- Email: [email protected]
Strength of sandwich beams with interface debondings
Composite Structures 17 ( 1991) 331-350
Strength of Sandwich Beams with Interface Debondings D a n Zenkert Department of Aeronautical Structures and Materials, The Royal Institute of
Technology, S-100 44 Stockholm, Sweden
ABSTRA CT The adhesive joint bonding the faces to the core material in a sandwich structure ensures that the loads are transferred between the components. However, debondings may arise either during the manufacturing process or due to overloading. These will reduce both the stiffness and the load bearing capacity of the structure. In the present paper, debondings in foam core sandwich beams are investigated assuming that cracks in the interface between the face and core are present. Stress intensity factors are found from an analytical model and compared to sohttions from several finite element calculations. Fracture toughness vahtes, determined from simple specimens, are used to predict the fracture loads for beams with simulated debondings subjected to four-point bending.
INTRODUCTION During the past decades, technical development has been heading towards more and more sophisticated and optimized structures in the pursuit of increased performance and reduced structural weight. Here, the sandwich concept has found an increased use due to its high stiffness and strength to weight ratio combined with thermal and acoustic insulation and the opportunity to utilize new materials. Sandwich structures may be designed so that each component is utilized to its ultimate limit. However, this trend makes quality control an even more important issue. It is of great importance not only to find flaws but also to quantitatively determine their influence on structural behaviour. The modus operandi of a sandwich construction is that the faces take up bending moments as tensile and compressive stresses and the core takes up transverse forces as shear stresses. The adhesive joint ensures that loads are transferred between the components. Debonding 331
Composite Structures 0263-8223/91/S03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
D. Zenkert
332
Pi T
pP V
i iii
I~1
!
' Lou t -J Fig. 1. Illustrationof the objectof study,a sandwichbeam in four-pointbendingwitha simulatedinteffacialdebondingsituated betweenan outer and inner support. will reduce the stiffness of the structure but also act as a crack that may initiate and cause premature failure. The aim of the present paper is to find the reduction in load bearing capacity of foam core sandwich beams with simulated debondings and to present methods that can predict this reduction. The object of study is a beam under four-point bending, containing a core/face debonding between one of the outer and inner supports, as illustrated in Fig. 1, The approach is to find fracture toughness data from simple specimens and to compute the stress intensity factors for the problem by means of analytical and FE analyses in order to predict the fracture loads. Related work Triantafillou and Gibson ~ used a simplified method to fmd the energy release rate of sandwich beams with interracial debondings under threepoint bending and used a double shear specimen to determine the fracture toughness. Their results are relatively good but the simplicity of the model gives some restrictions, for example, it does not account for the influence of the crack length for cracks longer than the core thickness. Carlsson et al. 2 presented a specimen for measuring the interracial fracture toughness, the so called cracked sandwich beam (CSB) along with closed form formulae for the calculation of the energy release rate. A solution to the problem of cracks situated at the mid-plane of the core is given by Zenkert. 3 The latter work showed that such cracks drastically reduce the load bearing capacity of foam core sandwich beams and the stress intensities can be found using a relatively simple analysis. The bi-materiai crack In linear elasticity, singular stress and strain fields appear at geometrical and material discontinuities. The singular stress field at the tip of a crack is well known and is widely used for the prediction of crack growth in
Strength of sandwich beams with interface debondings
333
components and structures. For a crack situated at the interface between two dissimilar media, the singular fields differ from those of a crack in an isotropic medium. The interface crack has been treated by Rice and Sih, 4 Erdogan 5 and England, 6 along with methods to compute stress intensity factors. The main difference from the isotropic case is that the strength of the singularity generally is a complex quantity that generates an oscillatory behaviour of the deformation close to the crack tip. C o m n i n o u 7 treated these oscillations by considering a frictionless contact near the crack tip. In a later paper, s she investigated a bi-material crack in a shear field, where the lack of symmetry produces a large contact zone at one of the crack tips. The same problem arises, as will be seen, for an interface crack in a sandwich beam, as the core material is mainly subjected to shear. A N A L Y T I C A L APPROACH The approach to analytically estimate the stress intensity of the crack tips in the problem, illustrated in Fig. 1, is to derive the energy release rate via the change in potential energy of the system due to crack extension. The derivation follows the same manner as in Refs 2 and 3. The assumptions made are that both crack tips exhibit the same stress intensity, that the beam remains symmetric about its centrepoint, point A in Fig. 2(a), and that Mode I stress intensity is negligible. Engineering beam theory including transverse shear deformation is used and hence it is assumed that deflections due to bending and shear can be superimposed. This is true if the faces are thin compared to the core. The energy release rate is computed as: G = -1/n.OU/Oa
(1)
where n is the number of equal crack tips and U is the potential energy of the system, - PAl2, where A is the difference in deflection between the load points B and E (cf. Fig. 2(a)).
Deflection analysis The right side of the beam is divided into four parts as shown in Fig. 2(a). The beam members AB, BC and DE all have bending stiffness D and shear stiffness S, while member CD has stiffnesses D' and S', computed below. The key problem is to treat the cracked part CD. First, the loads are divided into one symmetrical and one anti-symmetrical part as illustrated in Fig. 2(b). For the symmetrical part, the beam will still bend
334
D. Zenkert
IZ
k
l--.-
IP
l
I
P
I
I~. L3 ..I. A
B
P
L2
L I , 1.I.,
C (a)
D
.I E
=
PL21 [ "
] I PL2 + P
p
Pa
(b/ Fig. 2. The right part of the studied beam. (a) Definition and geometry of beam members used in the analytical approach. (b) Superposition of loads intoa symmetrical and an anti-symmetrical part for beam member CD. about its mid-plane since the end planes (points C and D) must remain straight and therefore its stiffness will be D. For the anti-symmetrical load the two beam parts will bend about their individual neutral axes and independently of each other. The load will hence be divided between the beam parts as:
P.p = a3 /3Olo~ + a/Stow P a3/3(1/D~ow+l/D,p)+a(1/S~ow+ 1/S,p)
(2)
where subscripts 'low' and 'up' refer to lower and upper beam parts, respectively. Now, the deflections, rotations and their derivatives with respect to crack length can be found for each point A - E . 3
Cross.sectional properties The bending stiffness for a symmetrical sandwich is computed as:
D = f Ez 2 dA --Eft3/6 + Eftfd2/2 + Ect3/12 JA
(3)
Strength of sandwich beams with interface debondings
335
where d =t~ + tf. As seen in the above expression, the second term is predominant and the two other terms only contribute 1-2% of the total stiffness for real sandwich constructions. The method to find the shear stiffness of a sandwich cross-section is by studying the shear stress distribution. Engineering beam theory is incomplete in the sense that plane cross-sections will remain plane. This has no effect for slender beams but here it must be taken into account. One can use the energy balance equation to compute the average shear angle, ~x:, by using:
½ T:Yx:= ½IA rx:(Z)" Yx:(z) dA, with );x: = Tz/S
(4)
where T. is the shear force at the studied cross-section and L-(z) the shear stress distribution. For a symmetrical sandwich with thin faces and Ef ,> E c, S equals i.~c/d2/to where/z~ is the core shear modulus. For part CD, the presence of the crack divides the cross-section into two beams. Their bending stiffnesses are computed using eqn (3) by integrating over their own neutral axes. A problem arises when estimating the shear stiffnesses of the beam parts. For a long crack the load sharing in eqn (2) is mainly governed by the beam bending stiffnesses and for short cracks mainly by beam shear stiffnesses. Since the shear stiffness for a single face would be/z e. tf/1.2, it would be greater than for the lower beam part but also greater than the shear stiffness for the uncracked beam parts. Therefore, for a short crack, all the load would be carded by the upper face and, hence, the total stiffness of the beam would increase. This is, of course, unrealistic. At the two ends, C and D, the cross-sections must be compatible with beam members BC and DE, respectively. The shear stiffness must then equal S at these two ends. In between, the beams are allowed to move relative to each other and sliding may take place. This sliding is here considered equivalent to a reduction in shear stiffness. The assumption made here, to make a feasible solution even for short cracks, is that the shear stiffness of the two beams will reduce to half that of the uncracked beam. In the transition region between the ends and the mid part, a shear lag effect is assumed to take place so that the shear stiffness varies between S and S/2 as a hyperbolic cosine function, which is the function for shear force variation in shear lag problems, 9 as S(x) = S/2.(cosh(x) - cosh(a/aree))/(cosh(a/aref) + S)
(5)
and the average value is found by integration from - a/are f to a/are ~. The
336
D. Zenken
reference value, aref, is chosen to get the best possible correlation with the FE analyses and was here found to be(t~ + 2tr)/5. The energy release rate can now be computed, as well as the stress intensity factor using eqn (6) with r~ = 3 - 4v i (Ref. 1 0 ) G, = 1/16[( rl + 1 )/,u I +(~:,_ + 1)//x2].K~l
(6)
The normalized stress intensity, or geometry factor, is defined as g = K,,/ rm, ~ a
(7)
where r~., is the shear stress in an undisturbed interface.
