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Development of transverse cracking in cross-ply laminates during fatigue tests
Composites Science and Technology 61 (2001) 1711–1721 www.elsevier.com/locate/compscitech
Development of transverse cracking in cross-ply laminates during fatigue tests J.-M. Berthelot*, A. El Mahi, J.-F. Le Corre Acoustics and Mechanics Institute, Composites and Mechanical Structures Group, Universite´ du Maine, 72085 Le Mans cedex 9, France Received 12 December 2000; accepted 8 May 2001
Abstract The purpose of this paper is to analyse the progressive development of transverse cracking in cross-ply laminates subjected to uniaxial fatigue tests. First, a finite-element analysis shows how an analytical model which considers a progressive shear through the thickness of the 0 layers can be used for evaluating the stress field in the case of progressive cracking across the specimen width. The paper then presents a simulation of the progression of the transverse cracking, based on statistical distributions along the specimen length for crack initiation and across the specimen width for crack growth across the specimen width. The simulation process is applied to glass-fibre/epoxy cross-ply laminates and the results obtained are compared with experimental results. # 2001 Published by Elsevier Science Ltd. All rights reserved. Keywords: A. Cross-ply laminates; B. Fatigue; C. Transverse cracking
1. Introduction The initiation and the development of transverse cracking in cross-ply laminates subjected to static uniaxial loading have been extensively studied in the literature (see Refs. [1–5] for a review). A generalized model has been developed [3] for evaluating the two-dimensional stress distribution in transverse cracked cross-ply laminates. This generalized model has then been extended [4] to evaluate the stress distribution in cross-ply laminates subjected to static tensile loading and containing transverse cracks and delamination initiated from the transverse crack tips. These models have been combined [5] with statistical distributions of strengths in the 90 layers in such a way to evaluate the progression of transverse cracking and delamination in cross-ply laminates subjected to static uniaxial loading. In the case of cross-ply laminates subjected to uniaxial tensile fatigue loading, experimental investigations [6– 10] show that the predominant damage mechanisms are the initiation and propagation of transverse cracking in the 90 layers. These damage mechanisms are observed [9] to occur during the first 80% of the lifetime of the * Corresponding author. E-mail address: [email protected] (J.-M. Berthelot).
test specimen. Then, this first stage is followed by longitudinal matrix crack splitting, local delamination at intersections of transverse and longitudinal cracks, and ultimately by fibre fracture of the 0 layers carrying the load applied to laminates. Because of the prevalence of transverse cracking and facility to observe it in transparent laminates, the development of the transverse cracking has been extensively studied in the case of glass-fibre laminates. For example an extensive experimental investigation has been implemented by Boniface and Ogin [11], in which they observed crack growth in a transparent [0/90/0] glass-fibre/epoxy laminate subjected to tensile fatigue loading. The behaviour of laminates subjected to fatigue loading is very complex, and in spite of various works on this subject, the mechanisms which are induced are not well understood. A fracture mechanics approach has been proposed by Ogin et al. [11–16], based on the concept of the stress-intensity factor at the tip of a growing transverse crack in the 90 layers. First, the stress-intensity factor is evaluated as a function of the stress in the transverse layer acting on the crack, using a shear-lag model for the load transfer between the 0 and 90 layers. Next, it is assumed that the growth rate of transverse cracks can be related to the stress-intensity factor by a Paris relationship. This model has been applied to predict
0266-3538/01/$ - see front matter # 2001 Published by Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00068-9
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the stiffness reduction during fatigue of a [0/90]s glassfibre laminate [12–14] and to fatigue growth in glassfibre and carbon-fibre laminates [11,15,16]. The strainenergy release rate has been used rather than the stress intensity factor by Wang et al. [17] and Guild et al. [18]. In a different way, various authors have attempted to develop models for predicting the fatigue life of crossply laminates in terms of static and fatigue properties of the laminate layers. Lee et al. [19] proposed a deterministic model to predict the fatigue life which was divided into two parts: an initial life associated with the development of the transverse cracking until a saturation state and next a residual life. The fatigue life until the saturation state was determined from the curves of stress vs. number of cycles measured on the 90 layers and the Young’s modulus-crack density evaluated on the basis of experimental investigations. Next, the residual life was estimated, without considering the type of damages induced, from the stress-cycles curves of the 0 layers and considering the average stress applied on them according to the residual modulus of the 90 layers after the saturation state. In a similar way, the stress-cycles number curves of off-axis layers were used by Gao [20] to develop a model for fatigue life of laminates. A statistical model for the evaluation of the residual strength and the fatigue life has been applied by Diao et al. [21–23] to describe the fatigue behaviour of cross-ply laminates. In this model it is assumed that the decreasing rate of the residual strength of the 0 layers, under cyclic loading of constant amplitude, is inversely proportional to a certain power of the residual strength itself according to the relation introduced by Halpin et al. [24]: 0 A max dr0 ¼ 1 ; ð1Þ dN 0
2. Progressive transverse cracking in fatigue tests 2.1. Introduction During fatigue tests, transverse matrix cracks in 90 layers initiate at the free edges of test specimen, then grow slowly across the specimen width as a function of cycle number (Fig. 1a). For a given cycle number, the matrix cracking pattern is characterized (Fig. 1b) by the edge of initiation (edge 1 or 2), the position xi along the length of the laminate and the length yi of each crack. So the analysis of the stress and strain fields is very complex, involving the resolution of a three-dimensional problem. Finite element analysis has been implemented in the case of regularly spaced cracks and some results have been considered previously in Ref. [26]. This analysis is completed in this section, considering the generalized model developed in Ref. [3] and drawing how this model can be used to evaluate the progression of transverse cracking during fatigue tests. 2.2. The elementary cell The elementary cell considered in the finite-element analysis is shown in Fig. 2. The geometric parameters of the cell are t0, the thickness of the 0 layers, t90, the thickness of the 90 layer and w, the specimen width. The transverse cracking is characterized by the distance 2l between two consecutive cracks and the length lc of the crack across the specimen width. Then the following reduced parameters can be introduced: the stacking parameter ¼
t0 ; t90
ð3Þ
r
the aspect ratio of transverse cracking where r0 is the residual strength of the 0 layers at N 0 cycles, max the maximum fatigue stress applied to the 0 layers and A and are two dimensionless constants derived from experimental investigations implemented on the 0 layers. The maximum fatigue stress applied to the 0 layers is related to the maximum fatigue stress max applied to laminate by the expression: 0 ¼ hðNÞmax ; max
ð2Þ
where h(N) is the stress redistribution function as a function of cycle number, according to the damage mechanisms induced in the laminate. Next, a statistic analysis of residual strength of the 0 layers is considered and related to the distribution of the static strengths. Finally, the stress redistribution function is derived in the case of progressive transverse cracking by using the shear lag analysis developed by Lee and Daniel [25].
a¼
l t90
;
ð4Þ
and the reduced crack length across the specimen width c¼
lc : w
ð5Þ
The laminates investigated are E-glass fibre/epoxy cross-ply laminates, with the following layer properties: EL ¼ 42:3 GPa; ET ¼ 13:2 GPa; GLT ¼ 4:4 GPa; GTT 0 ¼ 3:5 GPa; LT ¼ 0:30; ð6Þ relative to the layer directions ðL; T; T 0 Þ. Finite element analysis has been carried out for different values of the stacking parameter (= 1/3, 1, 2) and the transverse
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Fig. 1. Typical transverse crack distribution.
Fig. 2. Elementary cell.
cracking parameters (aspect ratio: 14a410 and reduced length across the specimen width: c=0.25, 0.5, 0.75). Boundary conditions corresponding to a tensile state were obtained by prescribing longitudinal displacements of the nodes of the extreme transverse planes of the elementary cell. 2.3. Displacement field Fig. 3. Deformed shape of the elementary cell.
