Characterisation of mode-I fracture resistance of adhesive layers with imperfections

In this work, a novel procedure to evaluate the inﬂuence of imperfections at the interface (such as voids and interfacial failure) on the fracture resistance of adhesive joints in mode-I debonding is proposed, based on an image-processing analysis of the crack surface. Its application to the characterisation of fracture resistance of aluminium DCB specimens bonded with an epoxy adhesive leads to a more accurate evaluation of the ‘eﬀective’ fracture resistance by taking into account the distribution of imperfections along the interface, therefore also conﬁrming that the typical oscillations and drops in the load-displacement curve can be attributed to the imperfections.

joints, in which the adhesive connection between two (or more) structural com- ponents is lost, is known as debonding or delamination.In order to assess the 7 resistance of an adhesive to debonding, it is necessary to perform tailored ex-8 periments on adhesive joints, from which relevant material parameters of the 9 adhesive can be determined for design purposes.

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Journal Pre-proof In this work, the attention will be focused on rate-dependent mode-I (nor- moisture absorption [3,4], substrate surface treatment [2,3], pressure applied

36
In particular, Heide-Jørgensen and Budzik [10,11] performed DCB experi-37 ments on joints with patterned interfaces [10] or with a a single interface dis-38 continuity [11] and developed an analytical model that can successfully describe Journal Pre-proof the process-zone size, the force-displacement curve is smooth and resembles a stable crack growth, while for larger voids, an oscillating behaviour related to the crack crossing sharp transitions, i.e. strong/weak adhesion zones, can be noticed.They correctly recognised that the effective fracture energy in that case can be misinterpreted as a property of the 'bulk' (without the 'voids') bondline.
Thus, they concluded that if the properties of a perfect/homogeneous material are known in advance, the difference between the 'measured' and 'perfect' properties could be related to size and distribution of voids inside the bondline.
Budzik et al. [12] also investigated mode-I fracture behaviour of widthvarying DCB specimens (arms), but with such specimens it is not possible to isolate the influence of the changing adhesive width on the load-bearing capacity of the adhesive joint as the width of the arms also changes simultaneously.
Moreover, the specimens studied by Budzik et al. in Ref. [12] had a regular change of width (defined by a function) along the specimen.
Although the last cited articles (Refs.[7][8][9][10][11][12]) shed some light on the effect of defects, a sistematic procedure to detect the presence and extract the distribution of interface defects on broken specimens and a rigorous quantitative evaluation of their effects on the structural response have not yet been provided and are the main research gaps addressed in this article.To this end, a postmortem analysis of broken DCB specimens, previously tested by the present authors (see Ref. [13]), is presented.It reveals a non-negligible part of the interface characterised by either patches of interfacial failure or by the presence of voids within the adhesive.In fact, the aim of the work presented in this article is not to investigate the reasons why defects can be present on the interface, but rather to study the relation between these defects and the characteristic bumps in the load-displacement curves that are found by many authors testing similar types of adhesives, which also result in oscillating R-curves [14][15][16][17].A second aim is to propose a multi-parameter material model that could take these effects into account, so that a more accurate determination of the fracture resistance of the adhesive is obtained.
To achieve the above aim, a novel method is presented, which is based on J o u r n a l P r e -p r o o f Journal Pre-proof some simple-to-implement image processing followed by a a procedure developed to extract the distribution and nature of defects so that these can be translated in a reduction of the effective width of the adhesive due to voids/interfacial failure.A multi-parameter material model that takes these effects into account is then used, so that a more accurate determination of the fracture resistance of the adhesive is obtained.Some interplay between the defects of the interface and its rate dependence can take place [11].However, as stated by Blackman et al. [18], the effects of material rate dependence could be negligible under quasi-static conditions and are not expected unless the strain rate varies across few orders of magnitude.
Therefore, also with a view to focussing on the original aspects of this present contribution, in this paper rate dependence will not be specifically considered and the focus will be on the experiments conducted at a single speed, namely at 0.1 mm/min.

