Reliability of composite structures — impact loading

Computers and Structures 76 (2000) 163±172

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Reliability of composite structures Ð impact loading L. Guillaumat LA.M.E.F.-E.N.S.A.M. Esplanade des Arts et MeÂtiers, 33405 Talence CeÂdex, France

Abstract The aim of this paper is to propose a method to study the response of laminated composite structures under impact loading. The importance of the mass, velocity of the striker and the dimensions of the composite structure is shown for di€erent responses such as the contact force between the striker and the composite structure, the displacement of the projectile, and the damage within the composite structure. The proposed method is based on an experimental design technique. Some examples are proposed to illustrate how the technique allows us to improve our understanding of the behaviour of impacted composite structures. 7 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Impact; Composite; Experimental design; Modelling; Damage

1. Introduction Design of structures is often made with a deterministic approach. However, this kind of design is not always ecient for composite materials because of the variability of their mechanical behaviour. In fact, some tools exist for designing structures taking into account the scatter of both mechanical characteristics of the material and loading. For this, it is necessary to well know all the loads applied to the structure. The three most important types of loading are accidental shocks, fatigue and vibrations. The aim of this paper is to study some responses of composite panels under low velocity impact loading. The analysis of the mechanical response of composite structures loaded with low velocity±high mass impacts is a multiparameter problem. Impact loading can be de®ned in terms of the geometry, mass and

E-mail address: [email protected] (L. Guillaumat).

speed of the impactor, the size of the composite plate, and the boundary conditions [1±6]. The main question is how to identify the critical parameters. One way of quantifying the in¯uence of factors on one or more responses is to use the experimental design method. This enables di€erent variables to be assessed and the most in¯uential ones to be identi®ed with a minimum of experiments. The evolution of responses as a function of variables is modelled by simple mathematical expressions (empirical polynomials) but these are not based on physical mechanisms. The aim of this paper is to present a tool for dimensioning composite structures. However, it should be emphasised that the results will only be valid within the experimental domain studied, and extrapolation is risky. Nevertheless, by identifying the above mentioned models, the optimisation of impact response of composite structures may be achieved. The in¯uence of plate dimensions as well as the in¯uence of the mass±velocity combination are studied because the critical parameter used to characterise an impact is the kinetic energy of the impactor [7].

0045-7949/00/$ - see front matter 7 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 9 ) 0 0 1 6 6 - 2

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L. Guillaumat / Computers and Structures 76 (2000) 163±172 Table 1 Two-variable factorial matrix Test No.

X1

X2

1 2 3 4

ÿ1 +1 ÿ1 +1

ÿ1 ÿ1 +1 +1

2. Experimental set-up

Fig. 1. Drop tower set-up.

The impact tests have been performed using drop weight set-ups (Fig. 1). An electric motor is used with a magnet to raise the mass. The drop height is determined by the position of a ®rst infra-red captor which stops the motor at the desired level. A second captor located on the set-up close to the specimen, activates a mechanism which avoids a second impact on the structure. The contact load between the striker and the composite specimen versus time is measured by means of an accelerometer and a piezoelectric captor which are attached to the drop weight (Fig. 2). Two laser captors provide the striker displacement and the de¯ection at the centre of the composite structure versus time. The tests have been ®lmed by a high speed camera system Camsys+ (2000±11,000 frames/s). The plate is put on

Fig. 2. Instrumented drop weight.

L. Guillaumat / Computers and Structures 76 (2000) 163±172 Table 2 Complete two-variable factorial matrix and the vector of the responses Y Test No.

X0

X1

X2

X1
X2

Y

1 2 3 4

1 1 1 1

ÿ1 +1 ÿ1 +1

ÿ1 ÿ1 +1 +1

+1 ÿ1 ÿ1 +1

Y1 Y2 Y3 Y4

two steel opposite supports. Two ¯exible sheets made of metal with a rubber end maintain the plate. These boundary conditions allow rotation and vertical movement.

