~ U E T T E R W Q R T H I N E M A N N

Int. J. Fatigue Vol. 17, No. 4, pp. 279-286, 1995 Copyright (~) 1995 Elsevier Science Limited

Printed in Great Britain, All rights reserved 0142-1123/95/$10.00

Mixed-mode fatigue and fracture behaviour of joints bonded with either filled or filled and toughened adhesive

X.X. Xu, A.D. Crocombe and P.A. Smith*

Department of Mechanical Engineering and *Department of Materials Science and Engineering, University of Surrey, Guildford, Surrey GU2 5XH, UK (Received 8 August 1994; revised 16 November 1994)

Mixed-mode fatigue and quasi-static tests have been carried out on joints bonded with either a filled or a filled and toughened adhesive. The global strain energy release rate G (GI or GI~ or GT) for crack growth in the adhesive parallel to the interface has been evaluated using the finite element method. The fatigue crack growth and quasi-static fracture under mixed-mode loading were controlled by a local mode I mechanism whereby filler particles fractured under the maximum tensile stress and the microcracks so formed subsequently linked up. Both the mode I and mode II components G~ and G. contributed to the fatigue and fracture processes in that they determined the local stress distributions around the crack tip. In the mixed-mode joints subject to fatigue load, AGt became a controlling factor in determining the fatigue crack growth rates as G~/G, increased. In the mixed- mode joints subject to quasi-static load, G~ was a close representation of the crack-driving force.

(Keywords: mixed mode; cohesive joint; fracture; particle cracking)

There is an increasing trend in the use of adhesives in industry, notably in automobile manufacturing t. Many practical adhesive applications involve the use of lap joint geometry, which leads to combination of in-plane tension and shear 2, i.e. mixed mode (mode I and mode II) loading. Hence a recent investigation 3 on the mode I fatigue behaviour of joints bonded with a filled and a filled and toughened adhesive has been extended to study the mixed-mode fatigue and fracture behaviour of joints bonded with the same adhesives.

In a joint experiencing only peel stresses, it is common to use mode I strain energy release rate range AG~ to correlate fatigue crack growth rate (FCGR) by means of the Paris law, i.e. da/ dN = k(AG~)', where da/dN is crack growth per cycle and k and m are material constants. In a joint subject to both peel and shear stresses, ranges of mode I, mode II and total strain energy release rate, AGI, AG~I and AGT, or some combination thereof, are all possible candidates as the characterizing parameters for fatigue crack-driving force. However, in early work on fatigue crack growth in mixed-mode joints, strain energy release rate G instead of strain energy release rate range AG was used to correlate FCGR in the joints. In the following paragraphs, reference will be made to the particular characterizing parameters where necessary.

Some workers believed that FCGR in mixed-mode joints was controlled by the mode I loading component. Hart-Smith 4 emphasized the importance of peel stress in fatigue crack initiation and propagation. Everett 5 applied clamping forces to generate mixed-mode effects

in crack lap shear (CLS) specimens. The FCGR was reduced drastically when the crack approached the clamped region where Gj decreased, and therefore it was suggested that G~ is the principal loading compo- nent causing the fatigue crack growth. Brussat et al. 6"7 suggested an empirical parameter AGe~-f = [1 + 2G~/ (G~ + GH)]AGI. In fact the parameter is a variation of mode I dominance, for AGef~ diminishes as AGz approaches zero.

Others have shown GT or AGT as a driving force for fatigue crack growth. Johnson and Mall 8 studied fatigue crack growth in T300/5208 graphite/epoxy composite bonded with either EC-3445 or FM-300 adhesive. Good correlation was found when the FCGR, da/dN, was plotted against total strain energy release rate GT, though da/dN could be equally well correlated by G I and GII individually. Schmueser 9 expressed a similar viewpoint, but he used AGT instead of GT as characterizing parameter in his study of cyclic debond- ing in the adhesive joints of various bondline thick- nesses. Liechti and Lin 1° ruled out G~ as a controlling factor by showing that FCGR still increased when G~ approached zero, and argued that AGT correlated the debonding rate da/dN in joints bonded with a rubber- toughened adhesive.

