Compressive Strength of Composite Laminates with Delamination-Induced Interaction of Panel and Sublaminate Buckling modes

Andrew T. Rhead 1 and Richard Butler 2 and Giles W. Hunt3

University of Bath, Bath, BA2 7AY, United Kingdom

Abstract

Compression After Impact (CAI) strength is critical to the safety and weight of carbon fibre aircraft. In this paper, the standard aerospace industry practice of using separate analyses and tests for panel buckling and CAI strength is challenged. Composite panels with a range of stacking sequences were artificially delaminated and subject to compression testing in a fixture that allowed local sublaminate and global panel buckling modes to interact. Compared to panels without delamination, interaction of buckling modes reduced panel buckling strains by up to 29%. Similarly, compared to delaminated panels restrained against panel buckling, interaction reduced delamination propagation strains by up to 49%. These results are the first to indicate that restriction of interaction during CAI testing is unconservative and therefore potentially unsafe. A novel integration of an analytical Strip model, for sublaminate buckling driven delamination propagation, and a Shanley model, for determining increased local strain due to sublaminate-buckling-induced panel curvature, is used to calculate the reduction in strength due to buckling mode interaction. Assuming a typical sublaminate post to pre-buckling stiffness ratio of 0.65, the difference in integrated model and experimental results is <11% - a level of accuracy will allow the integrated model to drive initial design studies.

1. Introduction

Owing to excellent specific strength and stiffness properties, Carbon Fibre Reinforced Plastic (CFRP) is replacing traditional aircraft construction materials (e.g. aluminium) as the material of choice for significant proportions of the structure of the latest generation of aircraft. However, pristine properties of CFRP are significantly degraded by impact damage in the form of delamination and intra-ply cracking caused,, for example bydropped tools and runway chippings . Impact damaged laminated CFRP structures typically show significantly reduced compressive strength owing to a propensity for delaminated sublaminates to buckling and drive delamination growth.

Compression After Impact (CAI) of coupons restrained against global buckling1-4 and larger structures with geometric stiffness e.g. stiffened panels5-9, is a topic that continues to see significant research. Work by the authors10,11 on compression of artificially delaminated panels has, despite efforts to restrain global panel buckling, shown that interaction of local sublaminate and global panel buckling modes caused both delamination propagation and panel buckling to occur at strains well below the expected values. Although the interaction of buckling modes of columns with through-the-width delaminations has been considered12-15, only limited research has been undertaken to characterize the behaviour involved in the interaction between enclosed sublaminate buckling and global panel buckling16-18. Analyses, where they exist, are typically Finite Element (FE) based and hence are computationally expensive, relying on significant processing power19-21. Few are experimentally10,11 verified and non-FE analyses are currently limited to delamination propagation following sublaminate buckling in flat unbuckled plates 10,11,22. More critically and because stiffness and strength issues are typically considered separately, current industry design practice for calculating ultimate capacity does not take interaction of buckling modes into account. Clearly, this separation could cause designers to overestimate ultimate strength and there is a need to incorporate models that consider interaction into future design methods.

This paper presents a new analytical and experimental results that show the deleterious effect that buckling interaction has on the strain at which both panel buckling and delamination propagation occur. Reductions in delamination propagation strains are demonstrated by a comparison of results for compression tests on large flat artificially delaminated panels that allow interaction of sublaminate and panel buckling modes and panels that were

1 Lecturer, Materials and Structures Research Centre, Department of Mechanical Engineering.2 Professor, Co-Director, Materials and Structures Research Centre, Department of Mechanical Engineering.3 Emeritus Professor, Materials and Structures Research Centre, Department of Mechanical Engineering.

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restrained against panel buckling11. Similarly, test results for panels without delaminations provide a baseline against which reductions in panel buckling strains can be measured.

A novel integration of the authors’ analytical Strip model10 for sublaminate driven delamination propagation and a Shanley model that accounts for increased local strains due to sublaminate-buckling-induced global panel curvature, is derived. The integrated model is shown to provide an estimate of the experimental delamination strain that is conservative and sufficiently accurate to be useful for initial design and optimization purposes.

2. Experimental compression testing of panels: panel design, test methods and results

From this section forward, groups of plies above a delamination that buckle locally under compression will be referred to a ‘sublaminate’. The full laminate, of which the sublaminate is a part, will be referred to as the ‘panel’.

