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Prediction of “Zed” section stringer pull-off loads
International Journal of Adhesion & Adhesives 23 (2003) 189–198
Prediction of ‘‘Zed’’ section stringer pull-off loads J.P. Sargenta,*, Q. Wilsonb a
BAE SYSTEMS Advanced Technology Centre—Sowerby, P.O. Box 5, FPC 267, Filton, Bristol BS34 7QW, UK b BAE SYSTEMS (Operations) Limited, Prestwick International Airport, Ayrshire, KA9 2RW, UK Accepted 19 December 2002
Abstract Closed-form analytical models are presented for predicting stringer pull-off loads for aircraft bonded stringer-skin construction in high-strength aluminium alloys. This was undertaken by modifying linear elastic fracture mechanics double cantilever beam analysis to accommodate simple single beams. The accuracy of the predictions was assessed by reference to experimental stringer pull-off strengths and to measured adherend displacements. Predictions of adherend displacement were also derived which included nonlinear adhesive properties by numerical solutions of the governing differential equations. Comparison of these with experimental results showed better agreement than when using a linear elastic foundation. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: A. Epoxy/epoxides; B. Aluminium and alloys; C. Fracture mechanics; C. Stress analysis
1. Introduction Bond design strength allowables for adhesively bonded stringers on aircraft wings have to take due account of the stresses which are present as a result of the wing loading, the effect of variability in bond quality, the presence of defects, the influence of operating temperature and any influence of adverse effects due to degradation from water and other contaminants. Wing loading stresses occur as a result of tension due to fuel and air pressure acting on the skin, shear due to wing box distortion, tension due to skin buckling, and shear stresses due to change in stringer end loads. Parameters that would be typically used to ensure that the skin/stringer bond has the necessary design strength would be derived from lap-shear and stringer pull-off tests. Closed-form algebraic solutions and finite element solution have been used extensively to derive the stress distribution present within the single lap-shear joint (see, for example, the review of Adams et al. [1]); however, there are far fewer analyses for pull-off loading of specimens with ‘‘L’’- or ‘‘Zed’’-type geometry which are similar to the stringer geometry used. Similarly, FE and *Corresponding author. E-mail address: [email protected] (J.P. Sargent).
closed-form algebraic strength prediction methods are reported for the lap-shear strengths based on the use of various failure criteria, but little is available for the other geometries. For these reasons, a closed-form algebraic analysis was undertaken which was suitable for predicting stringer pull-off strengths. This was based on analysis of Kanninen [2], Fernlund and Spelt [3] and Penado [4] which used linear elastic beam theory analysis of the double cantilever beam and strain energy release rate methods. However, rather than deriving values of strain energy release rates from dcb specimens with a known applied load, the analysis was modified to predict an unknown pull-off load by treating the stringer/skin combination as a single cantilevered beam using a critical value of the strain energy release rate.
2. Specimens and test methods Fig. 1 shows an example of the stringer pull-off specimen that was used to determine the strength for design calculations. Specimens comprised 50 mm lengths of ‘‘Zed’’ section stringers with the free flange removed, cut from stringer/skin panels. Alloy types were 7050-T76 clad sheet for the wing top surface skin, and 7050-T7651 for the stringer. Adherend surfaces were all pre-treated using standard procedures which included chromic
0143-7496/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0143-7496(03)00011-3
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J.P. Sargent, Q. Wilson / International Journal of Adhesion & Adhesives 23 (2003) 189–198
Fig. 1. Schematic diagram of the stringer pull-off specimen.
sulphuric etch and chromic acid anodise. The adhesive was AF 163-2K06 with EC 3960 spray applied adhesive bonding primer, both manufactured by 3M. Specimens had three different stringer thickness, representative of the inner (set 1 specimens), middle (set 2 specimens) and outer regions (set 3 specimens) of the wing top surface. For the purposes of this study, the original manufacturing spew fillets were removed and defects were introduced into the heel of the specimen by machining. This resulted in unbonded areas with lengths of up to 2.5 mm and gap-heights of approximately 0.76 mm. The testing machine was an Avery-Denison 7157 and tests were done under load control at a test rate of approximately 4.4 kN/min. A schematic diagram of the test fixture is shown in Fig. 2. Fig. 3 shows in schematic form the specimen geometry and dimensions defining the details of the loading arrangement used for measuring stringer pull-off loads. A few specimens were also made which were suitable for measuring the adherend displacement within the bonded section adjacent to the crack front. These were cut from larger stringer pull-off specimens to give specimens with widths between approximately 0.76 and 2 mm. Specimens were loaded using a similar geometry to that used for the larger specimens, but were instead tested in situ on the straining stage of an optical microscope using a ‘‘Minimat’’ miniature materials tester.1 Specimens were tested under displacement control at a test rate of approximately 0.04 mm/min. By taking digital photomicrographs (1280 pixels 1000 pixels) and correlating features from the photographs in the unloaded state with the same features in the loaded state using digital image correlation software2 it was possible to obtain vector displacement maps, from which the adherend displacement ðwðxÞÞ could be 1
MINIMAT miniature materials tester. Polymer Laboratories, UK. microDAC, Fraunhofer, IZM, Gustav-Meyer-Allee 25, D-13355 Berlin. 2
Fig. 2. Schematic diagram showing the testing fixture.
Fig. 3. Schematic diagram of the pull-off specimen defining geometry and dimensions.
measured. It was estimated that displacements could be obtained with an accuracy of better than 1/20th of a pixel, giving an equivalent accuracy of approximately 0.2 mm when using a 2.5 microscope objective. Tables 1 and 2 summarise the elastic moduli and the relevant geometric constants used to define the specimen dimensions and properties. Tensile stress–strain
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characteristics for the adhesive are shown in Fig. 4. These were obtained using the optical correlation method on small specimens strained on the stage of the optical microscope. This permitted simultaneous measurement of modulus and Poisson’s ratio both globally along the length of the specimen, and locally adjacent to the failure initiation site. Locally in the region of failure, strains of up to approximately 6–7% were recorded; these are shown by the dashed line in Fig. 4. The associated error bars represent the change in applied load at a fixed displacement during a strain measurement. It is believed this behaviour was due to sub-critical crack propagation. Measurements of GIc were made for AF163-2K06 with aluminium adherends using a double cantilever beam method. Fracture energies were measured using linear elastic analysis given in Ref. [5] by taking an average of results using a corrected beam theory analysis and an experimental compliance measurement method. This gave a value for GIc of approximately 1900 J/m2. Loading rate was approximately 0.1 mm/min. It was noted that the adhesive showed an R curve effect of increasing fracture energy with crack length, and also some non-linearity in the loading curve.