An example of the deformed shape of the elementary cell derived by finite element analysis is shown in Fig. 3, in the case of [0/902/0] laminates and for a transverse cracking state characterized by the value a ¼ 2:5 of the aspect ratio and the value c=0.5 of the reduced crack length across the specimen width. A quarter of the elementary cell has been suppressed for a better visualisation of the cracked part. The longitudinal displacement, related to the prescribed displacement uc, is reported through the laminate thickness in Fig. 4. The variations of the longitudinal displacement are given in the crack plane (x=l) and at a distance l/4 from the crack plane (x=3l/4), in the cracked band of the cell (y=w/4, Fig. 4a), and in the uncracked band (y=3w/4, Fig. 4b). The finite element investigation implemented for the different values of the stacking parameter and the
transverse cracking parameters shows that the longitudinal displacements can be evaluated by the longitudinal displacements considered in the generalized approach when the crack spans the whole width of the laminate [3]. So, in the cracked band, the longitudinal displacements in the laminate thickness can be expressed considering a parabolic variation of the longitudinal displacement across the 90 layer thickness: t90 2 2 u90 ðx; zÞ ¼ u 90 ðxÞ þ z A90 ðxÞ; 3
ð7Þ
and considering a progressive shear in the 0 layers: u0 ðx; zÞ ¼ u 0 ðxÞ þ fðzÞA0 ðxÞ;
ð8Þ
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where the progressive shear function is given by: sinht z fðzÞ ¼ cosht 1 þ : t90 t
2.4. Stress field ð9Þ
This function introduces the load transfer parameter t across the thickness of the 0 layers which can be evaluated [3] by the expression: t ¼
EL 1 : GLT a
ð10Þ
Variations derived from finite element analysis for the longitudinal normal stress through the thickness of laminates are shown in Figs. 5 to 7 in the cracked band and the uncracked band, for [0/902/0] laminates (Figs. 5 and 6) and [02/902/02] laminates (Fig. 7). Two values of the cracking aspect ratio (a=2.5, 1) are considered for the two laminates, with different values of the crack length across the specimen width (c=0.25, 0.5, 0.75) in
In expressions (7) and (8), u 0 ðxÞand u 90 ðxÞare the respective average values (estimated across the layer thicknesses) of the longitudinal displacements u0 ðx; zÞ and u90 ðx; zÞ in the 0 and 90 layers. Functions u 0 ðxÞ, u 90 ðxÞ, A0(x) and A90(x) are to be determined by solving the elasticity problem inside the elementary cell of a transverse cracked cross-ply laminate. Furthermore the finite element analysis shows that, in the uncracked band of the laminate (Fig. 4b), the longitudinal displacement is constant throughout the laminate width and can be evaluated by the classical laminate theory.
Fig. 4. Variation of the longitudinal displacement through the laminate thickness : (a) in the cracked band of the cell for y=w/4 and (b) in the uncracked band for y=3w/4.
Fig. 5. Variation of the longitudinal normal stress through the thickness of [0/902/0] laminates for a cracking aspect ratio equal to 2.5 (a=2.5) and for different values of the crack length: (a) c=0.25, (b) c=0.50 and (c) c=0.75.
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Fig. 6. Variation of the longitudinal normal stress through the thickness of [0/902/0] laminates for a cracking aspect ratio equal to 1 (a=1) and for a reduced crack length equal to 0.5 (c=0.50).
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Fig. 8. Variation of the average longitudinal stress 90 xx in the 90 layer along the length of [0/902/0] laminates in the cracked band (y=w/4), for a cracking aspect ratio equal to 2.5 (a=2.5).
layer along the laminate length in the cracked band. This average stress, evaluated through the thickness of the 90 layer, is related to the average stress 90 applied to the 90 layer, when the laminate is undamaged: 90 ¼ c
ET ; E0x
ð11Þ
where E0x is the Young’s modulus of the undamaged laminate given by the law of mixtures: E0x ¼
Fig. 7. Variation of the longitudinal normal stress through the thickness of [02/902/02] laminates for a reduced crack length equal to 0.5 (c=0.50) and for two values of the cracking aspect ratio: (a) a=2.5 and (b) a=1.
the case of [0/902/0] laminates (Fig. 5). In these figures, 90 variations of the longitudinal normal stress xx , related to the average stress c applied to laminates, are plotted as functions of the reduced transverse co-ordinate at the quarter way between two consecutive cracks (x=l/2). The results obtained show that the variations of the longitudinal normal stress are rather similar for the different values of the crack length : c=0.25 (Fig. 5a), c=0.5 (Fig. 5b) and c=0.75 (Fig. 5c). Next, Fig. 8 shows the variation of the average longitudinal stress in the 90
EL þ ET : 1þ
ð12Þ
In Figs. 5–8 the results obtained by the finite element analysis are compared in the cracked band with the results deduced from the analytical model [3] which considers the longitudinal displacements given in Eqs. (7) and (8). A good agreement is obtained between these results which shows that the stress field can be evaluated in the cracked band by the analytical model developed in Ref. [3], when the transverse crack spans the whole width of the laminate and considering a progressive shear of the 0 layers. Moreover, the results obtained by finite element analysis show that the longitudinal normal stress is constant in the thickness of each layer inside the uncracked band of the damaged laminate and can be evaluated by the laminate theory.
3. Process used for analysing the development of transverse cracking in fatigue 3.1. Band modelling Finite element analysis, implemented in the case of regularly spaced cracks and for different values of the cracking aspect ratio and of the crack length (previous section), shows that the displacement field and the stress
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distribution can be evaluated reasonably accurately by the analytical model considered previously in the cracked band and by the laminate theory in the uncracked band, in fact except near the crack tip where there is a stress concentration depending on the singularity sharpness of the crack. Based on these results, the fatigue test specimens (Fig. 1a) can be divided into bands across the width (Fig. 9) and the stress field along the length of each band can be evaluated by the analytical model considering a progressive shear in the thickness of the 0 layers. Next the initiation and the progression of the transverse cracking can be analysed by distributing statistically first an initiation criterion in the length of the test specimen and then a propagation criterion across its width, allowing to evaluate the cycle number necessary to initiate a crack at a point of the specimen edges or to propagate a crack from a band to the next one. Initiation and propagation criteria which are considered in the following subsections are based on stress approaches. The analysis could be extended to fracture mechanics approaches [17,18]. However there is some problem to obtain confident experimental values of the fracture parameters.
for two values of the ratio R between the minimum and maximum fatigue stresses: 0.1 and 0.5. Transverse cracks which span the full width of the test specimen are present after the first cycle at maximum fatigue stresses of 170 and 140 MPa. Next, the crack growth is similar within the few hundred cycles between observations: the new cracks created appear to span the full width indicating high crack rates. Slow crack growth is observed later when the average crack density has increased. At the maximum fatigue stress of 95 MPa, cracks are initiated at one of the free edges of the test specimen only after several hundred cycles. Next, cracks grow slowly across the width of the specimen over periods of a thousand cycles. As the crack density increases and the crack spacing decreases later in the fatigue test, the crack growth rate decreases. Finally, for a given value of the maximum fatigue stress, the value of the stress ratio R does not seem to have a determinant effect on the development across the specimen width of transverse cracking during fatigue tests. Only low values of the stress ratio will be considered hereafter.
3.2. Materials and experimental results
The strength distribution in the 90 layer can be derived from the curve relating the transverse crack density as a function of the average stress applied to laminate obtained in static tensile tests, using the process developed in Ref. [5]. Thus the experimental results obtained by Boniface and Ogin [11] for the laminate considered show that the distribution of the fracture stress of the 90 layer can be described by a pseudo-normal distribution with an expectation value 90 xxup equal to 68 MPa and a scattering of 30%. This strength distribution is distributed along the length and across the width of the test specimen for the evaluation of the development of the transverse cracking during fatigue tests.
