Mechanical and acoustical study of a structural bond: comparison theory/numerical simulations/experiment

European Journal of Mechanics A/Solids 25 (2006) 464–482

Mechanical and acoustical study of a structural bond:comparison theory/numerical simulations/experiment

Valentina Vlasie a,b,∗, Silvio de Barros b, Martine Rousseau a, Laurent Champaney b,Hugues Duflo c, Bruno Morvan c

a Laboratoire de Modélisation en Mécanique, Université Pierre et Marie Curie, UMR CNRS 7607,case 162, 4, place Jussieu, 75252 Paris, France

b Laboratoire d’Etudes Mécaniques des Assemblages, Université de Versailles, CNRS FRE 2481,45, avenue des Etats Unis, 78035 Versailles, France

c Laboratoire d’Acoustique Ultrasonore et d’Electronique, Université du Havre, UMR CNRS 6068,place Robert Schuman, 76610 Le Havre, France

Received 10 January 2005; accepted 10 September 2005

Available online 22 November 2005

Abstract

To carry out numerical simulations able to describe the behavior of a structural bond during a mechanical test, it is common touse a theory of damage. This approach consists in modeling the bonded zone by a surface distribution of springs with or withoutmass. The elasto-plastic with damage model and the numerical simulations are carried out with a finite element code. The use ofthis model requires two types of data: the critical energies in modes I and II measured during mechanical tests and the stiffnessesof the springs. These ones cannot be identified by mechanical measurements and, in this paper we propose an ultrasonic method tomeasure them. The ultrasonic approach and its experimental validation are first presented. Then, the mechanical model is detailed.The whole identification strategy is applied on aluminum/epoxy/aluminum samples.© 2005 Elsevier SAS. All rights reserved.

Keywords: Tri-layer structure; Mass/springs model; Ultrasound; Mechanical tests

1. Introduction

Adhesive bonding has become a powerful joining technique during the last few decades. Contrary to holes, rivets,clamps and screws which have a tendency to cause stress points in concentrated areas, adhesives tend to distribute theload over the entire bonded area. Although there are already many applications in aerospace, automotive and otherindustries, a better knowledge of the process of adhesion should allow to improve the tools of design and control.

From the mechanical point of view, we model the bonded zone by an interface that makes it possible to simplify thecalculations. The parameters of this model are the elastic stiffnesses and the critical energies of rupture. The identifica-tion of these interfacial parameters can be obtained only through tests on the entire structure and a theory/experiment

* Corresponding author.E-mail address: [email protected] (V. Vlasie).

0997-7538/$ – see front matter © 2005 Elsevier SAS. All rights reserved.doi:10.1016/j.euromechsol.2005.09.010

V. Vlasie et al. / European Journal of Mechanics A/Solids 25 (2006) 464–482 465

Fig. 1. Rheological model of the bonded joint.

comparison. As the stiffnesses of the interface have small influence on the global measured response of the assembly,they cannot be identified accurately. Moreover, the mechanical behavior of a thin adhesive layer can be different fromits bulk elastical properties. Thus, the stiffnesses cannot be calculated using the Lamé coefficients. The main interestof this work is to propose an ultrasonic method to determine the rheological parameters.

This paper is structured in two parts. In the first part we are interested in the ultrasonic wave propagation in ametal/adhesive/metal structure and particularly in the study of the cutoff-frequencies of guided modes. The aim isto develop an equivalent rheological model in which the adhesive layer is modeled by a geometrical interface witha uniform mass/springs distribution (Fig. 1). This ultrasonic approach allows us to evaluate the elastic behavior ofthe interface, i.e. the values of the spring stiffnesses and masses. The experimental validation is realized on sam-ples insonified at normal or oblique incidences, by short pulses. The signal transmitted through the structure is thenprocessed by a Fast Fourier Transform to give the resonance spectrum. The undoubling of Lamb modes and the ex-istence of a longitudinal vertical mode have been confirmed. The study shows a good agreement between theoreticaland experimental values of the stiffness coefficients and masses.