FINITE E L E M E N T ESTIMATES There are several methods to compute fracture mechanics parameters using the finite element (FE) method. A n obvious way is to perform two consecutive analyses with two slightly different crack lengths and compute the energy release rate via the change in potential energy of the external loads 2 and is used in this paper for validating the calculations. However, the stress intensities for different modes cannot be calculated and only the sum of the energy release rates for the two tips is obtained. Another approach is the so called Irwin's crack closure technique, used by, for example, Raju et al., ~ which is done by a single analysis but with several load cases. The method is to compute the nodal forces needed to close the crack by an increment, usually one element, and thereby finding the energy release rate. Smelser and Gurtin ~2 found that the standard J-integral can be used even for hi-material cracks provided that the bond line is straight. Yau and Wang t3 and Hong and Stern ~4 used contour integrals, of the same type as J, to find the stress intensity factors for bi-materiat bodies. A n important feature here is that the stress intensities for individual modes can be determined. Lin and Mar ~s used a special hybrid element designed for bi-material crack problems. The method used in the present paper, developed by Smelser t6 based on the results in Refs 14 and 15, utilizes the computed crack flank displacement data. It is a very straightforward method that can utilize an ordinary, but relatively fine, mesh and gives the stress intensities for different modes. First, one defines the crack opening as: Au
= u2 le-,,
- u , t e - - .~
(8)
where subscripts 1 and 2 refer to materials 1 and 2 as illustrated in Fig. 3 and u is the nodal displacement, being a complex quantity defined as
Strength of sandwich beams with interface debondings
337
u = u r + iu o. Now, the absolute value of Au can be written as: 16 1 ko tAul=~22 (A,+A_,)~ fr
(9)
where k = koei~ = k t + iklx, A; = 4(1 - v i ) / m i ( i = 1, 2) and r is the radial distance from the node pair to the crack tip. Here, k is defined as the stress intensity K/4r-~. The strength of the singularity at the crack tip 2 = 1 / 2 + i e = 2 0 ei6, with 2 0 2 = 1 / 4 + e 2, and 6 = a r c t a n 2 e . Here, e = 1 / 2 : r . l n y where y is the bi-material constant. ~6 The angle of complex stress intensity, fl, is fl = e . l n r - dP - 6 - ,-r / 2
(10)
where • is the angle of the crack opening displacement, • = arctan Auo/A,r. Using the above equations, every quantity, including the stress intensity factors, can be evaluated for each node pair on the crack flanks. In the vicinity of the crack tip, theoretically K is constant and independent of r, so that a plot of K versus r gives a mean value of the stress intensity. The energy release rate can now be computed from eqn (6) or by the relation given in Ref. 15. The attraction of this method is that it only requires a single FE calculation and that only the nodal displacements have to be computed. It separates the different modes and gives both the stress intensity factors and the energy release rates.
.atorial2 IY /r /
I
ibiateriail
i::?:i:?:?:i:!::i:ii:i:i:?:i:?::!!i:i::i:i:?:?:i:?::i:~. Fig. 3.
Definition of the bi-material crack.
A basic assumption made in the derivation of the above analysis is that the crack surfaces are traction free and hence the computation cannot be performed for a closed crack. The same assumption is valid for J-integral calculations. Another disadvantage with most of the above methods is that they are only derived for isotropic media.
338
D. Zenkert
MATERIALS TESTING
Interfacial fracture toughness The cracked sandwich beam (CSB) test specimen, illustrated in Fig. 4(a), was used to measure the fracture toughness for crack initiation and propagation in the interface. Here, L was 150 mm and the core thickness was 30 mm. T h e specimen was modelled with 240 eight node quadratic plane strain elements, of which 124 elements were in the zone surrounding the crack tip. T h e inner elements had a length of 0"05 mm. T h e nodes closest to the tip were moved to quarter point locations so that the elements would display a square root singularity. The computations were carried out with the F E program ABAQUS. A usable option in the FE code is that gap constraint can be put on the nodes, i.e. to ensure that elements on the crack surfaces do not overlap each other. If contact appears, the nodal forces can be computed. Due to the nature of the problem, such constraints were used on every node on the crack surfaces. The only contact between the crack surfaces was found near the free edge (cf. Fig. 4(a)) and since there is no contact close to the tip the above method for finding stress intensity factors could be used. T h e M o d e I stress intensity was found to be very small, less than 4% of the M o d e II stress intensity. A n attempt to find the influence of the contact
PI
,-- a - i
I........................
,I
/z '-
L
.... (a/
L
~'
PI
t
[tf
...............
--h
] t
2h
i
/
u.
f
i
f
/
i / L
/
f
I
/
/
8h (b)
/
j
~
/ I
/
.,* I L L J
l
Fig. 4. Geometry of test specimens used for fracture toughness tests where the dashed fines illustrate the crack p r o p ~ n directions. (a) For measuring interracial fracture toughness (CSB). (b) For measuring core material fracture toughness ( E ~ ) .
Strength of sandwich beams with inter]ace debondings
339
friction was made by introducing nodal forces, in the points of contact. perpendicular to the pressure corresponding to friction coefficients 0.125, 0.25 and 0.5, in the same manner as done by Gillespie et al. ~7 For the largest frictional forces the decrease in stress intensity was less than 1%. The energy release rate was also computed using the potential energy method by making two consecutive FE analyses with slightly different crack lengths (1 mm). This value was found to agree very well (within 1%) with the value computed from the stress intensity factor. As ABAQUS has the capability to evaluate the J-integral, this was also done for all the FE calculations. The difference between J and the computed G (from stress intensity factors) was less than 1%. Four different specimens were manufactured, with L = 150 mm, described in Table 1 along with the results from the FE analyses. The core materials used had three different densities, 60, 100 and 200 kg/m -~. designated H60, H100 and H200, of Divinycell, a rigid cross-linked PVC foam core manufactured by Diab Barracuda AB. The core materials and aluminium faces were treated as isotropic but an orthotropic GRP laminate was used on one of the specimens. The material data used for the latter were: E~ = 12 GPa, E, = 5 GPa, ~ 2 = 1.92 GPa, and v~2 = 0"3. A debonding was simulated by placing a 0.05 mm thick Teflon film in the interface before applying the adhesive. The aluminium faces were bonded to the core material using a two-component polyurethane, Sadofoss 2K 501, adhesive while the GRP faces were hand lay-up E-glass/polyester laminates in which the resin itself works as the adhesive joint. The beams were tested in an Instron Model 6025 Universal testing machine under a 2 mm/min prescribed displacement. Four specimens of each type were used. The cracks were seen to propagate in a stable way TABLE 1 Results from Analyses and Tests of the lnterfacial Fracture Toughness Specimen (Core Thickness Was 30 mm for all Specimens)
CSB No.
Face
tf (mm)
E c (MPa)
gFE
gilnct (MPa~m)
g ~ e (,~vlPaf-mm)a
1 2 3 4
Aluminium Aluminium GRP Aluminium
2.0 2-0 3-5 3.0
55 100 100 200
1.409 1-450 1-456 1.427
0"22 0-38 0-41 0"76
0"12 0"22 0"22 0-44
aThe K ~ re values are taken from Ref. 3.
340
D. Zenkert
along the interface and the initiation appeared to start at maximum load. As the crack propagated, the load gradually decreased. Hence, the maximum load was used to compute the fracture toughness given in Table 1.
Core material fracture toughness The so called end-notch flexure (ENF) test specimen, illustrated in Fig. 4(b), was used to find the fracture toughness for cracks within the core material subjected to pure shear. The specimens were manufactured by bonding two blocks of core material to each other with a polyester resin but with a 0"05 mm thick Teflon film to simulate the crack. The crack length was 50 mm and h was 30 mm. To avoid bending failure of the specimen, 2 mm faces of GRP material were bonded on each side. Stress intensity factors were obtained from FE analyses. The fracture toughness data included in Table 1 are taken from Ref. 3, where approximate relations between fracture toughness, elastic modulus and strength versus core density are given. The crack propagation angles were found to be approximately 80 ° from the crack direction, as illustrated in Fig. 4(b). This is the predicted angle if one uses the strain energy density criterion introduced by Sih. ~s As seen in Table 1. the fracture toughness of the interface, which corresponds to Mode II crack propagation, is higher than for the core for the E N F specimen. This is due to the fact that the crack may propagate in any direction whereas it is forced to propagate in the interface for the CSB specimen. In Ref. 3, large scatter in fracture toughness data was reported between different blocks of core material, even if the un-notched properties are relatively close. Differences up to 50% in fracture toughness have been reported ~9 for core blocks with the same density, although only very small differences could be found in the elastic and unnotched strength properties. This appeared to be due to the different average cell sizes between different blocks of core material.~9
B E A M ANALYSIS A N D TESTS The beam illustrated in Fig. l(a) was modelled with 408 eight node quadratic plane strain elements of which 124 elements were surrounding each crack tip. The length of the inner elements was 0.05 mm. The nodes close to the tips were moved to quarter point locations. Gap constraints were used on the nodes on the crack surfaces to ensure that no over-
Strength of sandwich beams with interface debondings
341
(a)
(b)
I (c) Fig. 5. Displacements of the beam from FE analysis, with P--1 N (case 1). (a) Deflection of beam magnified 100 times. (b) Enlargement of the zone around the crack magnified 100 times in the x-direction and 500 times in the z-direction. (c) Enlargement of the crack tip zones magnified 100 times.
lapping of the crack flanks occurred. The length between the inner supports was 400 mm and between the outer 900 mm for all cases except where otherwise mentioned. Also here the F E program A B A Q U S was used. The mesh and displacement of the beam (case 1 ) is shown in Fig. 5(a) and an enlargement of the crack in Fig. 5(b). As can be seen in the latter figure, the right part of the crack is partially closed. This is also shown in Fig. 5(c), which shows the displacements in the vicinity of the tips. Hence, the analysis could only be performed on the open left crack tip. A total of 23 different beam geometries was studied. These cases are described in Table 2 in terms of materials, thicknesses and crack lengths. The results are given as normalized stress intensity factors (eqn (7)) from
D. Zenkert
342
TABLE 2
Definition of Cross-Section and Material Properties for the Beam Cases Studied Analytically and Numerically (Computed Data for Each Case from the Analytical Model and FE Calculations in Normalized Stress Intensity Factors. In all Cases, L mwas 400 mm and Loutwas 900 mm, except where otherwise mentioned!