The materials considered here are the E-glass/epoxy [0/90/0] laminates studied in fatigue tests by Boniface and Ogin [11]. These laminates are constituted of a 90 layer of 0.52 thickness between two 0 layers of 0.3 mm thicknesses, with a volume fraction of fibres approximately 58% for each layer. Under static tensile loading, transverse cracking begins as the applied stress is increased above a threshold stress of about 105 MPa. Cracks initiate at one of the free edges of the test specimen and propagate across the thickness and width of the transverse layer almost instantaneously. The number of transverse cracks increases with increasing the applied stress, reaching a saturation density close to the failure of the specimen which corresponds to a crack spacing of the order of the thickness of the transverse layer. The development of the transverse cracking under static loading has been extensively analysed in Ref. [5]. During fatigue tensile loading, the crack development is studied by Boniface and Ogin for different values of the maximum fatigue stress: 170, 140 and 95 MPa and
Fig. 9. Dividing test specimen into bands across its width.
3.3. Strength distribution in 90 layer
3.4. Initiation criterion It has been reported previously that, during fatigue loading, transverse cracks are initiated and propagate across the full width of test specimen at the first cycle, when the maximum fatigue stress is higher than the threshold stress at which the first transverse crack appears in static test. When the maximum fatigue stress is lower than the threshold stress, cracks are initiated only after several hundred cycles and next grow slowly across the width of the specimen. Based on these results we have considered the following scheme for the crack initiation. In the case where the average longitudinal stress 90 xx in the 90 layer reaches at one point of the edges of the test specimen the fracture stress 90 xxu during the first cycle: 90 90 xx 5 xxu ;
ð13Þ
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a transverse crack is initiated at this point and propagated across the full width of the specimen. This process is repeated during the first cycle for every point at which relation (13) is verified, the new stress field in laminate being re-evaluated after each transverse cracking. Finally the process is similar to the one used in static tests [5], until the value max of the maximum fatigue stress applied to the laminate. At points of the test specimen where the fracture stress of the 90 layer is not reached, that is: 90 90 xx < xxu ;
ð14Þ
we have to introduce an initiation criterion function of the cycle number. For example, it may be advanced that a transverse crack will be initiated when the energy cumulated during fatigue cycles will be high enough, and thus it may be proposed that a crack is initiated at a point when the cumulated energy reaches a given value proportional to the fracture energy measured in a static test. Hence, the cycle number N necessary to observe a crack initiation at a point will be of the form: 90 2 90 2 xxmax N ¼ Ai xxu : 2E90 2E90
ð15Þ
It results that the initiation criterion is written as: 2 90 2 N 90 xxmax ¼ Ai xxu ;
ð16Þ
where Ai is a parameter which is characteristic of the 90 layer. In fact the experimental results show that the rate of development of the transverse crack initiation on the edges of the test specimen is high at the beginning of the fatigue test and decreases steeply when the cycle number increases. So a criterion of the form (16) leads to a decrease of the rate of development of crack initiation, but not enough steep. Thus the criterion (16) was modified and an initiation criterion of the following form was considered: m m N 90 ¼ Ai 90 xxmax xxu ;
ð17Þ
where the value of the parameter m (m/>2) has to be adjusted to the rate of the development of the cracks initiated on the edges of the test specimen. Moreover, the stress field inside the 90 layer is modified as a function of the development of the transverse cracking during fatigue cycles (Fig. 10). In this case, it is possible to consider the Palmgren–Miner cumulative damage criterion, commonly applied in the case of the fatigue of metals: X ni ¼ C; Ni i
ð18Þ
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Fig. 10. Fatigue stress at a point of the 90 layer, during the development of transverse cracking.
where ni is the number of cycles in a particular stress amplitude with value Si, Ni is the number of cycles of stress amplitude Si which would cause failure. The factor C is a material parameter which is somewhat less than unity. Considering the experimental results obtained by Boniface and Ogin for a fatigue stress ratio R=0.1 and the simulations we carried out, we have retained the following values for the parameters of the initiation criterion: Ai ¼ 3:9 cycles; m ¼ 8:
ð19Þ
These values associated to the criterion (17) lead to the curves of figure 11 giving the number of cracks initiated on one edge of the specimen as a function of the cycle number. The results are reported for the two values of the maximum fatigue stress: max ¼ 95 and 140 MPa, and for a fatigue stress ratio R=0.1. It has to be noted that it is difficult to deduce the number of initiated cracks from experimental investigations, due to the fact that an initiated crack is not easily observed if its length is two low. 3.5. Crack growth across the specimen width In the case of a homogeneous material, the crack growth during fatigue loading is generally described by the Paris law [27] which expresses the fatigue crack growth rate on the general form: dlc ¼ fðKmax ; KÞ; dN
ð20Þ
where Kmax is the maximum stress intensity factor and DK the stress intensity factor range at the tip of a growing crack. The Paris law was applied to the fatigue loading of composite materials, in particular by Ogin et al. [11–13,16] in the case of glass fibre/epoxy cross-ply laminates. The results which are obtained are scattered and it is difficult to deduce final conclusions from these results. In fact, simulations we carried out show that Paris law does not allow to describe properly the processes of crack propagation which are observed in experimental investigations. Boniface and Ogin [11] observed the crack growth of individual cracks during fatigue loading. The average crack growth rate is found to be independent of crack
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Fig. 12. . Propagation of transverse cracking across specimen width. Fig. 11. Evaluation of the number of cracks initiated on one edge of a test specimen as a function of cycle number.
length but to depend on the spacing between cracks. Transverse crack grows at a constant average rate across the width of specimen when the longitudinal spacing to the next nearest crack is constant, depending on the fatigue stress level. The crack growth rate decreases when the crack spacing decreases and is negligible when the crack spacing is low. Based on these results it can be concluded that the average crack growth rate is an increasing function of the average longitudinal normal stress 90 xxmax in the vicinity of the crack tip (Fig. 12), corresponding to the maximum fatigue stress. Thus we have considered a law of the form: n dlc ¼ Bp 90 xxmax ; dN
ð21Þ
for the crack growth rate, where the parameters Bp and n can be deduced from the experimental results observed on the growth of individual cracks. In the case of the laminates studied by Boniface and Ogin, the results reported in Ref. [11] lead to n=2. So in this case the crack growth rate is proportional to the energy dissipated in the vicinity of the crack tip. The value of the parameter Bp cannot be derived effectively from the results of Boniface and Ogin due to the fact that these results report the crack growth rate of individual cracks associated to the value of the spacing to the nearest crack, when Eq. (21) considers the normal stress induced by the two nearest cracks which overlap the crack under consideration (Fig. 12). Moreover the law considered in Eq. (21) leads to a crack growth rate which is independent of the fracture stress of the material at the point where the crack propagates, when Boniface and Ogin observe that there is a large scatter in growth rate at a particular spacing. Also the comparison of the results obtained by simulations we carried out with the experimental results of Boniface and Ogin shows that law (21) does not allow to describe properly the crack growth for the low values of cycle
number (values lower than 103 to 104 according to the value of the maximum fatigue stress). For these values the crack growth rate evaluated by Eq. (21) is too low, restricting the crack growth across the specimen width contrary to the experimental investigations. In such a way to promote the crack growth, we have considered a law of the form: 2 90 2 dlc 90 xxmax ¼ Bp xxup ; 90 dN xxu
ð22Þ
90 is the expectation value of the strength of the where xxup 90 90 layer, xxu the strength at the point where crack propagates and 90 xxmax the value of the average normal stress at this point associated to the maximum fatigue stress. Law described in Eq. (22) leads to crack growth all the more easy since the strength at a point is low. Thus this law makes easier the crack propagation when the cycle number is low or moderate. Simulations that we carried out lead to retain the following value for the parameter Bp:
Bp ¼ 8 106 mm cycle1 MPa2 :
ð23Þ
3.6. Simulation features To simulate the development of transverse cracking during fatigue tests, the test specimen is divided into bands across its width and divided into elements along its length. 90 The statistical distribution of the stress fracture xxu of the 90 layer is distributed in the whole specimen across its width and along its length. For a given state of the transverse cracking, the stress field in the 90 layer is evaluated using the analytical model considered in Section 2. In the case where expression (13) is verified at a point of the specimen edges during the first cycle, a crack is initiated and propagated across the full width of the test specimen. Then re-evaluating the new stress field after every transverse cracking, this process is continued until the maximum fatigue stress applied to laminate.
