Mode mixity for the fracture toughness obtained by climbing drum peel tests

Wind turbine blades include numerous interfaces, which are known to be critical regions for early failure. Thus, reliable testing of interfaces and a solid understanding of the tests used are of utmost importance. The climbing drum peel (CDP) test is a standardized test for characterization of adhesively bonded interfaces, which in literature has been referred to as a possible simple alternative to mode I double cantilever beam testing of composites. The current work uses finite element modeling of the CDP setup to show that it is actually a mixed mode test, and therefore one cannot use it to obtain the mode I fracture toughness. It can, however, be used to compare different interfaces, keeping in mind that the comparison is done for a mode mixity of approximately 40°, and that one adherend has to be thin (around 0.5mm).


Introduction
Wind turbine blades contain numerous interfaces and are one of the most critical locations for failure [1].Thus, it is of great importance to be able to test and model the interface strength (fracture toughness) to be used in the blade design.The properties of an interface are often defined by the fracture toughness calculated as the energy release rate.An interface with a crack considered in two dimensions (2D) can be loaded in opening (mode I), shearing (mode II), or a mixture of the two modes (mixed mode) at the crack tip.An interface is often weakest in mode I loading (e.g.[2]).Therefore, a mixed mode loading condition at the crack tip can lead to a higher measured fracture toughness than if the same interface was tested in mode I.If the fracture toughness experimentally measured in mixed mode loading conditions is used in place of the mode I fracture toughness in an analytical or numerical model, the results might thus be non-conservative and could have catastrophic consequences.
A common test approach for experimentally measuring the mode I fracture toughness of an interface is the double cantilever beam (DCB) test (ASTM D5528 [3]).In the DCB test it is either required to measure the crack length, which is not an easy task, or do a compliance calibration, which requires additional initial test steps [3].In addition, the crack front in a DCB test is closer to parabolically shaped than straight, even though a straight crack is assumed in the calculations of the fracture toughness [4,5].An alternative to the standard DCB test is the DCB test with uneven bending moments (DCB-UBM) [6].Using bending moments instead of forces eliminates the necessity to measure the crack length, and the DCB-UBM test can be used to test for any mode mixity by changing the ratio between the bending moments applied to the two adherends.However, this test requires a special-made fixture, and as for the DCB test, the 1293 (2023) 012032 IOP Publishing doi:10.1088/1757-899X/1293/1/012032 2 crack front is not straight.Thus, in some cases, it could be useful with a simpler test than the DCB and DCB-UBM tests to obtain the fracture toughness experimentally.
The climbing drum peel (CDP) test is a simple standard test (ASTM D1781 [7]) used in the sandwich community to measure the interface strength of an adhesive bond between the skin and core of a sandwich structure [7].Daghia and Cluzel [4] showed some experimental indications of the CDP test being more prone to have a straight crack front than the DCB test.They argued that it was due to the kinematics of the CDP test.In the standard for the CDP test, the peel torque is used to define the interface strength.The peel torque is geometry dependent and not a real material parameter.Thus, it can only be used to compare different cases with similar geometry and not as an input to a model or similar.Literature has shown that the energy release rate (fracture toughness) can be calculated by simple energy considerations for the CDP test [4,8,9].This idea was presented for sandwich structures [8,9] and later laminates [4] as an alternative to the DCB test for the mode I fracture toughness, which is interesting due to the simplicity of the CDP test.
Generally, the fracture toughness obtained from the CDP test is presented as the mode I fracture toughness G Ic [4,[8][9][10].However, Thouless and Jensen [11] have also shown that the mode mixity in a regular peel test varies for different peel angles and such a peel test is thus not actually a mode I test.Although a simple peel test is slightly different from the CDP test, the load is essentially applied in a similar manner, where the peel strip is pulled at an angle to the adherend.The mode mixity is also to some extent discussed in some of the later work by Daghia et al [12] on the double drum peel, but to the author's knowledge not for the CDP test.The finite element (FE) modelling in the current study will show that the fracture toughness obtained by the CDP test is actually mixed mode.Thus, it is important to properly understand the mode mixity in the CDP test before using it as asimple alternative or supplement to the DCB test.
In the current paper, a 2D FE model including all the components (steel bands, drum, flange, and test specimen) in between the actuator and the load cell in the test machine is presented.The FE model is used to estimate the fracture toughness using the J-integral obtained by a contour integral approach.The corresponding mode mixity is obtained from the stress intensity factors extracted from the J-integral by the interaction integral method.The results are compared to analytical calculations and experiments.