In the second part we present a mechanical model of the interface by using finite element method. The adhesivelayer is still described by a geometrical interface with a surface distribution of springs without masses (Fig. 1). Themechanical approach is complementary to the first one because it allows us to evaluate the critical energies of rupture.They are identified on two types of mechanical tests (End Noched Flexure mode and Mixed Mode Flexure) carried outon the same samples. These experimental values as well as the stiffnesses, calculated at low frequencies by ultrasonicmeasurements, are used to simulate the behavior of the same assembly under different loading conditions.

The tractions on interface (action of the upper face on the lower one) are denoted by:

t = tLz + tT x, (1)

and the associated kinematic variables are the displacement jumps at the interface, denoted by:

[u] = u+ − u− = [uL]z + [uT ]x. (2)

The elastic rheological behavior depends on a longitudinal stiffness KL and a transversal stiffness KT that are tobe identified:

t = K[u] ⇒{

tL = KL[uL],tT = LT [uT ]. (3)

2. The ultrasonic theoretical background

The study of guided waves propagation in multi-layers is of a considerable interest in view of various applicationsin non-destructive tests. The solutions of equations describing the wave propagation in such media are obtained byexpressing the displacements and the stresses in each layer, in terms of the wave amplitudes. By satisfying appropriateinterfacial conditions, we obtain the dispersion equation of guided waves as a function of geometrical and mechan-ical properties of layers. For a metal/adhesive/metal structure, we are confronted with the estimation of the bonded

466 V. Vlasie et al. / European Journal of Mechanics A/Solids 25 (2006) 464–482

Fig. 2. Tri-layer model geometry.

interfaces quality and with the detection of their damage. Commonly encountered bonding problems can be classifiedinto defects (delamination, cracks, etc.) and cohesive or adhesive weaknesses. Defects can be directly detected byconventional ultrasonic techniques, whereas weaknesses should be evaluated through the value of an acoustical spe-cific parameter. Many articles are concerned with the study of interfaces properties in a metal/ epoxy/metal tri-layerusing leaky Lamb waves. Thus, Nagy and Adler (1989), Jungman et al. (1991), Lowe and Cawley (1994), Kundu andMaslov (1997) have revealed experimentally that the reflection and transmission coefficients are not very sensitive tointerface properties. In order to model a thin adhesive layer, Jones and Whittier (1967) introduced boundary conditionsrelating stresses and displacements to longitudinal and transversal stiffnesses. That model was then developed by Pi-larski and Rose (1988) and by Rokhlin (1991), while Rokhlin and Wang (1991) introduced a viscoelastic rheologicalmodel (complex springs stiffnesses).

Several investigators were interested in the inverse problem. Lenoir et al. (1992) propose an experimental methodfor the resolution of the inverse problem in the case of a polystyrene/water/aluminum structure. Thus, three of the fourparameters which characterize the solid layers (longitudinal and transversal wave velocities, density, and thickness)may be evaluated.

In the following sections, we briefly present the development of the rheological model non-uniform in frequency(Section 2.3) introduced in a previous study (Vlasie and Rousseau, 2003) and its validation by ultrasonic measure-ments.

2.1. Tri-layer model

We consider a tri-layer structure composed of two identical metallic layers (S1, S3) joined by an adhesive layer (S2).Let ρi be the mass density of the ith medium (i = 1,2), cLi the corresponding longitudinal wave velocity, cT i thetransversal wave velocity, 2h the thickness of S1, S3, and d the thickness of S2. The geometry of the problem is shownon Fig. 2.

For plane deformations, the scalar potential and the single component of the vectorial potential are written as:{φ1 = [

A1L cos(kLz1(z + d/2)

) + B1L sin(kLz1(z + d/2)

)]ei(kxx−ωt),

ψ1 = [A1T cos

(kT z1(z + d/2)

) + B1T sin(kT z1(z + d/2)

)]ei(kxx−ωt),

(4)

for the S1-layer,{φ2 = [

A2LeikLz2z + A2L e−ikLz2z]

ei(kxx−ωt),

ψ2 = [A2T eikT z2z + A2T e−ikT z2z

]ei(kxx−ωt),

(5)

for the S2-layer, and{φ3 = [

A3L cos(kLz1(z − d/2)

) + B3L sin(kLz1(z − d/2)

)]ei(kxx−ωt),

ψ3 = [A3T cos

(kT z1(z − d/2)

) + B3T sin(kT z1(z − d/2)

)]ei(kxx−ωt),

(6)

for the S3-layer.In Eqs. (4)–(6), we introduced the following quantities: the common x-component of wave vectors kx , the

z-component of each wave vector kLzi , kT zi (i = 1,2), and the angular frequency ω.


