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Influence of anisotropy on the transient hygroscopic stresses in polymer matrix composites with cyclic environmental conditions
Composite Structures 55 (2002) 393–405 www.elsevier.com/locate/compstruct
Influence of anisotropy on the transient hygroscopic stresses in polymer matrix composites with cyclic environmental conditions A. Tounsi a, E. Adda Bedia a
a,*
, G. Verchery
b
Laboratoire de M ecanique et Mat eriaux, Universit e de Sidi Bel Abbes, BP 89 Cit e Ben M’hidi, 22000 Sidi Bel Abbes, Alg erie b Institut Sup erieur de l’Automobile et des Transports, BP 31, 58027 Nevers Cedex, France
Abstract The transient residual stresses in polymer matrix composites induced by cyclic variations in temperature and moisture are very important and must be taken into account in the design of composite materials, particularly aerospace structures e.g. aircraft. Aiming to reduce these stresses, we have studied the influence of anisotropy on the hygrothermal behaviour of the laminated plates. The objective, here, is to contribute to the optimal design of composite structures. The anisotropy is evaluated using the degree of anisotropy introduced from polar representation of tensors [1–5]. Various laminates, with controlled and random stacking sequences, were analysed under typical boundary conditions and with the same hygrothermal loading. The results show that the thermal and hygroscopic stresses vary differently with the degree of anisotropy in such a way that their superposition does not lead to a growth of the intern resulting stresses. The other obtained results show that the knowledge of the degree of anisotropy of the laminate permits us to predict the state of different stresses in the ply frame. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction One of the most important issues in the use of composite materials in the aeronautical structures is the degradation of the material due to moisture absorption through long years of service under cyclic environmental conditions. This problem has been extensively investigated to calculate the moisture content of composite materials exposed to air by varying temperature and relative humidity simulating long-term service. On the other hand, the residual stresses within the laminated structures induced by environmental conditions in terms of temperature and moisture were the object of rare studies [6,7]. These stresses are investigated in this work with some simplifying assumptions. First, the environmental loading is supposed to be cyclic and the basic cycle is described by varying the temperature and relative humidity. Secondly, the residual stresses due to the distribution of temperature across the plate and those due to the moisture concentration within the same plate are supposed to be uncoupled. This is to be permitted if moisture sorption and heat diffusion are not coupled. However, two circumstances generally authorise us to
*
Corresponding author. E-mail address: [email protected] (E.A. Bedia).
uncouple the analyses of moisture concentration and heat diffusion. First, in polymeric materials, these phenomena have very different characteristic times: their ratio of the diffusivities can be as high as six orders of magnitude. Secondly, for thicknesses of some millimetres, the variations of external temperature and moisture are slow compared to the time necessary to reach thermal equilibrium or steady state. Consequently, the stress components due to the change of temperature can be reasonably computed using the classical laminated plate theory [8] due to the uniform distribution of temperature across the plate at the thermal equilibrium. On the other hand, particular attention must be paid to the calculation of the transient stress components [6] due to the non-uniform moisture concentration. We note that all these hypotheses constitute the basis of many studies such as those presented by Springer [9–11], for the prediction of moisture diffusion with cyclic environmental conditions. Furthermore, the purpose of this work is to contribute to the design methodology by investigating and assessing the influence of anisotropy on the hygrothermal behaviour of the laminated plates. The polar representation of anisotropic properties is selected as a convenient tool for the study and can serve to explain the partial compensation of the two stresses which are originally different (stresses due to the moisture
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 6 1 - 1
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concentration and those due to distribution of the temperature within the plate). Finally, this contribution to the optimal design of materials composites permits us to find laminates with more resistant stacking sequences to the environmental condition.
2. Calculation of the moisture concentration The widespread and simple moisture diffusion model for the computation of the kinetic water absorption is Fick’s law. The analysis of the moisture content Cðz; tÞ in the composite material can be done with the hypothesis of an instantaneous thermal equilibrium, i.e. the temperature through the thickness of the material is equal to the given external time-dependent temperature. Then, the solution for a plate with thickness equal to h, exposed to the same moisture distribution on both faces, when described as one-dimensional Fickian problem with diffusivity D, satisfies the following partial differential equation and boundary conditions: oCðz; tÞ o2 Cðz; tÞ ¼ DðtÞ ; ot oz2 h ; t ¼ C0 ðtÞ and C 2 Cðz; oÞ ¼ 0:
ð1Þ
h ;t C 2
Fig. 1. Definition of the different strains involved in the calculation of the residual stresses in the ‘‘i ðx or yÞ’’ direction: (a) initial state; (b) free expansions; (c) actual final state.