2.1 Panel design

Three sets of panels are compared in this work, R, U and P, see Table 1. Panels were designed to establish whether the interaction between local sublaminate buckling and global panel buckling had an effect on the strain at which (i) delamination propagation and (ii) global panel buckling occurred. All panels were manufactured from Hexcel T700GC/M21 pre-preg CFRP plies with material properties E11 = 136 GPa, E22 = 8.9 GPa, G12 = 4.5 GPa, ν12 = 0.35, t = 0.25mm and G1C = 550 J/m2. Panels were cured in an auto-clave following the manufactures cure cycle.

Set of panels U were artificially delaminated by inserting a non-stick circle of polytetrafluoro-ethylene (PTFE) (0.0125mm thick and 39mm in diameter) at the panel (plan form) centre and desired depth during manufacture. Set of panels R have been presented previously10,11 and were artificially delaminated in the same way as set U. Set of panels P (Pristine) were not artificially delaminated, see Table 1. Throughout the text panels will be referred to using the sublaminate stacking sequences created by delamination; these are highlighted in bold on the stacking sequences in Table 2. An additional subscript suffix R, U or P will included to identify to which set of panels results belong.

Table 1. Description of panel size, state of delamination and amount of restraint against panel buckling.

Set of PanelsDelaminated? Gauge width and length (mm) Gauge area

(mm2)(Y/N) W L A

R (Restrained) Y 85 (Ø) 5674U (Unrestrained) Y 165 198 32670

P (Pristine) N 165 198 32670

Table 2. Laminate and sublaminate stacking sequences (sub-laminates in bold), thicknesses and in-plane and out-of-plane stiffness properties.

Stacking sequence(Sublaminate in bold)

Thicknesses (mm) In-plane stiffness (kN/mm) Out-of-plane stiffness (kNmm)

T T* A11* A11 A12 A22 D11 D12 D22

[02/90/±45/±45/90]s 4 0.75 70.8 231 73.5 231 486 56.1 214

[±30/0/902/0/90/0]S 4 0.5 41.1 301 35.4 237 422 87.2 216

[±45/02/-45/90/+45/90]S 4 0.5 21.3 231 73.5 231 370 125 194[±60/02/±30/∓30]S 4 0.5 9.08 320 81.1 128 328 102 280

All sets of panels used the same stacking sequences, see Table 2, thus allowing for direct comparison. All stacking sequences show a significant difference between panel and sublaminate Poisson’s ratios . Mismatches have previously been seen11 to affect sublaminate buckling and delamination propagation strains. This is a consequence of compressive (e.g. [±30] sublaminate) or tensile (e.g. [±60]) transverse strains being induced in the sublaminate. It is noted that [±30] and [±60] type panels are atypical designs for aerospace components. However, the development of tow steering techniques24-25 means similar stacking sequences may well play a role in future aerospace components.

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A [02/90] sublaminate, as opposed toa [02] sublaminate, as the latter previously10 displayed intra-ply splitting of surface plies during buckling of the sublaminate. This made comparison with analytical methods difficult.

Dimensions for sets of panels U and P are given in Fig. 1 and Table 1. Dimensions of R panels are given elsewhere10,11. Panels were sized such that during compression testing different amounts of restraint against panel buckling would be applied. Set of panels R were Restrained against panel buckling by an anti-buckling guide10,11. This allowed sublaminate buckling driven delamination propagation to be studied in relative isolation from the effects of panel deformation. Set of panels U were relatively Unrestrained, having a larger unsupported area than set R, to encourage interaction of local sublaminate and global panel buckling. Undelaminated set of panels P (Pristine) were also unrestrained during compression testing providing a benchmark against which panel buckling strains of set U could be compared. A comparison of gauge area (unsupported region) for U, R and P is given in Table 1.

Fig. 1 Test fixture and panel dimensions for testing of laminates unrestrained against global buckling.

2.2 Test methods

Figure 1 shows the fixture used for U and P panel tests. The fixture is a modified version of the one described in ASTM standard D713726. Both the height and width were increased to create a larger unsupported area. In a slight departure from ASTM D7137, loading edges were fully clamped to avoid edge failure. Solid red lines on the left hand side of Fig. 1 indicate the extent of 10mm horizontal regions of clamped support through which the compressive displacement is applied. Dashed vertical red lines indicate the position of the vertical knife edge simple supports. For all panels, an Instron 5585H test machine was used to apply compressive displacement in the 0o direction at a rate of 0.1 mm/min. To prevent adhesion between the PTFE disc and panel prior to testing to failure, all artificially delaminated panels were compressed until sublaminate buckling occurred and then reset. Tests to failure were halted when delaminations propagation was detected or, for panels that did not demonstrate sublaminate buckling, when global failure occurred. To ensure alignment of loading and panel axes and hence the application of pure axial compression, strains were recorded both prior to and during experiments by pairs of vertically aligned back-to-back strain gauges with the exception of the [02/90]U panel where a single pair was omitted, see Fig. 1(b) and Fig. 5(a).

