulk adhesive properties which were not considered here. The experimentally observed shift of 0.7 MHz was found to match when incremental failure area Ad was at around 55%. In the models described above the bond line thickness was kept constant, it was assumed that the surface was free of contaminants and the variation in temperature and moisture was not considered. Hence the frequency spectrum obtained from both the models does not match exactly with the experimental spectrum, however a similar trend could be observed. Both the models showed a negative shift in the frequency minimum but the individual contribution of the bulk adhesive properties and the interfacial degradation towards this shift is yet to be understood. It is reported by Rokhlin et al. [39] that the influence of interfacial degradation towards the spectrum shift is more significant when compared to that from bulk adhesive degradation. They have also shown that minor degradation in bulk adhesive properties tend to shift the spectrum minimum slightly towards higher frequency. This trend gets reversed when the adhesive is severely degraded. A similar trend was observed for a couple of samples with 10% and 20% PVA, where the degradation was relatively less. This phenomenon can be explained using Eq. (10), it can be seen that for the same fraction of interfacial damage area ‘Ad’ the transverse interfacial stiffness Kt is inversely proportional to the average distance between centers of circular disbonds ‘a’. It is

found that the stiffness Kt increases as the value of ‘a’ decreases; physically this implies that the stiffness reduction is more when larger disbonds are created rather than a number of small micro disbonds. For P10 and P20 samples the degradation is relatively less; as such small pores and micro disbonds are created at the interface (Fig. 6b). As the amount of PVA is increased (P30 and P40 samples) these pores and micro disbonds increase in their size (Fig. 6c) and decrease the value of Kt. Hence the spectrum shift can be related to the value of interfacial stiffness Kt. The more the decrease in the value of Kt the more is the spectrum shift towards lower frequency. It can thus be concluded that though both bulk adhesive degradation and interfacial degradation influences the spectrum shift; the latter contributed more towards it. It was difficult to separate their individual influences in this set of experiments but efforts in this regard are underway by creating different amounts of interfacial degradation using Teflon inserts. It is expected that the contribution from interfacial degradation towards the spectrum shift can be separated as the bulk adhesive properties are kept constant.

6. Ultrasonic determination interfacial properties of an adhesive joint The frequency shift in the reflection minimum for oblique incidence ultrasonic inspection was explained using two different models in the previous section; however the main objective of the current work was to characterize interfacial imperfections using ultrasonic signatures obtained from the experiment. The reflection from perfect interface is independent of frequency while that with spring boundary conditions (Eq. (9)) depends on it. The

1159

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

Fig. 15. Shift in the reflection minima due to variation of interfacial transverse stiffness (Kt).

interfacial spring stiffness can be determined using this frequency response [27]. Models have been developed and inversion procedures have been followed to determine the interfacial spring constants and bulk adhesive properties using ultrasonic signatures [25–28]. Adhesive bonds can deteriorate in severe working environments and ageing along the interface primarily due to accumulation of moisture in these regions. The distinctive feature of this region is that the ultrasonic signal reflected from the front and back of the interface layer is not separated in time domain and interfere. Ultrasonic spectroscopy may thus be useful in characterizing the interface imperfection. The frequencies at which the reflected minima occur are sensitive to interfacial condition and therefore can be used to monitor them [23]. Interfacial normal and transverse stiffness constants can be obtained experimentally using normal and oblique incidence methods and these values describe the imperfect interface uniquely. The ratio of normal and transverse interfacial stiffness has been predicted successfully by Baltazar et al. [28]. In the previous section the interface was modeled as an array of water filled disbonds; in doing so the properties of the adhesive and the substrates were measured independently and the interfacial spring stiffness value Kt was estimated using Eq. (10) and varied according to the interfacial damage area Ad. The shift in the frequency minima was estimated using the transfer matrix model and was found to be in accordance with the experimental observation. The present section involves the determination of interfacial stiffness Kt, using experimentally measured values of reflection minima using an inverse algorithm. The frequency response from an adhesive joint depends on adhesive bulk and interfacial properties and is often difficult to decouple the effects of these two [28]. The amplitude of reflection coefficient also depends on the impedance mismatch between the adhesive and the adherend; hence it is not purely an interfacial phenomenon since it is influenced by other factors like the attenuation in the adhesive layer, and the wave scattering at the interface. Hence these factors have to be considered while determining the interfacial and bulk layer properties using the inversion procedure.

