2.5.4. Embedded optical fiber curve analysis

Figure 2.5-6 and Figure 2.5-7 present the obtained strain profiles along the lap length for the embedded optical fiber of the second sample. Different measurement times were chosen and each profile corresponds to one of these moments before and during the crack propagation. Before the crack propagation (t<sub>B</sub>=58s), the resulting curves present a visible peak. This is supposed to correspond to the position of the initial crack front. During the test, this initial strain jump remains at the initial crack front position, but the strain profile is modified along the joint. The experimental results show a strain plateau with a length of up to 2cm at the beginning of the crack propagation (Figure 2.5-6). This is supposed to be linked to damage near the crack tip before its propagation and could thus be used as a measurement of the fracture process zone (FPZ) length. This length being bigger than 1.5 cm, discussion could be opened on the assumption commonly made in the linear elastic fracture mechanics theory (generally used to compute the critical toughness), that requires that the process zone size remains confined to the crack tip area and with a length being shorter than the adhesive thickness. Östlund and Nilsson [109] have further investigated this recommendation for crack propagation in a double cantilever beam specimen with a cohesive zone heading the crack tip. Crack propagation was analyzed both with beam theory and continuum theory. Letting PZI the process zone length, and h<sub>s</sub> the adherent thickness, the results of Östlund and Nilsson indicate that the ratio PZI/h<sub>s</sub> should not be greater than 10 as a reasonable limit for the applicability of beam theory. In our case, the adherent being 5 mm thick, and the process zone being supposed to be 15 mm long, this recommendation is fulfilled.

After the initial crack propagation (Figure 2.5-7), no clear indication of the position of the crack front is observed using optical fiber measurements. The use of embedded optical fiber may therefore require more complex strain state analysis tools to be able to monitor the crack front position.

<figure><img src="image_82.png" /><figcaption>Figure 2.5-6: Embedded distributed optical fiber strain measurements before crack propagation.</figcaption></figure>
<figure><img src="image_83.png" /><figcaption>Figure 2.5-7: Embedded distributed optical fiber strain measurements after crack propagation.</figcaption></figure>

### 2.5.5. Numerical validation of the crack length measurement

For the ENF test, a finite element model (FEM) has been built with the software Femap NX Nastran [110] using the same geometry and boundary conditions as the real one. It aimed at understanding the optical fiber strain measurements, having information on the strain state in the adhesive and validating the crack length measurement. The number of elements in the adhesive thickness is 3 and the mesh was set as in Figure 2.5-8 with linear quadrilateral elements (Q8). The model was 3D. The adhesive thickness was defined at 1 mm, with one element in the *z* direction. The boundary conditions definition was to block all the out of plane displacements (*z* direction) to simulate plane strain condition. A displacement in the *y* direction was applied at the center of the sample to simulate the load application point (*x* and *z* displacements were blocked). The displacement in the *y* direction were blocked at the position of the supports. A pure elastic behavior was considered for all materials. The Young modulus of the adhesive was set at 11 GPa, based on the adhesive datasheet and the steel modulus was defined at 210 GPa.

<figure><img src="image_84.png" /><figcaption>Figure 2.5-8: Finite element mesh and strain measurement area.</figcaption></figure>

A comparison of the strain profiles obtained from experimental and numerical works is carried out in Figure 2.5-9 and Figure 2.5-10. Two moments before crack propagation are analyzed: $t_A$ = 39 s and $t_B$ = 58 s.
For the time $t_a = 39$ s, Figure 2.5-9a shows that the model validates the assumption that the initial crack front position is correlated to the position of the local maximal strain measured at the surface of the specimen. Concerning the embedded measurement in the bondline, Figure 2.5-9b shows that, prior to the non-linear part of the load/displacement curve (before $t_A$), the internal behavior of the bonding is well modeled (with linear elastic modeling) with an initial peak at the position of the initial crack front, and a parabolical increase of the strain upstream of the crack front. It indicates that in the real case, the process zone creation has not started yet, and that the bondline is only elastically loaded, as supposed from the load/displacement curve shape.

For the time $t_A = 58$ s, the Figure 2.5-10a, presents the strain at the surface of the specimen obtained by FEM and the optical fiber. The results show that the numerical maximum strain location is situated right above the crack front. A slight shift of the position ($< 2mm$) of the maximal point of the strain measured by the exterior optical fiber (blue curves) can be noted just before the crack propagation. It indicates (compared to the numerical result) that, in the real case, the creation of the process zone has generated a shift of the maximal strain peak upstream of the crack front. Concerning the embedded measurement in the bondline, Figure 2.5-10b shows that the linear FEM continues to predict a fast strain increase (absolute value) upstream of the crack front with a similar shape than for $t_A$. On the contrary, the experimental result shows a strain plateau around the value -1500 µstrain. This is supposed to be linked to the local plasticity (or damage), and may thus correspond to the fracture process zone length, occurring in the bondline, which is not considered in the elastic FEM model.

For the strain profiles measured by the exterior optical fiber, it must be noted that thicker adherent, and consequently higher rigidity of the sample may create a flatter maximal strain peak. Because of the fiber measurement noise, it could involve higher error in the crack front position measurement.

<figure><img src="image_85.png" /><figcaption>Figure 2.5-9: Comparison of strain profiles along lap length: elastic FEM analysis vs experimental measurements at 39 s, a) exterior measurement, b) embedded measurement.</figcaption></figure>
<figure><img src="image_86.png" /><figcaption>Figure 2.5-10: Comparison of strain profiles along lap length: elastic FEM analysis vs experimental measurements at t<sub>B</sub>= 58 s, a) exterior measurement, b) embedded measurement.</figcaption></figure>

**2.6. MMB test investigations**

**2.6.1. Load vs time curve analysis and failure mode**

Finally, to validate the theoretical approach exposed in section 2.2.3, with mixed mode loading, an MMB test was conducted. The crack front position was monitored using only the exterior optical fiber.

The obtained load vs time curve for the MMB test is shown in Figure 2.6-1. Three stages can be observed. The first one is a linear elastic stage up to time point A (t<sub>A</sub> = 140 s). Once the load exceeds the point of non-linearity, damage appears in the bondline, extends and forms the fracture process zone (FPZ). This leads to a slope decrease, and an increase of the compliance. When the FPZ is fully developed, the load vs time curve reaches its maximum at time t<sub>B</sub>=165 s. The crack propagation is assumed to occur after the first load decrease. The failure mode for this test was cohesive in the bondline, similar to the DCB and ENF results.

<figure><img src="image_87.png" /><figcaption>Figure 2.6-1: MMB test load/displacement curve.</figcaption></figure>
2.6.2. Optical fiber analysis and crack propagation determination

2.6.2.1. Exterior optical fiber

The crack front position was monitored using only the exterior optical fiber. Figure 2.6-2 presents the strain measurements along the length of the sample for the optical fiber during the crack propagation. Each curve corresponds to a different time. It can be verified that a maximum strain value can be measured initially (corresponding to the initial crack front position) and that the position of this maximum is shifted during the test (related to the crack propagation).

<figure><img src="image_88.png" /><figcaption>Figure 2.6-2: Optical fiber continuous strain measurements between time 100s and 174s.</figcaption></figure>

At the beginning of the test, the position of the maximal strain value (at x=86 mm) corresponds to the initial crack length. A shift of the maximum of the curve can be observed, indicating a displacement of the crack tip after t=140 s.

The displacement of this peak is post-treated to obtain the crack propagation (Figure 2.6-3). A smoothing of the data with the use of local mean has been made to reduce the raw crack front computation noise (each raw data value is converted to average of the 5 data points centered on the original value).

<figure><img src="image_89.png" /><figcaption>Figure 2.6-3: Crack propagation curve for the MMB test.</figcaption></figure>
### 2.6.3. Crack length visual validation

The final crack front length was measured at mid-width of the sample as shown in Figure 2.6-4a for the MMB test. In this case, the result shows that the final crack length obtained by the optical fiber is 45.67 mm which is very close to the value measured with a caliper (45.5 mm).

<figure><img src="image_90.png" /><figcaption>Figure 2.6-4: a) Visual measurements of the crack length at the end of the MMB test, a) at mid width, b) on the border.</figcaption></figure>

In the MMB test, the crack length on the border of the sample, where visual inspection and DIC are normally carried out, was also measured as described in Figure 2.6-4b. The maximal difference between the crack front position at the center of the sample and on the edges is 4.5 mm which seems to be important in regards with the studied propagation distance of 45 mm.

## 2.7. Conclusion

This chapter has investigated the use of distributed optical fiber sensor to measure the crack propagation for steel-to-steel adhesively bonded joints in mode I, mode II and mixed mode (using DCB, ENF and MMB tests). It was shown that the high spatial resolution and the accurate strain measurement of the used optical fiber can greatly improve the monitoring of the crack propagation. The results obtained for the three tests are summarized in Table 2.7-1, showing a comparison of the visual measurement versus the optical fiber measurement. The results highlight a really good correlation between the optical fiber crack position measurement and the visual validation post-test, with an error equal or inferior to the millimeter.

<table><caption>Table 2.7-1: Crack length from the experimental investigations using exterior optical fiber and visual inspection.</caption><thead><tr><th></th><th>DCB</th><th>ENF</th><th>MMB</th></tr></thead><tbody><tr><td>Optical fiber measurement (mm)</td><td>122 ± 1</td><td>84 ± 1</td><td>46 ± 1</td></tr><tr><td>Post-failure visual measurement (mm)</td><td>123 ± 1</td><td>86 ± 1</td><td>46 ± 1</td></tr></tbody></table>

The exterior optical fiber proved to give access to consistent crack propagation data during the tests in mode I, II and mixed mode. The only limitation has been proved to be the acquisition frequency of the Luna interrogator in the case of uncontrolled crack propagation (ENF). As far
as the embedded optical fiber was concerned, it was not possible to use it as crack length sensor.
Additional development would be required for this.

Several advantages can be highlighted regarding the use of distributed optical fiber sensor:

- the central position of the exterior optical fiber is supposed to reduce error in crack front position compared to more commonly used methods such as DIC or visual inspection that are highly dependent on the crack front shape (both those methods are carried out on the flank of the sample).

- in regards with visual inspection, the use of the optical fiber seems more robust and independent of operators. In comparison with DIC, post-processing appears to be easier to carry out and several optical fibers could be used to improve crack propagation monitoring reliability.

It must be noted that these conclusions have been obtained with relatively thick steel-to-steel bonded samples. This methodology could be used on other samples with different materials (fiber reinforced composites or metals) or different samples shape. It may even be used for crack monitoring of full size CFRP reinforcement patches. It can also be highlighted that the final procedure does not require any embedment of the optical fiber, simplifying the realization of the samples.

The next chapter will study how the critical toughness can be computed using the optical fiber to monitor the crack propagation. This method will be compared to existing ones.
# Chapter 3: Critical toughness and cohesive law experimental determination methodology

This chapter intends to study the options to obtain the critical toughness with the help of crack length monitoring using optical fiber and DIC local surface displacement measurement on the edge of the assembly. The first part presents the realized investigations in mode I, mode II, and mixed-mode (DCB, ENF and MMB) on steel-to-steel bonded assembly using epoxy resin B. During these investigations, critical toughnesses were determined using different methods. For mode I and II, methodologies proposed by standards with slight modifications were applied in order to use the proposed crack length propagation measurement with continuous optical fiber. Then, a new methodology is tested to obtain the critical toughness and cohesive laws using J-integral method. The different methodologies are finally compared in order to identify the ones that will be used for the modeling.

## 3.1. Realized experimental investigations

### 3.1.1. Samples and experimental program

The specimens chosen for this analysis are similar to the ones described in part 2.3.1. The overall size of the samples remains the same in length, width and adhesive thickness. Only the adherent thickness was increased to 6 mm. The adhesive was a resin that can be used for infusion (called resin B) (Table 3.1-1, Appendix A4).

<table><caption>Table 3.1-1: Resin B mechanical properties in tension.</caption><thead><tr><td>Young modulus</td><td>Tensile strength</td><td>Poisson coefficient</td><td>Strain at failure</td></tr></thead><tbody><tr><td>3.2 GPa</td><td>58 MPa</td><td>0.43</td><td>2,5%</td></tr></tbody></table>

For each series of test (DCB, ENF, MMB), two samples were made with a pre-crack being made at each end of the samples, leading to two possible tests for one sample (number #1 or #2 was specified corresponding to the tested side of the specimen). The pre-crack was initiated with the same manner as described in part 2.3.1. The adhesive thickness was controlled with 1 mm insert at each end of the samples.

During the ENF investigations, specific tests were carried out related to the use of primer. Two sets of samples were made. One was prepared with primer. The second one was prepared without primer and was used as a reference.

The DCB (4 tests) and MMB (6 tests) tests were made using the samples with primer.

The samples were made following the preparation procedure already described previously. The steel plates have been sandblasted (rugosity was measured, $5 \mu m < Ra < 10 \mu m$), then solvent cleaned. The straightness of the substrate was checked after sandblasting to ensure a thickness of the adhesive as constant as possible along the sample length. The primer has been applied with a lint free cloth in thin layer (Figure 3.1-1a and b). The samples were then bonded (Figure 3.1-2). In comparison with the procedure described in chapter 2, because of the change in the resin choice, the resin application protocol was modified. The resin B viscosity being much lower than the one of the resin A, the resin application was made by "resin flow" technique.
The samples were prepared and set in place on the flank with the 1 mm gap between the adherents remaining empty. The resin was then introduced inside the sample by gravity at the center of the sample (Figure 3.1-3). This manufacturing process prevents from any bubble in the bondline and ensures good quality of the bondline (in particular, close to the pre-crack). Following the results obtained in chapter 2, an optical fiber was bonded on the surface of all the samples to monitor the crack propagation during the tests.

Table 3.1-2 summarizes the nomenclature of all the samples made for this study. Note that the sample V-2 broke during the manipulation of the sample and could not be studied.

<table><caption>Table 3.1-2: Fracture tests nomenclature.</caption><thead><tr><th></th><th>Samples with resin B with primer</th><th>Samples with resin B and without primer</th></tr></thead><tbody><tr><td>DCB test</td><td>DCB I -1 + DCB I -2<br>DCB II -1 + DCB II -2</td><td>-</td></tr><tr><td>ENF test</td><td>ENF V -1 + ENF V -2<br>ENF VI -1 + ENF VI -2</td><td>ENF III -1 + ENF III -2<br>ENF IV -1 + ENF IV -2</td></tr><tr><td>MMB test</td><td>MMB I -1 + MMB I -2<br>MMB II -1 + MMB II -2</td><td>-</td></tr></tbody></table>

<figure><img src="image_91.png" /><figcaption>Figure 3.1-1: a) Primer application with the lint free cloth, b) Primer applied in thin layer.</figcaption></figure>

<figure><img src="image_92.png" /><figcaption>Figure 3.1-2: ENF Samples bonding.</figcaption></figure>
<figure><img src="image_93.png" /><figcaption>Figure 3.1-3: Resin flow during the bonding process.</figcaption></figure>

### 3.1.2. Additional monitoring system: Digital Image Correlation

A DIC system was used in addition to the already described optical fiber monitoring system to measure the local displacement fields (local sliding, opening or rotation) at crack tip (initial crack front) on one flank of the specimens. The use of DIC was only limited to the local displacement measurement and it was not used for the crack position monitoring, as it was already performed by the optical fiber.

For each test, a black and white speckle is deposited on the specimen by non-homogeneous spraying of paint in three successive layers (white / black / white) with an approximative pattern size of around 0.2 mm.

The used optical set-up was the following: two 5 Mpx Basler acA2440-75um cameras installed with two Kowa LR1015WM52 lenses (50mm) at an angle of 15 degrees, placed at a certain distance of the sample depending on the required surface measurement. The acquisition rate was set at 5 Hz. The monitoring is made by a commercial system (CorreliSTC by Corelli Solution) to carry out digital image stereo-correlation on the flank of the specimen, at crack tip, as visible in Figure 3.1-4. The cameras were adjusted following the Corelli© software procedure (brightness / sharpness / calibration / synchronization). The distance between the sample and the cameras depends on the suited measurement area. This area called region of interest (ROI) remains in our case close to 6cm x 2cm around the initial crack location. The choice of the resolution is made following the recommendation of the Correli software speckling analysis tool that proposes the best mesh size compromise between spatial resolution and measurement accuracy (set at 16x16 pixels for all tests).

<figure><img src="image_94.png" /><figcaption>Figure 3.1-4: Example of speckle and initial crack front.</figcaption></figure>
## 3.2. Determination of critical toughnesses using standard procedures

### 3.2.1. Obtained failure mode

The failure mode of all the samples were registered to determine the weak link of the bonding. All the pictures of the ENF tests are visible in Appendix A1. For the DCB tests, the failure is fully cohesive in the adhesive layer (Figure 3.2-1). For each DCB test, two parts were visible in the failure pattern: a first one corresponding to the initial unstable crack propagation in the shape of a wave (crack front shape is also visible); and a second one, with a much more disturbed pattern for the stable crack propagation step. This strange behavior has already been described for thick (10mm) epoxy steel-to-steel bonding in [115] where the authors conclude that this phenomenon should be expected when adhesive's critical toughness is lower than the interfacial toughness.

<figure><img src="image_95.png" /><figcaption>Figure 3.2-1: DCB resin B fracture surface.</figcaption></figure>

For the ENF case, the bonding without primer (configuration 1) shows a perfect adhesive failure (the interface is thus considered as the weak point). When using the primer, the failure seems also to be adhesive. Nonetheless, it appears not clear if it is the primer/steel interface, or the primer/resin interface that has broken. The difference can hardly be made due to the really thin layer of primer.

For the MMB test, at the 50% mode ratio, the failure is mixed adhesive and cohesive (in adhesive layer). Whereas at the ratio 75%, the failure is 100% adhesive (Figure 3.2-2). These results tend to prove that for this resin, the analysis of the failure mode can give information on the loading mode ratio in the bondline when it is initially unknown.

The analysis of the failure mode allowed us validating the quality of the bonding procedure, as no bubble in the bondline are visible and as the initial crack front is well defined.

<figure><img src="image_96.png" /><figcaption>Figure 3.2-2: Resin B fracture surface for different modes.</figcaption></figure>
### 3.2.2. Obtained results in mode I (DCB investigations)

Following the ASTM standard test analysis, with the help of the optical fiber for crack length measurement, the three methods (BTM, MBTM and CCM), described in part 1.4.3, were used to compute the critical toughness $G_{Ic}$. The standard proposes two main critical value computation. The first one, which can be called the critical toughness at initiation, is computed with the value of the crack just before the crack propagation occurred, where the loading P is maximum. As expressed in part 1.4.3, this value is supposed to be the more conservative, compared to the second one, after crack propagation. The strain energy release rate curve after crack propagation should stabilize at a higher value of critical toughness (the propagation critical toughness).

#### 3.2.2.1. Load/displacement

The analysis of the load/displacement curves visible in Figure 3.2-3 shows that for the four performed tests, the load at failure ranges between 720N and 828N. The first part of the curves shows non-linearity for displacement of ~1 mm. This is supposed to be related to defaults in the samples manufacturing (adhesion of the normally unbonded part of the sample) but not to the material behavior.

The post-peak behavior seems to be very different between the four samples. For the tests I-1, II-1 and II-2, crack propagates continuously after a certain crack length, and tends to stabilize, while for the test I-2, an uncontrolled crack propagation pattern can be observed (series of violent load decrease during all the test), complexifying the determination of propagation critical toughness (Table 3.2-1).

