Analysis of the metal adhesively bonded double cantilever beam specimen

International Journal of Adhesion & Adhesives 61 (2015) 8–14

Contents lists available at ScienceDirect

International Journal of Adhesion & Adhesives journal homepage: www.elsevier.com/locate/ijadhadh

Analysis of the metal adhesively bonded double cantilever beam specimen A.B. de Morais n University of Aveiro, Department of Mechanical Engineering, RISCO research unit, Campus Santiago, 3810-193 Aveiro, Portugal

art ic l e i nf o

a b s t r a c t

Article history: Accepted 15 April 2015 Available online 30 April 2015

The double cantilever beam specimen is currently standardized for measuring the mode I fracture energy of adhesive joints. In addition, it has been increasingly employed to evaluate the adhesive tractionseparation law by the direct method, which involves crack tip separation measurements. The threedimensional finite element analyses here conducted showed that significant anticlastic deformations of the metal adherends compromise the accuracy of the direct method in the elastic domain. It was also seen that the adherend plane stress and adhesive uni-axial strain hypotheses are adequate for the typical specimen geometries. Finally, the new elastic crack length correction derived from a beam model can be used to predict accurately the initial specimen compliance, to obtain conservative fracture energy values and to gain additional insight into the adhesive fracture behaviour. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Metals Fracture mechanics Double cantilever beam Beam model

1. Introduction Owing to several advantageous characteristics and to the progress in adhesive formulations, adhesive joints are increasingly used in a wide variety of structural applications [1]. However, despite the vast research conducted, design of adhesive joints still faces considerable challenges right from the stress analysis stage [2–4]. Singularities at the adherend/adhesive interface and steep stress gradients are common in most joint configurations, posing difficulties to continuum mechanics based design approaches. Fracture mechanics is thus considered better suited for predicting joint strength if failure is actually dictated by the crack propagation stage. This failure mode is favoured by the preferential use of tough adhesives and by the frequent adoption of joint geometrical features that reduce stress gradients [1]. Moreover, one of the main limitations of traditional fracture mechanics, i.e. the need to assume a pre-existing crack, can be overcome by cohesive zone modelling (CZM) [5,6]. Considerable research has already been conducted on CZM of common structural adhesive lap-joints e.g. [7–11]. The basis of all formulations is the so-called tractionseparation sc–δc law. Most of the traction-separation laws used for pure mode I loadings include an initial hardening stage until a cohesive strength is attained [12]. The final softening stage ends when the energy dissipated is equal to the mode I fracture energy. Adhesive fracture energies have thus become particularly important properties for joint design. The majority of the experimental

work has focussed on mode I fracture [13,14], for which the double cantilever beam (DCB) (Fig. 1) and the tapered double cantilever beam (TDCB) are currently standardized [15,16]. Data analyses are often carried out within linear elastic fracture mechanics (LEFM), meaning that the mode I fracture energy is actually the adhesive mode I strain energy release rate GIC. However, fracture of tough adhesive may involve plastic zones around the crack tip that are too large for LEFM applicability. Therefore, the mode I fracture energy has also been designated as JIC and measured with J-integral based data reduction schemes [17–20]. This is supported by the observed dependence of the fracture energy on the bondline thickness, which can be correlated with the height of the adhesive plastic zone [17–20]. Accordingly, Pardoen et al. [18] proposed to view the perceived fracture energy as the sum of an intrinsic work of fracture with the energy of adhesive layer elastic–plastic deformations, only the former being a true material property. In spite of the above questions regarding the meaningfulness of the mode I fracture energy, the DCB specimen has been increasingly used to evaluate the traction-separation law of the adhesive [17–25]. The method employed, often designated as “direct”, requires measuring JI and δc until “crack initiation” i.e. the instant at which the initial crack actually starts to propagate. The derivative of the JI –δc curve yields the sc–δc law [17–25]. Obviously, this approach demands:


A representative “sharp” pre-crack created in the specimen e.g. by inserting thin release films and/or razor blades.

n

Tel.: þ 351 234 370830; fax: þ 351 234 370953. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.ijadhadh.2015.04.010 0143-7496/& 2015 Elsevier Ltd. All rights reserved.


Accurate measurements of the small δc that have been facilitated by the performance of recent optical methods [20–25].

A.B. de Morais / International Journal of Adhesion & Adhesives 61 (2015) 8–14

9

Fig. 1. The DCB specimen for adhesive joints.