Experimental tests and results
Experimental tests have been carried out on DCB specimens made of aluminium Al 6082-T6 bonded with Araldite R 2015 structural adhesive.The preparation of the specimens was done according to BS ISO 25217:2009 [19] and the technical data sheet provided by the manufacturer [20].The average value of the adhesive thickness for the 4 considered specimens was 0.15 mm with a coefficient of variation equal to 17%, which is less than the limit value reported in the standard (20%).Geometrical properties of the specimens are J o u r n a l P r e -p r o o f Journal Pre-proof reported in Figure 1 and Table 1.Young's modulus and Poisson's ratio of the aluminium read E = 70700 MPa and ν = 1/3.Loading blocks were used to attach the specimens to the tensile testing machine.Four debonding tests have been performed at cross-head displacement speed of 0.1 mm/min.The tests were performed at room temperature using a 30 kN electromechanical Instron testing machine.More details about the experimental set up can be found in [13].The individual load-displacement curves for 4 tests together with the corresponding average are given in Figure 2. Notice that, in order to produce a single average curve, the raw data from the test were postprocessed and an average computed for each cross-head displacement increment of 0.2 mm.This is why

J o u r n a l P r e -p r o o f
Journal Pre-proof load-displacement data for values of the cross-head displacement less than 0.2 mm is not given in the plot.As shown in [13], such load-displacement data result in R-curves where similar experimental scatter can be noticed.As reported in [13] and shown in Figure 3 for a representative specimen, post-mortem analysis of the broken specimens revealed several defects in the adhesive layer.These are either areas where interfacial failure has occurred, or areas with complete lack of adhesion, which we will refer to as 'voids' for simplicity.Although investigating the causes for these imperfections is outside the scope of this paper, the following observations can be made.Voids can be created if the adhesive layers on two adjacent plates do not completely connect after pressing the plates together due to insufficient thickness of the adhesive or trapped air.On the other hand, it is also possible that curing of the adhesive generates the voids due to shrinkage of the material.
After analysing the crack surfaces of all tests, it can be concluded that the voids are more frequent and larger than the interfacial-failure zones.The only explanation for occasional interfacial failure could be some grease or dirt that remained on the surface after cleaning, although such problems have not

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Journal Pre-proof been noticed during the preparation of the specimens.As explained in the introduction, voids and/or interfacial failure in the adhesive layer must have an influence on the computed fracture resistance, which is investigated in detail in the following section.

Evaluation of the effective fracture resistance by accounting for interface defects
In the following subsections, the difference between the effective and apparent fracture resistance will be defined depending on whether the defects on the interface are taken into account or not.

Apparent vs. effective fracture resistance
As discussed in the previous section, there are two principal types of defects at the interface, namely (a) interfacial failure and (b) voids.In general, it is reasonable to assume that both of them reduce the fracture resistance and therefore cause local bumps in the load-displacement curves.In fact, by assuming that the DCB arms act as Euler-Bernoulli beams clamped at the crack tip, the critical energy release rate can be approximated as [21] G where superscript E refers to Euler-Bernoulli beam theory, F is the applied load, v is the cross-head displacement and EI is the bending stiffness of a single DCB arm.From Equation (1), it follows that a reduction in the fracture resistance G c leads to a drop of the force F for the same values of the remaining parameters.As shown in [21], formula (1) for G E c , in which only the measured force-displacement data is required, gives a very accurate approximation of both G c and the work of separation Ω (area under the traction-separation law (TSL) of the interface) for a wide range of cases, i.e.
If the presence of defects at the interface is neglected and therefore, if it is assumed that only cohesive failure occurs over an interface that is fully covered