3. Test matrix A large number of experiments are frequently performed in order to measure mechanical, physical or chemical phenomena. The experimental design method enables test campaigns to be set up to minimise variance in results while reducing the number of tests. By analysing the results from such tests, empirical polynomial models can be established relating the variables of the mechanical system to the structural responses. The basis of this approach lies in the fact that the model variance …s2 † can be expressed as the product of the experimental variance and a term which depends only on the organisation of the tests (A) (1). var…Y † ˆ As2

…1†

2

s is estimated by repetition of one of the tests. The accuracy of this estimation improves with the number of repeats. This indicates that the variance of the model can be minimised by choice of an optimal experimental distribution of the tests. Setting up an experimental design method involves several steps. The nature and limit values of the variables must ®rst be ®xed. The degree of the polynomial must then be chosen. Thereafter, the matrix is established according to strict rules [8]. In order to facilitate the determination of the relative in¯uence of the di€erent parameters, the variables are expressed in centred, reduced co-ordinates. Their amplitude of variation is therefore normalised to the interval [ÿ1, +1]. The ®rst phase uses a factorial test matrix to study some basic responses of the composite structures under impact loading (Table 1). The model corresponding to this matrix is a ®rst order polynomial: Y ˆ b0 ‡ b1
X1 ‡ b2
X2 ‡ b12
X1
X2

…2†

165

The parameter X1 represents the drop velocity and X2 is the mass. The coecient b12 quanti®es the coupling between them. From the matrix shown in Table 1, a second matrix can be build allowing the calculation of bi and bij coef®cients (2) (Table 2). The coecients are calculated by multiplying the vector Y of all the responses with respect to the number of the test by the column corresponding to the selected coecient (for instance: column X1 for the coecient b1, (3)). The number in brackets are the elements of the column X1 (Table 2). b1 ˆ … ÿ 1 †
Y1 ‡ … ‡ 1 †
Y2 ‡ … ÿ 1 †
Y3 ‡ … ‡ 1 †
Y4 …3† For the second phase, a Doehlert matrix [9,10] is used to study the in¯uence of the plate dimensions. It is a test matrix known as a response surface. The basic principle is to select a position at one point in an experimental domain and to examine the neighbouring areas. These matrices have spherical symmetry. The factors are assumed to be quantitative and known. The Doehlert matrix has the advantage that it can be translated into the variables space with a minimum of new tests. The two variables in this second phase are the span (X1) and the width (X2). The test matrix associated with these variables requires nine tests (Fig. 3, Table 3): Test numbers 7, 8 and 9 are used to establish the experimental variance s2 (1). In addition, the associated polynomial is of second order. This assumes that the responses are possibly non-linear (4). Y ˆ b0 ‡ b1
X1 ‡ b2
X2 ‡ b11
X1
X1 ‡ b22
X2
X2 ‡ b12
X1
X2

Fig. 3. Test distribution according to Doehlert matrix.

…4†

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L. Guillaumat / Computers and Structures 76 (2000) 163±172

Table 3 Doehlert test matrix Test No.

1

2

3

4

5

6

7

8

9

X1 X2

1 0

0.5 0.866

ÿ0.5 0.866

ÿ1 0

ÿ0.5 ÿ0.866

0.5 ÿ0.866

0 0

0 0

0 0

All the coecients …bi † are calculated as shown previously.

4. Materials Two composites are used here. Both are composed of glass ®bres reinforced in an isophthalic polyester resin. All the plates are constituted of eight layers which give a total thickness of about 3.5 mm. The stacking sequence for the ®rst plate, called material 1, is: [0,90]4, and for the second plate called material 2: [03, 90]s. First, a product is spread on a table to avoid the bonding of the resin with it. The ®rst layer is put on a table and the resin is deposited by means of a roller. Further layers were similarly added. The resin polymerises at room temperature.

5. Test campaign In a preliminary study on the laminated material 1, the in¯uence of the mass was determined as well as the velocity on di€erent responses, such as contact time, contact force, central plate displacement, delaminated area between plies of di€erent orientations where appropriate, number (n1) of cracks parallel to the width of the plate and number (n2) of cracks parallel to its length (Fig. 4). This ®rst phase was not intended to establish models

of mechanical behaviour but particular attention was directed towards the combination of the two variables mass and velocity. From the test matrix (Table 1) (normalised variables), the organisation of the tests is de®ned (Table 4) for material 1 (physical variables). The second step used the Doehlert test programme matrix (Table 5) (physical variables) from test matrix (Table 3) (normalised variables) for material 2. The responses investigated are the contact time, the maximum contact load, the de¯ection of the plate centre and the delaminated area.

6. Results and analysis 6.1. Material 1 Fig. 5 shows a typical evolution of the contact load between the striker and the plate versus time measured by the accelerometer and the load captor. In fact, to obtain only the interaction between the striker and the structure, the accelerometer signal must be ®ltered (500 Hz) to eliminate the vibratory response of the drop weight. The ®lms given by the high speed camera show that the undulations on the curve load versus time (Fig. 5) can be attributed to a transverse eigen mode induced by the impacter during the impact loading. A ®nite element analysis con®rms this assumption by giving the ®rst transverse eigen mode at about 500 Hz (Fig. 6). Experimental results according to the experimental design for material 1 are presented in Table 6. From these data, the coecients bi and bij related to the mass and velocity variables can be determined together with their coupling from Eq. (2) (Table 7). It can be noticed that the coupling (b12) is signi®cant for all the variables related to the mechanical behaTable 4 Test programme

Fig. 4. Orientation of cracks with respect to the plate.