<table><caption>Table 3.2-1: DCB test nomenclature and crack propagation information.</caption><thead><tr><th></th><th>Initial crack propagation</th><th>Crack propagation stabilization</th></tr></thead><tbody><tr><td>DCB I-1</td><td>unstable</td><td>Stable (when displacement >4mm)</td></tr><tr><td>DCB I-2</td><td>unstable</td><td>Unstable</td></tr><tr><td>DCB II-1</td><td>unstable</td><td>Stable (when displacement >5mm)</td></tr><tr><td>DCB II-2</td><td>unstable</td><td>Stable (when displacement >3mm)</td></tr></tbody></table>

<figure><img src="image_97.png" /><figcaption>Figure 3.2-3: DCB tests load/displacement curves.</figcaption></figure>
3.2.2.2. Critical toughness

In the next paragraph, the computation of the critical toughness following the three methods (BTM, MBTM and CCM) will be described for the typical case of the test II-I. For this test, an initial unstable crack propagation is observed (Figure 3.2-4a), followed by a stable crack propagation. For the critical toughness determination, only the stable propagation is considered. The crack propagation curve obtained thanks to the exterior optical fiber is shown in Figure 3.2-4b.

<figure><img src="image_98.png" /><figcaption>Figure 3.2-4: a) II-I test load/time curve, and b) crack propagation curve.</figcaption></figure>

The two unstable crack propagations, occurring at t=73s and t=87s, are clearly visible on the
crack propagation curve (violent jump of the curve), and the reloading of the bondline and
stable crack propagation occurring after t=125s are also clearly visible.

Figure 3.2-5a displays the linear regression curve of C¹/³=f(a). This curve allows the computation of the Δ parameter (crack length compensation factor) in the MBTM. The obtained value is Δ=-2.92.

Figure 3.2-5b shows the linear regression curve of Ln(C)=f(Ln(a)) used in the CCM. The parameter value n=2.62 is obtained. It can be noted that, in both cases, these linear approximations are relatively accurate with a correlation coefficient R² superior to 0.94.

<figure><img src="image_99.png" /><figcaption>Figure 3.2-5: a) Linear approximation of the C¹/³=f(a) curve for the Δ computation, b) Linear approximation of Ln(C)=f(Ln(a)) for n computation.</figcaption></figure>
Based on the equations detailed in part 1.4.3 (equation 60 and 61) the BTM, MBTM and CCM toughnesses can be computed. The resulting curves are displayed in Figure 3.2-6. The crack initiation critical toughness is displayed with colored star at the beginning of the curves, while the stabilization values (mean) are displayed as colored line on the stabilized part of the curves. We can see, as expected, that the initiation values are much lower than the stabilization values. It can also be concluded, that the BTM values are much higher than the ones obtained with the two other theories, the MBTM being the more conservative for the stabilized value and for the initiation value.

<figure><img src="image_100.png" /><figcaption>Figure 3.2-6: Critical toughness in mode I at initiation and stabilization for the II-1 test.</figcaption></figure>

All the critical toughness computations performed for the DCB tests are summarized in the Table 3.2-2. Figure 3.2-7 compares average values of the critical toughness obtained for the BTM, MBTM and CCM methods.

<table><caption>Table 3.2-2: Standard critical toughness characterization with optical fiber crack length measurement.</caption><thead><tr><th>GIc (kJ/m²)</th><th>BTM at initiation</th><th>MBTM at initiation</th><th>CCM at initiation</th><th>BTM at stabilization</th><th>MBTM at stabilization</th><th>CCM at stabilization</th></tr></thead><tbody><tr><td>Test I-1</td><td>0.25</td><td>0.18</td><td>0.18</td><td>0.3</td><td>0.22</td><td>0.25</td></tr><tr><td>Test I-2</td><td>0.28</td><td>0.24</td><td>0.23</td><td>No stabilization</td><td>No stabilization</td><td>No stabilization</td></tr><tr><td>Test II-1</td><td>0.24</td><td>0.23</td><td>0.21</td><td>0.32</td><td>0.25</td><td>0.24</td></tr><tr><td>Test II-2</td><td>0.21</td><td>0.15</td><td>0.16</td><td>0.32</td><td>0.28</td><td>0.28</td></tr><tr><td>Average</td><td>0.25</td><td>0.20</td><td>0.20</td><td>0.31</td><td>0.25</td><td>0.26</td></tr><tr><td>Standard deviation /average (%)</td><td>10%</td><td>21%</td><td>20%</td><td>5%</td><td>12%</td><td>8%</td></tr></tbody></table>

The critical toughness at initiation is the lower critical toughness for all the tests by 20-30% independently of the chosen computation methods (Figure 3.2-7). The dispersion of the stabilized results for the BTM, MBTM and CCM remains inferior to 12%, while the dispersion for the critical toughness at initiation tends to be much higher (9%-21%). The limited number of tests allows limited confidence into the values of standard deviation. The differences in values of the BTM, MBTM and CCM is consistent with the description of the ASTM standard [87]. The difference between the MBTM and the CCM is minor at initiation and stabilization
values which indicates that the choice of one of the two methods should provide similar results for brittle adhesives (our case).

<figure><img src="image_101.png" /><figcaption>Figure 3.2-7: Critical toughness computation results in mode I.</figcaption></figure>

### 3.2.3. Obtained results in mode II (ENF investigations)

For the methodology of computation of the critical toughness in mode II, it has been detailed in part 1.4.4, that the MBTM and CCM can be applied [113] in case of controlled crack propagation. This was not the case in this study with all of tests ending with uncontrolled crack propagation. The analysis had therefore to be performed following the ASTM standard proposal. This ASTM standard [110] proposes approximations based on the measurement of the compliance of the ENF samples and the load at failure (as described in part 1.4.4).

The studied samples were tested with different initial crack length $a_0$ (between 59 mm and 90 mm). The three-point bending tests were performed up to the failure load value. The compliance was computed following the standard recommendations. The nomenclature of the specimens is as follows: specimens III and IV for configuration without primer; specimens V and VI for configuration with primer (Table 3.2-3).

<table><caption>Table 3.2-3: ENF test nomenclature.</caption><thead><tr><td></td><td>Surface preparation</td><td>a<sub>0</sub> (mm)</td></tr></thead><tbody><tr><td>ENF III-1</td><td>Without primer</td><td>73</td></tr><tr><td>ENF III-2</td><td>Without primer</td><td>90</td></tr><tr><td>ENF IV-1</td><td>Without primer</td><td>74</td></tr><tr><td>ENF IV-2</td><td>Without primer</td><td>91</td></tr><tr><td>ENF V-1</td><td>With primer</td><td>59</td></tr><tr><td>ENF VI-1</td><td>With primer</td><td>74</td></tr><tr><td>ENF VI-2</td><td>With primer</td><td>90</td></tr></tbody></table>

The post-test analysis of the sample has shown that the ENF VI-1 sample was already broken before the beginning of the test, thus this test was not considered in the analysis.

In the next paragraph, the computation of the critical toughness in mode II, $G_{IIc}$, is detailed.

#### 3.2.3.1. Load/displacement

Figure 3.2-8 displays the load/displacement curves for tests V-1, VI-1 and VI-2 (samples with primer) and III-1, III-2, IV-1 and IV-2 (sample without primer), corresponding to different initial crack length $a_0$. A difference in compliance between the samples is visible (difference in the initial slope of the curves). The uncontrolled crack propagation (fast decrease of the curve after the maximal load) and absence of non-linearity before the crack propagation are also visible.
<figure><img src="image_102.png" /><figcaption>Figure 3.2-8: ENF tests load/displacement curves with primer.</figcaption></figure>

<figure><img src="image_103.png" /><figcaption>Figure 3.2-9: ENF tests load/displacement curves without primer.</figcaption></figure>

### 3.2.3.2. Critical toughnesses

In the next paragraphs, the computation of the critical toughness for the specimens with and without the primer is detailed. The compliance was computed following the standard recommendations. The compliance value is computed as a linear regression of the load/displacement curve between the load P1=Pmax/10 and P2=Pmax/2. This is supposed to help to avoid non-linear part in the curve. Figure 3.2-10 displays the computed regression parameter of the C=f(a³) curve computed following the ASTM standard. It shows relatively good correlation between the linear approximation and the results (compared to result obtained without primer, Figure 3.2-11).
<figure><img src="image_104.png" /><figcaption>Figure 3.2-10: Linear approximation of the C=f(a³) curve with primer.</figcaption></figure>

<figure><img src="image_105.png" /><figcaption>Figure 3.2-11: Linear approximation of the C=f(a³) curve without primer.</figcaption></figure>

It is to be noted that a minimum of three tests is required. The critical toughness's results obtained for each series of ENF test are summarized in the Table 3.2-4.

<table><caption>Table 3.2-4: Critical toughness computation following the ASTM D7905 standard.</caption><thead><tr><th></th><th>Load at failure<br>P<sub>max</sub> (N)</th><th>Initial crack length<br>a<sub>0</sub> (mm)</th><th>Compliance<br>C</th><th>Slope of the linear<br>regression<br>m</th><th>Intercept of the<br>linear regression<br>A</th><th>Resulting<br>critical<br>toughness G<sub>IIc</sub></th></tr></thead><tbody><tr><td><strong>Resin B<br>without Primer</strong></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>III-2</td><td>6 680</td><td>73</td><td>0.00040</td><td rowspan="4">3.04x10<sup>10</sup></td><td rowspan="4">0.0003</td><td>1.73</td></tr><tr><td>III-1</td><td>7 450</td><td>90</td><td>0.00047</td><td>3.28</td></tr><tr><td>IV-1</td><td>8 090</td><td>74</td><td>0.00038</td><td>2.62</td></tr><tr><td>IV-2</td><td>6 260</td><td>91</td><td>0.00052</td><td>2.37</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>Mean</td><td>2.50</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>Standard<br>deviation</td><td>25%</td></tr><tr><td><strong>Resin B with<br>primer</strong></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>VI-1</td><td>8 700</td><td>74</td><td>0.00037</td><td rowspan="4">2.16 x10<sup>10</sup></td><td rowspan="4">0.00028</td><td>2.15</td></tr><tr><td>VI-2</td><td>7 710</td><td>90</td><td>0.00045</td><td>2.50</td></tr><tr><td>V-1</td><td>11 060</td><td>59</td><td>0.00034</td><td>2.21</td></tr><tr><td>V-2</td><td>unexpected failure</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>Mean</td><td>2.29</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>Standard<br>deviation</td><td>8%</td></tr></tbody></table>
The limited number of tests does not allow a high confidence into the values of standard
deviation. However, the results with primers seem to have a lower dispersion compared to the
results without primer.

The results show that the results with the primer (2.3 kJ/m²), and without any primer (2.5 kJ/m²) are similar (Figure 3.2-12). In addition, we observe that the primer does not decrease the critical toughness of the resin B interface on steel.

<figure><img src="image_106.png" /><figcaption>Figure 3.2-12: ASTM Critical toughness computation.</figcaption></figure>

**3.2.4. Obtained results in mixed mode (MMB investigations)**

3.2.4.1. Load/displacement

To obtain the final critical toughness ratio curve, a series of 4 MMB tests was performed. The load/displacement curves obtained for these tests are shown in Figure 3.2-13.

<figure><img src="image_107.png" /><figcaption>Figure 3.2-13: MMB load/displacement curves.</figcaption></figure>

3.2.4.2. Critical toughness

The critical toughness determination was realized following the standard ASTM D6671 [101] as described in the part 1.4.5. It is specified that in case of uncontrolled crack propagation, an approximation of G<sub>Ic</sub> and G<sub>IIc</sub> can be made using the maximal load P<sub>max</sub>, and the initial crack length a<sub>0</sub>. Nonetheless, it may cause an underestimation of G<sub>c</sub> value. In our case, crack propagation is very different for the 50% and 75% G<sub>IIc</sub>/G<sub>c</sub> ratios as visible in the load/displacement curves in Figure 3.2-13. The final results are summarized in Table 3.2-5.
<table><caption>Table 3.2-5: Critical toughness computation of the mixed mode test.</caption><thead><tr><th>Ratio (G<sub>II</sub>/G<sub>c</sub>)</th><th>50%</th><th>75%</th></tr></thead><tbody><tr><td><strong>Critical toughness (kJ/m<sup>2</sup>)</strong></td><td>0.18</td><td>0.22</td></tr><tr><td></td><td>0.17</td><td>0.28</td></tr><tr><td><strong>Average</strong></td><td>0.17</td><td>0.25</td></tr></tbody></table>

Figure 3.2-14 shows the resulting critical toughness in function of the mode ratio. It can be seen that the results are in accordance with the failure mode analysis, as, the critical toughness in mixed mode between 0% and 50% of mode II remains similar and corresponds to a cohesive failure. Then, between 50% and 100% of mode II, the critical toughness increases by a coefficient 10 up to more than 2kJ/m².

<figure><img src="image_108.png" /><figcaption>Figure 3.2-14: Critical toughness average value in function of the mode II ratio for the resin B.</figcaption></figure>

### 3.3. Determination of critical toughnesses and cohesive laws using J integral for mode I and II

Main methods (described in part 1.4.4, 1.4.5, 1.4.3) to obtain the critical toughness in mode I and II have been described previously: the beam theory method, the modified beam theory method and the compliance calibration method [113]. These methods have been adapted to the crack propagation monitoring with optical fiber (compare to the standard visual measurement) and will be in this part, compared to the J-integral analysis performed for the DCB and ENF tests.

#### 3.3.1. J-integral expression for DCB and ENF tests

For adhesive bonding loaded in mode II, Leffler et al. [114] proposed a formulation of the J-integral considering an integration path in the bondline, as shown in Figure 3.3-1. This path can be shrunk to a straight line in the y-direction at the crack tip, and $\sigma_{ij} \cdot n$ has to be null since the crack tips is a free surface.
<figure><img src="image_109.png" /><figcaption>Figure 3.3-1: Adhesive joint and J-integral integration path.</figcaption></figure>

From the equation (45), the equation (80) can then be obtained.

$$J = \int_{t_a} w(\varepsilon)dy \qquad (80)$$

With $w(\varepsilon)$, the strain energy density which can be divided in two parts, with the elongation, $\varepsilon_1$, and the shear deformation, $\varepsilon_2$, and the conjugated stresses $\sigma$ and $\tau$, resulting in the equation (81).

$$J = \int_{0}^{\varepsilon_1} \sigma dy + \int_{0}^{\varepsilon_2} \tau dx = J_I + J_{II} \qquad (81)$$

In the particular case of the ENF test, numerical analyses led by the authors showed that almost no normal stress appears in the adhesive layer except at the loading point. Thus, the equation can be simplified into equation (82).

$$J = \int_{0}^{\varepsilon_2} \tau dx \qquad (82)$$

In order to solve the equation (82), Leffler et al. considered a closed integration path as shown in Figure 3.3-2.

<figure><img src="image_110.png" /><figcaption>Figure 3.3-2: ENF J-integral integration path A+B+C+D.</figcaption></figure>

Based on this integration path, the expression of J can be partitioned as in equation (83).

$$0 = J_A + J_B + J_C + J_D \qquad (83)$$

With:

$$-J = J_B \qquad (84)$$

$$J_D = \int_{t_a} w(\varepsilon(x = l_0, y))dy - \int_{t_a} \tau \frac{du_y}{dx} |_{x=l_0,y} dy \qquad (85)$$

In the case of the distance $l_0$, between the crack tip and the loading point, being long enough, the adhesive can be assumed to be in a linear elastic state, and thus the shear in the thickness of
the adhesive can be considered as independent on the thickness. Moreover, $\frac{du_i}{dx} = \theta$ is equal to the local rotation at the loading point. This leads to the equation (86) for the expression of $J_D$.

$$J_D = w(\varepsilon(x = l_0))t_a + \tau(x = l_0) \cdot \theta(x = l_0) \cdot t_a \quad (86)$$

Due to horizontal integration paths for $J_A$ and $J_C$ only the second term contributes:

$$J_A = \int_0^{l_0} \tau \frac{du_u}{dx} dx \quad (87)$$

And

$$J_C = - \int_0^{l_0} \tau \frac{du_l}{dx} dx \quad (88)$$

With $u_u$ and $u_l$ the horizontal displacement of the upper and the lower interface.

In the case of the addition of $J_A + J_C$ the equation (89) can be obtained.

$$J_A + J_C = - \int_0^{l_0} \tau \frac{d(u_u - u_l)}{dx} dx \quad (89)$$

Alfredsson et al. in [118], based on Euler-Bernoulli beam theory, gives an expression of the shear stress in the adhesive layer for $a<x<l_0$.

$$\tau = \frac{Eh_s}{8} \frac{d^2(u_u-u_l)}{dx} + \frac{3}{8} \frac{P}{h_s b} \quad (90)$$

With P the load, $h_s$ the adherent thickness, b the sample width and E the adherent Young's modulus. This expression allows to directly compute the integrals as described in equation (91).

$$J_A + J_C = -\frac{Eh_s}{16} [U'(x = l_0)^2 - U'(x = 0)^2] - \frac{3}{8} \frac{P}{bh_s} [U(x = l_0) - U(x = 0)] \quad (91)$$

With $\frac{d(u_u-u_l)}{dx} = U'$ and $u_u - u_l = U$

At $x = 0$, Alfredsson et al. proposed the boundary condition (based on beam theory) [118]:

$$U'(x = 0) = - \frac{3Pa_0}{Ebh_s^2} \quad (92)$$

With $a_0$ the initial crack length.

Finally, the addition of all terms leads to the equation (93):

$$J = \frac{9 P^2 a_0^2}{16 Eb^2 h_s^3} - \frac{Eh_s}{16} [U'(x = l_0)^2] + \frac{3 P (\delta_s - U(x = l_0))}{8 bh_s} \\ + w(\varepsilon(x = l_0)) \cdot t_a + \tau(x = l_0) \cdot \theta(x = l_0) \cdot t_a \quad (93)$$

With $\delta_s = U(x = 0)$, the local sliding as explained in Figure 3.3-3.
<figure><img src="image_111.png" /><figcaption>Figure 3.3-3: ENF local sliding at the crack front.</figcaption></figure>

The first term of the equation (94) corresponds to the case of a rigid adhesive layer (beam theory) while the next ones give the influence of a flexible adhesive layer.

If a large distance between the crack tip and the loading point, $l_0$, is considered, $U(x = l_0)$ can be neglected compared to $U(x = 0) = \delta_s$. Moreover, $w(\varepsilon(x = l_0))$ can also be neglected compared to the strain energy at the crack tip. Finally, the two last terms of the equation (94) have a very low contribution that may be neglected. It is thus assessed to be around 0,1% by Leffler et al. [117].

Therefore, a simplified expression of the J-integral can be considered (equation 94):

$$J_{ENF} \approx \frac{9 (Pa_0)^2}{16 Eh_s^3 b^2} + \frac{3P\delta_s}{8h_s b} \quad (94)$$

For adhesive bonding loading in mode I (DCB tests), a simplified analytical expression of the J-integral was proposed by Högberg et al. [117] based on the same type of analysis led by Leffler et al. [114]. In that case, the considered integration path is described in the Figure 3.3-4. Högberg et al. act that, on this integration path, only the paths A and C contribute to $J_B$. Thus, the J-integral can be expressed directly using the adherends properties and the load following the equation (95). It must be specified that the first term corresponds to the beam theory considering a rigid adhesive, while the second term corresponds to the effect of the flexibility of the adhesive and the local rotation of the adherent.

$$J_{DCB} \approx \frac{12(Pa_0)^2}{Eb^2h_s^3} + \frac{P}{b}|\theta_{sup} - \theta_{inf}| \quad (95)$$

<figure><img src="image_112.png" /><figcaption>Figure 3.3-4: DCB J-integral integration path A+B+C.</figcaption></figure>

For our DCB and ENF tests, the digital image correlation system was used to measure the displacement fields at crack tip (local sliding and local rotation of the upper and the lower adherent, $\delta_s$ and $|\theta_{sup} - \theta_{inf}|$ respectively) on the side of the specimens, and thus to compute the J-integral.
### 3.3.2. Cohesive law expression for DCB and ENF tests

Rice produced two alternative expressions of J [118], [51]. He used the Barrenblatt approach which considers a cohesive zone ahead of the crack tips in which the restraining stress $\sigma(\delta_n)$ is represented as a function of the opening $\delta_n$. If the J-integral is evaluated along contour $\Gamma = \sum_1^3 \Gamma_i$ (example in Figure 3.3-5 for mode I loading) which runs along the cohesive zone, $\Gamma_2$ is upstream of the crack front. Therefore, there is no stress, $\sigma_{ij}$. n and dy are equal to zero. In addition, $\Gamma_1$ can be taken at distance long enough from the crack tip, and therefore $J_1 \approx 0$. On the contour $\Gamma_3$, dy = 0, $\sigma_{ij}$. n= $\sigma(\delta_n)$ because n = y and $d\Gamma = dx$. The equation (96) can then be obtained.

$$J = \oint_{\Gamma_3} \sigma(\delta_n) \frac{d\delta_n}{dx} dx \qquad (96)$$

This expression can be explained as follows. If the crack opening, or sliding, at each point in the cohesive zone increases by an amount $d\delta$, then the profile of the cohesive zone boundary extends of a distance $dx$. The quantity $\sigma(\delta_n)dx$ is the load over each infinitesimal area and $\sigma(\delta_n) d\delta_n dx$ is the energy absorbed for a separation increase of $d\delta_n$. Then J, in the equation (97), is defined as a rate of energy absorption, respecting the cohesive zone propagation.

$$J = \int_{0}^{\delta_t} \sigma(\delta_n) d\delta_n \qquad (97)$$

With $\delta_t$ the local opening at the crack tips which has to be measured.