Case
Face
tf (mm)
t~ (mm)
E~ (MPa)
a (ram)
gFE
gan
1
Aluminium Aluminium Aluminium Aluminium GRP Steel Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Alumilfium Alununlum Alunumum Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium
3.0 3.0 3.0 3.0 3.0 3"0 1.0 2.0 5-0 8-0 3-0 3.0 3.0 3-0 3"0 3"0 3.0 3.0 3.0 3-0 3-0 3.0 3.0
30 30 30 30 30 30 30 30 30 30 15 45 60 90 120 30 30 30 30 30 30 30 30
100 55 200 400 100 100 100 100 100 100 100 100 100 I00 100 100 100 100 100 100 100 100 100
30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 7.5 15 22-5 45 60 75 30 30
1.158 1-108 1.203 1-236 1.234 1-063 t.266 1.207 1.029 0-849 t.146 1.104 1-064 1.016 0.987 0.984 1.030 1.088 1-297 1.417 1-514 1.154 1.152
1.135 1.086 1.191 1.238 1.252 1.051 1.294 1-207 1.012 0.892 1.068 1.116 1.090 1.025 0.957 0-820 0~990 1-068 1.271 1-408 1.539 1-135 1.135
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 a 23 b
aLia 350 mm, Lout = 950 mm. bLia 450 mm, Lout= 850 mm. =
=
both the F E analyses a n d the analytical a p p r o a c h . T h e M o d e I stress intensity varied b e t w e e n 1 a n d 10% of the M o d e II for the different cases. This m e a n s that the total stress intensity, K ' - = K : f + K ~ , almost totally consists of the M o d e II c o m p o n e n t . T h e c o n t a c t zone varied b e t w e e n 0.1a a n d 0.5a, which is consistent with the results in Ref. 8. For case 1 in Table 2, an analysis was p e r f o r m e d using the potential energy m e t h o d . T h e total energy release rate f o u n d is the s u m for b o t h crack tips a n d it c o r r e s p o n d e d very closely (less t h a n 2% difference) with twice the value of G for the o p e n crack tip. H e n c e , o n e c a n assume that the energy release rate (G~ + Gn)is t h e s a m e for b o t h tips. T h e J-integral was evaluated for the o p e n tip a n d the difference b e t w e e n G a n d J was less
Strength of sandwich beams with interface debondings n_
343
-0.3
¢. m (D $1
-0.2
-0.1 o u
0
I
I
I
0.5
1.0
1.5
r (O = - n ) (mm) Fig. 6.
Contact pressure distribution in the vicinity of the closed crack tip.
than 1% in all the 23 cases. Friction coefficients of 0.125, 0.25 and 0.5 were used to find the influence of the contact for case 1, by applying nodal forces at the contact zone. The contact pressure for case 1 is shown in Fig. 6, and it exhibits a singular behaviour close to the tip. For the largest friction coefficient the decrease in stress intensity was only 0.5%. Another, rather rough approach was to lock all node pairs on the contact surface which produced a 25% decrease in stress intensity. Hence, the contact pressure at the closed crack tip has very little influence on the open tip. For the closed crack tip, however, one may assume that the friction due to the contact pressure has a larger influence on the stress intensity. A computation of case 1 was also performed for a crack situated at the interface of the lower face. One might assume some minor difference for this case since the face material along the crack has a tensile stress and not compressive as for the cases analysed above, but no difference was found in the stress intensity of the open crack tip. Also, a FE analysis was performed using a mesh with only half as many elements surrounding the crack tips, and less than 0.1% difference was found in the stress intensity. As seen from Table 2 the analytically computed values agree very well with the FE analyses. The faces are relatively thick compared to the core in cases 10 and 11 which explains the poor correspondence. In case 16 the crack is very short, which means that the beam parts produced by the crack are thick compared to their length and hence engineering beam theory is no longer valid. It is seen that the correspondence improves for thin faces and long cracks, which also were the assumptions for the analytical approach. The deflections at locations A - E were found to agree within 5% between FE and analytical computations.
D. Zenkert
344
Prediction of fracture loads As pointed out previously one can assume that the stress intensity is approximately the same at both crack tips, but the closed tip may have a slightly lower intensity due to friction. Even if it is possible to calculate the intensity at the open tip only (cf. Fig. 5), it was found that the total energy release rate was approximately twice that of the value for the open tip. As seen in Fig. 5(b) one might also assume that a major part of the stress intensity of the closed tip is Mode II. By using the strain energy density criterion t8 and comparing with the results in Ref. 3 the following reasoning seems justifiable. The open crack tip would like to propagate into the face material, just as the mid-plane crack is studied in Ref. 3 propagated upwards. But it cannot, since the face material has a very high toughness and the crack would therefore propagate along the interface. Crack initiation would occur as the stress intensity reaches the fracture toughness value found from the CSB tests. The closed crack tip, on the other hand, would want to propagate downward into the core material. Now, crack initiation would occur as the stress intensity reaches the fracture toughness value from the ENF tests. However, this crack tip is closed and subjected to a high contact pressure. If there is no. or little friction, this crack tip would first reach its critical value, but if there is much friction or even locking, the open tip would probably be the most critical. As a result of this, one must perform two predictions for failure of the two crack tips. Assuming no friction at the closed tip, which should produce a conservative estimate, the predicted critical remote shear stresses would be:
ropen--k'int/_ cr - - "~" l l c / ' ~ ~/-~"ff~ " " I " • gand r - c rd°sed___ KHc c o r e /,~--a.g
(11)
and the corresponding loads can be computed. Using the analytical model one can also compute the deflection between the supports for the computed fracture loads. The load bearing capacity of the same beam but without a crack, assuming core shear fracture, can be computed using the ultimate shear stress, -crrn° f l a w , data given by the manufacturer since this is the normal way in designing. Experimental results Four different beam geometries were manufactured, as described in Table 3. The same blocks of core materials as for the interracial fracture toughness tests were used with the exception of beam No. 2 where another H100 block was used with twice the thickness. This block had a slightly lower density which has been accounted for by interpolating the
Strength of sandwich beams with inte~we debondings
345
TABLE 3 Geometry and Materials Used for the Tested Beams (Distance bet~'een the Outer Supports, L,,u,, was 900 mm and bet~een the Inner. Lm. was 400 mm)
Beam No.
Face
tf (ram)
E¢ (MPa)
t,. (ram)
a (ram)
1 2 3 4
Aluminium Aluminium Aluminium GRP
2.0 3"0 3'0 3.8
55 10(I 200 85
30 60 30 30
30 30 60 45
relation between the fracture toughness, elastic modulus, etc., versus density? These debondings were simulated by placing a 0.05 mm thick Teflon film in the interface. The aluminium faces were bonded to the core material using a two-component polyurethane, Sadofoss 2K 501, adhesive while the GRP faces were hand lay-up E-glass/polyester laminates in which the resin acts as the adhesive joint. The beams were put in a four-point bending rig and tested in an Instron Model 6025 Universal testing machine under a 2 mm/min prescribed cross head displacement. The load supports were placed at 400 mm between the inner ones and 900 mm between the outer ones. Four specimens of each type were tested. The load-displacement (between the inner and outer supports) relation was measured throughout the test and was found to be very
Fig. 7.
Photograph of fracture zone in beam test (beam No. 4).
346
D. Zenkert
close to linear up to failure. For all beam types, the closed crack tip (right in Fig. 5(c))initiated and propagated the crack, which was unstable and initiated at peak load. The cracked zone in one of the test beams is shown in Fig. 7, where it is seen how the crack advances into the core material. The initial propagation angle was between 70 and 80 ° for all specimens. This is in good agreement with 83 ° predicted by the strain energy density criterion for pure Mode II loading. The measured peak load per unit width is included in Table 4, along with the ratio of this load to the design fracture load for an uncracked beam. This ratio corresponds to the reduction in load bearing capacity of the beam due to the presence of the crack. The scatter in the test results was small, with the maximum divergence from the average value given in Table 4. It is now clearly seen that the prediction of the fracture loads and displacements for the closed crack tip is in very good agreement with the experimental results. For beam type 2, the predicted fracture load is 20% higher than that measured. A probable reason for this result is that the fracture toughness for the core material is over estimated. TABLE 4
Average Predicted and Experimental Fracture Loads and Displacements at Fracture along with the Predicted Fracture Loads for Beams without Debondings (All Loads in N/mm and all Displacements in mm. A is the Standard Deviation of the Test Results) p,'.w
A (%)
w"".