A Limess/Correlated Solutions Digital Image Correlation (DIC) system employing a stereo pair of Photron SA3 cameras captured 3D surface displacement of panels in relation to their reference position under zero load throughout testing. Post-processing of DIC images with VIC 3D software27 allowed visualization of buckling modes and delamination growth, see Fig. 2. Post-test, panels were subject to non-destructive evaluation, using an Ultrasonic Sciences Limited C-scan system with a 35MHz probe.

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2.3 Experimental results

Figure 2 shows DIC images of the sequence of phenomena that occurred during loading of the [±30] U panel; sublaminate buckling (contrast Figs 2(a) and (b)) followed by panel buckling (Fig. 2(c)) and finally delamination propagation (Fig. 2(d)). This is typical of events observed in the [±45]U and [02/90]U panels. No sublaminate buckling, and hence no delamination propagation, was observed in the [±60]U panel. White circles in Fig. 2 show the extent of the original delamination. The uneven surface in the far-field region of Figs. 2(a)-(d) is a result of the inability of the system to resolve very small deflection changes at low-levels of load and the non-smooth surface of the painted sample.

Fig. 2 DIC images of the development of sublaminate buckling followed by delamination propagation in the [±30]UR unrestrained test: (a) Prior to buckling (25kN). (b) Initial sublaminate buckling modeshape (62kN). (c) Fully developed sublaminate buckle (90 kN). (d) Propagated delamination with significant panel deflection (146 kN). White circles indicate the extent of the artificial delamination. Colours (available online) represent out-of-plane displacement from an unloaded reference state.

Panel buckling strains for all panels are given in Fig. 3(a). In all cases experimental strains were identified by correlating qualitative events detected on DIC images (e.g. Figs. 2 and 4) with events on the average strain vs. load plots (Fig. 5). For example strain gauge readings bifurcate at approximately 100 kN in Fig. 5(b). Interaction of

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sublaminate and panel buckling strains in U panels and imperfect contact with boundaries in both U and P panels meant that panel buckling strains in all tests were approached asymptotically. Consequently, precise values for panel buckling cannot be reported and a range of strain values is given in Fig. 5. Where applicable, interaction of panel and sublaminate buckling modeshapes caused the sublaminates modes to form on the more highly compressed surface of the deformed panel.

Fig. 3 (a) Comparison of experimental panel buckling strains, both with (Exp. Delam) and without (Exp. Prist.) delamination, with numerical ABAQUS (FEA) and VICONOPT (VIC.) predictions for pristine panel buckling strains. (b) Comparison of delamination propagation strain for U and R panels. Shaded regions in experimental results account for uncertainty in experimental results owing to in (a) slow development of panel buckling and in (b) delamination propagation.

Fig. 4 (a) DIC image of out-of-plane displacement of the [02/90]U test at 92kN. A propagated delamination and a sublaminate buckle interacting with a panel buckling mode immediately before the conclusion of the test is visible. (b) Vertical cross-section through the horizontal mid-point of (a) showing out-of-plane displacement (solid line) and strain in the 0° direction (dashed line). (c) Post test C-scan of the [0 2/90]U panel showing horizontal propagation; the original artificial delamination indicated by a white circle.

Delamination propagation results for all panels are given in Fig. 3(b). To determine if delamination propagation had occurred, a circle with the diameter of the PTFE insert was overlaid onto DIC images such that it contained the fully formed sublaminate buckle . Propagation was then judged to have occurred if the buckled was, in subsequent images

(a) (b)

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at higher loads, no longer contained by the circle, contrast Figs 2 (c) and (d) and see Fig. 4. However, for slowly occurring growth of the buckled area it became difficult to ascribe a single load value to the point of propagation and hence a range of propagation strains is given. Post-test C-scans were also used to confirm delamination growth, e.g. see Fig. 4(c).

