A STUDY OF -FIBRE MICROBUCKLING IN MULTIDIRECTIONAL COMPOSITE LAMINATES P. Berbinau and C. Soutis Department o Aeronautics, Imperial College o Science, Technology & Medicine Prince Consort Rd, London SW7 BY, UK SUMMARY: The aim o this work is to study the compression ailure mechanisms in multidirectional CFRP composite laminates with internal -plies. In such laminates, as in unidirectional laminates, catastrophic ailure initiates by kinking o -plies at the ree-edges or manuacturing deects. T8/94C carbon ibre-epoxy laminates with a [(±/ ) ] s lay-up with equal to 3,45,6 or 75 are used here to study the eect o the supporting ply angle on the stress initiation o -ibre microbuckling. Experimental compressive strengths are compared to theoretical predictions obtained rom a ibre kinking model that incorporates interlaminar shear stresses developed at the ree edges at (/) interaces. Initial misalignment o the ibres and non-linear shear behaviour o the matrix are also included in the analysis. Interlaminar stresses do not appear to have a signiicant inluence on ibre microbuckling initiation and on the static compressive strength o the laminates. KEYWORDS: composite laminates, compression, microbuckling, kink bands, interlaminar edge stresses, compressive strength. INTRODUCTION The compressive strength o long, aligned ibre composites may be as low as 6% o their tensile strengths [1-4], thus it is recognised that the compressive strength is oten a design limiting parameter. An understanding o the compression ailure modes is thereore crucial to the development o improved composite materials. The critical mechanism in the compressive racture o unidirectional polymer matrix composite laminates is plastic microbuckling [5], with creation o a kink band. In carbon ibre-epoxy systems the kink band width is typically o the order o ten to iteen ibre diameters, and is inclined at an angle β=15-3 to the y-axis orthogonal to the loading x-axis. The through-thickness axis is z. Unidirectional composite laminates are rarely used in structural components but are usually combined with plies o varying orientation in a multidirectional laminate. However, it is the layers in such a laminate that carry most o the applied load, and, rom a structural-design viewpoint, it is important to know the ailure stress o these plies and how it relates to the stress o the unidirectional laminate on its own. In this paper the eect o the supporting ply orientation on ibre microbuckling initiation and inal ailure o [(±/ ) ] s T8/94C carbon
ibre-epoxy laminates is examined, with ± equal to either ±3, ±45, ±6 or ±75 (labelled L3, L45, L6 and L75, respectively). This particular lay-up has been selected to study more methodically the eects o the local constraint (supporting ply orientation) on ibre microbuckling initiation. The interlaminar shear stresses developed at the laminate ree edge, between the plies and the adjacent o-axis plies, are determined and incorporated into a ibre kinking model reported previously [6, 7]. The model is based on an initially misaligned ibre bundle on a non-linear oundation (resin) that is capable o inite delections. Analytical compressive strength predictions are compared with experimental measurements. EXPERIMENTS All laminates were autoclaved rom Toray T8 carbon ibres pre-impregnated with Hexcel Composites BSL 94C epoxy resin. The laminates were later inspected ultrasonically to conirm specimen quality. The unidirectional specimens (labelled L) were o gauge section 1mm x 1mm; this gauge length is suiciently short or anti-buckling guides not to be required. They were tested in compression using a modiied Celanese test rig [8, 9]. The multi-directional specimens [(±/ ) ] s (see Table 1) were o gauge section 116 mm x 3 mm, Table 1: Average compressive strength properties o T8/94C [(±/ ) ] s plates Laminate Angle E xx (GPa) G xy (GPa) σ (plate) (MPa) ε (%) σ ( -ply) (MPa) Notes: L 16 6. 143 1.1 143 L3 3 13 (11) (19.) 83 (995).88 1178 L45 45 81 (9) (3.6) 745 (8).95 198 L6 6 83 (87) (19.) 