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Synergism between layer cracking and delaminations in multidirectional laminates of carbon-fibre-reinforced epoxy
Composites Science and Technology 50 (1994) 343-354
SYNERGISM BETWEEN L A Y E R CRACKING A N D DELAMINATIONS IN MULTIDIRECTIONAL LAMINATES OF C A R B O N - F I B R E - R E I N F O R C E D EPOXY* H. Eggers, H. C. Goetting & H. B/iuml DLR-Institut fiir Strukturmechanik, PF 3267, D-38022 Braunschweig, Germany (Received 11 May 1992; revised version received 1 April 1993; accepted 20 April 1993) Abstract
the intensity of the three-dimensional stress field and the residual strength of the material in the vicinity of the crack tip but also by the spacings and directions of layer cracks in adjacent plies and the residual layer stiffnesses. Often, these aspects have been studied separately, as, for example:
Photographs of different crack modes are shown giving insight into the damage behaviour o f composites fabricated of T300/914C plies of 0.125mm thickness. An increase in the damage tolerance was observed for laminates stacked by quasi-unidirectional [dr2] angle plies. For cross-ply laminates under tension fatigue loading, test results, supplied for the spacings of layer cracks, are used to generate formulae for the residual layer stiffnesses. As an extension to the common energy release rate (ERR) concept an ERR tensor is established. The first invariant of this tensor and its mean values describe the energy release rate and its partitioning into different crack modes, respectively. From the complete set of invariants a first estimation of a crack condition for the delamination growth is proposed which includes the effect of fibre orientation.
--splitting of unidirectional laminates (UD laminates),1'2 --layer cracking in cross-ply laminates, 3-7 --residual stiffnesses for layers or laminates, 8-H and ---delaminations, 12,13 whereas the synergisms between layer cracking and delaminations were considered only to provide understanding of the mechanisms of damage growth. 1¢-17 To the authors' knowledge damage conditions have not yet been published which describe the delamination progress under consideration of layer cracks or fibre orientations. In multidirectional laminates (MD laminates) stacked from UD layers a delamination may 'jump' at the layer crack to the adjacent interface. These 'jumping delaminations' are often observed after fatigue loading. With nondestructive testing it is difficult to measure the correct size and depth of a jumping delamination because delaminations in different interfaces may shadow each other. This and the complexity of the damage modes have so far prevented numerical investigations of measured jumping delaminations. It is proposed to suppress complicated jumping delaminations in MD laminated by using quasi-unidirectional [+2] angle plies (QD plies) instead of U D plies.
Keywords: layer cracking, delamination, carbon-fibrereinforced laminate, damage condition, energy release rate, synergism 1 INTRODUCTION
Any composite structure is damaged locally to some extent. Damage becomes serious when it cannot be detected by inspection and when it may extend to a critical size during service life. In order to design a safe composite structure the components must be 'damage-tolerant'. This means that some unrecognizable local damage will progress so slowly that it can be detected by the next inspection before a critical size is achieved. Owing to the lack of reliable numerical tools for the analysis of damage propagation the aerospace industry qualifies safety-critical structures by extensive testing. The delamination progress is not only influenced by
2 DAMAGE BEHAVIOUR OF MULTIDIRECTIONAL LAMINATES In order to gain insight into the damage behaviour of layered composites, typical damage modes will be depicted which influence the analytical models. Only matrix cracking will be considered well below the rupture strength of the fibres.
* Presented at the AGARD Structures and Material Panel, 25-28 May 1992, Patras, Greece. Composites Science and Technology 0266-3538/94/$06.00 © 1994 Elsevier Science Publishers Ltd. 343
344
H. Eggers, H. C. Goetting, H. Biiuml
In MD laminates various types of cracks appear, which interfere in each other. In general, layer cracks in parallel to the fibres of the UD layers will be generated first at rather low load levels and/or number of load cycles. The layer cracks, formed side by side at narrow sequences, damage the interface and influence the subsequent delamination progress. In micrographs matrix cracks were never observed extending over a fraction of the layer thickness only. Once a micro-crack starts to propagate in the middle of a UD layer, the energy released at the crack tip increases rapidly and achieves a maximum value close to the interface. ~5"~7 Because the crack speed is roughly proportional to the ERR, the crack will split the layer almost instantaneously.4 As a consequence of the traction-free crack surface, the undamaged UD layer cell between two layer cracks is strained at the interfaces by shear and peeling stresses. The stresses with maxima close to the layer crack cause an extreme contraction at the edge of the layer.17 Layer cracks are therefore, occasionally accompanied by microdelaminations which release the peeling stresses significantly, whereas the magnitude of the shear stresses remains virtually unchanged. The mentioned combination of layer cracking and microdelaminations is typical of thin UD layers. A 'tree-shaped cracking' will not be considered, which is characteristic for UD layers of more than about 1 mm thickness. 5 Layer cracking seems to be a dynamic process. The sudden contraction of the layer at the crack causes not only micro-delaminations but also fibre cracking and debonding at the surface of adjacent layers. Micro-delaminations, expected on account of the discussed stress and displacement fields, are rarely found in micrographs (Fig. l(a)). Figure l(b) shows a photograph of the interface of the deplied upper layer. Is The partly broken and debonded fibres are ordered along lines which correspond exactly to cracks in the adjacent layer. These cracks damage the fibre/matrix interface even though a microdelamination might not be detected by micrographs. The local damage at the interface blunts the tip of a delamination crack. The delamination progress is therefore delayed perpendicular to the layer cracks. Because of the stepwise growth of the delamination front from one layer crack to the other, fairly long parts of the delamination front may form rather straight lines parallel to the layer cracks. This statement is supported by Fig. 2(a), which shows an ultrasonic C-scan for a notched MD laminate after tension-compression fatigue loading (T-C fatigue loading) (see Ref. 12, p. 204). In the figure, black dots mark locations where the depth of the delamination below the surface was measured after cutting the specimen. The corresponding numbers separated by a slash relate to the layers debonded by a delamination.
Photograph: H.J. Seifert
Fiber cracks T
a l :~.. ' ' •
• ,
U-.'.'. .]k.
.'o
, " • .. .. . .. . . . " " . .
.
..'
.".'"t90~
" "
". ." "
'
'.
.i ' " - ".'l
"
...___.__..__.---10° Actm = f(D) "-="I
Lay-
20 ~ m
Photograph: KI Schulte TM
Fig. 1. Micro-delaminations induced by layer cracks.
A delamination whose front is following a layer crack for a sufficient length may jump at this crack to the next interface and growth may continue there (jumping delamination). Under compressive loading a delamination tends to jump to adjacent interfaces located more and more closely to the laminate surface, such that local buckling of the debonded sublaminate is enforced. The X-radiograph in Fig. 2(b) shows a typical example for this behaviour. The black area in the middle of the picture is a circular artificial delamination of 11 mm diameter, which is generated by two layers of release foils below the third layer. Under T - C fatigue loading the delamination extends first into the 45° direction. Then it jumps over layer cracks at both sides of the delamination to the second interface between the 0° and 45° layers, where it progresses primarily in the 0° direction. Jumping delaminations are rather common in laminates fabricated from UD layers. An extreme example is given in Fig. 3. The left-hand side of the
345
Layer cracking and delaminations in reinforced epoxy a) U l t r a s o n i c C - s c a n
[0~/45/0~/-45/0,~,...]
r914C
Gr~t.,y
Fig. 3. Grating reflection photograph and a compiled view on the specimen edge. Hole • : 5 mm o', : 350 N / m m 2 N : 96000 LC M e a s u r e d by L. K i r s c h k e lzp2°4
b) X-ray radiograph
Del. O a, N
: 11 m m • 400 N/mrn 2 : 260000 LC
Fig. 2. Delaminations in [02/45/02/-45/0/90]s laminates after tension-compression fatigue loading, i[j, Layer numbers next to the delamination.
picture shows a grating reflection photograph. The distortion of the reflected grating lines is caused by out-of-plane deformations due to the opening of the edge delamination. Close to the edge the wavy lines indicate an alternating curvature of the upper sublaminate. Because of the angled view of the specimen, the crack opening at the edge and the jumps of the delamination from the left- to the right-hand side of the 90° layers are also visible. The right-hand side of Fig. 3 depicts a composite view on to the edge of the delaminated specimen. At a first glance, it is surprising that the delamination jumps at each layer crack to the adjacent interface. In order to understand this phenomenon it is assumed that the delamination is trapped solely in one interface (Fig. 4(a)). The upper sublaminate is about balanced and remains nearly plane by contraction under tension, whereas the lower sublaminate, stiffened by the 90° layers, is extremely unbalanced and curves strongly downwards. The difference of both curvatures generates the crack opening w. If the stiff 90 ° strips generated by the layer cracks are bonded alternately to the upper and lower sublaminate, both sublaminates are unbalanced and curve to the outside (Fig. 4(b)). For the same crack length the crack opening is about twice as large as before and, because the E R R is a quadratic function of the crack opening displacements, about four times more energy is released. Owing to the small misalignments of the fibres, QD layers will not be completely separated by layer cracks, so that jumps of the delamination to adjacent interfaces are avoided. The suppression of the more
H. Eggers, H. C. Goetting, H. Biiuml
346
a)
b) y F
F
F
[0 / -4s / o, / 45 / 02]
Fig. 4. Edge delaminations. (a) Trapped in one interface; (b) alternating between adjacent interfaces.
energy-rich jumping delaminations by QD layers may reduce or even avoid edge delaminations. In order to prove this assumption tension fatigue tests are in preparation for MD laminates stacked by QD and U D layers. The local buckling of UD layers at the surface of MD laminates is accompanied by layer cracks at both sides of the delamination (Fig. 5(a)). These cracks are as long as the delaminated zone and prevent a delamination growth perpendicular to the fibres.