3. Linear elastic analysis
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substrate. Defects were confined to a region beneath the web of the stringer, so it is assumed that the stringer side of the attached flange area does not behave as another beam which is cantilevered at the junction between cracked and uncracked adhesive. Note that in this instance, the effective crack length, a; is given by the debond length, +L1 ; h2 =2: The loading arrangement shown is statically determinate, and the load P acting on the specimen at the end of the cantilever beam can be determined from the applied load P3 and by consideration of equilibrium conditions, thus P ¼ P3 =ð1 þ L1 =L2 Þ:
ð1Þ
The differential equations relating the transverse deflection wðxÞ of a simple beam, width b; without shear deformation and partly supported on an elastic foundation is given by, for example, [2] d4 wðxÞ þ 4l4 wðxÞ ¼ 0 for x > 0; dx4 d4 wðxÞ ¼ 0 for xo0; dx4
ð2Þ
where l4=3k/Exbh3, Ex is the modulus of adherend in x direction. Following Kanninen [2] and Hete! nyi [6], it is assumed that a Winkler foundation is valid, in which case the transverse stress (sy ) is proportional to the transverse deflection of the beam at that point, i.e.
3.1. Stringer/skin pull-off specimens sy ¼ k For the loading arrangement shown in Fig. 3, failure was recorded as occurring cohesively at the heel of the specimen. This forms the basis of the analysis presented here. This loading geometry is similar to that found in a single cantilever beam arrangement (shown schematically in Fig. 5), with the skin serving as a single beam pivoting about the heel, and with the stringer serving as
wðxÞ b
ð3Þ
Table 1 Material parameters used for calculating stringer pull-off loads Material
Ex ; Ey (GPa)
n
G (Gpa)
Skin Stringer Adhesive
65.5 71 1.58
0.33 0.33 0.41
24.6 26.7 —
Fig. 4. Tensile stress/strain curve obtained for adhesive AF163-2K06.
Table 2 Nominal dimensions used in Fig. 3 for calculating stringer pull-off loads Specimen group
L1 (mm)
L2 (mm)
L3 (mm)
L4 (mm)
h (mm)
h1 (mm)
h2 (mm)
heff (mm)
h3 (mm)
t (mm)
Set 1 Set 2 Set 3 0.76 mm wide in situ specimen
5.3 5.0 4.7 7.3
27.7 28.0 28.3 25.7
33 33 33 33
24 24.3 24.5 25
2.64 2.64 2.64 2.64
4.83 3.56 2.24 3.2
3 2.4 1.75 2.05
o8 o6 o4
25.4 25.4 25.4 22
B0.15 B0.15 B0.15 0.083
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Fig. 5. Schematic diagram for the single cantilever beam used in the analysis.
using the prime notation to indicate differentiation, the boundary conditions are w00x ðaÞ ¼ 0; 3 w000 x ðaÞ ¼ 12:P=Ex bh ;
beam case for the strain energy release rate is trivially extracted using the compliance argument given by Penado [4], thus, in plane stress: GI ¼
w00x ðcÞ ¼ 0; w000 x ðcÞ ¼ 0: By matching the values of deflection and the first three derivatives at x ¼ 0; Fernlund and Spelt [3] give the solution to Eq. (2) for the nominal transverse stress: P sy ðxÞ ¼ 2 al2 elx b 1 1þ cos ðlxÞ sin ðlxÞ : ð4Þ la The stress at x ¼ 0 is therefore P 1 : sy ð0Þ ¼ 2 al2 1 þ b la
Following Fernlund and Spelt [3], using the virtual crack closure technique [7], the mode I component of the strain energy release rate, G I can be calculated from G1 ¼
b½sy ð0Þ 2 : 2k
ð6Þ
Expanding Eq. (6) gives the energy release rate for a single beam in terms of the foundation modulus k; thus, in plane stress: 2 32 GI ¼
7 6P2 a2 6 1 6 7 61 þ 0:25 7 : 2 3 5 b h Ex 4 1 a 3k 3 Exbh
ð7Þ
Solutions to Eq. (2) for the single beam case have also been derived by Kanninen [2] in solving the dcb case, but without the presence of an adhesive layer. Penado [4] used the Kanninen solutions, included an adhesive layer, and also gave corrections for shear distortion of the cracked portion of the beam adjacent to x ¼ 0 by introducing a correction to the beam deflection based on theory given by Timoshenko and Goodier [8]. The single
h Bh0:25 a
þ
1 E x h2 þ ; B2 h0:5 8Gxy a2
ð8Þ
0:25 ; Gxy is the shear modulus of where B ¼ 3k=Ex b adherend. The solutions given in Eqs. (7) and (8) may be compared with a simple beam solution (see, for example, [9]), where no corrections are made for shear distortion or the presence of an elastic foundation, thus GI ¼
ð5Þ
6P2 a2 b2 h3 E x 1þ 2
6P2 a2 : b2 h3 Ex
ð9Þ
For dcb specimens, it was suggested by Fernlund and Spelt [3], and Penado [4], that in an adhesively bonded specimen the foundation constant k is made up of a contribution from the adhesive layer, and a contribution whereby each adherend beam acts as an elastic foundation for the other. A Winkler foundation constant is introduced in order to model the elastic response of the adhesive and substrate to the bending of the cantilevered adherend. This can be imagined as a continuous array of independent linear springs, which simulates the transverse elasticity of the uncracked region. Penado [4] showed how this could be formulated by considering the components as if they were springs in series with one another. Fernlund and Spelt [3] followed the analysis of Kanninen [2] by modelling each of the two beams by assuming an adherend foundation stiffness k ¼ 2Ey b=h; whereas Penado [4] modelled each adherend using a foundation stiffness k ¼ 4Ey b=h: Penado [4] considered that the factor of 4 in the expression for the adherend contribution to the foundation constant gave better agreement with results from his FE modelling. This was included to accommodate the shear distortion of the cracked section of the cantilever beam. It was also considered by Penado [4] that the adhesive foundation modulus should incorporate a modification for plane
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strain, since the adhesive in-plane lateral strain was inhibited. Consideration of Fig. 3 shows that the pull-off load P3 is applied to the end of the stringer web. This suggests that an additional elastic foundation is required which is representative of the length of stringer web which contributes to the development of elastic stresses within the adhesive layer. In this instance, the continuity and shear within the web will result in a proportion of the stringer web length which moves bodily as a whole, and a proportion, which deforms differentially across its width. In terms of the development of stresses at the specimen heel, it is only the differential deformation that is important. Thus, if we assume the crack extends under constant load, then, by St. Venant’s principle, stress changes in the web adjacent to the adhesive will be determined at most by an elastic foundation that comprises a web length given approximately by the web width (h2 ). This would therefore mean that the web and attached flange contribution to the foundation reaction should be at most made up of a length heff Eh1 þ h2 : Following Kanninen’s [2] example, then to a first approximation:
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Fig. 6. Stringer pull-off loads and predictions for set 1 specimens.
k ¼ sy b=w ¼ ðEy eÞb=ðe heff Þ ¼ Ey b=heff This gives a revised foundation modulus in the stringer web area made up of contributions from the cantilevered adherend, the adhesive layer and the substrate web, thus: 1 k¼ ð10Þ h tð1 uyx uyxÞ heff þ þ CEy b Ea b Ey b
Fig. 7. Stringer pull-off loads and predictions for set 2 specimens.
where C ¼ 2 using Fernlund and Spelt analysis [3], C ¼ 4 using Penado analysis [4], heff is the effective web length, t=the thickness of adhesive layer, Ea=the modulus of adhesive, Ey the the modulus of adherend in y direction, uxy ¼ uyx=Poisson’s ratio of adhesive, u the Poisson’s ratio of adherend.