66 (768).93 13 L75 75 81 (86) (1.4) 653 (757).89 133 ( ) Predicted values, assuming elastic LPT and max stress criterion and were tested using an anti-buckling device to prevent bending during the test. All specimens were end-tabbed with glass/epoxy, and had strain gauges bonded on both aces. Compression tests were conducted in a screw-driven test machine at a displacement rate o.17 mm s -1. Full details on the experimental technique are given by Soutis [9]. Compressive strength data The stress-strain response o the T8/94C unidirectional [ 8 ] s laminate is linear almost up to ailure, with an elastic modulus E=16 GPa, a ailure stress σ=148-16 MPa, and an average ailure strain ε =-1% [1, 3]. In comparison, the average tensile strength or this material is 4 MPa, and a ailure strain close to 1.8%. Post ailure examination o the racture suraces using a scanning electron microscope revealed that ailure is by ibre kinking. The kink band angle β is approximately 15. The ibre orientation angle (rom their initial direction, x) within the band is φ=3, and the kink band width is W=5-6µm, approximately equal to ten ibre diameters (d=5.5µm). Failure o the [(±/ ) ] s laminates is sudden and mainly occurs within the gauge length; some specimens ailed near the grip (due to stress concentrations induced by the clamping orces) but results rom these tests were discarded. The longitudinal strain on the two aces o the specimen is initially the same, but as the applied load is increased the strains usually diverge, indicating bending. Final racture involves splitting parallel to the ibres, delamination between the and the plies and ibre
microbuckling. Fibre microbuckling in the layers is considered as the critical-damage mechanism, which causes catastrophic racture o the specimen. The investigation o plyinteraction phenomena by ractographic methods is very diicult because o the extensive post-ailure damage. Results or the average ailure stress or all successully tested specimens are presented in Fig. 1. They show that the ailure stress increases as the supporting 9 Stress (MPa) 8 7 6 5 5 1 15 5 3 35 4 45 5 55 6 65 7 75 8 85 9 Fig. 1: Experimental ailure stress or laminates [(±/ ) ] s ply orientation is reduced. At small angles the ibres are more aligned in the loading direction and oer higher bending resistance to the load. The ailure stress is maximum at 143 MPa [3] or a 1% laminate. Using the laminate plate theory and the measured laminate strength, the maximum stress developed in the plies within the multidirectional laminate was ound to be up to 17% lower than the stress in the unidirectional -laminate (Table 1). The lowest drop is 5% or =45, suggesting that or =45 the lateral support to the axial layers and the resistance to ibre microbuckling is optimum. The theoretical strengths or the [(±/ ) ] s laminates, predicted by the LPT and the maximum stress ailure criterion are higher (1-17%) than the measured values because the strain in the composite is assumed uniorm and the response linear elastic to ailure (Table 1). In the ollowing section, the ply interaction is analysed in terms o interlaminar shear stresses that develop at the ply interace. These stresses are then implemented in a recently developed ibre buckling model [6] that accounts or matrix non-linearities to estimate the ailure stress o the -ply and then determine the unnotched compressive strength o the various T8/94C [(±/ ) ] s laminates examined in this investigation. COMPRESSIVE STRENGTH PREDICTION The experimental work presented above shows that in multidirectional laminates dominated by internal plies, ailure occurs by in-plane microbuckling, and that the stress (or strain) level at which ibre microbuckling initiates is inluenced by the initial ibre waviness, the non-linear resin shear constitutive behaviour, and the orientation o the supporting plies adjacent to the layers. This is in agreement with results published earlier by other workers (see or instance Guynn et al [1]). On the other hand, in laminates were the surace plies are -plies, or in cross-ply laminates, ailure occurs rather by out-o-plane microbuckling [11]. The classical laminate plate theory [1] assumes that the state o stress within each lamina o a multidirectional laminate is planar. This assumption is not true in the vicinity o the ree
edge. Owing to stiness discontinuities between plies o dierent orientation, large interlaminar (through-thickness) stresses exist near the traction ree-edge regions in multidirectional composite panels (see or instance Pipes, Pagano [13]). Since ibre microbuckling initiates rom the specimen ree edge, where the ibre support is substantially reduced (as conirmed by experiments [1]), it becomes very important in the strength prediction o -dominated laminates to estimate the through-thickness edge stresses and include their inluence on the stability o the layers. This concept was put orward or the irst time by Berbinau [6], who urther showed that among the interlaminar stresses σ z, τ zy and