Accurate methods to compute JI, which may in turn require measurements of load-point crack-tip rotations [19,20,25], or simply application of beam theory based data reduction schemes [21–24] similar to those used for delamination of composites [16,26,27]. Furthermore, in order to validate traction-separation law measurements, experimental load–displacement curves have been compared with predictions of two-dimensional (2D) CZM [21–24]. It is thus clear that characterization of the mode I fracture of adhesive joints has reached a high level of sophistication in both experimental techniques and analysis methods. However, little attention has been paid to considerable threedimensional (3D) effects already reported [21,28–32]. In fact, Han and Siegmund [21] detected significant differences between the edge separations predicted by 3D CZM and those obtained from 2D plane strain CZM. This can be explained by the anticlastic deformations resulting from Poisson effects associated with the longitudinal bending of the adherends [29,30]. This causes nonuniform width-wise distribution of GI [30–32] that promote premature crack initiation at the centre of the specimen and end-up causing thumbnail-shaped crack fronts [28,32]. This phenomenon does not compromise steady-state propagation GIC measurements on laminated composites [33,34], as demonstrated in the 3D FEA of [35] which simulated the curved delamination propagation with CZM. However, the transition of the straight pre-crack to the thumbnail-shaped one creates additional difficulties in measuring the initiation GIC value [36]. As for evaluating the mode I tractionseparation law in composites, the very small separations and the low initiation GIC restrict the direct method to the characterization of the fibre bridging phenomenon [37,38] that has no parallel in adhesive fracture. Therefore, it seemed useful to investigate the 3D effects on the evaluation of the adhesive traction-separation law and on the 2D beam theory based data analysis methods commonly employed. The analyses described below considered the most desirable cohesive fracture mode [1] i.e. it was assumed that the starter crack was generated and subsequently propagated within the adhesive layer.

2. Three-dimensional finite element analyses As seen above, relatively few 3D FEA have been reported on the metal adhesively bonded DCB specimen. The present models were constructed with second-order 20-node reduced integration elements (C3D20R) of the ABAQUSs code. The material and loading y¼0 and z¼0 symmetry planes (Fig. 1) allowed the modelling of a quarter-specimen. Mesh refinement studies showed that accurate results could already be obtained by modelling the adhesive halflayer with a single layer of finite elements and the adherend by two layers of finite elements. The elements around the crack tip were 0.25 mm long and 0.83 mm wide. Loading consisted of applying a vertical displacement δ/2 (Fig. 1) to the adherend midthickness end node set, on which length-wise displacements were prevented. In view of the objectives of this work and the results

Fig. 2. Distributions of GI along half-width of steel adhesively bonded DCB specimens with dimensions (mm) h ¼7 (grey lines) and 13 (black lines), ha ¼ 0.1 (continuous lines) and 0.7 (dashed lines), a¼ 40 and 130 (vertically offset by one unit). GI values were normalized by the width-wise average.

that were actually obtained, the analyses adopted linear elastic behaviour for both adherends and adhesive. The virtual crack closure technique (VCCT) [39] was employed to compute GI along the straight crack front. The present analyses considered typical steel and aluminium (Al) adherend properties and geometries i.e. Young's modulus E¼ 210 and 70 GPa, Poisson ratio ν ¼0.3 and thickness h from 7 to 13 mm (Fig. 1). For the adhesive Ea ¼1.8 GPa and νa ¼0.35 were used with bondline thickness ha from 0.1 to 0.7 mm. Crack lengths a were varied between 40 and 130 mm, while the width was always b¼25 mm. Fig. 2 depicts the clearly non-uniform width-wise distribution of GI for steel adhesively bonded DCB specimens as a result of considerable adherend anticlastic deformations. Distributions for Al adherend specimens were similar but slightly more non-uniform. In fact, it can be seen that the degree of non-uniformity increases for decreasing adherend bending stiffness, especially by lowering the adherend thickness and, to less extent, by increasing the crack length. Thicker bondlines in turn reduce the effect of adherend anticlastic deformations on adhesive through-thickness strains εz (Fig. 1) and thus the non-uniformity of GI. Nonetheless, it seems clear that the non-uniformity is always significant enough to compromise the accuracy of traction-separation law evaluation by the direct method. Let us consider the most recent approach, based on optical measurements of crack tip separations at a specimen edge, applied to the most favourable case of Fig. 2, which corresponds to a stiff h¼13 mm adherend with a small a ¼40 mm initial crack. The edge GI is about 60% of the width-wise average value that would be measured by a beam theory or J-integral based analysis method. For the linear elastic adhesive behaviour adopted, GI correlates with δc2, and thus δc measured at the edge would be about 80% of the width-wise average δc. Therefore, the direct method would not provide an accurate linear elastic part of the traction-separation law. Naturally, it can be argued that the linear elastic range is only a small portion of the traction-separation law and that subsequent plastic deformations and damage could reduce the width-wise non-uniformity of energy dissipation through load redistribution. However, such phenomena do not prevent anticlastic deformations of the stiff adherends, which end-up causing the thumbnailshaped crack fronts observed in [28]. Therefore, evolution from the straight pre-crack to the thumbnail-shaped propagating crack will still bring about additional errors in traction-separation law measurements. Evidently, the true magnitude of such errors will have to be evaluated with 3D CZM.