- r'~"i
~,e~,,,l,,i,~,, p I'~,.,~.,,,.,.,i
w,~""",,,~,,
1
11.7
2 3 4
32-3 ,~7.~ 15.5
4.2 4.6 2.7 5-1
7.8 5"8 8-3 12.7
19.1 63.3 40.0 25-7
11-6 10.1 12.0 22.9
7.1 6-2 7.3 14.0
Beam No.
11.6 38.6 24.4 15.6
p ~,, tz,,,.-
p,.., p ~,, ;z....
19-2 94.5 99.0 49.5
061 0,34 0-28 0-31
DISCUSSION The relatively simple approach used for estimation of the stress intensity factors for the problem is found to give very good results. The agreement with the FE analyses is equally good for all the geometries and materials used. It is seen that the calculated values agree well when the basic assumptions are fulfilled and that the results are decreasingly accurate as they are violated. As seen in Table 2, the limits for the model are approximately that a > td2 and tJtf> 5. For practical purposes, thick faces are seldom used as a prime feature of sandwich constructions is low structural weight. Short cracks will give low stress intensity and will therefore have very little influence on the load bearing capacity. For very
Strength of sandwich beams with interface debondings
347
thin faces with the debonding situated near the face in compression, buckling may occur, but this has not been treated in the present analysis. It is shown that the two crack tips are quite different in that one is opened with a small Mode I stress intensity and that the other is closed with a high contact pressure. However, the FE calculations showed that the stress intensity of the closed crack tip is approximately the same as for the opened tip. It was also found that the opened crack is very little influenced by any frictional forces at the closed tip. How much the stress intensity of the closed tip is influenced by friction has not been established. The stress intensity is the same when the debonding is placed either at the interface close to the tensile or at the compressive face. The FE computations are probably quite accurate although some discretizing errors occur from the modelling. However, the mesh is relatively fine and the three different methods, using crack flank displacement data, the J-integral and potential energy calculation, gave the same results. The J-integral values were also close for different paths of integration. However, when computing the normalized stress intensity, g, the remote shear stress is included, which will not be accurately calculated if the faces are thick. A drawback with most methods, for computing stress intensity factors or energy release rates, is that the available theories are only valid for isotropic materials, with the exception of the compliance method, which has other drawbacks. It seems, however, when using orthotropic faces, that the crack flank displacement method yields fairly good results since it is insensitive to changes in the elastic properties of the stiffer material component. Problems could occur when using anisotropic cores, e.g. balsa wood or honeycomb. The two specimens used for fracture toughness tests gave rather different results. This could be anticipated as crack propagation is forced along the interface, for the CSB specimen, and not in the path of minimum energy, as for the ENF specimen. The values used for the ENF specimen are taken from tests with the same material but not the same blocks as used in the beam tests and may therefore be a bit inaccurate, due to scatter in fracture toughness data. For ordinary design purposes, a guaranteed minimum fracture toughness value must be used to give conservative estimates. A probable reason for the difference between the interfacial fracture toughness using the same core but different face materials, aluminium and GRP (cases CSB 2 and 3), is that the polyester resin used for the laminate which also is the adhesive, generally produces a better bond than the PUR adhesive and could therefore increase the initiation load. The prediction of fracture loads for beams in four-point bending containing simulated interfacial debondings was performed assuming
348
D. Zenkert
fracture initiation at either the open or the closed crack tip. For the open tip, fracture propagation was assumed along the interface and the toughness was taken from tests with the CSB specimen. For the closed tip, it was assumed that the stress intensity was the same as for the open tip and that there was no friction, which should give a conservative estimate and the toughness value was taken from the ENF specimen. The latter fracture mode occurred in the tests and as seen in Table 4, the prediction for the onset of crack propagation was quite accurately predicted both in terms of fracture load and displacement for this mode. The initial crack propagation angle also indicates that the stress intensity consists of almost pure Mode II. With other materials and manufacturing techniques, the amount of friction and the difference in fracture toughness between the two fracture modes may change so that crack initiation could occur at the open tip, as reported in Ref. 1. As this was not the case in this work, the accuracy of this prediction cannot be determined. Anyway, the prediction of this failure mode ought to be equally accurate. The aim of the present paper was to show that interracial debonding in foam core sandwich beams produces a large reduction in load bearing capacity, to quantify this reduction and to develop a model for its estimation. As seen in Table 4, the reduction for the four beam geometries used here is between 0.28 and 0-61. This is compared with the load bearing for transverse forces, assuming core shear fracture, i.e. the difference between the ultimate shear strength of the core and the critical remote shear stress from tests. There are, of course, other failure modes for sandwich structures, such as face yield, face tensile or compressive failure or face wrinkling that, depending on geometry and materials, could be more critical than core shear fracture. In any case, interracial debondings will have a drastic influence on the load bearing capacity of foam core sandwich beams.
CONCLUSIONS • An analytical model for the estimation of stress intensity factors for this problem has been developed that shows good agreement with FE analysis. • The fracture loads for beams with simulated debondings can be accurately predicted using fracture toughness values determined from simple test specimens. • Interface debondings have a drastic influence on the load bearing capacity of foam core sandwich beams.
Strength of sandwich beams with interfacedebondings
349
ACKNOWLEDGEMENTS This study was financially supported by the Swedish Board for Technical Development (STU) and the Nordic Fund for Technology and Industrial Development (NI). The author wishes to thank Dr Hans L. Groth for helpful suggestions during the work. Professor Jan B~icklund, Dr KarlAxel Olsson and Professor Anders F. Blom are gratefully acknowledged for their support. Special thanks are directed to Mr Johan Blomberg for help in making the test specimens and Diab Barracuda AB, Laholm, Sweden, for supplying the core material.
REFERENCES 1. Triantafillou, T. C. & Gibson, L. J., Debonding in foam core sandwich panels, to be published in Materials and Structures. 2. Carlsson, L. A., Sendlein, L. S. & Merry, S. L., Characterization of sheet/ core debonding of composite sandwich beams. Proc. First Int. Conference on Sandwich Constructions, Stockholm, 19-21 June 1989, ed. K. A. Olsson & R. P. Reichard. EMAS Ltd, UK. 3. Zenkert, D., Strength of sandwich beams with mid-plane debondings in the core. Comp. Struct., 15 (1990)279-99. 4. Rice, J. R. & Sih, G. C., Plane problems of cracks in dissimilar media. J. Appl. Mech., Trans. ASME, 32 (1965) 418-23. 5. Erdogan, F., Stress distribution in bonded dissimilar materials with cracks. J. Appl. Mech., 32, Trans. ASME, 87, Series E (1965) 403-10. 6. England, A. H., On stress singularities in linear elasticity. Int. J. Engng Sci., 9 (1971) 571-85. 7. Comninou, M., The interface crack. J. Appl. Mech., 44 (1977) 631-6. 8. Comninou, M., The interface crack in a shear field. J. Appl. Mech., 45 (1978) 287-90. 9. Megson, T. G. H., Aircraft Structures, for Engineering Students. Edward Arnold, London, 1972. 10. Hamoush, S. A. & Ahmad, S. H., Fracture energy release rate of adhesive joints. Int. J. Adhes. Adhesives, 9 (3)(1989) 171-7. 11. Raju, I. S., Crews, J. H. & Aminpour, M. A., Convergence of strain energy release rate components for edge-delaminated composite laminates. Engng Fract. Mech., 30 (1988) 383-96. 12. Smelser, R. E. & Gurtin, M. E., On the J-integral for bi-material bodies. Int. J. Fract., 13 (1977) 382-4. 13. Yau, J. F. & Wang, S. S., An analysis of interface cracks between dissimilar isotropic materials using conservation integrals in elasticity. Engng Fract. Mech., 20 (1984) 423-32. 14. Hong, C.-C. & Stem, M., The computation of stress intensity factors in dissimilar materials. J. Elast., $ ( 1978)21-34. 15. Lin, K. Y. & Mar, J. W., Finite element analysis of stress intensity factors for cracks at a bi-material interface. Int. J. Fract., 12 (1976) 521-31.
350
D. Zenken
16. Smelser, R. E., Evaluation of stress intensity factors for bi-material bodies using crack flank displacement data. Int. J. Fract., 15 (1979) 135-43. 17. Gillespie, J. W., Carlsson, L. A. & Pipes, R. B., Finite element analysis of the end notch flexure specimen for measuring mode II fracture toughness. Comp. Sci. Technol., 27 (1986) 177-97. 18. Sih, G. C., Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fract., 10 (1974) 305-21. 19. Zenkert, D., Poly(vinyl chloride) sandwich core materials: fracture behaviour under mode II and mixed mode conditions. Mater. Sci. Engng, A108 (1989) 233-40.