When the critical opening is achieved, at crack propagation, $J = J_c$, which is the area under the $\sigma(\delta_n)$ curve and can be seen as the rate of energy absorption in the cohesive zone.

<figure><img src="image_113.png" /><figcaption>Figure 3.3-5: Cohesive zone ahead of the crack tip.</figcaption></figure>

The relation between the J-integral expression and the cohesive law $\sigma(\delta_n)$, is expressed in the equation (98), with $\delta_n$ the local opening of the sample at the crack tip.

$$\sigma(\delta_n) = \frac{dJ_{DCB}}{d\delta_n} \qquad (98)$$

Consequently, with a limited data (load/displacement curve and local displacement measurement), the complete cohesive law shape should be obtained from the experimental data.

As for the DCB test, the relation between the J-integral expression and the cohesive law, $\tau(\delta_s)$ can be obtained for the ENF test. It is expressed in the equation (99). Following the same analysis in shear loading, with $\delta_s$ the local sliding of the sample at the crack tips.

$$\tau(\delta_s) = \frac{dJ_{ENF}}{d\delta_s} \qquad (99)$$
### 3.3.3. Analysis of the DIC strategy to monitor local displacement

The Leffler and Högberg expressions of the J-integral for ENF and DCB tests, require to measure the local displacement (sliding, adherent rotation and opening) at the crack tip.

Using the data provided by the DIC, the local measurements are used to obtain, for the J-integral computation, $|\theta_{sup} - \theta_{inf}|$, $\delta_n$ and $\delta_s$. Figure 3.3-6 presents the 4 positions of the required measurement points (A, B, C, D) to compute the local rotations. For the local opening and local sliding, only the points B and C should be required.

<figure><img src="image_114.png" /><figcaption>Figure 3.3-6: DIC measurement point on test samples.</figcaption></figure>

The local sliding is obtained following the equation (100).

$$ \delta_s = |DX_B| - |DX_C| \qquad (100) $$

With $DX_i$ the local displacement in the x direction of the point i compared to the initial position
The local opening is obtained following the equation (101).

$$ \delta_n = |DY_B| - |DY_C| \qquad (101) $$

With $DY_i$ the local displacement in the y direction of the point i compared to the initial position
Local rotations are obtained following the equation (102).

$$ |\theta_{sup} - \theta_{inf}| = \left| \frac{|DX_A| - |DX_B|}{Y_A - Y_B} - \frac{|DX_D| - |DX_C|}{Y_D - Y_C} \right| \qquad (102) $$

The measurements should be carried as close as possible to the adhesive layer and the initial
crack position. However, because of technical limitations of the use of the DIC, positions of
measurement points, A, B, C and D, may experimentally vary according to the type of tests and
also from one to another.

It must be noted that the use of DIC as the measurement device for these values may have some
limitation:

- DIC meshing may not be aligned with the different materials. Actually, most of the time, at the steel-adhesive interface, one element represents this interface. Therefore, the local displacement measurement at this point is not accurate,

- the definition of the exact numerical position of the experimental initial crack front is not easy. Figure 3.3-7 presents an ENF test speckle for which the initial crack front was marked with a thin black pen. The pencil thickness is large and thus introduces incertitude on the exact initial crack front position.

The next paragraphs will detail how these problematics were solved.
<figure><img src="image_115.png" /><figcaption>Figure 3.3-7: ENF test DIC speckle showing difficulties to have a precise initial crack front position based on the marking.</figcaption></figure>

Figure 3.3-8 presents the X displacements map for a DCB test prior to the crack propagation. The white line represents a chosen measurement profile in Correli software used to obtain the required X displacement values for the computation of $|\theta_{sup} - \theta_{inf}|$ and the Y displacement values for the computation of $\delta_n$.

<figure><img src="image_116.png" /><figcaption>Figure 3.3-8: DCB test DIC X displacement map.</figcaption></figure>

Figure 3.3-9 presents the obtained X displacement corresponding to the chosen profile. The figure shows that the curve is divided in two linear parts for the two adherents plus the bondline. It has been decided to use the maximum and minimum local displacement as the values for the calculation of the adherent rotations following the equation (103).

<figure><img src="image_117.png" /><figcaption>Figure 3.3-9: DCB test DIC X displacement profiles through the sample thickness.</figcaption></figure>
Figure 3.3-10 displays the X displacement map for an ENF test, prior to the crack propagation. The white line displays the measurement profiles where the local displacements through the thickness of the sample are registered. They are then used to obtain the required $\delta_s$ values. The Figure 3.3-11 presents the resulting profiles through the sample thickness.

Note also in Figure 3.3-10 that the initial crack is partially visible on the right side of the figure. However, its exact position is not clearly defined (on few elements).

<figure><img src="image_118.png" /><figcaption>Figure 3.3-10: Example of ENF test DIC X displacement map and local profiles.</figcaption></figure>

<figure><img src="image_119.png" /><figcaption>Figure 3.3-11: ENF test DIC X displacement profiles through sample thickness at the crack front position.</figcaption></figure>

Based on the assumption that the maximal displacement should occur at the bonded surface of the adherents, it has been decided to use the maximum (for the upper adherent) and minimum (for lower adherent) local displacements as the value of interfaces local displacements (local sliding, equation (100)).

The issue related with the determination of the exact position of the displacement measurement profile (which should be at the same position as the crack front) was studied assessing the impact of an error of this position on the critical J-integral, $J_c$. Figure 3.3-12 presents an example of the computation of J-integral curves obtained for one ENF test (V-1). The blue curve is set
as the reference at the initial crack position. The three other curves are obtained with measurement point shifted in the direction of the bondline from 0.7 to 3.3 mm.

<figure><img src="image_120.png" /><figcaption>Figure 3.3-12: ENF SV-1 test, J-integral computation depending of the error of position of the X displacement profile.</figcaption></figure>

It can be seen that higher is the error, lower are the local sliding measurement and the Jc value.
Figure 3.3-13 and Figure 3.3-14 present the resulting absolute values of the error in critical
toughness computation variation for some ENF and DCB tests. It can be seen that this error of
position always induces a diminution of the Jc values (conservative error).

<figure><img src="image_121.png" /><figcaption>Figure 3.3-13: Impact of the position of the profile on the Jc computation for ENF tests.</figcaption></figure>
<figure><img src="image_122.png" /><figcaption>Figure 3.3-14: Impact of the position of the profile on the Jc computation for DCB tests.</figcaption></figure>

### 3.3.4. Obtained results for DCB investigations

The J-integral analysis has been performed using the DIC data on the previously described tests. In mode I, the local rotations have been monitored, as well as the load and the local opening to obtained the *Jc* value. The cohesive laws are directly obtained by computation of a derivative curve of the polynomial approximation of degree 3 of the J-integral curve (equation 95) based on the equation (95). Figure 3.3-15 presents the resulting curve for three of the four tests and Table 3.3-1 the polynomial approximations of the J-integral curves. A problem with the DIC monitoring during the II-1 test preventing us from obtaining data for this test.

<figure><img src="image_123.png" /><figcaption>Figure 3.3-15: J-integral computation curves for the DCB tests.</figcaption></figure>

<table><caption>Table 3.3-1: DCB test J-integral polynomial approximation.</caption><tr><td>Test I-1</td><td>-277895x<sup>3</sup>+7268x<sup>2</sup></td></tr><tr><td>Test I-2</td><td>-344055x<sup>3</sup>+7615x<sup>2</sup></td></tr><tr><td>Test II-2</td><td>-507536x<sup>3</sup>+8957x<sup>2</sup></td></tr></table>

The curves show a direct increasing J-integral computation up to the failure value. The lack of stabilization of the curve shows that the failure is brittle with no plasticity in the bondline. The critical values *Jc* are obtained as the maximum of the curves. These values are reported in the Table 3.3-2.
<table><caption>Table 3.3-2: Critical toughness computation in mode I based on the J-integral curves.</caption><tr><td>Jc (kJ/m²)</td><td>0.38<br>0.28<br>0.27</td></tr><tr><td>Average</td><td>0.31</td></tr><tr><td>Standard deviation / Average</td><td>20%</td></tr></table>

The cohesive laws are then directly obtained by computation of a derivative curve of the polynomial approximation of the J-integral curves (equation 99), see Figure 3.3-16.

<figure><img src="image_124.png" /><figcaption>Figure 3.3-16: Cohesive laws in mode I based on the J-integral curves derivation.</figcaption></figure>

The curves obtained from the J-integral polynomial approximation show a lack of damage and nearly no plasticity, which is in accordance with the brittle behavior of the adhesive observed during the tests.

### 3.3.5. Obtained results for ENF investigations

The J-integral analysis has been performed with the DIC data for all the ENF tests. The local sliding was determined for each test. With the monitoring of the load, the J-integral curves are obtained for the tests without primer (Figure 3.3-17) and with primer (Figure 3.3-18) and the polynomial approximation of the J-integral curves are described in Table 3.3-3. A problem with the DIC monitoring during the V-2 test preventing us from obtaining data for this test. The obtained curve from the J-integral polynomial approximation was a polynomial of degree 3.

<figure><img src="image_125.png" /><figcaption>Figure 3.3-17: J-integral computation curves for the ENF tests without primer.</figcaption></figure>
<figure><img src="image_126.png" /><figcaption>Figure 3.3-18: J-integral computation curves for the ENF tests with primer.</figcaption></figure>

<table><caption>Table 3.3-3: ENF test J-integral polynomial approximation.</caption><thead><tr><th></th><th>Polynomial approximation</th></tr></thead><tbody><tr><td>Test III-1</td><td>-2949x<sup>3</sup>+661x<sup>2</sup></td></tr><tr><td>Test III-2</td><td>-2702x<sup>3</sup>+535x<sup>2</sup></td></tr><tr><td>Test IV-1</td><td>-6797x<sup>3</sup>+1093x<sup>2</sup></td></tr><tr><td>Test IV-2</td><td>-12019x<sup>3</sup>+1324x<sup>2</sup></td></tr><tr><td>Test V-1</td><td>-3394x<sup>3</sup>+728x<sup>2</sup></td></tr><tr><td>Test VI-1</td><td>-3839x<sup>3</sup>+765x<sup>2</sup></td></tr><tr><td>Test VI-2</td><td>-2645x<sup>3</sup>+679x<sup>2</sup></td></tr></tbody></table>

As for the DCB tests, the curves show a direct increasing J-integral computation up to the failure value. It shows that the failure is brittle. The critical values *Jc* are obtained as the maximum of the curves. These values are reported in Table 3.3-4.

<table><caption>Table 3.3-4: Critical toughness computation in mode II based on the J-integral curves.</caption><thead><tr><th></th><th>Without primer</th><th>With primer</th></tr></thead><tbody><tr><td rowspan="4">Jc (kJ/m²)</td><td>1.9</td><td>2.1</td></tr><tr><td>1.1</td><td>1.9</td></tr><tr><td>1.5</td><td>1.75</td></tr><tr><td>1.4</td><td></td></tr><tr><td>Average</td><td>1.5</td><td>2</td></tr><tr><td>Standard deviation</td><td>0.3</td><td>0.1</td></tr><tr><td>Standard deviation / Average</td><td>23%</td><td>5%</td></tr></tbody></table>

The computation of the cohesive laws from the J-integral curve is displayed on Figure 3.3-19. It shows that the three obtained curves are really similar in shape, and in term of maximal stresses.
<figure><img src="image_127.png" /><figcaption>Figure 3.3-19: Cohesive law in mode II based on the J-integral curves derivation.</figcaption></figure>

## 3.4. Conclusion and discussion

The tests performed during this study have shown that, in this particular case, the approximation of the CCM and MBTM methods proposed by the standard proved to be efficient to obtain crack initiation critical toughnesses for MMB and ENF tests with limited instrumentation and expertise. However, these initiation values are known as being low (or conservative) compared to “propagation” critical toughness. Therefore, their use for modeling might lead to too conservative prediction.

Despite being used for only one of the three test computation (DCB), the propagation critical toughness determination with optical fiber crack front measurement had a major advantage compared to standard visual crack front measurement (or DIC). It can be easily automated, leading to fast data analysis and critical toughness computation (few seconds) and avoiding any operator misjudgments. Despite the really good capacity of the optical fiber to follow crack propagation, its use might be limited in real critical toughness test application when uncontrolled crack propagation might be encountered with fragile adhesive (such as the resin B).

The J-integral approach also proved to be an efficient way to obtain the critical toughness with a different monitoring strategy and relatively easily. It must be pointed out that, despite being computed before the crack propagation occurs (as the critical toughness at initiation), the *Jc* computation leads to similar results to the ones obtained with the “propagation critical toughness” (CCM and MBTM) (Table 3.4-1 and Table 3.4-2). The J-integral derivation also proved to be effective in obtaining cohesive laws directly from tests. The obtained laws shapes are consistent with the brittle behavior of the studied adhesive.

<table><caption>Table 3.4-1: Critical toughness computation in mode I.</caption><thead><tr><th>Mode I critical toughness (kJ/m²)</th><th>G<sub>Ic</sub> (CCM)</th><th>G<sub>Ic</sub> (MBTM)</th><th>J<sub>Ic</sub> with primer</th></tr></thead><tbody><tr><td></td><td>0.25</td><td>0.22</td><td>0.38</td></tr><tr><td></td><td>No stabilisation</td><td>No stabilisation</td><td>0.28</td></tr><tr><td></td><td>0.24</td><td>0.25</td><td>0.27</td></tr><tr><td></td><td>0.28</td><td>0.28</td><td></td></tr><tr><td>Average</td><td>0.26</td><td>0.25</td><td>0.31</td></tr><tr><td>Standard deviation</td><td>8%</td><td>12%</td><td>20%</td></tr></tbody></table>
<table><caption>Table 3.4-2: Critical toughness computation in mode II.</caption><thead><tr><th>Mode II critical toughness<br>(kJ/m²)</th><th>G<sub>IIc</sub> Without<br>primer</th><th>G<sub>IIc</sub> with<br>primer</th><th>J<sub>IIc</sub> without<br>primer</th><th>J<sub>IIc</sub> with<br>primer</th></tr></thead><tbody><tr><td></td><td>1.73</td><td>2.15</td><td>1.9</td><td>2.1</td></tr><tr><td></td><td>3.28</td><td>2.50</td><td>1.1</td><td>1.9</td></tr><tr><td></td><td>2.62</td><td>2.21</td><td>1.5</td><td></td></tr><tr><td></td><td>2.37</td><td></td><td>1.4</td><td>1.75</td></tr><tr><td>Average</td><td>2.5</td><td>2.29</td><td>1.5</td><td>2</td></tr><tr><td>Standard deviation</td><td>25%</td><td>8%</td><td>23%</td><td>5%</td></tr></tbody></table>

Parts 3.2.2 and 3.2.3 have provided critical toughness values obtained based one the ASTM standard test procedure for the DCB and ENF test with the optical fiber crack propagation monitoring. These results will be used in chapter 4 as initial parameters of the indirect method for cohesive law modeling. The results obtained in part 3.2.4 for the MMB result will be used to fit mixed mode parameter of the model in chapter 4.

Part 3.3 with the J-integral analysis has provided cohesive laws in mode I and II. These laws will be implemented into the direct method of the cohesive analysis with some approximations to fit the data required by the FEM software used in chapter 4.

The results obtained in this chapter for the cohesive law calibration show that bilinear law should be sufficient to express the interface properties. These results are consistent with literature such as in [120] for DCB test or [123] for single-lap test which conclude that the cohesive law shape has little impact on the behavior and the failure load prediction of such bonded assemblies. For those reasons and for simplicity, only bilinear cohesive laws will be used in numerical models.
# Chapter 4: Description and evaluation of the adopted cohesive zone modeling strategy

In this chapter, two different methods are used to determine the cohesive laws that may be implemented in numerical FEM (finite element models) to design adhesively bonded assembly. The first method is a direct cohesive law calibration, based on the previously described J-integral analysis. The second method is an indirect method based on the critical toughness and relying on the iterative use of FEM to determine the cohesive law. For both methods, it was decided that the cohesive law represents the behavior of the entire assembly (adhesive and both interfaces). The load/displacements curves obtained for the three types of tests (DCB, ENF and MMB) presented in chapter 3 will be used for those determinations and to verify the suitability of the proposed methodologies.

First part will present more deeply the finite element modeling strategy and especially the mixed-mode cohesive zone modeling. The second part will describe more precisely the direct and indirect methods used in this work to determine the cohesive laws in mode I, mode II and in mixed mode. Then, both methods were applied on the results of the three types of tests (DCB, ENF and MMB) presented in chapter 3. In conclusion, the two approaches are compared and discussed.

## 4.1. Presentation of the used finite elements models

### 4.1.1. Cohesive elements

Cohesive elements are used to model delamination and adhesive failure by predicting both crack initiation and propagation. These elements can be included in the model at an interface or adhesive bond while allowing load to be transferred between the adherends. The formulation of these elements is based on the Cohesive Zone Model (CZM) associated with a local mechanical behavior that may describe ongoing processes close to the crack tip. This allows avoiding the issues related to stress singularities. This behavior is also referred as cohesive laws (traction-separation relationship).

It was chosen to use Simcenter FEMAP which is a finite element calculation software for pre/post processing for the Simcenter Nastran solver. FEMAP integrates the choice of solvers. In our case, Non-Linear (SOL402 - SAMCEF Solver) was used.

The cohesive law, called "Damage Interface" in SAMCEF, is assigned to specific elements called elements of interface (or cohesive elements), to which a thermodynamic potential $E_d$ (the density surface of elastic strain energy) is associated ( [115]). This parameter is more precisely developed in equation (103).

$$E_d = \frac{1}{2} \left( \frac{\langle \sigma_{33} \rangle^2}{K_I} + \frac{\langle \sigma_{33} \rangle^2}{K_I(1-d_1)} + \frac{\langle \sigma_{32} \rangle^2}{K_{II}(1-d_2)} + \frac{\langle \sigma_{31} \rangle^2}{K_{III}(1-d_3)} \right) \quad (103)$$

With:

- $d_1, d_2, d_3$: damages due to respectively peeling mode, plane shear mode, out-of-plane shear mode,
- $K_I, K_{II}, K_{III}$, the pseudo-stiffnesses (different from the stress intensity factor from the Griffith theory) for each mode.