4. Stringer pull-off results Figs. 6, 7 and 8 show the experimental results for the pull-off loads, P3 ; as a function of the debond length defined in Fig. 3, for specimen sets 1, 2 and 3 respectively. It was noted that whilst the set 1 specimens showed exclusively cohesive failure, sets 2 and 3 showed failure which was not 100% cohesive, with evidence for some pin-head sized adhesive failures which were not significantly clustered. Figs. 6–8 also shows stringer pull-off strength predictions for set 1–3 specimens which were derived using Eqs. (7) and (8). These are shown in plane strain (i.e. substituting Ex =ð1 n2 Þ for Ex and Ey =ð1 n2 Þ for Ey in
Fig. 8. Stringer pull-off loads and predictions for set 3 specimens.
Eqs. (7) and (8) above), and calculated for cases using the maximum effective web lengths (heff) given in Table 2. For comparative purposes, Figs. 6–8 also show the predictions using Eq. (7), in plane strain, where the adhesive alone contributes to the foundation modulus (i.e. h ¼ 0 and heff =0 in Eq. (10) above). Reference to Figs. 6–8 shows that, irrespective of choice of foundation modulus, predictions of strength are overestimated
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relative to a linear least mean squares fit for the experimental points. This was by amounts between B10% at best for the set 1 specimens using a foundation comprising an 8 mm foundation, and B30% at worst for the set 2 specimens using a foundation comprising adhesive alone. It should also be noted that for the current range of crack lengths and specimen geometry that there is little difference in the predictions based on either Eq. (7) or (8).
5. Non-linear properties and validation experiments Of particular concern in the above analysis is the seemingly rather arbitrary way in which the foundation modulus is established and the implications for predictions of the non-linearity shown in the stress/strain curve from Fig. 1. Fig. 9a and b show the predicted stress (sy ðxÞ) profiles calculated for the set 1 specimens at zero debond length based on the geometry shown in Fig. 3. These were calculated using Eq. (4) together with elastic foundations comprising (a), adhesive alone and (b), adhesive and an 8 mm effective web. Reference to the predicted stress profiles shows that the stress for either elastic solution exceeds the plastic plateau stress level of approximately 30 MPa shown in Fig. 4 over a significant portion of the bonded section of specimen. This is clearly physically unrealistic. By measuring adherend displacements as a function of applied load it was possible to assess the extent to which adhesive non-linearity and the choice of a particular foundation modulus influenced accuracy of the predictions. Fig. 10 shows an example of a vector displacement map for a small L-type specimen from which such measurements were made. This was obtained using the feature correlation software and loading arrangement stated in section 2 above. Fig. 11 shows the measured y component displacements for skin and web areas in the heel region adjacent to the crack front for another small L-type specimen (see Table 2 for specimen details) with load P ¼ 29:3 N/mm (P3 ¼ 35 N/mm). It should be noted that there is some
net rotation of the whole specimen, shown by the slope of the stringer adherend. Net displacement wðxÞ was obtained by subtracting skin displacement from the stringer displacement. This was then compared directly with predictions using Eqs. (3) and (4). Fig. 12a show this comparison for the cases where the foundation is assumed to be comprised of adhesive alone, and Fig. 12b shows the case where the foundation is assumed to be composed of adhesive and an 8 mm effective web. Fig. 13a and b shows the equivalent calculated stress distribution for the specimens from Fig. 12a and b. Reference to Fig. 13 shows that the predicted stress distribution is completely elastic along the whole bonded length (i.e. less than B30 MPa) only for the specimen with load P ¼ 8:7 N/mm (P3 ¼ 10:5 N/mm) for Fig. 13a, and for the specimens with loads less than 18 N/mm for Fig. 13b. All the other loads gave predicted stresses larger than 30 MPa.Comparison between the experimental results and predictions for wðxÞ would also suggest, tentatively, a better fit for the case where the foundation is composed of adhesive alone, i.e. Fig. 12a. Given a different adhesive with a larger plastic yield stress then it is likely that a better estimate of foundation modulus could be made. However, the most important point to note from the displacement graphs is that the discrepancy between experiment and theory increases as
Fig. 10. Correlation vector displacement map for stringer pull-off specimen. Vectors shown x5 : Vectors drawn at approximately 0.2 mm intervals.
Fig. 9. Predicted stress profiles based on Eq. (4) using foundation moduli, which include (a) adhesive alone, and (b) adhesive and an 8 mm effective web length.
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the predicted stress gets larger, irrespective of choice of foundation modulus. This suggests that the linear elastic analysis of the foundation for this tough adhesive with effectively short crack lengths is inadequate, and that the adhesive non-linearity has a more profound influence on predictions than the choice of a particular elastic foundation.
6. Linear beams resting on non-linear foundations Analysis of linear elastic beams resting on bilinear and non-linear foundations have been given by, for example, Kuo and Lee [10] and Farshad and Shahinpoor [11], and Beaufait and Hoadley [12]. A relatively simple expedient is adopted here, which is similar to that used by Beaufait and Hoadley [12] in their consideration of beams on soil
195
foundations. The adhesively supported beam is divided into discrete sections with different foundation modulus, with the governing differential equations being solved numerically using iterative techniques, ensuring continuity of boundary conditions between the sections. The beam was approximated as a partly free beam and a partly adhesively supported beam with five discrete sections. An initial trial solution was first estimated, and then successively better approximations to the stress distribution were obtained by adjusting the adhesive elastic modulus in each of the sections, such that the average stress over that section length with a chosen elastic foundation, for a particular strain, accorded with the stress/strain relationship shown in Fig. 4. A check on the accuracy of this approach was also made by ensuring that the area under the resulting stress distribution curve equalled P=b: Formally, this approach was based on the same governing differential Eq. (2) referred to earlier, with the same boundary conditions at x ¼ a and c: This used the general solution for x > 0: wðxÞ ¼ Ci cos ðlxÞ cosh ðlxÞ þ Ci cos ðlxÞ sinhðlxÞ þ Ci sin ðlxÞ cosh ðlxÞ þ Ci sinh ðlxÞ sin ðlxÞ and for xo0: wðxÞ ¼ Ci x3 þ Ci x2 þ Ci x þ Ci :
Fig. 11. Measured y component displacements for the in situ pull-off specimen with a load P ¼ 29:3 N/mm. Nominal crack front (x ¼ 0) located approximately at x pixel position 100.
At each boundary between the beam element sections the solutions for deflection and the first three derivatives were matched. This resulted in a set of simultaneous equations which were solved using numerical routines
Fig. 12. Measured (points) and predicted displacements wðxÞ (lines) calculated using linear elastic theory. (a) Using a foundation modulus comprising adhesive alone, and (b) using adhesive and an 8 mm effective web.