Strength of Sandwich Beams with Interface Debondings D a n Zenkert Department of Aeronautical Structures and Materials, The Royal Institute of
Technology, S-100 44 Stockholm, Sweden
ABSTRA CT The adhesive joint bonding the faces to the core material in a sandwich structure ensures that the loads are transferred between the components. However, debondings may arise either during the manufacturing process or due to overloading. These will reduce both the stiffness and the load bearing capacity of the structure. In the present paper, debondings in foam core sandwich beams are investigated assuming that cracks in the interface between the face and core are present. Stress intensity factors are found from an analytical model and compared to sohttions from several finite element calculations. Fracture toughness vahtes, determined from simple specimens, are used to predict the fracture loads for beams with simulated debondings subjected to four-point bending.
INTRODUCTION During the past decades, technical development has been heading towards more and more sophisticated and optimized structures in the pursuit of increased performance and reduced structural weight. Here, the sandwich concept has found an increased use due to its high stiffness and strength to weight ratio combined with thermal and acoustic insulation and the opportunity to utilize new materials. Sandwich structures may be designed so that each component is utilized to its ultimate limit. However, this trend makes quality control an even more important issue. It is of great importance not only to find flaws but also to quantitatively determine their influence on structural behaviour. The modus operandi of a sandwich construction is that the faces take up bending moments as tensile and compressive stresses and the core takes up transverse forces as shear stresses. The adhesive joint ensures that loads are transferred between the components. Debonding 331
Composite Structures 0263-8223/91/S03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
D. Zenkert
332
Pi T
pP V
i iii
I~1
!
' Lou t -J Fig. 1. Illustrationof the objectof study,a sandwichbeam in four-pointbendingwitha simulatedinteffacialdebondingsituated betweenan outer and inner support. will reduce the stiffness of the structure but also act as a crack that may initiate and cause premature failure. The aim of the present paper is to find the reduction in load bearing capacity of foam core sandwich beams with simulated debondings and to present methods that can predict this reduction. The object of study is a beam under four-point bending, containing a core/face debonding between one of the outer and inner supports, as illustrated in Fig. 1, The approach is to find fracture toughness data from simple specimens and to compute the stress intensity factors for the problem by means of analytical and FE analyses in order to predict the fracture loads. Related work Triantafillou and Gibson ~ used a simplified method to fmd the energy release rate of sandwich beams with interracial debondings under threepoint bending and used a double shear specimen to determine the fracture toughness. Their results are relatively good but the simplicity of the model gives some restrictions, for example, it does not account for the influence of the crack length for cracks longer than the core thickness. Carlsson et al. 2 presented a specimen for measuring the interracial fracture toughness, the so called cracked sandwich beam (CSB) along with closed form formulae for the calculation of the energy release rate. A solution to the problem of cracks situated at the mid-plane of the core is given by Zenkert. 3 The latter work showed that such cracks drastically reduce the load bearing capacity of foam core sandwich beams and the stress intensities can be found using a relatively simple analysis. The bi-materiai crack In linear elasticity, singular stress and strain fields appear at geometrical and material discontinuities. The singular stress field at the tip of a crack is well known and is widely used for the prediction of crack growth in
Strength of sandwich beams with interface debondings
333
components and structures. For a crack situated at the interface between two dissimilar media, the singular fields differ from those of a crack in an isotropic medium. The interface crack has been treated by Rice and Sih, 4 Erdogan 5 and England, 6 along with methods to compute stress intensity factors. The main difference from the isotropic case is that the strength of the singularity generally is a complex quantity that generates an oscillatory behaviour of the deformation close to the crack tip. C o m n i n o u 7 treated these oscillations by considering a frictionless contact near the crack tip. In a later paper, s she investigated a bi-material crack in a shear field, where the lack of symmetry produces a large contact zone at one of the crack tips. The same problem arises, as will be seen, for an interface crack in a sandwich beam, as the core material is mainly subjected to shear. A N A L Y T I C A L APPROACH The approach to analytically estimate the stress intensity of the crack tips in the problem, illustrated in Fig. 1, is to derive the energy release rate via the change in potential energy of the system due to crack extension. The derivation follows the same manner as in Refs 2 and 3. The assumptions made are that both crack tips exhibit the same stress intensity, that the beam remains symmetric about its centrepoint, point A in Fig. 2(a), and that Mode I stress intensity is negligible. Engineering beam theory including transverse shear deformation is used and hence it is assumed that deflections due to bending and shear can be superimposed. This is true if the faces are thin compared to the core. The energy release rate is computed as: G = -1/n.OU/Oa
(1)
where n is the number of equal crack tips and U is the potential energy of the system, - PAl2, where A is the difference in deflection between the load points B and E (cf. Fig. 2(a)).
Deflection analysis The right side of the beam is divided into four parts as shown in Fig. 2(a). The beam members AB, BC and DE all have bending stiffness D and shear stiffness S, while member CD has stiffnesses D' and S', computed below. The key problem is to treat the cracked part CD. First, the loads are divided into one symmetrical and one anti-symmetrical part as illustrated in Fig. 2(b). For the symmetrical part, the beam will still bend
334
D. Zenkert
IZ
k
l--.-
IP
l
I
P
I
I~. L3 ..I. A
B
P
L2
L I , 1.I.,
C (a)
D
.I E
=
PL21 [ "
] I PL2 + P
p
Pa
(b/ Fig. 2. The right part of the studied beam. (a) Definition and geometry of beam members used in the analytical approach. (b) Superposition of loads intoa symmetrical and an anti-symmetrical part for beam member CD. about its mid-plane since the end planes (points C and D) must remain straight and therefore its stiffness will be D. For the anti-symmetrical load the two beam parts will bend about their individual neutral axes and independently of each other. The load will hence be divided between the beam parts as:
P.p = a3 /3Olo~ + a/Stow P a3/3(1/D~ow+l/D,p)+a(1/S~ow+ 1/S,p)
(2)
where subscripts 'low' and 'up' refer to lower and upper beam parts, respectively. Now, the deflections, rotations and their derivatives with respect to crack length can be found for each point A - E . 3
Cross.sectional properties The bending stiffness for a symmetrical sandwich is computed as:
D = f Ez 2 dA --Eft3/6 + Eftfd2/2 + Ect3/12 JA
(3)
Strength of sandwich beams with interface debondings
335
where d =t~ + tf. As seen in the above expression, the second term is predominant and the two other terms only contribute 1-2% of the total stiffness for real sandwich constructions. The method to find the shear stiffness of a sandwich cross-section is by studying the shear stress distribution. Engineering beam theory is incomplete in the sense that plane cross-sections will remain plane. This has no effect for slender beams but here it must be taken into account. One can use the energy balance equation to compute the average shear angle, ~x:, by using:
½ T:Yx:= ½IA rx:(Z)" Yx:(z) dA, with );x: = Tz/S
(4)
where T. is the shear force at the studied cross-section and L-(z) the shear stress distribution. For a symmetrical sandwich with thin faces and Ef ,> E c, S equals i.~c/d2/to where/z~ is the core shear modulus. For part CD, the presence of the crack divides the cross-section into two beams. Their bending stiffnesses are computed using eqn (3) by integrating over their own neutral axes. A problem arises when estimating the shear stiffnesses of the beam parts. For a long crack the load sharing in eqn (2) is mainly governed by the beam bending stiffnesses and for short cracks mainly by beam shear stiffnesses. Since the shear stiffness for a single face would be/z e. tf/1.2, it would be greater than for the lower beam part but also greater than the shear stiffness for the uncracked beam parts. Therefore, for a short crack, all the load would be carded by the upper face and, hence, the total stiffness of the beam would increase. This is, of course, unrealistic. At the two ends, C and D, the cross-sections must be compatible with beam members BC and DE, respectively. The shear stiffness must then equal S at these two ends. In between, the beams are allowed to move relative to each other and sliding may take place. This sliding is here considered equivalent to a reduction in shear stiffness. The assumption made here, to make a feasible solution even for short cracks, is that the shear stiffness of the two beams will reduce to half that of the uncracked beam. In the transition region between the ends and the mid part, a shear lag effect is assumed to take place so that the shear stiffness varies between S and S/2 as a hyperbolic cosine function, which is the function for shear force variation in shear lag problems, 9 as S(x) = S/2.(cosh(x) - cosh(a/aree))/(cosh(a/aref) + S)
(5)
and the average value is found by integration from - a/are f to a/are ~. The
336
D. Zenken
reference value, aref, is chosen to get the best possible correlation with the FE analyses and was here found to be(t~ + 2tr)/5. The energy release rate can now be computed, as well as the stress intensity factor using eqn (6) with r~ = 3 - 4v i (Ref. 1 0 ) G, = 1/16[( rl + 1 )/,u I +(~:,_ + 1)//x2].K~l
(6)
The normalized stress intensity, or geometry factor, is defined as g = K,,/ rm, ~ a
(7)
where r~., is the shear stress in an undisturbed interface.