Under compressive loading the UD strip buckles like a beam, which is connected to the specimen body at both beam ends only. At the delamination front a Mode I crack is achieved. The size of the delaminated zone will be reduced by ~30%, when the outer UD layers are replaced by QD layers (Fig. 5(b)). Now the layer cracks are restrained by fibre misalignments and the length of the delaminated zone becomes shorter. Figure 6 shows the ERRs evaluated along the delamination front for
i¸¸/i ~i
(a) E 0.96 E Z
(.9
(/)
Mean stress Material Stack
/
0.64 0.48
/ k
:\ ~ / •
~ 0.16
(b)
c
/ ¢
¢
V "~-.
/
..[_G.~. i./N f
O. -8.
-451019o].
\ /
r--
Fig. 5. X-radiographs after tension-compression fatigue loading. ~2 Stress amplitude, +300N/mm2; no. of loads, 1800000. (a) [02/+45/02/-45/0/90],; (b) [±2/+45/±2/
/ 90]s
/
0.80
0.32
m
-550 N/mm 2 T300 / 914C [ 4- 2 / / 4 5 / 4 - 2 / - 4 5 / 0
-4. 0. 4. Delamination length S [ mm ]
8.
Fig. 6. ERRs at the delamination front. Calculated by Haug (see Ref. 12).
Layer cracking and delaminations in reinforced epoxy the specimen depicted in Fig. 5(b). In contrast to the previous example, only Mode II, III cracks were observed in combination with compressive peeling stresses. In the analysis intersections of the sublaminates were avoided by contact elements.12 The mentioned examples indicate, that MD laminates fabricated of QD instead of UD layers behave in a more damage-tolerant fashion because energy-rich jumping delaminations are suppressed. The residual stiffnesses and strengths of QD and UD layers are similar except for the shear resistance, which is significantly higher for QD layers. Therefore, common MD laminates may be stacked by QD layers without changing the global mechanical behaviour of the component significantly. This advantage is accompanied by minor increase of fabrication costs. It should be noted that it is impossible by numerical analysis to model each layer crack separately in a MD laminate. Generally, the damaged layers are homogenized and simulated by reduced stiffnesses. In this model the delaminations are locked in their own interfaces. This inherent error can be ignored only if the jumping delaminations are suppressed by structural devices. 3 LAYER CRACKING IN MULTIDIRECTIONAL LAMINATES The delamination growth is influenced primarily by the three-dimensional stress state in the vicinity of the crack tip, the residual strength at the interface and the layer cracking, which reduces the mean stiffnesses of the material. For conservative strain limits used for the design of aircraft structures, stiffness reductions caused by layer cracking are rather small. However, to extend the design limit for damage-tolerant structures, realistic residual stiffnesses must be considered. Therefore, the damage behaviour was studied for the inner UD layers of cross-ply laminates, in order to achieve realistic data for the modelling. More detailed informations are given in Ref. 17.
347
Cross-ply laminates were tested under tensile fatigue loading at ambient temperature. The specimen configurations, variations of stacking sequences, load levels and number of load cycles are depicted in Fig. 7. Each specimen was cycled with R = 0.1 under load control in a servo-hydraulic testing machine. After the proposed number of load cycles the specimens were dismounted, impregnated with di-iodinemethane to increase contrast and X-rayed. In the X-ray photographs the cracks seemed to be of higher density at both edges. Therefore, one specimen was cut into half and contrasted and X-rayed again. The new X-ray photograph showed the same crack concentration at the centre line of the specimen as at the edges. The check verified that the visible crack density at the edges is representative for the whole specimen. In the X-ray photographs the coordinates of the cracks were therefore, measured only along both edges. For the measurements a light microscope was used combined with a stepper-motor controlled manipulator, which generated and stored the coordinates automatically. In specimens with one 90° layer the cracks did not appear in the X-ray photographs. In order to obtain a first estimate of the crack distances, a few micrographs were taken after 100000 load cycles on [0,/901/0,] laminates. For the chosen test configurations the measured mean values of the crack distances are listed in Table 1. The measured mean crack distances, plotted for one configuration in Fig. 8, decrease with the number of load cycles until approximately constant values for the characteristic damage state (CDS) are achieved. ~4 For the CDS the crack distances seem to be independent of the load level. This phenomenon is in contradiction to the elasticity theory and cannot be explained solely by material degradations. There are some indications that the change of the slope might be accompanied by a change of the damage mode from successive layer cracking to micro-delamination growth. H The transition point to the CDS is defined by a damage condition described in Section 4. The relations of crack distances A~ and A2 between
F
-'~.--2o,~,,Y
Material : T300 / 914C S t r e n g t h of UD-pJies O'llU=t : 1600, N/ram 2 O/uit : 50. N / m m 2 Stacking sequence
Ply t h i c k n e s s
F
~
~
Load F/Fu, Load frequency Stress ratio log (N)
Fig. 7. Tension fatigue tests on cross-ply laminates.
: [ On ] 9 0 . / 0, ] n = 1,2,4 m = 1...4 : 0.125 m m : : : :
0.5 ... 0.9 5 Hz 0.1 3 ... 6.6
H. Eggers, H. C. Goetting, H. Biiuml
348
Table 1. Measured mean crack distances in cross-ply laminates Stack
Load
No. of LC/1000
F/F. 5
10
20
50
100
200
500
1000
2000
4000
Measured mean crack distances A (mm)
[0,/902/0,]
0-90
0-47 0-64 0"82
0-49
(2.49) (1.16) 0.61 0.85 0.84 0.69 1-25 1-07 0.93
0.50 0-66 0.80
0.22 0-45 0-57 0-76
[o,/9o,/o,]
0-70 (4.81) (1.72) 0-71 (2-68) 1.24 1.04 0-80 1"37 1.10 0.92
0-56 0.62 0"85
[o2/9o,/o2]
0.84
[o,/9o3/o,] [o,/9o,/o,] [o,/9o,/o,] [o,/9ojo,]
0-73 0.73 0-87
0"58 0.74 0"87
0.56 0-75
0-80
[o,/9o/o,] [o,/9o,/o,] [o,/9o~/o,] [o,/9o/o,1 [o2/902/o2] [0J903/o=] [0/9o,/o=]
[o2/9o~/o2] [o2/9o,/o~1 [o2/90~/o2] [o2/9o/o~1
0.69
0.56 0.70
0.51 0.67
0.66
0-54 0-67 0.74
0.47 0.63 0-69
0.19 0.40 0.57 0.63
0.96 1-04
0.65 0.74 0.82
0.47 0.62 0.79
0.43 0.62 0-72
1-62 1.43
0.95 1.01 1-00
0-67 0.79 0.91
0.55 0-72 0.89
0-61 1-63
0-60 1.21
0.50 0.90
0-46 0.71
0.61
0.53
0.83 1.43
0-79 1.20
0.78 1-05
0.77 0.92
0.76 0.81
0-73
0.66
[02/904/02]
IO,/9o2/o,1 [o,/9o2/o,1 [o,/9o,/o,1 [o,/9o,/o,]
0.41 0.54 0.72
0.64 0.76 0.96 0-75
[o2/9o/o~1
0.46 0.56 0.72
0.70 0.50
0.89
0-70 0.50
0-97
0.63 0.85
0.39
0.73
0.67
0.71
Material: T300/914C. Outliers in brackets.
~ 2. E E
i
N
i
I
i
Successive l a y e r cracking
o
0.5. ~u.
•
0 . 7 - ~u.
I
< e--
¢J
Micr~:lelamination
U C
_
growth O. I0 3
.2
.3
.5
104
.2
.3
.5
10s
.2 .3
I I t
.s
1~
.2 .3
.5
No. of load cycles log(N) Fig. 8. Crack distances as a function of number of load cycles. Material, T300/914C; stack, [04/904/04]; stress ratio, 0-1
Layer cracking and delaminations in reinforced epoxy
a)
b)
2.0
max A A(max 0
1.7
X t~
E t5 < r-
349
2' A2 [mm]
!
1.0
•
-
m
•
•
6
•
r-
"-~ 0.5
•
m
•
•
min A A(max 0
w
I""
/:.-:".".!.il ~."
.'.
gO=
40.
max A-- ',, "
E
mean A A(max 0
4 0 n
L" I " "'
L.~-Av-t'~- A2 - ~ [~l A
x. 'x.