Fig. 3. Geometry of the rheological model.

Moreover, we define the wave numbers: kLi = ω/cLi and kT i = ω/cT i (i = 1,2), which satisfy the dispersionequations: k2

Li = k2x + k2

Lzi and k2T i = k2

x + k2T zi . The amplitudes AiL, AiT , BiL, BiT (i = 1,3) are real while A2L,

A2T are complex (A2L, A2T are the complex conjugated numbers).The boundary conditions at the upper and lower interfaces are the free surface conditions, whereas we have to

fulfill a perfect contact condition at the bonded interfaces. Thus, we obtain the dispersion equation of guided modesin the structure. Its numerical resolution gives the dispersion curves which depend on the parameter d/2h.

2.2. Rheological model

Here we consider the same tri-layer structure as before, but the S2 adhesive layer is now replaced by the geometricalinterface z = 0. The adhesion process is then modeled by a uniform distribution of longitudinal and transversal springswith inertia. We denote Km

L and KmT their stiffnesses respectively, and m their common mass. The geometry of the

problem is shown on Fig. 3.The potentials are given by:{

φ1 = [A1L cos

(kLz1(z + h)

) + B1L sin(kLz1(z + h)

)]ei(kxx−ωt),

ψ1 = [A1T cos

(kT z1(z + h)

) + B1T sin(kT z1(z + h)

)]ei(kxx−ωt),

(7)

for the S1-layer, and by{φ3 = [

A3L cos(kLz1(z − h)

) + B3L sin(kLz1(z − h)

)]ei(kxx−ωt),

ψ3 = [A3T cos

(kT z1(z − h)

) + B3T sin(kT z1(z − h)

)]ei(kxx−ωt),

(8)

for the S3 layer.The boundary conditions are the free surface conditions at the upper and lower interfaces and the mass/springs

conditions (Jones and Whittier, 1967; Vlasie and Rousseau, 2003) at the z = 0 interface. We notice that the dispersionequation of guided modes, and consequently the dispersion curves, depend on stiffnesses and masses.

2.3. Comparison between the tri-layer model and the rheological model

The comparison between the two models shows an identical behavior of the dispersion curves. In particular, it ispossible to obtain some informations by comparing the cutoff-frequencies expressions. In a previous paper (Vlasieand Rousseau, 2003), we have presented mathematical calculations which show that:

(i) if cot(�ωnL2d/4h) > 0, then:

�KmLexact = �ωnL1

zL

cot

(�ωnL2

d

4h

),

mLexact = 2

�ωnL1(�ωnL1/KmL + zL cot(�ωnL2d/4h))

,

(9)


(a) (b)

Fig. 4. KmL

versus �ω for different values of d/2h.

(ii) if cot(�ωnL2d/4h) < 0, then:

KmLexact = −�ωnL1

zL

tg

(�ωnL2

d

4h

),

mL exact = −2

�ωnL1(−�ωnL1/KmL + zL tg(�ωnL2d/4h))

.

(10)

We used here the dimensionless quantities �ω = 2keh, ke = ω/ce (ce = 1500 m/s), KmL = 4Km

L h/(ρ1c2L1), Km

T =4Km

T h/(ρ1c2T 1), m = m/(2ρ1h) and the notations nLi = kLi/ke, nT i = kT i/ke (i = 1,2), zL = (ρ1cL1)/(ρ2cL2),

zT = (ρ1cT 1)/(ρ2cT 2).

Remark. If we replace the subscript L by T , we obtain the corresponding relations for the equivalent transversalstiffnesses and masses.

