Tensile fracture characterization of adhesive joints by standard and optical techniques

Engineering Fracture Mechanics 136 (2015) 292–304

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Tensile fracture characterization of adhesive joints by standard and optical techniques C.J. Constante a, R.D.S.G. Campilho a,⇑, D.C. Moura b a Departamento de Engenharia Mecânica, Instituto Superior de Engenharia do Porto, Instituto Politécnico do Porto, Rua Dr. António Bernardino de Almeida, 431, 4200-072 Porto, Portugal b Instituto de Telecomunicações, Departamento de Engenharia Eletrotécnica e de Computadores, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal

a r t i c l e

i n f o

Article history: Received 4 December 2014 Accepted 6 February 2015 Available online 14 February 2015 Keywords: Aluminium alloys R-curves J-integral Fracture toughness Double-Cantilever Beam

a b s t r a c t The use of adhesive joints has increased in recent decades due to its competitive features compared with traditional methods. This work aims to estimate the tensile critical strain energy release rate (GIC) of adhesive joints by the Double-Cantilever Beam (DCB) test. The J-integral is used since it enables obtaining the tensile Cohesive Zone Model (CZM) law. An optical measuring method was developed for assessing the crack tip opening (dn) and adherends rotation (ho). The proposed CZM laws were best approximated by a triangular shape for the brittle adhesive and a trapezoidal shape for the two ductile adhesives. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The developments in adhesives technology made possible the use of adhesive bonding in many fields of engineering, such as automotive and aeronautical, because of higher peel and shear strengths, and ductility. As a result, bonded joints are replacing fastening or riveting [1]. More uniform stress fields, capability of fluid sealing, high fatigue resistance and the possibility to join different materials are other advantages of this technology. However, stress concentrations exist in bonded joints along the bond length owing to the gradual transfer of load between adherends and also the adherends rotation in the presence of asymmetric loads [2]. A large amount of works addresses the critical factors affecting the integrity of adhesive joints, such as the parent structure thickness, adhesive thickness, bonding length and geometric modifications that reduce stress concentrations [3–5]. A large number of predictive techniques for bonded joints is currently available, ranging from analytical to numerical, using different criteria to infer the onset of material degradation, damage or even complete failure. Initially, stresses were estimated by analytical expressions as those of Volkersen [6], which had a lot of embedded simplifying assumptions, and the current stresses were compared with the allowable material strengths. Many improvements were then introduced, but these analyses usually suffered from the non-consideration of the material ductility. Fracture mechanics-based methods took the fracture toughness of materials as the leading parameter. These methods included more simple energetic or stressintensity factor techniques that required the existence of an initial flaw in the materials [7,8]. More recent numerical techniques, such as CZM, combine stress criteria to account for damage initiation with energetic, e.g. fracture toughness, data to

⇑ Corresponding author. Tel.: +351 939526892; fax: +351 228321159. E-mail address: [email protected] (R.D.S.G. Campilho). http://dx.doi.org/10.1016/j.engfracmech.2015.02.010 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.

C.J. Constante et al. / Engineering Fracture Mechanics 136 (2015) 292–304

Nomenclature ! v bottom !

direction vector of the bottom curve direction vector of the top curve a crack length a0 initial crack length aeq equivalent crack length arccos inverse cosine function B width C compliance C0, C1, C2, C3 polynomial constants Ci experimentally measured initial compliance d calibration length for the optical method E Young’s modulus Ef corrected flexural modulus G shear modulus GC critical strain energy release rate GI tensile strain energy release rate GIC tensile critical strain energy release rate GIIC shear critical strain energy release rate h adherend thickness LT total length of the specimen mbottom slope of the bottom curve mtop slope of the top curve P load pi (i = 1, 2, . . . 8) points for the optical method Pu load per unit width q quadratic approximation function R correlation factor tA adhesive thickness tCT current tA value at the crack tip A tAi initial value of tA tn current tensile traction t0n cohesive strength in tension t0s cohesive strength in shear U strain energy u x-coordinate displacement v y-coordinate displacement x1, x2, x3 x coordinates of the points for the fitting procedure y1, y2, y3 y coordinates of the points for the fitting procedure a1, a2, a3 constants for the cubic equation of aeq b1, b2, b3 fitting polynomial coefficients D crack length correction d displacement dn crack tip opening dnc tensile end-opening at failure ef tensile failure strain ho crack tip adherends rotation hp adherends rotation at the specimen’s free ends rf tensile failure strength ry tensile yield stress CBBM Compliance-Based Beam Method CBT Corrected Beam Theory CCD Charge-Coupled Device CCM Compliance Calibration Method CZM Cohesive Zone Model DCB Double-Cantilever Beam ENF End-Notched Flexure LEFM Linear Elastic Fracture Mechanics LVDT Linear Variable Differential Transducer