Test No.

Drop height (m)

Mass (kg)

1 2 3 4

0.5 1.5 0.5 1.5

1 1 2 2

L. Guillaumat / Computers and Structures 76 (2000) 163±172

167

Table 5 Doehlert test programme matrix

Width (mm) Span (mm)

Centre

Step

No. 1

No. 2

No. 3

No. 4

No. 5

No. 6

No. 7

No. 8

No. 9

250 250

150 150

400 250

325 379.9

175 379.9

100 250

175 120

325 120

250 250

250 250

250 250

Table 6 Results from impact test on laminate 1 Test No.

Load (N)

De¯ection (mm)

Contact time (ms)

Delaminated area (cm2)

n1

n2

1 2 3 4

1000 1500 1266 1900

8.4 13.3 12.9 22.1

10.0 10.8 15.0 15.5

0 0.58 1 2.28

20 22 27 32

37 70 90 106

viour of the structure (load, de¯ection and damage). Therefore, it is risky with these responses, to consider only the kinetic energy as the main parameter. In addition, Fig. 7 shows that the damage observed is made up essentially of delaminations at several interfaces and cracks oriented parallel to the width and length of the panel (Fig. 4). Moreover, from the ®lms of the camera (Fig. 8), we can establish a chronological evolution of the damage during the impact loading (Fig. 9). At point no. 1 the matrix cracking begins. At point no. 2 the delamination appears. Later, we observe the evolution of the delamination area as following: point 3 Ð 0.75 cm2, point 4 Ð 1.46 cm2, point 5 Ð 1.83 cm2. 6.2. Material 2 The in¯uence of the plate dimensions on the mechanical responses under impact loading is very important. For instance, for a large span (no. 2, Table 5) the evolution of the load versus time shows several loading and unloading (Fig. 10a). We can observe a loss of contact between the striker and the structure each time the load is equal to zero (between two peaks). But, for a small span, the response of the system is very di€erent (Fig. 10b). The oscillations corresponding to the transverse eigen mode are very small. Moreover, the duration of the test (6 ms) is smaller than the previous one (25 ms). In fact, the ¯exural sti€ness of the structure for a small span is higher than for a large one, and consequently, the response is faster. For this part of the study, the damage of the material is the same for all the plates as discussed previously. First, we observe the cracking of the matrix (Fig. 11) then, the delamination appears and grows during the test (Fig. 12). Furthermore, the delamina-

tion is always inside the area where the matrix cracks are located, assuming that a coupling exists between them. A 3D analysis of the delamination has been achieved using the de-ply technique. This technique allows direct observation of delaminations at each interface in the composite [11,12]. We use a solution of zinc iodide to penetrate the delaminations. The laminate is exposed to temperatures in the range of 1508C for a period of 1±2 h in order to vaporise the solvent. During this period, the zinc iodide crystallises and is deposited on the delamination surfaces. In order to realise a partial pyrolysis of the matrix, the composite is baked at about 3508C for about 1 h. After that, all the laminae can be separated with a sharp blade and examined under a microscope (Fig. 13a). From all the observations of each interface of the laminate we can realise a 3D picture of the delamination. The assembling (Fig. 13b) shows that the delamination exist nearly at

Fig. 5. Relationship between contact force and time from both accelerometer and load sensor.

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L. Guillaumat / Computers and Structures 76 (2000) 163±172

Fig. 6. Calculation of the transverse vibration mode of the panel induced by the impact loading.

Fig. 9. Chronological evolution of the damage the material 1.

Fig. 7. Laminate 1 damage.

Fig. 8. Material damage versus time.