J o u r n a l P r e -p r o o f
Journal Pre-proof with adhesive, the bumps in the load-displacement curve could be effectively attributed to a local change of the material properties of the adhesive, i.e. to a local, variable reduction of its fracture resistance.As a consequence, bumpy R-curves, in which the fracture resistance can significantly change during crack propagation, are obtained.However, the fracture resistance computed in this way is not an inherent material property of the adhesive, because it is affected by the presence, distribution and nature of defects at the interface.Therefore, as anticipated in the Introduction, this will be referred to as the apparent fracture resistance and denoted by G app c .Instead, the effective fracture resistance of the adhesive, G eff c , will be introduced by accounting for the reduced size of the interface area (due to the presence of defects).
For DCB specimens, whose width b is constant along the length of the specimen, we can introduce a coordinate x along the direction of crack propagation.
To each coordinate x there corresponds a straight line orthogonal to the x axis, of length b, and on this line we can identify the portion of the line affected by some defect, in the form of either a void or interfacial failure.In this way, an effective width, denoted by b eff , whose size depends on the width of the defects at any given point along the length of the DCB, so that b eff = b eff (x), can be introduced.
Indicating with Π the total potential energy, from the derivation of Griffith's fracture propagation criterion (see e.g.[21]) it is easy to verify that it can be written in the usual way also when b is not constant for each value of the current crack length a.Therefore, if the constant b is replaced with the effective width b eff , function of x, and assuming that the crack front is still orthogonal to the longitudinal direction of the specimen and therefore is fully defined for each value of a, then G c can be replaced with the effective fracture resistance G eff c .If instead it is assumed that b is constant and equal to the width of the specimen, meaning that the interface defects are neglected, then G c should be replaced with the apparent fracture

J o u r n a l P r e -p r o o f
Journal Pre-proof resistance G app c .Therefore, by setting x = a, it can be written The effective fracture resistance G eff c , by definition, should be an inherent property of the material but, as such, it can still be characterised by some scatter, both from specimen to specimen in the same test conditions, and within the interface of a single specimen.Therefore, G eff c will in general be a function of The effective fracture resistance can be defined from Equation (3) as where ζ(a) = b eff (a)/b is a factor with limit values 0 and 1 corresponding to no adhesive and no voids over the width, respectively.Therefore, in order to determine G eff c we need to determine the apparent fracture resistance (for which standards and well-established procedures exist, e.g.Ref. [19]) and the effective respectively.On the other hand, the portion of the width with voids will be denoted by b v .In addition, we will assume that the fracture resistance in the case of interfacial failure, G i c , is smaller than that in the case of cohesive failure, G eff c .By introducing a coefficient α, this can be written as Note that the value of coefficient α could change along the interface because the lack of adhesion that triggers interfacial failure could be caused by variable type and degree of local imperfections on the adherend's surface.However, not only is it impossible to determine the distribution α(a) a-priori, but it also cannot be done in an accurate manner by analysing the crack surfaces of the broken specimens, which will be discussed in more detail in Section 3.3.

J o u r n a l P r e -p r o o f
Journal Pre-proof Therefore, for the sake of simplicity, we will assume that α is a constant.By separating the contributions of the inetfacial and cohesive failure in Equation (3), we obtain whereby it follows that Therefore, in order to determine b eff , we will need to measure the amount of voids and interfacial failure on each plate over the width for any position along the interface.The procedure for extracting these data from the broken specimens is proposed in the next subsection.

Defect-data extraction procedure by image processing of fracture surfaces
The proposed procedure for identifying and quantifying defects at the interface is based on the photographs of the broken specimens' crack surfaces.
Interfacial failure completely removes the adhesive from one part of the specimen, leaving the aluminium surface open.When photographed under the light, this surface will shine and clearly distinguish itself from the matt surface of the adhesive where cohesive failure has occurred.Likewise, the inner surface of voids, unlike the surface of the broken adhesive where cohesive failure occurred, has a glossy finish that also shines when photographed under the light.Therefore, the same technique can be applied to identify both interfacial failure and voids.
The proposed procedure for extracting the defect data from the photographs is described by the three columns in Figure 3 for a representative specimen.In the first column (Figure 3  Mathematica and the code is available for download from the link provided in the "Supplementary data" section. By comparing the top and the bottom surfaces in Figure 3, it can be noticed that in the first (upper) part of the interface (for x between 40 and approximately 90 mm), interfacial failure is dominant.For the rest of the interface surface (for x > 90 mm), it can be noticed that the voids are mostly dominant because the white patterns are nearly symmetrical.However, because of the aforementioned imprecisions, a non-negligible amount of artificial interfacial failure is obtained around the voids.