puted at each point zk of the thickness where the moisture concentration C is given by the computer code ‘‘W8GAIN’’. The value of k varies from 0 to N , where N is the number of plies and subplies (in counting, the subplies are considered as a ply). In all considered laminates we have taken N ¼ 144. The free expansions are given by the following equation: eðKÞ ¼ ax DT þ bx cðzK ; tÞ; x ¼ ay DT þ by cðzK ; tÞ; eðKÞ y
¼ C0 ðtÞ;
ð4Þ
ð2Þ
ðKÞ eS
ð3Þ
The non-mechanical stresses are then easily computed in the ply frame
When the environmental conditions are cyclic, the functions DðtÞ and C0 ðtÞ are cyclic functions, with the period T . The solution of Eq. (1) with the identical boundary conditions on both faces of the plate specified in Eq. (2) has been obtained by using the program ‘‘W8GAIN’’, based on the step by step integration method, developed and published by Springer [10] which is programmed in FORTRAN 77 on a workstation ‘‘Apollo DN 4500’’ (processor 68030 with a Floating Point Accelerator). This program has been used in order to validate the analytical method suggested by Adda Bedia EA and Verchery G [12–14] which also permits the solution of such a type of problem.
nðKÞ
ri
¼ 0:
ðKÞ
¼ Qij ej ;
i; j ¼ x; y; s:
ð5Þ
In the laminate frame defined in Fig. 2, these stresses become rnðkÞ a , a ¼ 1; 2; 6, and the non-mechanical inplane stress components Na and moment components Ma are Z h=2 Z h=2 Nan ¼ rna dz and Man ¼ zrna dz; h=2
a ¼ 1; 2; 6:
h=2
ð6Þ
3. Calculation of the non-mechanical stresses due to the moisture concentration distribution The moisture concentration distribution inside the plate gives rise to residual stresses. Such stresses can be computed using the method presented by Benkeddada [6], which is an extension of the method given by Tsai [8] for uniform moisture concentration distributions. Time t being given, the moisture concentration distribution is given by the computer code ‘‘W8GAIN’’. The first step is to compute the on-axis free expansions ex and ey defined in Fig. 1. These expansions are com-
Fig. 2. Laminate frame ð1; 2Þ and local frame ðx; yÞ of one of the plies.
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
Here Nan and Man are computed assuming a linear distribution of the non-mechanical stresses rnðkÞ between a the points zk and zkþ1 ; k ¼ 0; . . . ; N 1: Nan ¼
N 1 X
aðkÞ a
ðz2kþ1 z2k Þ þ bðkÞ a ðzkþ1 zk Þ; 2
aðkÞ a
ðz3kþ1 z3k Þ ðz2kþ1 z2k Þ þ bðkÞ a 3 2
k¼0
Man
¼
N 1 X k¼0
ð7Þ
with aðkÞ a ¼
ranðkþ1Þ rnðkÞ a ; zkþ1 zk
bka ¼ rnðkÞ aðkÞ a a zk ;
5. Validation of the program ð9Þ
The non-mechanical strains are deduced using the usual strain–stress relation in laminated plates [8]. 0n ¼ ½afN n g þ ½bfM n g; e ð10Þ t fk n g ¼ ½b fN n g þ ½dfM n g: are then computed at The non-mechanical strains enðkÞ a each point zk n enðkÞ ¼ e0n a a þ zk ka :
ð11Þ
In the ply frame, these non-mechanical strains become nðkÞ ei . The residual strains are then defined by the difference between the non-mechanical strains and the free expansions: rðkÞ
ei
ðkÞ
nðkÞ
¼ ei
ei :
ð12Þ rðkÞ
Then we can obtain the residual stresses ri frame as follows: rðkÞ
ri
rðkÞ
¼ Qij ej :
in the ply ð13Þ
rðkÞ
Each stress ri can be expressed by the summation of two stresses which are originally different rðkÞ
ri
rðkÞ
rðkÞ
¼ rig þ rit ;
convenient for many applications. Its special application is the definition of the degree of anisotropy of a material. It is defined as the relative deviation in stiffness between the material and its isotropic part (the isotropic part of a material should be defined from polar components, but for a laminate, it can more simply be seen as the homogeneous material with properties identical to the in-plane properties of a quasi-isotropic laminate). The degree of anisotropy so defined is positive and is always less than 1.
ð8Þ
ðkÞ rnðkÞ ¼ aðkÞ a a z þ ba :
zk < z < zkþ1 ;
395
ð14Þ
rðkÞ rig
is the hygroscopic stress calculated with a where null distribution of temperature within the plate (DT ¼ 0). rðkÞ rit is the thermal stress calculated with a null distribution of moisture concentration within the plate ðCðzK ; tÞ ¼ 0Þ.
4. Polar representation method This method was developed by Verchery and his coworkers [1–5]. It uses the representation of tensors by parameters which are the so-called polar components. For plane elastic stiffness, six parameters are required. These polar components give the key to compare various anisotropic materials together, using a relative deviation defined from the polar components of the materials to be compared. This relative deviation is
The validation of this program has been achieved in three steps. The first consists in testing the computer code ‘‘W8GAIN’’ developed and published by Springer [10], used for the calculation of the moisture concentration inside the plate which is submitted to the cyclic environmental conditions, wherein we have checked the same results obtained by Adda Bedia EA and Verchery G [12–14] who have used the same computer code. The second step shows that the evolution of the transverse transient stress rrðkÞ obtained by a part of the program in y charge of calculating stresses and strains has the same tendencies with those of Hahn and Kim [7]. The calculations have been made for three different graphite/epoxy laminates exposed to a constant moist environment: ½08T ; ½0; 903 s and ½04; 904 T . The appeared discrepancies are very probably due to the analytical model for the swelling strains which takes into account a threshold value of the concentration below which the swelling strains are zero [7]. The same results compared to those of Benkeddad A [6] using the same laminates confirm the validation of this part of the program which nearly has given identical results. This is due to the used method which follows the same step described by Benkeddad A [6]. The third step consists in validating a part of the program which is in charge of the calculation of the degree of anisotropy where we have checked with those of laminates which we already know.