Materials and specimens
Figure 1 shows an illustration of the specimen geometry used for the CDP experiments and FE analyses in the current study.The peel strip and the adherend were both made from the same unidirectional non-crimp fabric based glass fiber composite with a small amount of backing, where the peel strip is one ply of the adherend.The glass fiber composite was modeled as an isotropic material with a Young's Modulus of E p = E a = 40 GPa and Poisson's ratio of ν p = ν a = 0.3.A pre-crack was made in the specimens at the end that was attached to the drum, as also illustrated in Figure 1.The steel used to model the steel bands, flange, and drum (see Figure 2) was modeled with a Young's modulus of E s = 210 GPa and a Poisson's ratio of ν s = 0.3.

Climbing drum peel test experiments
The CDP tests were carried out according to the ASTM D1781 standard [7].Figure 2 shows the experimental setup for the CDP test of the glass fiber test specimens considered in this study.The peel strip at the uncracked end of the specimen was attached to the upper grip of the test machine where the load cell was also placed.The peel strip at the pre-cracked end of the specimen was then inserted into the drum with flanges and steel bands.The steel bands were attached to the actuator of the test machine with a lever-type bar, as indicated in Figure 2. The tests were carried out in a universal testing machine (Instron 88R1362) with a 5kN load cell.The lower grip of the actuator holding the lever bar with the steel bands was pulled in displacement control with a speed of 25 mm/min.The load in the load cell was measured throughout the test.The diameter of the drum, flange, and the steel band thickness were 101.6 mm, 126.89 mm, and 0.11 mm, respectively.The average specimen weight was 388 g, and the drum and flange had a combined weight of 3332 g.When the drum rotates, the peel strip is gradually bent upwards to eventually wrap around the drum.This is reflected through a plateau with a small overshoot (c) in load, observed in all the experiments, followed by a stable plateau load (d).The plateau is commonly referred to as the "winding stage" in literature, since this is where the drum is "winding" up the peel strip prior to any crack propagation.After this stage, the force increases steadily through (e) until delamination initiates (f).The second load plateau (f) is where the delamination progresses in a steady state manner [4,9].

Analytical estimation of fracture toughness from experiments
The ASTM D1781 standard [7] approach was made for testing the adhesion at the bonding interface between the skin and core of a sandwich plate.In the standard, the average peel torque, T, is estimated as the final output of the test by Eq. 1.
where F d and F w are the forces measured by the load cell during winding (plateau (d) in Figure 3) and debonding (plateau (f) in Figure 3), respectively, and w is the specimen width.The radii r 1 and r 2 are related to the drum and flange diameters r d1 and r d2 and the thicknesses of the peel strip and steel bands h p and h s as follows: However, ashighlighted in literature [4,9], the peel torque is not a real material parameter, and can only be used for comparing different bonding solutions.Daghia and Cluzel [4] suggested using the test for estimating the fracture toughness of monolithic laminates, which can be calculated as in Eq. 3 (see [4] for details).In Eq. 3, ∆E debond is the change in dissipated energy during debonding and ∆A is the corresponding debonded area.Note that Daghia and Cluzel originally stated the fracture toughness as G Ic , but the current work will show that it is not pure mode I, and therefore it is instead stated as the total mixed mode fracture toughness G c .This value is calculated for the experiments and compared to the corresponding predictions of the FE model.

Finite element model
The current section outlines the FE model developed.The Abaqus model files can be downloaded from [13].

