aramide fibre core) using four-point bending tests

International Journal of Fatigue 32 (2010) 1739–1747

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

Modelling the fatigue behaviour of composites honeycomb materials (aluminium/aramide fibre core) using four-point bending tests A. Abbadi a,b,*, Z. Azari a, S. Belouettar b, J. Gilgert a, P. Freres c a

Laboratoire de Fiabilité Mécanique, Ecole des Ingénieurs de Metz, Ile du Saulcy, F-57045 Metz, France Centre de Recherche Public Henri Tudor, 29, Avenue John F. Kennedy, L-1855 Luxembourg, GD of Luxembourg, Luxembourg c Euro-Composites, S.A. Zone Industrielle, L-6401 Echternach, GD of Luxembourg, Luxembourg b

a r t i c l e

i n f o

Article history: Received 2 May 2009 Received in revised form 8 January 2010 Accepted 15 January 2010 Available online 20 January 2010 Keywords: Structural materials Fatigue Sandwich Stiffness degradation Damage

a b s t r a c t Composite Sandwich Materials are being increasingly used in high-performance structural applications because of their high stiffness and low weight characteristics. Presently, the long-term performance of such structures, especially under fatigue loading, is not enough studied. The aim of this paper is to address such fatigue behaviour by using a fatigue model verified by experimentation. The fatigue model is based on the fatigue modulus concept (degradation of stiffness) which is proposed for core-dominated behaviour and for two directions cells (L and W). Two non-linear cumulative damage models (L and W) derived from the chosen stiffness degradation equation, are examined in context with the linear Miner’s damage summation and compared with available experimental results. Ó 2010 Published by Elsevier Ltd.

1. Introduction The use of sandwich structure continues to increase rapidly due to the wide fields of their application, for instance: satellites, aircraft, ships, automobiles, rail cars, wind energy systems, and bridge construction to mention only a few. The sandwich composites are multi-layered materials made by bonding stiff, high strength skins facings to low density core material (Fig. 1). The main benefits of using the sandwich concept in structural components are the high stiffness and low weight ratios. In order to use these materials in different applications, the knowledge of their static and fatigue behaviours [1] are required and a better understanding of the various failure mechanisms under static and fatigue loadings conditions is necessary and highly desirable. From phenomenological point of view, fatigue damage can be evaluated, in the global sense by stiffness, residual strength, dissipated energy or other mechanical properties [2–6]. Fatigue modulus concept for fatigue life prediction of composite materials is proposed by Hwang and Han [7]. It is suggested that the changes in stiffness might be an appropriate measure of fatigue damage. Many investigators have examined the effectiveness of the stiffness degradation in composite materials as a measure of accumulated damage [8]. To test the change in Young’s modulus of

* Corresponding author. Laboratoire de Fiabilité Mécanique, Ecole des Ingénieurs de Metz, Ile du Saulcy, F-57045 Metz, France. Tel.: +33 686607574. E-mail address: [email protected] (A. Abbadi). 0142-1123/$ - see front matter Ó 2010 Published by Elsevier Ltd. doi:10.1016/j.ijfatigue.2010.01.005

material, the damage development of composite materials can be described by stiffness degradation of materials in fatigue behaviour investigation [9]. As residual strength, stiffness and life are affected by fatigue damage, only residual stiffness can be monitored nondestructively [10]. Residual strength decreases slowly with the increase of the number of cycles until a stage close to the end of life of the specimen, where it begins to change rapidly until complete destruction [11]. On the contrary, stiffness exhibits greater changes during fatigue specifically at the early stage of fatigue life of specimen [12]. Much important work on residual stiffness has been done by Reifsnider et al. [13]. The residual stiffness as a parameter to describe the degradation behaviour and to predict the fatigue life is selected by Wu et al. [14]. There is an interesting feature in stiffness degradation approach that only limited amount of data is needed for obtaining reasonable results [15]. Kin [16] reported that the reduction of bending strength of foam cored sandwich specimen is caused by the stiffness reduction of foam due to ageing of polyurethane foam during fatigue cycles. Shenoi [17] investigated the static and flexural fatigue characteristics of foam core polymer composite sandwich beams. Failure modes relate to both core shear and skin failure. Jen and Chang [18] analysed the fourpoint bending fatigue strengths of aluminium honeycomb sandwich beams with cores of various relative densities. In their study, the debonding of the adhesive between the face sheet and the core was identified to be the major failure mode. Kulkarni et al. [19] observed the fatigue failure of foam core sandwich composites under flexural loading. The crack propagation rate was used to predict the