We observe that the stiffnesses and masses depend on the frequency, i.e. the rheological model must be adaptedaccording to the considered frequency range (which depends on the thickness of the adhesive layer). Then, we talkabout a rheological model non-uniform in frequency (Fig. 4(a)).

For d/2h = 10−2 we observe a small frequency variation of KmL , whereas for d/2h = 3.510−1 this variation is more

important. In this case, we successively use the expressions (9) and (10) to define KmL . So, two branches appear on

both sides of dotted lines. Between these dotted lines the stiffness tend to 0 or ∞ and consequently it is not possible todefine an equivalent rheological model inside this zone. This result is natural because the central frequency, indicatedby the solid line, corresponds to a cutoff-frequency of a free adhesive layer of thickness d .

For low frequencies or for d/2h sufficiently small (�ωnL2d/4h < 1), a first order expansion of the expressions (9)and (10) allows us to obtain Km

Lapp = 2((2hρ2c2L2)/(dρ1c

2L1)) and mLapp = (ρ2d)/(ρ12h). In this case, we recover the

definitions of the stiffnesses and masses used in the literature (Pilarski and Rose, 1988; Rokhlin, 1991) and we talkabout a rheological model uniform in frequency (Fig. 4(b)).

2.4. Ultrasonic experimental results

2.4.1. Preparation and description of samplesWe made aluminum/epoxy/aluminum samples with different relative thicknesses. The epoxy resin is composed of

one mole of diglycidylether of bisphenole A (DGEBA) and two moles of diamino diphenyl methane (DDM). Thissolid mixing is melted, degassed (in order to eliminate all bubbles and to ensure a good quality of the samples), andthen introduced between the two aluminum plates and put in an oven at 80 degrees Celsius for twelve hours.


Fig. 5. Experimental set-up.

2.5. Experimental procedure

For the inspection of adhesively bonded joints, the most used non-destructive testing methods are:

(a) the emission/reception (pulse-echo) technique, which allows for inspection with one-sided access. Goglio andRossetto (1999) used this method to detect the presence or absence of adhesive between the two aluminum platesand to identify zones of poor adhesion.

(b) the transmission technique (Vine et al., 2002), efficient in detecting disbondings. It requires a simultaneous appli-cation of both transducers on the two sides of the sample.

(c) the acousto-ultrasonic technique (Laperre and Thys, 1994; Tiwari et al., 1991), based on the transmission of anultrasonic pulse through the structure, from an emitter to a receiver both placed on the same surface.

(d) the double transmission technique (Nagy and Adler, 1989), based on one ultrasonic transducer used to generatethe incident ultrasonic pulse and to pick up the double-transmitted echo from a perpendicular plane reflectorplaced at the back-side.

(e) the laser ultrasonic technique (McKie and Addison, 1993; Heller et al., 2000).(f) the air-coupling technique (Yano et al., 1987; Castaings and Cawley, 1996).

For the experimental investigations performed at LAUE-Le Havre, we applied the transmission technique (b). Thesamples are immersed in a water-filled tank and fixed at the end of steams, screwed in a bar lying on the edge ofthe tank. The transducers are immersed at the end of steams screwed in a second bar placed at a variable distancefrom the first one. A pulse generator sends pulses to the emitter; its bandpass is approximately 500 KHz–3 MHz.The amplitude of the pulse is 300 V, its duration is about 1 µs and its periodicity is 10 ms. The emitter, excited bypulses, insonifies the structure at incidence angle θ with respect to the normal. The receiver, placed symmetricallywith respect to the normal, receives the transmitted signal, which is then sent to the digital oscilloscope LECROY9310 M after amplification (Fig. 5).

This signal (Fig. 6) is essentially composed of specularly reflected echoes (which correspond to the reflectionson each face of the structure) and a “tail” corresponding to the free elastic response of the structure (which givesthe resonance frequencies of the target) (Delestre et al., 1986). The signal is averaged by summation over about 200sweeps in order to reduce the stochastic noise.