v top

293

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estimate damage propagation [9,10]. This allows to consider the distinct ductility of adhesives and to gain accuracy in the predictions. All of these fracture toughness-dependent analyses rely on an accurate measurement of GIC and the shear critical strain energy release rate (GIIC). CZM in particular can accurately predict damage growth disregarding the bonded structures’ geometry if the fracture laws are correctly estimated [11]. These laws are based on the values of cohesive strength in tension and shear, t0n and t0s , respectively, and also GIC and GIIC. These parameters cannot be directly related with the material properties measured as bulk, since they account for constraint effects (for adhesive joints, caused by the adherends). The estimation of these fracture parameters is generally accomplished by performing pure tension or shear tests. Regarding GIC, the DCB test is the most suitable, due to the test simplicity and accuracy [12]. As described by Suo et al. [13], in the presence of largescale plasticity, J-integral solutions can also be employed for accurate results, in contrast to LEFM-based solutions. The J-integral is a relatively straight-forward technique, provided that the analytical solution for a given test specimen exists for the determination of GIC or GIIC. The most prominent example is the DCB specimen, for which J-integral solutions are available. It is also possible to estimate the tensile CZM law. Since the values of strength and toughness of adhesive layers vary as discussed, it is mandatory that they are estimated with accuracy. These parameters that cannot be directly related with the material properties measured as bulk, since constraint effects must be accounted for. The DCB test is the most suitable to measure GIC due to the test simplicity and accuracy. The conventional and standardized GIC estimation methods are based on Linear-Elastic Fracture Mechanics (LEFM) and rely on the continuous measurement of the crack length (a) during the test. However, it is known that GIC of ductile adhesives is not accurately characterized by LEFM methods, as discussed in the work of Kafkalidis and Thouless [14]. In recent years, methods that do not need measurement of a were developed [15,16], additionally including the plasticity effects around the crack tip. As an alternative, in the presence of large-scale plasticity, J-integral solutions are recommended [17]. For tensile loading, J-integral solutions are available for the DCB test, either for loading by pure bending moments or the standardized and moment-free tensile pulling. With this technique, it is also possible to estimate the cohesive law of the adhesive layer. Carlberger and Stigh [18] computed the cohesive laws of adhesive layers (epoxy adhesive Dow BetamateÒ XW1044-3) in tension and shear using the DCB and End-Notched Flexure (ENF) tests, respectively, considering 0.1 6 tA 6 1.6 mm, tA being the adhesive thickness. The value of ho was measured by an incremental shaft encoder and dn by two Linear Variable Differential Transducers (LVDT). The analysis of Ji et al. [19] used a J-integral technique applied to the DCB specimen to study the influence of tA on t0n and GIC for a brittle epoxy adhesive (LoctiteÒ Hysol 9460). The analysis methodology relied on the measurement of GIC by an analytical J–integral method, requiring the measurement of the adherends rotation at the specimen free ends (hp). For the measurement of rotation, two digital inclinometers with a 0.01° precision were attached at the free end of each adherend. A charge-coupled device (CCD) camera with a resolution of 3.7  3.7 lm/pixel was also used during the experiments to measure dn, necessary for correlation with the load (P) and hp for the definition of the tensile strain energy release rate (GI). The current tensile traction (tn) vs. dn laws (or CZM laws) were obtained by differentiation of the GI–dn data. The obtained results showed that the value of GIC increases with tA up to the maximum considered value (1 mm). On the contrary, for tA = 0.09 mm, t0n is approximately 3 times the bulk tensile strength of the adhesive, while the increase of tA leads to a reduction of t0n up to near the bulk adhesive strength (for tA = 1 mm). A study regarding the tA effect on the interfacial value of GIC of laminated composites was presented by the same authors [20], considering the same experimental analysis procedure. The same tA effect on GIC was found, although t0n increased with tA, oppositely to the previous study. This difference was justified by the different external constraints. Ouyang et al. [21] derived a concise form of the J-integral for the pure tensile fracture behaviour of DCB specimens with dissimilar adherends, considering the inclinometer method to estimate hp. A different study was published [22] regarding the mixed-mode fracture behaviour of the Single-Leg Bending test using an identical procedure to measure hp and digital image correlation to measure the crack-tip displacements. Campilho et al. [16] developed a methodology for the DCB test geometry that enables obtaining GIC and the CZM law by a J-integral methodology. The procedure consisted on an automated image processing technique that estimated the required parameters during the test. The GIC measurements of the adhesive SikaforceÒ 7888 were consistent with the literature data and the CZM law confirmed the ductile characteristics of the adhesive. The objective of this work is the estimation of GIC of adhesive joints between aluminium adherends by the DCB test, considering adhesives with different degrees of ductility. The J-integral is used to estimate GIC since it takes into account the plasticity of the adhesives. To calculate the J-integral, an optical measuring method is used, developed for assessing dn and ho during the tests, since these parameters are necessary to calculate GIC. This procedure is supported by a MatlabÒ subroutine for the automatic extraction of these values. On the other hand, the cohesive laws of the adhesives are obtained by the direct method. The J-integral results are also compared with traditional methods for evaluating GIC (methods that require the measurement of the crack length and methods based on an equivalent crack) and the corresponding conclusions are drawn.