The term $\frac{\langle\sigma_{33}\rangle^2}{K_I}$ designates the strain energy due to the compression forces. It is considered that compressive stresses tend to close cracks which is why no damage variable is associated with the compression in the peel direction. The term $E_{d_I} = \frac{\langle\sigma_{33}\rangle^2}{K_I(1-d_I)}$ designates the strain energy due to the peeling forces. This term is associated with the variable of damage $d_I$ which transcribes the degradation of the mechanical properties of the material when the damage evolves according to mode I. Terms $E_{d_{II}} = \frac{\langle\sigma_{32}\rangle^2}{K_{II}(1-d_{II})}$ and $E_{d_{III}} = \frac{\langle\sigma_{31}\rangle^2}{K_{III}(1-d_{III})}$ designate the strain energies associated with the in-plane and out-of-plane shear loads and express the decrease of the mechanical properties in shear when damage is due to modes II and III. The pseudo-stiffnesses are the ratio between the peel or shear stiffnesses divided by the numerical thickness of the cohesive element.

In FEMAP, cohesive damage in mode I, II or III for bi-linear cohesive law will always be equal ($d_i = d$), even in pure mode I. Indeed, it is based on an equivalent thermodynamic force $Y_{eq}$, that is based on the loadings of each mode (equation (104)).

$$Y_{eq} = (Y_I^{\alpha} + (\gamma_{II}Y_{II})^{\alpha} + (\gamma_{III}Y_{III})^{\alpha})^{\frac{1}{\alpha}} \quad (104)$$

$Y_I, Y_I$ and $Y_{III}$ can then be defined by differentiating the density surface of strain energy from strain compared to the three variables of damage. They are analogous to energy release rate. They are obtained with:

- $Y_i$ is the thermodynamic force for each mode $i$, and is equal to $Y_i = \frac{\partial E_d}{\partial d_i}$,

- $\gamma_{II} = \frac{G_{Ic}}{G_{IIc}}, \gamma_{III} = \frac{G_{Ic}}{G_{IIIc}}$, is a ratio of the critical toughness which reflects the state of stress in a point of the structure,

- $\alpha$ is the coupling parameter, that must be calibrated. This can be carried out in our case using the MMB tests.

In the model used in FEMAP solver, the elastic energy absorbed before damage is described using only one parameter $y0s$ (in mode I) and $y0s$. $\frac{G_{IIc}}{G_{Ic}}$ (in mode II). This explains why only the stiffness and this parameter $y0s$ are needed, and not the interface strength for each mode. Thus, in pure mode, it is equal to the area below the first linear part (for bilinear law) of the curve, as shown in Figure 4.1-1.
<figure><img src="image_128.png" /><figcaption>Figure 4.1-1: Graphic showing all the parameters defining the bilinear cohesive law in mode I.</figcaption></figure>

It must be noted that, in this description, the shear strength is dependent on the peel strength. If the stiffnesses are fixed, it can be described by equation (105).

$$ \tau_c = \sigma_c \sqrt{\frac{G_{IIc}K_{II}}{G_{Ic}K_I}} \qquad (105) $$

In the FEMAP software (and more specifically for the SOL 402 solver), the parameters (apart from purely numerical parameters such as numerical damping) that are needed for the bilinear law for transverse isotropic material for which the influence of the mode III ($G_{IIc}=G_{IIIC}$) can be neglected, are:

- $K_I$ and $K_{II}$, the pseudo-stiffnesses for mode I and II,

- $G_{Ic}$ and $G_{IIc}$, the fracture toughnesses for mode I and II,

- $y0s$, the thermodynamic force,

- $\alpha$, a mixed mode coupling coefficient.

### 4.1.2. Finite element model hypothesis

It was chosen that the cohesive laws in mode I and II describe the global behavior of the bonded assembly, i.e. the adhesive and its interfaces with both adherends. Thus, cohesive elements using these laws are placed directly between the two adherends. A picture of the implementation of the cohesive elements in a model of ENF test is given in Figure 4.1-2. The geometric thickness of the cohesive elements is set to zero to simplify the creation of the contact condition of the unbonded area. Contact condition is not needed in the part with cohesive elements as this type of elements does not allow compression below their initial geometric thickness. The contact of the debonded area was therefore not modeled. In the part with no cohesive elements (initial crack length) for ENF and MMB tests, contact conditions had to be applied on the surfaces of the adherends. The default parameters from FEMAP were used defining a minimal distance between nodes. Friction was not modeled.
<figure><img src="image_129.png" /><figcaption>Figure 4.1-2: Cohesive element implementation on the bonded area, contact definition of the initially not bonded area.</figcaption></figure>

The models are presented in Figure 4.1-3 for DCB test, in Figure 4.1-4 for ENF test and in Figure 4.1-5 for MMB test. The used geometries for the model are the geometries described in part 2.2. For all these models, the mesh size in the crack propagation direction, which is the most important and the one that will be studied in the mesh convergence analysis, is 1mm. In all numerical models (DCB, ENF and MMB), only one 3D element is modeled in the width and its width is 1 mm. The cohesive elements are defined with a layer of one element between the two adherents of zero thickness (green elements on Figure 4.1-3). To apply plane strain conditions to the models, the displacements in the width were blocked, this was applied for each model (DCB, ENF, MMB). To validate that this choice does not interfere with the quality of the prediction of the model, full 3D models have been made for ENF and DCB tests.

<figure><img src="image_130.png" /><figcaption>Figure 4.1-3: DCB model representing the adhesive layer with cohesive elements only.</figcaption></figure>

<figure><img src="image_131.png" /><figcaption>Figure 4.1-4: ENF model representing the adhesive layer with cohesive elements only.</figcaption></figure>
<figure><img src="image_132.png" /><figcaption>Figure 4.1-5: MMB model representing the adhesive layer with cohesive elements only.</figcaption></figure>

The applied loading for all models was realized through displacement. The displacement values were determined from the experimental investigations. The boundary conditions were applied accordingly with the experimental investigations with the same boundary conditions (same fixed displacement).

For the DCB model:

- x displacements are blocked at the load applications points,

- y and z displacements are blocked, on the sliding pivot,

For the ENF model:

- y and z displacements are blocked, on the sliding pivot,

- x and y displacements are blocked, on the left support,

For the MMB model:

- same boundary conditions as the ENF model are applied for the left support and the sliding pivot.

- beam elements were added in order to model the loading frame and get the correct mode ratio. The beam elements used a rigid material (E = 1.10¹⁰ MPa) and were linked to the adherends with rigid elements. At the load application points, y displacements are defined as the load of the sample.

The values for the time steps were densified around the crack propagation time to help converging and obtain more data during propagation (especially at the failure loads). The first timesteps had also to be really small for convergence reasons. All the model hypotheses were valid for our specific case (thin adhesive, thick adherend layers). The described models will be used for both the direct and indirect methods.

### 4.1.3. Mode I and II cohesive laws determination methods

#### 4.1.3.1. Indirect method

The indirect method consists in testing a large number of cohesive laws until the numerical and experimental load/displacements curves coincide. This method can be performed purely numerically. More precisely:

- the toughness is obtained experimentally from tests,

- the mesh sensitivity is assessed for the models,

- the cohesive law in mode II is determined,

- the cohesive law in mode I is determined,

- Finally, the resulting cohesive laws in mode I and II are implemented in MMB models and the sensibility to the coupling parameter is assessed to determine its value.
4.1.3.2. Direct method

In the direct method, the cohesive laws are determined experimentally in pure mode I (DCB test) and pure mode II (ENF test), based on the J-integral computation from the local displacement measurements by DIC. From these laws, the parameters (in mode I: K<sub>I</sub>, G<sub>Ic</sub> and y0s and in mode II: K<sub>II</sub> and G<sub>IIc</sub>) needed by FEMAP for the numerical finite elements models can be directly obtained. However, as stated in previous part, in the FEMAP software, there is a relation between stiffness and critical stresses (peel or shear) based on the thermodynamic force that fixes y0s. To overcome this issue, it was chosen to fix the critical stress based on measured laws and calculate the resulting stiffness.

Though the experimentally determined cohesive laws are used in the finite elements models, it
is important to validate the numerical simulations results in mode I and II through a comparison
of the obtained load/displacement curves (experimental and numerical).

In mixed mode, no cohesive law was measured during the MMB test. Only the load/displacement curves as well as the critical toughness Gc related to the studied mode ratio are known. The MMB tests are modeled numerically with a coupling parameter (α) that drives the cohesive laws in mixed mode. In FEMAP, the default value is 1. This parameter must be calibrated until numerical and experimental load/displacement curves coincide, and more specifically, the failure load. Figure 4.1-6 summarizes the overall method.

<figure><img src="image_133.png" /><figcaption>Figure 4.1-6: Direct method flowchart.</figcaption></figure>
## 4.2. Cohesive zone model determination using indirect method

### 4.2.1. Calibration method

The inverse method to determine the cohesive law using Femap and its particular implementation is described in Figure 4.2-1. Several steps are carried out:

- measure the toughness experimentally from DCB and ENF tests, $G_{Ic}$ and $G_{IIc}$,

- assess the mesh sensitivity for all models,

- determine the cohesive law in mode II, with the only given value being $G_{IIc}$, by:

- evaluating the pseudo-stiffness sensitivity for ENF models ($K_{II}$),

- evaluating the critical shear stress sensitivity for ENF models,

- determining the cohesive law in mode II with optimal pseudo-stiffness and critical shear stress (determination of the parameter $y_0$),

- determine the cohesive law in mode I, with the only given value being $G_{Ic}$, by:

- evaluating the stiffness sensitivity in DCB models ($K_I$),

- evaluating the critical peel stress sensitivity in DCB models,

- determining the cohesive law in mode I with optimal pseudo-stiffness and critical peel stress and updated $y_0$. Indeed in FEMAP, one value will be dependent on all the others (either the stiffness in mode I or the critical peel stress),

- implement the resulting cohesive laws in mode I and II in MMB models and evaluate the sensitivity of the coupling parameter ($\alpha$),

- determine the optimal coupling parameter $\alpha$ in MMB models for mixed mode. In our case, the determination of the law in mode II was done in a first step as the data was available. The same procedure with the cohesive law in mode I being obtained firstly can also be done.
<figure><img src="image_134.png" /><figcaption>Figure 4.2-1: Indirect method flowchart.</figcaption></figure>

As explained previously, in Femap, the interface strengths $\tau_c$ and $\sigma_c$ of the different modes are not independent. Thus, the complete method consists in determining preliminary stiffnesses in mode I and II, and then calibrating the interface strength in mode I. After this, the parameter $y_0$ is set (as well as interface strength and stiffness in mode I). Then, the calibration of the interface strength in mode II is carried out modifying stiffness in mode II.
### 4.2.2. Cohesive laws reference parameters

The critical toughness was obtained from the experiments and is set as the average values obtained in part 3.2.2.

$$G_{Ic} = 0.25 \text{ kJ/m}^2 \text{ and } G_{IIc} = 2.15 \text{ kJ/m}^2$$

Based on TAST results (Appendix A2), initial values of $\tau_c = 35 \text{ MPa}$ and $K_{II} = 1100 \text{ MPa/mm}$ were chosen (close to TAST test results) for the interface strength in pure mode II. Preliminary values of $\sigma_c = 50 \text{ MPa}$ and $K_I = 18500 \text{ MPa/mm}$ were chosen for pure mode I loading based on Bulk test (Appendix A4). These values are initial values. During the sensitivity study, they will be modified in order to obtain the best fitting cohesive laws.

### 4.2.3. Model results in mode II

#### 4.2.3.1. Mesh convergence study

To validate that the model is mesh independent, a mesh convergence study was done by changing the mesh size in the crack propagation direction. The reference mesh uses an element size of 1 mm in the crack propagation direction. The other mesh sizes that were tested were 0.5 mm and 2 mm. In each case, the element size in the x-direction was increased or decreased by twofold for all elements (not just in the crack propagation part). The mesh with element size of 1 mm for the ENF model with an initial crack size of 74 mm is presented in Figure 4.2-2.

<figure><img src="image_135.png" /><figcaption>Figure 4.2-2: Mesh of an ENF for an element size of 1 mm.</figcaption></figure>

The load/displacement curves for the three mesh sizes can be seen in Figure 4.2-3. In order to quantitatively compare the different mesh sizes, the failure loads (here the maximum load) were compared. For the ENF tests, the failure load is defined as the maximum stress of the first linear part, before the first decrease. The 1 mm and 0.5 mm mesh showed no difference of the failure load prediction compared to the 2 mm mesh. The 1 mm mesh will be used for the assembly characterization described in the next parts.
<figure><img src="image_136.png" /><figcaption>Figure 4.2-3: Load/displacement curves for different mesh sizes for the ENF model with an initial crack length of 74 mm.</figcaption></figure>

#### 4.2.3.2. Stiffness sensitivity analysis

The sensitivity analysis was performed on the model with a mesh size of 1 mm and an initial crack length of 74 mm based on one of the performed tests. The cohesive parameters used in this analysis are:

- $G_{IIc}$ = 2.15 kJ/m², obtained from the ENF tests results,

- $\tau_c$ = 35 MPa,

- $K_{II}$ = 800 MPa/mm, 1100 MPa/mm and 1500 MPa/mm.

The load/displacement curves for each tested stiffness are shown in Figure 4.2-4. This figure highlights that the stiffness barely influences the results and especially the failure force, the difference being inferior to 1% for all the three models. The stiffness for the cohesive law was $K_{II}$ = 1500 MPa/mm since the slope before crack propagation was the closest to the experimental results.

<figure><img src="image_137.png" /><figcaption>Figure 4.2-4: Comparison between experimental and numerical load/displacement curves for different cohesive stiffness for ENF models.</figcaption></figure>
#### 4.2.3.3. Critical shear stress sensitivity analysis

The critical shear stress sensitivity analysis was also performed on the model with a mesh size of 1mm and an initial crack length of 74mm. The cohesive parameters used for this analysis are as follows:

- $G_{IIc} = 2.15 \text{ kJ/m}^2$

- $K_{II} = 1500 \text{ MPa/mm}$

- $\tau_c = 31 \text{ MPa}, 35 \text{ MPa}, 50 \text{ MPa}, 57 \text{ MPa} \text{ and } 70 \text{ MPa}$

The comparison between the numerical and the experimental load/displacement curves for each critical shear stress is shown in Figure 4.2-5.

<figure><img src="image_138.png" /><figcaption>Figure 4.2-5: Comparison between experimental ($a_0 = 74 \text{ mm}$) and numerical load/displacement curves for different critical shear stress for ENF models.</figcaption></figure>

The comparison of failure forces (to the experimental average failure load of 8.60 kN) for each critical shear stress is presented in Table 4.2-2. The results show a high dependency to the value of the critical shear stress. Based on these results and the lowest difference between experimental failure and numerical prediction, the critical shear stress of $\tau_c = 35 \text{ MPa}$ was kept for the cohesive law in mode II.

<table><caption>Table 4.2-1: Comparison of experimental and numerical failure forces for different critical shear stress for ENF model with an initial crack length of 74mm.</caption><thead><tr><th>τ<sub>c</sub> (MPa)</th><th>31</th><th>35</th><th>50</th><th>57</th><th>70</th></tr></thead><tbody><tr><td>Failure force (kN)</td><td>8.576</td><td>8.834</td><td>9.580</td><td>9.791</td><td>10.151</td></tr><tr><td>Difference to exp failure force (%)</td><td>-1.46</td><td>1.50</td><td>10.08</td><td>12.50</td><td>16.64</td></tr></tbody></table>

#### 4.2.3.4. Resulting cohesive law in mode II

The final cohesive parameters for the indirect cohesive law in mode II is given in Table 4.2-2 and the cohesive law is shown in Figure 4.2-6.

$$ \frac{\text{Fracture toughness}}{G_{IIc} = 2.15 \text{ kJ/m}^2} \quad \frac{\text{Cohesive stiffness}}{K_{II} = 1500 \text{ MPa/mm}} \quad \frac{\text{Critical shear stress}}{\tau_c = 35 \text{ MPa}} $$

<caption>Table 4.2-2: Cohesive parameters in mode II obtained for indirect method for Resin B steel-steel for indirect method.</caption>
<figure><img src="image_139.png" /><figcaption>Figure 4.2-6: Cohesive parameters in mode II obtained for indirect method for resin B steel-to-steel bonding.</figcaption></figure>

This cohesive law was implemented in the other two ENF models with initial crack lengths of 59 mm and 90 mm. The comparison of load/displacement curves for both experimental and numerical tests is shown in Figure 4.2-7. The numerical and experimental failure forces for each crack length are compared in Table 4.2-3. The results show that the cohesive law obtained for a model with an initial crack length $a_0 = 74$ mm can well predict the general behavior of ENF models with initial crack lengths $a_0 = 59$ mm and $a_0 = 90$ mm. The load at failure's prediction error remains under 6%. The unstable crack propagation for the specimen with an initial crack length $a_0 = 59$ mm is well represented by the model whereas it is not the case for the specimen with an initial crack length $a_0 = 90$ mm, for which a stable crack propagation is predicted by the model. However, it is important to underline that the failure forces are well predicted by the models for both initial crack lengths.

<figure><img src="image_140.png" /><figcaption>Figure 4.2-7: Comparison of experimental and numerical load/displacement curves obtained for ENF tests and models, with initial crack lengths of 59 mm and 90 mm.</figcaption></figure>

<table><caption>Table 4.2-3: Comparison of failure forces obtained numerically and experimentally for ENF tests and models with different initial crack lengths.</caption><thead><tr><th>Crack length (mm)</th><th>59</th><th>74</th><th>90</th></tr></thead><tbody><tr><th scope="row">Numerical failure force (kN)</th><td>10.453</td><td>8.834</td><td>7.648</td></tr><tr><th scope="row">Experimental failure force (kN)</th><td>11.060</td><td>8.703</td><td>7.711</td></tr><tr><th scope="row">Difference (%)</th><td>-5.49%</td><td>1.51%</td><td>-0.82%</td></tr></tbody></table>
The cohesive law, calibrated with the model with an initial crack length of 74 mm, seems thus validated with the other two models presenting different initial crack lengths. Indeed, errors on the failure force are less than 6%. However, it must be noted that only three tests were used for comparison and only one for each crack length. More tests would have allowed to improve the estimations of the parameters by the indirect method.

#### 4.2.3.5. Comparison with 3D model

A comparison was made with a 3D model and a mesh size of 2mm in the x-direction. Instead of boundary conditions on nodes as in the 2D model, an effort was made to be as close as possible to the experiment and thus, cylinders were used to model the supports. The 3D model is shown in Figure 4.2-8.

<figure><img src="image_141.png" /><figcaption>Figure 4.2-8: 3D Model of the ENF (the contact zones are highlighted).</figcaption></figure>

The comparison of the load/displacement curves is shown in Figure 4.2-9. It can be seen that the difference between the 2D and 3D models is quite low, the difference of 2D and 3D numerical failure load being less than 3%. The hypothesis made for the 2D, i.e. plane strain model can therefore be considered as verified. Moreover, the planar crack propagation in the 3D model is illustrated in Figure 4.2-10 showing the cohesive element damage during crack propagation. Figure 4.2-11 presents the resulting experimental flat crack front shape also observed during one ENF test. Both results are therefore in agreement.
<figure><img src="image_142.png" /><figcaption>Figure 4.2-9: Load/displacement curves for the 2D and 3D model of the ENF test.</figcaption></figure>

<figure><img src="image_143.png" /><figcaption>Figure 4.2-10: Plot of the cohesive element damage during the ENF test showing a planar crack propagation.</figcaption></figure>

<figure><img src="image_144.png" /><figcaption>Figure 4.2-11: Crack front observations on an ENF sample at the end of the test.</figcaption></figure>
### 4.2.4. Model results in mode I

#### 4.2.4.1. Convergence Study

The same analysis of mesh convergence as for the ENF model has been performed for the DCB model. Three mesh sizes in the crack propagation direction were tested: 2 mm, 1 mm and 0.5 mm. Note that the initial crack length was set to 74 mm for this analysis. The resulting load/displacement curves for the three mesh sizes are compared in the Figure 4.2-12.

<figure><img src="image_145.png" /><figcaption>Figure 4.2-12: Load/displacement curves for different mesh sizes for the DCB model.</figcaption></figure>

The three curves show minimum error between each mesh size from 2mm to 0.5mm. The comparison of predicted failure forces for each mesh size is presented in Table 4.2-4.