The above studies and similar ones are valuable in that they have provided a starting point for further investigation of fatigue crack growth in mixed-mode joints. It is felt that, apart from considerations of the degree of the empirical fit, the fracture mechanism needs to be determined before one can conclude whether AGT, or AG~ or AG~I or some combination

279

280 Xu et a l.

of them can best correlate FCGR in adhesive joints. Also, any characterizing parameter should apply to mode I joints as well as mixed-mode joints. The aim of the current study of mixed-mode fatigue behaviour and the comparison with the previous mode I results 3, obtained under similar conditions, is to find a physical interpretation of fatigue crack growth in adhesive joints under mixed-mode loading.

Apart from the mixed-mode fatigue tests, quasi- static tests have also been carried out in this study. As with fatigue loading, effort has been made to find a characterizing parameter for crack-driving force in joints loaded quasi-statically. Trantina xl studied a problem of mixed-mode crack extension (either interfa- cial or in the centre of bondline) in aluminium- epoxy-aluminium specimens with a slant crack. G1 and GII were obtained by utilizing results of finite element analysis for homogeneous materials. He acknowledged that, as a failure criterion, GT = Gc (where Gc is critical strain energy release rate) is not appropriate for the adhesive system. Johnson and MangalgirP 2 suggested that fracture in adhesive joints is controlled by an equation, (PI/PIc) 2+ (PII / PHc) 2 = 1, where P is load and 'c' in subscript indicates a critical condition. Mall and Kochhar 13 found that mode I, mode II and mixed-mode cohesive fracture in composite bonded joints was controlled by a criterion, G~/G~c + GH/GIIc = 1. Hamoush and Ahmad TM also conducted mode I, mode II and mixed- mode fracture tests of joints bonded with one adhesive. Their results showed that interfacial failure in the three types of specimen was controlled by the criterion GT = Gc in spite of different initial crack lengths, thickness of the adhesive layer and loading mode. Sancaktar 2 proposed a criterion for mixed-mode brittle fracture in adhesive joints in terms of stress intensity factors/( i and/(11. In view of the different approaches proposed so far, more experimental work is necessary to further the understanding in this area.

SPECIMENS AND E X P E R I M E N T A L PROCEDURES

A mixed-mode beam geometry, shown in Figure 1, has been chosen for the current study. The geometry, which is loaded in three-point bending, offers obvious simplicity in manufacturing and testing. Mild steel

substrates were machined to size, bead blasted, and given three ultrasonic baths in an inhibisol solution to remove dirt and oil. The substrates then received a silane treatment. A solution of 1 ml of silane dissolved in 100 ml of distilled water was poured onto the bond area of the substrates. After 3-4 min the substrates were stood vertically to dry in an oven preheated to 40 °C.

Two adhesives, designated A and B, were used. Adhesive A is filled and adhesive B is filled and toughened by rubber. The adhesives are the same as those used in our previous study of mode I fatigue behaviour in double cantilever beam joints 3. Cartridges containing the adhesives were preheated in the oven in order to minimize the amount of air in the adhesive by reducing its viscosity. This practice also made it easier to apply the adhesives in the joint manufacturing. A bondline thickness of around 0.2 mm was achieved by inserting paper spacers between the substrates at the ends of the shorter substrate. An initial pre-crack of 20 mm in length was introduced by sandwiching a piece of thin polyimide film between the two spacers so that it protruded 5 mm into the adhesive layer (see Figure 1). Bonding was completed within 20 s. The bonded specimens were put in the oven at 40 °C to cure for 24 h. The cured joints were stored in a desiccator before testing.