2. Experimental work 2.1. Materials The material selected for the adherends is a laminated high-strength aluminium alloy sheet (AA6082 T651) cut by precision disc cutting into specimens of 140  25  3 mm3. The mechanical properties of this material are available in the lit-

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erature in the reference of Campilho et al. [23], giving the following bulk values: Young’s modulus (E) of 70.07 ± 0.83 GPa, tensile yield stress (ry) of 261.67 ± 7.65 MPa, tensile failure strength (rf) of 324 ± 0.16 MPa and tensile failure strain (ef) of 21.70 ± 4.24%. Three structural adhesives, ranging from brittle to highly ductile, were considered: the brittle epoxy AralditeÒ AV138, the ductile epoxy AralditeÒ 2015 and the high strength and ductile polyurethane SikaforceÒ 7888. The mechanical and toughness properties of these adhesives were obtained in previous works by the authors by experimental testing [16,24,25]. Bulk specimens were tested in a servo-hydraulic machine to obtain E, ry, rf and ef. The DCB test was selected to obtain GIC and the ENF test was used for GIIC. The collected data of the adhesives is summarized in Table 1. 2.2. Joint geometries Fig. 1 represents the geometry of the DCB specimens. The dimensions of the specimens are: total length LT = 160 mm, initial crack length a0  55 mm, adherend thickness h = 3 mm, width B = 25 mm and tA = 1 mm. The fabrication of the specimens involved grit blasting the bonding surfaces with corundum sand, cleaning with acetone and assembly in a steel mould. To achieve the different values of tA uniformly throughout the adhesive layer, calibrated spacers were inserted between the adherends. A sharp pre-crack at the specimens’ free edge was induced by a 0.1 mm thick razor blade between calibrated bars. Curing was performed at room temperature. The spacers were removed and the adherends sides were sprayed with white brittle paint, to allow an easy identification of a, and a printed scale was glued in both adherends to aid the a measurement or input data for the digital correlation technique [26]. Eighteen specimens were tested (six for each configuration) at room temperature in an electro-mechanical testing machine (Shimadzu AG-X 100) with a load cell of 100 kN. Each test was fully documented using an 18 MPixel digital camera with no zoom and fixed focal distance to approximately 100 mm. This procedure allowed obtaining the values of dn and ho, necessary for the J-integral method. The correlation of the mentioned parameters with the load–displacement (P–d) data was done by the time elapsed since the beginning of each test. 3. Methods to determine GIC In the work of Giovanola and Finnie [27] it is claimed that LEFM methods are inaccurate to estimate the critical strain energy release rate (GC) in presence of ductile adhesives, although some expressions consider correction factors to account for plasticity (e.g. the methods depicted in the standards ASTM D3433-99:2005 and BS 7991:2001). In this work, four methods were considered to consider plasticity effects: the Compliance Calibration Method (CCM), the Corrected Beam Theory (CBT), the Compliance-Based Beam Method (CBBM), all three accounting for the damage zone ahead of the crack tip (also

Table 1 Properties of the adhesives AralditeÒ AV138, AralditeÒ 2015 and SikaForceÒ 7888 [16,24,25].

* a b

Property

AV138

2015

7888

Young’s modulus, E (GPa) Poisson’s ratio, m Tensile yield stress, ry (MPa) Tensile failure strength, rf (MPa) Tensile failure strain, ef (%) Shear modulus, G (GPa) Shear yield stress, sy (MPa) Shear failure strength, sf (MPa) Shear failure strain, cf (%) Toughness in tension, GIC (N/mm) Toughness in shear, GIIC (N/mm)

4.89 ± 0.81 0.35* 36.49 ± 2.47 39.45 ± 3.18 1.21 ± 0.10 1.56 ± 0.01 25.1 ± 0.33 30.2 ± 0.40 7.8 ± 0.7 0.20a 0.38a

1.85 ± 0.21 0.33* 12.63 ± 0.61 21.63 ± 1.61 4.77 ± 0.15 0.56 ± 0.21 14.6 ± 1.3 17.9 ± 1.8 43.9 ± 3.4 0.43 ± 0.02 4.70 ± 0.34

1.89 ± 0.81 0.33* 13.20 ± 4.83 28.60 ± 2.0 43.0 ± 0.6 0.71b – 20* 100* 1.18 ± 0.22 8.72 ± 1.22

Manufacturer’s data. Estimated in reference [23]. Estimated from Hooke’s law.

Fig. 1. Geometry of the DCB specimens.

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known as fracture process zone), and the J-integral. By these methods, the effects of root rotation caused by the expected long plastic zone beyond the crack tip are taken into account. 3.1. Conventional methods The classical reduction schemes to estimate GIC are usually based on compliance calibration or the beam theory. The CCM is based on the Irwin-Kies equation [28]

GIC ¼

P2 dC ; 2B da

ð1Þ

where C = d/P is the specimen’s compliance. Regarding the definition of the C = f(a) function for application of Eq. (1), two methodologies are possible to follow [29]: (1) parameter calibration by separate tests or (2) by the concurrent measurement, during a DCB test, of the P–d–a data, and further data reduction to obtain the C = f(a) function for the specific specimen under analysis. The first mentioned technique has the requirement of further tests and could introduce errors in the measured values of GIC on account of heterogeneity of material properties between specimens (although this is more critical for materials such as wood). In this research, the C = f(a) relationship was estimated individually for each specimen from the P–d–a data from the respective test by using cubic polynomials (C = C3a3 + C2a2 + C1a + C0) to fit the C = f(a) curves, leading to