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In the opposite case, at each point of the test specimen, the cycle number which is necessary to initiate a crack is calculated using expressions (17) and (18) or the cycle number necessary to propagate a crack from a band to the next one using expression (22). The cycle number necessary to reach the next state of transverse cracking corresponds to the minimum of cycle numbers calculated at each point of the test specimen. The process is continued by considering the new state of the stress field in the 90 layer corresponding to the new state of transverse cracking, until a given cycle number. 3.7. Results derived from simulation The numerical process considered previously for analysing the initiation and propagation of transverse cracking has been implemented by using matlab software. Typically simulations were carried out on test specimens 5 cm long and 2 cm wide. The specimen was divided into 500 elements along its length and 20 bands across its width. The strength distribution in the 90 layer used for the simulations was the distribution considered in Section 3.3. It has to be noted that, in such a way to have acceptable computation times for a simulation, it is necessary to use an efficient model for analysing the stress distribution in the cracked laminate at each step of the simulation process. The model considered here associates acceptable computation times with a good evaluation of the stress distribution in the cracked laminate. Figs. 13 and 14 show the development of the transverse cracking deduced from simulation for the values 95 and 140 MPa of the maximum fatigue stress, by using the initiation law (17) with the values of parameters Ai and m given by expression (19) and the propagation law (22) with the value (23) of the parameter Bp. For the value of the maximum fatigue stress equal to 95 MPa, very few cracks are initiated during the first cycles (Fig. 11), then the initiated cracks propagate little for a cycle number lower than 1000 (Fig. 13). Next the number of initiated cracks increases (Fig. 11) and cracks propagate more rapidly across the specimen width (Fig. 13). Finally, the development of the initiation of transverse cracks decreases steeply for high cycle number, and a saturation state of transverse cracking is reached at about 105 to 106 cycles. In the case of a maximum fatigue stress equal to 140 MPa, transverse cracks are initiated since the first cycle (Fig. 14). Then the number of initiated cracks increases steeply (Fig. 11) and the cracks propagate first slowly across the specimen width (cycle number lower than a few hundred). When the cycle number increases, cracks propagate faster across the specimen width and then the development of crack initiation decreases, and a satura-
Fig. 13. Development of the transverse cracking derived from the simulation for a maximum fatigue stress equal to 95 MPa.
tion state is reached for about 104 cycles. This saturation state corresponds to a crack density slightly higher than the crack density observed for the saturation state obtained when the maximum fatigue life is equal to 95 MPa. The results of Figs. 13 and 14 deduced from the simulations are in good agreement with the results derived by Boniface and Ogin [11] from experimental investigation. 3.8. Stiffness reduction The development of the transverse cracking being evaluated by simulation, it is easy to estimate the stiffness reduction of cross-ply laminate during fatigue cycles. Fig. 15 shows the results obtained for the Young’s modulus reduction as a function of cycle number. According to the high value of the ratio of layer thicknesses (t0/t90), the modulus reduction stays limited. Furthermore, it is interesting to note that it is possible to evaluate the behaviour law of damaged laminate relating the stress as function of strain after a given cycle number, by applying to the damaged specimen and from the damage induced by fatigue the process considered in Ref. [5].
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Fig. 14. Development of the transverse cracking derived from the simulation for a maximum fatigue stress equal to 140 MPa.
cell corresponding to the case of regularly spaced cracks. Different values of the parameters of the progressive transverse cracking (crack length, cracking aspect ratio, layer thicknesses) have been considered. The results obtained show that the displacement field and the stress distribution can be well evaluated in the cracked band by an analytical model considering progressive shear in the thickness of the 0 layers and by laminate theory in the uncracked band, except near the crack tip. Based on these results, a process of simulation of the development of the transverse cracking during fatigue tests has been implemented by dividing the test specimen into bands across its width. Then the values of the fracture stress of the 90 layer were distributed along the length and across the width of the test specimen according to the strength distribution of the 90 layer evaluated in static tests. For every crack state, the stress field in the cross-ply laminate has been estimated in each band by the previous analytical model considering a progressive shear of the 0 layers. An initiation criterion of the transverse cracks on the free edges of the test specimen has been proposed and distributed along the specimen length. The growth rate of the transverse cracks across the specimen width has been considered as a function of the energy dissipated at the crack tip during fatigue cycles, with a value all the more high as the fracture stress of 90 layer is low at the point where the crack propagates. These concepts have been applied to the analysis of the development of transverse cracking in glass-fibre/epoxy crossply laminates, and the comparison between the results derived from the process of simulation and the experimental results shows a good agreement which validates these concepts. Initiation and propagation criteria used in the present work are based on simple stress approaches associated with statistical distribution of these criteria inside the laminate into consideration. The analysis can easily be extended to more sophisticated criteria.
References
Fig. 15. Evaluation of the Young’s modulus reduction as a function of cycle number.
4. Conclusions During fatigue tests, transverse matrix cracks are initiated at the free edges of test specimen and grow progressively across the specimen width as function of cycle number. A finite element analysis has been implemented in the case of a three-dimensional elementary
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J.-M. Berthelot et al. / Composites Science and Technology 61 (2001) 1711–1721 [7] Charewicz A, Daniel IM. Damage mechanisms and accumulation in graphite/epoxy laminates. In: Hahn HT, editor. Composite materials: fatigue and fracture, ASTM STP 907. PA: American Society for Testing and Materials, 1984. p. 274–97. [8] Jamison RD, Schulte K, Keneth K, Reifsnider KL, Stinchcomb WW. Characterisation and analysis of damage mechanisms in tension–tension fatigue of graphite/epoxy laminates. In: Wilkins DJ, editor. Effects of Defects in Composite Materials, ASTM STP 836. American Society for Testing and Materials:PA, 1984. p. 21–55. [9] Daniel IM, Lee JW, Yaniv G. Damage mechanisms and stiffness degradation in graphite/epoxy composite. Proceedings of ICCM6 and ECCM-2, Elsevier Applied Science, London, 1987; 412938. [10] Reifsnider KL. Damage and damage mechanisms. In: Reifsnider KL, editor. Fatigue of Composite Materials. London: Elsevier and Science Publishers, 1990. p. 11–77. [11] Boniface L, Ogin SL. Application of the Paris equation to the fatigue growth of transverse ply cracks. Journal of Composite Materials 1989;23:735–54. [12] Ogin SL, Smith PA, Beaumont PWR. Matrix cracking and stiffness reduction during the fatigue of a (0/90)s GFRP laminate. Composites Science and Technology 1985;22:21–31. [13] Ogin SL, Smith PA, Beaumont PWR. A stress intensity factor approach to the fatigue growth of transverse ply cracks. Composites Science and Technology 1985;24:47–59. [14] Ogin SL, Smith PA. Fast fracture and fatigue growth of transverse ply cracks in composite laminates. Scripta Metallurgica 1985;19:779–84. [15] Ogin SL, Boniface L, Bader MG. Fatigue growth of transverse ply cracks in (0/902)s glass fibre reinforced and (0/903)s carbon fibre reinforced plastic laminates. Proceedings of Conference on Fibre Reinforced Composites. Inst. Mech. Eng. 1986; 173-8. [16] Ogin SL, Smith PA. A model for matrix cracking in cross-ply laminates. ESA Journal 1987;11:45–60.