J o u r n a l P r e -p r o o f
Journal Pre-proof In the next subsection we will first present a procedure for determining the apparent fracture resistance and then, using the extracted defect-distribution data, we will compute the effective fracture resistance and determine an optimal value of coefficient α.

Approximate calculation of the local values of the effective fracture resistance
In order to compute the apparent fracture resistance we will use a datareduction scheme based on the equivalent crack length, which does not require the experimental measurement of the crack length.The equivalent crack length is the value of the crack length that, using the measured values of the applied load F and displacement v, satisfies the simple beam deflection formula for a cantilever beam.In the case of Euler-Bernoulli beam theory, the equivalent crack length will be denoted by a E eq and is defined by the following equivalent relationships [21]: By doing this, we are assuming that DCB arms are clamped at an 'equivalent' crack tip, which does not correspond to the actual crack tip.In fact, because at the actual crack tip both relative separation and rotations of the arms can occur before failure, for the same values of F and v, the equivalent crack length will be always larger than the actual one.Assuming that the equivalent crack length is a function of the actual one, i.e. a E eq = a E eq (a), for a case with prescribed crosshead displacement v, the applied load can be written as F = F (v, a E eq (a)).Note that function a E eq (a) cannot be determined without measuring the actual crack length.Therefore, in the present work we will only assume that this functional dependency exist, but it is not known.Since using Equation ( 7) 2 we have: J o u r n a l P r e -p r o o f

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It was shown in Ref. [21] that da E eq /da is extremely close to unity for a wide range of cases in a DCB test.In other words, although the difference ∆a E = a − a E eq (a) is in general not negligible, it can be considered constant across the crack length.Therefore, replacing v from Equation ( 7) 1 , we can write: and the equivalent crack can be approximated as Assuming that ∆a E can be determined (or at least approximated), the effective fracture resistance can be computed as Substituting this result in Equation ( 4) gives the same result as combining Equations ( 1) and ( 7) 1 , which confirms that G app c = G E c and the apparent fracture resistance can be therefore computed as Plotting function (12) would give the effective R-curve, which is expected to be smoother than the apparent one, because the bumps in the load-displacement curve can be compensated by taking into account the reduced effective width of the adhesive on the interface.As ∆a E in Equation ( 11) is unknown, obtaining the effective R-curve is not straight forward.However, a relatively simple method for estimating the value of ∆a E is proposed below.
Using load-displacement data of the representative specimen, produced for each increment of displacement ∆v = 0.01 mm, the equivalent crack length a E eq and the corresponding value of G E c have been computed for each available pair (F, v) using Equations ( 7) and ( 1), respectively.The first part of the loaddisplacement plot and the equivalent apparent R-curve (G E c -a E eq plot) are given in Figure 4.Note that G E c represents the apparent fracture resistance only when the crack is propagating, i.e. when ȧ > 0. However, because the experimental

J o u r n a l P r e -p r o o f
Journal Pre-proof crack-length measurements are not available, determining whether the crack propagates or not has to be done in an indirect way.Virtual experiments with perfect interface (such as those performed in [21]) show that crack propagation will start when a plateau of the equivalent apparent R-curve plotted for all available (F, v) pairs is reached.Before that, G E c computed for (F, v) pairs with ȧ = 0 will increase very rapidly and obviously does not represent the apparent fracture resistance.Because in Figure 4(b) the plateau of the apparent R-curve is not a straight line (due to defects on the interface), the start of crack propagation has been assumed at the point with the smallest value of a E eq where the incremental change of G E c with respect to a E eq changes sign (i.e.dG E c /da E eq ≈ 0).Note that the cross-head displacement of that point is larger than that corresponding to F max , which is shown in Figure 4(a).This is due to the development of the damage-process zone in front of the crack tip before crack propagation [22].
Thus, for the representative specimen, the first point on R-curve plateau corresponds to a E eq (a 0 ) = 50.24mm, which from Equation (11) gives ∆a E = 10.24 mm.Note that ∆a E is not the length of the damage-process zone (DPZ) because in the concept of equivalent crack length, besides damage softening,