Reflection coefficients due to impedance mismatch were determined and their effect on the total reflection coefficient was eliminated. The energy loss at the interface due to scattering is accounted for by imaginary parts in the interfacial stiffness constants; literary evidences have shown that the imaginary parts in the complex springs produce the same effect on the reflection spectra as attenuation (al and at) in the thin embedded layer and therefore cannot be separated easily [27,28]. This can be accomplished by performing measurements in a broad frequency range which results in decoupling the effect of the interfacial stiffness from the layer bulk properties. Due to lack of such an inspection facility it was assumed that attenuations in the adhesive layer are known a priori and used for reconstruction of interfacial spring constants. 6.1. Inversion algorithm In determining the properties using the inversion algorithm a procedure similar to that given by Rokhlin et al. [27] was followed. The acoustic response from a layer depends on elastic modulus ‘E’, shear modulus ‘G’, thickness ‘h’, density ‘q’, shear and longitudinal wave attenuations at and al. The properties of the interface are described by normal (KN) and transverse (KT) complex interfacial spring constants. Baltazar and Wang [28] have suggested non dimensional parameters to describe all unknown variables which can be reconstructed from measured normal and angular reflection spectra. From the measured normal reflection spectrum four parameters can be determined.

Zn ¼

Zl ; Zs

Hl ¼

hxo ; Vl

KN ¼

KN

xo zl

;

al

ð15Þ

where Zl and Zs denotes the acoustic impedance in a layer and the substrate, ‘h’ is the thickness of the layer, xo is the angular frequency taken as 1 MHz for convenience as a normalization factor, V l is the longitudinal velocity in the layer, K N is the non dimensional complex normal interfacial spring constant, and al is the longitudinal attenuation. An additional set of four parameters can be determined using oblique incidence inspection.

1160

Cl ¼

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

Vl ; V ls

Ct ¼

Vt ; V ts

KT ¼

KT

xo zt

;

at

ð16Þ

where V ls and V ts are the longitudinal and shear wave velocities in the substrates respectively, K T is the complex transverse interfacial spring constant, zt is the impedance of the transverse wave in the layer and at is the transverse wave attenuation in the adhesive. Once these non dimensional parameters are determined, dimensional parameters can be obtained using the following equations [27]:

q¼

Z n qs ; Cl

h¼

Hl C l V ls

xo

;

k þ 2l ¼ qC 2l V 2ls ;

l ¼ qC 2t V 2ts

ð17Þ

The parameters defined in Eq. (15) were first obtained by the normal reflection spectrum using a least-squares optimization procedure; they were then used to find other parameters from the oblique reflection spectrum. The non dimensional parameters were treated as independent variables in a multidimensional space and an error function for the same was estimated.

eðXÞ ¼ min

f2 X

ðjRt ðXÞj  jRe ðX o ÞjÞ2

reconstruction. When the thin adhesive layer and the substrates have the same properties the position and amplitude of the reflection minima will depend only on the interface stiffness which is zero for complete disbond and infinity for a perfect bond. Lavrentyev and Rokhlin [23] have modeled the reflection signature from normal incidence of longitudinal wave and have proposed that the frequency minima shift can be used to estimate the normal interfacial stiffness. The experimental spectra obtained from normal incidence ultrasonic inspection did not show significant shift from a healthy to degraded joint, however the shift was observed in the oblique reflection spectra (Fig. 16). Therefore, the sensitivity of the obliquely incident waves to the degradation at the interface is considered much stronger. A similar observation can be found in the model proposed by Lavrentyev and Rokhlin [24] to characterize adhesive bond degradation due to moisture accumulation at the interface, where it was shown that formation of very thin water-filled interfacial microdisbonds produced very little effect on longitudinal wave reflection at normal incidence. This was supported by the infrared measurements performed by Nguyen et al. [40].

ð18Þ

f1

6.3. Reconstruction of interfacial stiffness

where Re ðX o Þ and Rt ðXÞ are spectra for experimental and theoretical signals, ‘X’ is an iterated set of non dimensional parameters and ‘Xo’ is the actual set of material properties. f1 and f2 are the bounds of the frequency range in which minimization is performed. The frequency bounds are selected based on the transducer bandwidth and the resonance frequency of the adhesive layer. The least-square algorithm may converge to one of the minima of the error function. The initial guesses and accuracy of the measurement will affect the convergence of the algorithm. 6.2. Experiment Adhesive joint specimens with different percentages of PVA were subjected to normal and oblique incidence ultrasonic inspection using a 20 MHz contact transducer. The experimental ultrasonic signature was deconvolved from the reference signal obtained from the bottom of the CFRP adherend-air interface, this removes the beam effects of the transducers from the experimental spectra and a plane wave approximation may be used in