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[17] Wang ASD, Chou PC, Lei SC. A stochastic model for the growth of matrix cracks in composite laminates. Journal of Composite Materials 1984;18:239–54. [18] Guild FJ, Ogin SL, Smith PA. Modelling of 90 ply cracking in crossply laminates, including three-dimensional effects. Journal of Composite Materials 1993;27:646–67. [19] Lee JW, Daniel IM, Yaniv G. Fatigue life prediction of cross-ply composite laminates. In: Lagace PA, editor. Composite Materials: Fatigue and Fracture, Vol. 2, ASTM STP 1012. PA: American Society for Testing and Materials, 1989. p. 19–28. [20] Gao Z. A cumulative damage model for fatigue life of composite materials. Journal of Reinforced Plastics and Composites 1994; 13:128–41. [21] Diao XX, Ye L, Mai YW. A statistical model for residual strength and fatigue life of composite laminates. Composites Science and Technology 1995;54:329–36. [22] Diao X, Ye L, Mai YW. Fatigue life prediction of composite laminates using a stress redistribution function. Journal of Reinforced Plastics and Composites 1996;3:249–66. [23] Diao XX, Ye L, Mai YW. Statistical fatigue life prediction of cross-ply laminates. Journal of Composite Materials 1997;31: 1442–60. [24] Halpin JC, Jerina KL, Johnson TS. Characterisation of composites for the purpose of reliability evaluation. In: Whitney JM, editor. Analysis of the Test Methods fot High Modulus Fibres and Composites, ASTM STP 521. PA: American Society for Testing and Materials, 1973. p. 5–64. [25] Lee JW, Daniel IM. Progressive transverse cracking of cross-ply laminates. Journal of Composite Materials 1990;24:1225–43. [26] Berthelot J-M, Leblond P, El Mahi A, Le Corre J-F. Transverse cracking of cross-ply laminates: Part 2. Progressive widthwise cracking. Composites 1996;27A:989–1001. [27] Paris PC, Gomez MP, Anderson WE. The trend in engineering. University of Washington, Seattle, 1961.
Development of transverse cracking in cross-ply laminates during fatigue tests J.-M. Berthelot*, A. El Mahi, J.-F. Le Corre Acoustics and Mechanics Institute, Composites and Mechanical Structures Group, Universite´ du Maine, 72085 Le Mans cedex 9, France Received 12 December 2000; accepted 8 May 2001
Abstract The purpose of this paper is to analyse the progressive development of transverse cracking in cross-ply laminates subjected to uniaxial fatigue tests. First, a finite-element analysis shows how an analytical model which considers a progressive shear through the thickness of the 0 layers can be used for evaluating the stress field in the case of progressive cracking across the specimen width. The paper then presents a simulation of the progression of the transverse cracking, based on statistical distributions along the specimen length for crack initiation and across the specimen width for crack growth across the specimen width. The simulation process is applied to glass-fibre/epoxy cross-ply laminates and the results obtained are compared with experimental results. # 2001 Published by Elsevier Science Ltd. All rights reserved. Keywords: A. Cross-ply laminates; B. Fatigue; C. Transverse cracking
1. Introduction The initiation and the development of transverse cracking in cross-ply laminates subjected to static uniaxial loading have been extensively studied in the literature (see Refs. [1–5] for a review). A generalized model has been developed [3] for evaluating the two-dimensional stress distribution in transverse cracked cross-ply laminates. This generalized model has then been extended [4] to evaluate the stress distribution in cross-ply laminates subjected to static tensile loading and containing transverse cracks and delamination initiated from the transverse crack tips. These models have been combined [5] with statistical distributions of strengths in the 90 layers in such a way to evaluate the progression of transverse cracking and delamination in cross-ply laminates subjected to static uniaxial loading. In the case of cross-ply laminates subjected to uniaxial tensile fatigue loading, experimental investigations [6– 10] show that the predominant damage mechanisms are the initiation and propagation of transverse cracking in the 90 layers. These damage mechanisms are observed [9] to occur during the first 80% of the lifetime of the * Corresponding author. E-mail address: [email protected] (J.-M. Berthelot).
test specimen. Then, this first stage is followed by longitudinal matrix crack splitting, local delamination at intersections of transverse and longitudinal cracks, and ultimately by fibre fracture of the 0 layers carrying the load applied to laminates. Because of the prevalence of transverse cracking and facility to observe it in transparent laminates, the development of the transverse cracking has been extensively studied in the case of glass-fibre laminates. For example an extensive experimental investigation has been implemented by Boniface and Ogin [11], in which they observed crack growth in a transparent [0/90/0] glass-fibre/epoxy laminate subjected to tensile fatigue loading. The behaviour of laminates subjected to fatigue loading is very complex, and in spite of various works on this subject, the mechanisms which are induced are not well understood. A fracture mechanics approach has been proposed by Ogin et al. [11–16], based on the concept of the stress-intensity factor at the tip of a growing transverse crack in the 90 layers. First, the stress-intensity factor is evaluated as a function of the stress in the transverse layer acting on the crack, using a shear-lag model for the load transfer between the 0 and 90 layers. Next, it is assumed that the growth rate of transverse cracks can be related to the stress-intensity factor by a Paris relationship. This model has been applied to predict
0266-3538/01/$ - see front matter # 2001 Published by Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00068-9
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the stiffness reduction during fatigue of a [0/90]s glassfibre laminate [12–14] and to fatigue growth in glassfibre and carbon-fibre laminates [11,15,16]. The strainenergy release rate has been used rather than the stress intensity factor by Wang et al. [17] and Guild et al. [18]. In a different way, various authors have attempted to develop models for predicting the fatigue life of crossply laminates in terms of static and fatigue properties of the laminate layers. Lee et al. [19] proposed a deterministic model to predict the fatigue life which was divided into two parts: an initial life associated with the development of the transverse cracking until a saturation state and next a residual life. The fatigue life until the saturation state was determined from the curves of stress vs. number of cycles measured on the 90 layers and the Young’s modulus-crack density evaluated on the basis of experimental investigations. Next, the residual life was estimated, without considering the type of damages induced, from the stress-cycles curves of the 0 layers and considering the average stress applied on them according to the residual modulus of the 90 layers after the saturation state. In a similar way, the stress-cycles number curves of off-axis layers were used by Gao [20] to develop a model for fatigue life of laminates. A statistical model for the evaluation of the residual strength and the fatigue life has been applied by Diao et al. [21–23] to describe the fatigue behaviour of cross-ply laminates. In this model it is assumed that the decreasing rate of the residual strength of the 0 layers, under cyclic loading of constant amplitude, is inversely proportional to a certain power of the residual strength itself according to the relation introduced by Halpin et al. [24]: 0 A max dr0 ¼ 1 ; ð1Þ dN 0
2. Progressive transverse cracking in fatigue tests 2.1. Introduction During fatigue tests, transverse matrix cracks in 90 layers initiate at the free edges of test specimen, then grow slowly across the specimen width as a function of cycle number (Fig. 1a). For a given cycle number, the matrix cracking pattern is characterized (Fig. 1b) by the edge of initiation (edge 1 or 2), the position xi along the length of the laminate and the length yi of each crack. So the analysis of the stress and strain fields is very complex, involving the resolution of a three-dimensional problem. Finite element analysis has been implemented in the case of regularly spaced cracks and some results have been considered previously in Ref. [26]. This analysis is completed in this section, considering the generalized model developed in Ref. [3] and drawing how this model can be used to evaluate the progression of transverse cracking during fatigue tests. 2.2. The elementary cell The elementary cell considered in the finite-element analysis is shown in Fig. 2. The geometric parameters of the cell are t0, the thickness of the 0 layers, t90, the thickness of the 90 layer and w, the specimen width. The transverse cracking is characterized by the distance 2l between two consecutive cracks and the length lc of the crack across the specimen width. Then the following reduced parameters can be introduced: the stacking parameter ¼
t0 ; t90
ð3Þ
r
the aspect ratio of transverse cracking where r0 is the residual strength of the 0 layers at N 0 cycles, max the maximum fatigue stress applied to the 0 layers and A and are two dimensionless constants derived from experimental investigations implemented on the 0 layers. The maximum fatigue stress applied to the 0 layers is related to the maximum fatigue stress max applied to laminate by the expression: 0 ¼ hðNÞmax ; max
ð2Þ
where h(N) is the stress redistribution function as a function of cycle number, according to the damage mechanisms induced in the laminate. Next, a statistic analysis of residual strength of the 0 layers is considered and related to the distribution of the static strengths. Finally, the stress redistribution function is derived in the case of progressive transverse cracking by using the shear lag analysis developed by Lee and Daniel [25].