10

A.B. de Morais / International Journal of Adhesion & Adhesives 61 (2015) 8–14

Another interesting result of the present 3D FEA concerns the stress state in the adhesive. As expected, the stiff adherends were found to restrain the in-plane deformations of the adhesive, thereby generating significant longitudinal sx and transverse sy stresses in addition to the through-thickness sz (Fig. 1). It is worth reminding that a similar adhesive layer restraint exists in tensile tests on butt joints [1]. This uni-axial strain hypothesis has also been recently applied to analyse asymmetric DCB specimens [29,30], and is expected to hold in the vicinity of the crack tip [15]. The difficulties posed to stress analysis by the crack tip singularity can be circumvented by plotting the sx/sz and sy/sz ratios. Assuming εx,εy E0 in the adhesive, it can be shown from Hooke's law that those stress ratios should be νa/(1 νa). This was seen to be the prevalent condition in the specimens analysed, even for the least adhesive deformation restraining configuration considered i.e. Al adherends with h¼ 7 mm, ha ¼ 0.7 mm and a¼130 mm (Fig. 3). The above stress ratios only depart from the theoretical value in small regions where sz changes from peel to compressive stress, thus being very small. Obviously, all stresses in the adhesive layer vanish at sufficient distance from the crack tip, but this zone is not of interest here and thus is not depicted in Fig. 3. A fully inplane restrained adhesive layer if favourable for obtaining conservative adherend independent mode I fracture energies. Moreover, such condition will also prevail in the adhesive plastic regime, as the adherend-to-adhesive stiffness mismatch increases. Finally, the significant anticlastic deformations suggest the adherends are essentially under plane stress, especially if a parallel is drawn with delamination of composites (see discussion in Section 1). Nevertheless, some of the specimen geometries analysed had crack length-to width ratios small enough to raise the possibility of the plate theory plane strain assumption being more appropriate. In fact, 2D plane strain FEA of DCB specimens are not uncommon [12,23]. Local peel stresses transmitted by the adhesive could also cause a stiffer adherend bending response, especially when the peel stress is a high fraction of the bending stress. This effect could be more significant than in delamination of composites because of the thick adherends often employed. The adherend plane stress hypothesis is particularly important for the application of beam models and is thus quantitatively evaluated below.

experimental and analytical studies [21–24,30,31,40,41]. The main results of beam models are predictions of the specimen compliance C¼ δ/P, δ designating the load-point displacement and P the applied load (Fig. 1), and GI. The early analysis by Mostovoy et al. [42] adopted simple beam theory with a plane strain correction that predicts

C=

GI =

(

8 a3 + h2a

3.1. Current status Beam theory based data reduction methods are recommended in existing standards [15,16] and have been extensively used in

Fig. 3. Adhesive sx/sz (dark grey lines) and sy/sz (black lines) stress ratios along the y¼ 0 (continuous lines) and y¼ 11.4 mm (dashed lines) planes compared to the theoretical in-plane restrained value (continuous light grey line).

(1)

12P 2a2 ⎛ h2 ⎞ ⎜1 + ⎟ 2 3 3a2 ⎠ Eb h ⎝

(2)

Blackman et al. [40] found that the above expressions lacked accuracy when compared to the so-called corrected beam theory (CBT) developed by Williams [26] for delamination of composites. According to CBT [16,26],

C=

GI =

8 (a + ΔI )3 (3)

Ebh3 12P 2 (a + ΔI )2 Eb2h3

=

3Pδ 2b (a + ΔI )

(4)

where ΔI 40 is a correction for crack tip rotation and deflection. Further corrections are available if the specimen is loaded via bonded load-blocks or when the specimen undergoes large displacements, but these cases are not considered here. ΔI can be determined from the intercept of a C1/3 versus a plot constructed from (P, δ, a) data gathered throughout the DCB test [16,40]. Eq. (3) can then be used to back-calculate an estimate of the adherend modulus, which is expected to be close to Young's modulus E, despite the inevitable errors associated with crack propagation monitoring. There is also a reference value for ΔI, which was derived from a beam on elastic foundation analysis [26] and adapted to the DCB specimen with isotropic adherends as follows [16,40]:

ΔI = h 3. Beam analysis of the metal adhesively bonded DCB specimen

)