Interfacial stiffness properties of the adhesive joint were computed from the experimental normal and oblique reflection spectra. Since the normal reflectivity spectra was not sensitive to interfacial degradation the normal stiffness KN was assumed to be infinite and the transverse spring stiffness constant was determined. The imaginary part of the complex interfacial spring stiffness corresponds to energy loss due to ultrasonic scattering at the interface; however its effect is not so significant to be considered, hence only the real part of spring constant Kt was considered. Fig. 17 shows the variation of interfacial stiffness for adhesive joint samples with different amounts of PVA. The interfacial transverse stiffness Kt shows a significant decrease with increase in degradation, depending on the value of ‘Kt’ a qualitative assessment can be made as to whether a bond is good or bad. However literary evidence [25,29] show that there is threshold limit for the value of Kt (Kt = 3  1015 N/m3) above which the ultrasonic reflection signature does not depend on the spring stiffness value, hence it becomes difficult to reconstruct

Fig. 16. Spectra of the reflected signal form oblique incidence ultrasonic inspection (20 MHz).

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

1161

Fig. 17. Variation of interfacial transverse stiffness Kt with different percentage of PVA.

the interfacial properties in the frequency range below 15 MHz. At a value below this threshold; frequency minimum shift can be detected. 7. Conclusions Experimental and analytical investigations were carried out on Adhesive joints made of unidirectional CFRP substrates and epoxy adhesive. Oblique incidence ultrasonic inspection was performed using a pitch catch mode experimental setup where the angle of incidence was chosen to be beyond the first critical angle so that only the shear wave component is incident on the interface. The amplitude of reflection of shear waves from the adhesive layer increased with an increase in degradation, a shift in the frequency minimum towards lower frequencies was observed in degraded samples. This shift can be attributed to the changes in the pulse due to an increase in the defects at the interface and in the bulk adhesive. The bond strength was found to be related to the percentage of interfacial failure which in turn was proportional to the shift in reflection minima observed. This phenomenon was cross verified using a transfer matrix model; where the adhesive layer was treated as a viscoelastic material and the degradation was taken into account by decreasing the value of relaxation parameter. The model showed a similar trend to that observed in the experiments. Another model where the interface was treated as an array of liquid filled disbonds was used to examine the influence of interfacial degradation on the frequency minimum shift. Both models showed a similar trend, the mismatch between the experimentally observed spectra and the theoretical spectra is attributed to sensitivity of the experimentally observed reflection minima to various factors like variation in bondline thickness, surface roughness, and environmental conditions like moisture and temperature. The spectrum shift can be attributed to the interfacial stiffness Kt. The more the decrease in the value of Kt the more is the spectrum shift towards lower frequency. Both bulk adhesive degradation and interfacial degradation influences the spectrum minimum shift but the contribution of latter was found to be more. It was difficult to separate their individual influences in this set of preliminary experiments, but efforts in this regard are underway

by creating different amounts of interfacial degradation while maintaining constant bulk adhesive properties. These results need to be correlated to environmental degradation and further work is being carried out to accomplish this objective. An inversion algorithm was used to determine the value of interfacial transverse stiffness (Kt) using the experimental spectra obtained from oblique incidence inspection. A significant decrease in the value of Kt was observed with an increase in degradation, using which a qualitative assessment can be made as to whether the bond is good or bad.