<table><caption>Table 4.2-4: Failure forces for different mesh sizes for DCB model.</caption><thead><tr><th></th><th>0.5</th><th>1</th><th>2</th></tr></thead><tbody><tr><th scope="row">Mesh size cohesive (mm)</th><td></td><td></td><td></td></tr><tr><th scope="row">Failure forces (kN)</th><td>0.609</td><td>0.609</td><td>0.617</td></tr><tr><th scope="row">Difference to converged mesh (%)</th><td>0.00%</td><td>0.06%</td><td>1.29%</td></tr></tbody></table>

The DCB models have a lower mesh sensitivity than ENF models. The mesh size for the following used characterization was set to an element size of 1mm as the computation time difference between 1mm and 2mm mesh size was not significant. This allowed using similar mesh for DCB and ENF models.

The next step in the indirect method for mode I is to calibrate either the critical peel stress or the cohesive stiffness since the parameter y0s was already set in FEMAP from the calibration of the mode II cohesive law.
#### 4.2.4.2. Stiffness sensitivity analysis

This stiffness sensitivity analysis was performed with a mesh size of 1mm. The used cohesive parameters are:

- $G_{IC} = 0.25 \text{ kJ/m}^2$,

- $\sigma_c = 35 \text{ MPa}$, This initial value is obtained from the final value for the cohesive law in mode II,

- $K_I = 13\,000 \text{ MPa/mm}, 18\,560 \text{ MPa/mm}$ (preliminary value), $23\,000 \text{ MPa/mm}$ and $75\,000 \text{ MPa/mm}$.

In this analysis, the critical peel stress was taken as constant, as well as the cohesive parameters in mode II. Because of the elastic energy absorbed before damage $y0s$ (in mode I) and $y0s$. $\frac{G_{IIc}}{G_{IC}}$ (in mode II) being fixed and the relation described by the equation 105, the parameter $y0s$ must be updated for each stiffness. This means that the critical shear stress and therefore the mode II cohesive law has to be updated for each mode I cohesive law tested. The final cohesive laws in mode I and II will be summarized in the conclusion.

The load/displacements curves for each mode I cohesive stiffness is shown in Figure 4.2-13. It can be observed that the cohesive stiffness in mode I barely influences the results, the failure forces difference being below 1% for these four results. Thus, the calibration of the mode I cohesive law will only be based on the critical peel stress and the cohesive stiffness will be calculated to maintain a constant value for the parameter $y0s$ from the mode II cohesive law.

<figure><img src="image_146.png" /><figcaption>Figure 4.2-13: Numerical load/displacement curves for different cohesive stiffness for DCB models.</figcaption></figure>

#### 4.2.4.3. Critical peel stress sensitivity analysis

This sensitivity analysis was performed with a mesh size of 1mm. The used cohesive parameters are:

- $G_{IC} = 0.25 \text{ kJ/m}^2$,

- $y0s = 0.047 \text{ kJ/m}^2$, which will fix the stiffness in mode I, it is obtained using the equation (105).

- $\sigma_c = 35 \text{ MPa}$ ($K_I = 12\,900 \text{ MPa/mm}$), $\sigma_c = 50 \text{ MPa}$ ($K_I = 26\,350 \text{ MPa/mm}$) and $\sigma_c = 70 \text{ MPa}$ ($K_I = 51\,650 \text{ MPa/mm}$).
The comparison of load/displacement curves is presented in Figure 4.2-14 for the different critical peel stresses. It can be observed that doubling the critical peel stress barely impacts the failure forces. Furthermore, the difference to experimental failure load is reduced when increasing the critical peel stress. Yet, choosing a value greater than 70 MPa has no physical meaning. Thus, the choice was made to keep a critical peel stress of 35 MPa, equal to the critical shear stress, for the resulting cohesive law.

<figure><img src="image_147.png" /><figcaption>Figure 4.2-14: Comparison of load/displacement curves for different critical peel stresses for DCB models.</figcaption></figure>

Table 4.2-5 presents the final difference between the failure load obtained experimentally and numerically in mode I for different critical peel stresses $\sigma_c$. The difference shows that the failure load prediction remained lower than the experimental results (Table 4.2-5). This is supposed to be related to the chosen critical toughness value based on the standard results being conservative. The same analysis performed with the results obtained with the J-integral method may provide better results.

<table><caption>Table 4.2-5: Comparison of experimental and numerical failure forces for different critical peel stresses for DCB models.</caption><thead><tr><th>σ<sub>c</sub> (Mpa)</th><th>Numerical<br>σ<sub>c</sub> = 35</th><th>Numerical<br>σ<sub>c</sub> = 50</th><th>Numerical<br>σ<sub>c</sub> = 70</th></tr></thead><tbody><tr><td>Failure load (kN)</td><td>0.647</td><td>0.661</td><td>0.670</td></tr><tr><td>Difference to experimental failure load (%)</td><td>-15.00%</td><td>-13.24%</td><td>-12.03%</td></tr></tbody></table>

A comparison of the numerical model with a critical peel stress of 35 MPa and experimental load/displacement curves is shown in Figure 4.2-15. It can be seen that, despite being conservative on the failure load (-15%), the crack propagation is well represented on its globality. However, it must be noted that the model does not represent the uncontrolled crack propagation for some of the tests.
<figure><img src="image_148.png" /><figcaption>Figure 4.2-15: Comparison of experimental and numerical load/displacement curves of DCB resin B.</figcaption></figure>

The final cohesive parameters for the indirect cohesive law in mode I following the equation (105) are given in Table 4.2-6. The cohesive law is shown in Figure 4.2-16.

<table><caption>Table 4.2-6: Cohesive parameters in mode I for indirect method.</caption><thead><tr><th>Fracture toughness</th><th>Cohesive stiffness</th><th>Critical peel stress</th></tr></thead><tbody><tr><td>G<sub>Ic</sub> = 0.25 kJ/m<sup>2</sup></td><td>K<sub>I</sub> = 12 900 MPa/mm</td><td>σ<sub>c</sub> = 35 MPa</td></tr></tbody></table>

<figure><img src="image_149.png" /><figcaption>Figure 4.2-16: Resulting indirect cohesive law in mode I for steel-to-steel bonded assembly with resin B.</figcaption></figure>

#### 4.2.4.4. Comparison to 3D model

A 3D model was created to assess the hypothesis of the 2D model which considers a constant crack pattern along the width, but also plane strain conditions. The mesh of the model can be seen in Figure 4.2-17 and the element size in the crack propagation direction is 2mm to save computation time. The nodes along the width for the section at x=0 are all pinned at the bottom while a rigid element includes all the top nodes of the same section. The loading is applied on an independent node from this same rigid element. As the computation time is quite long, the model was not run until the end but during a long enough time to obtain a sufficient propagation.
<figure><img src="image_150.png" /><figcaption>Figure 4.2-17: 3D DCB model.</figcaption></figure>

The cohesive parameters for this simulation were taken from the final cohesive law in mode I. The load/displacement curve comparison between 2D and 3D models (both with same element size in the crack propagation direction) is shown in Figure 4.2-18. It can be seen that the propagation is almost identical in both models, although the 3D curve is smoother. The difference of failure forces is around 4% which validates the hypothesis made in 2D model. Furthermore, the crack propagation pattern is shown in Figure 4.2-19 by considering the damage in the bondline as the crack front shape. It can be compared to experimental observations after failure presented in Figure 3.2-1 that are similar. Thus, a 3D model can be interesting to confirm fracture surfaces but is not necessary for predicting failure forces in this case.

<figure><img src="image_151.png" /><figcaption>Figure 4.2-18: Comparison of load/displacement curves for DCB model in 2D and 3D with a 2mm mesh size.</figcaption></figure>
<figure><img src="image_152.png" /><figcaption>Figure 4.2-19: Fracture surface along the width for the 3D DCB model.</figcaption></figure>

### 4.2.5. Model results in mixed mode

For the MMB modeling, both mode I and mode II laws obtained previously by the indirect method are used (Figure 4.2-20). The mesh to model the MMB test was the same than the one chosen for the DCB and ENF convergence meshing study, with a mesh size in the crack propagation direction of 1mm. The applied boundary conditions are the same than the one described in the part 4.1.2.

<figure><img src="image_153.png" /><figcaption>Figure 4.2-20: Indirect cohesive laws obtained in mode I and mode II.</figcaption></figure>

Figure 4.2-21 presents the comparison between the experimental (part 3.2.4) and the numerical results for a mode ratio of 0.7. A series of computation were performed to find the best fit of
the coupling parameter $\alpha$ by varying its value between 0 to 1. The result of this analysis is a final parameter $\alpha$ chosen equal to 0.37. This value was chosen to get the smallest difference between numerical and experimental failure loads as described in Table 4.2-7.

<figure><img src="image_154.png" /><figcaption>Figure 4.2-21: Comparison of load/displacement curves for MMB experimental test and model with a 1mm mesh size, a mixed mode ratio of 0.7 and different values of the coupling parameter $\alpha$.</figcaption></figure>

Figure 4.2-22 presents the same model results performed with the same cohesive laws, and the same values of the coupling parameter, compared to the experimental results for a mode ratio of 0.5.

<figure><img src="image_155.png" /><figcaption>Figure 4.2-22: Comparison of load/displacement curves for MMB experimental test and model with a 1mm mesh size, a mixed mode ratio of 0.5 and different values of the coupling parameter $\alpha$.</figcaption></figure>
The curves show that the model can reproduce well the maximal load, rigidity and crack propagation. The resulting errors on the maximal load for each mode ratio and each value of the coupling parameter are given in Table 4.2-7.

<table><caption>Table 4.2-7: Comparison of experimental and numerical failure loads for MMB test for a mode mixity of 50% and 70% and for different coupling parameter values.</caption><thead><tr><th></th><th colspan="4">MMB 70% mode II</th><th colspan="4">MMB 50% mode II</th></tr></thead><tbody><tr><th>Coupling parameter α</th><td>1</td><td>0.41</td><td>0.37</td><td>0.33</td><td>1</td><td>0.41</td><td>0.37</td><td>0.33</td></tr><tr><th>Failure load (kN)</th><td>2.811</td><td>1.884</td><td>1.739</td><td>1.571</td><td>1.693</td><td>1.21</td><td>1.12</td><td>1.02</td></tr><tr><th>Difference to experimental<br>failure load (%)</th><td>53.58</td><td>2.94</td><td>-4.98</td><td>-14.16</td><td>51.80</td><td>9.25</td><td>0.71</td><td>8.10</td></tr></tbody></table>

In conclusion, this study shows that the cohesive laws determination by using the indirect method (for brittle adhesive) can represent well the behavior of adhesively steel-to-steel bonding assembly with thin adhesive layer (1mm) and thicker adherends, in the full range of studied modes for standard tests. The generalization of this assumption to more complex geometries and loadings will be assessed in the next chapter to validate that this method can be used for full size bonded assembly.

### 4.3. Cohesive zone model determination using direct method

In the direct method, the cohesive law is measured experimentally in pure mode I (DCB test) and pure mode II (ENF test), based on the J-integral method and the local displacements measurements by DIC. These laws are then directly used in numerical DCB and ENF models with the FEM software FEMAP and no parameter needs to be calibrated in these cases. However, in mixed mode, no cohesive law was measured during the MMB test.

The available experimental results from the ENF test and the numerical cohesive law in pure mode II are plotted in Figure 4.3-1. Likewise, the available experimental results from the DCB test and the numerical cohesive law in pure mode I are plotted in Figure 4.3-2. As the experimental laws are monotonous, it was challenging to introduce a bilinear law based on these results. A bilinear law in the direct method was designed to have similar critical peel and shear stress compared to the experimental cohesive laws (average values). The analysis of part 4.2.2 and 4.2.3 proved that the stiffness had little to no impact on the general behavior. Therefore, its definition is not considered as critical.

Furthermore, in this direct method, it is important to underline that the critical toughness value is based on the value obtained with the J-integral method and consequently, is slightly different to the one used in the indirect method. The cohesive law values used for the direct modeling are summarized in the Table 4.3-1. The fracture toughnesses and critical stresses were kept as close as possible to experimental results and the fitting was mainly made using the stiffness parameter.

<table><caption>Table 4.3-1: Cohesive parameters using direct method for mode I and II.</caption><thead><tr><th></th><th>Fracture toughness G<sub>c</sub> (kJ/m<sup>2</sup>)</th><th>Critical stress (MPa)</th><th>Stiffness K (MPa/mm)</th></tr></thead><tbody><tr><th>Mode I</th><td>0.316</td><td>60</td><td>8426</td></tr><tr><th>Mode II</th><td>1.960</td><td>53</td><td>1060</td></tr></tbody></table>
<figure><img src="image_156.png" /><figcaption>Figure 4.3-1: Experimental and proposed numerical cohesive laws in pure Mode II for ENF test.</figcaption></figure>

<figure><img src="image_157.png" /><figcaption>Figure 4.3-2: Experimental and proposed numerical cohesive laws in pure Mode I for DCB test.</figcaption></figure>

### 4.3.1. Model results in mode II

Based on the mesh sensitivity analysis performed in the part 4.2.3, the 1mm element size was used. The numerical and experimental load/displacement results are plotted for different initial crack length ($a_0 = 59 \text{ mm}, 74 \text{ mm}, 90 \text{ mm}$) in Figure 4.3-3. It can be seen that the stiffness as well as the failure loads are quite close between numerical and experimental results. As the difference in failure loads is below 10% for all the tests, the models using direct cohesive laws can be considered to replicate properly the behavior of bonded assembly in pure mode II.
<figure><img src="image_158.png" /><figcaption>Figure 4.3-3: Numerical and experimental ENF load/displacement curves for different initial crack lengths.</figcaption></figure>

<table><caption>Table 4.3-2: Comparison of failure loads in mode II obtained numerically by the direct method and experimentally by ENF tests.</caption><tr><td>Crack length (mm)</td><td>59</td><td>74</td><td>90</td></tr><tr><td>Numerical failure force (kN)</td><td>11.235</td><td>9.340</td><td>7.936</td></tr><tr><td>Experimental failure force (kN)</td><td>11.06</td><td>8.703</td><td>7.711</td></tr><tr><td>Difference (%)</td><td>1.58%</td><td>7.31%</td><td>2.91%</td></tr></table>

### 4.3.2. Model results in mode I

The comparison of load/displacement curves obtained experimentally and numerically is given in Figure 4.3-4. It shows that the direct method predicts a more accurate failure load compared to the indirect method. This implies a more optimistic crack propagation part of the curve. This is supposed to be directly correlated to the higher critical toughness of the cohesive law, which is 0.316 kJ/m² for the direct method, and 0.25 kJ/m² for the indirect one. The values of failure loads are presented in Table 4.3-3 and tend to prove that the value measured with the J-integral computation leads to a more accurate failure prediction than the one obtained with the standard.

<figure><img src="image_159.png" /><figcaption>Figure 4.3-4: Numerical direct method and experimental DCB load/displacement curves.</figcaption></figure>
<table><caption>Table 4.3-3: Comparison of numerical and experimental failure loads for the DCB direct method.</caption><thead><tr><th></th><th>I-1</th><th>I-2</th><th>II-1</th><th>II-2</th></tr></thead><tbody><tr><td>Experimental failure force (kN)</td><td>0.759</td><td>0.717</td><td>0.837</td><td>0.733</td></tr><tr><td>Numerical failure force (kN)</td><td></td><td></td><td>0.752</td><td></td></tr><tr><td>Difference to mean experimental (%)</td><td></td><td></td><td>-1.23%</td><td></td></tr></tbody></table>

### 4.3.3. Model results in mixed mode

The cohesive laws determined in pure mode I and II were then implemented in MMB models with an element size of 1 mm in the crack propagation direction. This part will present the calibration of the coupling parameter $\alpha$ for both mixed mode ratio 50% and 70% of mode II. An initial value of the coupling parameter was set at $\alpha = 0.37$ based on the indirect method results.

Results of the coupling parameter calibration are shown in Figure 4.3-5 (for a ratio of 50% of mode II) and Figure 4.3-6 (for a ratio of 70% of mode II). Comparison of numerical and experimental failure loads is presented in Table 4.3-4 (for 50% of mode II) and Table 4.3-5 (for 70% of mode II). The coupling parameter allowing to obtain the closest average failure loads was 0.35. This is close to the parameter found in the indirect method. This parameter seems to be adapted for both mode ratios.

<figure><img src="image_160.png" /><figcaption>Figure 4.3-5: Comparison of numerical and experimental load/displacement curves for MMB test with 50% of mode II, for different values of the coupling parameter, using direct cohesive law.</figcaption></figure>

<table><caption>Table 4.3-4: Comparison of experimental and numerical failure loads for MMB test with 50% of mode II, for different values of the coupling parameter, using direct cohesive law.</caption><thead><tr><th>Coupling parameter α</th><th>1</th><th>0.41</th><th>0.37</th><th>0.35</th><th>0.33</th></tr></thead><tbody><tr><td>Failure load (kN)</td><td>1.932</td><td>1.290</td><td>1.184</td><td>1.135</td><td>1.068</td></tr><tr><td>Difference to experimental failure load (%)</td><td>73.30</td><td>15.68</td><td>6.20</td><td>1.75</td><td>-4.23</td></tr></tbody></table>
<figure><img src="image_161.png" /><figcaption>Figure 4.3-6: Comparison of numerical and experimental load/displacement curves for MMB test with 70% of mode II, for different values of the coupling parameter values, using direct cohesive law.</figcaption></figure>

<table><caption>Table 4.3-5: Comparison of experimental and numerical failure loads for MMB test 70% of mode II, for different values of the coupling parameter, using direct cohesive law.</caption><tr><td>Coupling parameter α</td><td>1</td><td>0.41</td><td>0.37</td><td>0.35</td><td>0.33</td></tr><tr><td>Failure load (kN)</td><td>3.109</td><td>1.967</td><td>1.799</td><td>1.713</td><td>1.621</td></tr><tr><td>Difference to experimental failure load (%)</td><td>69.87</td><td>7.46</td><td>-1.67</td><td>-6.38</td><td>-11.42</td></tr></table>

### 4.3.4. Resulting cohesive law parameters

The final parameters for the calibration of cohesive laws using the direct method for the assembly characterization are given in Table 4.3-6.

<table><caption>Table 4.3-6: Resulting parameters of cohesive laws using direct method.</caption><thead><tr><th></th><th>Fracture toughness G<sub>c</sub><br>(kJ/m<sup>2</sup>)</th><th>Critical stress σ<sub>c</sub><br>(MPa)</th><th>Stiffness K<br>(MPa/mm)</th><th>Coupling<br>parameter</th></tr></thead><tbody><tr><td>Mode I</td><td>0.316</td><td>60</td><td>8426</td><td rowspan="2">0.35</td></tr><tr><td>Mode II</td><td>1.960</td><td>53</td><td>1060</td></tr></tbody></table>

## 4.4. Discussion

Figure 4.4-1 and Figure 4.4-2 present the final cohesive laws in mode I and II obtained in this chapter using both direct and indirect studied methods. The cohesive laws measured in this part are only valid for the considered assembly with its specific adhesive, adherends, interfaces but also its specific adhesive thickness.

The shape of the direct and indirect laws are significatively different. The intial rigidity are similar, but the difference in the critical toughnesses and the critical stresses obtained for both methods, generate difference in the obtained cohesive law shape. From a physical point of view, the direct method seems more accurate, as brittle behavior has been observed and analyzed in chapter 3 (the area under the curve for the damaged part is lower, which corresponds well to a brittle fracture).
The results obtained with indirect method proved that results from standards (with the help of optical fiber) provided usable data. The indirect approach remains a valid method to obtain the cohesive laws as its implementation in the numerical models proved to lead to results close to the experimental ones, in terms of load/displacement curves for the pure mode I, pure mode II and mixed mode tests.