In order to isolate the different resonances, the time signal was gated by different windows (between 5.3 µs and52 µs). The resonance spectrum (Fig. 7) is calculated by performing a Fast Fourier Transform on the gated signal.The spectrum is stored into the hard-disk memory of a microcomputer via an IEEE interface bus. A microcomputercontrols the angular position of the sample. The manipulations are made for all angles θ between 0◦ and 15◦ with anangular step �θ = 0.2◦. Thus we obtain the dispersion curves of the tri-layer structure.

2.5.1. Theory/experiment comparisonFor the numerical applications, we take the sound velocity in water as 1500 m/s, the characteristics of the alu-

minum as cL1 = 6380 m/s, cT 1 = 3100 m/s, ρ1 = 2800 kg/m3, and those of the epoxy resin cL2 = 2622 m/s,cT 2 = 1356 m/s, ρ2 = 1159 kg/m3. For the evaluation of the adhesive thickness, we use one of the two particularmodes, called vertical modes (V L or V T ). The study of these low frequency modes was already made in a previous


Fig. 6. Amplitude spectrum of the received signal.

Fig. 7. Resonance spectrum.

work Vlasie (2003). Their cutoff-frequency is given by a simple relation. For instance, the cutoff-frequency of theV L mode is: �ω = (1/nL1)

√2/νL (1 − 1/3νL), where νL = (ρ1/ρ2)(cL1/cL2)

2 d2h

= 1/KmLapp. This expression allows

to estimate the value of the adhesive thickness d . We studied several samples with d/2h varying from 5 × 10−2 to6.5 × 10−1. In this paper, we present only one of them.

On Figs. 8 and 9, we plot the theoretical and experimental dispersion curves obtained for this specimen.Considering that the ratio ρeau/ρ1 � 1, we observe a good agreement theory/experiment for all frequencies be-

tween 0 and 3 MHz even though the theoretical curves correspond to a non-immersed tri-layer and the experimentalcurves to an immersed structure.

Previous studies (Couchman et al., 1978; Guyott and Cawley, 1988; Vlasie and Rousseau, 2003) show that theguided modes of the metal/adhesive/metal structure appear as doublets centered around the Lamb modes of a freemetal layer. In particular, the cutoff-frequencies of each doublet are located on both sides of cutoff-frequencies of thefree metal layer. In Table 1 we compare the theoretical and experimental values of stiffnesses and masses. The firstcolumn indicates the considered mode. Let us take as example the transversal mode of an aluminum plate ( �ω = 6.49).The cutoff-frequencies of the corresponding doublet ( �ωth = 6.08 or �ωexp = 6.10) and ( �ωth = 6.55 or �ωexp = 6.66) arenoted respectively “−” and “+”. Moreover, certain modes were observed in experiments but their cutoff-frequency isnot in the table �ω = 11.79 (fexp = 0.94 MHz) and �ω = 17.30 (fexp = 1.38 MHz). These are the Lamb modes of a freeadhesive layer.

If we compare the theoretical and experimental values of KmL,T and mL,T , we observe a good agreement which

confirms the quality of the sample and validates the rheological model non-uniform in frequency.For the mechanical model of the interface presented in Section 3, the elastic characteristics of the interface are

those evaluated with the ultrasonic method for low frequency ranges (Kmapp, Km

app).
L T


Fig. 8. Theoretical dispersion curves.

Fig. 9. Experimental dispersion curves.

Fig. 10. Resonances spectrum at 00.

3. The mechanical theoretical background

Cohesive-zone models have been extensively used for the non-linear incremental analysis of interface debondingin the last few years see Corigliano (1993), Allix and Corigliano (1996), Allix et al. (1998), Mi et al. (1998), Allix
and Corigliano (2000), Alfano and Crisfield (2001).