Model geometry
The developed 2D FE model includes all the parts in between the actuator and load cell of the test machine, i.e. test specimen, drum, flange, and steel bands as illustrated in Figure 4 (a).The steel band, flange, drum, and specimen are modeled as individual instances only connected through local tie constraints and contact definitions, as detailed below.Note that since it is modeled in 2D, the flange overlaps with the test specimen.However, it has no influence on the results since contact is not modeled between these parts.Modeling it this way makes it possible to avoid going into 3D while still realistically modeling the test setup.

Loads and constraints
The loads and constraints are applied in two analysis steps as shown in Figure 4.The gravitational loads on the drum and specimen were applied in an initial load step (Figure 4 (a)) to ensure that the full gravitational load was applied before applying the load to the steel bands since loads are gradually applied over an analysis step in Abaqus.The weight of the drum is applied to its center (F drum g ) and the specimen weight is applied as an evenly distributed load across the whole specimen (P spec g ).Both of the loads due to gravity are applied acting in the negative x-direction and calculated from the weight using a gravitational constant of 9.82 m/s 2 , which gives F drum g = 32.72N, F spec g = 3.810N , and In the gravity step (Figure 4 (a)) the steel band was constrained in the x-direction to avoid excessive movement of the part, which led to the model not converging.A prescribed displacement was applied to the steel bands in a second load step as shown in Figure 4 (b).In the load step (b), a displacement is applied to the steel band instead of the constraint in the x-direction, as also seen in Figure 4.For both load steps, the specimen strip was gripped on the right-hand side and the steel band was constricted in the y-direction.
The end of the steel band was fixed to the flange using a tie constraint (tie 1), and the end of the peel strip of the specimen was fixed to the drum with a tie constraint (tie 2), as indicated in Figure 4 (a).Two hard contacts were modeled between the contacting regions of the steel band and flange and the drum and specimen.The contact between the steel band and flange allowed for separation when the load is applied and the drum is rotated.The rotations of the drum and flange were locked together using a kinematic constraint applied to a reference point in their center.

Mesh
The mesh used in the FE model in the regions of interest is shown in Figure 5.The mesh was refined in the contact regions to obtain stable convergence of the solution and avoid overlapping of the interfaces.The crack tip mesh was refined in a circular region around the crack tip with the mesh becoming increasingly finer towards the crack tip.The model had a total of 85658 linear plane stress elements (type CPS4R, CPS3) and it took roughly 5 hours to run the analysis on 16 cores of 1 CPU on a computational cluster.The analysis time could be shortened by increasing the maximum time step increment, which was set to 0.001s to obtain high resolution in the output results.However, for the purpose of this work, high resolution in the output data was desired and the analysis time was not an issue.Therefore, only limited effort was put into reducing the analysis time of the model.