&
\ f/f.~ = 5 % \
max A~_L A(max f)--
min A - -
.~ 0.0 10 3 L o a d c y c l e s
e=..~,, [ MPa ]
O
z
min A~ o
~-'- [ o./9o.,/o.]
~tMCM
A, [ram]
n
I
Fig. 9. Characteristic crack lengths measured at the CDS.
three successive cracks were studied. Each couple of adjacent crack distances marks a point in an AIA2 plane. The plane is subdivided into squares of 0.05 mm side length. Within each square the cluster of dots was counted and the cluster sum, plotted as a square column normal to the A1A 2 plane, forms a frequency surface. The surface was smoothed and lines of equal frequencies were generated (Fig. 9(b)). With respect to the CDS, the frequency surface yields some remarkable results (Fig. 9(a)). - - t h e mean crack distance marks the peak of the frequency surface: mmean ~ A ( m a x f )
(1)
- - T h e extreme crack distances are limited by: 0.5. A . . . . ~ A -< 1.5. A . . . .
Figure 10 depicts the mean crack distances in 90 ° layers as a function of the number of loading cycles and the mean strain, e±, normal to the fibres. The crack distances are plotted for four 90 ° layers with a total thickness of 0.5 mm. For each layer less, the crack distances must be reduced by 0.146 mm. The range of the undamaged state (UDS) is rather large. For an aircraft component, designed for safe life, the design strain for static loading is often limited to e ~ 0.3%. The actual service load is lower and is a function of the number of load cycles. Relevant values for load sequences are plotted in Fig. 10, taken from the Transport Aircraft Wing Standard ('I"~IST). 19 Both curves indicate that for a design limit of e-< 0.3% the structure remains undamaged during its service life, justifying the common practice in aircraft
(2)
- - A Gaussian distribution can be used to describe the frequency of the crack distances at the CDS.
Table 2. Approximation for mean crack distances in crossply laminates ACDS
In Table 2 an approximation is given for the mean crack distances in cross-ply laminates. Except for some outliers, enclosed in brackets in Table 1, the approximation is rather accurate (standard deviation 11%). For the CDS, the formula yields the following mean crack distances for different stacking sequences (n -< 4):
Stacking
Acos (mm)
[0,/901/0,] [0./90J0.] [0./903/0.] [0./904/0.]
0-24 0.38 0.53 0-67
Am =za4 + C t , . . ( m - 4 )
for0-
Standard deviation of the approximation error: 11% A cbs = 0.674 mm C,,. = 0.146 mm ell,, = 1.2%
Bo = -0.037 B, = 1.637 B2
=
0.225
Am : Mean crack distance m: No. of 90° layers (layer thickness 0.125 mm) N: No. of load cycles e~: Mean strain in the 90° layers (Design strain e± -< 0.3%) Material: T300/914C.
350
H. Eggers, H. C. Goetting, H. Biiuml Am = A4 + 0.146
(m - 4)
for [ 90°l-layers
1.2
t
o
I •-~
.6
=
.3
:~
.2
.0
Design limit
~~
~
. .~
TWIST 19
~
^ ~~
-- ~'~
New layer crack l
0
.'.~_'.':1? n :
• .rc.~.'...[ I.'. t- .. z I I
1.*- Ac -~ 2. A c - ~
~
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
No. of load cycles log(N)
Fig. 10. Mean crack distancesin cross-plylaminatesafter tensionfatigueloading•Stressratio, 0.1. design to cover the strength degradation caused by dynamic service loading by the choice of a conservative static design strain of e-< 0.3%. Figure 10 allows also an assessment of the effect, when the design strain for static loading increases. If the design limit is elevated to e =0-5% the first cracking under the standardized load sequence TWIST will be expected after about 3 x 1 0 7 load cycles and after about 109 load cycles a mean crack distance of approximately 1.0 mm is achieved in a single 90° layer. The formula in Table 2 describes the mean crack distances in the UD layers of arbitrarily stacked cross-ply laminates. It can be used to determine the residual stiffnesses for UD layers in dependence of the layer thickness, load level and number of load cycles (see Appendix).
K
Fig. 11. Notation for the specific ERR. where G is the E R R at the tip of a growing layer crack; Ac, Fc are the new crack spacing and the cross-section of the layer, respectively; H ikjt is the stiffness tensor of the undamaged material; ni are the components of the unit vector normal to the crack surface; and e,k is the strain tensor (Fig. 11). The denominator of the integrand describes the specific elastic energy of the undamaged layer referred to strain components normal to the crack surface only. The defined specific E R R g is a nondimensional value, because the total energy released by layer cracks is normalized by the corresponding total strain energy for the undamaged layer. In Fig. 12 the specific E R R g is plotted versus the normalized crack spacing A c / A c o s . For the crack spacing at the CDS, maximum values were achieved for g independently of layer thickness and loading. If this holds for different materials and mixed loading conditions, max g(A) can be used to define the crack spacing A c o s independently of tests. In the next step the energy released at the delamination front will be observed more closely. According to the fracture theory the E R R is defined 1
G = lim - - . a,~0 2. Aa
4 GENERALIZED CRACK CONDITIONS
In MD laminates layer cracking and delamination interfere and intensify each other. Generally, only delaminations have been explicitly modelled in numerical analyses, whereas layer cracking is either neglected or comprehended by reduced stiffnesses. This process homogenizes the material with the effect that layer cracking becomes insignificant for delamination growth. This is not true in reality and some ideas will therefore be discussed to overcome this shortcoming. In order to describe the layer cracking, a specific energy release rate g is defined by:
nj. Aui• da
a
[02/90,102]
0.234
Acos=
0.2
~
(5)
0.674
o.1 ~ I
o9
[o2/90/o2] Aces = 0236
I
,½. ~.~----7j. I:l ei 0.5
with: l~I ij = H i k # . ~'i = e i k .
~// ~,
"-.. ~.......
~.7 ~" --. "~ -~ "~'~
I
l g ( A ) = A c . Fc "
0
n k
1.0
1.5
Normalized crack distance
n k . nt
(4)
3.0
Ac/AcDs
Fig. 12. Specific ERR for cracked UD Layers.
Layer cracking and delaminations in reinforced epoxy where Aa and Au~ are the incremental crack extension and the corresponding crack opening displacements, respectively; and o~i . nj are the tractions in front of the crack surface. Numerically, the integration is difficult to execute because the stresses are singular at the crack tip. The crack closure method 2° is therefore, used for finite element solutions, where the integral in eqn (5) is replaced by:
G = ~ S, . AU` ;=, 2. AFc = G, + (31 + (33
(6)
351
of a stress tensor, when the generalized ERR-tensor: G~k = lim arc--,0
s,. a G + & . au, 4. AFc
(9)
is introduced with the invariants.
11 = Gii 12 = (--G/k . Gki + G, . Gkk)/2 /3 =
(10)
Ia,/I
regards the stress tensor, a deviator may be defined by:
As
S~ and U; are the components for the forces and crack opening displacements at the element nodes, referred to the axes of the moving trihedral, and AFc is the area of the new crack surface generated by an elementwise crack extension. In eqn (6) each term of the sum describes the split ERRs Gi for the different crack modes. These values are not invariant like G but depend on the chosen coordinate system. Only the mean values extr. G~ are therefore, used in a damage condition:
DI(G, extr. G/; G o G/c) - 1
(7)
In eqn (7), D~ is a function of the variables G and extr. G~, whereas the material constants G o G/c, separated from the variables by a semicolon, are used to normalize D~. For homogeneous material the mean axes are spanned by the moving trihedral at the crack front (Fig. 13). Problems arise for composites where the crack front is directed by damage in adjacent layers, so that the moving trihedral declines from mean directions. In order to generate the mean axes, nodal forces and displacements are transformed to a rotated coordinate system. The basic vectors, e~(pk), depend on the rotational angles p,, and the extrema of the ERRs. aO,(pj,) ap----~ = 0 (8) define the mean directions. This scheme follows exactly the procedure for generating the mean stresses
G ° = Gq - 31. 11.6q
(11)
with the invariants:
17=0 1D= ½. G ° . G~D
(12)
Indices which appear twice in a product will be summed from 1 . . . . ,3. The solution of the characteristic equation: /~,3 -- I1 . ~ , 2 + 1 2 . ~ , - - / 3 = 0
(13)
defines the mean values A,-= extr. G~ for the split ERRs and: (G,j -
. a,).