a¼
l t90
;
ð4Þ
and the reduced crack length across the specimen width c¼
lc : w
ð5Þ
The laminates investigated are E-glass fibre/epoxy cross-ply laminates, with the following layer properties: EL ¼ 42:3 GPa; ET ¼ 13:2 GPa; GLT ¼ 4:4 GPa; GTT 0 ¼ 3:5 GPa; LT ¼ 0:30; ð6Þ relative to the layer directions ðL; T; T 0 Þ. Finite element analysis has been carried out for different values of the stacking parameter (= 1/3, 1, 2) and the transverse
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Fig. 1. Typical transverse crack distribution.
Fig. 2. Elementary cell.
cracking parameters (aspect ratio: 14a410 and reduced length across the specimen width: c=0.25, 0.5, 0.75). Boundary conditions corresponding to a tensile state were obtained by prescribing longitudinal displacements of the nodes of the extreme transverse planes of the elementary cell. 2.3. Displacement field Fig. 3. Deformed shape of the elementary cell.
An example of the deformed shape of the elementary cell derived by finite element analysis is shown in Fig. 3, in the case of [0/902/0] laminates and for a transverse cracking state characterized by the value a ¼ 2:5 of the aspect ratio and the value c=0.5 of the reduced crack length across the specimen width. A quarter of the elementary cell has been suppressed for a better visualisation of the cracked part. The longitudinal displacement, related to the prescribed displacement uc, is reported through the laminate thickness in Fig. 4. The variations of the longitudinal displacement are given in the crack plane (x=l) and at a distance l/4 from the crack plane (x=3l/4), in the cracked band of the cell (y=w/4, Fig. 4a), and in the uncracked band (y=3w/4, Fig. 4b). The finite element investigation implemented for the different values of the stacking parameter and the
transverse cracking parameters shows that the longitudinal displacements can be evaluated by the longitudinal displacements considered in the generalized approach when the crack spans the whole width of the laminate [3]. So, in the cracked band, the longitudinal displacements in the laminate thickness can be expressed considering a parabolic variation of the longitudinal displacement across the 90 layer thickness: t90 2 2 u90 ðx; zÞ ¼ u 90 ðxÞ þ z A90 ðxÞ; 3
ð7Þ
and considering a progressive shear in the 0 layers: u0 ðx; zÞ ¼ u 0 ðxÞ þ fðzÞA0 ðxÞ;
ð8Þ
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where the progressive shear function is given by: sinht z fðzÞ ¼ cosht 1 þ : t90 t
2.4. Stress field ð9Þ
This function introduces the load transfer parameter t across the thickness of the 0 layers which can be evaluated [3] by the expression: t ¼
EL 1 : GLT a
ð10Þ
Variations derived from finite element analysis for the longitudinal normal stress through the thickness of laminates are shown in Figs. 5 to 7 in the cracked band and the uncracked band, for [0/902/0] laminates (Figs. 5 and 6) and [02/902/02] laminates (Fig. 7). Two values of the cracking aspect ratio (a=2.5, 1) are considered for the two laminates, with different values of the crack length across the specimen width (c=0.25, 0.5, 0.75) in
In expressions (7) and (8), u 0 ðxÞand u 90 ðxÞare the respective average values (estimated across the layer thicknesses) of the longitudinal displacements u0 ðx; zÞ and u90 ðx; zÞ in the 0 and 90 layers. Functions u 0 ðxÞ, u 90 ðxÞ, A0(x) and A90(x) are to be determined by solving the elasticity problem inside the elementary cell of a transverse cracked cross-ply laminate. Furthermore the finite element analysis shows that, in the uncracked band of the laminate (Fig. 4b), the longitudinal displacement is constant throughout the laminate width and can be evaluated by the classical laminate theory.
Fig. 4. Variation of the longitudinal displacement through the laminate thickness : (a) in the cracked band of the cell for y=w/4 and (b) in the uncracked band for y=3w/4.
Fig. 5. Variation of the longitudinal normal stress through the thickness of [0/902/0] laminates for a cracking aspect ratio equal to 2.5 (a=2.5) and for different values of the crack length: (a) c=0.25, (b) c=0.50 and (c) c=0.75.
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Fig. 6. Variation of the longitudinal normal stress through the thickness of [0/902/0] laminates for a cracking aspect ratio equal to 1 (a=1) and for a reduced crack length equal to 0.5 (c=0.50).
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Fig. 8. Variation of the average longitudinal stress 90 xx in the 90 layer along the length of [0/902/0] laminates in the cracked band (y=w/4), for a cracking aspect ratio equal to 2.5 (a=2.5).
layer along the laminate length in the cracked band. This average stress, evaluated through the thickness of the 90 layer, is related to the average stress 90 applied to the 90 layer, when the laminate is undamaged: 90 ¼ c
ET ; E0x
ð11Þ
where E0x is the Young’s modulus of the undamaged laminate given by the law of mixtures: E0x ¼
Fig. 7. Variation of the longitudinal normal stress through the thickness of [02/902/02] laminates for a reduced crack length equal to 0.5 (c=0.50) and for two values of the cracking aspect ratio: (a) a=2.5 and (b) a=1.
the case of [0/902/0] laminates (Fig. 5). In these figures, 90 variations of the longitudinal normal stress xx , related to the average stress c applied to laminates, are plotted as functions of the reduced transverse co-ordinate at the quarter way between two consecutive cracks (x=l/2). The results obtained show that the variations of the longitudinal normal stress are rather similar for the different values of the crack length : c=0.25 (Fig. 5a), c=0.5 (Fig. 5b) and c=0.75 (Fig. 5c). Next, Fig. 8 shows the variation of the average longitudinal stress in the 90
EL þ ET : 1þ
ð12Þ
In Figs. 5–8 the results obtained by the finite element analysis are compared in the cracked band with the results deduced from the analytical model [3] which considers the longitudinal displacements given in Eqs. (7) and (8). A good agreement is obtained between these results which shows that the stress field can be evaluated in the cracked band by the analytical model developed in Ref. [3], when the transverse crack spans the whole width of the laminate and considering a progressive shear of the 0 layers. Moreover, the results obtained by finite element analysis show that the longitudinal normal stress is constant in the thickness of each layer inside the uncracked band of the damaged laminate and can be evaluated by the laminate theory.