Crack
The J-integral was calculated in Abaqus using a contour integral.The J-integral is evaluated along a chosen number of contours around the crack tip, following the local circular mesh.The value of the J-integral extracted from each contour converges a distance away from the crack tip.Therefore, the values at the 16th (outhermost) contour, where a stable value was achieved, was used.The interaction integral approach was used to obtain the mode I and II stress intensity factors (K I and K II ) to estimate the mode mixity.The mode mixity is in the current study calculated by Eq. 4. Note that it is required to consider a length scale for the mode mixity if the two interfaces are not the same material (bi-material) [14].However, in the current study the material of the peel strip and adherend is similar and thus the expression in Eq. 4 is sufficient.
In the experimental setup the upper specimen end is pinned (allowing for rotations), which means that the specimen rotates a few degrees during the test.Apparently, this rotation affects the values of K I and K II predicted by the FE model due to the implementation in Abaqus.When calculating the mode mixity for the CDP setup with a pinned upper end based on K I and K II from Abaqus, the value differs significantly from the corresponding mode mixity calculated analytically based on Suo & Hutchinson [14].However, if the upper specimen end was gripped, allowing for no rotation, the mode mixity based on K I and K II from Abaqus matched the analytical solution well.As we do not see a reason for why the FE results should not match the analytical solution, the above indicated an issue with the values obtained from Abaqus for the pinned specimen end boundary condition.To investigate this interpretation, an alternative stress based measure for the mode mixity calculated from the stresses σ 12 and σ 22 [15] in front of the crack tip was plotted as in Figure 6.The plot is not meant as an alternative way to obtain the mode mixity, but simply a means to look into the above issue.Based on Figure 6, it is seen that for the stress based mode mixity there is no difference for the two different boundary conditions.Therefore, a difference is not expected for the mode mixity based on K I and K II .Thus, the rotation of the specimen in the model apparently causes an issue in the way K I and K II are calculated in Abaqus.Since the issue is caused by the rotation, and the two cases should be equivalent regarding mode mixity, the results for the clamped specimen end are used.It should be noted that the predicted fracture toughness (J-integral) is unaffected by the change in boundary conditions.
As an additional check, the bending moment and normal forces were calculated from σ 11 extracted in the peel strip a distance of 0.5 mm away from the crack tip in the FE model, and the mode mixity was estimated based on Suo and Hutchinson [14] for verification.The stresses in the peel strip were extracted in a coordinate system in the tangent direction of the deformed strip.At the considered location σ 22 and σ 12 from the FE model were close to zero and also assumed to be zero in the analytical mode mixity calculations.

Parameter dependency
The main geometric parameter that can vary from test to test is the length of the steel bands.
To check for dependency on the results, analyses were run for an initial steel band length of 155, 280, and 530mm.This had no significant effect on the results.Another parameter that changes during the test is the crack length.The initial crack length was the same for all the test specimens but the crack length will change during the test as delamination progresses, and should still give a similar value for the fracture toughness.To look further into this, the initial crack lengths of 10, 15, and 20mm were modeled and compared.
The current study focuses on a unidirectional glass fiber composite.However, for completeness, the effect of material stiffness on the mode mixity was checked by running the Figure 6: Ratio of σ 12 over σ 22 compared for fixed and pinned boundary conditions in the specimen end gripped by the test machine analysis for a unidirectional carbon fiber composite material with a stiffness of 153GPa [16].

Results
In the following, the FE predictions of the displacement of the overall test setup, the fracture toughness, and the mode mixity are presented.The force-displacement curve predicted by the FE model is compared to experiments.In addition, the fracture toughness obtained by the J-integral in the FE model is compared to the fracture toughness predicted analytically using the experimental data.Finally, the effect of the initial crack length is discussed.

Displacement of CDP test setup
The force measured by the load cell (load cell force) is plotted as a function of the cross-head displacement in Figure 7 (a) comparing one representative experiment (solid line) and the FE model (dashed line).The load cell force for the FE model is the x-direction reaction force in the right-hand side grip of the specimen in Figure 4. Furthermore, the displacement obtained from the FE model has been offset to fit the experimental cross-head displacement.They are initially offset since it is necessary to tighten the steel bands in the experimental setup, whereas the steel bands are already tightened from the start in the FE model.This also explains why the curve for the FE model starts at the winding stage.It is seen that delamination experimentally occurs around a load cell force of F lc = 230 N. Figure 7 (b) shows a close-up of the experiment during delamination where a small gap is visible near the crack tip during propagation.
Figure 8 shows the deformation of the test specimen near the crack tip at four different load levels.The small insert plot indicates where on the FE force-displacement curve the image is from.As seen in Figure 8 (a) the specimen is slightly tilted (less than 1 • ) as a result of the applied gravitational loads before any load is applied to the steel bands.Applying further load the peel strip starts to wind around the drum and at the end of the winding stage is nicely wrapped around the drum as seen in Figure 8 (b).Loading up to the experimental load of delamination to initiate, the peel strip separates partly from the drum close to the crack tip as seen in Figure 8 (c).This is in agreement with observations from the literature [17] and was also observed in the experiments of the current study (Figure 7 (b)).However, it is difficult to measure the exact curvature in the experiments due to the presence of the flange.At the stage in (c) the tilt of the specimen is roughly 2 • , which is also in line with experimental observations where the measured angle was somewhere in between 2 • and 2.5 • as observed by the authors for shows how increasing the load further increases the space without contact between the drum and the peel strip.Thus, it is expected that the extent of non-conformance of the peel strip to the drum shape increases with increasing interface strength.However, as will be shown in the following results, the fracture toughness predicted by the FE and analytical models match almost perfectly even though the peel strip does not conform perfectly to the drum shape during propagation.