The results obtained with the direct method proved that the J-integral analysis has provided higher quality data. Yet, this method is more difficult to use, because of the local displacements measurement required to monitor local sliding, opening and rotation. In our case, the DIC was used, but these data could be obtained with much more simple sensor such as LVDT or inclinometer.

Regarding numerical prediction, one major drawback of the indirect method could be the computation time (high number of simulations). The validation of the directly computed cohesive laws requires actually a relatively limited number of models, computation and post process.

Though both approaches lead to rather different cohesive laws, both matched well with experimental results. This is consistent with literature such as in [120] for DCB test or [123] for single-lap test that conclude that the cohesive law shape has little impact on the behavior and the failure load prediction for such bonded assemblies.

The final cohesive model that has been fitted to mode I and II experimental results, should be able to predict correctly the behavior for other types of loading and assembly. The analysis has also shown that the proposed approaches relying on the formulation proposed in FEMAP and a coupling parameter seemed to be satisfactory. The coupling parameter, $\alpha$, obtained for the direct and indirect method are really similar (respectively 0.35 and 0.37) and the comparisons between experimental and numerical results are very close.

<figure><img src="image_162.png" /><figcaption>Figure 4.4-1: Final DCB cohesive laws.</figcaption></figure>
<figure><img src="image_163.png" /><figcaption>Figure 4.4-2: Final ENF cohesive laws.</figcaption></figure>

In the next chapter, the direct and indirect cohesive laws will be tested to predict the behaviors of full size specimens more representative of real applications.
# Chapter 5: Robustness appraisal of the proposed methodology

As the validity of the developed modeling approach (using finite element models and cohesive zone model) was carried out using small scale fracture mechanics investigations in the previous chapter, it was decided to apply it on more realistic samples to verify its adequacy. Thus, real-scale tests with loading conditions closed to reality (most commonly encountered loading situations for ship decks) were realized. In order to assess the efficiency of the proposed methodology and to investigate its robustness, it was decided to compare its performance with an alternative design methodology relying on both continuous and fracture mechanics, the coupled stress-energy-criteria method.

First, the experimental campaign and the obtained results are described. The coupled stress-energy criteria method is then carried out. Thirdly, the developed methodology is applied using both direct and indirect approaches. The final discussion compares the different methods and their ability to predict the studied bonded samples' behavior.

## 5.1. Experimental campaign at real scale and under most commonly encountered loading situations

### 5.1.1. Studied samples and instrumentation

To assess the robustness of the different design methodologies (the direct, indirect method, and coupled criteria) for full scale bonded reinforcement, two mostly encountered type of loadings have been chosen. Figure 5.1-1 presents those two load cases. The first one is a typical shear loading of the bonded patch which can be encountered on ship decks. The second one is a three-point bending (bending/tension) test which could be encountered on pressure loaded tanker.

The materials used for the sample manufacturing are medium carbon steel (C45 steel with a yield stress between 600 and 800 MPa, from datasheet) to ensure that no plasticity in the steel will occur during the test, and the adhesive is the resin B previously studied. The design and geometry of the specimens are shown in Figure 5.1-1.

<figure><img src="image_164.png" /><figcaption>Figure 5.1-1: Full-size test geometries for tension and bending/tension loadings.</figcaption></figure>

The bonding procedure is similar to the one described in chapter 3. The geometry is similar for bending and tension tests. The total length is of 1600 mm with a 1000 mm long bonded patch. The thickness of one of the substrates is 20 mm. The other one (the reinforcement corresponding
to the thinner substrate) is 6 mm thick. No crack initiations are made in these samples. These
choices have been made to ensure that:

- the patch is long enough to express the behavior of a full-size patch (transition length should be small compared to the patch length),

- the 20 mm substrate length is long enough to have some distance between the border of the patch and the clamp of the test machine for the tension test, or between the border of the patch and the support for the bending test (20 cm),

- the 20 mm substrate has a standard thickness which can be encountered on ship deck.

This change in shape has introduced much more leakage during the bonding procedure. Thus, the samples were made in two steps with two castings of the resin (using the same batch). Post curing (70°C during 16h) was made after the second casting. The thickness of the bonding layer (1mm) was controlled using 3D printed plastic pieces maintaining the steel plates in place during the casting. The thickness of the adhesive has been controlled after curing, showing that, in the first 20 cm at each border of the bonded patch, the thickness has not change more than ±0.2 mm. The center section was not measured.

The instrumentation of the specimens is shown in Figure 5.1-2. A set of ten strain gauges were used to obtain local strain measurements on the surface of the specimen. The gauges 1/2/9/10 were used to monitor the bending and tension in the 20 mm substrate due to the off-centered disposition of the 6 mm substrate. The gauges 3/4/7/8 were used to monitor the strain at the border of the patch and obtain information concerning the crack initiation and the corresponding failure load. The gauges 5 and 6 were used to monitor the strain difference in the substrate and the reinforcement. Finally, the continuous optical fiber, placed in the center of the reinforcement substrate (exterior optical fiber), was used to monitor the strain all along the reinforcement length and the crack propagation when it was slow enough. These data have been used to compare the experimental behavior to the ones obtained with the modeling.

<figure><img src="image_165.png" /><figcaption>Figure 5.1-2: Full size tension and bending samples instrumentation.</figcaption></figure>
5.1.2. Test protocols

Two sets of three tests were performed:

- 3 tension tests made on a 2500kN Instron test machine (1% error in measurement, 1% error in repeatability). The samples were clamped on a length of 10 cm on each side, leading to a 140 cm tested length (Figure 5.1-3a, b). The used load cell was the one from the machine,

- 3 three-point bending tests made on a 300kN Losen test machine. The test frame was set at a length of 148 cm (maximal test frame allowable length) (Figure 5.1-4). The used load cell was adapted to the expected load during the test (10kN, precision ± 0.145%).

<figure><img src="image_166.png" /><figcaption>Figure 5.1-3: a) Initial tension test without instrumentation, b) Tension sample lower end with optical fiber and gauge instrumentation.</figcaption></figure>
<figure><img src="image_167.png" /><figcaption>Figure 5.1-4: Bending test with instrumentation.</figcaption></figure>

On Figure 5.1-4, a vice clamp was used at one edge of the bonded area. It was placed to remove the mode I loading at one end of the patch for the tension and bending tests. This allowed ensuring that the crack initiates on the other side of the patch, where the DIC setup was placed. In each case, the clamp was put in place next to the gauges 7 and 8 leading the crack initiation to start in the gauges 3 and 4 area. The tests were conducted up to the total debonding of the patch in tension and in bending.

The analysis of the failure surfaces (visible in Figure 5.1-5) shows that the failure was similar for the six tests and 100% adhesive (Appendix A6). The failure pattern reveals a parabolic crack front shape that corresponds to the propagation. Based on the analysis performed in chapter 4 on the failure pattern for DCB, ENF and MMB tests, this figure tends to attest that mixed mode ratio ($G_{II}/G_c$) was obtained for the two sets of tests, with at least a ratio of 50%.

<figure><img src="image_168.png" /><figcaption>Figure 5.1-5: Adhesive failure and crack front shape of the samples.</figcaption></figure>

### 5.1.3. Strain measurement and load at failure

The next part presents the load/displacement curves and the strain gauges measurements curves for the tension and bending tests.

#### 5.1.3.1. Tension test

Figure 5.1-6 shows the three load/displacement curves obtained for the three traction tests. It can be observed that the curves are initially linear. A second nonlinear part can be seen which might be correlated to the crack propagation occurring in the bondline (in a stable manner, progressive loss of rigidity). Then, the crack propagates violently on the whole bondline length.
In fact, the load/displacement curve of the three tests shows a violent decrease corresponding to the stiffness loss by the debonded adherent (abrupt loss of rigidity around the displacement of 2 mm). For the analysis of these curves and the modeling, the patch is considered to be broken when the crack initiates, i.e. at the first point of non linearity. However, the identification of this point on the curve is not an easy task. The results obtained are summarized in Table 5.1-1 with relatively low confidence.

<figure><img src="image_169.png" /><figcaption>Figure 5.1-6: Tension test load/displacement curves.</figcaption></figure>

To solve this issue, the gauges bonded just near the crack initiation on the sample (gauges 3 and 4, at 3 cm of the border of the adherent) are used as crack time initiation measurement devices. When the gauge curve decreases, the crack is considered to have propagated under the gauges. Figure 5.1-7 displays the gauges 3 and 4 measurements for the three tension tests. It can be seen that during the loading of the bondline, the measurements are really similar for the three tests. At the beginning of the test, an offset of the strain values is visible. It is related to the initial loading of the sample which is not perfectly straight (tightening of the clamps). This may have an impact on the patch's general behavior (small initial torsion or bending). Disbonding occurred in the patch for relatively similar strain measurement of around 350 µstrain (three tests' mean value) at a load level of around 170kN (three tests' mean value). The precise values are given in Table 5.1-1. These values will be used as comparison with the model results to validate the load at failure predictions.
<figure><img src="image_170.png" /><figcaption>Figure 5.1-7: Gauges 3 and 4 strain measurement for the three samples in tension.</figcaption></figure>

<table><caption>Table 5.1-1: Strain and load at failure for the tension test.</caption><thead><tr><th></th><th>test 1</th><th>test 2</th><th>test 3</th></tr></thead><tbody><tr><td>Gauge 3 max (µm/m)</td><td>404</td><td>327</td><td>375</td></tr><tr><td>Gauge 4 max (µm/m)</td><td>400</td><td>282</td><td>366</td></tr><tr><td>Mean gauge 3 and 4 max (µm/m)</td><td>402</td><td>303</td><td>371</td></tr><tr><td>Load at break (kN) using mean gauge data</td><td>170</td><td>148</td><td>181</td></tr><tr><td>Load at failure (kN) using load/displacement non-linearity</td><td>160(±10)</td><td>140 (±10)</td><td>160 (±10)</td></tr></tbody></table>

### 5.1.3.2. Bending test

Figure 5.1-8 shows the three load/displacement curves obtained for the three bending tests. It can be seen that the curves are initially linear then nonlinear when the crack propagates. The load/displacement curves show an abrupt decrease corresponding to the stiffness loss by the debonded adherent. The same analysis as the tension test is performed on the failure time, and the gauges 3 and 4 are used as crack time initiation measurement devices (Figure 5.1-9). Disbonding occurred in the patch for around 440 µstrain (mean value) at a load level of around 570 daN (mean value). The precise values are described in Table 5.1-2. These values will be compared with the modelling predictions.
<figure><img src="image_171.png" /><figcaption>Figure 5.1-8: Load/displacement curves of the bending tests.</figcaption></figure>

<figure><img src="image_172.png" /><figcaption>Figure 5.1-9: Gauges 3 and 4 strain measurements during the bending tests.</figcaption></figure>

<table><caption>Table 5.1-2: Strain and load at failure of the bending tests.</caption><thead><tr><th></th><th>test 1</th><th>test 2</th><th>test 3</th></tr></thead><tbody><tr><td>Gauge 3 (µm/m)</td><td>506</td><td>427</td><td>398</td></tr><tr><td>Gauge 4 (µm/m)</td><td>454</td><td>396</td><td>441</td></tr><tr><td>Mean gauge 3 and 4 (µstrain)</td><td>480</td><td>411</td><td>420</td></tr><tr><td>Load at failure (daN) using gauge measurement</td><td>500</td><td>605</td><td>601</td></tr><tr><td>Load at failure (daN) using load/displacement non-linearity</td><td>450(±10)</td><td>440(±10)</td><td>450(±10)</td></tr></tbody></table>

## 5.2. Stress/energy criteria application to full size model

To check the capabilities of the direct and indirect methodology, the load at failure prediction capability has to be assessed. In our case, it is compared to another method of the literature: the coupled stress energy method. This method has, compared to the cohesive element methods,
limited capacity to express the local behavior of the assembly, but can provide good predictive failure capacities.

This criterion consists in combining the stress and the energy while bypassing the problems:

- of stress concentration at patch border for the stress approaches,

- necessity to have an initial crack to compute the energy approach.

In the coupled criteria the damaged length ($a_d$) predicted by the stress criterion is compared to the critical crack length ($a_c$), the minimal crack length required for the crack to propagate predicted by the energy criterion. For a design loading, the coupled stress-energy approach states that the failure (crack initiation and propagation) will occur when both damage and critical lengths become equal as illustrated in the Figure 5.2-1.

<figure><img src="image_173.png" /><figcaption>Figure 5.2-1: Stress- energy coupled criteria.</figcaption></figure>

### 5.2.1. Hypothesis for FE model

X-symmetry was taken into account to model only half of the samples. The material properties were defined as linear elastic for the steel substrates with a Young modulus of 210 GPa, and the adhesive properties were defined as linear elastic with a Young modulus of 3.3 GPa (using bulk test results for resin B - Appendix A4). The applied loading for both models (tension test and bending test) was set from the average experimental failure load. The boundary conditions are shown in Figure 5.2-2 for the tension test. Regarding the bending test, the boundary conditions are described in Figure 5.2-3.

The boundary conditions were applied accordingly with the experimental investigations (same fixed displacement).

For the tension model:

- at the load applications points, y and z displacements were blocked,

- at the symmetry plane, x displacements were blocked.

For the bending model:

- on the sliding pivot, y and z displacements are blocked,

- at the symmetry plane, x displacements were blocked.

The applied loading for both models was realized using the average experimental failure load obtained with the gauges.

The values for the time steps were defined constant during the loading. No convergence issue was observed.
<figure><img src="image_174.png" /><figcaption>Figure 5.2-2: Finite element model boundary conditions for tension test.</figcaption></figure>

<figure><img src="image_175.png" /><figcaption>Figure 5.2-3: Finite element model boundary conditions for bending test.</figcaption></figure>

Compared to the previous finite element models described in chapter 4, the 1 mm thick adhesive layer was modelled using 4 elements in the thickness, as visible in Figure 5.2-4. It was made to fit the geometry of the experiments and of the cohesive models.

<figure><img src="image_176.png" /><figcaption>Figure 5.2-4: Stress analysis meshing.</figcaption></figure>

### 5.2.2. Stress criteria and mesh sensitivity

To be in accordance with the failure mode obtained experimentally, it must be noted that the stress and energy analysis are performed at the resin/steel interface in the model. The stress will be computed in the first element in the thickness and at a centered position in width (where stress is maximal). The strain energy will be considered only at the interface layer to reduce computation time. This approximation is made following the results obtained at chapter 3 (adhesive failure mode).

When using a stress criterion, the predicted failure load is the load at which the monitored stress reaches the critical stress. Two types of critical stresses can be used:

- the stress in the bulk of the adhesive, in case of cohesive failure,

- the interfacial stress, in case of adhesive failure, obtained through standardized test, such as TAST tests.
The stress analysis can be performed at different locations, and on different stresses. The choice of the studied stress is also really important and highly impacts the obtained results. The state of the art has shown that the peel stress, shear stress, and maximal principal stress have been used to predict bonding failure in different application (part 1.3.1). These three stresses will therefore be studied during the mesh sensitivity analysis.

First, a mesh sensitivity analysis was performed on the tension test model. The basic mesh is shown in Figure 5.2-4. The mesh is refined in the first few centimeters of the crack propagation length (in x-direction). The mesh sensitivity results performed for tension test are plotted for the shear stress, peel stress and maximum principal stress, respectively in Figure 5.2-5, Figure 5.2-6 and Figure 5.2-7, in the first element in the thickness of the adhesive.

<figure><img src="image_177.png" /><figcaption>Figure 5.2-5: Mesh sensitivity analysis for the tension test: shear stress.</figcaption></figure>

<figure><img src="image_178.png" /><figcaption>Figure 5.2-6: Mesh sensitivity analysis for the tension test: peel stress.</figcaption></figure>
<figure><img src="image_179.png" /><figcaption>Figure 5.2-7: Mesh sensitivity analysis for the tension test: principal stress.</figcaption></figure>

The mesh sensitivity analysis shows that for the peel stress and the maximum principal stress, the result does not converge around the border of the model (< 2 mm) and stresses tend to infinity as the mesh size decreases because of the stress concentration. Thus, no analysis can be performed using these values. In the case of shear stress, the model is being considered to converge for a mesh size of 0.25mm, except for the initial element. This stress will therefore be used for the stress/energy criteria analysis, with the exception of the first element.

The experimental campaign performed in the chapter 3 has shown that adhesive failure was obtained. In this case, defining the stress criteria from bulk results seems to be inadequate. Therefore, it was decided to adopt the shear strength obtained from TAST test as critical value. The obtained average shear stress at failure was 30.4 MPa (see Appendix A2). The Figure 5.2-8 shows the mesh sensitivity analysis performed for the bending model.

<figure><img src="image_180.png" /><figcaption>Figure 5.2-8: Mesh sensitivity analysis for the bending test: shear stress.</figcaption></figure>

In the case of bending test, for shear stresses, the model seems to converge for a mesh size of 1mm. Thus, it was decided, because of the low difference in computation time, to keep the 0.5 mm mesh.

Figure 5.2-9 presents the final shear stress curves that will be used for the coupled criteria definition (obtained for a mesh size of 0.25 mm for the tension model, and 0.5mm for the bending test). It can be seen that, despite the two types of loading being really different (tension and bending), the final shear stress curves are close.
<figure><img src="image_181.png" /><figcaption>Figure 5.2-9: Tension and bending model shear stress.</figcaption></figure>

### 5.2.3. Energy criteria

This criterion is based on the measurement of the energy release rate (ERR, described in part 1.3.3). The Virtual Crack Closure Technique (VCCT) can be used to determine it using numerical finite element approach. Several methods can be performed within the VCCT but only the two-step method will be presented here.

The local energy release rate equation used for the VCCT in 2D without any mode III contribution (neglected) is reminded in equation (106).

$$G_t = \frac{1}{2da} [F_{c,x}(u_d - u_c) + F_{c,y}(v_d - v_c)] = G_{II} + G_I \quad (106)$$

In this criterion, the ERR is used and compared to the critical ERR, $G_c$, of the interface. For a certain level of loading, when $G_t = G_c$, a characteristic length $a_c$ will thus be defined. If the length of the initial crack $a_0$ is greater than the critical crack length $a_c$, the crack propagates.

This criterion can be also used in such a way that the ERR is assessed for a step of propagation starting with an initial crack $a_0$. If the available energy at the crack tip is greater than the critical ERR, then the crack will propagate.

Alternatively, the ERR can be evaluated as the crack propagates from the initial position to a large distance as illustrated in Figure 5.2-10. Thus, in order to use that criterion, it is necessary to calculate the ERR for different crack lengths.

<figure><img src="image_182.png" /><figcaption>Figure 5.2-10: Illustration of the energy criteria with a critical interface ERR $G_c$ and a characteristic length $a_c$.</figcaption></figure>
A drawback of the method is the necessity to know the initial crack dimension. This is in general related to inspection techniques.

When the loading induces mode mixity (when the crack is loaded in mode I, II and III simultaneously), the total critical ERR can be defined in several ways. The one that was chosen here (making the assumption of mode I and II only) is the Benzeggagh-Kenane (B-K) criteria. It considers values based on tests in mode I, mode II and mixed mode and interpolates the values with a Power law as shown in Figure 5.2-11. The function used for the interpolation is detailed in equation (107).

$$G_t = G_{Ic} + (G_{IIc} - G_{Ic}) \left(\frac{G_{II}}{G_t}\right)^{\eta} \qquad (107)$$

where $\eta$ is the only parameter that needs to be characterized. In the presence of mode III, the last term with the power would be $(\frac{G_{II}+G_{III}}{G_t})^\eta$.

<figure><img src="image_183.png" /><figcaption>Figure 5.2-11: Example of the definition of the critical ERR with the B-K criterion for a resin C.</figcaption></figure>

### 5.2.3.1. Elaboration of VCCT in FEMAP

In order to calculate the mode I and II contributions, the VCCT method was used. This method is not available in NASTRAN and was developed using FEMAP API, and PYTHON CODE. Several simulations were performed with the same model by unzipping one node by one node at the adhesive/steel interface. Since it is a tedious task to run each model and post-process it, a small routine (an API) was created in FEMAP to automate it. The routine was written in the VBA language and used in the software as a custom tool. The pseudo-codes main steps for the routine is detailed in the Appendix A5.