Table 1Theory/experimental comparison (d/2h = 3.5 × 10−1)

Mode �ωth fexp (MHz) �ωexp KmL,Tth

KmL,Texp

mL,Tth mL,Texp

V L 2.51 0.20 2.51 0.39 0.39 0.14 0.14T (6.49−) 6.08 0.49 6.10 0.22 0.22 0.04 0.04T (6.49+) 6.55 0.53 6.66 0.18 0.17 – –T (12.98−) 12.77 1.02 12.79 0.89 0.88 0.03 0.03T (19.48+) 19.72 1.57 19.73 2.16 2.16 – –L(26.73−) 25.69 2.05 25.75 0.66 0.65 0.03 0.03L(26.73+) 27.09 2.16 27.12 0.50 0.49 – –T (32.46−) 31.70 2.53 31.74 0.41 0.39 0.01 0.01T (38.95+) 39.03 3.11 39.04 1.05 1.05 – –L(40.09−) 39.41 3.14 39.43 1.52 1.52 0.02 0.02L(40.09+) 40.68 3.25 40.84 1.21 1.18 – –

Fig. 11. Interface model.

In the cohesive-zone approach, the description of a state of damage along an interface relies upon the definitionof a traction-separation law incorporating the dependence of the surface tractions on the corresponding displacementdiscontinuities [u] = u+ − u− and the damage criterion to be met for the cohesive process zone to grow and the crackadvance. In the simplest one-dimensional case, the damage onset and decohesion propagation conditions involve onlythe single-mode displacement or energy release rate component; on the contrary, when considering the mixed-modecase these conditions have to properly account for the interaction of the pure-mode contributions.

In this last case the work of separation per unit fracture area does actually result from the interplay of the I and IIpure-mode contributions, that are not independent in that they evolve together as a consequence of the interaction ofthe traction-displacement jump relationships in two directions.

In what follows we discuss briefly the cohesive-zone model used in this work. A more exhaustive presentation ofthis model can be found in Champaney and Valoroso (2004).

3.1. 1D model

The adhesive joint considered here consists of two elastic bodies (adherents) joined by a plane adhesive layer, withthickness assumed to be negligible compared to that of the joined bodies and to its in-plane dimensions. These featuresenable the adhesive layer to be conveniently schematized as an interface, i.e. as a zero-thickness surface entity whichensures displacement and stress transfer between the adherents, see Fig. 11.

Assuming that the displacement jump [u] = u+ − u− at the interface in one direction is small in the usual sense,the elastic damage model for the interface can be derived based on a stored energy function ψ defined as:

ψ([u],D) = (1 − D)

1K[u]2, (11)

2


Fig. 12. 1D model: loading and unloading.

where D is a scalar variable measuring the damage of the interface as a loose of stiffness. The associated interfacetraction in the direction of the jump is then:

t = ∂ψ

∂[u] = (1 − D)K[u]. (12)

The damage driving force Y is the work conjugate variable of the damage D. It is classically defined by:

Y = − ∂ψ

∂D= 1

2K[u]2. (13)

The damage evolution is subject to the classical loading/unloading conditions:

f (Y ) � 0, D � 0, f (Y )D = 0, (14)

f (Y ) = Y − Y , D ∈ [0,1], (15)

where the damage threshold Y is defined by:⎧⎨⎩

Y = G0 if D = 0,

Y = G0 + (Y f − G0)D1/N if D ∈]0,1[,

Y = maxτ∈[0,T ] Y(τ) if D = 1.

(16)

This damage evolution law, depicted in Fig. 12, is the same as the one used for the modeling of interface degradationin composite materials by Allix and Ladevèze (1992). In Eq. (16), Yf is calculated such that the locally dissipatedenergy is Gc:

+∞∫0

YD dt = Gc ⇒ Yf = 1

N

((N + 1)Gc − G0

). (17)

3.2. Mixed mode model

Consider the following form of the stored energy function:

ψ([u],D) = (1 − D)

[1

2KL

⟨[uL]⟩2+ + 1

2KT [uT ]2

]+ 1

2KL

⟨[uL]⟩2−, (18)

where D ∈ [0,1] denotes a scalar damage variable in the usual sense, and the symbols 〈·〉+ and 〈·〉− stand for thepositive and negative part of 〈·〉, defined as 〈x〉± = 1/2(x ± |x|).