= 0
=1
(14)
the mean directions. For a delamination in a layered composite, the crack extends selfsimilar at the interface. The axis ~ = e l therefore, remains unchanged and the components G~(0~= 1, 2) must be set to zero in eqn (9). The ERR-tensor (eqn (9)) is an extension to the common theory. The mean values Z/coincide with the split ERRs Gi and the first invariant is identical to the total ERR G. However, eqn (9) has the advantage that two more invariants are available for generating more complex crack conditions:
02(1/, X,; 1/o X/c) <- S ,
(15)
The crack condition (15) is normalized by the characteristic values 1/o )~ic, such that the normalized crack resistance: SR = 1 (16) //
// ////.7
...... //---
#
-.-
~)--~'7/
J
- 0- - / - - - - / -
/
/
/ t
]/ Fig° 13. M o v i n g t r i h e d r a l at t h e c r a c k f r o n t .
is achieved for homogeneous material. In Fig. 14 an example is given for a damage condition, which depends on the invariants 11 and V~2 only. In order to normalize /1 the critical ERR 11c=0.185N/mm is chosen measured for a Mode I crack in a notched UD laminate stressed perpendicular to the fibres (Ref. 2, p. 211). The values depicted in Fig. 14 were achieved for: --layer cracks in notched UD laminates stressed by
352
H. Eggers, H. C. Goetting, H. Biiuml
Layercracksin [or]l-laminatesafter O ~tensilel~°ading~. ]_
1.
v OoO
\\8 .5
oN
"~"tlu_ ( ~~O ~
~
0°_<=-<90°
Embeddeddelaminationsin • [ _ 2//45/4"2/-45/0/90]rlaminates aftercompressiveloading(Fig.6).2
"
FDS
168 mm -, ~7 Edgedelaminationsin [021190t2/O2]laminatesaftertensile . , j~loading ~
~% ~ . I
.0 .0
o 0.2-I1 / Ilc
Fig. 14. Interaction diagram for matrix cracking. Material, T300/914C; reference ERR 1,¢, 0-185 N/ram.
tension loading off-axis to the fibres (Ref. 2, p. 211), -----embedded delaminations in a MD laminate under compressive loading (Ref 12, pp. 338 and 353), and -----edge delaminations in a cross-ply laminate under tension loading (E. Haug, unpublished data). The scatter of the results is rather large. Nevertheless, a first approximation for a crack condition will be proposed: D3 = 0-2. 1-L+ 1-5. V~2 = SR
Ilc
Sac - 0.8
11
SR=SRc + Cs. ~
"x;~-~\
[-Inhibited
I Qrowth
ACDSi
~-~.
Ai
=
Aic0s, ~tu = 90°,
0.
a~ = 135°,
\/
n= 8
Sac + Cs
Sac O Z
~ Preferred growth
(19)
Cs is a small but positive constant determined by tests; the indices i = l, u mark the lower and upper layer next to the delamination, respectively; and 2n -> 8 is a sufficiently large even integer number. The remaining parameters are depicted in Fig. 15, where for a specific parameter combination the resistance SR
.N
delamination
cos2"(~;- a0
i=l,u
I Cracks below the ----~delamin~tion I ~ . t ~ x tz,,~ !
(18)
where the index C indicate a conservative limit. In eqn (17) the left-hand side is influenced only by the stresses and the characteristic parameter for the homogeneous material. Heterogeneous effects like layer cracking can be considered by a change of the resistance Sac. A first proposal might be:
(17)
The solid line in Fig. 14 marks the least-squares line defined by eqn (17) for SR = 1. For stress points below the dashed line (SR = 0.8), a damage state remains frozen (FDS). Because different test configurations are used to define this limit, a conservative condition
~
for the onset of damage growth may be defined by:
_90° -45°
90° 135°
Slope of the crack front
Fig. 15. Normalized crack resistance according to eqn (19).
270°
Layer cracking and delaminations in reinforced epoxy is plotted. Because the crack resistance is elevated for a crack front, which is about parallel to the fibres, the delamination will grow in preference of the fibre direction. In order to verify eqn (19) tests are necessary, in which not only the delamination front but also the crack spacing in the adjacent layers have to be measured. 5 CONCLUSIONS Formulae for stiffnesses reductions have been developed for multiply cracked U D layers e m b e d d e d in MD laminates. For service loading the reduction is rather small and barely influences the delamination process. Because of the still significant residual stiffnesses more complex damage patterns were prevented like jumps of the delamination to adjacent interfaces. In order to avoid delaminations spread over several interfaces, MD laminates were stacked only from Q D layers ([+2] angle plies). These laminates are considerably more damage-tolerant than those stacked from U D layers only. Because of the limited accuracy of manufacture, the angle of 4-2° may be too small to ensure a misalignment between the U D layers. If the industry will not provide Q D prepregs the angle should be increased to 4-5 ° . In order to generate more invariant crack conditions an E R R tensor has been introduced. The tensor defines not only the c o m m o n E R R and its subdivision for different crack modes but also two more invariants describing E R R s of higher order. On the basis of these invariants an approximation is given for a conservative crack condition for the delamination growth.
REFERENCES 1. Bergmann, H. W. et al., Mechanical properties and damage mechanisms of carbonfibre-reinforced composites, Tension loading. DLR Report No. DFVLR-FB 85-45. German Aerospace Establishment, K61n-Porz, 1985. 2. Eggers, H., Kirschke, L. & Zick, R., Initiation and propagation of cracks in notched UD-laminates of carbonfibre-reinforced epoxy under static testile load. DLR Report No. DFVLR-FB 86-30. German Aerospace Establishment, K61n-Porz, 1986. 3. Peters, P. W. M., Quen'igbildung in 0/90/0 CFKLaminaten. Z. Werkstofftechnik, 18 (1987), 313-22. 4. Bonfiace, L. & Ogin, S. L., Application of Paris equation to the fatigue growth of transverse ply cracks. J. Comp. Mat., 23 (1989) 735-54. 5. Ohira, H., Difference in the failure process of CFRP and GFRP-cross-ply laminates. In Int. Conf. Comp. Mat. 8, June 1991, Hawaii. 6. Nairn, J. A., The strain energy release rate of composite microcracking: A variational approach. J. Comp. Mat., 23 (1989) 1106-29. 7. Han, Y. M. & Hahn, H. T., A simplified analysis of
8. 9. 10. 11.
12.
13.
14. 15.
16.
17.
18.
19.
20.
353
transverse ply cracking in cross-ply laminates. J. Comp. Sci. & Technol., 31 (1988) 165-77. Talreja, R., Transverse cracking and stiffness reduction in composite laminates. J. Comp. Mat., 19 (1985) 355-75. Laws, N. & Dvorak, G. J., Progressive transverse cracking in composite laminates. J. Comp. Mat., 22 (1988) 900-16. Nuismer, R. J. & Tan, S. C., Constitutive relations of a cracked composite lamina. J. Comp. Mat., 22 (1988) 306-21. Eggers, H. & Goetting, H. C., Damage mechanisms and constitutive relations for cracked UD-layers in cross-ply laminates. In Proc. Int. Conf. Spacecraft Structures and Mechanical Testing, 24-26 April 1991, Noordwijk, The Netherlands. Bergmann, H. W. et al., Mechanical properties and damage mechanics of carbonfibre-reinforced composites, Compression loading. DLR Report No. DFVLRFB 88-41. German Aerospace Establishment, K61nPorz, 1988. Schiitze, R. & Goetting, H. C., Einflul3 der Randdelaminationsbehinderung in multidirektionalen CFK-Laminaten unter Zugschwellbelastung. ZFW, 10 (1986) 267-72. Reifsnider, K. L. & Talug, A., Analysis of fatigue damage in composite laminates. Int. J. Fatigue, Jan. (1980) 3-11. Wang, A. S. D., Kishore, N. N. & Li, C. A., Crack development in graphite-epoxy cross-ply laminates under uniaxial tension. Comp. Sci. & Technol., 24 (1985) 1-31. Kim, T. W., Kim, H. J. & Im, S., Delamination crack originated from transverse cracking in cross-ply composite laminates under extension. Int. J. Solid Structures, 27 (1991) 1925-41. Bergmann, H. W. & Block, J. et al., Fracture/damage mechanics of composites, Static and fatigue properties. DLR Report No. DLR-Mitt. 92-03. German Aerospace Establishment, K61n-Porz, 1993. Schulte, K. & Baron, CH., Load and failure analyses of CFRP-laminates by means of electrical resistivity measurements. Comp. Sci. & Technol., 36 (1989) 63-76. Schiitz, H. et aL, Standardisierter Einzelflugbelastungsablauf fiir Schwingfestigkeitsversuche an Tragfl/ichenbauteilen von Transportflugzeugen. LBF Report No. FB-106 (1973). O'Brien, T. K., Mixed mode strain energy release rate effects on edge delaminations of composites. ASTM STP 836. American Society for Testing and Materials, Philadelphia, PA, 1984, pp. 125-42.
APPENDIX Formulae for the residual stiffnesses of a cross-ply laminate damaged by layer cracks were developed by Nuismer and Tan. 1° These formulae were simplified for a single layer strained at the interface only, in order to generate the following constitutive equations for a multiply cracked U D layer (Ref. 17, p. 250): oi = ( C 0 + tanh(yk). C,/-Cks~ (ej - % . AT) Yk Ck, / " for k -- 1 and i, j = 1-3, 5 or k = 6 and i, ] = 4, 6
H. Eggers, H. C. Goetting, H. Biiuml
354 with:
point of the material; A, D are the crack spacing and the layer thickness, respectively; and : A
a / 3 . C4,
C is the stiffness matrix for the undamaged layer; AT is the temperature difference in relation to the glass
ql~ = [ OxxOyyOzzOyzOxzOxy ]
is the stress vector with similar arrangements for the strain vector ~ and for the temperature coefficients a. The x and z axes point in fibre direction and normal to the layer, respectively.