Fatigue tests were carried out using an Instron 1341 servohydraulic machine under load control. Tests were carried out at a load ratio of 0.2 and a frequency of 2 Hz. Maximum fatigue loads of 400 N and 450 N were used for the joints bonded with adhesive A and 350 N and 400 N for the joints bonded with adhesive B. Constant load amplitude was maintained for all the tests with one exception, where one joint bonded with adhesive B was loaded first at a maximum load of 350 N and then at 400 N. During testing, the room temperature was kept at around 20 °C. A fine scale was attached to the side parallel to the adhesive layer. A travelling microscope was used to measure the crack length and to observe the crack profile without interrupting the cyclic loading.

Quasi-static tests were carried out on an Instron 6025 testing machine at a cross-head speed of 0.1 mm rain -1 and a room temperature of 21 °C. The test data were logged using a Macintosh LC personal computer.

Fracture surfaces of fatigued and quasi-statically

Crack , /

~ - i~: :z : : ;~:~ ~ ........ \ Spacer

I_..., L/2=50mm

P,A

Adhesive

L

r I

Bond line thickness = 0.2 ram, thickness of specimen = 8 mm

I mm

Figure 1 Geometry and loading configuration of mixed-mode specimens

Mixed-mode fatigue and fracture behaviour of bonded joints 281

Y I I I I I I I I I ] I ~ I I 1 1 I I [ l / • ' 1 ] 1 l ' l I I I ] I I

. . . . . . . . - "J,_] .N:' 1-t - . , ' " " ' , l l l l l l l l l l l l l l l l l l ' ] ' ,

\

~ 1 1 1 1 1 1 1 I l l l t / ! ' ' ' = ~ " ' '

F i g u r e 2

# X

Mesh for finite element analysis: (a) mesh of whole model; (b) mesh details around crack tip 'o'

fractured specimens were observed using an S100 scanning electron microscope (SEM) at an accelerating voltage of 10 kV. The samples were gold-coated before the SEM observation.

FINITE E L E M E N T ANALYSIS

There is no analytical solution for G of the current specimen geometry; therefore a finite element analysis was carried out using ANSYS finite element code. Plane strain conditions were assumed and an 8-noded quadrilateral isoparametric mesh was generated (Figure 2a). The 0.2 mm thick adhesive bondline was divided into four layers around the crack tip (Figure 2b). The size of the smallest elements around the crack tip is 0.05 mm. Further away from the crack tip, the adhesive elements change gradually to rectangles with an aspect ratio of 5.

To evaluate strain energy release rates associated with the crack growth in an adhesive joint, it is necessary to know the Young's modulus of the adhesive. From tensile tests it was known that the stress-strain behaviour of adhesive B exhibits a strong dependence on strain rate while that of adhesive A does not (Figure 3) is. The influence of strain rate on adhesive modulus in the joints bonded with adhesive

40

3 0

20

Adhesive A, fitted for various rates

/ / /

/ Adhesive B, 8 .33x t0 1 O/dee c

Adhesive B, 1.67x10 "2 Q/dsec

Adhesive B, 3.33x10 -4 °/dsec

5 1 0 1 5 2 0 2 5

S t r a i n , %

F i g u r e 3 Stress-strain curves for adhesives A and B at various s t r a in r a t e s ts

B was evaluated in an iterative manner as described in our previous study of mode I joints a. The strain rate in the mixed mode specimen was evaluated over a 'gauge length' centred at the crack tip with a length equal to half of the bondline thickness. The strain rate evaluated for the joints bonded with adhesive B at a fatigue frequency of 2 Hz is 0.06 (% s -1) and the corresponding elastic modulus is 1400 MPa. An experimental value for E of 3500 MPa is used for the joints bonded with adhesive A.

Values for G were evaluated employing a virtual crack growth technique. GI and Gu were evaluated based on nodal forces and displacements in the Y and X directions in a global Cartesian coordinate system (see Figure 2b) respectively. The evaluations were made at crack lengths a ranging from 20 mm to 50 mm. The difference between the global G values at various crack lengths for the joints bonded with adhesives A and B is negligible. Therefore only the curves determ- ined for joints bonded with the adhesive B are presented in Figure 4.