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Fig. 13. Predicted stress distribution sy ðxÞ calculated using linear elastic theory. (a) Using a foundation modulus comprising adhesive alone, and (b) using adhesive and an 8 mm effective web.
Fig. 14. (a) Predicted displacement wðxÞ calculated using non-linear elastic analysis. (b) Approximate stress distribution sy ðxÞ (see text for details). Shaded lines in (a) and (b) correspond to each beam element used in the analysis. Shown for the small stringer pull-off specimen at load P ¼ 29:3 N/mm.
given in MATHCAD. This gave values for the constants (Ci ; i ¼ 1y24) at each boundary and at x ¼ a and c: Fig. 14a and b shows, respectively, the resulting displacement and stress distribution calculated for the specimen from Fig. 11 at a load P ¼ 29:3 N=mm: Fig. 15 shows a comparison between these revised predicted displacement profiles (wðxÞ) and the experimental displacements for each of the loading cases shown in Fig. 12a. Whilst not perfect, clearly agreement is much better here using this non-linear analysis than when using the previous linear elastic analysis.
7. Mixed-mode loading Fig. 15. Comparison between predicted (lines) and measured (points) displacements wðxÞ for a small stringer pull-off specimen as a function of applied load. Predictions derived using non-linear elastic analysis. (Compare with the equivalent linear elastic approximation derived earlier and shown in Fig. 12a.).
Prediction of pull-off loads (Eqs. (7) and (8)) and beam displacement has so far been restricted to analysis of the transverse deformation, i.e. just mode I loading. However, since the adherends are asymmetric there will
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Fig. 16. (a) Measured y component skin and stringer displacements (uy ) either side of the adhesive bond line for the in situ pull-off specimen with a load P ¼ 14:1 N/mm. Nominal crack front (x ¼ 0) located approximately at x pixel position 140. (b) Same as in (a) but showing the measured x component skin and stringer displacements (ux ). The dotted line in both cases shows the net adherend displacement that occurred across the adhesive bond line.
inevitably be a shear or mode II component present. In addition to the y displacement components (uy ) which were used to measure the transverse skin and stringer deformation, vector displacement fields like that shown in Fig. 10 can also be use to give the x displacement components (ux ). Fig. 16a and b shows, respectively, the uy and ux skin and stringer displacements as functions of x for regions which were adjacent to the adhesive for the specimen from Fig. 10 at a load P of 14.1 N/mm. Together with a knowledge of the adhesive thickness, then these displacements can be used to measure the applied adhesive shear strain using exy ¼ 1=2ðqux =qy þ quy =qxÞ: For an adhesive bond line thickness of approximately 20 pixels, and at a position given by the nominal crack front (pixel position B140), then this gave applied tensile and shear strains of approximately 2%. It may also be noted that the component @uy =@x was small in comparison to @ux =@y; and that the mode I/ mode II ratio changed significantly with distance from the crack front, with the mode I loading reducing more rapidly than the mode II loading. It may be mentioned that at larger loads when the crack front position was less well defined and characterised by a zone of damage which extended along the bond line, then the applied mode ratio would change appreciably over this damaged zone region.
8. Conclusions and discussion The optical correlation method for deriving measured displacements was found to be a particularly useful tool for validation of the analysis and for comparing predictions with experimental results. This was because displacement and strain field maps could be obtained quickly and accurately without the need for special fixtures or stability requirements, and with the minimum
of surface preparation (light polishing with 800 grit emery paper). The main practical limitations in the implementation used here involved the relatively slow image capture rates as a result of the relatively large file sizes, which implied low applied strain rates. It was noted that some time-dependent sub-critical crack growth occurred when testing the tensile test specimen in Fig. 4, and also when testing the small-scale specimens at larger loads than those shown in Fig. 11. AF163-2K is known to support loads at very large applied strains (B100%) by the formation of adhesive and monofilament bridges [13]. Given a full knowledge of the stress/strain relationship for this adhesive covering rate effects and larger applied strains beyond those measured in Fig. 4, then it is envisaged that the adherend displacement modelling using this non-linear analysis could be extended to predict the stress and displacement profiles for larger applied loads than those shown in Fig. 15. Given that the adhesive displayed significant nonlinearity it would seem likely that this was one of the more important factors responsible for the discrepancy between the measured stringer pull-off strengths and those predicted using the linear elastic fracture mechanics analysis. Note should also be made that since this tough adhesive also displayed large damage zones, which extended beyond the stringer web, then a physically more correct analysis may also have to consider the contribution that a second beam, as represented by the stringer web/attached flange area, would have both on the net beam displacement and the contribution to the energy release rates. However, it is also likely that given specimens with similar crack lengths and dimensions to those used here, but which used a more brittle adhesive which displayed less nonlinearity, the analysis might then be more successful in predicting stringer pull-off loads.
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The predictions of pull-off strength were also based on the use of a mode I fracture energy, and ignored any consequences of the mixed mode nature of the loading. Given that an appreciable shear component would result in a larger critical energy release rate, then it could be argued that this would then have given rise to an overestimate of the pull-off strength with the present mode I analysis. It is clear, however, that a more comprehensive and accurate treatment of the stringer pull-off test specimens should also consider the mixed mode nature of the test. References [1] Adams RD, Comyn J, Wake WC. Structural adhesive joints in engineering. London: Chapman and Hall; 1997. [2] Kanninen MF. An augmented double cantilever beam model for studying crack propagation and arrest. Int J Fract 1974;10(3):415. [3] Fernlund G, Spelt JK. Mixed mode energy release rates for adhesively bonded beam specimens. J Compos Technol Res JCTRER 1994;16(3):234.
[4] Penado FE. A closed form solution for the energy release rate of the double cantilever beam specimen with an adhesive layer. J Compos Mater 1993;27(4):383. [5] Adhesion, Adhesives and Composites Website, http:///www.me.ic.ac.uk/materials/AACgroup/index.html, Imperial College of Science, Technology and Medicine. [6] Het!enyi M. Beams on elastic foundation. Ann Arbor: The University of Michigan Press; 1946. [7] Rybicki EF, Kanninen MF. A finite element calculation of stress intensity factors by a modified crack closure integral. Eng Fract Mech 1977;9:931. [8] Timoshenko SP, Goodier JN. Theory of elasticity. New York: McGraw-Hill; 1970. [9] Charalambides M, Kinloch AJ, Wang Y, Williams JG. On the analysis of mixed-mode failure. Int J Fract 1992;54:269. [10] Kuo YH, Lee SY. Deflection of nonuniform beams resting on a nonlinear elastic foundation. Comput Struct 1994;51:513. [11] Farshad M, Shahinpoor M. Beams on bilinear elastic foundations. Int J Mech Sci 1972;14:441. [12] Beaufait FW, Hoadley PW. Analysis of elastic beams on nonlinear foundations. Comput Struct 1980;12:669. [13] Sargent JP. Microextensometry, the peel test, and the influence of adherend thickness on the measurement of adhesive fracture energy. Int J Adhes Adhes 1998;18:215.