J o u r n a l P r e -p r o o f
Journal Pre-proof linear elastic behaviour (which also results in relative displacement and rotations of DCB arms at the crack tip) is neglected in front of the crack tip.Therefore, the DPZ should be certainly shorter than ∆a E .
Using this result it is possible to compute the apparent and the effective R-curve with approximated crack length a on the horizontal axis.In Figure 5,  It can be noticed from Figure 5 that, at some locations, no value of coefficient α can capture the shape of the normalised apparent R-curve.This indicates that either the void-extraction procedure is not sufficiently accurate or there are effects other than defects in the adhesive layer that can alter fracture resistance of the adhesive.It is expected that, by employing a more reliable method for obtaining the defect-distribution data, such as 3D [2,23] or X-ray [6] scans, an even better correlation between ζ(a) and normalised G app c (a) would be obtained.
The normalised effective fracture resistance could be computed by dividing the normalised apparent fracture resistance by ζ(a) (both plotted in Figure 5).
However, this is not done here because these two plots are still not close enough, which does not reduce, but instead amplifies the fluctuations in the normalised effective R-curve.
Instead of adding the normalised effective R-curve to Figure 5, which would make the figure less clear, the results are given in Table 2.It can be noticed that, independently from the value of α, the coefficient of variation (CV) for the normalised effective fracture resistance is higher than for the normalised apparent fracture resistance, which, considering our hypothesis that G eff c (unlike ) is an inherent material property, cannot be an acceptable result.It is obvious that a representative effective R-curve (G eff c (a)) cannot be obtained in this way due to all inaccuracies mentioned earlier.Moreover, for α = 1, the average of G eff c (a) is less than max(G app c (a)), which should not be the case if

J o u r n a l P r e -p r o o f
Journal Pre-proof   we assume that the average value of G eff c (as an inherent material property) is representative for the entire interface.
Therefore, b eff cannot be obtained without making any a-priori assumptions on the relationship between G app c and G eff c , and/or the value of coefficient α.
As for the latter, from now on we will assume that α = 0.5 because the results of Figure 5 show that this assumption is reasonable and sufficiently accurate.
Moreover, as it will be shown in Section 3.4, introducing this assumption will significantly simplify the expression for b eff .Also, because we previously concluded that the effective R-curve cannot be accurately determined using the available data, we will now focus our attention on obtaining the average value of G eff c that will be representative for the entire interface or, more generally, for a specific type of adhesive.

Calculation of a weighted average of the effective fracture resistance
Assuming that α = 0.5, Equation ( 6) can be rewritten as The amount of adhesive on each plate can be then defined as for the top and bottom plate, respectively.Thus, Equation ( 14 where G app c.i and ζ m.i are the apparent fracture resistance and mean defect distribution determined at point i (i = 0, 1, 2, ..., n) that corresponds to position a = a 0 + i in mm (because in our case adjacent points are 1 mm apart).Although through image-processing ζ m.i can obtained for the entire interface (i.e.n = 197), as explained earlier, the number of points n for which G app c.i can be computed will typically be less than 197 because of the final dynamic collapse.
This small discrepancy will be here neglected and, for the sake of simplicity, different values of n will be used in Equations ( 21) and (22).Our preliminary analyses have shown that this simplification has a negligible influence on the final values of ζ m .
For the representative specimen, ζ m = 0.79 obtained from Equation (22) with n = 197 and gives the average effective adhesive width b eff = 20.06mm.
This means that for this specimen, 21% of the interface has defects.From (21) it follows that Ḡapp c = 0.282 N/mm, with n = 180.It should be emphasised that in order to obtain this result, the actual crack length a has been estimated using the approach presented in the previous subsection for each measured (F, v) pair.
For the same dataset, the apparent fracture resistance has been computed using the formula for G E c (Equation ( 1 In order to account for shear deformability of the DCB arms, instead of using Equation ( 1), G app c can be computed using the Enhanced-Simple-Beam-Theory (ESBT) data-reduction scheme [24] as where µA s is the shear stiffness of DCB arms computed as the product of the shear modulus µ, cross-sectional area A as the shear-correction coefficient k s = 5/6, α * = µA s /EI, while the equivalent crack length is defined as with a E eq defined in (7).For the representative specimen, values of G app c computed using Equations ( 1) and ( 23) are essentially the same.This is in line with the results presented in Ref. [24], where it was confirmed that taking into account shear strains has a very small effect on the computed fracture resistance using the concept of equivalent crack length, even for arms that are more shear-deformable than the aluminium ones used in the present work (e.g.arms made of composite materials).For an even more sophisticated definition of the equivalent crack length see Ref. [25].