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Aluminium skin Honeycomb core Aluminium skin Glue Fig. 1. Description of the honeycomb sandwich structure.

fatigue life. Azouaoui et al. [20] investigated the evaluation of impact fatigue damage in glass/epoxy composite laminate. The fatigue characteristic of two cellular foam core materials is tested by Burman and Zenkert [21]. The bending fatigue behaviour of pure epoxy and 3D woven sandwich composites is studied by Judawisastra et al. [22]. Panel with aramide fibre core were selected. The results of the stiffness degradation correlated well with the mechanical properties of the sandwich panel. The purpose of this study is to develop analytical models describing the flexural behaviour of honeycomb composite core (aramide fibre) sandwich under cyclic fatigue. The stiffness degradation approach allows the assessment of the fatigue damage. Two non-linear cumulative damage models for (L and W) oriented cells derived from the chosen stiffness degradation equations are examined assuming linear Miner’s damage summation. Predicted results are compared to the available experimental results. It is worth it to note that the proposed models are not compared with other existing works, since there is no similar investigation in the literature with the considered material (Sandwich with aluminium skins and Nomex core with the same core geometry, dimension and density).

During fatigue cycling, the stress/strain curve changes, causing a reduction of fatigue modulus. The fatigue modulus at a specific load cycle, n, is represented on the stress/strain curve by a line drawn from the origin to the resultant strain at the applied stress level. The rate of decrease of fatigue modulus can be related to an empirical power-law function of the form AnC. The theoretical decrease in modulus from an initial static value can be expressed as:

Gf ðnÞ ¼ G0  AnC

ð1Þ

where Gf(n) and G0 are the transient fatigue modulus and instantaneous static modulus, respectively, and A and C are material constants to be determined from experimental study. This is achieved by converting the experimental deflection response into fatigue modulus term (see Section 4). Assuming a normal distribution, the constants are determined by using curve fitting techniques. Clark et al. [28] modified this model because it is applied to the secondary region of the fatigue modulus response, i.e. n P nif . Therefore the cycle number, n, must be replaced by ðn  nif ). After closer examination of the secondary region of the experimental deflection response, the power-law function shown in Eq. (1) was deemed to be less satisfactory than an exponential function. Thus Equation was modified and expressed as:

Gf ðnÞ ¼ G0

where n 6 nif

Gf ðnÞ ¼ G0  Aeðnnif ÞC

where n P nif

) ð2Þ

where nif is the number of cycles to initiate damage. The equation used by Clark et al. [28] to predict the number of cycles at failure for different applied stress level, r was given by:

Nf ¼ nif þ

ln½Bð1  rÞ C

ð3Þ

2. Fatigue modelling

where B ¼GA0 , r isequal to the ratio of applied stress to ultimate static stress, r ¼ ssua , and Nf is the number of cycles at failure.

2.1. General

2.2. Cumulative damage model

Fatigue damage in composites and other related materials has often been modelled by using the reduction in strength or stiffness. With strength-degradation approaches, the residual strength is determined from a static test after fatigue cycling, so that a series of tests are required to determine a single strength degradation curve. The Strength Life Equal Rank Assumption (SLERA) is often assumed, which states that the rank in static strength is equal to the rank in fatigue life. This assumption appears to be valid for a wide range of fibre-reinforced plastics where the scatter in fatigue data is primarily due to variations in static strength, provided that the failure mode does not change [23]. A statistical distribution of strengths determined from static tests of undamaged specimens until failure is often used to relate the static scatter to the scatter of residual strengths in fatigue [23–25]. Stiffness degradation methods have the advantage of being able to allow measurement of effective stiffness during cycling without destruction of the specimen, so that a stiffness degradation curve can be obtained from a single test. Smaller numbers of specimens are required and average results can be used rather than the use of a full statistical analysis. However, stiffness can be defined in different ways. Usually, it is taken as a modulus term where the reduction in stiffness can be measured from the linear portions of stress/strain graphs at different cycle numbers. Yang et al. [26] assumed that the degradation rate is a power-law function of the number of load cycles. This approach is mainly suitable for fibre-dominated response where the stress/strain curve is linear. Hwang and Han [27] introduced a model based on the fatigue modulus concept, an approach particularly suitable for resin-dominated behaviour.