From Figure 4, it is seen that G~, GII and G-r ( = GI + Gu) increase monotonically as the crack grows until the crack length reaches 45 mm. At a -- 50 mm, there is a drop in the driving force. This, presumably, is caused by the redistribution of the shear force around the central loading point (see Figure 1). The mode mixity or the ratio of GI/Gn against the crack length is also shown in Figure 4. It is seen that the mode mixity varies in the range of 1.12 to 1.53. The variation in the mode mixity with crack length in the current specimen geometry is caused by the shear forces associated with the three- point bending. This effect appears to have been overlooked by Williams ~6 in the analysis of a similar geometry of homogeneous materials, leading to a premature conclusion of constant mode mixity (GI/ Gn or KI/Ku) independent of crack length in three- point bend specimens. Schmueser 9 mentioned that the mode mixity in his CLS joints was sensitive to the number of through bondline thickness elements. It

282 Xu et a l.

320

240

160 d

80

~ GT

J / ~ cm /

/ J - - - o G I/Gn /

/ / / / "

~ J 400 N

1.5

I

0 ~ ~ L 0.5

15 25 35 45 55

Crack length, mm

Figure 4 G~, Gn, GT and G] /G. as functions of crack length in mixed-mode joints at a load of 400 N

has been found that the calculated mode mixity in the joints used here did not vary much when the number of the through bondline thickness elements around the crack tip was doubled.

E X P E R I M E N T A L RESULTS

Fatigue tests Plots of log(da/dN)-log(AGT) were generated for

the mixed-mode joints bonded with adhesives A and B. The F C GR data have been fitted to the Paris law using a least-square regression and Paris parameters, m and log(k), have been obtained. There is no obvious dependence of F C GR on maximum fatigue load for the joints. For example, the variation in m is within 5% at loads of 400 N and 450 N for the joints bonded with adhesive A, and the da/dN data of the joint bonded with adhesive B at loads of 350 N and 400 N were completely in a line on the Paris plot. As a result, no distinction was made between FCGRs obtained at different maximum loads. Figure .5 shows the FCGRs plotted against the total strain energy

l0 ° . . . . . . . . . . . . . . . . . l0 °

Adhesive B

iog(da/dN)=4.00,10-t t+3.5 l , log( f iGx ) o

+ \ 2 2 o o o o

~ l0 4 10-4

log(dMdN)=9.43 * 10 t t+3.07, lOg(AGT)

Adhesive A tO i ~ ~ i l l l ~ l ~ I L + + + ~ 10 6

10 tO0 1000

A G T , J / m 2

Figure 5 Log-log plot of FCGR against AGT in mixed-mode joints bonded with adhesives A and B

release rate range, A G T. At a given AGT, FCGR in the joints bonded with adhesive B is consistently higher than that in the joints bonded with adhesive A. Figure 5 also shows that at first the fatigue cracks, especially in the joints bonded with adhesive B, behaved as if they grew from a threshold. Those data corresponding to the initial crack growth were excluded from the linear regression. It was observed that when a crack was initiated in the fatigue tests, it grew toward the longer substrate (i.e. upper substrate in Figure 1) at an angle of about 40 ° to the plane of the precrack. The crack grew subsequently in a 'zigzag' way before the specimen finally failed.

Quasi-static tests Figure 6 shows typical load against cross-head

displacement curves for the joints under mixed-mode quasi-static loading. The average peak load was 694 _+ 35 N for the joints bonded with adhesive B and 893 -+ 54 N for the joints bonded with adhesive A. A similar trend was found in the mode I quasi-static tests, where double cantilever beam joints bonded with adhesive A also failed at a higher load than those bonded with adhesive B. 3 The broken quasi-static mixed-mode specimens showed a similar initiation angle to that of the fatigue specimens.