In all tests propagation direction was principally controlled by panel buckling/bending. The modeshape of the buckled sublaminate and the fibre angle of plies adjacent to the delamination having a secondary influence. The [±30]U sublaminate buckle, for example, has a rectangular shape with a transverse orientation but delamination propagation initiates in the top left corner, see Fig. 2(d). [±45]U panel showed a similar propagation event. The [02/90]U panel buckled sublaminate caused propagation at 90 to the applied compression This was a consequence of a pronounced panel buckling modeshape and the orientation of the lower ply in the sublaminate, see Fig. 4.

Fig. 5 Load vs strain plots for (a) [02/90]UR (b) [±30]UR (c) [±45]UR and (d) [±60]UR panels. Inset shows schematic of strain gauges and circular delamination. Dotted lines represent the average strain gauge values. Red crosses indicate failure and/or delamination propagation.

3. Theoretical methods and results

In the following, variables relating to the sublaminate have a superscript suffix *. The aim of this section is to derive a model that can predict the far-field strain at which delamination propagation occurs when local in-plane strains, in the region of the sublaminate, are magnified by bending of the panel following sublaminate buckling. Figure 6 demonstrates the effect of sublaminate and panel buckling interaction on the local strain applied to a delaminated sublaminate. The blue line tracking the vertical axis in Fig. 6 makes an assumption (thin-film) that the panel remains flat following sublaminate buckling. This situation has been the subject of models previously developed by the authors10,11 and others1. A consequence of this assumption is that local in-plane strain applied to the sublaminate, ε*, matches that applied in the far-field to the panel, ε. On the right hand side of Fig. 6 (red curve) it is assumed once sublaminate buckling occurs, and sublaminate stiffness is lost, then, owing to a shift in its neutral axis, the panel starts to bend. This causes a local increase in ε* in comparison to ε. This increase is captured by modifying a Shanley model18,28 for eccentric loading of a beam to account for the post-buckling response of a plate with all edges clamped. The resulting local strain ε* is then input into the author’s Strip Model for delamination propagation and a strain below which delamination will not occur is established.

(a) (b) (c) (d)

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3.1 Derivation of increased local strain ε*

Following the classic formulation29 for an eccentric strut with buckling strain εC, it is assumed that the mid-point deflection δ, see Fig. 7, caused by eccentric axial strain is given by,

δ=e (sec( π2 √ ε

εC ))(1)

where e is the off-set in the panel neutral axis, caused by an asymmetric stiffness loss about the panel mid-plane following sublaminate buckling, calculated using a moment balance about the mid-plane of the panel,

e=(T−t¿

2 )( A11¿(1−k )A11+ A11 ¿(k−1))

(2)

where T is the panel thickness, t* is the sub-laminate thickness, A11* is the longitudinal axial stiffness of the

sublaminate in the 0o fibre direction, k is the post-buckling stiffness reduction factor for the sublaminate (where k = 1 means full stiffness is retained by the sublaminate and k = 0 means zero stiffness is retained) and A11 is the longitudinal axial stiffness of the panel in the 0o fibre direction.

Fig. 6 Equilibrium diagram showing induced local bending strain with increasing axial panel strain. Left hand diagrams and vertical line represent the assumption of a flat panel; no global deformation occurs following sublaminate buckling. Right hand diagrams and curves represent the integrated Strip and Shanley model: panel bending occurs following sublaminate buckling and an additional strain (ε*- ε) is induced at the boundary of the delamination.

Eq. (1) is based on beam mechanics and thus ε < εC for all deflections δ. As this paper deals with applications to panels, which have post-buckling stiffness and thus allow for ε > εC, a modification is required to Eq.(1) and the

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assumption is made that the beam buckling asymptote ε = εC can be replaced by a post-buckling asymptote30,31 for an isotropic panel simply-supported on all edges.

ε CP=εC+ π 4

4 ( δL )

2

(3)

where L is the panel gauge length (see Table 1) and εC is the panel buckling strain. It is noted that improved accuracy may be achieved by tailoring the post-buckling asymptote to both the support conditions and stacking sequence of the panel under investigation. However, this would come at the cost of additional computation and loss of a simple general form. Earlier work32 also indicates that the shape of the post-buckling asymptote is relatively insensitive to these considerations. Substitution of the asymptote from Eq. (3) into Eq. (1) gives

δ=e (sec( π2 √ ε

εC+π4

4 ( δl )

2))(4)

In this paper, calculation of εC is undertaken in both the infinite strip program VICONOPT33 and using S4 shell elements in the ABAQUS Finite Element Analysis (FEA) program34 (see Fig. 3(a) for comparison of results). Boundary conditions representative of those found in the experimental set-up were used i.e. clamped loading edges and simply-supported sides. Determination of εC using both packages is trivial and thus further details are omitted.