Fracture toughness and mode mixity comparisons
Figure 9 shows the fracture toughness as a function of the load cell force predicted by the FE model using the J-integral and the analytical approach estimating the critical energy release rate G c from Eq. 3. The curve with circle markers in Figure 9 shows the analytically predicted fracture toughness using the winding force obtained from the FE model (F F E w = 173.4N).A very good agreement is obtained between the fracture toughness predicted by the J-integral in the FE model and the corresponding analytical prediction.This confirms the applicability of the simple analytical approach to estimate the fracture toughness in the CDP test.A couple of papers [4,17] suggest that if the peel strip shape does not conform to the shape of the drum, the analytical expression for the fracture toughness may not apply.However, our analysis shows that for the case considered here, one can still use the analytical expression even if the peel strip does not conform to the drum.
The winding force is slightly higher for the FE prediction than the experimentally obtained value (see Figure 7 (a)), which affects the predicted fracture toughness.The curve with triangle markers in Figure 9 shows the analytical prediction using the experimentally measured winding force (F exp w = 165.3N).It is seen that a difference in the winding force of 8.1N between the FE model and experiment leads to a constant difference of 66.2 J m 2 in the fracture toughness.At a load cell force of 230N (where experimental delamination occurs), this gives a difference of 14.4% between the FE-predicted (463.2J m 2 ) and the experimentally measured (533.1 J m 2 ) fracture toughness obtained from Eq. 3. The difference in winding force could indicate that some additional effects like permanent deformation, cracking, and similar effects occur in the just before the end of the winding stage, (c) at the approximate load required for experimental delamination to initiate, and (d) at a higher applied load for demonstration.Videos of the deformation can be downloaded from [13] experiment, but are not taken into account in the FE model.This is not a problem for the analytical approach when estimating the fracture toughness based on the experiments, since these effects are "included" in the experimentally measured winding force and subtracted when the fracture toughness is estimated by Eq. 3. Nevertheless, the FE model is expected to give applicable results for the mode mixity in the CDP test.The J-integral and mode mixity predicted by the FE model are shown in Figure 10 as a function of the displacement (a) and load cell force (b).The dashed grey line shows the point at which experimental delamination occurred corresponding to F ls = 230N.It is seen from Figure 10 that the J-integral increases linearly with the force whereas it looks more parabolically increasing with the displacement.Another thing that can be seen from Figure 10 (a) is that the mode mixity initially (during the winding stage) experiences a few jumps whereafter it slowly stabilizes.Looking at the same mode mixity and J-integral plotted as a function of the load cell force in Figure 10 (b), it is clear that the mode mixity quickly stablizes after the J-integral begins to increase.At the time of experimental delamination, the mode mixity is −38.1 • .In the load range of 200-300N the mode mixity varies from −39.7 • and −36.9  predicted by the FE model using the approach of Suo and Hutchinson [14] with the normal force (N=279.6N)and bending moment (M=-390.1N•mm) in the peel strip extracted from the FE model 0.5mm away from the crack tip, and the remaining calculated from force and moment balance gives a mode mixity of -39 • .These results are also in a similar range as those obtained by Daghia et al [12] for the double drum peel test and Thouless and Jensen [11] for the regular peel test.
If changing the material to a unidirectional carbon fiber composite with a significantly higher stiffness than the glass fiber composite, a mode mixity varying between −40.8 • and −37.6 • is obtained for a load cell force of 200N and 300N.Thus, the effect of material stiffness on the mode mixity (for a similar peel strip and adherend material) is limited when considering the same geometry.