The average strain energy release rates at the maximum loads of the specimens tested quasi-statically are listed in Table 1. The mode I strain energy release rates at the maximum loads from the previous study 3 are also listed in the table for comparison. It is seen that under the mixed-mode loading, the total strain energy release rate GT is about 100% and 50% higher than that under pure mode I loading for joints bonded with adhesives A and B respectively.

Fractography The fracture surfaces of the broken specimens

showed that all the joints failed cohesively. SEM photographs of the broken surfaces of the mixed- mode fatigue joints bonded with the two adhesives are presented in Figures 7 and 8. As in the case of

900

750

600

z

4 5 0

3 O 0

150

/ / / \

/ / \

:~ Adhesive A

~/ Adhesive B ~f

0 i _ _ i i . ,

0 0.1 0.2 0.3 0.4 0.5

Cross head displacement, mm

Figure 6 Quasi-static load-displacement curves of mixed-mode joints

/ /

/

Mixed-mode fatigue and fracture behaviour of bonded joints

Table 1 M a x i m u m s t ra in ene rgy re lease ra tes of the jo in t s b o n d e d wi th adhes ives A and B

Adhes ive A B

283

Mode I GIM = Gx~a = 212 N m -1 Mixed m o d e GtM = 2 2 6 N m I , G n ~ = 2 0 2 N m 1,GaM = 4 2 8 N m

"M' in the subscr ip ts ind ica tes the va lues at the m a x i m u m load

GIM = GTM = 166 N m ' GIM = 136 N m J,G I~M = 121 N m I,G.rM = 2 5 7 N m t

Figure7 Frac tu re surface of mixed m o d e - j o i n t b o n d e d wi th adhes ive A , cyclic load ing

Figure 8 Fracture surface of mixed-mode joint bonded with adhes ive 13, cyclic l oad ing

previous mode I fatigue tests, no striations were found on the surfaces, The characteristic of the fatigue fracture surface of the joints bonded with adhesive A is that of small debonded particles combined with scattered larger flat facets (Figure 7). The fracture surface of the joints bonded with adhesive B is dominated by flat facets (Figure 8). Similar features were seen on the fracture surfaces of the joints tested quasi-statically.

DISCUSSION

Failure in the mixed-mode joints The fracture surfaces of the mixed-mode joints and

those of mode I joints reported in previous work -~ are similar, although it seems that more voids (i.e. debonded particles) were observed in the mixed-mode joints bonded with adhesive A. It is likely that this is due to the extra shear introduced by the mode II loading component. The similar fracture appearances suggest similar local mode I fatigue crack growth mechanisms. In the joints bonded with adhesive B, microcracking of filler particles and the subsequent linkage of the formed microcracks dominated the fatigue crack growth process. In the joints bonded with adhesive A, microcracking of large particles and microvoiding around smaller particles both occurred but the dominant event was still the propagation and linkage of microcracks. In the joints subject to fatigue loading, the repeated microcracking (microvoiding) and microcrack linkage continued until the catastrophic failure, which should be similar to that experienced in the joints subject to quasi-static loading. Such a mechanism is characteristic of brittle materials, where failure is generally triggered by brittle second-phase particles under local tensile stresses.

Over the years, a number of studies have been carried out to investigate the cracking in homogeneous brittle materials under mixed-mode loading. Despite different approaches, various criteria gave similar predictions for the direction in which a pre-existing crack grows. For instance, Erdogan and Sih ~7 proposed that fracture occurs in the direction perpendicular to the maximum tensile stress. In the present study, the fatigue crack headed away from the pre-crack plane at an angle of about 40 ° . To investigate this further, a vector plot of the maximum principal stresses was generated for the pre-cracked mixed-mode joint bonded with adhesive B, as shown in Figure 9. The plot shows clearly that if a crack is to propagate in a brittle manner, it should do so toward the upper substrate, with the crack path making an angle of about 40 ° with the precrack plane. A similar plot was generated for the joint bonded with adhesive A, and the predicted cracking angle is close to the prediction based on Figure 9. The agreement between the experimental observation and the finite element analy- sis suggests that the cracks grow in a local mode I manner in the mixed-mode joints. This also suggests that at the mode mixity corresponding to the initial crack length, the global parameters GI or Gn is unlikely to be a controlling parameter in the mixed- mode failure processes unless Gj or G~ closely represent the local controlling parameter. G~ and GH did make a contribution to the mixed-mode fatigue and fracture processes in that their combination