J o u r n a l P r e -p r o o f
Journal Pre-proof  Table 3: Apparent and effective values of parameters defining the bi-linear TSL for the case of the representative specimen.The apparent parameters were identified using DCB PAR software [28].
J o u r n a l P r e -p r o o f Journal Pre-proof data ζ m is not expected for a number of reasons.A very small discrepancy can be expected due to the assumption that ∆a E = a − a E eq (a) is constant, which is nearly but not strictly true.Another reason for expecting some discrepancy is that the 2D defect-distribution data is lumped on a 1D function ζ m , in which the actual 2D distribution of defects and the distinction between voids and interfacial failure are both averaged out.Furthermore, even if we wanted to employ the full data on the 2D distribution of defects and on their nature, the 1D beam model used would not capture its effects.
On the other hand, if all potential sources of error were excluded, there would still be another reason for the above-mentioned discrepancy in Figure 5.This is investigated in this section, by analysing some suitably constructed 'virtual experiments'.More precisely, using a numerical model in which a variable width of the adhesive layer along the interface is given as an input, the relationship between the apparent and effective fracture resistance defined in Equation ( 4  Based on these observations it can be concluded that obtaining the effective fracture resistance cannot be done accurately using expression (4) even using perfectly accurate data from virtual experiments, in which all the issues mentioned at the start of this section do not arise.

Conclusions
In this work, the influence of imperfections in the adhesive layer (consisting of voids and interfacial failure) on the fracture resistance of adhesive joints in

Crack length a 0 DCB 1
resistance, rate-dependent behaviour, cohesive-zone models Nomenclature α Fraction of the contribution of the intefacial-failure fracture resistance with respect to the cohesive-failure fracture resistance α * Ratio between the shear and bending stiffness of a single DCB arm Average value of G app c determined at point i Π Total potential energy σ max Peak interface traction in the bi-linear TSL ζ Defect-distribution coefficient ζ b Defect-distribution coefficient on the bottom face ζ m Mean value of the defect-distribution coefficient ζ t Defect-distribution coefficient on the top face ζ m.i Mean value of the defect-distribution coefficient determined at point i A Cross-sectional area of a single DCB arm a Initial crack length A s Shear-corrected cross-sectional area of a single DCB arm J o u r n a l P r e -p r o o f Journal Pre-proof a eq Equivalent crack length based on Timoshenko beam theory according to ESBT a E eq Equivalent crack length based on Euler-Bernoulli beam theory a max Maximal value of the crack length b Width of a DCB specimen b i Total area of interfacial failure on both plates per unit of length b db area of defects per unit of length on the bottom face b dt area of defects per unit of length on the top face b dv area of voids per unit of length b ef f Effective width of the adhesive layer b ib area of interfacial failure per unit of length on the bottom face b it area of interfacial failure per unit of length on the top face c Height of the loading block for a DCB specimen d Thickness of a single DCB arm E Young's modulus of the bulk material F Applied load F max Maximal value of the applied load G c Critical energy release rate G app J o u r n a l P r e -p r o o f Journal Pre-proof i Counter for the points along the interface in which ζ m.i and G app c.i are determined k Stiffness of the linear-elastic part of the bi-linear TSL k s Shear-correction coefficient L Length of a DCB specimen n Total number of points along the interface in which ζ m.i and G app c.i are determined t Bondline thickness v Cross-head displacement of a DCB specimen x Co-ordinate along the direction of crack propagation Adhesives are nowadays applied to join structural components in a variety of 2 industries, such as automotive, aerospace and civil engineering.Thus, a proper 3 design of adhesively bonded structures must also include an assessment of their 4 resistance to failure.The most common and critical failure mode of adhesive 5