FINITE E L E M E N T ESTIMATES There are several methods to compute fracture mechanics parameters using the finite element (FE) method. A n obvious way is to perform two consecutive analyses with two slightly different crack lengths and compute the energy release rate via the change in potential energy of the external loads 2 and is used in this paper for validating the calculations. However, the stress intensities for different modes cannot be calculated and only the sum of the energy release rates for the two tips is obtained. Another approach is the so called Irwin's crack closure technique, used by, for example, Raju et al., ~ which is done by a single analysis but with several load cases. The method is to compute the nodal forces needed to close the crack by an increment, usually one element, and thereby finding the energy release rate. Smelser and Gurtin ~2 found that the standard J-integral can be used even for hi-material cracks provided that the bond line is straight. Yau and Wang t3 and Hong and Stern ~4 used contour integrals, of the same type as J, to find the stress intensity factors for bi-materiat bodies. A n important feature here is that the stress intensities for individual modes can be determined. Lin and Mar ~s used a special hybrid element designed for bi-material crack problems. The method used in the present paper, developed by Smelser t6 based on the results in Refs 14 and 15, utilizes the computed crack flank displacement data. It is a very straightforward method that can utilize an ordinary, but relatively fine, mesh and gives the stress intensities for different modes. First, one defines the crack opening as: Au
= u2 le-,,
- u , t e - - .~
(8)
where subscripts 1 and 2 refer to materials 1 and 2 as illustrated in Fig. 3 and u is the nodal displacement, being a complex quantity defined as
Strength of sandwich beams with interface debondings
337
u = u r + iu o. Now, the absolute value of Au can be written as: 16 1 ko tAul=~22 (A,+A_,)~ fr
(9)
where k = koei~ = k t + iklx, A; = 4(1 - v i ) / m i ( i = 1, 2) and r is the radial distance from the node pair to the crack tip. Here, k is defined as the stress intensity K/4r-~. The strength of the singularity at the crack tip 2 = 1 / 2 + i e = 2 0 ei6, with 2 0 2 = 1 / 4 + e 2, and 6 = a r c t a n 2 e . Here, e = 1 / 2 : r . l n y where y is the bi-material constant. ~6 The angle of complex stress intensity, fl, is fl = e . l n r - dP - 6 - ,-r / 2
(10)
where • is the angle of the crack opening displacement, • = arctan Auo/A,r. Using the above equations, every quantity, including the stress intensity factors, can be evaluated for each node pair on the crack flanks. In the vicinity of the crack tip, theoretically K is constant and independent of r, so that a plot of K versus r gives a mean value of the stress intensity. The energy release rate can now be computed from eqn (6) or by the relation given in Ref. 15. The attraction of this method is that it only requires a single FE calculation and that only the nodal displacements have to be computed. It separates the different modes and gives both the stress intensity factors and the energy release rates.
.atorial2 IY /r /
I
ibiateriail
i::?:i:?:?:i:!::i:ii:i:i:?:i:?::!!i:i::i:i:?:?:i:?::i:~. Fig. 3.
Definition of the bi-material crack.
A basic assumption made in the derivation of the above analysis is that the crack surfaces are traction free and hence the computation cannot be performed for a closed crack. The same assumption is valid for J-integral calculations. Another disadvantage with most of the above methods is that they are only derived for isotropic media.
338
D. Zenkert
MATERIALS TESTING
Interfacial fracture toughness The cracked sandwich beam (CSB) test specimen, illustrated in Fig. 4(a), was used to measure the fracture toughness for crack initiation and propagation in the interface. Here, L was 150 mm and the core thickness was 30 mm. T h e specimen was modelled with 240 eight node quadratic plane strain elements, of which 124 elements were in the zone surrounding the crack tip. T h e inner elements had a length of 0"05 mm. T h e nodes closest to the tip were moved to quarter point locations so that the elements would display a square root singularity. The computations were carried out with the F E program ABAQUS. A usable option in the FE code is that gap constraint can be put on the nodes, i.e. to ensure that elements on the crack surfaces do not overlap each other. If contact appears, the nodal forces can be computed. Due to the nature of the problem, such constraints were used on every node on the crack surfaces. The only contact between the crack surfaces was found near the free edge (cf. Fig. 4(a)) and since there is no contact close to the tip the above method for finding stress intensity factors could be used. T h e M o d e I stress intensity was found to be very small, less than 4% of the M o d e II stress intensity. A n attempt to find the influence of the contact
PI
,-- a - i
I........................
,I
/z '-
L
.... (a/
L
~'
PI
t
[tf
...............
--h
] t
2h
i
/
u.
f
i
f
/
i / L
/
f
I
/
/
8h (b)
/
j
~
/ I
/
.,* I L L J
l
Fig. 4. Geometry of test specimens used for fracture toughness tests where the dashed fines illustrate the crack p r o p ~ n directions. (a) For measuring interracial fracture toughness (CSB). (b) For measuring core material fracture toughness ( E ~ ) .
Strength of sandwich beams with inter]ace debondings
339
friction was made by introducing nodal forces, in the points of contact. perpendicular to the pressure corresponding to friction coefficients 0.125, 0.25 and 0.5, in the same manner as done by Gillespie et al. ~7 For the largest frictional forces the decrease in stress intensity was less than 1%. The energy release rate was also computed using the potential energy method by making two consecutive FE analyses with slightly different crack lengths (1 mm). This value was found to agree very well (within 1%) with the value computed from the stress intensity factor. As ABAQUS has the capability to evaluate the J-integral, this was also done for all the FE calculations. The difference between J and the computed G (from stress intensity factors) was less than 1%. Four different specimens were manufactured, with L = 150 mm, described in Table 1 along with the results from the FE analyses. The core materials used had three different densities, 60, 100 and 200 kg/m -~. designated H60, H100 and H200, of Divinycell, a rigid cross-linked PVC foam core manufactured by Diab Barracuda AB. The core materials and aluminium faces were treated as isotropic but an orthotropic GRP laminate was used on one of the specimens. The material data used for the latter were: E~ = 12 GPa, E, = 5 GPa, ~ 2 = 1.92 GPa, and v~2 = 0"3. A debonding was simulated by placing a 0.05 mm thick Teflon film in the interface before applying the adhesive. The aluminium faces were bonded to the core material using a two-component polyurethane, Sadofoss 2K 501, adhesive while the GRP faces were hand lay-up E-glass/polyester laminates in which the resin itself works as the adhesive joint. The beams were tested in an Instron Model 6025 Universal testing machine under a 2 mm/min prescribed displacement. Four specimens of each type were used. The cracks were seen to propagate in a stable way TABLE 1 Results from Analyses and Tests of the lnterfacial Fracture Toughness Specimen (Core Thickness Was 30 mm for all Specimens)
CSB No.
Face
tf (mm)
E c (MPa)
gFE
gilnct (MPa~m)
g ~ e (,~vlPaf-mm)a
1 2 3 4
Aluminium Aluminium GRP Aluminium
2.0 2-0 3-5 3.0
55 100 100 200
1.409 1-450 1-456 1.427
0"22 0-38 0-41 0"76
0"12 0"22 0"22 0-44
aThe K ~ re values are taken from Ref. 3.
340
D. Zenkert
along the interface and the initiation appeared to start at maximum load. As the crack propagated, the load gradually decreased. Hence, the maximum load was used to compute the fracture toughness given in Table 1.
Core material fracture toughness The so called end-notch flexure (ENF) test specimen, illustrated in Fig. 4(b), was used to find the fracture toughness for cracks within the core material subjected to pure shear. The specimens were manufactured by bonding two blocks of core material to each other with a polyester resin but with a 0"05 mm thick Teflon film to simulate the crack. The crack length was 50 mm and h was 30 mm. To avoid bending failure of the specimen, 2 mm faces of GRP material were bonded on each side. Stress intensity factors were obtained from FE analyses. The fracture toughness data included in Table 1 are taken from Ref. 3, where approximate relations between fracture toughness, elastic modulus and strength versus core density are given. The crack propagation angles were found to be approximately 80 ° from the crack direction, as illustrated in Fig. 4(b). This is the predicted angle if one uses the strain energy density criterion introduced by Sih. ~s As seen in Table 1. the fracture toughness of the interface, which corresponds to Mode II crack propagation, is higher than for the core for the E N F specimen. This is due to the fact that the crack may propagate in any direction whereas it is forced to propagate in the interface for the CSB specimen. In Ref. 3, large scatter in fracture toughness data was reported between different blocks of core material, even if the un-notched properties are relatively close. Differences up to 50% in fracture toughness have been reported ~9 for core blocks with the same density, although only very small differences could be found in the elastic and unnotched strength properties. This appeared to be due to the different average cell sizes between different blocks of core material.~9
B E A M ANALYSIS A N D TESTS The beam illustrated in Fig. l(a) was modelled with 408 eight node quadratic plane strain elements of which 124 elements were surrounding each crack tip. The length of the inner elements was 0.05 mm. The nodes close to the tips were moved to quarter point locations. Gap constraints were used on the nodes on the crack surfaces to ensure that no over-
Strength of sandwich beams with interface debondings
341
(a)
(b)
I (c) Fig. 5. Displacements of the beam from FE analysis, with P--1 N (case 1). (a) Deflection of beam magnified 100 times. (b) Enlargement of the zone around the crack magnified 100 times in the x-direction and 500 times in the z-direction. (c) Enlargement of the crack tip zones magnified 100 times.
lapping of the crack flanks occurred. The length between the inner supports was 400 mm and between the outer 900 mm for all cases except where otherwise mentioned. Also here the F E program A B A Q U S was used. The mesh and displacement of the beam (case 1 ) is shown in Fig. 5(a) and an enlargement of the crack in Fig. 5(b). As can be seen in the latter figure, the right part of the crack is partially closed. This is also shown in Fig. 5(c), which shows the displacements in the vicinity of the tips. Hence, the analysis could only be performed on the open left crack tip. A total of 23 different beam geometries was studied. These cases are described in Table 2 in terms of materials, thicknesses and crack lengths. The results are given as normalized stress intensity factors (eqn (7)) from
D. Zenkert
342
TABLE 2
Definition of Cross-Section and Material Properties for the Beam Cases Studied Analytically and Numerically (Computed Data for Each Case from the Analytical Model and FE Calculations in Normalized Stress Intensity Factors. In all Cases, L mwas 400 mm and Loutwas 900 mm, except where otherwise mentioned!