2.2.1. General damage model At a constant frequency and environmental condition, fatigue damage, D accumulates from an initial damage state, usually equal to zero, at zero cycles to a final failure value, usually equal to unity. For constant loading

D¼0

where n ¼ 0

 ð4Þ

D ¼ 1 where n ¼ Nf

2.2.2. Cumulative damage equations Different forms of the cumulative damage parameter, D, can be chosen depending on the degree of linearity of the degradation response. Two models are investigated. The first model is linear, based on ‘‘number of cycles”, whilst the second a model based on changes of ‘‘modulus”. Damage is assumed to initiate when fatigue damage is first observed, i.e. at n = nif. At n = Nf the damage is equal to one. For the purpose of all the cumulative damage models investigated it is assumed that:

DðnÞ ¼ 0

where n 6 nif

DðnÞ ¼ 1 where nif 6 n 6 Nf

 ð5Þ

(A) Model I: The most basic linear cumulative damage model is that proposed by Miner [29] which was originally derived from energy considerations. It states that the amount of damage at a given number of cycles is the ratio of the current cycle number to the number of cycles to cause fatigue failure. In this case, the damage model occurs after the initiation of damage and can be expressed as:

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DðnÞ ¼

ðn  nif Þ ðNf  nif Þ

ð6Þ

(B) Model II: The damage function is defined in terms of the fatigue modulus (Clark et al.) as:

G0  Gf ðnÞ DðnÞ ¼ G0  Gf ðN R Þ

Failure strength (MPa)

Tensile strength (MPa)

Maximum elongation (%)

70,000

268

367

13

Table 3 Mechanical properties of the cores [Euro-composites].

ðnnif ÞC

e eðNR nif ÞC

Young’s modulus (MPa)

ð7Þ

By manipulation of Eq. (2), Eq. (7) becomes:

DðnÞ ¼

Table 2 Properties of sandwich facings.

where n P nif

ð8Þ

In this paper, the model given by (Eq. (7)) is used to check its validity in sandwich honeycomb core for both direction of cells (L and W). 3. Experimental characterisation

Core

Fibre aramide core (ECA)

Cell size (mm) Density (kg/m3) Shear resistance (L, direction) (MPa) Shear modulus (L, direction) (MPa) Shear resistance (W, direction) (MPa) Shear modulus (W, direction) (MPa) Compression resistance (MPa)

3.2 48 1.32 50 0.56 30 2.1

3.1. Material specimens The Honeycomb sandwich panels are provided by Euro-Composites (Luxembourg) and intended for the aircraft industry. The geometrical dimensions of the specimen are shown in Table 1. The faces of a thickness equal to 0.60 mm are made of aluminium (AlMg3), the core structure is made from aramide fibres (ECA) folded and glued together forming a hexagonal cell structure. As in the standard lay-out for commercial honeycombs, the assembly of the structure produces some cell walls with double thickness. In the tested configuration these double thickness walls were parallel to the specimen longitudinal axis. The honeycomb core is an opened cell with a density of 48 kg/m3 and a cell size of 3.2 mm. The geometrical and mechanical properties of the panels are depicted in tables (Tables 2 and 3). Fig. 2. Sketch of the four-point bending test and specimen dimensions.

3.2. Fatigue rig Fatigue tests were carried out through a four-point bending testing fixture device schematically shown in Fig. 2. Such device, designed and built expressly for these tests, was connected to a servo-hydraulic universal testing machine INSTRON model 4302 controlled by an INSTRON electronic unit. The electronic unit performs the test control and the data acquisition. Another PC equipped with a NI acquisition device was used to acquire the load and stroke signals. The design of the fixture device allows the inner supports to rotate around the neutral axis of the specimen. When a damaged specimen is tested, the specimen deformation is asymmetric due to the different bending stiffness of the two portions (with and without defect) of the specimen. The rotation of the inner supports allows the testing device to adapt the testing conditions, following the asymmetric displacement of the specimen. In this way it is possible to obtain that, during the whole loading process, the same loads are applied on both the inner supports and consequently to keep the four-point bending condition. To apply the loading signal, a Labview program has been set which enables to introduce all points of the cycle. The portions of specimen between the inner and the outer supports are loaded with a constant shear force. The bending moment raises linearly from the outer support to the inner one. The portion of specimen between the two inner supports is loaded with a constant bending moment