The constitutive equations for the interface traction vector t and the damage driving force are obtained in the usualway as:


t = ∂ψ

∂[u] , (19)

Ym = − ∂ψ

∂D= 1

2KLδ2, (20)

where δ represent an equivalent displacement defined as:

δ =(⟨[uL]⟩2+ + KT

KL

[uT ]2)1/2

. (21)

The energy release rate for the two modes are:⎧⎪⎪⎪⎨⎪⎪⎪⎩

YI = 1

2KL

⟨[uL]⟩2+ = 1

1 + β2Ym (mode I participation),

YII = 1

2KT [uT ]2 = β2

1 + β2Ym (mode II participation),

(22)

where β accounts for the mode mixity:

β =√

KT

KL

β and β = [uT ]〈[uL]〉+ ∈ [0,+∞]. (23)

Here β is the mixed mode ratio.The damage evolution is subjected to the classical loading/unloading conditions:

f (Ym) � 0, D � 0, f (Ym)D = 0, (24)

f (Ym) = Y − Y m, D ∈ [0,1], (25)

where the damage threshold Y m is defined by:⎧⎪⎨

⎪⎩Y

m = Y 0m if D = 0,

Y m = Y 0

m + (Y

fm − Y 0

m

)D1/N if D ∈]0,1[,

Y m = maxτ∈[0,T ] Y(τ) if D = 1.

(26)

The damage onset is obtained according to the following criterion:(YI

G0I

)α

+(

YII

G0II

)α

= 1, (27)

which gives:

Y 0m = (1 + β2)G0IG0II

[Gα0II + (β2G0I)α]1/α

. (28)

For the delamination propagation, the well-known ellipse criterion is assumed:(GI

GcI

)α

+(

GII

GcII

)α

= 1, (29)

where the released energies are defined as:⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

GI =+∞∫0

YID dt,

GII =+∞∫

YIID dt = β2GI.

(30)

0


(a) (b)

(c) (d)

Fig. 13. Tests. (a) Pure mode I, (b) pure mode II, (c) mixed mode: 60% mode I, (d) mixed mode: 30% mode I.

Accordingly, the propagation of decohesion takes place for:

GI = GcIGcII

[GαcII + (β2GcI)α]1/α

, (31)

where GI can be computed from the relationship:

GI = 1

1 + β2

+∞∫0

Y mD dt. (32)

According to the damage evolution law, the expression of the parameter Yfm follows as:

Yfm = 1

N

((N + 1)GI − Y 0

m

). (33)

It can seen that the interface model takes into account the modification of the mixed mode ratio during the loadingpath. Fig. 13 presents the behavior of the model for pure mode I and mode II and for mixed mode (with two differentmixities).

This model has been implemented in the Finite Element Code CAST3M, where it can be used for simulation ofdamage evolution in adhesively bonded joints.

3.3. Mechanical tests

The parameters of the interface model are the undamaged stiffnesses (KL and KT ), the activation energies for eachpure mode (G0I and G0II), the critical energies (GcI and GcII), and the exponent of the ellipse criterion for activationand propagation (α). The exponent α is typically set to 2 (elliptic criterion). In a previous work (Champaney and


Fig. 14. Classical crack propagation tests.

Fig. 15. ENF test.

Fig. 16. ENF results.


Fig. 17. MMF test.

Fig. 18. MMF results.

Valoroso, 2004), Champaney and Valoroso have shown that the activation energies have a very small influence on theglobal response. In the following computations, they have been set to 40% of the critical energies.

The stiffnesses of a thin layer of adhesive cannot be easily derived from the elastic properties of the adhesiveitself (Tong and Steven, 1999). They cannot be identified from mechanical tests on adhesively bonded assembliesbecause they have a small influence on the global response of the assembly. As shown in Section 2, KL and KT canbe identified however from acoustical tests.


Fig. 19. ENF test.

Fig. 20. ENF numerical and test results.

The activation energies G0i and the critical energies Gci can be identified directly from classical crack propagationtest results. The tests depend on the solicitation mode used to propagate the crack. The double-cantilever beam (DCB)and the end-notched flexure (ENF) are pure mode I and pure mode II tests, respectively. We can also have mixed-modetests like the mixed-mode flexure (MMF). These tests are presented schematically in Fig. 14.