Fig. 7 Schematic of induced moment Mx due to offset e arising from interaction of local sublaminate and panel buckling modes.

As bending of the delaminated panel and thus deflection of the panel does not occur until sublaminate buckling occurs (i.e. ε = εC*) an initial condition δ = 0 at ε = εC*

needs to be applied. This results in an initial ‘effective’ imperfection,

δ 0=e¿¿(5)

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Note that as δ = 0 at ε = εC* so εCP= εC for the initial imperfection, see Eq.(4).

From Engineer’s bending theory, the strain applied to the edge of the delaminated sublaminate is given by,

ε ¿=ε+M x

2 EI(T−t ¿ )

(6)

where Mx is the applied moment and classical lamination theory gives

EI=D11−D12

2

D22(7)

Here D11 is the bending stiffness of the panel aligned with the 0o fibre direction, D12 is the bending stiffness that couples curvature perpendicular to 0o fibre direction to Mx and D22 the bending stiffness of the panel perpendicular to the 0o. (It is assumed that transverse moments are zero and that bending orthotropy exists i.e. D13 = D23 = 0.) Note that the term (T-t*)/2 in Eq. (6) implies the additional strain is computed at the sublaminate mid-plane.

Maximum ε* will occur at the mid-length of the panel and it is assumed this is representative of the strains in the 0 o

fibre direction at the sublaminate boundary parallel to the loading edges of the panel. The deflection δ and moment Mx at the this point can be related to the in-plane loads (Nx) by the following equation,

M x=N x δ=EA εδ(8)

where classical lamination theory for composite panels gives

EA=A11−A12

2

A22(9)

Here A22 is the in-plane stiffness of the panel perpendicular to the 0o fiber direction and A12 couples strain perpendicular to the 0o fiber direction to Nx. Applying Eq.(8) with the imperfection δ0 derived in Eq. (5) gives,

ε ¿=ε (1+(T−t ¿) EA2 EI (δ−δ0 ))

(10)

Rearranging Eq.(4) for ε and substituting into Eq.(10) gives a function for the local in-plane strain ε* in terms of δ,

ε ¿=( 2π

arc sec( δe ))

2 (εC+ π 4

4 ( δl )

2)(1+(T−t ¿) EA2EI (δ−δ0 ))

(11)

Note that eq.(11) is transcendental in δ and thus cannot be inverted to find δ for a specific value of ε*. Instead, δ must be steadily increased until the required ε* is reached.

3.2 Derivation of far-field threshold strain εth for delamination propagation

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In order to determine the far-field threshold strain, εth, below which delamination propagation will not occur, the equation for local in-plane strain ε* developed in Eq.(11) is substituted into the Strip Model10,11 given by

G=A11

¿

2¿¿

(12)

where *11A is the axial stiffness of the sublaminate and εC* is the sublaminate buckling strain. εC* is a 2D circular

plate buckling strain calculated using VICONOPT33 (calculation of buckling strain is also possible via other methods such as FEA10,11). A clamped circular boundary is assumed35,36 and only material properties given in section II-A are required.

To determine εth*, δ is increased in Eq.(11), thereby increasing the local strain ε* in Eq.(12), until Strain Energy

Release Rate (SERR) G equals GIC the SERR required to cause Mode I fracture of the matrix. At this point δ = δth, the threshold deflection required to cause delamination propagation. The far-field threshold strain εth is then derived by setting δ = δth in Eq.(4) and inverting to find the required ε.

The following brief notes are made about the Strip Model with full derivations available in other publications10. In the Strip Model, G which is developed by the post-buckled sublaminate during compression, is established through a comparison of bending and membrane energies in the sublaminate prior to and following infinitesimal delamination propagation. The Strip Model implicitly includes an assumption that the panel remains flat thus preventing double accounting of additional strain due to panel bending. GIC is a lower bound to a mixed mode condition, i.e. GIC < GIIC

< GIIIC and has previously36 been shown to produce lower bound values of εth*.

3.3 Theoretical results

Figure 8(a) shows integrated model results for the [02/90]U, [±30]U and [±45]U panels. The left hand boundary of each shaded region relates to use of k = 0.35 in Eq. (2). The right hand bound is given by k = 0.65. These values of k are typical bounds for post-buckled orthotropic laminates31,32. Intervening values of k produce strains within the shaded area in Fig. 8(a). Corresponding experimental results from Figs. 3 are indicated by filled circles.