Ebh3

⎡ ⎛ 2.36 (1 + ν ) ⎞2⎤ 2 (1 + ν ) ⎢ 3 − 2⎜ ⎟⎥ ⎝ 1 + 2.36 (1 + ν ) ⎠ ⎥⎦ 11 ⎢⎣

(5)

It is must be noted that Eq. (5) does not consider the effect of the adhesive layer, while tough adhesives are known to form large plastic zones ahead of the crack tip that are believed to increase significantly C and GI. This effect can be taken into account by computing an effective crack length ae from measured C values [22,23], a procedure that also avoids the need of monitoring crack propagation. Moreover, ae can be viewed as the sum of the true crack length a with a correction for the fracture process zone (FPZ) and elastic deformations of the adhesive and adherend. Therefore, comparison between the linear elastic ΔI with the steady-state propagation correction determined experimentally can provide useful insight into the adhesive fracture behaviour. Furthermore, an accurate elastic ΔI can be employed to obtain conservative GIC values that are more consistent with a LEFM perspective. In fact, as seen above, by including considerable adhesive layer elastic– plastic deformations, the perceived fracture energy may overestimate the true intrinsic work of fracture [18]. Such conservative GIC values can be computed from the first form of Eq. (4), which involves the P2(a þ ΔI)2 term. Finally, an accurate elastic ΔI can also be used to predict the initial load–displacement response of the DCB specimen, thereby providing an additional check of experimental results. Here the low modulus of structural adhesives and the bondline thickness close to 1 mm sometimes employed raise the question of the effect

A.B. de Morais / International Journal of Adhesion & Adhesives 61 (2015) 8–14

11

of the adhesive layer on ΔI. Williams and Hadavinia [43] developed cohesive zone models for DCB specimens that can be applied to derive a crack length correction for linear elastic adhesive layers

E + 11G

ΔI = h

Eha 1 + 6 3Ea h

(6)

In addition, by neglecting transverse shear, Williams and Hadavinia [44] also proposed

ΔI = h 4

E′ha 6E′a h

(7)

where E′¼E/(1  ν ) and E′a ¼ Ea/(1  ν designate the plane strain moduli of adherend and adhesive, respectively. The beam model developed below presents a new solution. 2

2 a )

3.2. The beam on elastic adhesive layer model We begin by analysing an infinitesimal element of a DCB specimen adherend at xZa (Figs. 1 and 4). The shear force V and the bending moment M act on the cross-sections, while the lower surface is subjected to the normal stress sa transmitted by the adhesive layer. This can therefore be viewed as a beam on elastic extensional foundation problem. The sa stresses do induce local effects on the adherend beam displacements and cross-section rotations. Those effects were modelled by Williams [26] through extensional and rotational foundations in the composite DCB specimen. However, the author showed in [45] that the rotational foundation should be discarded, because it would imply shear stresses that cannot exist in the DCB specimen. Moreover, the metal adherends of the DCB specimen considered in the present work are much stiffer than the polymeric adhesives. Therefore, it seems reasonable to assume that the vertical beam displacements w are practically identical to those experienced by the adhesive layer. Assuming small uniform εz E2w/ha (Fig. 1) strains in the adhesive layer,

σa =

(8)

⎛ d2w 1 dV ⎞ ⎟ + M = EI ⎜ 2 k ⎝ dx V AG dx ⎠

(9)

where G is the adherend shear modulus, I¼ bh3/12 the second moment of area, A ¼bh the cross-section area and kV ¼5/6 the rectangular cross-section transverse shear factor. We can now combine Eqs. (8) and (9) with the vertical force and moment equilibrium equations of the infinitesimal element depicted in Fig. 4 i.e.

dV dM = − bσa , =V dx dx

(10)

This leads to the differential Eq.

dx 4

− 2λ12

d2w + λ 24 w = 0 dx2

(11)

where

λ1 =

are extensional foundation parameters. The λ1/λ2 ratio dictates the form of the solution of Eq. (11). We consider hereafter the λ1/λ2 o 1 case, which applies to the majority of the metal adhesively bonded DCB specimens. The solution of Eq. (11) can be written as follows:

(

)

w = e−μ1x A1 cos μ2 x + A2 sin μ2 x

6Eas , λ2 = 5Ghha

4

24Eas Eh3ha

(12)

(13)

which includes the integration constants A1 and A2 and the stress decay and oscillation parameters

μ1 =

λ 22 + λ12 , μ2 = 2

λ 22 − λ12 2

(14)

We can now use Eq. (13) in Eqs. (8)–(10) to obtain

M (x) μ1x e = A1 ⎡⎣2μ1μ2 sin μ2 x + μ22 − μ12 cos μ2 x⎤⎦ EI + A2 ⎡⎣ μ22 − μ12 sin μ2 x − 2μ1μ2 cos μ2 x⎤⎦