τ zx, only the shear stress components (τ zy, τ zx ) are likely to inluence the in-plane movement o the ibres. The stacking sequence o the laminate was chosen so that σ z under compressive loading was compressive, making thereore no contribution to edge delamination or ibre (in-plane) microbuckling. It should be noted that although some existing models [14,15] included through-thickness stresses, they did not incorporate explicitly interlaminar ree-edge stresses nor studied their eect on the in-plane microbuckling o ibres. Briely, the general structure o the Berbinau and Wol theoretical approach [6,7] is to calculate irst the interlaminar shear stresses τ zy and τ zx, here by using the Puppo & Evensen model [16]. Then the shear stresses are incorporated into a general microbuckling equation [6, 7] or a ibre located at the specimen ree-edge in a -ply; equilibrium conditions are applied to relate the compression load to the deormation o the ibres and obtain the -ply ailure stress. Finally, this value is used with the maximum stress ailure criterion to predict the laminate compressive strength. In the present paper, the non-linear matrix response is considered by expressing the composite shear modulus as a unction o shear strain, G(γ). Inluence o interlaminar stresses The interlaminar stresses arising during the compression loading o a [(±/ ) ] s laminate are calculated by the Puppo & Evensen method [16] extended to laminates with an arbitrary number o layers [6]. A program was written using the Mathematica sotware [17] or this purpose. Results are presented in Fig. (a), where the maximum values o the interlaminar Interlaminar stresses (MPa) 1 5-5 -1-15 - -5-3 -35-4 -45-5 Interace 5 5 1 15 5 3 35 4 45 5 55 6 65 7 75 8 85 9 Interace or 3 Tau zx Tau zy Middle plane Sigma z 1 3 4 5 middle plane (a) (b) Fig. : (a) Maximum interlaminar edge stresses (strain =-1%). (b) Interace numbering
stresses are plotted as a unction o the orientation o the supporting plies. The ply interace where a given stress reaches its maximum value, is also indicated. Interaces are numbered as in Fig. (b), starting rom a surace ply. For between 6 and 65 both shear stress components are very small, suggesting that the instability eect on the neighbouring layers is not signiicant. The stress magnitude is directly proportional to the applied compressive strain ε. The stress results presented in Fig. (a) correspond to ε =-1%, which is close to the ailure strain measured or the T8/94C laminates, Table 1. The normal stress σ z has been determined by using the approximate solution o Pipes and Pagano [18]. Delamination initiation at the ree edge can then be predicted by using the Tsai-Wu quadratic ailure criterion [1, 19], τ + τ = τ (1) zx zy IL where τ IL is the interlaminar shear strength o the composite (ILSS=15 MPa). For 1 5, edge delamination is expected to occur, as shown on Fig. 3, at an applied compressive strain o.95%< ε <.88%. 14 1 ε =.95% Stress (MPa) 1 8 6 4 ε =.88% Maximum interlaminar stress ILSS 1 3 4 5 6 7 8 9 Fig. 3: Delamination criterion or an applied compressive strain between.88% and.95% -ibre microbuckling model The next step in the present strength prediction is to incorporate the interlaminar shear stresses into a general microbuckling equation or a -ibre (modelled as a Euler beam on a nonlinear oundation). In order to predict the onset o ibre microbuckling, it is convenient to model the initial ibre waviness by a sine unction v (x) o amplitude V and hal-wavelength λ. Under the inluence o the applied compressive stress, the sine unction v (x) will deorm into a new sine wave unction v(x) o amplitude V and hal-wavelength λ. We thus have: x v ( x) = V sin π λ and ( ) vx x = V sin π λ () Assuming small delections and a constant axial orce P, the equilibrium equation o the delected ibre is [, 1]: ( ) ( ) EI d 4 v v P dv p A G( ) d v v 4 + + γ = (3) dx dx dx