References [1] R.D. Adams, B.W. Drinkwater, Non destructive testing of adhesively bonded jonts, NDT&E Int. 30 (2) (1997) 93–98. [2] W.A. Dukes, A.J. Kinloch, Non destructive testing of bonded joints, an adhesion science viewpoint, Nondestr. Test. (1974) 324–326. [3] E. Segal, J.L. Rose, Non destructive testing of adhesive bond joints, in: R.S. Sharpe (Ed.), Research Techniques in Nondestructive Testing, vol. 4, Academics press, London, 1980 (Chapter 8). [4] R.D. Adams, P. Cawley, Defects types and non destructive testing techniques for composites and bonded joints, Constr. Build. Mater. 3 (4) (1989) 170–183. [5] A.O.K. Nieminen, J.L. Koenig, Macroscopic and modern microscopic NDE methods for adhesive bond structures, Int. J. Adhes. Adhes. 11 (1) (1991) 5–10. [6] Shou Yang, G.U. Lan, Ronald F. Gibson, Non destructive detection of weak joints in adhesively bonded composite structures, Compos. Struct. 51 (2001) 63–71. [7] R.J. Schliekelmann, Non destructive testing of adhesive bonded metal to metal joints 1, Nondestr. Test. April (1972) 79–86. [8] S.I. Rokhlin, M. Hefetz, M. Rosen, An elastic interface wave guided by a thin film between two solids, J. Appl. Phys. 52 (1981) 2847–2851. [9] S.I. Rokhlin, An ultrasonic bridge for the study of viscoelastic properties of thin interface films, J. Acoust. Soc. Am. 73 (1983) 1619–1623. [10] S.I. Rokhlin, Ultrasonic evaluation of environmentally degraded adhesive joints, in: D.O. Thompson, D.E. Chimenti (Eds.), Review of Progress in QNDE, vol. 8B, Plenum Press, New York, 1986, pp. 1301–1308. [11] S.I. Rokhlin, D. Marom, Study of adhesive bonds using low-frequency obliquely incident ultrasonic waves, J. Acoust. Soc. Am. 80 (1986) 585–590. [12] A. Pilarski, J.L. Rose, A transverse-wave ultrasonic oblique-incidence technique for interfacial weakness detection in adhesive bonds, J. Appl. Phys. 63 (1987) 300–307. [13] P.B. Nagy, L. Adler, Interface characterisation by true guided modes, in: D.O. Thompson, D.E. Chimenti (Eds.), Review of Progress in QNDE, vol. 8B, Plenum Press, New York, 1988, pp. 1417–1424. [14] W. Wang, S.I. Rokhlin, Evaluation of interfacial properties in adhesive joints of aluminium alloys using angle-beam ultrasonic spectroscopy, J. Adhes. Sci. Technol. 5 (1991) 647–666. [15] C.C.H. Guyott, P. Cawley, Evaluation of the cohesive properties of adhesive joints using ultrasonic spectroscopy, NDT Int. 21 (4) (1988) 233–240.

1162

R.L. Vijaya Kumar et al. / Ultrasonics 53 (2013) 1150–1162

[16] P.N. Dewen, P. Cawley, The practical application of ultrasonic spectroscopy for the measurement of the cohesive properties of adhesive joints, NDT&E Int. 25 (2) (1992) 65–75. [17] N.Y. Chung, S.I. Park, M.D. Lee, C.H. Park, Ultrasonic detection of interface crack in adhesively bonded DCB joints, Int. J. Automot. Technol. 3 (4) (2002) 157– 163. [18] A.K. Moidu, A.N. Sinclair, J.K. Spelt, A new ultrasonic technique for the interfacial characterization of adhesive joints, IEEE Ultrason. Symp. (1996) 757–760. [19] C.J. Brotherhood, B.W. Drinkwater, S. Dixon, The detectability of kissing bonds in adhesive joints using ultrasonic techniques, Ultrasonics 41 (2003) 521–529. [20] S.I. Rokhlin, L. Wang, B. Xie, V.A. Yakovler, L. Adler, Modulated angle beam ultrasonic spectroscopy for evaluation of imperfect interfaces and adhesive bonds, Ultrasonics 42 (2004) 1037–1047. [21] S.I. Rokhlin, Evaluation of the curing of structural adhesives by ultrasonic interface waves, correlation with strength, J. Compos. Mater. 17 (1983) 15–25. [22] S.I. Rokhlin, M. Hefets, M. Rosen, An ultrasonic interface wave method for predicting the strength of adhesive bonds, J. Appl. Phys. 52 (4) (1981) 2847– 2851. [23] A.I. Lavrentyev, S.I. Rokhlin, Ultrasonic spectroscopy of imperfect contact interfaces between a layer and two solids, J. Acoust. Soc. Am. 103 (2) (1998) 657–664. [24] A.I. Lavrentyev, S.I. Rokhlin, Models for ultrasonic characterization of environmental degradation of interfaces in adhesive joints, J. Appl. Phys. 76 (8) (1994) 4643–4650. [25] S.I. Rokhlin, M. Rosen, An ultrasonic method for the evaluation of interface elastic properties, Thin Solid Films 89 (1982) 143–148. [26] A.I. Lavrentyev, S.I. Rokhlin, Determination of elastic moduli, density, attenuation and thickness of a layer using ultrasonic spectroscopy at two angles, J. Acoust. Soc. Am. 102 (6) (1997) 3467–3477. [27] L. Wang, B. Xie, S.I. Rokhlin, Determination of embedded layer properties using adaptive time–frequency domain analysis, J. Acoust. Soc. Am. 111 (6) (2002) 2644–2653. [28] A. Baltazar, L. Wang, B. Xie, S.I. Rokhlin, Inverse ultrasonic determination of imperfect interfaces and bulk properties of a layer between two solids, J. Acoust. Soc. Am. 114 (3) (2003) 1424–1434. [29] P.A. Dickstein, S. Girshovich, Y. Sternberg, A.N. Sinclair, H. Leibovitch, Ultrasonic feature based classification of the interfacial condition in composite adhesive joints, Res. Nondestr Eval. (1990) 207–224. [30] O.Y. Kwon, S.H. Lee, Acousto-ultrasonic evaluation of adhesively bonded CFRPaluminium joints, NDT&E Int. 32 (1992) 153–160. [31] K.S. Kim, J.S. Yoo, Y.M. Yi, C.G. Kim, Failure mode and strength of unidirectional composite single lap bonded joints with different bonding methods, Compos. Struct. 72 (2006) 477–485.