11 mal)
debonding and, in particular, on some challenges posed by the presence 12 of defects within the adhesively bonded interface and how to address them to 13 obtain a more accurate evaluation of the fracture resistance.Defects on the 14 interface, such as voids or areas of poor adhesion (including interfacial failure), 15 can significantly decrease the bearing capacity of the adhesive joint locally and, 16 in turn, alter the computed values of the fracture resistance.Sometimes they 17 may also lead to unstable crack propagation on part of the crack path.18 Nucleation of voids in adhesive layers and their effects on structural be-19 haviour of adhesive joints have been studied by several authors in the last 50 20 years.Possible causes investigated include trapped air within the adhesive [1, 2], 21

23 shinkage [ 5 ] 25 to 27 Smith [ 7 ]
. The effects include reduction in durability and in T-peel and hon-24 eycomb peel strengths[4] and could depend on the adhesive thickness according Lißner et al.[6] who noticed that the thinner the adhesive, the larger the 26 volume fraction of voids.By using edited photographs of the fracture surfaces, created an overlay of the upper and lower adhesive regions in order to 28 identify the failure mechanism (cohesive or interfacial) at the interface of DCB 29 specimens, determine the percentage of interface defects and correlate them to 30 the measured fracture resistance using nonlinear regression, but without incor-31 porating the defects in a numerical model of the DCBs.32 Patterned interfaces, in which regular patterns of voids or weak areas are 33 intentionally introduced in the bondline, can be used to investigate the influ-34 ence of heterogeneities (imperfections) at the interface on the stability of crack 35

39a
sudden drop of force, i.e. reduction in the joint's load carrying capacity as 40 the crack approaches the void.When the size of voids is much smaller than 41 J o u r n a l P r e -p r o o f

Figure 1 :
Figure 1: Geometry of DCBs tested; dimensions are reported in Table1.

Figure 2 :
Figure 2: Experimental individual load-displacement curves for the 4 tested specimens and the corresponding average curve.
width of the adhesive (i.e.coefficient ζ).In this work, we will in particular focus on the latter.It should be noted that ζ(a) in Equation (4) takes into account both interfacial failure and voids.The portion of the width affected by interfacial failure, b i must take into account interfacial failure on each of the plates, so that b i = b it + b ib , where indices t and b refer to the top and bottom plate, (a)), the surfaces of the broken adhesive on the top and bottom plate are shown.These images have been obtained by cropping out from the original photos everything except the interface surface and then transforming this quadrilateral area to a perfect rectangle with length-to-width aspect ratio 197:25.4.The upper edge of this rectangle corresponds to the initial J o u r n a l P r e -p r o o f Journal Pre-proof crack tip, whereas the lower edge corresponds to the free end of the DCB.The positions of the initial crack tip and the free end of the specimen correspond to a 0 = 40 mm and a max = L − b/2 = 237 mm, respectively, while the width of the interface is b = 25.4 mm.Note that the value of a max mm has been rounded to integer for the sake of simplicity, as the resulting error is only 0.13%.

Figure 3 :
Figure 3: Example of extraction of defect-distribution data on a representative specimen: the original (a) and the processed photos (b) of the broken adhesive surfaces for both plates, the obtained 2D defect-distribution (c) including voids (black areas) and interfacial failure on the top (red areas) and bottom surface (blue areas) and the longitudinal distribution of the defects (d).

Figure 4 :
Figure 4: Load-displacement plot (a) and the equivalent R-curve (b) for the representative specimen.
comparison between the normalised apparent R-curve and the distribution of the normalised effective width of the adhesive, ζ(a) along the interface is shown for the entire range of possible values of coefficient α between its limits 0 and 1.It can be noticed that the extracted defect data resembles the shape of the normalised apparent R-curve fairly well, but it is strongly influenced by the value of α.

Figure 5 :
Figure 5: Distribution of the normalised apparent fracture resistance (G E c (a)/ max(G E c )) and the mean defect-distribution data (ζm(a) = b eff (a)/b) along the interface, with the latter additionally filtered using a 9-point average, for the representative specimen.
G app c (a) = G E c (a)/ max(G E c )) and normalised effective fracture resistance (computed as G eff c (a) = G app c (a)/ζ(a)) for different values of coefficient α.