(

(

)

)

V (x ) μ1x ⎡ ⎤ = A1 ⎣μ1 μ 22 + μ12 cos μ 2 x − μ 2 μ 22 + μ12 sin μ 2 x ⎦ e EI ⎡ ⎤ + A 2 ⎣μ 2 μ 22 + μ12 cos μ 2 x + μ1 μ 22 + μ12 sin μ 2 x ⎦

(

)

(

2Eas w ha

where Eas ¼ Ea(1  νa)/(1  νa 2νa2) is the uni-axial strain adhesive layer modulus, in view of the stress state observed in the above FEA. The effect of the adhesive layer on the specimen compliance is thereby taken into account and its relevance is assessed below. The present model adopts Timoshenko's beam theory and thus

d 4w

Fig. 4. Internal shear force and bending moment acting on an infinitesimal adherend element of a DCB specimen, together with the lower surface normal stress transmitted by the adhesive layer.

(

)

(

(15)

)

)

(16)

The boundary conditions V(a)¼P and M(a) ¼Pa (Fig. 1) enable evaluation of A1 and A2 and subsequently of the crack tip beam displacement and rotation

w (a) =

P

(

EI μ22 + μ12

)

⎛ 2μ1 ⎞ ⎜a + ⎟ ⎜ ⎟ μ22 + μ12 ⎠ ⎝

⎛ dw ⎞ P ⎟ ⎜ =− ⎝ dx ⎠x = a EI μ22 + μ12

(

)

(17)

⎛ 3μ12 − μ22 ⎞ ⎜2μ1a + ⎟ ⎜ ⎟ μ22 + μ12 ⎠ ⎝

(18)

Adding the displacement of the pre-cracked x ra region (Fig. 1), the load-point displacement can be obtained from

⎡ ⎛ dw ⎞ Pa3 ⎤ ⎥ ⎟ δ = 2 ⎢w (a) − a ⎜ + ⎝ dx ⎠x = a ⎢⎣ 3EI ⎥⎦

(19)

It is worth remarking that the contribution of transverse shear in the pre-cracked region, which is equal to 2Pa/kVAG, is not being neglected. In fact, (dw/da)x ¼ a includes (  P/kVAG), which is multiplied by (  2a). Therefore, the specimen compliance C ¼δ/P can be expressed as follows:

C=

3 ⎛ 2μ1 ⎞ 2μ1 6μ22 − 2μ12 2 ⎜ ⎟ + a+ 2 ⎜ ⎟ 3EI ⎝ 3EI μ 2 + μ 2 3 μ2 + μ12 ⎠

(

2

1

)

(20)

The second term of the right-hand side of Eq. (20) can be neglected, while in the first term one can readily identify ΔI of CBT

12

A.B. de Morais / International Journal of Adhesion & Adhesives 61 (2015) 8–14

Eqs. (3) and (4). Using Eqs. (12) and (14) we quickly arrive at

ΔI = h

E + 10G

Eha 6Eas h

(21)

that is very similar to ΔI obtained in [45] for the composite DCB specimen, which proved to be at least as accurate as Eq. (5). Actually, by adapting the model developed in [45], it can be shown that Eq. (21) also holds for the λ1/λ2 41 case. It is important to note that Eq. (21) takes into account the effects of both the adherend and adhesive layer, being similar to Eq. (6) derived in [43]. The (Eha/6Eash)1/2 term of Eq. (21) is also similar to Eq. (7) obtained in [44]. However, as seen above, the FEA does not support the plane strain hypothesis for both adherends and adhesive implicit in Eq. (7). Moreover, the first term of Eq. (21), associated with adherend transverse shear deformations, is not negligible. Such deformations are taken into account by Eq. (5), which is used below for comparison purposes because it provides a reference for assessing the adhesive layer effect and because it was adopted in [16]. By noting that E ¼2G(1 þ ν) and for the typical ν ¼0.3, Eq. (21) without the (Eha/6Eash)1/2 term predicts ΔI ¼0.51h, while according to Eq. (5) ΔI ¼ 0.66h. However, the (Eha/6Eash)1/2 term of Eq. (21) is dominant for the majority of the specimens analysed in Section 2, whereas Eq. (5) ignores adhesive layer properties. In both cases, it is clear that ΔI can be a very significant fraction of a, given the thick adherends often used to avoid adherend yielding.

Fig. 5. Compliances predicted by CBT with ΔI given by Eq. (5) (dashed lines) and (21) (continuous lines), and by FEA (triangular points) for Al adhesively bonded DCB specimens with ha (mm) ¼ 0.1 (grey lines and points) and 0.7 (black lines and points). C values were normalized by uncorrected beam theory predictions.