[32] S. Yang, L. Gu, R.F. Gibson, Non destructive detection of weak joints in adhesively bonded composite structures, Compos. Struct. 51 (2001) 63–71. [33] J.A. Schroeder, T. Ahmed, B. Choudhry, S. Shepard, Non destructive testing of structural composites and adhesively bonded composite joints: pulsed thermography, Compos. A 33 (2002) 1511–1517. [34] I.L. Baciu, I. Crivelli, Viscanti, A. Langella, V. Luprano, R. Teti, Quality evaluation of thermoplastic composite material single lap joints, Intelligent Production Machines and Systems, Pub. Elselvier Ltd., UK, 2006. pp. 258–263. [35] R.L. Vijayakumar, M.R. Bhat, C.R.L. Murthy, Nondestructive evaluation of adhesively bonded carbon fiber reinforced composite lap joints with varied bond quality, in: D.O. Thompson, Dale.E. Chementi (Eds.), Review of Progress in Quantitative Nondestructive Evaluation, vol. 31B, American Institute of Physics, Melville, New York, 2012, pp. 1276–1283. [36] A. Pilarski, J.L. Rose, Ultrasonic oblique incidence for improved sensitivity in interface weakness determination, NDT Int. 21 (1988) 241–246. [37] A. Pilarski, J.L. Rose, K. Balasubramaniam, The angular and frequency characteristics of reflectivity from a solid layer embedded between two solids with imperfect boundary conditions, J. Acoust. Soc. Am. 87 (1990) 532–542. [38] S.I. Rokhlin, W. Wang, Y.J. Wang, J.L. Hammill, Evaluation of weak interfaces in adhesive joints, in: Proceedings of the 1989 ASNT Fall Conference, 1989, pp. 41–50. [39] S.I. Rokhlin, A.I. Lavrentyev, B. Li, Research Ultrasonic evaluation of environmental durability of adhesive joints, Nondestr. Eval. 5 (1993) 90–109. [40] T. Nguyen, E. Byrd, C. Lin, Dale Bentz, A novel technique for measurement of water at the interface between a metal and an opaque polymeric film, Adv. Compos. Mater. 19 (1991) 1051–1056. [41] Y.C. Angel, J.D. Achenbach, Reflection and transmission of elastic waves by a periodic array of cracks, J. Appl. Mech. 52 (1985) 33–41. [42] R.B. Thompson, C.J. Fiedler, The effect of crack closure on ultrasonic scattering measurements, in: D.O. Thompson, D.E. Chimenti (Eds.), Review of Progress in QNDE, vol. 3A, Plenum, New York, 1984, pp. 207–215. [43] J.M. Baik, R.B. Thompson, Ultrasonic scattering from imperfect interfaces: a quasi-static model, J. Nondestr. Eval. 4 (1984) 177–196. [44] J.D. Achenbach, Z.L. Li, Analysis of bond-stiffness deterioration due to distributed disbonds, in: D.O. Thompson, D.E. Chimenti (Eds.), Review of Progress in QNDE, vol. 12B, Plenum, New York, 1993, pp. 1445–1452. [45] Michael J.S. Lowe, Matrix techniques for modeling ultrasonic waves in multilayered media, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42 (4) (1995) 525–542. [46] S.I. Rokhlin, Y.J. Wang, Analysis of boundary conditions for elastic wave interaction with an interface between two solids, J. Acoust. Soc. Am. 89 (1991) 503–515. [47] A. Pilarski, Ultrasonic evaluation of the adhesion degree in layered joints, Mater. Eval. 43 (1985) 765–770.