GIC ¼

P2 ð3C 3 a2 þ 2C 2 a þ C 1 Þ: 2B

ð2Þ

Beam theories were also used to measure GIC. Using the CBT, GIC is obtained using [30]

GIC ¼

3Pd ; 2Bða þ jDjÞ

ð3Þ

where D is a crack length correction for crack tip rotation and deflection, obtained as specified in the standard ISO 15024. The CBBM is a relatively straightforward but robust method, based on an equivalent crack length (aeq), and it only depends on the specimen’s compliance during the test. Applied to the DCB test specimen, it gives

! 1 : GIC ¼ 2 þ B h h2 Ef 5G 6P2 2a2eq

ð4Þ

In the expression, aeq is an equivalent crack length estimated from the current specimen compliance and taking into consideration the damage zone, Ef is a corrected flexural modulus to account for stress concentrations at the crack tip and stiffness variability between specimens, and G is the shear modulus of the adherends. The evaluation of aeq is presented with some detail in Appendix A. 3.2. J-integral method In the proposed technique, the CZM law is measured by the direct method. Under this scope, the path-independence of the J-integral can be used to extract relations between the specimen loads and the cohesive law of the crack path [31]. Based on the fundamental expression for J defined by Rice [32], it is possible to derive an expression for the value of GI applied to the DCB specimen from the concept of energetic force and also the beam theory for this particular geometry, as follows (the following formulae are developed assuming that the J-integral gives a measurement of GI) [33,34]:

GI ¼ 12

ðP u aÞ2 3

Eh

þ Pu ho

or GI ¼ Pu hp ;

ð5Þ

where Pu represents the applied load per unit width at the adherends’ edges. A schematic representation of dn, ho and hp, required by this method, is given in Fig. 2. The values of ho and hp are measured between the adherends’ cross-sections at

Fig. 2. DCB specimen under loading, with description of the analysis parameters, and estimation of the cohesive law.

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297

the crack tip and loading point, respectively, considering the rotations suffered by these sections on account of the applied loading. In this work, the first expression of (5) is considered, using ho instead of hp, due to a simpler extraction of the parameter by the optical method. The J-integral can be calculated along an arbitrary path encircling the start of the adhesive layer, giving [31]:

GI ¼

Z

dnc

tn ðdn Þddn ;

ð6Þ

0

where dnc is the tensile end-opening at failure of the cohesive law (measured at the initial crack tip). GIC can be considered the value of GI at the beginning of crack growth. Thus, GIC is given by the steady-state value of GI, at a dn value of dnc [19]. The tn(dn) curve can be easily obtained by differentiation of Eq. (1) with respect to dn

tn ðdn Þ ¼

dGI : ddn

ð7Þ

As a result, the procedure of an experiment is to measure the history of P, a, dn and ho. The cohesive law in tension can then be estimated by plotting GI in Eq. (5) as a function of dn, polynomial fitting of the obtained curve and differentiation [31]. 3.2.1. Continuous measurement of dn and ho by the optical method A numerical algorithm was developed, based on digital image processing and tracking reference points by the software, to give estimated measurements of ho and dn. The optical method requires the identification of 8 points, from p1 to p8. Fig. 3a shows points p1 to p6, which allow the following: points p3 and p4 enable measuring the current tA value at the crack tip (tCT A ) during loading in image units (pixels), and points p1 and p5 on the top specimen and points p2 and p6 on the bottom specimen will enable the estimation of ho. Fig. 3b shows points p7 and p8, which identify a straight line segment in the image for which the length (d) is known in real world units (mm), to convert the measured data with points p1 to p6 from pixels to mm, as it will be detailed further in this work. Initially, the eight points are identified manually in the first picture of a test. To be noticed that points p1 to p6 are printed with a distinct colour (although this is not perceptible in Fig. 3), which helps finding their correct locations. Starting from the points in the first picture, the points of the following pictures are automatically tracked with an algorithm in MatlabÒ. Full details of the point tracking algorithm are presented in a previous work [16]. With this procedure, all 8 points locations are easily found for all pictures taken during a DCB test. 3.2.1.1. ho calculation. The value of ho is obtained by the angle between the tangents to the horizontal curves of the 2 scales closest to the adhesive, measured at the crack tip (Fig. 4). The curvature of the top adherend is first computed by fitting a quadratic function to points p1, p3 and p5. The quadratic function q is defined as

Fig. 3. Points taken by the optical method: (a) points p1 to p6 to measure ho and dn and (b) points p7 and p8 to convert the measured data from pixels to mm.

Fig. 4. Calculation of ho. Quadratic functions were fitted to points p1, p3, p5 and p2, p4, p6, representing the curvature of the top and bottom specimen, respectively, while the straight lines show the tangents to the curves at the crack tip (corresponding to 10 mm in the scales).