3. Process used for analysing the development of transverse cracking in fatigue 3.1. Band modelling Finite element analysis, implemented in the case of regularly spaced cracks and for different values of the cracking aspect ratio and of the crack length (previous section), shows that the displacement field and the stress
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distribution can be evaluated reasonably accurately by the analytical model considered previously in the cracked band and by the laminate theory in the uncracked band, in fact except near the crack tip where there is a stress concentration depending on the singularity sharpness of the crack. Based on these results, the fatigue test specimens (Fig. 1a) can be divided into bands across the width (Fig. 9) and the stress field along the length of each band can be evaluated by the analytical model considering a progressive shear in the thickness of the 0 layers. Next the initiation and the progression of the transverse cracking can be analysed by distributing statistically first an initiation criterion in the length of the test specimen and then a propagation criterion across its width, allowing to evaluate the cycle number necessary to initiate a crack at a point of the specimen edges or to propagate a crack from a band to the next one. Initiation and propagation criteria which are considered in the following subsections are based on stress approaches. The analysis could be extended to fracture mechanics approaches [17,18]. However there is some problem to obtain confident experimental values of the fracture parameters.
for two values of the ratio R between the minimum and maximum fatigue stresses: 0.1 and 0.5. Transverse cracks which span the full width of the test specimen are present after the first cycle at maximum fatigue stresses of 170 and 140 MPa. Next, the crack growth is similar within the few hundred cycles between observations: the new cracks created appear to span the full width indicating high crack rates. Slow crack growth is observed later when the average crack density has increased. At the maximum fatigue stress of 95 MPa, cracks are initiated at one of the free edges of the test specimen only after several hundred cycles. Next, cracks grow slowly across the width of the specimen over periods of a thousand cycles. As the crack density increases and the crack spacing decreases later in the fatigue test, the crack growth rate decreases. Finally, for a given value of the maximum fatigue stress, the value of the stress ratio R does not seem to have a determinant effect on the development across the specimen width of transverse cracking during fatigue tests. Only low values of the stress ratio will be considered hereafter.
3.2. Materials and experimental results
The strength distribution in the 90 layer can be derived from the curve relating the transverse crack density as a function of the average stress applied to laminate obtained in static tensile tests, using the process developed in Ref. [5]. Thus the experimental results obtained by Boniface and Ogin [11] for the laminate considered show that the distribution of the fracture stress of the 90 layer can be described by a pseudo-normal distribution with an expectation value 90 xxup equal to 68 MPa and a scattering of 30%. This strength distribution is distributed along the length and across the width of the test specimen for the evaluation of the development of the transverse cracking during fatigue tests.
The materials considered here are the E-glass/epoxy [0/90/0] laminates studied in fatigue tests by Boniface and Ogin [11]. These laminates are constituted of a 90 layer of 0.52 thickness between two 0 layers of 0.3 mm thicknesses, with a volume fraction of fibres approximately 58% for each layer. Under static tensile loading, transverse cracking begins as the applied stress is increased above a threshold stress of about 105 MPa. Cracks initiate at one of the free edges of the test specimen and propagate across the thickness and width of the transverse layer almost instantaneously. The number of transverse cracks increases with increasing the applied stress, reaching a saturation density close to the failure of the specimen which corresponds to a crack spacing of the order of the thickness of the transverse layer. The development of the transverse cracking under static loading has been extensively analysed in Ref. [5]. During fatigue tensile loading, the crack development is studied by Boniface and Ogin for different values of the maximum fatigue stress: 170, 140 and 95 MPa and
Fig. 9. Dividing test specimen into bands across its width.
3.3. Strength distribution in 90 layer
3.4. Initiation criterion It has been reported previously that, during fatigue loading, transverse cracks are initiated and propagate across the full width of test specimen at the first cycle, when the maximum fatigue stress is higher than the threshold stress at which the first transverse crack appears in static test. When the maximum fatigue stress is lower than the threshold stress, cracks are initiated only after several hundred cycles and next grow slowly across the width of the specimen. Based on these results we have considered the following scheme for the crack initiation. In the case where the average longitudinal stress 90 xx in the 90 layer reaches at one point of the edges of the test specimen the fracture stress 90 xxu during the first cycle: 90 90 xx 5 xxu ;
ð13Þ
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a transverse crack is initiated at this point and propagated across the full width of the specimen. This process is repeated during the first cycle for every point at which relation (13) is verified, the new stress field in laminate being re-evaluated after each transverse cracking. Finally the process is similar to the one used in static tests [5], until the value max of the maximum fatigue stress applied to the laminate. At points of the test specimen where the fracture stress of the 90 layer is not reached, that is: 90 90 xx < xxu ;
ð14Þ
we have to introduce an initiation criterion function of the cycle number. For example, it may be advanced that a transverse crack will be initiated when the energy cumulated during fatigue cycles will be high enough, and thus it may be proposed that a crack is initiated at a point when the cumulated energy reaches a given value proportional to the fracture energy measured in a static test. Hence, the cycle number N necessary to observe a crack initiation at a point will be of the form: 90 2 90 2 xxmax N ¼ Ai xxu : 2E90 2E90
ð15Þ
It results that the initiation criterion is written as: 2 90 2 N 90 xxmax ¼ Ai xxu ;
ð16Þ
where Ai is a parameter which is characteristic of the 90 layer. In fact the experimental results show that the rate of development of the transverse crack initiation on the edges of the test specimen is high at the beginning of the fatigue test and decreases steeply when the cycle number increases. So a criterion of the form (16) leads to a decrease of the rate of development of crack initiation, but not enough steep. Thus the criterion (16) was modified and an initiation criterion of the following form was considered: m m N 90 ¼ Ai 90 xxmax xxu ;
ð17Þ
where the value of the parameter m (m/>2) has to be adjusted to the rate of the development of the cracks initiated on the edges of the test specimen. Moreover, the stress field inside the 90 layer is modified as a function of the development of the transverse cracking during fatigue cycles (Fig. 10). In this case, it is possible to consider the Palmgren–Miner cumulative damage criterion, commonly applied in the case of the fatigue of metals: X ni ¼ C; Ni i
ð18Þ
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Fig. 10. Fatigue stress at a point of the 90 layer, during the development of transverse cracking.
where ni is the number of cycles in a particular stress amplitude with value Si, Ni is the number of cycles of stress amplitude Si which would cause failure. The factor C is a material parameter which is somewhat less than unity. Considering the experimental results obtained by Boniface and Ogin for a fatigue stress ratio R=0.1 and the simulations we carried out, we have retained the following values for the parameters of the initiation criterion: Ai ¼ 3:9 cycles; m ¼ 8:
ð19Þ
These values associated to the criterion (17) lead to the curves of figure 11 giving the number of cracks initiated on one edge of the specimen as a function of the cycle number. The results are reported for the two values of the maximum fatigue stress: max ¼ 95 and 140 MPa, and for a fatigue stress ratio R=0.1. It has to be noted that it is difficult to deduce the number of initiated cracks from experimental investigations, due to the fact that an initiated crack is not easily observed if its length is two low. 3.5. Crack growth across the specimen width In the case of a homogeneous material, the crack growth during fatigue loading is generally described by the Paris law [27] which expresses the fatigue crack growth rate on the general form: dlc ¼ fðKmax ; KÞ; dN
ð20Þ
where Kmax is the maximum stress intensity factor and DK the stress intensity factor range at the tip of a growing crack. The Paris law was applied to the fatigue loading of composite materials, in particular by Ogin et al. [11–13,16] in the case of glass fibre/epoxy cross-ply laminates. The results which are obtained are scattered and it is difficult to deduce final conclusions from these results. In fact, simulations we carried out show that Paris law does not allow to describe properly the processes of crack propagation which are observed in experimental investigations. Boniface and Ogin [11] observed the crack growth of individual cracks during fatigue loading. The average crack growth rate is found to be independent of crack
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Fig. 12. . Propagation of transverse cracking across specimen width. Fig. 11. Evaluation of the number of cracks initiated on one edge of a test specimen as a function of cycle number.