J o u r
n a l P r e -p r o o f Journal Pre-proof where for a given a, b dt = b v +b it and b db = b v +b ib represent the total amount of defects (voids and interfacial failure) on the top and bottom plate, respectively.
) finally becomes b eff (a) = ζ t (a) + ζ b (a) 2 b = ζ m (a)b, (16) where ζ m (a) is the distribution of the mean amount of adhesive on both plates at position a.Note that in our defect-data extraction procedure ζ t (a) and ζ b (a) correspond to the amount of white pixels (see Figure 3(b)) on the top and bottom plate, respectively, which makes them easy to obtain, but as such, they contain no information about the type of the defect (void or interfacial failure).However, obtaining the effective adhesive width from Equation (16) without this information is possible only because we previously assumed that α = 0.5.Taking the average of both sides of Equation (4) over the entire specimen c (a)ζ m (a)da (18) is the weighted average of G eff c , of G app c and a max is the maximal value of the crack length for which experimental (F, v) data is available.Note that usually a max does not J o u r n a l P r e -p r o o f Journal Pre-proof reach to the end of the specimen (a max < L − b/2) because the last part of the crack propagation is unstable due to final dynamic collapse.As G app c are ζ m are not true functions in the practial implementation of the procedure, but lists of values computed at discrete points (co-ordinates x), their averages can be approximated as Ḡapp c )).Then, using linear interpolation, values of G app c.i have been computed for values of the actual crack length a = a 0 + i, where i = 0, 1, 2, ..., 180.Finally, the average value of the effective fracture resistance follows from (17) as Ĝeff c = 0.357 N/mm.Note that this value is within 1.5% of the maximum computed value of the apparent fracture resistance J o u r n a l P r e -p r o o f Journal Pre-proof max(G E c ) = 0.352 N/mm.Although this procedure for obtaining Ĝeff c is relatively simple, it could be simplified even more by avoiding the estimation of the actual crack length and compute the average of the apparent fracture resistance not with respect to the crack length a (or coordinate x), but with respect to the cross-head displacement v (as the experiments were displacement-controlled).Therefore, for the sake of simplicity, Ḡapp c can be computed as the average of all values of G app c computed from (1) for each v − F pair after the peak load, i.e. during crack propagation.By taking the average of all values of G app c computed from (1) during crack propagation for an increment of the cross-head displacement ∆v = 0.01 mm, Ḡapp c = 0.285 N/mm has been obtained, which is only approximately 1% higher than the value obtained from (21).Thus, the procedure for computing Ḡapp c can be simplified without significantly affecting the accuracy.
width of the interface by a factor ζ m (where ζ m < 1), increases the values of σ max and k by a factor 1/ζ m .From there, it can be easily shown that by reducing the width of the adhesive, characteristic values of the relative displacements δ c and δ 0 (see Figure 6(a)) remain unchanged.Therefore, the effective values of Ω, σ max and k for the case of interface with defects can be obtained by dividing the apparent parameters by ζ m .In Table 3, a comparison between the apparent and the effective parameters is given for the representative specimen with ζ m = 0.79.Note that DCB PAR [28] computes Ḡapp c not according to (21), but using values of G app c computed for equidistant cross-head displacements during crack propagation, as explained at the end of Section 3.4.As for the displacement input parameters, DCB PAR provides values of δ 0 = 5.41 • 10 −4 mm and δ c = 0.014 mm, which are both apparent and effective parameters, as previously noted.

Figure 7 :
Figure 7: A comparison between the results of DCB-test simulations using the apparent and effective interface parameters, and the experimental results for the representative specimen.
) J o u r n a l P r e -p r o o f Journal Pre-proof the load-displacement response to the presence of voids and in turn the stability of the crack propagation.

Figure 10 :
Figure 10: Normalised R-curves of the apparent fracture resistance for Case A using σmax = 20, 40 and 50 MPa.

Table 1 :
Dimensions of the DCB specimens according to Figure 1.

Table 2 :
Average values (AVG), standard deviation (SD) and coefficient of variation (CV) for the normalised effective width of the adhesive ζ, normalised apparent fracture resistance