SYNERGISM BETWEEN L A Y E R CRACKING A N D DELAMINATIONS IN MULTIDIRECTIONAL LAMINATES OF C A R B O N - F I B R E - R E I N F O R C E D EPOXY* H. Eggers, H. C. Goetting & H. B/iuml DLR-Institut fiir Strukturmechanik, PF 3267, D-38022 Braunschweig, Germany (Received 11 May 1992; revised version received 1 April 1993; accepted 20 April 1993) Abstract
the intensity of the three-dimensional stress field and the residual strength of the material in the vicinity of the crack tip but also by the spacings and directions of layer cracks in adjacent plies and the residual layer stiffnesses. Often, these aspects have been studied separately, as, for example:
Photographs of different crack modes are shown giving insight into the damage behaviour o f composites fabricated of T300/914C plies of 0.125mm thickness. An increase in the damage tolerance was observed for laminates stacked by quasi-unidirectional [dr2] angle plies. For cross-ply laminates under tension fatigue loading, test results, supplied for the spacings of layer cracks, are used to generate formulae for the residual layer stiffnesses. As an extension to the common energy release rate (ERR) concept an ERR tensor is established. The first invariant of this tensor and its mean values describe the energy release rate and its partitioning into different crack modes, respectively. From the complete set of invariants a first estimation of a crack condition for the delamination growth is proposed which includes the effect of fibre orientation.
--splitting of unidirectional laminates (UD laminates),1'2 --layer cracking in cross-ply laminates, 3-7 --residual stiffnesses for layers or laminates, 8-H and ---delaminations, 12,13 whereas the synergisms between layer cracking and delaminations were considered only to provide understanding of the mechanisms of damage growth. 1¢-17 To the authors' knowledge damage conditions have not yet been published which describe the delamination progress under consideration of layer cracks or fibre orientations. In multidirectional laminates (MD laminates) stacked from UD layers a delamination may 'jump' at the layer crack to the adjacent interface. These 'jumping delaminations' are often observed after fatigue loading. With nondestructive testing it is difficult to measure the correct size and depth of a jumping delamination because delaminations in different interfaces may shadow each other. This and the complexity of the damage modes have so far prevented numerical investigations of measured jumping delaminations. It is proposed to suppress complicated jumping delaminations in MD laminated by using quasi-unidirectional [+2] angle plies (QD plies) instead of U D plies.
Keywords: layer cracking, delamination, carbon-fibrereinforced laminate, damage condition, energy release rate, synergism 1 INTRODUCTION
Any composite structure is damaged locally to some extent. Damage becomes serious when it cannot be detected by inspection and when it may extend to a critical size during service life. In order to design a safe composite structure the components must be 'damage-tolerant'. This means that some unrecognizable local damage will progress so slowly that it can be detected by the next inspection before a critical size is achieved. Owing to the lack of reliable numerical tools for the analysis of damage propagation the aerospace industry qualifies safety-critical structures by extensive testing. The delamination progress is not only influenced by
2 DAMAGE BEHAVIOUR OF MULTIDIRECTIONAL LAMINATES In order to gain insight into the damage behaviour of layered composites, typical damage modes will be depicted which influence the analytical models. Only matrix cracking will be considered well below the rupture strength of the fibres.
* Presented at the AGARD Structures and Material Panel, 25-28 May 1992, Patras, Greece. Composites Science and Technology 0266-3538/94/$06.00 © 1994 Elsevier Science Publishers Ltd. 343
344
H. Eggers, H. C. Goetting, H. Biiuml
In MD laminates various types of cracks appear, which interfere in each other. In general, layer cracks in parallel to the fibres of the UD layers will be generated first at rather low load levels and/or number of load cycles. The layer cracks, formed side by side at narrow sequences, damage the interface and influence the subsequent delamination progress. In micrographs matrix cracks were never observed extending over a fraction of the layer thickness only. Once a micro-crack starts to propagate in the middle of a UD layer, the energy released at the crack tip increases rapidly and achieves a maximum value close to the interface. ~5"~7 Because the crack speed is roughly proportional to the ERR, the crack will split the layer almost instantaneously.4 As a consequence of the traction-free crack surface, the undamaged UD layer cell between two layer cracks is strained at the interfaces by shear and peeling stresses. The stresses with maxima close to the layer crack cause an extreme contraction at the edge of the layer.17 Layer cracks are therefore, occasionally accompanied by microdelaminations which release the peeling stresses significantly, whereas the magnitude of the shear stresses remains virtually unchanged. The mentioned combination of layer cracking and microdelaminations is typical of thin UD layers. A 'tree-shaped cracking' will not be considered, which is characteristic for UD layers of more than about 1 mm thickness. 5 Layer cracking seems to be a dynamic process. The sudden contraction of the layer at the crack causes not only micro-delaminations but also fibre cracking and debonding at the surface of adjacent layers. Micro-delaminations, expected on account of the discussed stress and displacement fields, are rarely found in micrographs (Fig. l(a)). Figure l(b) shows a photograph of the interface of the deplied upper layer. Is The partly broken and debonded fibres are ordered along lines which correspond exactly to cracks in the adjacent layer. These cracks damage the fibre/matrix interface even though a microdelamination might not be detected by micrographs. The local damage at the interface blunts the tip of a delamination crack. The delamination progress is therefore delayed perpendicular to the layer cracks. Because of the stepwise growth of the delamination front from one layer crack to the other, fairly long parts of the delamination front may form rather straight lines parallel to the layer cracks. This statement is supported by Fig. 2(a), which shows an ultrasonic C-scan for a notched MD laminate after tension-compression fatigue loading (T-C fatigue loading) (see Ref. 12, p. 204). In the figure, black dots mark locations where the depth of the delamination below the surface was measured after cutting the specimen. The corresponding numbers separated by a slash relate to the layers debonded by a delamination.
Photograph: H.J. Seifert
Fiber cracks T
a l :~.. ' ' •
• ,
U-.'.'. .]k.
.'o
, " • .. .. . .. . . . " " . .
.
..'
.".'"t90~
" "
". ." "
'
'.
.i ' " - ".'l
"
...___.__..__.---10° Actm = f(D) "-="I
Lay-
20 ~ m
Photograph: KI Schulte TM
Fig. 1. Micro-delaminations induced by layer cracks.
A delamination whose front is following a layer crack for a sufficient length may jump at this crack to the next interface and growth may continue there (jumping delamination). Under compressive loading a delamination tends to jump to adjacent interfaces located more and more closely to the laminate surface, such that local buckling of the debonded sublaminate is enforced. The X-radiograph in Fig. 2(b) shows a typical example for this behaviour. The black area in the middle of the picture is a circular artificial delamination of 11 mm diameter, which is generated by two layers of release foils below the third layer. Under T - C fatigue loading the delamination extends first into the 45° direction. Then it jumps over layer cracks at both sides of the delamination to the second interface between the 0° and 45° layers, where it progresses primarily in the 0° direction. Jumping delaminations are rather common in laminates fabricated from UD layers. An extreme example is given in Fig. 3. The left-hand side of the
345
Layer cracking and delaminations in reinforced epoxy a) U l t r a s o n i c C - s c a n
[0~/45/0~/-45/0,~,...]
r914C
Gr~t.,y
Fig. 3. Grating reflection photograph and a compiled view on the specimen edge. Hole • : 5 mm o', : 350 N / m m 2 N : 96000 LC M e a s u r e d by L. K i r s c h k e lzp2°4
b) X-ray radiograph
Del. O a, N
: 11 m m • 400 N/mrn 2 : 260000 LC
Fig. 2. Delaminations in [02/45/02/-45/0/90]s laminates after tension-compression fatigue loading, i[j, Layer numbers next to the delamination.
picture shows a grating reflection photograph. The distortion of the reflected grating lines is caused by out-of-plane deformations due to the opening of the edge delamination. Close to the edge the wavy lines indicate an alternating curvature of the upper sublaminate. Because of the angled view of the specimen, the crack opening at the edge and the jumps of the delamination from the left- to the right-hand side of the 90° layers are also visible. The right-hand side of Fig. 3 depicts a composite view on to the edge of the delaminated specimen. At a first glance, it is surprising that the delamination jumps at each layer crack to the adjacent interface. In order to understand this phenomenon it is assumed that the delamination is trapped solely in one interface (Fig. 4(a)). The upper sublaminate is about balanced and remains nearly plane by contraction under tension, whereas the lower sublaminate, stiffened by the 90° layers, is extremely unbalanced and curves strongly downwards. The difference of both curvatures generates the crack opening w. If the stiff 90 ° strips generated by the layer cracks are bonded alternately to the upper and lower sublaminate, both sublaminates are unbalanced and curve to the outside (Fig. 4(b)). For the same crack length the crack opening is about twice as large as before and, because the E R R is a quadratic function of the crack opening displacements, about four times more energy is released. Owing to the small misalignments of the fibres, QD layers will not be completely separated by layer cracks, so that jumps of the delamination to adjacent interfaces are avoided. The suppression of the more
H. Eggers, H. C. Goetting, H. Biiuml
346
a)
b) y F
F
F
[0 / -4s / o, / 45 / 02]
Fig. 4. Edge delaminations. (a) Trapped in one interface; (b) alternating between adjacent interfaces.
energy-rich jumping delaminations by QD layers may reduce or even avoid edge delaminations. In order to prove this assumption tension fatigue tests are in preparation for MD laminates stacked by QD and U D layers. The local buckling of UD layers at the surface of MD laminates is accompanied by layer cracks at both sides of the delamination (Fig. 5(a)). These cracks are as long as the delaminated zone and prevent a delamination growth perpendicular to the fibres.