Case
Face
tf (mm)
t~ (mm)
E~ (MPa)
a (ram)
gFE
gan
1
Aluminium Aluminium Aluminium Aluminium GRP Steel Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Alumilfium Alununlum Alunumum Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium
3.0 3.0 3.0 3.0 3.0 3"0 1.0 2.0 5-0 8-0 3-0 3.0 3.0 3-0 3"0 3"0 3.0 3.0 3.0 3-0 3-0 3.0 3.0
30 30 30 30 30 30 30 30 30 30 15 45 60 90 120 30 30 30 30 30 30 30 30
100 55 200 400 100 100 100 100 100 100 100 100 100 I00 100 100 100 100 100 100 100 100 100
30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 7.5 15 22-5 45 60 75 30 30
1.158 1-108 1.203 1-236 1.234 1-063 t.266 1.207 1.029 0-849 t.146 1.104 1-064 1.016 0.987 0.984 1.030 1.088 1-297 1.417 1-514 1.154 1.152
1.135 1.086 1.191 1.238 1.252 1.051 1.294 1-207 1.012 0.892 1.068 1.116 1.090 1.025 0.957 0-820 0~990 1-068 1.271 1-408 1.539 1-135 1.135
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 a 23 b
aLia 350 mm, Lout = 950 mm. bLia 450 mm, Lout= 850 mm. =
=
both the F E analyses a n d the analytical a p p r o a c h . T h e M o d e I stress intensity varied b e t w e e n 1 a n d 10% of the M o d e II for the different cases. This m e a n s that the total stress intensity, K ' - = K : f + K ~ , almost totally consists of the M o d e II c o m p o n e n t . T h e c o n t a c t zone varied b e t w e e n 0.1a a n d 0.5a, which is consistent with the results in Ref. 8. For case 1 in Table 2, an analysis was p e r f o r m e d using the potential energy m e t h o d . T h e total energy release rate f o u n d is the s u m for b o t h crack tips a n d it c o r r e s p o n d e d very closely (less t h a n 2% difference) with twice the value of G for the o p e n crack tip. H e n c e , o n e c a n assume that the energy release rate (G~ + Gn)is t h e s a m e for b o t h tips. T h e J-integral was evaluated for the o p e n tip a n d the difference b e t w e e n G a n d J was less
Strength of sandwich beams with interface debondings n_
343
-0.3
¢. m (D $1
-0.2
-0.1 o u
0
I
I
I
0.5
1.0
1.5
r (O = - n ) (mm) Fig. 6.
Contact pressure distribution in the vicinity of the closed crack tip.
than 1% in all the 23 cases. Friction coefficients of 0.125, 0.25 and 0.5 were used to find the influence of the contact for case 1, by applying nodal forces at the contact zone. The contact pressure for case 1 is shown in Fig. 6, and it exhibits a singular behaviour close to the tip. For the largest friction coefficient the decrease in stress intensity was only 0.5%. Another, rather rough approach was to lock all node pairs on the contact surface which produced a 25% decrease in stress intensity. Hence, the contact pressure at the closed crack tip has very little influence on the open tip. For the closed crack tip, however, one may assume that the friction due to the contact pressure has a larger influence on the stress intensity. A computation of case 1 was also performed for a crack situated at the interface of the lower face. One might assume some minor difference for this case since the face material along the crack has a tensile stress and not compressive as for the cases analysed above, but no difference was found in the stress intensity of the open crack tip. Also, a FE analysis was performed using a mesh with only half as many elements surrounding the crack tips, and less than 0.1% difference was found in the stress intensity. As seen from Table 2 the analytically computed values agree very well with the FE analyses. The faces are relatively thick compared to the core in cases 10 and 11 which explains the poor correspondence. In case 16 the crack is very short, which means that the beam parts produced by the crack are thick compared to their length and hence engineering beam theory is no longer valid. It is seen that the correspondence improves for thin faces and long cracks, which also were the assumptions for the analytical approach. The deflections at locations A - E were found to agree within 5% between FE and analytical computations.
D. Zenkert
344
Prediction of fracture loads As pointed out previously one can assume that the stress intensity is approximately the same at both crack tips, but the closed tip may have a slightly lower intensity due to friction. Even if it is possible to calculate the intensity at the open tip only (cf. Fig. 5), it was found that the total energy release rate was approximately twice that of the value for the open tip. As seen in Fig. 5(b) one might also assume that a major part of the stress intensity of the closed tip is Mode II. By using the strain energy density criterion t8 and comparing with the results in Ref. 3 the following reasoning seems justifiable. The open crack tip would like to propagate into the face material, just as the mid-plane crack is studied in Ref. 3 propagated upwards. But it cannot, since the face material has a very high toughness and the crack would therefore propagate along the interface. Crack initiation would occur as the stress intensity reaches the fracture toughness value found from the CSB tests. The closed crack tip, on the other hand, would want to propagate downward into the core material. Now, crack initiation would occur as the stress intensity reaches the fracture toughness value from the ENF tests. However, this crack tip is closed and subjected to a high contact pressure. If there is no. or little friction, this crack tip would first reach its critical value, but if there is much friction or even locking, the open tip would probably be the most critical. As a result of this, one must perform two predictions for failure of the two crack tips. Assuming no friction at the closed tip, which should produce a conservative estimate, the predicted critical remote shear stresses would be:
ropen--k'int/_ cr - - "~" l l c / ' ~ ~/-~"ff~ " " I " • gand r - c rd°sed___ KHc c o r e /,~--a.g
(11)
and the corresponding loads can be computed. Using the analytical model one can also compute the deflection between the supports for the computed fracture loads. The load bearing capacity of the same beam but without a crack, assuming core shear fracture, can be computed using the ultimate shear stress, -crrn° f l a w , data given by the manufacturer since this is the normal way in designing. Experimental results Four different beam geometries were manufactured, as described in Table 3. The same blocks of core materials as for the interracial fracture toughness tests were used with the exception of beam No. 2 where another H100 block was used with twice the thickness. This block had a slightly lower density which has been accounted for by interpolating the
Strength of sandwich beams with inte~we debondings
345
TABLE 3 Geometry and Materials Used for the Tested Beams (Distance bet~'een the Outer Supports, L,,u,, was 900 mm and bet~een the Inner. Lm. was 400 mm)
Beam No.
Face
tf (ram)
E¢ (MPa)
t,. (ram)
a (ram)
1 2 3 4
Aluminium Aluminium Aluminium GRP
2.0 3"0 3'0 3.8
55 10(I 200 85
30 60 30 30
30 30 60 45
relation between the fracture toughness, elastic modulus, etc., versus density? These debondings were simulated by placing a 0.05 mm thick Teflon film in the interface. The aluminium faces were bonded to the core material using a two-component polyurethane, Sadofoss 2K 501, adhesive while the GRP faces were hand lay-up E-glass/polyester laminates in which the resin acts as the adhesive joint. The beams were put in a four-point bending rig and tested in an Instron Model 6025 Universal testing machine under a 2 mm/min prescribed cross head displacement. The load supports were placed at 400 mm between the inner ones and 900 mm between the outer ones. Four specimens of each type were tested. The load-displacement (between the inner and outer supports) relation was measured throughout the test and was found to be very
Fig. 7.
Photograph of fracture zone in beam test (beam No. 4).
346
D. Zenkert
close to linear up to failure. For all beam types, the closed crack tip (right in Fig. 5(c))initiated and propagated the crack, which was unstable and initiated at peak load. The cracked zone in one of the test beams is shown in Fig. 7, where it is seen how the crack advances into the core material. The initial propagation angle was between 70 and 80 ° for all specimens. This is in good agreement with 83 ° predicted by the strain energy density criterion for pure Mode II loading. The measured peak load per unit width is included in Table 4, along with the ratio of this load to the design fracture load for an uncracked beam. This ratio corresponds to the reduction in load bearing capacity of the beam due to the presence of the crack. The scatter in the test results was small, with the maximum divergence from the average value given in Table 4. It is now clearly seen that the prediction of the fracture loads and displacements for the closed crack tip is in very good agreement with the experimental results. For beam type 2, the predicted fracture load is 20% higher than that measured. A probable reason for this result is that the fracture toughness for the core material is over estimated. TABLE 4
Average Predicted and Experimental Fracture Loads and Displacements at Fracture along with the Predicted Fracture Loads for Beams without Debondings (All Loads in N/mm and all Displacements in mm. A is the Standard Deviation of the Test Results) p,'.w
A (%)
w"".