3.3. Comparison with 3D finite element analyses The 3D FEA of Section 2 are here used to assess the accuracy of CBT Eqs. (3) and (4) with ΔI given either by Eq. (5), derived by Williams [26], or by Eq. (21), obtained from the present beam analysis. Figs. 5 and 6 present the results for the specimen compliance C normalized by the simple uncorrected beam theory value, C ¼8a3/Ebh3, which is computed from Eq. (3) with ΔI ¼0. Therefore, where CBT predictions are concerned, the values plotted in Figs. 5 and 6 are actually (1 þ ΔI/a)3. It can be seen that the adhesive layer does influence the specimen compliance, especially for thicker bondlines and stiffer adherends. In fact, using ΔI from Eq. (5) introduces significant errors in Eq. (3) with ha ¼0.7 mm e.g. errors from  6% to  16% for a ¼70 mm. Instead, Eq. (21) allows quite accurate specimen compliance predictions, even for the small a ¼40 mm crack length (Figs. 5 and 6). Naturally, for larger crack lengths the effect of the adhesive layer becomes smaller, as does the value of ΔI relative to a. Figs. 7 and 8 plot GI values computed from Eq. (4) with ΔI of Eqs. (5) and (21), together with width-wise average values obtained from the FEA and VCCT. Again, values presented were normalized by the simple uncorrected beam theory value, GI ¼12a2/Eb2h3, computed from Eq. (4) with ΔI ¼0. This means CBT predictions plotted in Figs. 7 and 8 are (1 þ ΔI/a)2. It is clear that the above comments regarding specimen compliance predictions also apply to GI. This validates the adherend plane stress/in-plane restrained adhesive layer hypothesis of the present beam model, as well as the crack length correction ΔI (21) derived. Moreover, the latter is clearly better suited for the metal adhesively bonded DCB specimens than the one adapted from Williams [26] analysis of the composite DCB specimen. Nevertheless, as discussed in Section 3.1, ΔI (21) is only strictly applicable to the initial elastic loading stage of the DCB specimen. Computation of GIC values using CBT requires crack length corrections determined as stipulated in [16], for which ΔI (21) is a useful reference.

Fig. 6. Compliances predicted by CBT with ΔI given by Eq. (5) (dashed lines) and (21) (continuous lines), and by FEA (triangular points) for steel adhesively bonded DCB specimens with ha (mm) ¼ 0.1 (grey lines and points) and 0.7 (black lines and points). C values were normalized by uncorrected beam theory predictions.

Fig. 7. Strain energy release rates predicted by CBT with ΔI given by Eq. (5) (dashed lines) and (21) (continuous lines), and by FEA (triangular points) for Al adhesively bonded DCB specimens with ha (mm) ¼ 0.1 (grey lines and points) and 0.7 (black lines and points). GI values were normalized by uncorrected beam theory predictions.

A.B. de Morais / International Journal of Adhesion & Adhesives 61 (2015) 8–14

13

provides an approximate traction-separation law by combining cohesive zone modelling with parameter optimization in order to achieve best fits to experimental load–displacement curves [46]. However, the importance of the portion of the load–displacement curve up to crack initiation and the associated 3D effects pose considerable challenges. The alternative could be therefore other test methods ensuring that separations measured at the edges are representative across the specimen width.

References

Fig. 8. Strain energy release rates predicted by CBT with ΔI given by Eq. (5) (dashed lines) and (21) (continuous lines), and by FEA (triangular points) for steel adhesively bonded DCB specimens with ha (mm) ¼ 0.1 (grey lines and points) and 0.7 (black lines and points). GI values were normalized by uncorrected beam theory predictions.

4. Conclusions and outlook The present study involved three-dimensional (3D) finite element analyses (FEA) and the development of a beam on elastic adhesive layer model of the metal adhesively bonded double cantilever beam (DCB) specimen. The main conclusions were the following:


The significant adherend anticlastic deformations compromise














the accurate evaluation of the elastic part of the adhesive traction-separation law based on separations measured at the specimen edges. The adherend plane stress and adhesive uni-axial strain hypotheses are adequate for the typical specimen geometries, including the relatively small crack lengths sometimes employed. The new crack length correction derived from the beam model enabled accurate compliance and strain energy release rate predictions for ideal adhesive linear elastic behaviour. The correction takes into account the effects of the adhesive layer and of adherend transverse shear deformation, and both proved to be relevant. From a practical viewpoint, the present beam model: Provides an additional check of the accuracy of DCB test results through comparison of predicted and experimental initial specimen compliance values. Evidently, this requires previous measurements of the adhesive modulus and Poisson ratio. Alternatively, the present model can be used to back-calculate accurate estimates of the uni-axial strain adhesive elastic modulus from the initial specimen compliance. Gives further insight into the adhesive fracture behaviour through comparison of the elastic crack length correction with the one obtained from the monitoring of crack propagation. In fact, the large fracture process zones of tough adhesives are expected to lead to steady-state propagation crack length corrections significantly larger than the elastic corrections predicted. In turn both crack length corrections are likely to be similar for brittle adhesives. Can be used to obtain conservative fracture energy values that neglect the effects of large plastic zones, which may be inconsistent with linear elastic fracture mechanics (LEFM).