where E, A, and I are the ibre modulus, cross-section area and second moment o inertia, respectively. p is the distributed transverse orce normal to the ibre, Fig. 4. Several researchers have developed microbuckling models with various expressions or p (see or instance [14,, -4], and also [] or a comprehensive review on this topic). In Eqn 3, the composite shear modulus G is not constant but is a unction o the shear strain γ. That means that the non-linear shear response o the composite system is ully considered in the model. For the T8/94C system the shear stress-strain curve τ(γ) has been experimentally determined [3] and can be very well approximated (see [1]) by: τγ ( ) = τ G ult 1 Exp τ 1 ult γ (4) Eqn 4 is used to derive an analytical expression or the slope G(γ) o the τ γ curve, which appears in the dierential equation (3). For the T8/94C system, the ultimate shear stress τ ult is ound experimentally to be 18 MPa and the elastic shear modulus G 1 is 6 GPa. y x -ply W τ zx τ zy -ply x -ply W -V (a) (b) (c) Fig. 4: (a) Interlaminar shear stresses. (b) Edge ibre in a -ply. (c) FBD or a ibre. E D G E W y y M P Q m p q P+dP M+dM Q+dQ In the current work, p is expressed in terms o the interlaminar shear stresses τ zx and τ zy, which inluence the stability o the axial plies. Consider an initially wavy ibre located at the edge o the ply (y=w/), Fig. 4(b). It is subjected to a normal component p and a tangential component q along its length, Fig. 4(c). As a irst approximation, or the kinking deormation under consideration, it is plausible to set q=. The normal component p can be expressed in terms o τ zx and τ zy as:! ps = d τ n!! i+ τ n! j (5) () ( zx zy ) x where d is the ibre diameter, n! is the unit normal to the ibre,! i and! j are the unit vectors in the x and y directions, respectively. Due to symmetry, only a ibre wavelength needs to be considered. The orientation o the normal n! varies along the ibre length. It can be shown (see Berbinau et al [1]) that or a V value small compared to λ, the τ zx component in Eqn 5 can be neglected, and the normal stress p experienced by a ibre at the edge can be approximated by: d zy pv ( ) = d zy W zy W d v V dy W τ τ τ (6) The shear strain γ can be approximated by the slope o the curve (v-v )(x). The applied load
P on a ibre can be expressed in terms o the stress σ developed on the -ply, the ibre cross-section area A and the ibre volume raction V. We have: γ ( v) dv dx and P ( ) A σ V (7) Substituting equations 6 and 7 into Eqn 3 yields the ollowing dierential equation: ( ) A σ o ply dv d zy ( ) τ d ( v v ) EI d 4 v v 4 dx d v A G dv v + V dx dy W dx dx = (8) Eqn 8 is non linear and gives the compressive stress σ in the ply in terms o the maximum amplitude V o the buckled ibre, ibre properties and interlaminar shear stress τ zy. Strength results In the current analysis, Eqn 8 is solved numerically by using the Mathematica sotware [17]. The input data required are the ibre diameter d, ibre volume raction V, the interlaminar shear stress τ zy, the non-linear shear response, and the initial ibre amplitude V, which is calculated rom the microbuckling hal-wavelength λ and the initial misalignment angle φ. The initial ibre waviness may be measured rom a microscopic observation o the -ply; or the T8/94C system φ was ound equal to [1-4]. The hal-wavelength λ is o the order o 1-15 ibre diameters [1-4]. In Fig. 5, the maximum ibre amplitude, V, versus ply stress is presented or a T8/94C [(± 3/ ) ] s laminate. V increases slowly with increasing applied stress σ and then grows dramatically ast. Failure o the ply due to ibre V V σ * σ (MPa) Fig. 5: Microbuckling amplitude V vs. the stress σ on a -ply ([(3/-3/ ) ] s laminate). microbuckling occurs when the ibre amplitude V starts to increase asymptotically, Fig. 5. For =3, the critical stress σ o * is 961 MPa compared to 1178 MPa, estimated rom the measured strength data and the laminate plate theory. For a 1% laminate and λ=1 d, Eqn 8 predicts σ o * =968 MPa, 3% lower than the average experimental strength. It should be noted
that σ o * is strictly the critical stress at which ibre microbuckling initiates and not inal ailure stress. However, since inal ailure is considered as initiation controlled rather than propagation, σ o * could be taken as the -ply strength within the multidirectional laminate, which here appears to be a conservative estimate. In Table the -ply critical stress is presented or dierent (orientation o the supporting plies) and λ (ibre waviness) values. It can be seen that the interlaminar shear stress has a small inluence on the strength o the ply and is almost independent o the o-axis orientation. This is in good agreement with previous [9, 1] and present experimental evidence. Hence the o-axis plies orientation angle seems to have a small inluence on the ibre microbuckling initiation stress (or strain); the initial ibre waviness and non-linear shear response are more important parameters. Table : Predicted ailure stress σ * o the -ply in MPa, Eqn 8. O-axis ply orientation, λ=1 d Fibre wavelength λ=15 d λ= d 968 743 657 3 961 737.5 637 45 961 73 635.5 6 975 748 673 75 + 