while the shear force is zero. The maximum value of the bending moment is in the portion between the two inner supports. 3.3. Fatigue test results Fatigue tests were performed at a room temperature under direct load control, while the load cycling amplitudes were chosen on the basis of the static test results. The test load was sinusoidal with a frequency f = 2 Hz and a load ratio R = 0.1 and a constant amplitude loading. With such load ratio the face in contact with the outer supports was submitted to time varying in-plane tensile stresses, while the other face that is in contact with the inner supports was submitted to time varying in-plane compressive stresses. In this way it has been possible to evaluate separately the effect of a tensile and a compressive stress field on the fatigue behaviour of the tested specimen. The value of the specimen bending stiffness was monitored during all the tests in order to gather information about the possible reduction of the sandwich structural properties with fatigue cycling. Fatigue data were generated at load levels of 100%, 90%, 80%, 70%, 65% and 60% of the static ultimate load. The core density is used, 48 kg/m3 (aramide fibres) of three specimens within each configuration (Fig. 3). The data acquisition performed monitored the stiffness variation during the tests using the output

Table 1 Specimen dimensions. L (mm)

b (mm)

h (mm)

hc (mm)

tf (mm)

L2 (mm)

L1 (mm)

d = hc + tf (mm)

500

250

10

8.80

0.60

420

210

9.40

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For these type of sandwich structures, both W and L configurations failed in shear with a crack propagating through the thickness of the core (details are shown on Fig. 5). The failure propagation is always in the diagonal direction in the case of the L configuration and horizontal for the W one. In both cases, cracks or micro defects appear before any macro size crack is formed. Crack propagation in one direction, subjected to an opening load during half of load cycle, is unaffected by the cracks growing in the other diagonal direction since these cracks are closed. Hence, cracks initiate from the crack tip of the horizontal macro crack: one growing upwards during half the load cycle and the other one growing downwards during the other half of the load cycle. Notice here that the final crack length is independent of the maximum load and loading ratio. We also noticed a subsequent shear buckling (Fig. 5) of the vertical cell walls in the centre region between the inner and outer support from the first load cycles as well as the formation of several clusters of small horizontal cracks in the cell walls formed within separate cell as shown in Fig. 5. The fatigue cracks formation in the L configuration is similar to those in the W configuration. However, the number of observed micro cracks was significantly less important and the failure was more abrupt. In both cases, the fracture pattern of a diagonal crack is the same (Fig. 5) and not affected by the number of cycles to failure. The angle between the crack and the horizontal axis was 30°. This is due to the final rupture of the sandwich which was caused by a buckling followed by a shearing.

signals of the load cell and the deflection of the hydraulic piston. However, since the fatigue threshold and maximum applicable load vary with the different configurations and with the different load ratios the levels were slightly adjusted accordingly. The fatigue life of the specimens is characterized as the number of cycles to ultimate failure. The number of cycles from crack initiation to final fracture was in all cases short when compared to the fatigue life. The variation of the stiffness during the major part of the lifetime was insignificant. The stiffness did not decrease below 90% of the original stiffness until just before final failure. During this last part of the fatigue life the degradation was more pronounced due to the crack formation further discussed below. The fatigue curves in terms of load level versus the number of cycles, shown in (Fig. 4a), illustrate a qualitative comparison between the fatigue lifetime of sandwich composites made of aramide fibres cores in both configuration L and W. Fig. 4 shows that the fatigue life limit is the same and equals 60% of the applied load for both directions. Moreover, the material shows more resistance in L than in W direction. Indeed, for an applied load of 100% of ultimate static load the number of cycles for failure is about 102 cycles for W direction, while it is greater and close to 104 cycles for L direction. After extrapolation of the two (L and W configurations) fatigue curves, one notices the intersection of these curves for a load level of 0.5 (Fig. 4b) and a lifetime of 1.07  107 cycles. The following discussions regarding the fatigue failure processes are only based on visual inspection of the free sides of the beams.