Prediction of ‘‘Zed’’ section stringer pull-off loads J.P. Sargenta,*, Q. Wilsonb a
BAE SYSTEMS Advanced Technology Centre—Sowerby, P.O. Box 5, FPC 267, Filton, Bristol BS34 7QW, UK b BAE SYSTEMS (Operations) Limited, Prestwick International Airport, Ayrshire, KA9 2RW, UK Accepted 19 December 2002
Abstract Closed-form analytical models are presented for predicting stringer pull-off loads for aircraft bonded stringer-skin construction in high-strength aluminium alloys. This was undertaken by modifying linear elastic fracture mechanics double cantilever beam analysis to accommodate simple single beams. The accuracy of the predictions was assessed by reference to experimental stringer pull-off strengths and to measured adherend displacements. Predictions of adherend displacement were also derived which included nonlinear adhesive properties by numerical solutions of the governing differential equations. Comparison of these with experimental results showed better agreement than when using a linear elastic foundation. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: A. Epoxy/epoxides; B. Aluminium and alloys; C. Fracture mechanics; C. Stress analysis
1. Introduction Bond design strength allowables for adhesively bonded stringers on aircraft wings have to take due account of the stresses which are present as a result of the wing loading, the effect of variability in bond quality, the presence of defects, the influence of operating temperature and any influence of adverse effects due to degradation from water and other contaminants. Wing loading stresses occur as a result of tension due to fuel and air pressure acting on the skin, shear due to wing box distortion, tension due to skin buckling, and shear stresses due to change in stringer end loads. Parameters that would be typically used to ensure that the skin/stringer bond has the necessary design strength would be derived from lap-shear and stringer pull-off tests. Closed-form algebraic solutions and finite element solution have been used extensively to derive the stress distribution present within the single lap-shear joint (see, for example, the review of Adams et al. [1]); however, there are far fewer analyses for pull-off loading of specimens with ‘‘L’’- or ‘‘Zed’’-type geometry which are similar to the stringer geometry used. Similarly, FE and *Corresponding author. E-mail address: [email protected] (J.P. Sargent).
closed-form algebraic strength prediction methods are reported for the lap-shear strengths based on the use of various failure criteria, but little is available for the other geometries. For these reasons, a closed-form algebraic analysis was undertaken which was suitable for predicting stringer pull-off strengths. This was based on analysis of Kanninen [2], Fernlund and Spelt [3] and Penado [4] which used linear elastic beam theory analysis of the double cantilever beam and strain energy release rate methods. However, rather than deriving values of strain energy release rates from dcb specimens with a known applied load, the analysis was modified to predict an unknown pull-off load by treating the stringer/skin combination as a single cantilevered beam using a critical value of the strain energy release rate.
2. Specimens and test methods Fig. 1 shows an example of the stringer pull-off specimen that was used to determine the strength for design calculations. Specimens comprised 50 mm lengths of ‘‘Zed’’ section stringers with the free flange removed, cut from stringer/skin panels. Alloy types were 7050-T76 clad sheet for the wing top surface skin, and 7050-T7651 for the stringer. Adherend surfaces were all pre-treated using standard procedures which included chromic
0143-7496/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0143-7496(03)00011-3
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Fig. 1. Schematic diagram of the stringer pull-off specimen.
sulphuric etch and chromic acid anodise. The adhesive was AF 163-2K06 with EC 3960 spray applied adhesive bonding primer, both manufactured by 3M. Specimens had three different stringer thickness, representative of the inner (set 1 specimens), middle (set 2 specimens) and outer regions (set 3 specimens) of the wing top surface. For the purposes of this study, the original manufacturing spew fillets were removed and defects were introduced into the heel of the specimen by machining. This resulted in unbonded areas with lengths of up to 2.5 mm and gap-heights of approximately 0.76 mm. The testing machine was an Avery-Denison 7157 and tests were done under load control at a test rate of approximately 4.4 kN/min. A schematic diagram of the test fixture is shown in Fig. 2. Fig. 3 shows in schematic form the specimen geometry and dimensions defining the details of the loading arrangement used for measuring stringer pull-off loads. A few specimens were also made which were suitable for measuring the adherend displacement within the bonded section adjacent to the crack front. These were cut from larger stringer pull-off specimens to give specimens with widths between approximately 0.76 and 2 mm. Specimens were loaded using a similar geometry to that used for the larger specimens, but were instead tested in situ on the straining stage of an optical microscope using a ‘‘Minimat’’ miniature materials tester.1 Specimens were tested under displacement control at a test rate of approximately 0.04 mm/min. By taking digital photomicrographs (1280 pixels 1000 pixels) and correlating features from the photographs in the unloaded state with the same features in the loaded state using digital image correlation software2 it was possible to obtain vector displacement maps, from which the adherend displacement ðwðxÞÞ could be 1
MINIMAT miniature materials tester. Polymer Laboratories, UK. microDAC, Fraunhofer, IZM, Gustav-Meyer-Allee 25, D-13355 Berlin. 2
Fig. 2. Schematic diagram showing the testing fixture.
Fig. 3. Schematic diagram of the pull-off specimen defining geometry and dimensions.
measured. It was estimated that displacements could be obtained with an accuracy of better than 1/20th of a pixel, giving an equivalent accuracy of approximately 0.2 mm when using a 2.5 microscope objective. Tables 1 and 2 summarise the elastic moduli and the relevant geometric constants used to define the specimen dimensions and properties. Tensile stress–strain
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characteristics for the adhesive are shown in Fig. 4. These were obtained using the optical correlation method on small specimens strained on the stage of the optical microscope. This permitted simultaneous measurement of modulus and Poisson’s ratio both globally along the length of the specimen, and locally adjacent to the failure initiation site. Locally in the region of failure, strains of up to approximately 6–7% were recorded; these are shown by the dashed line in Fig. 4. The associated error bars represent the change in applied load at a fixed displacement during a strain measurement. It is believed this behaviour was due to sub-critical crack propagation. Measurements of GIc were made for AF163-2K06 with aluminium adherends using a double cantilever beam method. Fracture energies were measured using linear elastic analysis given in Ref. [5] by taking an average of results using a corrected beam theory analysis and an experimental compliance measurement method. This gave a value for GIc of approximately 1900 J/m2. Loading rate was approximately 0.1 mm/min. It was noted that the adhesive showed an R curve effect of increasing fracture energy with crack length, and also some non-linearity in the loading curve.