Although the present work was confined to the linear elastic domain, it indicates that new approaches are needed for evaluating the adhesive traction-separation law from the metal adhesively bonded DCB specimen. The so-called inverse method

[1] Adams RD. Structural adhesive joints in engineering. Berlin, Heidelberg: Springer; 2013. [2] da Silva LFM, Neves PJC, Adams RD, Spelt JK. Analytical models of adhesively bonded joints – Part I: literature survey. Int J Adhes Adhes 2009;29:319–30. [3] da Silva LFM, Neves PJC, Adams RD, Wang A, Spelt JK. Analytical models of adhesively bonded joints – Part II: comparative study. Int J Adhes Adhes 2009;29:331–41. [4] He X. A review of finite element analysis of adhesively bonded joints. Int J Adhes Adhes 2011;31:248–64. [5] Dugdale DS. Yielding steel sheets containing slits. J Mech Phys Solids 1960;8:100–4. [6] Barenblatt GI. Mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 1962;7:55–129. [7] Campilho RDSG, de Moura MFSF, Domingues JJMS. Modelling single and double-lap repairs on composite materials. Compos Sci Technol 2005;65:1948–58. [8] Campilho RDSG, Banea MD, Neto JABP, da Silva LFM. Modelling of single-lap joints using cohesive zones models: effect of the cohesive parameters on the output of the simulations. J Adhes 2012;88:513–33. [9] Campilho RDSG, Banea MD, Neto JABP, da Silva LFM. Modelling adhesive joints with cohesive zone models: effect of the cohesive law shape of the adhesive layer. Int J Adhes Adhes 2013;44:48–56. [10] Hell S, Weißgraeber P, Felger J, Becker W. A coupled stress and energy criterion for the assessment of crack initiation in single lap joints: a numerical approach. Eng Fract Mech 2014;117:112–26. [11] May M, Voß H, Hiermaier S. Predictive modeling of damage and failure in adhesively bonded metallic joints using cohesive interface elements. Int J Adhes Adhes 2014;49:7–17. [12] Alfano M, Furgiuele F, Leonardi A, Maletta C, Paulino GH. Mode I fracture of adhesive joints using tailored cohesive zone models. Int J Fract 2009;157:193– 204. [13] Banea MD, da Silva LFM, Campilho RDSG. Temperature dependence of the fracture toughness of adhesively bonded joints. J Adhes Sci Technol 2010;24:2011–26. [14] Chaves FJP, da Silva LFM, de Moura MFSF, Dillard DA, Esteves VHC. Fracture mechanics tests in adhesively bonded joints: a literature review. J Adhes 2014;90:955–92. [15] ASTM D3433-99. Standard test method for fracture strength in cleavage of adhesives in bonded metal joints. West Conshohocken, PA: ASTM International; 2012. [16] ISO 25217:2009. Adhesives – determination of the mode 1 adhesive fracture energy of structural adhesive joints using double cantilever beam and tapered double cantilever beam specimens. Geneva, Switzerland: International Organization for Standardization. [17] Andersson T, Stigh U. The stress-elongation relation for an adhesive layer loaded in peel using equilibrium of energetic forces. Int J Solids Struct 2004;41:413–34. [18] Pardoen T, Ferracin T, Landis CM, Delannay F. Constraint effects in adhesive joint fracture. J Mech Phys Solids 2005;53:1951–83. [19] Carlberger T, Stigh U. Influence of layer thickness on cohesive properties of an epoxy-based adhesive – an experimental study. J Adhes 2010;86:816–35. [20] Ji G, Ouyang Z, Li G, Ibekwe S, Pang S-S. Effects of adhesive thickness on global and local mode-I interfacial fracture of bonded joints. Int J Solids Struct 2010;47:2445–58. [21] Han J, Siegmund T. Cohesive zone model characterization of the adhesive Hysol EA-9394. J Adhes Sci Technol 2012;26:1033–52. [22] de Moura MFSF, Gonçalves JPM, Magalhães AG. A straightforward method to obtain the cohesive laws of bonded joints under mode I loading. Int J Adhes Adhes 2012;39:54–9. [23] Dias GF, de Moura MFSF, Chousal JAG, Xavier J. Cohesive laws of composite bonded joints under mode I loading. Compos Struct 2013;106:646–52. [24] Valoroso N, Sessa S, Lepore M, Cricrì G. Identification of mode-I cohesive parameters for bonded interfaces based on DCB test. Eng Fract Mech 2013;104:56–79. [25] Campilho RDSG, Moura DC, Banea MD, da Silva LFM. Adherend thickness effect on the tensile fracture toughness of a structural adhesive using an optical data acquisition method. Int J Adhes Adhes 2014;53:15–22. [26] Williams JG. End corrections for orthotropic DCB specimens. Compos Sci Technol 1989;35:367–76.