Effect of crack length
Figure 11 shows (a) the J-integral and (b) the mode mixity as a function of the load cell force predicted by the FE model.They are seen to be equivalent.Thus, the initial crack length has no effect on the results of neither the fracture energy nor the mode mixty, as also assumed in the   experimental method.However, as seen from the predicted force-displacement curves in Figure 12, increasing the crack length extends the winding stage.This is logical, as a longer peel strip has to be winded up before delamination will occur.
Figure 12: Effect of crack length on the force-displacement curves predicted by the FE model

Using the CDP test as fracture toughness test
The results presented throughout Section 5 show several important points to take into account when considering using the CDP test as an alternative to a DCB test, some of which are listed below.
• The CDP test is not pure mode I, rather the mode mixity during delamination was −38 • for the material combination considered in the current study.This might vary slightly for other geometry and material combinations but is expected to reach a nearly constant mode mixity during delamination • The simple analytical energy considerations for calculating the fracture toughness match well with the FE model and are thus sufficient to use for estimating the interfacial strength from experiments • Even though a gap is observed between the peel strip and the drum in the FE model, the fracture toughness matches well with the analytical calculations.Thus the conformity to the drum circle shape does not seem to affect the results for the case considered in the current study • The fracture toughness and mode mixity as a function of the load cell force is not dependent on the crack length • Estimation of the mode mixity for a specific test setup would be necessary, and to be feasible as part of a standard test it would likely require an analytical approach for doing so instead of FE.Inspiration could perhaps be obtained from the work by Daghia et al [12] on the double drum peel test.
Considering the above points the CDP test can be an alternative to the DCB test for comparing interfaces as long as one remembers that it will provide the mixed mode fracture toughness rather than mode I.However, in the CDP there are limitations to how thick the peel strip can be, which is not an issue in a DCB test.Nevertheless, the simplicity of the CDP test and that it is not necessary to measure the crack length could definitely make it a useful alternative or supplement to obtain the interface strength of laminates for some cases -despite not being a mode I test.To ensure that the measured fracture toughness from the CDP test is in agreement with DCB tests, mixed mode DCB testing could be carried out for the same mode mixity as observed in he CDP test.This is, however, ongoing work.

Conclusion
The current paper presented a finite element (FE) model to predict the fracture toughness and mode mixity of the climbing drum peel (CDP) test.The predicted load-displacement curve was shown to agree well with the experiments.The fracture toughness calculated by the J-integral in the FE model matched the analytical calculations.It was shown that the CDP test for the geometry and material used in the current study lead to a mode mixity of -38 • .Thus, the CDP is a mixed mode test rather than mode I as suggested in some literature and therefore one cannot use it to obtain the mode I fracture toughness.It can, however, be used to compare different interfaces.However, one has to be aware that one will test at a given mode mixity, likely in the range of -37 • to -40 • depending on the material.

Figure 1 :
Figure 1: Specimen geometry (all numbers are in millimeters)

Figure 4 :
Figure 4: Model load steps, boundary conditions, and dimensions

Figure 5 :
Figure 5: Mesh used in the finite element model

Figure 7 :
Figure 7: (a) Experimental and FE load-displacement curves compared and (b) close-up photo of the experiment during delamination

Figure 8 :
Figure 8: Deformation of peel strip in FE model at (a) no load but after gravitation step, (b)just before the end of the winding stage, (c) at the approximate load required for experimental delamination to initiate, and (d) at a higher applied load for demonstration.Videos of the deformation can be downloaded from[13]

Figure 9 :
Figure 9: Comparison between the fracture energy calculated by the FE model and analytically

Figure 11 :
Figure 11: J-integral (a) and mode mixity (b) as function of the force, as predicted by the FE model for different crack lengths