### 5.2.3.2. Energy criteria definition

The B-K criterion was used to define a critical ERR for all mode ratios $G_{II}/G_t$. This law has been experimentally obtained and already discussed in chapter 3. Figure 5.2-12 presents the final B-K criterion fitting on the experimental data. The results of the resin B were quite particular (compared to Figure 5.2-11) as the results in mixed mode were just equal or slightly
below the results in pure mode I up to relatively high $G_{II}/G_t$ ratio (~0.75). The curve was kept as nearly linear until reaching this mode ratio.

<figure><img src="image_184.png" /><figcaption>Figure 5.2-12: Resin B fracture test results and B-K criterion fitting.</figcaption></figure>

The results of the VCCT for the converged mesh is plotted in Figure 5.2-13 for the tension model, and in Figure 5.2-14 for the bending model.

<figure><img src="image_185.png" /><figcaption>Figure 5.2-13: VCCT energy release rate for the tension model.</figcaption></figure>

<figure><img src="image_186.png" /><figcaption>Figure 5.2-14: VCCT energy release rate for the bending model.</figcaption></figure>
It can be seen that the ERR continuously increases except from the first to the second element. The analysis of the first element must be avoided because of this issue. In the following, the analysis will be done starting from the second element. Figure 5.2-15 presents the resulting (from the VCCT) mixed mode ratio along the bondline. It can be seen that the mode I loading is predominant initially (and should not be underestimated) and that the crack propagates mainly in mode II. It can also be stated that, despite the two tests being totally different, the energy at the border of the bondline seems to be close.

<figure><img src="image_187.png" /><figcaption>Figure 5.2-15: VCCT $G_{II}/G_{tot}$ mode ratio for the tension and bending model.</figcaption></figure>

### 5.2.4. Load at failure prediction analysis using coupled stress-energy criteria

Using the shear stress profiles, the shear strength (value from TAST), the energy release rate profiles and the critical toughnesses obtained with fracture mechanics tests for the converged mesh, a numerical failure load can be determined. The coupled criterion at the experimental failure load tests is shown in Figure 5.2-16 for tension test and in Figure 5.2-17 for bending test. It can be noted that depending on the applied load:

- the criteria can intersect each other, without any of them being fulfilled (example: at average experimental failure load),

- the criteria can intersect each other with only one of the criteria being fulfilled (example at 1.15 time the average experimental failure load). In this case, the stress criterion is not fulfilled, and the failure is not predicted to happen,

- the first load where the two criteria are fulfilled, is 1.22 times the average experimental failure load (Figure 5.2-16) in tension, and 1.13 time the average failure load in bending (Figure 5.2-17).
<figure><img src="image_188.png" /><figcaption>Figure 5.2-16: Tension test coupled stress-energy criterion computation for:
a) experimental failure load, b) 1.15 * experimental failure load, c) 1.22 * experimental failure load and d) 1.4 * experimental failure load.</figcaption></figure>

<figure><img src="image_189.png" /><figcaption>Figure 5.2-17: Bending test coupled stress-energy criterion computation for: a) experimental failure load and b) 1.13 * experimental failure load.</figcaption></figure>
The shape of the energy criteria also provides information on the propagation characteristics. Once the coupled criterion is fulfilled, the crack propagates. Then:

- if the energy criterion decreases at one point, the load should be increased again to continue to propagate, which implies a controlled crack propagation,

- if the energy criterion is always increasing at each crack propagation step (for a fixed load), the remaining energy stored by the bondline is sufficient to allow the propagation to continue. It led to an uncontrolled propagation.

For the tension and bending test, this result is consistent with the visual observations during the test and the load/displacement curves. Actually, the crack initially propagates slowly along a few centimeters and then violently up to the total disbonding.

**5.3. Application of the developed methodologies: direct and indirect method failure prediction**

This part presents the analysis of the load/displacement curves and the local measurement analysis of the gauges for the tension and bending test obtained by the model both for direct and indirect method. The results are compared to the experimental results to validate if the models predict correct load at failure values, and good bonded patch behavior.

**5.3.1. Hypothesis for FE model**

The cohesive elements are introduced in the model at the interface between the steel substrate and the steel reinforcement. It was decided to have no volume elements for the adhesive (Figure 5.3-1). This assumption is made to simplify the modeling as in chapter 4. Consequently, the additional momentum generated by the offset of the 1 mm bondline has to be modeled by adding thickness of the cohesive elements. The mesh was changed in one direction compared to the stress energy model. In order to reduce the total number of elements (and therefore the computation time), the number of elements was reduced in the center of the sample (dividing the total number of elements by 3.5).

<figure><img src="image_190.png" /><figcaption>Figure 5.3-1: Cohesive model meshing.</figcaption></figure>
The cohesive properties were taken from the chapter 4 results. The parameters are reminded in Table 5.3-1 and Table 5.3-2.

<table><caption>Table 5.3-1: Resulting parameters of cohesive law for resin B steel-to-steel assembly characterization using direct method.</caption><thead><tr><th></th><th>Fracture toughness G<sub>c</sub><br>(kJ/m<sup>2</sup>)</th><th>Critical stress σ<sub>c</sub><br>(MPa)</th><th>Stiffness K<br>(MPa/mm)</th><th>Coupling<br>parameter</th></tr></thead><tbody><tr><td>Mode I</td><td>0.316</td><td>60</td><td>8426</td><td rowspan="2">0.35</td></tr><tr><td>Mode II</td><td>1.960</td><td>53</td><td>1060</td></tr></tbody></table>

<table><caption>Table 5.3-2: Resulting parameters of cohesive law for resin B steel-to-steel assembly characterization using indirect method.</caption><thead><tr><th></th><th>Fracture toughness G<sub>c</sub><br>(kJ/m<sup>2</sup>)</th><th>Critical stress σ<sub>c</sub><br>(MPa)</th><th>Stiffness K<br>(MPa/mm)</th><th>Coupling<br>parameter</th></tr></thead><tbody><tr><td>Mode I</td><td>0.25</td><td>35</td><td>12 900</td><td rowspan="2">0.37</td></tr><tr><td>Mode II</td><td>2.15</td><td>35</td><td>1 500</td></tr></tbody></table>

### 5.3.2. Load/displacement curves

Figure 5.3-2a and b display the obtained experimental load/displacement curves for the bending and tension tests compared to FE cohesive models. The overall stiffness is well modeled for bending tests while the difference in the stiffness for the tension tests is around 25%.

<figure><img src="image_191.png" /><figcaption>Figure 5.3-2: a) Bending test load/displacement curves, b) Tension test load/displacement curves.</figcaption></figure>

The error in stiffness, depending on the type of loading, can be due to two possible issues. Firstly, it could be related to an approximation or error in the modeling hypothesis which could be translated by a wrong behavior of the model. Secondly, it could be due to an error with the load or displacements measurement during the tests. The meshing and cohesive behavior being the same for the two models, and as good stiffness is observed for the bending test, this may be related to possible sliding in the clamps during the tension tests. This introduces higher displacement than in the model where the displacements are mimicking a perfect clamping at the end of the sample. The analysis of the gauge measurements in the next part will validate this assumption.
### 5.3.3. Strain measurements analysis during tension tests

#### 5.3.3.1. Gauges data analysis

Based on Figure 5.1-2, the gauges results that are compared to the modeling results are:

- the gauges 3 and 4 that give informations on the capacity of the modeling:

- to express the transition length impact on the strain in the 6 mm adherent. If the transition length predicted by the model is too long or too short, the gauges measurement will indicate smaller or bigger values compared to experimental values,

- to give informations on the propagation of the crack at the beginning of the patch failure,

- to predict the same maximum values as the experimental ones.

- the gauges 1 and 5 related to the global patch behavior during the loading and the crack propagation.

The gauges 7 and 8 will not be described as they were on the side of the vice clamp and as this behavior is not modeled.

Figure 5.3-4, Figure 5.3-4 and Figure 5.3-5 present the gauges 1 and 5 measurement (as function of the load) compared to the results obtained through the developed finite element model at the same position (3 mm from the patch end and in the center of the patch).

<figure><img src="image_192.png" /><figcaption>Figure 5.3-3: Experimental and numerical gauge 1 measurements for the tension tests,</figcaption></figure>
<figure><img src="image_193.png" /><figcaption>Figure 5.3-4: Zoom on the non-linear part of the curves for gauge 1 measurements for the tension tests.</figcaption></figure>

<figure><img src="image_194.png" /><figcaption>Figure 5.3-5: Experimental and numerical gauge 5 data for the tension test.</figcaption></figure>

The gauge 1 experimental curves are similar to the numerical ones with the same initial slope during the loading of the patch, with close first point of non-linearity for the direct model (Figure 5.3-4, highlighted crosses) and close maximal strains related to crack initiation for both models. A similar loss of stiffness (non-linear part) during the crack propagation is also observed for both models. Thus, the results show that the predictions of the direct and indirect methods are accurate and that the overall behavior of the patch, in the 20 mm adherent, is correctly modeled.

Table 5.3-3 gives the different values obtained for each model and compares them to the experimental ones. It can be observed that the indirect model predicts a non-linearity at a higher strain and load level than the direct model (20% more).
<table><caption>Table 5.3-3: Several characteristic values determined through experimental and numerical analyses using the developed cohesive zone approaches (direct and indirect methods) for gauges 1 and 5, for the tension tests.</caption><thead><tr><th></th><th>Initial slope<br>(µm/kNm)</th><th>Strain (µm/m) at<br>stiffness change<br>(gauge 1)</th><th>Strain (µm/m) at half-<br>length debonding<br>(gauge 5)</th><th>Load (kN) at half-<br>length debonding<br>(gauge 5)</th></tr></thead><tbody><tr><td>Test 1</td><td>5.58</td><td>1021</td><td>640</td><td>216</td></tr><tr><td>Test 2</td><td>5.78</td><td>1086</td><td>592</td><td>188</td></tr><tr><td>Test 3</td><td>5.79</td><td>1071</td><td>701</td><td>223</td></tr><tr><td>Experimental average</td><td>5.71</td><td>1063</td><td>641</td><td>209</td></tr><tr><td>Direct method</td><td>6.15</td><td>1042</td><td>598</td><td>192</td></tr><tr><td>Average difference<br>with direct method (%)</td><td>(+8%)</td><td>(-2%)</td><td>(-5%)</td><td>(-8%)</td></tr><tr><td>Indirect method</td><td>6.15</td><td>1176</td><td>730</td><td>231</td></tr><tr><td>Average difference<br>with indirect method<br>(%)</td><td>(+8%)</td><td>(+10%)</td><td>(+13%)</td><td>(+10%)</td></tr></tbody></table>

Figure 5.3-6 presents the results for the gauge 3. The gauge 4 results are similar and are not presented in this document. The initial slopes of the experimental and model results are similar. Table 5.3-4 presents the load at failure predicted for the tension test by the direct method (difference of -4% with experimental results) and the indirect method (difference of +20% with experimental results). This load is defined when the gauge 3 strain measurement is maximal. The results prove that the cohesive law shape (direct method) and the critical toughness value (indirect method) seem to influence mainly the failure load prediction but not the general behavior of the patch.

<figure><img src="image_195.png" /><figcaption>Figure 5.3-6: Experimental and numerical gauge 3 measurements for the tension test.</figcaption></figure>
<table><caption>Table 5.3-4: Several characteristic values determined through experimental and numerical analyses using the developed cohesive zone approaches (direct and indirect methods) for gauge 3, for the tension tests.</caption><thead><tr><th></th><th>Initial slope (µm/kNm)</th><th>Strain at failure (µm/m)</th><th>Failure load (kN)</th></tr></thead><tbody><tr><td>Test 1</td><td>2.23</td><td>354</td><td>167</td></tr><tr><td>Test 2</td><td>2.27</td><td>325</td><td>148</td></tr><tr><td>Test 3</td><td>1.91</td><td>372</td><td>181</td></tr><tr><td>Experimental average</td><td>2.13</td><td>348</td><td>165</td></tr><tr><td>Direct method</td><td>2.01</td><td>328</td><td>169</td></tr><tr><td>Difference with direct method (%)</td><td>(6%)</td><td>(-6%)</td><td>(+2%)</td></tr><tr><td>Indirect method</td><td>1.99</td><td>368</td><td>197</td></tr><tr><td>Difference with indirect method (%)</td><td>(7%)</td><td>(+8%)</td><td>(+19%)</td></tr></tbody></table>

### 5.3.3.2. Optical fiber data analysis during tension test

Figure 5.3-7 displays the experimentally obtained optical fiber strain measurement for the tension test n°3 (the two other tests show similar surface strain measurements), and for different times before and after the crack propagation (from t = 61.11 s to t = 71.05 s). The absciss of the figure corresponds to the distance from the border of the reinforcement (0 is the border). The blue line shows the position of the gauge 3 and the black line corresponds to the strain profile obtained with the finite element model on the 6 mm adherent surface when numerical maximal strains are measured at the position of gauge 3 (fixed as the experimental crack initiation time).

<figure><img src="image_196.png" /><figcaption>Figure 5.3-7: Experimental optical fiber strain measurements for tension test n°3 and numerical strain profile results.</figcaption></figure>

The curves show that the use of gauge 3 (gauge 4 gives similar results) as crack initiation measurement seems to be consistent. As highlighted by the blue line (gauge 3 position), the strain at this position is maximal just before the beginning of the crack propagation and is similar to the one obtained with the gauge 3 and 4 (~375 µstrain) at crack propagation. A shift of the strain curve is visible after the crack initiation and the measurement of this shift can provide crack propagation curve.
Finally, it can be seen that the finite element model strain profile and experimental one, before
crack propagation (orange curve, t = 64.97 s), are nearly identical. It validates that the general
behavior of the patch and, in particular, the transition length, is well modeled.

All the comparisons between the experimental results and the numerical ones allow to have
high confidence into the capacity of the model, and the cohesive methodology (with a limited
impact of the cohesive law shape) to be able to predict general bonded patch behaviors.

**5.3.4. Strain measurements analysis during bending tests**

5.3.4.1. Gauges data analysis

Figure 5.3-8 and Figure 5.3-9 present the gauge 1 and 5 measurements (as function of the load) compared to the results obtained with FE modelling. Table 5.3-5 presents the obtained load and strains at half debonding of the patch measured by the gauges 1 and 5.

The two methods (direct and indirect) predict:

- good global behavior of the patch (slope of the gauge 1 and 5 curves during the loading of the patch),

- underestimated local strains at failure (especially the direct method) (Table 5.3-5)

The direct model predicts lower strain at failure than the indirect one. These different results
are consistent with the load at failure obtained from the two models. The results are also
consistent with the tension test results and indicate that the overall behavior of the patch is
correctly modeled. Again, the shape of the cohesive law does not seem to influence the general
behavior, except the load at failure value.

<figure><img src="image_197.png" /><figcaption>Figure 5.3-8: Experimental and numerical gauge 1 measurements for the bending tests.</figcaption></figure>
<figure><img src="image_198.png" /><figcaption>Figure 5.3-9: Experimental and numerical gauge 5 measurements for the bending tests.</figcaption></figure>

<table><caption>Table 5.3-5: Several characteristic values determined through experimental and numerical analyses using the developed cohesive zone approaches (direct and indirect methods) for gauge 1 and 5, for the bending tests.</caption><thead><tr><th></th><th>Strain (µm/m) at failure (gauge 1)</th><th>Strain (µm/m) at half debonding (gauge 5)</th><th>Load (kN) at half debonding (gauge 5)</th></tr></thead><tbody><tr><td>Test 1</td><td>576</td><td>1822</td><td>5.87</td></tr><tr><td>Test 2</td><td>595</td><td>1910</td><td>6.15</td></tr><tr><td>Test 3</td><td>645</td><td>2210</td><td>7.03</td></tr><tr><td>Experimental average</td><td>605</td><td>1980</td><td>6.35</td></tr><tr><td>Direct method</td><td>483</td><td>1480</td><td>5.2</td></tr><tr><td>Difference with direct method (%)</td><td>(-19%)</td><td>(-25%)</td><td>(-18%)</td></tr><tr><td>Indirect method</td><td>560</td><td>1720</td><td>6</td></tr><tr><td>Difference with indirect method (%)</td><td>(-7%)</td><td>(-13%)</td><td>(-6%)</td></tr></tbody></table>

Figure 5.3-10 presents the obtained results for the gauge 3 and Table 5.3-6 presents the results extracted from these data, in the same manner as for the tension tests. The initial slope and load at failure of the experiments show limited difference for the three tests. Model results are similar to experimental ones (direct and indirect methods). The difference of the load at failure predicted by the direct and indirect methods is by 10%, the direct model remaining more conservative. Conclusions are similar to the ones obtained with the gauges 1 and 5.
<figure><img src="image_199.png" /><figcaption>Figure 5.3-10: Experimental and numerical gauge 3 measurements for the bending tests.</figcaption></figure>

<table><caption>Table 5.3-6: Several characteristic values determined through experimental and numerical analyses using the developed cohesive zone approaches (direct and indirect methods) for gauge 3, for the bending tests.</caption><thead><tr><th></th><th>Strain (µm/m) at break</th><th>Load at break<br>(kN)</th></tr></thead><tbody><tr><td>Test 1</td><td>450</td><td>5.0</td></tr><tr><td>Test 2</td><td>417</td><td>6.1</td></tr><tr><td>Test 3</td><td>438</td><td>6.0</td></tr><tr><td>Experimental<br>average</td><td>435</td><td>5.7</td></tr><tr><td>Direct method</td><td>403</td><td>5.15</td></tr><tr><td>Difference with<br>direct method (%)</td><td>(-7%)</td><td>(-9%)</td></tr><tr><td>Indirect method</td><td>437</td><td>5.65</td></tr><tr><td>Difference with<br>indirect method<br>(%)</td><td>(0.5%)</td><td>(0.8%)</td></tr></tbody></table>

These results demonstrate that the overall behavior of the patch in bending is correctly modeled using the two methods (direct and indirect) with an accurate load at failure prediction.

### 5.3.4.2. Optical fiber data analysis during bending test

Figure 5.3-11 displays the experimentally obtained optical fiber strain measurements for the bending test n°2 (the two other tests show similar surface strain measurements), and for different times before and after the crack propagation (from t = 38 s to t = 88.59 s). The absciss of the figure corresponds to the distance from the border of the reinforcement (0 is the border). The blue line shows the position of the gauge 3 and the black line corresponds to the strain profile obtained with the finite element model on the 6 mm adherent surface when numerical maximal strain is measured at the position of gauge 3 (fixed as the experimental crack initiation time, t = 71.73).
<figure><img src="image_200.png" /><figcaption>Figure 5.3-11: Experimental optical fiber strain measurements for the bending test n°2 and numerical strain profile results.</figcaption></figure>

The curves show that the use of gauge 3 (gauge 4 shows similar results) as crack initiation measurement seems to be consistent. As highlighted by the blue line, the strain at this position is maximal just before the beginning of the crack propagation. Moreover, a shift of the strain curve is visible after the crack initiation and the measurement of this shift can provide crack propagation curve.

Finally, it can be seen that the finite element model strains profile and experimental measurements before crack propagation (green curve, t = 71.73 s) are nearly identical. It validates that the general behavior of the patch is correctly modeled.

All the comparisons between the experimental results and the numerical ones allow to have high confidence into the capacity of the model, and the cohesive methodology (direct and indirect methods), to be able to predict general bonded patch characteristics under monotonous loading.

### 5.3.5. Crack front shape

The crack front shape analysis performed in chapter 2 and 3 (for different resins) has shown that this shape is dependent on the loading mode. Mode I generates parabolic crack front shape and mode II generates straight crack front shape. These results are in accordance with Griffith theory as described in [124]. The description of the process zone shape for mode II loading shows no difference between plane strain and plane stress. On the contrary, for mode I loading, the process zone for plane strain behavior is much more condensed around the crack tips compared to plane stress loading. Thus, to ensure that the modeling of the bonded patch describes precisely the bonding behavior, a validation of the correct mode ratio in the bondline (represented by a correct crack front shape) is required.