14

A.B. de Morais / International Journal of Adhesion & Adhesives 61 (2015) 8–14

[27] ISO 15024:2001. Fibre-reinforced plastic composites – determination of mode I interlaminar fracture toughness, GIC for unidirectionally reinforced materials. Geneva, Switzerland: International Organization for Standardization. [28] Imanaka M, Takagishi M, Nakamura N, Kimoto M. R-curve characteristics of adhesives modified with high rubber content under mode I loading. Int J Adhes Adhes 2010;30:478–84. [29] Budzik MK, Jumel J, Shanahan MER. Formation and growth of the crack in bonded joints under mode I fracture: substrate deflection at crack vicinity. J Adhes 2011;87:967–88. [30] Jumel J, Budzik MK, Shanahan MER. Beam on elastic foundation with anticlastic curvature: application to analysis of mode I fracture tests. Eng Fract Mech 2011;78:3253–69. [31] Khoshravan M, Mehrabadi FA. Fracture analysis in adhesive composite material/aluminum joints under mode-I loading; experimental and numerical approaches. Int J Adhes Adhes 2012;39:8–14. [32] Jiang Z, Wan S, Zhong Z, Li M, Shen K. Determination of mode-I fracture toughness and non-uniformity for GFRP double cantilever beam specimens with an adhesive layer. Eng Fract Mech 2014;128:139–56. [33] Davidson BD. An analytical investigation of delamination front curvature in double cantilever beam specimens. J Compos Mater 1990;24:1124–37. [34] de Morais AB. Double cantilever beam testing of multidirectional laminates. Compos Part A 2003;34:1135–42. [35] de Morais AB, de Moura MF, Gonçalves JPM, Camanho PP. Analysis of crack propagation in double cantilever beam tests of multidirectional laminates. Mech Mater 2003;35:641–52. [36] de Morais AB, de Moura MFSF. Assessment of initiation criteria used in interlaminar fracture tests of composites. Eng Fract Mech 2005;72:2615–27.

[37] Sorensen BF, Jacobsen TK. Large-scale bridging in composites: R-curves and bridging laws. Compos Part A: Appl Sci Manuf 1998;29:1443–51. [38] Sorensen BF, Jacobsen TK. Determination of cohesive laws by the J-integral approach. Eng Fract Mech 2003;70:1841–58. [39] Krueger R. The virtual crack-closure technique: history, approach and applications. 2002 ICASE Report No. 2002-10. NASA/CR-2002-211628. [40] Blackman BRK, Kinloch AJ, Paraschi M, Teo WS. Measuring the mode I adhesive fracture energy, GIC, of structural adhesive joints: the results of an international round-robin. Int J Adhes Adhes 2003;23:293–305. [41] Álvarez D, Blackman BRK, Guild FJ, Kinloch AJ. Mode I fracture in adhesivelybonded joints: a mesh-size independent modelling approach using cohesive elements. Eng Fract Mech 2014;115:73–95. [42] Mostovoy S, Crosley PB, Ripling EJ. Use of crack-line-loaded specimens for measuring plane strain fracture toughness. J Mater 1967;2:661–81. [43] Williams JG, Hadavinia H. Analytical solutions for cohesive zone models. J Mech Phys Solids 2002;50:809–25. [44] Williams JG, Hadavinia H. Elastic and elastic–plastic correction factors for DCB specimens. In: Proceedings of the 14th European conference on fracture (ECF 14) – fracture mechanics beyond 2000, vol. 3. Krakow, Poland: EMAS Publishing; 2002. p. 573–92. [45] de Morais AB. Mode I cohesive zone model for delamination in composite beams. Eng Fract Mech 2013;109:236–45. [46] Silva FGA, Xavier J, Pereira FAM, Morais JJL, Dourado N, Moura MFSF. Determination of cohesive laws in wood bonded joints under mode I loading using the DCB test. Holzforschung 2013;67:913–22.