Due to their boundary conditions, which are simpler than in mode I tests, ENF and MMF tests were performed first.DCB tests will be developed in the future. Aluminum/epoxy/aluminum samples with 0.5 mm thickness were testedusing a traction/compression machine MTS 816 with 7500 kgf load capacity. The characteristics of the aluminum areE = 75000 MPa and ν = 0.3. Fig. 15 shows the dimensions of the specimens.

Fig. 16 shows two results of ENF tests developed at ESPCI-Paris.Fig. 17 presents a MMF test and Fig. 18 shows these results of mixed-mode tests. The inclination at the beginning

of each curve corresponds to a given initial crack length a. The structure is more or less stiff, depending on the lengthof the initial crack.

3.4. Numerical results

Numerical simulations and tests results must be compared in order to evaluate the parameters of the mechanicalinterface model.


Fig. 21. MMF test.

Fig. 22. MMF numerical and test results.

The finite-element (FE) mesh showed in Fig. 19 and used to simulate an ENF test is composed by 528 quadraticelements with eight nodes (3 elements in the thickness of each plate) and 74 quadratic elements of interface.

Fig. 20 shows the comparison between an experimental curve and the result of the numerical model after identi-fication of the damage parameters in mode II (G0II and GcII). Just after the start of the crack, the effects of dynamicpropagation (not represented in this quasi-static model) do not allow for the correct representation of the structureanswer.

To simulate a MMF test, the FE mesh showed in Fig. 21 is used. It is composed of 504 quadratic elements witheight nodes (3 elements in the thickness of each plate) and 77 quadratic elements of interface.

Fig. 22 shows the comparison between a experimental curve and the result of the numerical model after iden-tification of the rest of the damage parameters (G0I and GcI). Just after the start of the crack, the effects of greatdisplacements of the lower plate (not represented in this model) do not allow for the correct representation of thestructure answer.

Finally, the elastic characteristics of the bonded interface that were identified by ultrasonic measurements and itsdamage characteristics that were identified in mechanical tests in mode II (ENF) and mixed-mode (MMF) are found as:

α = 2, KT = 760 N/mm3, KL = 810 N/mm3,

GcI = 0.02, G0I = 0.4 × GcI,

GcII = 0.09, G0I = 0.4 × GcII.


Fig. 23. ENF numerical and test results for different initial crack lengths.

Fig. 24. MMF numerical and test results for different initial crack lengths.

Figs. 23 and 24 present the simulations and the experimental results for the same assembly with different initialcrack lengths for ENF and MMF tests. One can see that the identified model can simulate the behavior of the assemblyunder different loading conditions.


4. Conclusion

This paper proposes an interface model which simulates the damage evolution of a structural joint alu-minum/epoxy/aluminum. Several parameters have been identified. Some of them are easily obtained by mechanicaltests, whereas the stiffnesses of the interface cannot be derived from a mechanical approach. The originality of thispaper is to propose an ultrasonic approach in order to reach these stiffnesses.

The first part of the paper presents the construction of the rheological model used for the acoustical identification.We compared the cutoff-frequencies of the ultrasonic guided modes for a tri-layer model with a mass/springs modelof the adhesive layer. Thus, it is possible to define longitudinal and transversal stiffnesses and mass functions of thefrequency. The comparison theory/ultrasonic measurements enables us to validate the model and to evaluate the elasticparameters.

The second part of the paper presents the mechanical cohesive-zone model and the mechanical tests. The testsin mode II or mixed mode are then carried out on the different samples in order to obtain the critical energies.All parameters measured either by ultrasonic experiments or by mechanical tests are used for simulations in theFinite Element Code CAST3M, where the mechanical model is implemented. There is a good agreement between themechanical tests and the simulation results.

Acknowledgements

This project was supported by the French Ministry of Research under contract ACI (Action Concerté IncitativeJeunes Chercheurs). The authors thank Freddy Martin from ESPCI (Ecole Supérieure de Physique et de ChimieIndustrielles de Paris) for the preparation of the samples.

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Documents

Mechanical and acoustical study of a structural bond: comparison theory/numerical simulations/experiment