The Shanley model relies on a quadratic approximation30 in δ (the sublaminate buckling induced deflection of the overall panel) to calcuate the post-buckling asymptote, see Fig. 6 and Eq. (3). This approximation relies on the experimental parameter L, the gauge length of the panel in the 0° direction. For the results in Fig. 6, L = 198mm. However, as experimental boundary conditions are often imperfect, thus making the true panel gauge length uncertain, the effect of L on the integrated model results for the [±45]U is explored in Fig. 7(b).

(a) (b)

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Fig. 8 (a) SERR as a function of far-field strain for the [±30], [02/90] and [±45] sublaminates. The left hand bound of each shaded region is given by k = 0.35 and the right by k = 0.65. Points 1-3 show indicate lower bound sublaminate buckling strains and points 1'-3' indicate corresponding lower bound experimental εth

results for [±45]U, [02/90] U and [±30] U, respectively. (b) SERR as a function of ε for the [±45]U panel employing the with k = 0.65. Curves indicate the effect of varying panel length in 50mm increments from 50mm to 200mm.

4. Discussion

4.1 Panel buckling

Figure 3(a) shows results for experimental and numerical prediction of εC for sets of panels U and P. Good correlation is shown between VICONOPT and FEA results with strains matching to within 3.5%. However, experimental support conditions were not the perfect support conditions applied in the VICONOPT and FEA analyses meaning εC was over predicted by up to 29%. Buckling strains for U panels were over predicted by as much as 43%. A comparison of experimental buckling strains for U and P panels indicates that the interaction of sublaminate and panel buckling modes results in a reduction in panel buckling strain of up to 26% with a typical reduction of 10-15%. Despite the lack of sublaminate buckling, εC for [±60]U was 82.6% of the [±60]P result indicating that the presence of delamination with or without sublaminate buckling is destabilising. The above highlights the care which must be taken when relying on computational methods and separate tests for buckling and damage tolerant performance.

4.2 Compression after impact strength

No delamination propagation occurred in the [±60]U and [±60]R tests and so no further discussion of these tests is undertaken. Figure 4(b) shows that local strain levels at the edge of the buckled sublaminate in the [0 2/90]U coupon are significantly greater than the far-field at the point of failure. This is also true of the other tests and lends credence to the basis of the integrated model.

Figure 7(a) shows that εC* was reasonably well predicted for all panels. However, it should be noted that εC* for the [±30]U panel was the least well predicted owing to the significant contact between the sublaminate and panel, extension-twist coupling and high Poisson’s ratio exhibited by the sublaminate. Neither FEA nor VICONOPT analyses were fully able to capture the effect of these phenomena.

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Figure 3(b) shows that εth is up to 49% lower for U panels compared to R panels. Differences in εth were smallest for where panel buckling and thus interaction with sublaminate modeshapes was previously detected in the R panels10. For example in the [02/90]U panel the buckling of 0 surface plies significantly reduces laminate stiffness on one side of the panel thereby destabilising it. If these panels were fully restrained against buckling (e.g. using a sandwich construction) then it is likely even higher propagation strains than those in Fig. 3(b) would result. Hence, the drop in εth when interaction of buckling modes occurs could be even higher than was demonstrated. Assuming that a ‘no growth’ design concept is followed then comparison of Fig. 3 results show , that small panel tests, where panel buckling is restrained, should be used to determine damage tolerant strain allowables. .

Figure 8(a) shows delamination propagation strain results from the integrated model produce a conservative lower bound approximation to the maximum average strain results from the [02/90]U, [±30]U and [±45]U tests. Results for k = 0.65 (k = 0.35) differ from experimental values by a maximum of 11% (17%). Theoretical results are also shown to place the laminates in the same order of strength as the experiments.

Figure. 8(b) shows the effect of varying panel length on calculation of εth. Increasing panel length is shown to have a nonlinear effect on εth. The loss in εth when L is varied from 150mm to 200mm is considerable smaller than that caused by a variation from 50mm to 100mm. As such, sizing of test panels should be more closely linked to panel lengths seen in parent aircraft structures whether these be formed by rib/stiffener bays or nodes in the buckling of the full structure. Fig. 8(b) also shows that as resin fracture toughness GIC increases e.g. with next generation resin-matrices or inclusion carbon nanotubes, small panel tests will become less and less representative of larger structures. Equally small panels are more representative of the CAI strength of structures made with older resin-matrices with lower fracture toughnesses.