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02

x1

31

2

x1

32

2

x1

3

B6 7C 6 7 6 7 q@4 x3 5A ¼ b1 4 x3 5 þ b2 4 x3 5 þ b3 ; x5

x5

ð8Þ

x5

where bi are the coefficients of the polynomial and xi are the x coordinates of point pi. The coefficients are found by fitting the quadratic function to the y coordinates of the points, q([x1, x2, x3]T) = [y1, y2, y3]T, in the least squares sense. The first derivative of the quadratic function at p3 yields the slope of the top curve (mtop) at the crack tip,

mtop ¼ 2b1 x3 þ b2 ;

ð9Þ

which is then used to define a direction vector ~ v top = (1, mtop). The same process is repeated for points p2, p4 and p6, yielding the slope of the tangent to the bottom curve at the crack tip (mbottom) and its direction vector ~ v bottom = (1, mbottom). Finally, ho is obtained by measuring the angle between the two vectors:

h0 ¼ arccos



 ~ v top  ~ v bottom : j~ v top jj~ v bottom j

ð10Þ

3.2.1.2. dn calculation. Initially, tCT A is calculated in real world units (mm) by the following expression

tCT A ¼ d

jp3  p4 j : jp7  p8 j

ð11Þ

A length of d = 15 mm was used for all trials (illustrated in Fig. 3). The pixel size was on average 0.021 mm and, thus, the estimated maximum error of the image acquisition process is ±0.011 mm. Finally, dn can be defined as i dn ¼ tCT A  tA ;

ð12Þ

where tiA is the initial value of tA. 4. Results The DCB tests were accomplished by the mentioned testing procedure. By visual inspection of the failed specimens all failures were cohesive in the adhesive layer with no signs of plasticity in the adherends. 4.1. GIC calculation by the conventional methods The values of GIC were calculated by the previously described techniques. Fig. 5 shows the experimental P–d curves for the specimens bonded with the adhesive AralditeÒ 2015, showing the overall agreement between specimens bonded with a given adhesive. One of the required steps to obtain GIC by the CCM is the estimation of dC/da during the test. This is a very critical step, as the GIC measurement is affected by a large amount by this parameter [29]. For a correct measurement, the C = f(a) curve should span between the beginning of crack growth and before catastrophic failure of the specimens, where a sudden drop of stiffness occurs. Fig. 6 gives an example, for a specimen bonded with the adhesive AralditeÒ 2015, of the C = f(a) curve and 3rd degree polynomial approximation. Fig. 7 presents the experimental R-curves, relating the evolution of GI with a, for one representative tested specimen for each one of the adhesives. The CBBM enables plotting the R-curve for each specimen directly from the P– d curve, relating GI with aeq instead of a. Notwithstanding the method, the R-curve ideally gives a constant value of GI over the entire propagation phase, although experimentally some fluctuations occur due to different adhesive mixing, adhesion

200

P [N]

160 120 80 40 0

0

5

10

15

20

25

δ [mm] Fig. 5. Experimental P–d curves obtained for the adhesive AralditeÒ 2015.

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0.16

y = -3.9146E-07x3 + 9.9848E-05x2 - 5.6026E-03x + 1.0871E01 R² = 9.9448E-01

C [mm/N]

0.12 0.08 0.04 0 40

50

60

70

80

90

a [mm] C=f(a) points

Polinomial (C=f(a) points)

Fig. 6. Example of C = f(a) curve for a specimen bonded with the adhesive AralditeÒ 2015 and polynomial approximation.

0.5

0.8 0.6

GI [N/mm]

GI [N/mm]

0.4 0.3 0.2

0.2

0.1 0

0 45

(a)

0.4

50

55 60 a or aeq [mm] CCM

CBT

65

70

CBBM

45

50

55 60 a or aeq [mm] CCM

(b)

CBT

65

70

CBBM

2.0

GI [N/mm]

1.6 1.2 0.8 0.4 0.0 45

(c)

50

55 60 a or aeq [mm] CCM

CBT

65

70

CBBM

Fig. 7. Comparison of representative R-curves for each of the adhesives: (a) AralditeÒ AV138, (b) AralditeÒ 2015 and (c) SifaforceÒ 7888.

issues, defects and crack arrest phenomena [35]. These curves show the crack propagation with a steady-state value of GI [36]. The initial phase of the CBBM curves relates to the increase of GI up to the onset of crack propagation and allows the visualization of aeq before damage, which is slightly bigger than a0 because aeq accounts for the damage zone [16]. For the presented specimens, a0 was measured at 47.26 mm (AralditeÒ AV138, (a)), 46.68 mm (AralditeÒ 2015, (b)) and 46.31 mm (SikaforceÒ 7888, (c)). This can be visualized in Fig. 7 in the CCM and CBT plots at the initial stabilization of GI, which corresponds to crack initiation. The corresponding values of aeq, by the same order, are 49.85 mm, 49.90 mm and 53.89 mm (calculated directly from the P–d data by the first drop of P in the P–d curve). The visible shifting in the CBBM R-curve, corresponding to bigger a0 values than the CCM and CBT, is explained by the use of an equivalent crack to account for local plasticity, longer than the real value of a measured during the tests and used in the other two methods. Because of this, the difference between a0 and aeq increases with the adhesive plasticity, which is also visible in the R-curves of Fig. 7. Despite this fact, the three methods provide comparable GIC results. Table 2 summarizes the GIC (N/mm) values of all specimens for the three adhesives and by the four methods (the J-integral results are discussed in the following Section). A coherent CCM R-curve was not attained for specimen 4 with the adhesive AralditeÒ AV138 because of difficulties in the adjustment of the C = f(a) polynomial function. An identical scenario was found for specimen 1 with the adhesive SikaforceÒ 7888, while specimen 6 of this adhesive exhibited an adhesive failure, thus making the results not applicable to the study. The agreement between specimens of the same adhesive is good between the CCM, CBT and CBBM, although in some specimens it was not possible to obtain the CCM prediction. This can be accredited to difficulties in the estimation of the poly-