Under compressive loading the UD strip buckles like a beam, which is connected to the specimen body at both beam ends only. At the delamination front a Mode I crack is achieved. The size of the delaminated zone will be reduced by ~30%, when the outer UD layers are replaced by QD layers (Fig. 5(b)). Now the layer cracks are restrained by fibre misalignments and the length of the delaminated zone becomes shorter. Figure 6 shows the ERRs evaluated along the delamination front for
i¸¸/i ~i
(a) E 0.96 E Z
(.9
(/)
Mean stress Material Stack
/
0.64 0.48
/ k
:\ ~ / •
~ 0.16
(b)
c
/ ¢
¢
V "~-.
/
..[_G.~. i./N f
O. -8.
-451019o].
\ /
r--
Fig. 5. X-radiographs after tension-compression fatigue loading. ~2 Stress amplitude, +300N/mm2; no. of loads, 1800000. (a) [02/+45/02/-45/0/90],; (b) [±2/+45/±2/
/ 90]s
/
0.80
0.32
m
-550 N/mm 2 T300 / 914C [ 4- 2 / / 4 5 / 4 - 2 / - 4 5 / 0
-4. 0. 4. Delamination length S [ mm ]
8.
Fig. 6. ERRs at the delamination front. Calculated by Haug (see Ref. 12).
Layer cracking and delaminations in reinforced epoxy the specimen depicted in Fig. 5(b). In contrast to the previous example, only Mode II, III cracks were observed in combination with compressive peeling stresses. In the analysis intersections of the sublaminates were avoided by contact elements.12 The mentioned examples indicate, that MD laminates fabricated of QD instead of UD layers behave in a more damage-tolerant fashion because energy-rich jumping delaminations are suppressed. The residual stiffnesses and strengths of QD and UD layers are similar except for the shear resistance, which is significantly higher for QD layers. Therefore, common MD laminates may be stacked by QD layers without changing the global mechanical behaviour of the component significantly. This advantage is accompanied by minor increase of fabrication costs. It should be noted that it is impossible by numerical analysis to model each layer crack separately in a MD laminate. Generally, the damaged layers are homogenized and simulated by reduced stiffnesses. In this model the delaminations are locked in their own interfaces. This inherent error can be ignored only if the jumping delaminations are suppressed by structural devices. 3 LAYER CRACKING IN MULTIDIRECTIONAL LAMINATES The delamination growth is influenced primarily by the three-dimensional stress state in the vicinity of the crack tip, the residual strength at the interface and the layer cracking, which reduces the mean stiffnesses of the material. For conservative strain limits used for the design of aircraft structures, stiffness reductions caused by layer cracking are rather small. However, to extend the design limit for damage-tolerant structures, realistic residual stiffnesses must be considered. Therefore, the damage behaviour was studied for the inner UD layers of cross-ply laminates, in order to achieve realistic data for the modelling. More detailed informations are given in Ref. 17.
347
Cross-ply laminates were tested under tensile fatigue loading at ambient temperature. The specimen configurations, variations of stacking sequences, load levels and number of load cycles are depicted in Fig. 7. Each specimen was cycled with R = 0.1 under load control in a servo-hydraulic testing machine. After the proposed number of load cycles the specimens were dismounted, impregnated with di-iodinemethane to increase contrast and X-rayed. In the X-ray photographs the cracks seemed to be of higher density at both edges. Therefore, one specimen was cut into half and contrasted and X-rayed again. The new X-ray photograph showed the same crack concentration at the centre line of the specimen as at the edges. The check verified that the visible crack density at the edges is representative for the whole specimen. In the X-ray photographs the coordinates of the cracks were therefore, measured only along both edges. For the measurements a light microscope was used combined with a stepper-motor controlled manipulator, which generated and stored the coordinates automatically. In specimens with one 90° layer the cracks did not appear in the X-ray photographs. In order to obtain a first estimate of the crack distances, a few micrographs were taken after 100000 load cycles on [0,/901/0,] laminates. For the chosen test configurations the measured mean values of the crack distances are listed in Table 1. The measured mean crack distances, plotted for one configuration in Fig. 8, decrease with the number of load cycles until approximately constant values for the characteristic damage state (CDS) are achieved. ~4 For the CDS the crack distances seem to be independent of the load level. This phenomenon is in contradiction to the elasticity theory and cannot be explained solely by material degradations. There are some indications that the change of the slope might be accompanied by a change of the damage mode from successive layer cracking to micro-delamination growth. H The transition point to the CDS is defined by a damage condition described in Section 4. The relations of crack distances A~ and A2 between
F
-'~.--2o,~,,Y
Material : T300 / 914C S t r e n g t h of UD-pJies O'llU=t : 1600, N/ram 2 O/uit : 50. N / m m 2 Stacking sequence
Ply t h i c k n e s s
F
~
~
Load F/Fu, Load frequency Stress ratio log (N)
Fig. 7. Tension fatigue tests on cross-ply laminates.
: [ On ] 9 0 . / 0, ] n = 1,2,4 m = 1...4 : 0.125 m m : : : :
0.5 ... 0.9 5 Hz 0.1 3 ... 6.6
H. Eggers, H. C. Goetting, H. Biiuml
348
Table 1. Measured mean crack distances in cross-ply laminates Stack
Load
No. of LC/1000
F/F. 5
10
20
50
100
200
500
1000
2000
4000
Measured mean crack distances A (mm)
[0,/902/0,]
0-90
0-47 0-64 0"82
0-49
(2.49) (1.16) 0.61 0.85 0.84 0.69 1-25 1-07 0.93
0.50 0-66 0.80
0.22 0-45 0-57 0-76
[o,/9o,/o,]
0-70 (4.81) (1.72) 0-71 (2-68) 1.24 1.04 0-80 1"37 1.10 0.92
0-56 0.62 0"85
[o2/9o,/o2]
0.84
[o,/9o3/o,] [o,/9o,/o,] [o,/9o,/o,] [o,/9ojo,]
0-73 0.73 0-87
0"58 0.74 0"87
0.56 0-75
0-80
[o,/9o/o,] [o,/9o,/o,] [o,/9o~/o,] [o,/9o/o,1 [o2/902/o2] [0J903/o=] [0/9o,/o=]
[o2/9o~/o2] [o2/9o,/o~1 [o2/90~/o2] [o2/9o/o~1
0.69
0.56 0.70
0.51 0.67
0.66
0-54 0-67 0.74
0.47 0.63 0-69
0.19 0.40 0.57 0.63
0.96 1-04
0.65 0.74 0.82
0.47 0.62 0.79
0.43 0.62 0-72
1-62 1.43
0.95 1.01 1-00
0-67 0.79 0.91
0.55 0-72 0.89
0-61 1-63
0-60 1.21
0.50 0.90
0-46 0.71
0.61
0.53
0.83 1.43
0-79 1.20
0.78 1-05
0.77 0.92
0.76 0.81
0-73
0.66
[02/904/02]
IO,/9o2/o,1 [o,/9o2/o,1 [o,/9o,/o,1 [o,/9o,/o,]
0.41 0.54 0.72
0.64 0.76 0.96 0-75
[o2/9o/o~1
0.46 0.56 0.72
0.70 0.50
0.89
0-70 0.50
0-97
0.63 0.85
0.39
0.73
0.67
0.71
Material: T300/914C. Outliers in brackets.
~ 2. E E
i
N
i
I
i
Successive l a y e r cracking
o
0.5. ~u.
•
0 . 7 - ~u.
I
< e--
¢J
Micr~:lelamination
U C
_
growth O. I0 3
.2
.3
.5
104
.2
.3
.5
10s
.2 .3
I I t
.s
1~
.2 .3
.5
No. of load cycles log(N) Fig. 8. Crack distances as a function of number of load cycles. Material, T300/914C; stack, [04/904/04]; stress ratio, 0-1
Layer cracking and delaminations in reinforced epoxy
a)
b)
2.0
max A A(max 0
1.7
X t~
E t5 < r-
349
2' A2 [mm]
!
1.0
•
-
m
•
•
6
•
r-
"-~ 0.5
•
m
•
•
min A A(max 0
w
I""
/:.-:".".!.il ~."
.'.
gO=
40.
max A-- ',, "
E
mean A A(max 0
4 0 n
L" I " "'
L.~-Av-t'~- A2 - ~ [~l A
x. 'x.
&
\ f/f.~ = 5 % \
max A~_L A(max f)--
min A - -
.~ 0.0 10 3 L o a d c y c l e s
e=..~,, [ MPa ]
O
z
min A~ o
~-'- [ o./9o.,/o.]