3. Linear elastic analysis
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substrate. Defects were confined to a region beneath the web of the stringer, so it is assumed that the stringer side of the attached flange area does not behave as another beam which is cantilevered at the junction between cracked and uncracked adhesive. Note that in this instance, the effective crack length, a; is given by the debond length, +L1 ; h2 =2: The loading arrangement shown is statically determinate, and the load P acting on the specimen at the end of the cantilever beam can be determined from the applied load P3 and by consideration of equilibrium conditions, thus P ¼ P3 =ð1 þ L1 =L2 Þ:
ð1Þ
The differential equations relating the transverse deflection wðxÞ of a simple beam, width b; without shear deformation and partly supported on an elastic foundation is given by, for example, [2] d4 wðxÞ þ 4l4 wðxÞ ¼ 0 for x > 0; dx4 d4 wðxÞ ¼ 0 for xo0; dx4
ð2Þ
where l4=3k/Exbh3, Ex is the modulus of adherend in x direction. Following Kanninen [2] and Hete! nyi [6], it is assumed that a Winkler foundation is valid, in which case the transverse stress (sy ) is proportional to the transverse deflection of the beam at that point, i.e.
3.1. Stringer/skin pull-off specimens sy ¼ k For the loading arrangement shown in Fig. 3, failure was recorded as occurring cohesively at the heel of the specimen. This forms the basis of the analysis presented here. This loading geometry is similar to that found in a single cantilever beam arrangement (shown schematically in Fig. 5), with the skin serving as a single beam pivoting about the heel, and with the stringer serving as
wðxÞ b
ð3Þ
Table 1 Material parameters used for calculating stringer pull-off loads Material
Ex ; Ey (GPa)
n
G (Gpa)
Skin Stringer Adhesive
65.5 71 1.58
0.33 0.33 0.41
24.6 26.7 —
Fig. 4. Tensile stress/strain curve obtained for adhesive AF163-2K06.
Table 2 Nominal dimensions used in Fig. 3 for calculating stringer pull-off loads Specimen group
L1 (mm)
L2 (mm)
L3 (mm)
L4 (mm)
h (mm)
h1 (mm)
h2 (mm)
heff (mm)
h3 (mm)
t (mm)
Set 1 Set 2 Set 3 0.76 mm wide in situ specimen
5.3 5.0 4.7 7.3
27.7 28.0 28.3 25.7
33 33 33 33
24 24.3 24.5 25
2.64 2.64 2.64 2.64
4.83 3.56 2.24 3.2
3 2.4 1.75 2.05
o8 o6 o4
25.4 25.4 25.4 22
B0.15 B0.15 B0.15 0.083
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Fig. 5. Schematic diagram for the single cantilever beam used in the analysis.
using the prime notation to indicate differentiation, the boundary conditions are w00x ðaÞ ¼ 0; 3 w000 x ðaÞ ¼ 12:P=Ex bh ;
beam case for the strain energy release rate is trivially extracted using the compliance argument given by Penado [4], thus, in plane stress: GI ¼
w00x ðcÞ ¼ 0; w000 x ðcÞ ¼ 0: By matching the values of deflection and the first three derivatives at x ¼ 0; Fernlund and Spelt [3] give the solution to Eq. (2) for the nominal transverse stress: P sy ðxÞ ¼ 2 al2 elx b 1 1þ cos ðlxÞ sin ðlxÞ : ð4Þ la The stress at x ¼ 0 is therefore P 1 : sy ð0Þ ¼ 2 al2 1 þ b la
Following Fernlund and Spelt [3], using the virtual crack closure technique [7], the mode I component of the strain energy release rate, G I can be calculated from G1 ¼
b½sy ð0Þ 2 : 2k
ð6Þ
Expanding Eq. (6) gives the energy release rate for a single beam in terms of the foundation modulus k; thus, in plane stress: 2 32 GI ¼
7 6P2 a2 6 1 6 7 61 þ 0:25 7 : 2 3 5 b h Ex 4 1 a 3k 3 Exbh
ð7Þ
Solutions to Eq. (2) for the single beam case have also been derived by Kanninen [2] in solving the dcb case, but without the presence of an adhesive layer. Penado [4] used the Kanninen solutions, included an adhesive layer, and also gave corrections for shear distortion of the cracked portion of the beam adjacent to x ¼ 0 by introducing a correction to the beam deflection based on theory given by Timoshenko and Goodier [8]. The single
h Bh0:25 a
þ
1 E x h2 þ ; B2 h0:5 8Gxy a2
ð8Þ
0:25 ; Gxy is the shear modulus of where B ¼ 3k=Ex b adherend. The solutions given in Eqs. (7) and (8) may be compared with a simple beam solution (see, for example, [9]), where no corrections are made for shear distortion or the presence of an elastic foundation, thus GI ¼
ð5Þ
6P2 a2 b2 h3 E x 1þ 2
6P2 a2 : b2 h3 Ex
ð9Þ
For dcb specimens, it was suggested by Fernlund and Spelt [3], and Penado [4], that in an adhesively bonded specimen the foundation constant k is made up of a contribution from the adhesive layer, and a contribution whereby each adherend beam acts as an elastic foundation for the other. A Winkler foundation constant is introduced in order to model the elastic response of the adhesive and substrate to the bending of the cantilevered adherend. This can be imagined as a continuous array of independent linear springs, which simulates the transverse elasticity of the uncracked region. Penado [4] showed how this could be formulated by considering the components as if they were springs in series with one another. Fernlund and Spelt [3] followed the analysis of Kanninen [2] by modelling each of the two beams by assuming an adherend foundation stiffness k ¼ 2Ey b=h; whereas Penado [4] modelled each adherend using a foundation stiffness k ¼ 4Ey b=h: Penado [4] considered that the factor of 4 in the expression for the adherend contribution to the foundation constant gave better agreement with results from his FE modelling. This was included to accommodate the shear distortion of the cracked section of the cantilever beam. It was also considered by Penado [4] that the adhesive foundation modulus should incorporate a modification for plane
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strain, since the adhesive in-plane lateral strain was inhibited. Consideration of Fig. 3 shows that the pull-off load P3 is applied to the end of the stringer web. This suggests that an additional elastic foundation is required which is representative of the length of stringer web which contributes to the development of elastic stresses within the adhesive layer. In this instance, the continuity and shear within the web will result in a proportion of the stringer web length which moves bodily as a whole, and a proportion, which deforms differentially across its width. In terms of the development of stresses at the specimen heel, it is only the differential deformation that is important. Thus, if we assume the crack extends under constant load, then, by St. Venant’s principle, stress changes in the web adjacent to the adhesive will be determined at most by an elastic foundation that comprises a web length given approximately by the web width (h2 ). This would therefore mean that the web and attached flange contribution to the foundation reaction should be at most made up of a length heff Eh1 þ h2 : Following Kanninen’s [2] example, then to a first approximation:
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Fig. 6. Stringer pull-off loads and predictions for set 1 specimens.
k ¼ sy b=w ¼ ðEy eÞb=ðe heff Þ ¼ Ey b=heff This gives a revised foundation modulus in the stringer web area made up of contributions from the cantilevered adherend, the adhesive layer and the substrate web, thus: 1 k¼ ð10Þ h tð1 uyx uyxÞ heff þ þ CEy b Ea b Ey b
Fig. 7. Stringer pull-off loads and predictions for set 2 specimens.
where C ¼ 2 using Fernlund and Spelt analysis [3], C ¼ 4 using Penado analysis [4], heff is the effective web length, t=the thickness of adhesive layer, Ea=the modulus of adhesive, Ey the the modulus of adherend in y direction, uxy ¼ uyx=Poisson’s ratio of adhesive, u the Poisson’s ratio of adherend.