5. Conclusions

Analytic and experimental investigations have been carried out to determine, for the first time, the effect on panel buckling and delamination propagation strains of the interaction of sublaminate and panel buckling modes. Experimentally, interaction of buckling modes is shown to reduce panel buckling strains by up to 29% and delamination propagation strains by up to 49%. Panel buckling strains are seen to reduce owing to a loss of panel axial and bending stiffness initiated by local buckling of the sublaminate. Reduced delamination propagation strains are a consequence of increased local strains in the buckled sublaminate caused by interaction with a global panel buckling modeshape. These reductions indicate that the current industry practice of defining allowable strains using separate tests for damage tolerance and panel buckling is unconservative and potentially unsafe. Analytical results show this situation may worsen as future generations of resin systems produce better fracture toughness.

Assuming a typical sublaminate post-buckling to pre-buckling stiffness ratio of 0.65, then in a panel where local sublaminate and global panel buckling modes are allowed to interact, it has been shown that a novel integration of Strip propagation and Shanley models can conservatively predict, with an accuracy of 11%, the applied (far-field) strain required to cause propagation of a delamination. The high computational efficiency of the integrated model means it is well suited to initial design and optimization for compression after impact strength.

AcknowledgmentsRichard Butler holds a Royal Academy of Engineering/GKN Aerospace Research Chair. The help of Dr. Wenli Liu (now at Cranfield University) in obtaining finite element results is gratefully acknowledged.

References

1Chai, H., Knauss, W.G., Babcock, C.D., “Observation of damage growth in compressively loaded laminates,” Experimental Mechanics, Sept 1983, p.329-337.2Sanchez-Saez, S., Barbero, E., Zaera, R., Navarro, C., “Compression after impact of thin composite laminates,” Comp. Sci. Tech., Vol. 65, 2005, pp.1911-1919.3Nilsson, K. -F., Asp, L. E., Alpman, J. E., Nystedt, L., “Delamination buckling and growth for delaminations at different depths in a slender composite panel,” Int. J. Solids and Structures, Vol.38, 2001, pp.3039-3071.4Nilsson, K. -F., Thesken, C., Sindelar, P., Giannakopoulos, A. E., Storåkers B., “A theoretical and experimental investigation of buckling induced delamination growth”, J. Mech. Phys. Solids, Vol.41, No.4, 1993, pp.749-782.