length but to depend on the spacing between cracks. Transverse crack grows at a constant average rate across the width of specimen when the longitudinal spacing to the next nearest crack is constant, depending on the fatigue stress level. The crack growth rate decreases when the crack spacing decreases and is negligible when the crack spacing is low. Based on these results it can be concluded that the average crack growth rate is an increasing function of the average longitudinal normal stress 90 xxmax in the vicinity of the crack tip (Fig. 12), corresponding to the maximum fatigue stress. Thus we have considered a law of the form: n dlc ¼ Bp 90 xxmax ; dN
ð21Þ
for the crack growth rate, where the parameters Bp and n can be deduced from the experimental results observed on the growth of individual cracks. In the case of the laminates studied by Boniface and Ogin, the results reported in Ref. [11] lead to n=2. So in this case the crack growth rate is proportional to the energy dissipated in the vicinity of the crack tip. The value of the parameter Bp cannot be derived effectively from the results of Boniface and Ogin due to the fact that these results report the crack growth rate of individual cracks associated to the value of the spacing to the nearest crack, when Eq. (21) considers the normal stress induced by the two nearest cracks which overlap the crack under consideration (Fig. 12). Moreover the law considered in Eq. (21) leads to a crack growth rate which is independent of the fracture stress of the material at the point where the crack propagates, when Boniface and Ogin observe that there is a large scatter in growth rate at a particular spacing. Also the comparison of the results obtained by simulations we carried out with the experimental results of Boniface and Ogin shows that law (21) does not allow to describe properly the crack growth for the low values of cycle
number (values lower than 103 to 104 according to the value of the maximum fatigue stress). For these values the crack growth rate evaluated by Eq. (21) is too low, restricting the crack growth across the specimen width contrary to the experimental investigations. In such a way to promote the crack growth, we have considered a law of the form: 2 90 2 dlc 90 xxmax ¼ Bp xxup ; 90 dN xxu
ð22Þ
90 is the expectation value of the strength of the where xxup 90 90 layer, xxu the strength at the point where crack propagates and 90 xxmax the value of the average normal stress at this point associated to the maximum fatigue stress. Law described in Eq. (22) leads to crack growth all the more easy since the strength at a point is low. Thus this law makes easier the crack propagation when the cycle number is low or moderate. Simulations that we carried out lead to retain the following value for the parameter Bp:
Bp ¼ 8 106 mm cycle1 MPa2 :
ð23Þ
3.6. Simulation features To simulate the development of transverse cracking during fatigue tests, the test specimen is divided into bands across its width and divided into elements along its length. 90 The statistical distribution of the stress fracture xxu of the 90 layer is distributed in the whole specimen across its width and along its length. For a given state of the transverse cracking, the stress field in the 90 layer is evaluated using the analytical model considered in Section 2. In the case where expression (13) is verified at a point of the specimen edges during the first cycle, a crack is initiated and propagated across the full width of the test specimen. Then re-evaluating the new stress field after every transverse cracking, this process is continued until the maximum fatigue stress applied to laminate.
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In the opposite case, at each point of the test specimen, the cycle number which is necessary to initiate a crack is calculated using expressions (17) and (18) or the cycle number necessary to propagate a crack from a band to the next one using expression (22). The cycle number necessary to reach the next state of transverse cracking corresponds to the minimum of cycle numbers calculated at each point of the test specimen. The process is continued by considering the new state of the stress field in the 90 layer corresponding to the new state of transverse cracking, until a given cycle number. 3.7. Results derived from simulation The numerical process considered previously for analysing the initiation and propagation of transverse cracking has been implemented by using matlab software. Typically simulations were carried out on test specimens 5 cm long and 2 cm wide. The specimen was divided into 500 elements along its length and 20 bands across its width. The strength distribution in the 90 layer used for the simulations was the distribution considered in Section 3.3. It has to be noted that, in such a way to have acceptable computation times for a simulation, it is necessary to use an efficient model for analysing the stress distribution in the cracked laminate at each step of the simulation process. The model considered here associates acceptable computation times with a good evaluation of the stress distribution in the cracked laminate. Figs. 13 and 14 show the development of the transverse cracking deduced from simulation for the values 95 and 140 MPa of the maximum fatigue stress, by using the initiation law (17) with the values of parameters Ai and m given by expression (19) and the propagation law (22) with the value (23) of the parameter Bp. For the value of the maximum fatigue stress equal to 95 MPa, very few cracks are initiated during the first cycles (Fig. 11), then the initiated cracks propagate little for a cycle number lower than 1000 (Fig. 13). Next the number of initiated cracks increases (Fig. 11) and cracks propagate more rapidly across the specimen width (Fig. 13). Finally, the development of the initiation of transverse cracks decreases steeply for high cycle number, and a saturation state of transverse cracking is reached at about 105 to 106 cycles. In the case of a maximum fatigue stress equal to 140 MPa, transverse cracks are initiated since the first cycle (Fig. 14). Then the number of initiated cracks increases steeply (Fig. 11) and the cracks propagate first slowly across the specimen width (cycle number lower than a few hundred). When the cycle number increases, cracks propagate faster across the specimen width and then the development of crack initiation decreases, and a satura-
Fig. 13. Development of the transverse cracking derived from the simulation for a maximum fatigue stress equal to 95 MPa.
tion state is reached for about 104 cycles. This saturation state corresponds to a crack density slightly higher than the crack density observed for the saturation state obtained when the maximum fatigue life is equal to 95 MPa. The results of Figs. 13 and 14 deduced from the simulations are in good agreement with the results derived by Boniface and Ogin [11] from experimental investigation. 3.8. Stiffness reduction The development of the transverse cracking being evaluated by simulation, it is easy to estimate the stiffness reduction of cross-ply laminate during fatigue cycles. Fig. 15 shows the results obtained for the Young’s modulus reduction as a function of cycle number. According to the high value of the ratio of layer thicknesses (t0/t90), the modulus reduction stays limited. Furthermore, it is interesting to note that it is possible to evaluate the behaviour law of damaged laminate relating the stress as function of strain after a given cycle number, by applying to the damaged specimen and from the damage induced by fatigue the process considered in Ref. [5].
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Fig. 14. Development of the transverse cracking derived from the simulation for a maximum fatigue stress equal to 140 MPa.
cell corresponding to the case of regularly spaced cracks. Different values of the parameters of the progressive transverse cracking (crack length, cracking aspect ratio, layer thicknesses) have been considered. The results obtained show that the displacement field and the stress distribution can be well evaluated in the cracked band by an analytical model considering progressive shear in the thickness of the 0 layers and by laminate theory in the uncracked band, except near the crack tip. Based on these results, a process of simulation of the development of the transverse cracking during fatigue tests has been implemented by dividing the test specimen into bands across its width. Then the values of the fracture stress of the 90 layer were distributed along the length and across the width of the test specimen according to the strength distribution of the 90 layer evaluated in static tests. For every crack state, the stress field in the cross-ply laminate has been estimated in each band by the previous analytical model considering a progressive shear of the 0 layers. An initiation criterion of the transverse cracks on the free edges of the test specimen has been proposed and distributed along the specimen length. The growth rate of the transverse cracks across the specimen width has been considered as a function of the energy dissipated at the crack tip during fatigue cycles, with a value all the more high as the fracture stress of 90 layer is low at the point where the crack propagates. These concepts have been applied to the analysis of the development of transverse cracking in glass-fibre/epoxy crossply laminates, and the comparison between the results derived from the process of simulation and the experimental results shows a good agreement which validates these concepts. Initiation and propagation criteria used in the present work are based on simple stress approaches associated with statistical distribution of these criteria inside the laminate into consideration. The analysis can easily be extended to more sophisticated criteria.
References
Fig. 15. Evaluation of the Young’s modulus reduction as a function of cycle number.
4. Conclusions During fatigue tests, transverse matrix cracks are initiated at the free edges of test specimen and grow progressively across the specimen width as function of cycle number. A finite element analysis has been implemented in the case of a three-dimensional elementary
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