284 Xu et a l.

The thicker line repr~ents a crack with its tip at "o ' .

10 o . , . . . . . . . . . . . . . . . 10 °

10 -2

10 -4

10 2

O 0

/ • Mode AG log(k) m 1 AG/ 6.68,10 I~ 5.26 Mix AGI 2.03.I0 9 2.80 Mix AGT 9.43,10 "u 3.07

100

AG, J/m 2

10 "4

10 -6 j , , , . . . . 10 -6

10 1000

Figure 10 Log-log plot of mode I and mixed-mode FCGR against various strain energy release rate range, joints bonded with adhesive A

Figure 9 Vector plot of the maximum principal stress in vicinity of crack tip "o" in the mixed-mode joint bonded with adhesive B (thicker line represents crack plane)

determined the local stress distribution, which is asymmetric to the bondline.

Fatigue crack growth As mentioned already, for the mixed-mode loading

there is an apparent fatigue threshold. However, the corresponding FCGRs of about 10 -5 mm cycle -1 are high compared with the rate of order 10 -s mm cycle -~ at which a mode I fatigue crack approached the threshold 3. It was thought that the apparent threshold was caused by the friction between the upper and lower surfaces of the crack due to the initial crack deflection. As the crack grew the mode mixity G~/GI1 increased and friction had progressively less influence. Consequently the FCGR increased and the linear log(da/dN)-log(AG) curves were obtained.

Since the fatigue mechanism is similar to that in the mode I joints, it is reasonable to conceive that the mixed-mode FCGR data can be brought in line with those of the mode I joints by a common controlling parameter. The mixed-mode FCGRs in the joints bonded with adhesive A, based on AGT, are therefore compared with the mode I data (where AGT = AGI) from previous work 3 in Figure 10. It is seen that at a given AGT the fatigue crack grows faster under mode I loading. This has ruled out AGT as a unifying parameter for FCGR in the joints. The same FCGRs are also plotted against AG~ in the figure. It is seen that the two fitted lines cross each other around AGI = 180 J m -2. At lower AGI, FCGR in the mixed- mode joints is higher than that in the mode I joints, while around AG1 = 180 J m -2 and above, the FCGRs of both kinds of joint effectively mixed with each other. Figure 4 shows that G~ becomes progressively higher as fatigue cracks grow in the mixed-mode joints. It is therefore suggested that the FCGR of the joints bonded with adhesive A is controlled by AGx as G1 becomes a dominant loading mode. In other words, AG~ becomes representative of the local controlling parameter in the fatigue process.

Similar to the joints bonded with adhesive A, at a

g i v e n A G T (in the mode I case, A G T = A G I ) the FCGRs in the joints bonded with adhesive B are lower under mixed-mode loading than those under mode I loading (Figure 11). Thus AGT has been excluded as the unifying parameter in the joints bonded with adhesive B. At a given AGI, the data points from the mixed-mode and mode I joints seem close to each other (Figure 11). The two fitted lines of log(daMN) versus log(AGl) cross each other around AGI = 160 J m -2. Around and above this value, the FCGRs of both kinds of joint effectively mix with each other. This suggests that AG~ becomes the controlling parameter as mode I loading component G I dominates.