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Table 2 Values of GIC (N/mm) for the three adhesives obtained by all methods. Adhesive

AralditeÒ AV138

AralditeÒ 2015

SikaforceÒ 7888

Specimen

CCM

CBT

CBBM

J-integral

CCM

CBT

CBBM

J-integral

CCM

CBT

CBBM

J-integral

1 2 3 4 5 6

0.200 0.219 0.193 – 0.189 0.195

0.237 0.241 0.215 0.291 0.237 0.206

0.231 0.247 0.234 0.310 0.254 0.217

0.224 0.252 0.231 0.329 0.237 0.197

0.403 0.387 0.576 0.399 0.590 0.619

0.431 0.484 0.484 0.471 0.676 0.632

0.444 0.467 0.492 0.460 0.709 0.663

0.437 0.434 0.494 0.456 0.665 0.712

– 1.277 1.166 1.413 0.707 –

1.433 1.580 1.163 1.538 0.780 –

1.423 1.789 1.208 1.557 0.791 –

1.246 1.533 1.220 1.418 0.783 –

Average Deviation

0.199 0.012

0.238 0.030

0.249 0.033

0.245 0.045

0.496 0.110

0.530 0.099

0.539 0.116

0.533 0.123

1.141 0.306

1.299 0.332

1.354 0.379

1.240 0.286

nomial functions used to fit the C = f(a) curves, required for dC/da, which resulted in a non-conform R-curve. Between methods, the correspondence is also quite good, especially when comparing the CBT and CBBM. With a similar justification, the CCM values show a bigger deviation.

4.2. GIC calculation by the J-integral The J-integral methodology followed the technique described in Section 3.2. GIC was calculated by the first equation of (5), which considered ho instead of hp to estimate GI. A full description of the procedure to obtain ho is presented in Section 3.2.1.2 and enabled plotting the ho-testing time curve with one data point every 5 s. Fig. 8 shows a ho-testing time curve for a specimen bonded with the adhesive SikaforceÒ 7888, with emphasis to the raw curve, the 4th degree fitting law and the adjusted law, considering ho(testing time = 0) = 0. The fitting procedure by a polynomial curve removes the experimental measurement noise and the adjustment at the test beginning cancels any initial misalignment between the measurement scales. Depending on the specimen under analysis, different order polynomials were considered by selecting the best correlation factor, R (this also applies to the forthcoming fitting data). Compared to the work of Ji et al. [19], which used an inclinometer to measure ho, a different but still accurate approach is thus proposed that enables obtaining ho without the requirement of using physical sensors in the specimens. The dn-testing time plot was also obtained to estimate the GI–dn curve and consequently the cohesive law by differentiation (equation (7)). Fig. 9 shows this curve for the same specimen of Fig. 8. In this figure, the adjusted curve was shifted upwards by 0.05 mm for clarity, since the adjustment to make dn(testing time = 0) = 0 was minimal and the curves would overlap. Fig. 10 shows a representative GIC–dn curve for each tested adhesive. The AV138 curve is shown on secondary axis (to the right) to prevent overlap with the 2015 curve. The polynomial approximations, used to estimate the CZM laws by Eq. (7), are also presented. The three adhesives give an identical curve shape, constituted by three distinct regions: initially, the increase of GI with dn is slow (theoretically, the curve should begin with zero slope, corresponding to the CZM law beginning with zero stresses), secondly, a region of approximate linear increase of GI appears and, finally, the curve tends to a steady-state value of GI. Equally to the work of Ji et al. [19], the value of GIC is obtained at the beginning of crack propagation (peak value in the P–d curve) and is defined by the steady-state GI value in the GI–dn curve. The shape of these curves is similar to the ones reported in the literature (e.g. Campilho et al. [16] and Ji et al. [19]) although, depending on the adhesive, the proportion between these three parts may vary. It can be found that the curve slope is higher for the adhesive AralditeÒ AV138 than for the other two adhesives, whose slope is roughly similar (the difference between the adhesive AralditeÒ AV138 and the other two is not directly visible on account of the different y-axes). This is directly related to the adhesives’ stiffness, i.e., the stiffer the adhe-

0.08 y = 5.5396E-11x4 - 1.3775E-08x3 + 1.2098E-06x2 + 9.5667E-05x + 2.4644E-02 R² = 9.9524E-01

θo [rad]

0.06

0.04

0.02

0 0

40

80

120

160

Testing time [s] Raw curve

Adjusted curve

Polynomial (Raw curve)

Fig. 8. Plot of ho – testing time for a specimen bonded with the adhesive SikaforceÒ 7888: raw curve, polynomial approximation and adjusted polynomial curve.