4. Stringer pull-off results Figs. 6, 7 and 8 show the experimental results for the pull-off loads, P3 ; as a function of the debond length defined in Fig. 3, for specimen sets 1, 2 and 3 respectively. It was noted that whilst the set 1 specimens showed exclusively cohesive failure, sets 2 and 3 showed failure which was not 100% cohesive, with evidence for some pin-head sized adhesive failures which were not significantly clustered. Figs. 6–8 also shows stringer pull-off strength predictions for set 1–3 specimens which were derived using Eqs. (7) and (8). These are shown in plane strain (i.e. substituting Ex =ð1 n2 Þ for Ex and Ey =ð1 n2 Þ for Ey in
Fig. 8. Stringer pull-off loads and predictions for set 3 specimens.
Eqs. (7) and (8) above), and calculated for cases using the maximum effective web lengths (heff) given in Table 2. For comparative purposes, Figs. 6–8 also show the predictions using Eq. (7), in plane strain, where the adhesive alone contributes to the foundation modulus (i.e. h ¼ 0 and heff =0 in Eq. (10) above). Reference to Figs. 6–8 shows that, irrespective of choice of foundation modulus, predictions of strength are overestimated
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relative to a linear least mean squares fit for the experimental points. This was by amounts between B10% at best for the set 1 specimens using a foundation comprising an 8 mm foundation, and B30% at worst for the set 2 specimens using a foundation comprising adhesive alone. It should also be noted that for the current range of crack lengths and specimen geometry that there is little difference in the predictions based on either Eq. (7) or (8).
5. Non-linear properties and validation experiments Of particular concern in the above analysis is the seemingly rather arbitrary way in which the foundation modulus is established and the implications for predictions of the non-linearity shown in the stress/strain curve from Fig. 1. Fig. 9a and b show the predicted stress (sy ðxÞ) profiles calculated for the set 1 specimens at zero debond length based on the geometry shown in Fig. 3. These were calculated using Eq. (4) together with elastic foundations comprising (a), adhesive alone and (b), adhesive and an 8 mm effective web. Reference to the predicted stress profiles shows that the stress for either elastic solution exceeds the plastic plateau stress level of approximately 30 MPa shown in Fig. 4 over a significant portion of the bonded section of specimen. This is clearly physically unrealistic. By measuring adherend displacements as a function of applied load it was possible to assess the extent to which adhesive non-linearity and the choice of a particular foundation modulus influenced accuracy of the predictions. Fig. 10 shows an example of a vector displacement map for a small L-type specimen from which such measurements were made. This was obtained using the feature correlation software and loading arrangement stated in section 2 above. Fig. 11 shows the measured y component displacements for skin and web areas in the heel region adjacent to the crack front for another small L-type specimen (see Table 2 for specimen details) with load P ¼ 29:3 N/mm (P3 ¼ 35 N/mm). It should be noted that there is some
net rotation of the whole specimen, shown by the slope of the stringer adherend. Net displacement wðxÞ was obtained by subtracting skin displacement from the stringer displacement. This was then compared directly with predictions using Eqs. (3) and (4). Fig. 12a show this comparison for the cases where the foundation is assumed to be comprised of adhesive alone, and Fig. 12b shows the case where the foundation is assumed to be composed of adhesive and an 8 mm effective web. Fig. 13a and b shows the equivalent calculated stress distribution for the specimens from Fig. 12a and b. Reference to Fig. 13 shows that the predicted stress distribution is completely elastic along the whole bonded length (i.e. less than B30 MPa) only for the specimen with load P ¼ 8:7 N/mm (P3 ¼ 10:5 N/mm) for Fig. 13a, and for the specimens with loads less than 18 N/mm for Fig. 13b. All the other loads gave predicted stresses larger than 30 MPa.Comparison between the experimental results and predictions for wðxÞ would also suggest, tentatively, a better fit for the case where the foundation is composed of adhesive alone, i.e. Fig. 12a. Given a different adhesive with a larger plastic yield stress then it is likely that a better estimate of foundation modulus could be made. However, the most important point to note from the displacement graphs is that the discrepancy between experiment and theory increases as
Fig. 10. Correlation vector displacement map for stringer pull-off specimen. Vectors shown x5 : Vectors drawn at approximately 0.2 mm intervals.
Fig. 9. Predicted stress profiles based on Eq. (4) using foundation moduli, which include (a) adhesive alone, and (b) adhesive and an 8 mm effective web length.
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the predicted stress gets larger, irrespective of choice of foundation modulus. This suggests that the linear elastic analysis of the foundation for this tough adhesive with effectively short crack lengths is inadequate, and that the adhesive non-linearity has a more profound influence on predictions than the choice of a particular elastic foundation.
6. Linear beams resting on non-linear foundations Analysis of linear elastic beams resting on bilinear and non-linear foundations have been given by, for example, Kuo and Lee [10] and Farshad and Shahinpoor [11], and Beaufait and Hoadley [12]. A relatively simple expedient is adopted here, which is similar to that used by Beaufait and Hoadley [12] in their consideration of beams on soil
195
foundations. The adhesively supported beam is divided into discrete sections with different foundation modulus, with the governing differential equations being solved numerically using iterative techniques, ensuring continuity of boundary conditions between the sections. The beam was approximated as a partly free beam and a partly adhesively supported beam with five discrete sections. An initial trial solution was first estimated, and then successively better approximations to the stress distribution were obtained by adjusting the adhesive elastic modulus in each of the sections, such that the average stress over that section length with a chosen elastic foundation, for a particular strain, accorded with the stress/strain relationship shown in Fig. 4. A check on the accuracy of this approach was also made by ensuring that the area under the resulting stress distribution curve equalled P=b: Formally, this approach was based on the same governing differential Eq. (2) referred to earlier, with the same boundary conditions at x ¼ a and c: This used the general solution for x > 0: wðxÞ ¼ Ci cos ðlxÞ cosh ðlxÞ þ Ci cos ðlxÞ sinhðlxÞ þ Ci sin ðlxÞ cosh ðlxÞ þ Ci sinh ðlxÞ sin ðlxÞ and for xo0: wðxÞ ¼ Ci x3 þ Ci x2 þ Ci x þ Ci :
Fig. 11. Measured y component displacements for the in situ pull-off specimen with a load P ¼ 29:3 N/mm. Nominal crack front (x ¼ 0) located approximately at x pixel position 100.
At each boundary between the beam element sections the solutions for deflection and the first three derivatives were matched. This resulted in a set of simultaneous equations which were solved using numerical routines
Fig. 12. Measured (points) and predicted displacements wðxÞ (lines) calculated using linear elastic theory. (a) Using a foundation modulus comprising adhesive alone, and (b) using adhesive and an 8 mm effective web.