It is interesting to note that although toughened by rubber, adhesive B did not show an advantage in terms of resistance to fatigue crack growth in the joints (Figure 5). A similar result was found in the fatigue tests of mode I DCB specimens 3. The higher FCGR in the joints bonded with adhesive B can be explained by the lower adhesive strength and thus the longer process zone, which contains more cracked filler particles.

10 ° i0 o

10 2

l0 -4

10 -6

10

Mode I I ! Mix(T) 0 ~ ~,

O I O ~ t* 0 ° ~1o~ ~ ~

0 ~ Mode AG log(k) rn ' AG[ 3.22.10 "IQ 391 Mix AGT 1.24.10"9 307 Mix AGT 4.00"10 "]] 3 51

10 2

I0 4

10 6

100 I000

A G , J / m 2

Figure 11 Log-log plot of mode I and mixed-mode FCGR against various strain energy release rate range, joints bonded with adhesive B

Mixed-mode fatigue and fracture behaviour of bonded joints 285

Usually, the slopes of log(da/dN)-log(AG) curves, or Paris parameters, m, in adhesives are higher than those in aluminium and steel. The latter range from 1.5 to 3? 8 Higher slopes mean that a small change in applied load would result in a large change in FCGR. In our study of mode I fatigue crack growth in the joints bonded with adhesives A and B, m is 5.26 for joints bonded with adhesive A and 3.91 for joints bonded with adhesive B. In comparison m is 3.07 and 3.51 for the mixed-mode joints bonded with adhesive A and B respectively (Figure 5). However, these values were obtained when fatigue failure was cohesive. If interfacial failure occurs, rn values could be higher, even when fatigue cracks are subject to a mixed-mode load. For instance, m ranges from 7.5 to 9.7 for the interfacial fatigue cracks in the primed CLS joints studied by Schemueser 9. The present results indicate that it is possible for m values in mixed-mode adhesive joints to be similar or close to those in metallic structures as long as a proper adhesive is selected and a good manufacturing procedure is adopted.

Quasi-static fracture In an effort to find a controlling parameter for

fracture performance in mixed-mode joints subject to quasi-static loading, an equivalent strain energy release rate, GEQv, is suggested. GEov should reflect the contributions of both G~ and G[I and only at two extremes, i.e. Gn = 0 or G~ = 0, should GEQV reduce to G~ or Gu. GT meets these requirements; however, the fact that the maximum GT in the mixed-mode joints is much higher than that in the mode I joints has ruled out GT as a controlling parameter. Another simple option is a weighted combination of G~ and G u :

- - Q- a 1 GEOv -- 1 ~- ~ I ~- 1_~_ 6 G I I (1)

where ~l is the mode mixity GI/GIt. In quasi-static tests, joints will fail only when the controlling parameter reaches a critical value. From Equation (1), a failure condition at the maximum load can be derived:

Gl Gt l

GIM + GriM 1 (2)

where GIM and GUM are the maximum quasi-static energy release rates for pure mode I and mode II loading respectively. Equation (2) limits the perform- ance of mixed-mode joints. This equation is similar to the one given by Mall and Kochhar ~3.

The mode mixity GJGu is 1.12 for the quasi-static joints, ignoring subcritical crack growth. Making use of Equation (1) and the data in Table 1, GEov for the mixed-mode joints bonded with adhesive A was evaluated as 215 J m -2, almost the same as a i m in the mode I joints bonded with the same adhesive. A GEov value of 129 J m -2 was obtained for the mixed- mode joints bonded with adhesive B. The value is a little lower than GIM in the mode I joints. Considering the data in Table 1, GIM values in the mixed-mode joints seem close to those in the mode I joints. The GEov approach does not offer a superior quantification. This means that G~ is a good approximation of the characterizing parameter for crack-driving in the mixed- mode and mode I joints subject to quasi-static load.