L, configuration

W, configuration

y x

Fig. 3. Cells configuration (L and W).

1.2

Load level, (Fapp /Fsta, max)

1.1

(b)

Alu/Fibre, density 48 kg/m3 , L direction 3 Alu/Fibre, density 48 kg/m , W direction

1.0 0.9

L

0.8 0.7

W 0.6 0.5 1000

1.2

1.0 0.9

100000

Cycles to failure, N f

1000000

y=4,02x-0,1284 -0,08378

0.8

y=1,952x

0.7 0.6 0.5

10000

3

Alu/Fibre, density 48 kg/m , L direction Power fit , L direction Power fit , W direction Alu/Fibre, density 48 kg/m 3, W direction

1.1

Load level, (Fapp /Fsta, max)

(a)

1000

NR=1,075E07 r=Fapp/Fsta,max=0,5 10000

100000

1000000

Cycles to failure, Nf

Fig. 4. Fatigue curves in terms of load level versus cycles to failure for both directions L and W.

1E7

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L, direction

W, direction

W, direction

L, direction

Fig. 5. Failure modes of the aramide fibres cores in the W and L-directions.

Displacement (mm)

Max displacement

Min displacement Time (s) Fig. 6. Evolution of displacement (mini and maxi).

3.4. Deflection response

4. Application of models

During fatigue testing, the min and the max displacements evolution was recorded (Fig. 6). For core shear failures, the beam deforms slowly at a constant rate until near failure. Close to the end of the beams life, the rate of deflection starts to increase rapidly in an exponential manner till failure. This region is associated with a sudden loss of strength and stiffness. A typical deflection response is shown in Fig. 7.

4.1. Initiation of fatigue damage

Deflection at midspan (mm)

10

beam Alu/Fibre, load=1328N, W direction, Nf =20896 Cycles maximum deflection

8

initiation offatigue damage

6

4

0

5000

10000

15000

20000

Numbre of Cycles, n Fig. 7. Fatigue deflection versus number of cycles response.

The initiation of fatigue damage has been identified in the first time by Clark et al. In practice, it has been observed that the initiation of fatigue damage occurs where the rate of deflection versus number of cycles response just starts to increase near final failure of the beam for both directions L and W, (Figs. 6 and 7). The number of cycles to fatigue damage initiation corresponds to about 5% average drop of the curve representing the fatigue shear modulus (see Section 4.2). This number of cycles to fatigue damage initiation for different level load was obtained experimentally. The main observation is that at higher load levels, 80–100% from ultimate static load, the number of cycles taken until the initiation of fatigue damage expressed as a percentage of the total fatigue life, is much greater than for lower load levels. In cases where the applied load exceeds 80% of the ultimate static load, initiation of fatigue damage occurs at a percentage higher than 70% of the number of cycles to failure, whereas for a lower load levels, 60–70% from ultimate static load, the initiation of fatigue damage occurs at lower percentage of the total fatigue life, typically 57–66%. 4.2. Determination of fatigue modulus from experimental results To compare the theoretical stiffness degradation models (L and W) with experimental results, it is important to reproduce the

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overall deflection data into a change in stiffness. In the experimental studies is assumed that the skins of the sandwich carry all the tensile/compressive stress and the sandwich core the shear stress. The fatigue results of a sandwich, where core shear is the observed failure mode, can be treated solely by the change in core shear modulus with number of cycles [30]. The total deflection of sandwich in four-point bending (Fig. 8) is the summation of bending component, dbend and shear component, dshear . Where the fatigue failure is dominated by a core shear, it probable assumed that the shear component alone degrades. The fatigue modulus can be calculated from measured deflections, d for a sandwich beam supported in four-point bending under a load, F the equation of total deflection is given by:

FL2 8A1 GðnÞ

ð10Þ

and

dbend ¼

ð13Þ

The fatigue modulus degradation curves determinate directly from experimental data using Eq. (13) for both direction (L and W) and for three load level are shown in Fig. 9a and b. These curves show, first that the core shear modulus remain unchanged and equal to the static modulus until the initiation of fatigue damage (zone 1) for both direction (L and W). Second, it rapidly decreases in a short number of cycles until failure (zone 2). At lower load levels, the rate of decrease of the fatigue modulus increases as function of number of cycles. 4.3. Determination of fatigue model degradation parameters