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5Riccio, A., Raimondo, A., Fragale, S., Camerlingo, F., Gambino, B., Toscano, C., Tescione, D., “Delamination buckling and growth phenomena in stiffened composite panels under compression. Part I: An experimental study” Journal of Composite Materials, Vol. 48, No.23, 2013, pp.2843-2855.6Riccio, A., Raimondo, A., Di Caprio, F., Scaramuzzino, F., “Delaminations buckling and growth phenomena in stiffened composite panels under compression. Part II: a numerical study,” Journal of Composite Materials, Vol. 48, No.23, 2013, pp.2857-2870.7Greenhalgh, E., Meeks, C., Clarke, A., Thatcher, J., “The effect of defects on the performance of post-buckled CFRP stringer-stiffened panels,” Composites: Part A, Vol. 34, 2003, pp.623-633.8Greenhalgh, E., Clarke, A., Millson, B., Thompson. R., “Investigation of impact damage in a CFRP wingbox under load,” 41st AIAA Structures, Structural Dynamics, and Materials Conference and Exhibit, Atlanta, GA, U.S.A., April 2000.9Kontis, N., “Damage tolerance of composite stiffened structures,” Ph.D. Thesis, Mechanical Engineering Dept., Bath Univ., Bath, UK, 2008.10Butler, R., Rhead, A. T., Liu, W., Kontis, N., “Compressive strength of delaminated aerospace composites” Phil. Trans. R. Soc. Lond. A. Vol.370, No.1965, 2012, pp.1721-2026.11Rhead, A. T., Butler, R., Baker, N., Liu, W., “The influence of surface ply fibre angle on the compressive strength of composite laminates containing delamination,” Aeronautical J., Vol. 116, No. 1186, 2012, pp.1315-1330.12Ovesy, H.R., Kharazi, M., “Compressional stability behaviour of composite plates with through-the-width and embedded delaminations by using first order shear deformation theory,” Computers and Structures, Vol. 89, 2011, pp.1829–1839.13Hunt, G.W., Hu, B., Butler, R., Almond, D.P., Wright, J.E., “Nonlinear modeling of delaminated struts,” AIAA Journal, Vol.42, No.11, Nov 2004, pp.2364-2372.14Hwang, S. -F., Liu. G. -H., “Buckling behaviour of composite laminates with multiple delaminations under uniaxial compression,” Composite Structures, Vol.53, 2001, pp.235-243.15Aslan, Z., Sahin, M., “Buckling behavior and compressive failure of composite laminates containing multiple large delaminations,” Comp. Struct., Vol. 89, 2009, pp.382-390.16Damghani, M., Kennedy, D., Featherston, C., “Global buckling of composite plates containing rectangular delaminations using exact stiffness analysis and smearing method,” Computers and Structures, Vol. 134, No.3, 2014, pp.32–47.17Ovesy, H.R., Taghizadeh, M., Kharazi, M., “Post-buckling analysis of composite plates containing embedded delaminations with arbitrary shape by using higher order shear deformation theory,” Composite Structures Vol. 94, No.3, 2012, pp.1243–1249.18Rhead A. T., Butler R., Hunt, G.W.,"Compressive Strength Following Delamination Induced Interaction of Panel and Sublaminate Buckling," 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Waikiki, Hawaii, April 2012.19Gasco, F., Feraboli. P., “A crack length control scheme for solving nonlinear finite element Equations in stable and unstable delamination propagation analysis,” Composite Structures, Vol.117, 2014, pp.267-273.20Saavedra, K., Allix. O., Gosselet, P., “On a multiscale strategy and its optimization for the simulation of combined delamination and buckling,” Int. J. Numerical Methods in Engineering, Vol.91, 2012, pp.772-798.21Bouvet, H., Castaniéa, B., Bizeulb, M., Barraua, J., “Low velocity impact modelling in laminate composite panels with discrete interface elements”, Int. J. Sols. Struct., Vol.46, No.14-15, 2009, pp.2809-2821.22Suemasu, H., Ichiki. M., “Analytical study on low compressive strength of composite laminates with impact damage,” Composite Structures, Vol.104, 2013, pp.169-175.23Kim, B. C, Potter, K., Weaver, P., “Continuous tow shearing for manufacturing variable angle tow composites,” Compos Part A: Appl. Sci. Manuf., Vol. 43, No. 8, 2012, pp.1347-1356. 24Liu W., Butler R., "Buckling Optimization of Variable Angle Tow Panels Using Exact Strip Models," 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Waikiki, Hawaii, April 2012.25Wu, Z., Raju, G., Weaver P. M., “Buckling of VAT plates using energy methods,” 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Waikiki, Hawaii, April 2012.26ASTM international, “Standard test method for compressive residual strength properties of damaged polymer matrix composite plates,” ASTM D7137/D 7137M-07, 2007.27Correlated Solutions, “VIC-3D system”, http://www.correlatedsolutions.com/vic-3d accessed 11/07/2014.28Shanley, F. R., “Inelastic column theory,” Journal of the Aeronautical Sciences, Vol.14, No.5, May 1947, pp.261-268.

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29Case, J., Chilver, A., H., Ross, C., T., “Strength of materials and structures,” Edward Arnold, Hodder & Stoughton, London, 1993, pp.363-365.30Thompson, J.M.T., Hunt, G.W., “Elastic Instability Phenomena,” John Wiley & Sons Ltd, Chichester, 1984, pp.170-173.31Koiter W.T., Pignataro M. “A general theory for the interaction between local and overall buckling of stiffened panels”. Delft University of Technology Report, WTHD-83, 1976.

32Weaver P.M., Diaconu C.G. “Approximate solution and optimum design of compression-loaded, postbuckled laminated composite plates”. AIAA J, Vol.43, No.4, pp 906-914, 2005.

33Williams, F.W., Kennedy, D., Butler, R., Anderson, M.S., “VICONOPT – Program for exact vibration and buckling analysis or design of prismatic plate assemblies,” AIAA Journal, Vol. 29, No. 11, 1991, pp. 1927 -1928.34ABAQUS, ABAQUS Analysis User’s Manual 2009, Version 6.9. Dassault Systèmes Simulia Corp, Providence, RI, USA 2009.35Rhead, A. T., Butler, R., Baker, N., “Analysis and Compression Testing of Laminates Optimised for Damage Tolerance,” Appl. Compos. Mater., Vol.18, No.1, 2010, pp.85-100.

36Rhead, A. T., Butler, R., “Compressive static strength model for impact damaged laminates,” Composites Science and Technology., Vol.69, No.14, 2009, pp.2301-2307.

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