Figure 5.3-12 presents a comparison of the obtained experimental fracture surface and the numerical model damage pattern for a tension test. The same result was obtained for the bending test. The failure was 100% adhesive at the substrate interface. On the figure on the left, some “micro” cracking is occurring inside the bondline up to the interface with the 6mm reinforcement plate (no propagation on the other interface). The shape of these cracks is used
as a measurement of the crack shape in the bondline during the crack propagation. It is
compared to the damage distribution in the bondline during the crack propagation of the tension
modeling, which represents the numerical crack front shape.

<figure><img src="image_201.png" /><figcaption>Figure 5.3-12: Experimental crack front shape and FE method crack front shape prediction.</figcaption></figure>

Figure 5.3-12 shows that the crack front shape obtained numerically for the direct and indirect method analysis is in accordance with the experimental results. However, it can be seen that the predicted process zone length by the direct and indirect methodologies are quite different. Studying the shape of the cohesive laws described in the conclusion of the chapter 4, these results are consistent as the energy stored by the damaged part of the cohesive is much smaller in the direct cohesive laws than in the indirect ones.

The relatively flat shape of the parabolic front shape tends to indicate, compared to the chapter 3 results, that the loading mode is mainly mode II (as expected from the loading of the sample). The mode ratio cannot be directly experimentally quantified, but gives informations on the correct interpretation of the experimental results by the model. The bending test has shown similar results and conclusions than the tension test.

**5.4. Conclusion**

In this chapter, real-scale tests with loading conditions close to reality (most commonly encountered loading situations on ship deck) were carried out. Two types of loading were performed on the same geometry: bending (3 specimens) and tension (3 specimens) loadings.

The three modeling methodologies (direct, indirect and coupled criteria) chosen in the chapter 1 were tested and the patch's behavior and maximal capacity were assessed. The monitoring of the tests with strain gauges and optical fiber was used as reference values. It has helped to validate the capacity of the cohesive element model (with the direct and indirect method) to predict the failure load of the assembly and its local and global behavior. The application of the coupled stress-energy method was also assessed. The definition of each criterion was made using experimental data from adapted and described characterization tests. Final predictions were done using finite element methods and the presented methodologies.

Table 5.4-1 and Figure 5.4-1 present the final comparison of the failure loads predicted by the three methods and the experimental results. They show that, despite the relatively low number of tests that could be made (especially critical toughness determination), the methodologies that have been initially defined were able to predict failure load of bonded patch with good accuracy. The only methodology which proved to have conservative results (and the better accuracy) was the direct one.
<table><caption>Table 5.4-1: Failure load prediction of direct, indirect methodologies and coupled stress-energy criteria compared to experimental results for real-scale investigations.</caption><thead><tr><th></th><th>Load at break<br>Bending (kN)</th><th>Load at break<br>Tension (kN)</th></tr></thead><tbody><tr><td>Experimental average</td><td>5.7</td><td>166</td></tr><tr><td>Direct method</td><td>5.2</td><td>169</td></tr><tr><td>Difference with direct method (%)</td><td>(-9%)</td><td>(+2%)</td></tr><tr><td>Indirect method</td><td>5.6</td><td>197</td></tr><tr><td>Difference with indirect method (%)</td><td>(-2%)</td><td>(+19%)</td></tr><tr><td>Coupled criteria</td><td>6.32</td><td>203</td></tr><tr><td>Difference with coupled criteria (%)</td><td>(+13%)</td><td>(+22%)</td></tr></tbody></table>

<figure><img src="image_202.png" /><figcaption>Figure 5.4-1: Normalized failure load predictions of direct, indirect and coupled stress-energy criteria methods compared to experimental results.</figcaption></figure>
# General conclusion

This study was part of the Strength Bond project dedicated to the development of a design methodology for bonded reinforcements of corroded steel plates like those that can be found in offshore applications. The absence of effective failure criteria has led to a tendency to oversize bonded joints in industrial applications. A better understanding of the failure of bonded joint and the development of reliable design methodologies are necessary to increase confidence and to lead to more efficient application of composite bonded repairs in offshore applications. In order to design adhesively bonded joints, the mechanical state of the adhesive within the joint (stress, strain, energy) should be well determined and a reliable failure criterion, adapted to the obtained failure mode, should be chosen.

In the first chapter dedicated to the state of the art of bonded assemblies, two main strategies have been detailed for mode I and II failure criteria. The first is based on a stress or strain criterion in relation with continuous mechanics. The second is based on an energetic criterion in accordance with fracture mechanics. Both criteria have advantages and drawbacks. For mixed-mode approaches, the coupled stress-energy criteria can be used as well as the cohesive zone models. This last approach was more deeply studied in this work. Regarding the determination of the mechanical state in a bonded assembly, two main strategies can be considered: the use of analytical methods, or numerical methods. Analytical methods can be an efficient way to obtain fast results that may be valuable in the preliminary design stage, or in case of simple joint design. They allow a good understanding of the behavior of the bonded joints and an identification of the driving parameters. On the other hand, numerical methods allow considering more complex and detailed geometries and the introduction of nonlinear material behavior that could affect the properties of the assembly. It was chosen in this work to rely on the use of finite element modeling so that the developed strategy may be easily applied to alternative geometries. To carry out the design process using cohesive zone approach and finite elements, several steps are needed. First, the critical toughnesses and the cohesive laws must be determined. This is in general done through fracture mechanics investigations. Then, the finite element model can be built and its robustness verified through mesh sensitivity analysis and comparison with experimental results. Finally, it can be applied to different cases.

The second chapter was dedicated to the improvement of the existing fracture mechanics investigations, studying the capacity of distributed optical fiber to monitor crack propagation in bonded assemblies. Resin A (epoxy) was used during those investigations. Several strategies have been assessed through experimental investigations, analytical and numerical modelling. The use of externally surface bonded optical fiber on the adherends proved to be successful for the studied geometries (it was compared to post-failure measurements). The crack monitoring process was settled for mode I, mode II and mixed mode investigations. This technique has several advantages as it is rather easy to settle, it is non-intrusive, it can measure mid-width crack propagation and can be rather easily automated. The use of embedded optical fiber has also been studied but the results were less satisfactory and more investigations seem to be required.

The third chapter presented the realized fracture mechanics investigations using developed methodology relying on continuous optical fiber. It aimed to study the different proposed methodologies for the critical toughness determination, and to present a methodology able to provide cohesive laws from experimental local measurements using digital image correlation.
Those investigations were realized using resin B (epoxy). In the case of uncontrolled crack propagation, the used of continuous optical fiber system was not able to monitor properly the crack (this was the case for ENF and MMB tests), and consequently, only conservative critical initiation toughness values could be determined through standardized approaches. When controlled crack propagation occurred, the continuous optical fiber proved to be able to improve the critical propagation toughness determination (done for DCB tests). The use of the J-integral method also proved to be successful, providing closed results to the critical propagation toughness in mode I. It allowed also to determine cohesive laws shape for mode I and mode II.

Fourth chapter aimed at presenting the chosen finite element modelling strategy relying on the use of FEMAP software. Cohesive laws are implemented in this software and mode coupling is carried out through a coupling parameter that needs to be determined. This software allowed to study two different methodologies to determine the complete cohesive laws (including the coupling parameters) and to compare the numerical expectations with the experimental measurements. The first method is called direct method and consists in using the mode I and II cohesive laws determined through the J-integral approach (described in chapter 3). The second method, called the indirect method, relies on the use of the numerical model to determine all the cohesive parameters (mode I and II laws and the coupling parameter). Both approaches gave satisfactory results in comparison with experimental measurements for the DCB, ENF and MMB tests (presented in chapter 3). The obtained coupling parameters were closed, but the cohesive laws were rather different. Though the cohesive law shape does not seem to have an impact on the studied quasi-static tests, it is important to underline that it could have more influence in the case of durability investigations (creep or fatigue for example). This should be analyzed more deeply.

The last chapter was dedicated to the application of the developed numerical approaches to large-scale experiments on adhesively bonded joints. Samples more representative of industrial applications were studied and tested. The developed numerical methodology was applied with both the direct and indirect method to assess numerical predictions and verify their consistency with experimental measures. The coupled stress-energy approach was also carried out. It proved to be rather difficult as several strong hypotheses had to be settled (choice of the stress and its location, selection of the stress criteria). The obtained results were satisfactory and the direct method proved to be the closest to the experimental results. Table 6.4-1 summarizes the main contribution and drawbacks of each studied methodology. The direct method proved to be the most precise, but requires more instrumentation. The indirect method requires more computation time and less instrumentation. The coupled stress-energy approach is highly dependent upon strong hypotheses particularly related to the stress criteria application.
<table><caption>Table 6.4-1: Comparison of the design methodologies studied.</caption><thead><tr><th></th><th>Test required</th><th>Model computation time (standard test)</th><th>Model computation time (full size test)</th><th>Model data</th><th>Failure prediction</th></tr></thead><tbody><tr><td>Direct method</td><td>DCB/ENF/MMB With special instrumentation (DIC)</td><td>2 days</td><td>4 days</td><td>Complete patch behavior and load at failure</td><td>Conservative</td></tr><tr><td>Indirect method</td><td>DCB/ENF/MMB With standard instrumentation</td><td>5-8 days</td><td>4 days</td><td>Complete patch behavior and load at failure</td><td>Non conservative</td></tr><tr><td>Coupled criteria</td><td>TAST/DCB/ENF/MMB</td><td>Ø</td><td>5-6 days</td><td>Load at failure</td><td>Non conservative</td></tr></tbody></table>

These results prove that the developed methodologies (direct and indirect methods) can be applied to adhesively bonded composite steel joints.

It would be interesting to carry out similar experiments using ductile adhesive to verify the adequacy of the methodology in this case. In addition, more investigations related to the process zone could be done as the continuous optical fiber measurements seems to be highly sensitive to it. This could maybe provide an alternative method to obtain cohesive laws.

To be extended to adhesively bonded composite patches, it will yet be needed to verify the accuracy of the methodology when multiple layers are present in a single cohesive zone. In fact, in this case, a glass fiber reinforced polymer layer is often inserted between the steel surface and the carbon fiber reinforced polymer, to limit galvanic corrosion risks and to soften the bonded interface.
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# Appendix

## A1: ENF specimens with primer – observations after failure.

<figure><img src="image_203.png" /><figcaption>Figure A1-1: ENF failure mode with primer.</figcaption></figure>

<figure><img src="image_204.png" /><figcaption>Figure A1-2: ENF failure mode without primer.</figcaption></figure>
A2: TAST test of the resin B with primer.

<figure><img src="image_205.png" /><figcaption>Figure A2-1: Resin B TAST test with primer - load/displacement curves (provided by NavalGroup).</figcaption></figure>
## A3: Surface preparation analysis

Surface preparation is of great importance in bonded repairs as described in paragraph 1.1.2. It would be a great advantage to be able to qualify the level of surface preparation carried out on site before bonding to avoid any unanticipated low interface properties. In this application case, samples which have been deliberately degraded (corrosion and pollution) and with several surface preparation methodologies were tested. The quality of these preparations was then assessed using methods that can be implemented on site and that are complementary to more commonly used methods. Two main aspects were investigated: the detection of the residual presence of pollutants through the use of a portable infrared spectrometer and the determination of surface energy after preparation using contact angle and wettability measurement equipment. The objective is to compare the obtained measurements to the critical toughness measurements performed in the previous parts to see if correlation can be made directly with the wettability, chemical properties and mechanical properties.

### A3.1 Studied specimens and initial pollution

Fourteen specimens were used for this study and are metallic plates of S235 steel (yield strength ~250MPa). Each specimen was treated as follows:

- initial sandblasting to remove scale and residual oxidation from the manufacturing process (crude steel),

- ageing for two days in an outdoor environment (humidity > 95%) to create a surface corrosion layer,

- pollution of the surface of the specimens with: diesel (four specimens), hydraulic oil (four specimens) or a paraffinic corrosion protection spray (four specimens). Two specimens were left unpolluted and serve as a reference,

- after the pollution of each sample, two more days of ageing in an outdoor environment (humidity > 90%) were observed to recreate a surface corrosion layer.

The resulting specimen are presented Figure A3.1-1 A and B prior and after treatment.

<figure><img src="image_206.png" /><figcaption>Figure A3.1.1 a) Surface preparation specimens before ageing and pollution. b) Specimens after ageing and pollution.</figcaption></figure>
## A3.2 Surface preparations

Four types of surface treatment were carried out for each type of pollution:

- Preparation (A): detergent cleaning,

- preparation (B): detergent cleaning, follows by a sandblasting and then follows by the application of an anti-corrosion primer,

- preparation (C): detergent cleaning, follows by a sandblasting, follows by a solvent cleaning and then follows by the application of an anti-corrosion primer,

- preparation (D): detergent cleaning, follows by a sandblasting and then follows by a solvent cleaning.

The pollution by diesel, hydraulic oil, or paraffin spray used as corrosion protection, were respectively named (1), (2) and (3) while the pollution-free samples were named (4). The different surface preparations of each specimen are summarized in Table A3.2-1 and visible in Figure A3.2-1.

<figure><img src="image_207.png" /><figcaption>Figure A3.2.1: Surface preparation specimens after surface preparation.</figcaption></figure>

<table><caption>Table A3.2.1: Studied surface preparations.</caption><thead><tr><th>Type of surface treatment</th><th>Detergent cleaning</th><th>Sand blasting</th><th>Solvent cleaning</th><th>Anti-corrosion primer</th><th>Specimens considered</th></tr></thead><tbody><tr><td>A</td><td>X</td><td></td><td></td><td></td><td>4 specimens :<br>3 polluting agents (1, 2, 3) +<br>Unpolluted reference (4)</td></tr><tr><td>B</td><td>X</td><td>X</td><td></td><td>X</td><td>3 specimens :<br>3 polluting agents (1, 2, 3)</td></tr><tr><td>C</td><td>X</td><td>X</td><td>X</td><td>X</td><td>3 specimens :<br>3 polluting agents (1, 2, 3)</td></tr><tr><td>D</td><td>X</td><td>X</td><td>X</td><td></td><td>4 specimens :<br>3 polluting agents (1, 2, 3) +<br>Unpolluted reference (4)</td></tr></tbody></table>

## A3.3 Analysis of the presence of pollutants by infrared spectrometer

In order to investigate the ability of the different surface preparations to remove the initially deposited pollutants, an Agilent Technology 4300 infrared spectrometer was used [125]. The spectra obtained for all measurements are shown in Figure A3.3-1, Figure A3.3-2, and Figure A3.3-3. For reasons of clarity, an offset on the absorption values has been artificially introduced in Figure A3.3-1 and Figure A3.3-2 to facilitate the comparison of the obtained spectra.
Figure A3.3-1 shows a comparison of the absorption curves obtained for the specimens with type (A) surface preparation (specimens 1A, 2A, 3A, 4A) and the specimens with type (D) surface preparation (specimen 4D). This figure shows the presence of rust (dome between 3000 and 3500 cm-1) on specimens 1A, 2A, 3A and 4A, as well as the presence of more or less marked pollution (peaks between 2800 and 3000 cm-1, and peaks between 1200 and 1700 cm-1) for the specimens 1A, 2A and 3A. As a result, cleaning with detergent alone does not seem sufficient to remove completely rust and pollutants.

<figure><img src="image_208.png" /><figcaption>Figure A3.3.1: Influence of initial pollution on infrared spectrometer measurements for type (A) surface preparation.</figcaption></figure>

Figure A3.3-2 shows the absorption curves for the reference specimens (4D) before and after the primer application. As it can be seen, the presence of primer does not seem to affect the measurements in this wavelength range.

<figure><img src="image_209.png" /><figcaption>Figure A3.3.2: Influence of the presence of primer on infrared spectrometer measurements.</figcaption></figure>

Finally, Figure A3.3-3 shows a comparison of the absorption curves obtained for the specimens with the different pollutions. It can be observed that the surface preparation of type (B), (C) and (D) (with a sandblasting, followed or not by a solvent cleaning) give similar infrared spectra, independently of the pollution applied. As a result, it is set that sandblasting allows removing rust (which becomes undetectable in the FTIR analysis), but also all the pollution, since it is no longer detectable in the FTIR analysis after sandblasting, whether or not solvent degreasing has been carried out. It therefore seems that solvent cleaning after sandblasting is not necessary to eliminate residual pollutants.
<figure><img src="image_210.png" /><figcaption>Figure A3.3.3: Comparison of the infrared spectra obtained for the three surface preparation methods studied and the different type pollutions.</figcaption></figure>
## A3.4 Surface energy measurements

In this study, surface energy measurements were carried out using:

- the WCA "Water Contact Angle" technique [127]. This technique was carried out using a thermoregulated tensiometer (Kruss DSA100), available at the MIT Laboratory of the Gustave Eiffel University, and for three reference liquids (Water, Ethylene Glycol, Glycerol) deposited in drop on the studied surfaces (Figure A3.4.1a). It should be noted that some of the used liquids (Ethylene glycol and Glycerol) led to a dissolution of the primer. The measurements made with these liquids could therefore not be processed and analyzed,

- spread measurements using wettability measurement inks (Table A3.4.1). The used test inks have a measuring range of 31 to 67 mJ/m² with a step between each ink of 2mJ/m² (any values outside this range cannot be precisely measured).

<figure><img src="image_211.png" /><figcaption>Figure A3.4.1: a) Contact angle measurement b) Use of spread measuring inks.</figcaption></figure>

The results obtained with these two methods are summarized in Table A3.4.1.

<table><caption>Table A3.4.1: Presentation of the performed surface energy measurements.</caption><thead><tr><th>Specimens</th><th>Wettability ink (mJ/m²)</th><th>Contact angle measurement (mJ/m²)</th></tr></thead><tbody><tr><td>A<br>With (1,2,3) and without initial pollution(4)<br>B et C</td><td>27 - 35</td><td></td></tr><tr><td>With (1,2,3) and without initial pollution(4)</td><td>36 - 44</td><td>40 ± 12</td></tr><tr><td>D without pollution</td><td>67+</td><td>90 ± 10</td></tr></tbody></table>

With surface treatment B, C, D, it appears that it is not possible to clearly differentiate between polluted test specimens.

It appears also that sandblasting followed by cleaning with solvent (preparation D) increases significantly the surface energy. The comparison of the surface energies obtained for the type C and D surface preparations shows that the treatment with an anti-corrosion primer of the sandblasted surface strongly reduces the wettability (reducing from - 30% to - 60%).

## A3.5 Surface energy analysis: Conclusions

### Surface preparation analysis:

Within the framework of the optimization of bonding patch over steel, the carried-out investigations focused on the comparison of different surface preparation methods. The evaluation of the obtained surfaces is carried out using two different methodologies.

The selected pollutants represent a sample of probable pollution on a ship or offshore unit, plus a product used to maintain a corrosion-free surface.

The study showed that cleaning with detergent does not remove the rust, nor the pollutants since they were clearly detected with the FTIR spectrometer, and surface energy of the steel remains very low. This preparation is insufficient. The study did not highlight specific interest of solvent
cleaning after grit blasting as neither FTIR nor surface energy measurement showed significant
differences.

The obtained results indicate that solvent cleaning plus sandblasting removes all measurable
traces of rust and pollutant, and increase the surface energy of the specimens. The application
of the primer significantly degrades the energy of the prepared surfaces. It should be checked
that this degradation remains sufficiently low so as not to diminish the capacities of the interface
between the patch and the steel substrate. Methods FTIR, WCA, and wettability inks are
available in portable format, and seem to bring an interesting information on the quality of the
surface preparation. However no direct relation between the measured wettability and critical
toughness can be made. In this series of tests, the impact of the primer was not measured as
damageable for the toughness compared to bare steel specimens, where high decrease of the
wettability was measured.