ð9Þ

where

dshear ðnÞ ¼

FL2 8A1 dshear ðnÞ

FL32 96D

ð11Þ

G(n) is the shear fatigue modulus, n the number of cycles, L2 the distance between the fixed support and D and A1 are the beam’s flexural rigidity and the beam’s effective cross sectional area, respectively, both defined in Appendix A. The shear deflection which is dependent of the cycle number can thus be calculated directly from experimental data after subtraction of the static bending deflection which is supposed independent of the number of cycles assuming core shear degradation.

dshear ðnÞ ¼ dtotal ðnÞ  dbend

To determine the experimental parameters A and C of the stiffness degradation models (Eq. (2)) for both direction (L and W), experimental modulus degradation curves of the secondary region (zone 2), are plotted as a modulus decrease versus number of cycles. Because the fatigue damage occurs, in the secondary region, after the initiation of fatigue damage, the number of cycles, for the purpose of fitting the parameters, is given by (n  nif ). The exponential equation used to determine the parameters is:

Gðn  nif Þ ¼ Aeðnnif ÞC

ð14Þ

An example of this method is shown in Fig. 10 only for W direction. Parameters of Eq. (14) were calculated using least square fit tech-

Shear Modulus Degradation, G0 -G f (n) [MPa]

dtotal ðnÞ ¼ dbend þ dshear ðnÞ

GðnÞ ¼

ð12Þ

Once the shear deflection is known the effective core shear modulus GðnÞ can be calculated as:

20

Alu/Fibre, F=1033N, direction W, N f =126444, n if =72980, r=0,7

18

Experimentally data Exponential Fit

16 14 12

G0-Gf (n)=0,74e0,00006(n-nif)

10 8 6 4 2 0

0

10000

20000

30000

40000

50000

60000

Number of cycles after damage initiation, (n-n if ) Fig. 8. Sketch of the four-point bending test.

40

Spicemen 1 Alu/Fiber, r=0.7 Spicemen 2 Alu/Fiber, r=0.65 Spicemen 3 Alu/Fiber, r=0.6

Direction W

35 Core Shear Failure

30 Zone1 Zone2

25

20

15

1000

10000

100000

Number of Cycles, n

(b) Shear Fatigue Modulus, Gf (n) [MPa]

Shear Fatigue Modulus, Gf (n)[MPa]

(a)

Fig. 10. Exponential fit to fatigue modulus degradation curve for W direction.

60

Spicemen 1 Alu/fiber, r=0.75 Spicemen 2 Alu/fiber, r=0.85 Spicemen 3 Alu/fiber, r=0.95

Direction L

Core Shear Failure

50 Zone1

Zone 2 40

30

0

100000 200000 300000 400000 500000 600000

Number of cycles, n

Fig. 9. Fatigue modulus degradation curves (a – W direction and b – L direction).

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Applied load level r

Cycles to fatigue damage initiation nif

Cycles to failure Nf

1 0.9 0.9 0.8 0.8 0.7 0.7 0.65 0.6 0.6

3348 5705 5378 17,088 17,836 72,980 103,490 313,655 369,810 424,711

3775 8415 7588 24,950 34,060 126,444 113,314 473,928 630,630 742,071

Exponential fatigue model degradation parameters A

C

B

1.84 1.5 1.65 1.85 1.14 0.74 0.9 0.602 0.19 0.14

0.0011 5.1E04 6.7E04 2.8E04 1.37E04 6E05 6.2E05 2E05 1.6E05 1.48E05

16.30 20 18.18 16.21 26.31 40.54 33 49.83 157.9 214.28

nique for different applied load levels, r. These parameters are shown in Table 4. The exponential parameter, A determines the value of G0  Gf ðnÞ at damage initiation and grows with the increase in the load level r. Whilst the exponent C determines the rate of damage after initiation, grows with the increase in the load level r and have an influence on the rate of damage. The exponential coefficient, A was found to vary with the exponent C in a non-linear manner; see Fig. 11. For L direction, the secondary region (zone 2) of the shear fatigue modulus fitted by an exponential function was found to be less satisfactory than a third order polynomial function. The parameters of this function were shown in Table 5. This finding means that the use of an exponential curve fitting is note the general trend but is material dependent. The polynomial equation used to fit the parameters is:

Gðn  nif Þ ¼ a1 þ a2 ðn  nif Þ þ a3 ðn  nif Þ2 þ a4 ðn  nif Þ3

2.0 1.8

ð15Þ

where a1, a2, a3 and a4 are material constants to be determined from experimental data (see Table 5). In general, we believe that the specimen geometry has an important impact on the fatigue modulus degradation. Therefore, specific studies are needed to analyse the impact of each configuration, which is out of the aim of this paper. The equation adopted for the prediction of the number of cycles at failure for L direction and for different applied load level, r is given by:

1  r ¼ a1 þ a2 ðN f  nif Þ þ a3 ðNf  nif Þ2 þ a4 ðNf  nif Þ3

4.4. S–N curves The number of cycles to failure experimentally measured and calculated using the exponential and polynomial shear modulus degradation approach Eqs. (3) and (16) for direction W and L, respectively is shown in Fig. 13a and b. The trend of theoretical results in terms of number of cycles to failure was estimated using a non-linear power function fit for both directions (W and L). It is clear that the models show good correlation with the experimental fatigue data at lower load levels. However, a discrepancy is observed at higher load levels. Thus, the proposed models need a deeper investigation to check its validity at higher load levels. 4.5. Cumulative damage equation The two different cumulative damage curves obtained from Eqs. (6) and (7) for direction W and L, respectively for each for two load

Experimental Data Logarithm Fit

Parametre "A"

1.6 1.4 1.2 1.0

A=0,3627ln(C)+4,3112

0.8 0.6 0.4 0.2 0.0 0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

ð16Þ

Modulus degradation, Gðn  nif Þ for L direction is shown in Fig. 12. Parameters of Eq. (15) were calculated using least square fit technique for different applied load levels, r. These parameters are shown in Table 5.

0.0012

Parametre "C"

Shear Modulus degradation, G0 -Gf (n) [MPa]

Table 4 Experimental fatigue results and damage model parameters (W direction).

25

Alu/Fibre, Direction L, Nf =82721, nif=49300, r=0,95 Experimental Data Polynomial Fit

20

G0-Gf(n)=a1 + a2*(n-nif) + a3*(n-nif)^2 + a3*(n-nif)^3 a1 1,8641 a2 1,2912E-4 a3 -1,39272E-8 a4 9,34284E-13

15 10 5 0 0

5000

10000 15000 20000 25000 30000 35000

Number of cycles after damage initiation, (n-n if )

Fig. 11. Variation of fatigue modulus degradation parameter ‘‘A” versus parameter ‘‘C” (determined from experimental data).

Fig. 12. Polynomial fit to fatigue modulus degradation curve for L direction.

Table 5 Experimental fatigue results and damage model parameters (L direction). Applied load level r

Cycles to fatigue damage initiation nif

Cycles to failure Nf

Polynomial fatigue model degradation parameters

1 0.95 0.9 0.85 0.8 0.75 0.7 0.65

41,100 49,300 127,873 200,353 304,239 508,841 318,405 1,307,799

60,552 82,721 161,294 271,532 322,178 770,972 521,976 1,923,234

G0 – Gf(n) = 2.96 + 4.33E4(n – nif)–3.86E8(n – nif)2 + 2.35E12(n – nif)3 G0–Gf(n) = 1.86 + 1.29E4(n – nif)–1.39E8(n – nif)2 + 9.34E13(n – nif)3 Indentation failure G0 – Gf(n) = 6.24 + 1.59E4(n – nif)–8.6E9(n – nif)2 + 1.69E13(n – nif)3 Indentation failure G0–Gf(n) = 8.92 + 1.91E4(n – nif)–8.07E9(n – nif)2 + 9.28E14(n – nif)3 G0–Gf(n) = 8.6–1.4E5(n – nif) + 6.49E11(n – nif)2–9.59E18(n – nif)3

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A. Abbadi et al. / International Journal of Fatigue 32 (2010) 1739–1747

Applied Load level, r

(b)

Experimental data Model based on exponential fatigue Modulus degradation Power function Fit W direction

0.9

0.8

Core Shear Failure

0.7

r=1.887Nf-0.084

Experimentally data Model based on polynomial fatigue Modulus degradation Power Fuction Fit L direction

1.0

Applied Load level, r

(a)

0.9 Core Shear Failures

0.8 r=4.486Nf-0.136

0.7

0.6

0.6 10000

100000

1000000

100000

Number of cycles to failure, N f

1000000

Number of cycles to failure, Nf

Fig. 13. Non-Linear S–N curve based on fatigue modulus degradation (a – W direction, b – L direction).