CONCLUDING REMARKS

As in the earlier work on fatigue crack growth in mode I joints, the mixed-mode fatigue crack growth mechanism in the joints bonded with either the filled or filled and toughened adhesive involved the microcracking and subsequent linkage of the formed microcracks. It has been suggested that this mechanism is controlled by local tensile stresses to which both Gt and Gu made contributions. The suggestion has been supported by the experimental observation and the FEA prediction of the initial directions of the crack growth. AGT cannot bring the fatigue crack growth rates in the mixed-mode joints in line with those in the mode I joints. However, AG~ seems to correlate FCGRs in both the mixed-mode and mode I joints when G~ becomes dominant. The slopes of the mixed- mode log(da/dN)-log(2~G) curves in the present study are similar to those in metallic materials. This means that adhesive structures could be as safe as metallic ones if pure mode I loading can be avoided. In the mixed-mode quasi-static joints, the possibility of exploring an equivalent strain energy release rate GEov as crack-driving force was discussed. However, the GEov approach did not yield data better than using Gt values, which are rather similar between the mode I and mixed-mode joints. For the experimental data available so far, G~ seems to be a good approxi- mation of the crack-driving force in the mixed-mode joints subject to quasi-static load.

ACKNOWLEDGEMENTS

The authors would like to acknowledge DRA for support of this work through a research contract (no.CB-RAE-9-4-2064-135-RAE) and to thank Mr Ray Gilmore and Dr Steve Shaw of DRA (RAE) for useful discussion. The assistance from Dr Lynn Boniface of the Department of Materials Science and Engineering, University of Surrey, with the fatigue testing is greatly appreciated.

REFERENCES

1 Lawley, E., Int. J. Adhesion Adhesives 1990, 10, 22l 2 Sancaktar, E. 'Engineering Materials Handbook', Vol.3:

Adhesives and Sealants, American Society of Metals, 1990, pp. 501-520

3 Xu, X.X., Crocombe, A.D. and Smith, P.A. lnt.J.Fatigue 1994, 16, 469

4 Hart-Smith, L.J. 'Analysis and design of Advanced Com- posite Bonded Joints', NASA CR-2039, 1973

5 Everett, R.A.Jr, Adhesives Age 1983, 26, 24 6 Brussat, T.R., Chiu, S.T. and Mostovoy, S. 'Fracture

Mechanics for Structural Adhesive Bonds', Air Force Materials Laboratory, AFML-TR-77-163, 1977

7 Brussat, T.R, and Chiu, S.T.J.Eng.Mater. Technol. 1978, 100, 39

8 Johnson, W.S. and Mall, S. in 'Delamination and Debonding of Materials' (Ed. W.S. Johnson), ASTM STP 876, American Society for Testing and Materials, Philadelphia, 1975, pp. 189-199

9 Schmueser, D .W.J . Adhesion 1991, 36, 1 10 Liechti, K.M. and Lin, C. in "Structural Adhesives in

Engineering', Proc.Inst.Mech.Eng. 1986, pp. 83-9l 11 Trantina, G.G.J.Compos.Mater. 1972, 6, 371 12 Johnson, W.S. and Mangalgiri, P.D. 'Influence of the Resin

on Interlaminar Mixed Mode Fracture', NASA Technical Memorandum 87571, National Technical Information Ser- vice, 1985

13 Mall, S. and Kochhar, N.K. 'Structure Adhesives in Engineer- ing. Proc. Inst. Mech. Eng. 1986, 71

2 8 6 Xu et a l.

14 Hamoush, S.A. and Ahmad, S.H. Int. J. Adhesion Adhesives 1989, 9, 171

15 Richardson, G., Crocombe, A.D. and Smith, P . A . J . • Adhesion, submitted

16 Williams, J.G. Proc. lnst. Mech. Eng. 1990, 204, 209

17 Erdogan, F. and Sih, G.C.J. Basic Eng., Trans. Am. Soc. Mech. Eng. 1963, 85, 519

18 'Damage Tolerant Design Handbook', Battelle Metal and Ceramics Information Center, Columbus, OH, 1972