Table 7 Representation of coecients related to each variable Load (N) b0 b1 b2 b12

1416 283 166 33

De¯ection (mm)

Contact time (ms)

Delaminated area (cm2)

n1

n2

14.18 3.52 3.33 1.1

12.8 0.3 2.4 ÿ0.1

0.97 0.47 0.67 0.18

25 1.7 4.2 0.7

75 12 22 4.3

L. Guillaumat / Computers and Structures 76 (2000) 163±172

169

6.2.1. De¯ection of the plate centre (cm) (5) Y ˆ 2:41 ÿ 0:69X1 ‡ 1:85X2 ‡ 0:38X 21 ‡ 0:43X 22 ÿ 0:73X1 X2

Fig. 10. Evolution of the contact load between striker and composite 2 versus time for: (a) the plate with respect to test 2 and (b) the plate with respect to test 5.

all the interfaces. Moreover, the area increases towards the back surface of the laminate and is the largest in the furthest interface from the impact surface. The delamination propagated in the ®bre direction of the layer below the interface. These phenomena are well known [13,14]. The following results of the polynomials calculated from the Doehlert test campaign (7.4 kg and 0.5 m height) for material no. 2 were obtained.

…5†

We can observe that the coecient b2 (1.85) is higher than b1 (0.69). It means that the de¯ection is rather dependent on the span. The coupling between the two variables can be considered as signi®cant because the value of the coecient b12 (0.73) is close that one of b1. Fig. 14 illustrates this point. 6.2.2. Maximum contact load (V) (6) Y ˆ 3:23 ‡ 0:8
X1 ÿ 3:48
X2 ÿ 0:16
X 21 ‡ 1:92
X 22 ‡ 0:02
X1
X2

…6†

This response is related to the previous one. In fact, if the span is small the load will be very high because

Fig. 11. Evolution of the matrix cracking versus time (material 2).

Fig. 12. Evolution of the delamination versus time (material 2).

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L. Guillaumat / Computers and Structures 76 (2000) 163±172

Fig. 13. 3D observation of the delamination by the de-ply technique. (a) A photograph of one ply with a mark, (b) assembling of all the marks.

Fig. 14. Model of the de¯ection of the plate centre versus span and width.

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171

Fig. 15. Multi-impact loading on the same structure. (a) Comparison between the ®rst and second impact (b) Responses from second impact up to the sixth.

the ¯exural sti€ness of the structure is high. That is why minus sign appears in front of X2. Furthermore, the coupling is insigni®cant (0.02). 6.2.3. Contact time (ms) (7) Y ˆ 24 ÿ 7:96
X1 ‡ 20:76
X2 ‡ 4:75
X 21 ‡ 5:11
X 22 ÿ 6:78
X1
X2

…7†

The analysis of this polynomial shows that the contact time depends more on the span …b2 ˆ 20:76† than on the width …b1 ˆ 7:96). As shown previously, the increase of the span decreases the ¯exural sti€ness of the structure which explains why the striker remains longer on it. 6.2.4. Delamination area (cm2) (8) Y ˆ 4:37 ‡ 2:06:X1 ÿ 13:41:X2 ÿ 0:86:X 21 ‡ 12:77:X 22 ‡ 3:56:X1 :X2

…8†

This response is a damage mode of the structure. Therefore, it has the same evolution as the contact load: there is a minus sign with respect to X2. Finally, the results presented here show that with larger span and smaller width, the structure is less damaged by the impact loading for the same couple, mass±velocity. The ¯exural sti€ness of the plate decreases more of the kinetic energy of the drop weight which is transformed into elastic energy. In addition, 10 impact tests have been performed on

the same structure. Fig. 15a shows the responses after the ®rst and the second impact. We can notice that only the ®rst peak is di€erent according to the number of shocks. Actually, when the striker touches the plate, the ®rst time the load increases suddenly, while for the second time, the load increases with a slower rate: this is due to the fact that the local sti€ness decreases after the ®rst impact because the local damage remains constant with the following impacts (Fig. 15b). This result demonstrates that the local damage induced by the impact loading does not have a large in¯uence on the global behaviour of the plate.

7. Conclusions The mechanical behaviour of composite structures varies widely according to their dimensions. For all the plates, the load versus time exhibits some undulations which are attributed to a transverse eigen mode of vibration. The delamination grows only after the development of the matrix cracking, suggesting a coupling between them. A 3D observation of the damage by a de-ply technique shows that the delamination exists nearly at each interface of the material. The experimental design method enables: 1. quantitative informations to be obtained on the in¯uence of impact parameters, and coupling between variables to be detected, 2. empirical models for impact responses to be established. The coupling coecients for mass and velocity for laminated composites are signi®cant for most re-

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L. Guillaumat / Computers and Structures 76 (2000) 163±172

sponses studied. This implies that the energy parameter will not always be sucient to describe an impact loading.

[5] [6]

Acknowledgements The author would like to thank Albugues L., Haramburu E., Cahuzac C., Lahourcade P.J., and Pataki H. for their support for this work and Couach-Plascoa for providing composites specimens for the experimental phase.

[7] [8] [9] [10]

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