~tMCM
A, [ram]
n
I
Fig. 9. Characteristic crack lengths measured at the CDS.
three successive cracks were studied. Each couple of adjacent crack distances marks a point in an AIA2 plane. The plane is subdivided into squares of 0.05 mm side length. Within each square the cluster of dots was counted and the cluster sum, plotted as a square column normal to the A1A 2 plane, forms a frequency surface. The surface was smoothed and lines of equal frequencies were generated (Fig. 9(b)). With respect to the CDS, the frequency surface yields some remarkable results (Fig. 9(a)). - - t h e mean crack distance marks the peak of the frequency surface: mmean ~ A ( m a x f )
(1)
- - T h e extreme crack distances are limited by: 0.5. A . . . . ~ A -< 1.5. A . . . .
Figure 10 depicts the mean crack distances in 90 ° layers as a function of the number of loading cycles and the mean strain, e±, normal to the fibres. The crack distances are plotted for four 90 ° layers with a total thickness of 0.5 mm. For each layer less, the crack distances must be reduced by 0.146 mm. The range of the undamaged state (UDS) is rather large. For an aircraft component, designed for safe life, the design strain for static loading is often limited to e ~ 0.3%. The actual service load is lower and is a function of the number of load cycles. Relevant values for load sequences are plotted in Fig. 10, taken from the Transport Aircraft Wing Standard ('I"~IST). 19 Both curves indicate that for a design limit of e-< 0.3% the structure remains undamaged during its service life, justifying the common practice in aircraft
(2)
- - A Gaussian distribution can be used to describe the frequency of the crack distances at the CDS.
Table 2. Approximation for mean crack distances in crossply laminates ACDS
In Table 2 an approximation is given for the mean crack distances in cross-ply laminates. Except for some outliers, enclosed in brackets in Table 1, the approximation is rather accurate (standard deviation 11%). For the CDS, the formula yields the following mean crack distances for different stacking sequences (n -< 4):
Stacking
Acos (mm)
[0,/901/0,] [0./90J0.] [0./903/0.] [0./904/0.]
0-24 0.38 0.53 0-67
Am =za4 + C t , . . ( m - 4 )
for0-
Standard deviation of the approximation error: 11% A cbs = 0.674 mm C,,. = 0.146 mm ell,, = 1.2%
Bo = -0.037 B, = 1.637 B2
=
0.225
Am : Mean crack distance m: No. of 90° layers (layer thickness 0.125 mm) N: No. of load cycles e~: Mean strain in the 90° layers (Design strain e± -< 0.3%) Material: T300/914C.
350
H. Eggers, H. C. Goetting, H. Biiuml Am = A4 + 0.146
(m - 4)
for [ 90°l-layers
1.2
t
o
I •-~
.6
=
.3
:~
.2
.0
Design limit
~~
~
. .~
TWIST 19
~
^ ~~
-- ~'~
New layer crack l
0
.'.~_'.':1? n :
• .rc.~.'...[ I.'. t- .. z I I
1.*- Ac -~ 2. A c - ~
~
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
No. of load cycles log(N)
Fig. 10. Mean crack distancesin cross-plylaminatesafter tensionfatigueloading•Stressratio, 0.1. design to cover the strength degradation caused by dynamic service loading by the choice of a conservative static design strain of e-< 0.3%. Figure 10 allows also an assessment of the effect, when the design strain for static loading increases. If the design limit is elevated to e =0-5% the first cracking under the standardized load sequence TWIST will be expected after about 3 x 1 0 7 load cycles and after about 109 load cycles a mean crack distance of approximately 1.0 mm is achieved in a single 90° layer. The formula in Table 2 describes the mean crack distances in the UD layers of arbitrarily stacked cross-ply laminates. It can be used to determine the residual stiffnesses for UD layers in dependence of the layer thickness, load level and number of load cycles (see Appendix).
K
Fig. 11. Notation for the specific ERR. where G is the E R R at the tip of a growing layer crack; Ac, Fc are the new crack spacing and the cross-section of the layer, respectively; H ikjt is the stiffness tensor of the undamaged material; ni are the components of the unit vector normal to the crack surface; and e,k is the strain tensor (Fig. 11). The denominator of the integrand describes the specific elastic energy of the undamaged layer referred to strain components normal to the crack surface only. The defined specific E R R g is a nondimensional value, because the total energy released by layer cracks is normalized by the corresponding total strain energy for the undamaged layer. In Fig. 12 the specific E R R g is plotted versus the normalized crack spacing A c / A c o s . For the crack spacing at the CDS, maximum values were achieved for g independently of layer thickness and loading. If this holds for different materials and mixed loading conditions, max g(A) can be used to define the crack spacing A c o s independently of tests. In the next step the energy released at the delamination front will be observed more closely. According to the fracture theory the E R R is defined 1
G = lim - - . a,~0 2. Aa
4 GENERALIZED CRACK CONDITIONS
In MD laminates layer cracking and delamination interfere and intensify each other. Generally, only delaminations have been explicitly modelled in numerical analyses, whereas layer cracking is either neglected or comprehended by reduced stiffnesses. This process homogenizes the material with the effect that layer cracking becomes insignificant for delamination growth. This is not true in reality and some ideas will therefore be discussed to overcome this shortcoming. In order to describe the layer cracking, a specific energy release rate g is defined by:
nj. Aui• da
a
[02/90,102]
0.234
Acos=
0.2
~
(5)
0.674
o.1 ~ I
o9
[o2/90/o2] Aces = 0236
I
,½. ~.~----7j. I:l ei 0.5
with: l~I ij = H i k # . ~'i = e i k .
~// ~,
"-.. ~.......
~.7 ~" --. "~ -~ "~'~
I
l g ( A ) = A c . Fc "
0
n k
1.0
1.5
Normalized crack distance
n k . nt
(4)
3.0
Ac/AcDs
Fig. 12. Specific ERR for cracked UD Layers.
Layer cracking and delaminations in reinforced epoxy where Aa and Au~ are the incremental crack extension and the corresponding crack opening displacements, respectively; and o~i . nj are the tractions in front of the crack surface. Numerically, the integration is difficult to execute because the stresses are singular at the crack tip. The crack closure method 2° is therefore, used for finite element solutions, where the integral in eqn (5) is replaced by:
G = ~ S, . AU` ;=, 2. AFc = G, + (31 + (33
(6)
351
of a stress tensor, when the generalized ERR-tensor: G~k = lim arc--,0
s,. a G + & . au, 4. AFc
(9)
is introduced with the invariants.
11 = Gii 12 = (--G/k . Gki + G, . Gkk)/2 /3 =
(10)
Ia,/I
regards the stress tensor, a deviator may be defined by:
As
S~ and U; are the components for the forces and crack opening displacements at the element nodes, referred to the axes of the moving trihedral, and AFc is the area of the new crack surface generated by an elementwise crack extension. In eqn (6) each term of the sum describes the split ERRs Gi for the different crack modes. These values are not invariant like G but depend on the chosen coordinate system. Only the mean values extr. G~ are therefore, used in a damage condition:
DI(G, extr. G/; G o G/c) - 1
(7)
In eqn (7), D~ is a function of the variables G and extr. G~, whereas the material constants G o G/c, separated from the variables by a semicolon, are used to normalize D~. For homogeneous material the mean axes are spanned by the moving trihedral at the crack front (Fig. 13). Problems arise for composites where the crack front is directed by damage in adjacent layers, so that the moving trihedral declines from mean directions. In order to generate the mean axes, nodal forces and displacements are transformed to a rotated coordinate system. The basic vectors, e~(pk), depend on the rotational angles p,, and the extrema of the ERRs. aO,(pj,) ap----~ = 0 (8) define the mean directions. This scheme follows exactly the procedure for generating the mean stresses
G ° = Gq - 31. 11.6q
(11)
with the invariants:
17=0 1D= ½. G ° . G~D
(12)
Indices which appear twice in a product will be summed from 1 . . . . ,3. The solution of the characteristic equation: /~,3 -- I1 . ~ , 2 + 1 2 . ~ , - - / 3 = 0
(13)
defines the mean values A,-= extr. G~ for the split ERRs and: (G,j -
. a,).