- r'~"i
~,e~,,,l,,i,~,, p I'~,.,~.,,,.,.,i
w,~""",,,~,,
1
11.7
2 3 4
32-3 ,~7.~ 15.5
4.2 4.6 2.7 5-1
7.8 5"8 8-3 12.7
19.1 63.3 40.0 25-7
11-6 10.1 12.0 22.9
7.1 6-2 7.3 14.0
Beam No.
11.6 38.6 24.4 15.6
p ~,, tz,,,.-
p,.., p ~,, ;z....
19-2 94.5 99.0 49.5
061 0,34 0-28 0-31
DISCUSSION The relatively simple approach used for estimation of the stress intensity factors for the problem is found to give very good results. The agreement with the FE analyses is equally good for all the geometries and materials used. It is seen that the calculated values agree well when the basic assumptions are fulfilled and that the results are decreasingly accurate as they are violated. As seen in Table 2, the limits for the model are approximately that a > td2 and tJtf> 5. For practical purposes, thick faces are seldom used as a prime feature of sandwich constructions is low structural weight. Short cracks will give low stress intensity and will therefore have very little influence on the load bearing capacity. For very
Strength of sandwich beams with interface debondings
347
thin faces with the debonding situated near the face in compression, buckling may occur, but this has not been treated in the present analysis. It is shown that the two crack tips are quite different in that one is opened with a small Mode I stress intensity and that the other is closed with a high contact pressure. However, the FE calculations showed that the stress intensity of the closed crack tip is approximately the same as for the opened tip. It was also found that the opened crack is very little influenced by any frictional forces at the closed tip. How much the stress intensity of the closed tip is influenced by friction has not been established. The stress intensity is the same when the debonding is placed either at the interface close to the tensile or at the compressive face. The FE computations are probably quite accurate although some discretizing errors occur from the modelling. However, the mesh is relatively fine and the three different methods, using crack flank displacement data, the J-integral and potential energy calculation, gave the same results. The J-integral values were also close for different paths of integration. However, when computing the normalized stress intensity, g, the remote shear stress is included, which will not be accurately calculated if the faces are thick. A drawback with most methods, for computing stress intensity factors or energy release rates, is that the available theories are only valid for isotropic materials, with the exception of the compliance method, which has other drawbacks. It seems, however, when using orthotropic faces, that the crack flank displacement method yields fairly good results since it is insensitive to changes in the elastic properties of the stiffer material component. Problems could occur when using anisotropic cores, e.g. balsa wood or honeycomb. The two specimens used for fracture toughness tests gave rather different results. This could be anticipated as crack propagation is forced along the interface, for the CSB specimen, and not in the path of minimum energy, as for the ENF specimen. The values used for the ENF specimen are taken from tests with the same material but not the same blocks as used in the beam tests and may therefore be a bit inaccurate, due to scatter in fracture toughness data. For ordinary design purposes, a guaranteed minimum fracture toughness value must be used to give conservative estimates. A probable reason for the difference between the interfacial fracture toughness using the same core but different face materials, aluminium and GRP (cases CSB 2 and 3), is that the polyester resin used for the laminate which also is the adhesive, generally produces a better bond than the PUR adhesive and could therefore increase the initiation load. The prediction of fracture loads for beams in four-point bending containing simulated interfacial debondings was performed assuming
348
D. Zenkert
fracture initiation at either the open or the closed crack tip. For the open tip, fracture propagation was assumed along the interface and the toughness was taken from tests with the CSB specimen. For the closed tip, it was assumed that the stress intensity was the same as for the open tip and that there was no friction, which should give a conservative estimate and the toughness value was taken from the ENF specimen. The latter fracture mode occurred in the tests and as seen in Table 4, the prediction for the onset of crack propagation was quite accurately predicted both in terms of fracture load and displacement for this mode. The initial crack propagation angle also indicates that the stress intensity consists of almost pure Mode II. With other materials and manufacturing techniques, the amount of friction and the difference in fracture toughness between the two fracture modes may change so that crack initiation could occur at the open tip, as reported in Ref. 1. As this was not the case in this work, the accuracy of this prediction cannot be determined. Anyway, the prediction of this failure mode ought to be equally accurate. The aim of the present paper was to show that interracial debonding in foam core sandwich beams produces a large reduction in load bearing capacity, to quantify this reduction and to develop a model for its estimation. As seen in Table 4, the reduction for the four beam geometries used here is between 0.28 and 0-61. This is compared with the load bearing for transverse forces, assuming core shear fracture, i.e. the difference between the ultimate shear strength of the core and the critical remote shear stress from tests. There are, of course, other failure modes for sandwich structures, such as face yield, face tensile or compressive failure or face wrinkling that, depending on geometry and materials, could be more critical than core shear fracture. In any case, interracial debondings will have a drastic influence on the load bearing capacity of foam core sandwich beams.
CONCLUSIONS • An analytical model for the estimation of stress intensity factors for this problem has been developed that shows good agreement with FE analysis. • The fracture loads for beams with simulated debondings can be accurately predicted using fracture toughness values determined from simple test specimens. • Interface debondings have a drastic influence on the load bearing capacity of foam core sandwich beams.
Strength of sandwich beams with interfacedebondings
349
ACKNOWLEDGEMENTS This study was financially supported by the Swedish Board for Technical Development (STU) and the Nordic Fund for Technology and Industrial Development (NI). The author wishes to thank Dr Hans L. Groth for helpful suggestions during the work. Professor Jan B~icklund, Dr KarlAxel Olsson and Professor Anders F. Blom are gratefully acknowledged for their support. Special thanks are directed to Mr Johan Blomberg for help in making the test specimens and Diab Barracuda AB, Laholm, Sweden, for supplying the core material.
REFERENCES 1. Triantafillou, T. C. & Gibson, L. J., Debonding in foam core sandwich panels, to be published in Materials and Structures. 2. Carlsson, L. A., Sendlein, L. S. & Merry, S. L., Characterization of sheet/ core debonding of composite sandwich beams. Proc. First Int. Conference on Sandwich Constructions, Stockholm, 19-21 June 1989, ed. K. A. Olsson & R. P. Reichard. EMAS Ltd, UK. 3. Zenkert, D., Strength of sandwich beams with mid-plane debondings in the core. Comp. Struct., 15 (1990)279-99. 4. Rice, J. R. & Sih, G. C., Plane problems of cracks in dissimilar media. J. Appl. Mech., Trans. ASME, 32 (1965) 418-23. 5. Erdogan, F., Stress distribution in bonded dissimilar materials with cracks. J. Appl. Mech., 32, Trans. ASME, 87, Series E (1965) 403-10. 6. England, A. H., On stress singularities in linear elasticity. Int. J. Engng Sci., 9 (1971) 571-85. 7. Comninou, M., The interface crack. J. Appl. Mech., 44 (1977) 631-6. 8. Comninou, M., The interface crack in a shear field. J. Appl. Mech., 45 (1978) 287-90. 9. Megson, T. G. H., Aircraft Structures, for Engineering Students. Edward Arnold, London, 1972. 10. Hamoush, S. A. & Ahmad, S. H., Fracture energy release rate of adhesive joints. Int. J. Adhes. Adhesives, 9 (3)(1989) 171-7. 11. Raju, I. S., Crews, J. H. & Aminpour, M. A., Convergence of strain energy release rate components for edge-delaminated composite laminates. Engng Fract. Mech., 30 (1988) 383-96. 12. Smelser, R. E. & Gurtin, M. E., On the J-integral for bi-material bodies. Int. J. Fract., 13 (1977) 382-4. 13. Yau, J. F. & Wang, S. S., An analysis of interface cracks between dissimilar isotropic materials using conservation integrals in elasticity. Engng Fract. Mech., 20 (1984) 423-32. 14. Hong, C.-C. & Stem, M., The computation of stress intensity factors in dissimilar materials. J. Elast., $ ( 1978)21-34. 15. Lin, K. Y. & Mar, J. W., Finite element analysis of stress intensity factors for cracks at a bi-material interface. Int. J. Fract., 12 (1976) 521-31.
350
D. Zenken
16. Smelser, R. E., Evaluation of stress intensity factors for bi-material bodies using crack flank displacement data. Int. J. Fract., 15 (1979) 135-43. 17. Gillespie, J. W., Carlsson, L. A. & Pipes, R. B., Finite element analysis of the end notch flexure specimen for measuring mode II fracture toughness. Comp. Sci. Technol., 27 (1986) 177-97. 18. Sih, G. C., Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fract., 10 (1974) 305-21. 19. Zenkert, D., Poly(vinyl chloride) sandwich core materials: fracture behaviour under mode II and mixed mode conditions. Mater. Sci. Engng, A108 (1989) 233-40.