C.J. Constante et al. / Engineering Fracture Mechanics 136 (2015) 292–304

0.20

301

y = 3.7790E-10x4 - 7.6265E-08x3 + 5.1503E-06x2 + 2.2968E-04x + 2.5201E-03 R² = 9.8974E-01

δn [mm]

0.15

0.10

0.05

0.00

0

40

80

120

160

Testing time [s] Raw Curve

Adjusted curve

Polynomial (Raw curve)

Fig. 9. Plot of dn – testing time for a specimen bonded with the adhesive SikaforceÒ 7888: raw curve, polynomial approximation and adjusted polynomial curve.

0.3

1.6 GI in secondary axis

0.25

GI [N/mm]

1.2 0.2 0.15

0.8

0.1 0.4 0.05 0 0

0.04

0.08

0 0.12

δn [mm] 2015

7888

AV138

Polinomial Polynomial(AV138) approximations

Fig. 10. Representative GIC–dn laws for each tested adhesive and respective polynomial approximations.

sive is, the higher is the slope. On the other hand, the value of dn corresponding to the GI–dn curve stabilization (giving the GIC measurement) increases with the adhesive ductility, which will reflect on the bigger value of dnc in the CZM laws. On account of this, the adhesive SikaforceÒ 7888 largely differentiates from the two epoxy adhesives, especially the AralditeÒ AV138. Moreover, the adhesives AralditeÒ 2015 and SikaforceÒ 7888 exhibit a gradual slope reduction near to GIC because of the large damage zone development induced by the ductility of these adhesives. Contrarily, this effect is minimal for the adhesive AralditeÒ AV138. Table 2 summarizes the GIC results for all specimens and compares them against the conventional methods. The results show a very good match and the maximum percentile deviations of the average values to the other methods, weighted against the J-integral values and not considering the CCM (because of the polynomial fitting issues, which give this method the largest deviations) are as follows: +2.81% compared to the CBT (AralditeÒ AV138), 1.15% compared to the CBBM (AralditeÒ 2015) and 9.16% compared to the CBBM (SikaforceÒ 7888). Differentiation of the GIC–dn curves, according to Eq. (7), gives the tn–dn or CZM laws of the adhesive layer. Fig. 11 compares representative tn–dn or CZM laws for each tested adhesive (corresponding to the GI–dn curves of Fig. 10). In this figure, triangular (AralditeÒ AV138) or trapezoidal simplified CZM laws (AralditeÒ 2015 and SikaforceÒ 7888) are overlapped to the real curves as the most suited shapes for input in numerical simulations using the CZM technique. Owing to the brittleness of the adhesive AralditeÒ AV138, it is best modelled by a triangular law, while the other two adhesives present a lengthier steady-state region that is more suitably modelled by a trapezoidal law. This behaviour is consistent with the previous characterization of the adhesives, which showed that the adhesive AralditeÒ AV138 has a linear r–e curve up to failure, while the adhesives AralditeÒ 2015 and SikaforceÒ 7888 have minor and significant plasticization prior to failure, respectively [16,24,25]. These findings agree with the work of Carlberger and Stigh [18] regarding a ductile epoxy adhesive, in which a clear and significant steady-state region of tn appears at the onset of plasticization, followed by quick failure. 4.3. Further discussions The presented results by the four tested methods agreed quite well, although some inconsistencies were found between the CCM and the other methods for the three adhesives, which is inputted to the polynomial fitting difficulties required to obtain the derivative of C with respect to a. Comparison with available data in the literature also provides validation of the

302

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tn [MPa]

30

20

10

0 0

0.02

0.04

0.06

0.08

δn [mm] AV138

2015

7888

Simplified CZM laws

Fig. 11. Representative tn–dn or CZM laws for each tested adhesive and simplified CZM laws.