= 0
=1
(14)
the mean directions. For a delamination in a layered composite, the crack extends selfsimilar at the interface. The axis ~ = e l therefore, remains unchanged and the components G~(0~= 1, 2) must be set to zero in eqn (9). The ERR-tensor (eqn (9)) is an extension to the common theory. The mean values Z/coincide with the split ERRs Gi and the first invariant is identical to the total ERR G. However, eqn (9) has the advantage that two more invariants are available for generating more complex crack conditions:
02(1/, X,; 1/o X/c) <- S ,
(15)
The crack condition (15) is normalized by the characteristic values 1/o )~ic, such that the normalized crack resistance: SR = 1 (16) //
// ////.7
...... //---
#
-.-
~)--~'7/
J
- 0- - / - - - - / -
/
/
/ t
]/ Fig° 13. M o v i n g t r i h e d r a l at t h e c r a c k f r o n t .
is achieved for homogeneous material. In Fig. 14 an example is given for a damage condition, which depends on the invariants 11 and V~2 only. In order to normalize /1 the critical ERR 11c=0.185N/mm is chosen measured for a Mode I crack in a notched UD laminate stressed perpendicular to the fibres (Ref. 2, p. 211). The values depicted in Fig. 14 were achieved for: --layer cracks in notched UD laminates stressed by
352
H. Eggers, H. C. Goetting, H. Biiuml
Layercracksin [or]l-laminatesafter O ~tensilel~°ading~. ]_
1.
v OoO
\\8 .5
oN
"~"tlu_ ( ~~O ~
~
0°_<=-<90°
Embeddeddelaminationsin • [ _ 2//45/4"2/-45/0/90]rlaminates aftercompressiveloading(Fig.6).2
"
FDS
168 mm -, ~7 Edgedelaminationsin [021190t2/O2]laminatesaftertensile . , j~loading ~
~% ~ . I
.0 .0
o 0.2-I1 / Ilc
Fig. 14. Interaction diagram for matrix cracking. Material, T300/914C; reference ERR 1,¢, 0-185 N/ram.
tension loading off-axis to the fibres (Ref. 2, p. 211), -----embedded delaminations in a MD laminate under compressive loading (Ref 12, pp. 338 and 353), and -----edge delaminations in a cross-ply laminate under tension loading (E. Haug, unpublished data). The scatter of the results is rather large. Nevertheless, a first approximation for a crack condition will be proposed: D3 = 0-2. 1-L+ 1-5. V~2 = SR
Ilc
Sac - 0.8
11
SR=SRc + Cs. ~
"x;~-~\
[-Inhibited
I Qrowth
ACDSi
~-~.
Ai
=
Aic0s, ~tu = 90°,
0.
a~ = 135°,
\/
n= 8
Sac + Cs
Sac O Z
~ Preferred growth
(19)
Cs is a small but positive constant determined by tests; the indices i = l, u mark the lower and upper layer next to the delamination, respectively; and 2n -> 8 is a sufficiently large even integer number. The remaining parameters are depicted in Fig. 15, where for a specific parameter combination the resistance SR
.N
delamination
cos2"(~;- a0
i=l,u
I Cracks below the ----~delamin~tion I ~ . t ~ x tz,,~ !
(18)
where the index C indicate a conservative limit. In eqn (17) the left-hand side is influenced only by the stresses and the characteristic parameter for the homogeneous material. Heterogeneous effects like layer cracking can be considered by a change of the resistance Sac. A first proposal might be:
(17)
The solid line in Fig. 14 marks the least-squares line defined by eqn (17) for SR = 1. For stress points below the dashed line (SR = 0.8), a damage state remains frozen (FDS). Because different test configurations are used to define this limit, a conservative condition
~
for the onset of damage growth may be defined by:
_90° -45°
90° 135°
Slope of the crack front
Fig. 15. Normalized crack resistance according to eqn (19).
270°
Layer cracking and delaminations in reinforced epoxy is plotted. Because the crack resistance is elevated for a crack front, which is about parallel to the fibres, the delamination will grow in preference of the fibre direction. In order to verify eqn (19) tests are necessary, in which not only the delamination front but also the crack spacing in the adjacent layers have to be measured. 5 CONCLUSIONS Formulae for stiffnesses reductions have been developed for multiply cracked U D layers e m b e d d e d in MD laminates. For service loading the reduction is rather small and barely influences the delamination process. Because of the still significant residual stiffnesses more complex damage patterns were prevented like jumps of the delamination to adjacent interfaces. In order to avoid delaminations spread over several interfaces, MD laminates were stacked only from Q D layers ([+2] angle plies). These laminates are considerably more damage-tolerant than those stacked from U D layers only. Because of the limited accuracy of manufacture, the angle of 4-2° may be too small to ensure a misalignment between the U D layers. If the industry will not provide Q D prepregs the angle should be increased to 4-5 ° . In order to generate more invariant crack conditions an E R R tensor has been introduced. The tensor defines not only the c o m m o n E R R and its subdivision for different crack modes but also two more invariants describing E R R s of higher order. On the basis of these invariants an approximation is given for a conservative crack condition for the delamination growth.
REFERENCES 1. Bergmann, H. W. et al., Mechanical properties and damage mechanisms of carbonfibre-reinforced composites, Tension loading. DLR Report No. DFVLR-FB 85-45. German Aerospace Establishment, K61n-Porz, 1985. 2. Eggers, H., Kirschke, L. & Zick, R., Initiation and propagation of cracks in notched UD-laminates of carbonfibre-reinforced epoxy under static testile load. DLR Report No. DFVLR-FB 86-30. German Aerospace Establishment, K61n-Porz, 1986. 3. Peters, P. W. M., Quen'igbildung in 0/90/0 CFKLaminaten. Z. Werkstofftechnik, 18 (1987), 313-22. 4. Bonfiace, L. & Ogin, S. L., Application of Paris equation to the fatigue growth of transverse ply cracks. J. Comp. Mat., 23 (1989) 735-54. 5. Ohira, H., Difference in the failure process of CFRP and GFRP-cross-ply laminates. In Int. Conf. Comp. Mat. 8, June 1991, Hawaii. 6. Nairn, J. A., The strain energy release rate of composite microcracking: A variational approach. J. Comp. Mat., 23 (1989) 1106-29. 7. Han, Y. M. & Hahn, H. T., A simplified analysis of
8. 9. 10. 11.
12.
13.
14. 15.
16.
17.
18.
19.
20.
353
transverse ply cracking in cross-ply laminates. J. Comp. Sci. & Technol., 31 (1988) 165-77. Talreja, R., Transverse cracking and stiffness reduction in composite laminates. J. Comp. Mat., 19 (1985) 355-75. Laws, N. & Dvorak, G. J., Progressive transverse cracking in composite laminates. J. Comp. Mat., 22 (1988) 900-16. Nuismer, R. J. & Tan, S. C., Constitutive relations of a cracked composite lamina. J. Comp. Mat., 22 (1988) 306-21. Eggers, H. & Goetting, H. C., Damage mechanisms and constitutive relations for cracked UD-layers in cross-ply laminates. In Proc. Int. Conf. Spacecraft Structures and Mechanical Testing, 24-26 April 1991, Noordwijk, The Netherlands. Bergmann, H. W. et al., Mechanical properties and damage mechanics of carbonfibre-reinforced composites, Compression loading. DLR Report No. DFVLRFB 88-41. German Aerospace Establishment, K61nPorz, 1988. Schiitze, R. & Goetting, H. C., Einflul3 der Randdelaminationsbehinderung in multidirektionalen CFK-Laminaten unter Zugschwellbelastung. ZFW, 10 (1986) 267-72. Reifsnider, K. L. & Talug, A., Analysis of fatigue damage in composite laminates. Int. J. Fatigue, Jan. (1980) 3-11. Wang, A. S. D., Kishore, N. N. & Li, C. A., Crack development in graphite-epoxy cross-ply laminates under uniaxial tension. Comp. Sci. & Technol., 24 (1985) 1-31. Kim, T. W., Kim, H. J. & Im, S., Delamination crack originated from transverse cracking in cross-ply composite laminates under extension. Int. J. Solid Structures, 27 (1991) 1925-41. Bergmann, H. W. & Block, J. et al., Fracture/damage mechanics of composites, Static and fatigue properties. DLR Report No. DLR-Mitt. 92-03. German Aerospace Establishment, K61n-Porz, 1993. Schulte, K. & Baron, CH., Load and failure analyses of CFRP-laminates by means of electrical resistivity measurements. Comp. Sci. & Technol., 36 (1989) 63-76. Schiitz, H. et aL, Standardisierter Einzelflugbelastungsablauf fiir Schwingfestigkeitsversuche an Tragfl/ichenbauteilen von Transportflugzeugen. LBF Report No. FB-106 (1973). O'Brien, T. K., Mixed mode strain energy release rate effects on edge delaminations of composites. ASTM STP 836. American Society for Testing and Materials, Philadelphia, PA, 1984, pp. 125-42.
APPENDIX Formulae for the residual stiffnesses of a cross-ply laminate damaged by layer cracks were developed by Nuismer and Tan. 1° These formulae were simplified for a single layer strained at the interface only, in order to generate the following constitutive equations for a multiply cracked U D layer (Ref. 17, p. 250): oi = ( C 0 + tanh(yk). C,/-Cks~ (ej - % . AT) Yk Ck, / " for k -- 1 and i, j = 1-3, 5 or k = 6 and i, ] = 4, 6
H. Eggers, H. C. Goetting, H. Biiuml
354 with:
point of the material; A, D are the crack spacing and the layer thickness, respectively; and : A
a / 3 . C4,
C is the stiffness matrix for the undamaged layer; AT is the temperature difference in relation to the glass
ql~ = [ OxxOyyOzzOyzOxzOxy ]
is the stress vector with similar arrangements for the strain vector ~ and for the temperature coefficients a. The x and z axes point in fibre direction and normal to the layer, respectively.