(a)

1.0

Model I Miner: r=0.7 & r=0.9 Model II Exponential: r=0.7 Model II Exponential: r=0.9

(b)

Direction W

1.0

Damage, D(n)

Damage, D(n)

Direction L

0.8

0.8 0.6 0.4

0.6 0.4 0.2

0.2 0.0

Model I Miner: r=0.85 & r=0.95 Model II Polynomial: r=0.85 Model II Polynomial: r=0.95

0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.0

Normalised life After Initiation of Fatigue Damage

0.2

0.4

0.6

0.8

1.0

Normalised life After Initiation of Fatigue Damage

Fig. 14. Cumulative damage models (a – W direction, b – L direction).

(a)

Alu/Fibre, r=0.9 Experimantal data Exponential Model

Alu/Fibre, r=0.95 Experimental data Polynomial Model

0.8

0.6 0.4

L Direction

0.6 0.4 0.2

0.2 0.0

1.0

W direction

Damage, D

0.8

Damage, D

(b)

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Normalised Life After Initiation of Fatigue Damage

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Normalised Life After Initiation of Fatigue Damage

Fig. 15. Comparison between the experimental and models (a – W direction, b – L direction).

levels are represented schematically in Fig. 14a and b. It is observed that the first model (Miner’s rule) is linear as expected. While, the models based on fatigue modulus for both direction (W and L) are non-linear and exhibit the same behaviour with damage initiation between 0–0.3 and 0–0.6 of normalized life for W and L direction, respectively. The non-linear models display a flatting out of the damage curve at higher and lower load levels for both directions. Fig. 15a and b show the comparison between the theoretical models based on shear modulus degradation with experimental data for both direction (L and W). These curves show a good correlation between the theoretical models and experimental results for W direction (Fig. 15a) and L direction (Fig. 15b).

5. Conclusion Fatigue tests in four points bending were performed on an aramide fibre core of sandwich structure for both directions (W and L). The fatigue test results were presented in standard S/N diagrams. The lifetime of the L configuration (Fig. 4a) is larger than that in the W direction at constant load level. After extrapolation of the two (L and W configurations) fatigue curves, one notices the intersection of these curves for a load level of 0.5 (Fig. 4b) and a lifetime of 1.07  107 cycles. Several equation of shear modulus degradation G, have been proposed. In this paper, an equation taking into account different displacement of plate is used. The number of cy-

A. Abbadi et al. / International Journal of Fatigue 32 (2010) 1739–1747

cles from crack initiation to final fracture was in all cases short when compared to the fatigue life. The variation of the stiffness during the major part of the lifetime was insignificant. We also noticed a subsequent shear buckling of the vertical cell walls in the core. The angle between the crack and the horizontal axis was 30°. This is due to the final rupture of the sandwich which was caused by a buckling followed by a shearing. The degradation of shear modulus, G, was determined according to fatigue life and the material parameters A and C were identified. The application of the exponential model gave a good result when the cells are in W direction. While for L direction exponential model was found to be less satisfactory than a third order polynomial function. This finding means that the use of an exponential curve fitting is not the general trend but is material dependent. Appendix A. Static sandwich beam equations The beam length is a with beam width b and core depth tc. Other dimensions are defined in Fig. 2. Skin properties: skin modulus and stresses (tension or compression) are Ef, and r1, r2, respectively. Top and bottom skin thickness are tf. The distance between the neutral axes of the two facing is d = tc + tf. Core properties: ultimate core shear stress, sc and core shear modulus, Gc. Core thickness, tc. Sandwich beam flexural rigidity, i.e. beam stiffness: for two faces with equal thickness: 2

Ef btf d 2

D¼

Core shear rigidity:

A¼

Gc bd tc

2

Skin direct tension or compression failure: use ordinary beam theory (four-point bending)

r1;2 ¼

FðL2  L1 Þ 2tc bd

F: applied load. Coreshear failure: assume shear stress constant through depth of core (four-point bending)

sc ¼

F 2bd

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