obtained results, although measurement of GIC is not independent of the adherend materials and thickness [37]. For the adhesive AralditeÒ AV138, the value of GIC = 0.3459 ± 0.045 N/mm was found for DCB specimens with steel adherends (DIN St33) [38]. Some data is also available regarding the adhesive AralditeÒ 2015 [39]. The value of GIC = 0.43 ± 0.02 N/ mm was obtained in DCB joints with [0]16 unidirectional carbon/epoxy composite adherends. Finally, for the adhesive SikaforceÒ 7888, GIC = 1.103 ± 0.405 N/mm was obtained for jute reinforced natural fibre composite adherends [16] and GIC = 1.095 ± 0.1955 N/mm was estimated for 4 mm thick aluminium adherends [40]. Thus, for all of the three tested adhesives, the obtained results compare reasonably well with the available data. By performing a comparison of the different methods for obtaining GIC, the more significant difference is the requirement to measure a continuously during the test for the CCM and CBT. On average, about 100 images required analysis, which is highly time consuming because of uncertainties on the correct position of a. Apart from this difficulty, the CCM needs the calculation of dC/da, which in turn is based on polynomial fitting of the C = f(a) curve during crack growth. The estimation of GI during propagation is highly sensitive to the chosen polynomial and, without the comparative analysis with the other methods, it is difficult to know which polynomial is most suited. The CBBM surpasses this limitation as it does not require measurement of a. By the J-integral, with the current implemented procedure, it is necessary to measure dn and ho throughout the test, but the method allows extracting the CZM law of the adhesive layer in tension, which is a major advantage for posterior strength prediction by CZM modelling. On account of the automated procedure implemented in MatlabÒ, this is straightforward, but the posterior analysis is relatively time consuming (e.g. polynomial derivation of the GI–dn law to obtain the CZM law). The correct shape of the CZM law is highly dependent on the dn measurement, and thus this parameter shall be measured with sufficiently high resolution, either with mechanical sensors or optical means, as it is the present case. When applied in real structures, the three adhesives behave differently. The results obtained showed that the SikaForceÒ 7888 is the most ductile and the AralditeÒ AV138 the most brittle. As it is known, the adhesive properties highly influence the joint strength. However, a stronger adhesive does not necessarily give a higher joint strength. In fact, a strong yet brittle adhesive attains high peak stresses locally at the edges, but it does not allow the stress redistribution to the interior. As a result, the average shear stress at failure is small. This is the expected behaviour of the adhesive AralditeÒ AV138. On the other hand, highly ductile and low modulus adhesives generally have a low strength. However, they are able to distribute stresses more uniformly along the joint by plastic strain (due to the low stiffness), which makes their joints much more resistant than with high strength and brittle adhesives [41]. The adhesive AralditeÒ 2015 fits into this category on account of the moderate ductility for structural adhesives and a tensile and shear strength lower than the AralditeÒ AV138. The adhesive SikaForceÒ 7888 combines high strength with large ductility, which gives it an advantage in comparison to the other two adhesives. Actually, in bonded joints, the adhesive reaches high shear stresses in the joint and has a greater capacity of stress redistribution in the adhesive layer after the elastic limit is reached. Moreover, the fatigue strength of bonded joints is typically lower when using brittle adhesives. This is explained by the more uniform distribution of stresses and higher energy damping of ductile adhesives [1]. Ductile adhesives have a greater ability to withstand cleavage and peel efforts and hence the preference of their use for joining thin plates [1]. 5. Conclusions This work presented the estimation of GIC of adhesive joints between aluminium adherends by the DCB test, considering adhesives with different degrees of ductility. The J-integral was used to estimate GIC and to obtain the CZM law of the adhesives in tension. GIC was also compared with standard characterization techniques. All tested methods are suited to capture GIC for ductile adhesives by considering root rotation effects that are large on account of the adhesive characteristics. The GIC values by the four tested methods agreed quite well between themselves and with available data in the literature. However, some inconsistencies were found between the CCM and the other methods for the three adhesives. This is inputted to poly-

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303

nomial fitting difficulties in the CCM since, in this approach, the C = f(a) relationship was obtained specifically for each specimen instead of conducting separate tests. Comparing all methods regarding time requirements as defined for this research, the CCM and CBT imply measurement of a continuously during the tests, which is not necessary for the CBBM. However, as previously mentioned, the CCM can also rely on separate tests to establish the C = f(a) relationship. The J-integral is more time consuming because of the necessity to measure ho and dn during the test. Actually, if only GIC is requested, the method only requires measurement of ho. However, the measurement of dn gives the possibility to obtain the cohesive law of the adhesive layer by differentiation of the GI = f(dn) curve, which is a clear advantage for subsequent strength predictions by advanced methods such as CZM modelling. The tensile CZM laws of the adhesives, enabled by the J-integral technique, showed the high strength and brittleness of the adhesive AralditeÒ AV138, which can be accurately modelled with a triangular CZM law. The adhesives AralditeÒ 2015 and SikaforceÒ 7888 have a lower tensile strength but higher ductility (increasing difference from the AralditeÒ 2015 to the SikaforceÒ 7888), and the curve shaped resembled the trilinear trapezoidal law. Full characterization of the adhesives requires the shear CZM laws. With suitable mixed-mode damage initiation and propagation criteria, strength prediction of bonded joints under generic geometric and loading conditions with CZM modelling is enabled. Acknowledgments The authors would like to thank SikaÒ Portugal for supplying the adhesive SikaforceÒ 7888. Appendix A. Evaluation of aeq The compliance of the specimen (C = d/P) can be obtained from the strain energy of the DCB specimen due to bending, U (including shear effects), and the Castigliano theorem, giving

C¼

8a3 3

EBh

þ

12a : 5BhG

ðA:1Þ

The value of aeq can be calculated from Eq. (A.1), replacing a by aeq and considering the values of C experimentally measured during the DCB test. Since Eq. (A.1) is based on the Beam Theory, and some phenomena are not accounted for in this formulation (such as stress concentrations at the crack tip), the value of Ef is considered instead of E. This parameter is estimated from Eq. (A.1) considering a0 and the experimentally measured initial compliance, Ci, as

 1 12ða0 þ jDjÞ 8ða0 þ jDjÞ3 Ef ¼ C i  : 3 5BhG Ef Bh

ðA:2Þ

Application of equation (A.1), with aeq instead of a and Ef replacing E, results in a cubic equation of aeq as follows

a1 a3eq þ a2 aeq þ a3 ¼ 0; with a1 ¼

8 3

Bh Ef

;

a2 ¼

12 ; 5BhG

a3 ¼ C;

ðA:3Þ

which can be solved analytically by software to obtain aeq. Detailed explanations of the method and the final expression of aeq can be found in reference [15]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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