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Fig. 13. Predicted stress distribution sy ðxÞ calculated using linear elastic theory. (a) Using a foundation modulus comprising adhesive alone, and (b) using adhesive and an 8 mm effective web.
Fig. 14. (a) Predicted displacement wðxÞ calculated using non-linear elastic analysis. (b) Approximate stress distribution sy ðxÞ (see text for details). Shaded lines in (a) and (b) correspond to each beam element used in the analysis. Shown for the small stringer pull-off specimen at load P ¼ 29:3 N/mm.
given in MATHCAD. This gave values for the constants (Ci ; i ¼ 1y24) at each boundary and at x ¼ a and c: Fig. 14a and b shows, respectively, the resulting displacement and stress distribution calculated for the specimen from Fig. 11 at a load P ¼ 29:3 N=mm: Fig. 15 shows a comparison between these revised predicted displacement profiles (wðxÞ) and the experimental displacements for each of the loading cases shown in Fig. 12a. Whilst not perfect, clearly agreement is much better here using this non-linear analysis than when using the previous linear elastic analysis.
7. Mixed-mode loading Fig. 15. Comparison between predicted (lines) and measured (points) displacements wðxÞ for a small stringer pull-off specimen as a function of applied load. Predictions derived using non-linear elastic analysis. (Compare with the equivalent linear elastic approximation derived earlier and shown in Fig. 12a.).
Prediction of pull-off loads (Eqs. (7) and (8)) and beam displacement has so far been restricted to analysis of the transverse deformation, i.e. just mode I loading. However, since the adherends are asymmetric there will
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Fig. 16. (a) Measured y component skin and stringer displacements (uy ) either side of the adhesive bond line for the in situ pull-off specimen with a load P ¼ 14:1 N/mm. Nominal crack front (x ¼ 0) located approximately at x pixel position 140. (b) Same as in (a) but showing the measured x component skin and stringer displacements (ux ). The dotted line in both cases shows the net adherend displacement that occurred across the adhesive bond line.
inevitably be a shear or mode II component present. In addition to the y displacement components (uy ) which were used to measure the transverse skin and stringer deformation, vector displacement fields like that shown in Fig. 10 can also be use to give the x displacement components (ux ). Fig. 16a and b shows, respectively, the uy and ux skin and stringer displacements as functions of x for regions which were adjacent to the adhesive for the specimen from Fig. 10 at a load P of 14.1 N/mm. Together with a knowledge of the adhesive thickness, then these displacements can be used to measure the applied adhesive shear strain using exy ¼ 1=2ðqux =qy þ quy =qxÞ: For an adhesive bond line thickness of approximately 20 pixels, and at a position given by the nominal crack front (pixel position B140), then this gave applied tensile and shear strains of approximately 2%. It may also be noted that the component @uy =@x was small in comparison to @ux =@y; and that the mode I/ mode II ratio changed significantly with distance from the crack front, with the mode I loading reducing more rapidly than the mode II loading. It may be mentioned that at larger loads when the crack front position was less well defined and characterised by a zone of damage which extended along the bond line, then the applied mode ratio would change appreciably over this damaged zone region.
8. Conclusions and discussion The optical correlation method for deriving measured displacements was found to be a particularly useful tool for validation of the analysis and for comparing predictions with experimental results. This was because displacement and strain field maps could be obtained quickly and accurately without the need for special fixtures or stability requirements, and with the minimum
of surface preparation (light polishing with 800 grit emery paper). The main practical limitations in the implementation used here involved the relatively slow image capture rates as a result of the relatively large file sizes, which implied low applied strain rates. It was noted that some time-dependent sub-critical crack growth occurred when testing the tensile test specimen in Fig. 4, and also when testing the small-scale specimens at larger loads than those shown in Fig. 11. AF163-2K is known to support loads at very large applied strains (B100%) by the formation of adhesive and monofilament bridges [13]. Given a full knowledge of the stress/strain relationship for this adhesive covering rate effects and larger applied strains beyond those measured in Fig. 4, then it is envisaged that the adherend displacement modelling using this non-linear analysis could be extended to predict the stress and displacement profiles for larger applied loads than those shown in Fig. 15. Given that the adhesive displayed significant nonlinearity it would seem likely that this was one of the more important factors responsible for the discrepancy between the measured stringer pull-off strengths and those predicted using the linear elastic fracture mechanics analysis. Note should also be made that since this tough adhesive also displayed large damage zones, which extended beyond the stringer web, then a physically more correct analysis may also have to consider the contribution that a second beam, as represented by the stringer web/attached flange area, would have both on the net beam displacement and the contribution to the energy release rates. However, it is also likely that given specimens with similar crack lengths and dimensions to those used here, but which used a more brittle adhesive which displayed less nonlinearity, the analysis might then be more successful in predicting stringer pull-off loads.
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The predictions of pull-off strength were also based on the use of a mode I fracture energy, and ignored any consequences of the mixed mode nature of the loading. Given that an appreciable shear component would result in a larger critical energy release rate, then it could be argued that this would then have given rise to an overestimate of the pull-off strength with the present mode I analysis. It is clear, however, that a more comprehensive and accurate treatment of the stringer pull-off test specimens should also consider the mixed mode nature of the test. References [1] Adams RD, Comyn J, Wake WC. Structural adhesive joints in engineering. London: Chapman and Hall; 1997. [2] Kanninen MF. An augmented double cantilever beam model for studying crack propagation and arrest. Int J Fract 1974;10(3):415. [3] Fernlund G, Spelt JK. Mixed mode energy release rates for adhesively bonded beam specimens. J Compos Technol Res JCTRER 1994;16(3):234.
[4] Penado FE. A closed form solution for the energy release rate of the double cantilever beam specimen with an adhesive layer. J Compos Mater 1993;27(4):383. [5] Adhesion, Adhesives and Composites Website, http:///www.me.ic.ac.uk/materials/AACgroup/index.html, Imperial College of Science, Technology and Medicine. [6] Het!enyi M. Beams on elastic foundation. Ann Arbor: The University of Michigan Press; 1946. [7] Rybicki EF, Kanninen MF. A finite element calculation of stress intensity factors by a modified crack closure integral. Eng Fract Mech 1977;9:931. [8] Timoshenko SP, Goodier JN. Theory of elasticity. New York: McGraw-Hill; 1970. [9] Charalambides M, Kinloch AJ, Wang Y, Williams JG. On the analysis of mixed-mode failure. Int J Fract 1992;54:269. [10] Kuo YH, Lee SY. Deflection of nonuniform beams resting on a nonlinear elastic foundation. Comput Struct 1994;51:513. [11] Farshad M, Shahinpoor M. Beams on bilinear elastic foundations. Int J Mech Sci 1972;14:441. [12] Beaufait FW, Hoadley PW. Analysis of elastic beams on nonlinear foundations. Comput Struct 1980;12:669. [13] Sargent JP. Microextensometry, the peel test, and the influence of adherend thickness on the measurement of adhesive fracture energy. Int J Adhes Adhes 1998;18:215.