977 751.5 68 9 + 979 753.5 683.5 Note: + For >7, out-o-plane ibre microbuckling could occur and a dierent model is required. Once the ailure stress o the axial ply is known, the compressive strength o the multidirectional laminate can be determined by using the laminate plate theory and the maximum stress ailure criterion or simply by the stiness ratio method, given below: σ lam * N σ = n NE 11 k = 1 ( k) ( k) Ex (9) where σ Lam is the laminate strength, σ o * is the reduced strength o the lamina, N is the total number o plies in the laminate, E 11 is the ply stiness in the ibre direction, n is the number o plies o a given orientation and E x the modulus o a -ply along the load x-axis. Compression Strength (Mpa) 1 9 8 7 6 5 4 Theory Experiments 5 1 15 5 3 35 4 45 5 55 6 65 7 75 8 85 9 Fig. 6: Comparison o the experimental and theoretical compressive strengths or [(±/ ) ] s
The calculated -ply strength, σ o *, takes into account the ply interaction eect as well as the non linear response o the resin material and the initial ibre waviness. Varying rom to 9, the theoretical compressive strength σ Lam as a unction o the angle or the T8/94C [(/-/ ) ] s laminate is obtained by ollowing the above procedure. The results are summarised in Fig. 6 and compared to experimental strength measurements. The static compressive strength o the [(±3/ ) ] s laminate is higher than the other lay-ups with >3 due to higher bending stiness. However, under atigue loading the interlaminar shear stresses may introduce edge delamination, Fig. 3, which would reduce the support to the layers, causing ibre microbuckling and premature laminate ailure. In this case, the laminate with ±6 o-axis plies is expected to perorm better due to lower interlaminar stresses. CONCLUSIONS Experiments show that the static compressive ailure o unidirectional and multidirectional unnotched [(±/ ) ] s T8/94C carbon ibre-epoxy laminates is controlled by -ibre microbuckling. Average ailure strains ε av or each o the our angles (3,45,6,75 ) examined are scattered between -.9 % and -1 %. The present tests do not reveal any obvious trend as to the variation o the ailure strain with the angle, hence the o-axis layers have only a small inluence on the compressive ailure stress (or strain). Specimens with =45 appear to provide better resistance to ibre microbuckling with an average ailure strain o -.95%. The -ply stress in a such laminate is 13 MPa compared to 143 MPa measured or the 1% laminate. The 1% strength reduction can be due to ply interaction (edge eects) and probably manuacturing deects (interlaminar voids, resin rich regions). Soutis et al [8, 9] and Guynn et al [1] obtained similar experimental results. The modiied Berbinau-Wol [6, 7] microbuckling model incorporates the non-linear resin shear response, ibre imperections (waviness) and ibre content. In the model the eect o the interlaminar shear stress τ zy that develops at the laminate edges is included explicitly to account or ply interaction. It is revealed that the τ zy stress component can damage the stability o the axial plies but the eect o resin sotening and ibre waviness are more signiicant. The instantaneous tangent shear modulus G rather than the elastic modulus controls the ibre microbuckling initiation o initially misaligned ibres. Theoretical predictions are accurate to within 1-15%. It is worth pointing out here that out-o-plane microbuckling is the ailure mode or =9 [11], and that experiments [5] suggest that it could also be the ailure mode or > 7. In act, the present model hints at such a mode competition, since above =6 the τ zy stress can be shown to hinder the in-plane microbuckling o the ibres [6]. Simultaneously, out-o-plane ibre movement is acilitated by the act that the bending stiness o the (±) plies about the! y axis decreases as increases [5]. Since the present ailure model is based on in-plane movement o the ibres, it might not be valid above =7. More experimental observations and theoretical modelling on the ailure o [(±/ ) ] s laminates with >7 are thereore needed. A theoretical and experimental investigation o the inluence o the interlaminar stresses near the ree edge on -ibre microbuckling initiation under compression-compression atigue loading will also be o interest. Matrix cracking and edge delamination may then trigger ibre microbuckling at lower applied strains, and the selection o the supporting ply orientation may become more critical than in static loading.
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