6. Material and laminated plates used The material used for the present studies is a graphite/ epoxy, in which properties are summarised in Tables 1–3 and the thickness of the laminated plate is 12.7 mm. Two categories of multilayer composite plates are used in this work: Laminates with controlled and random stacking sequences. 6.1. Controlled stacking sequences laminated plates Three types of laminates with controlled stacking sequences are used in this work.
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Table 1 Properties of material [11] Diffusivity, mm2 /s
D ¼ 0:57 expð4993=T Þ, T : temperature (K)
Moisture concentration at the surface as a function of relative humidity in atmosphere, %
C0 ¼ 0:015H, H: relative humidity (%)
Table 2 Mechanical properties of material [8] Ex (GPa)
Ey (GPa)
mx
Gxy (GPa)
ax ð106 K1 Þ
ay ð106 K1 Þ
bx
by
181
10.3
0.28
7.17
0.02
22.5
0
0.6
Table 3 Strength data of material [8] Material
X (MPa) X 0 (MPa)
Y (MPa) Y 0 (MPa) S (MPa)
Graphite/ epoxy
1500
40
1500
246
temperature [8] and room temperature is DT ¼ 100 °C. The variations in temperature and relative humidity retained are shown in Figs. 3–5 as a function of the reduced time, tr ¼ t/period. These environmental conditions are similar to those described by Springer [9] to simulate the 20-yr service life of a military aircraft. The temperature and humidity were assumed to vary in a cyclical way. For the 6-day period (144 h), normal atmospheric conditions are active for 123 h, followed by
68
6.1.1. Unidirectional laminates Since all unidirectional laminates have the same hygrothermal behaviour, we have considered only the laminates of type ½0rT , where r is a repetition index, equal to 36. 6.1.2. Cross-ply laminates The selected symmetric stacking sequences are: ð½0=90=0=90=0=90=0=90=0rÞs, noted under condensed shape ½0=90rs and ½þh; hps; h ¼ 20°; 30°; 45°. With: r ¼ 2; p ¼ 1; 4; 6; 9; 18 and 36.
Fig. 3. Ambient relative humidity before and after the flight versus the reduced time: tr ¼ t=ð6 hÞ, t=ð6 daysÞ, t=ð6 weeksÞ.
6.1.3. Quasi-isotropic laminates The selected stacking sequences are: ½60= 60=0= 60= 0=60=0=60= 60r and ð½0= þ 45=90= 45pÞs, noted, respectively, under condensed shape ½þ60= 60=0r and ½0=45=90ps with r ¼ 1;2;4;8 and 16; p ¼ 1;2;3;6;9 and 18. 6.2. Random stacking sequences laminated plates Various 18-ply symmetric laminates were selected at random with a uniform distribution of the angles. A sample of 20 random sequences was studied. It was found that the in-plane degree of anisotropy varies from 5.99% to 42.22%. The extreme cases are shown below:
Fig. 4. Constant relative humidity before and after the flight versus the reduced time: tr ¼ t=ð6 hÞ, t=ð6 daysÞ, t=ð6 weeksÞ.
½35=8=65=140=118=128=75=126=86=132=74=15=7=2= 75=141=41=27=S ;
eA ¼ 5:99%;
½23=26=170=13=142=1=21=167=51=33=112=71=6=18= 81=31=148=28=S ;
eA ¼ 42:22%:
7. Results The results below are given under the following assumptions. The difference DT between the stress-free
Fig. 5. Ambient temperature before and after the flight versus the reduced time: tr ¼ t=ð6 hÞ, t=ð6 daysÞ, t=ð6 weeksÞ.
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
a flight for 11 h, and normal atmospheric conditions once more for 10 h. In Fig. 3, the relative humidity of the environment was assumed to be reduced as the temperature increases, except during flight, when the relative humidity decreases to zero. In Fig. 4, the relative humidity of the environment was assumed to be constant (82%), except during flight when the relative humidity decreases to zero. In order to facilitate the interpretation of results we have considered two moist middles. Humid environment of type A or B. These two middles are defined as follows: Environment of the type A: In this middle the plate has been submitted of these two faces in cyclic environmental conditions illustrated in Figs. 3 and 5. Environment of the type B: In this middle the plate has been submitted of these two faces to the cyclic environmental conditions represented in Figs. 4 and 5. We present some examples with different cyclic environmental conditions (only on the relative humidity).
397
The transient hygroscopic stresses are computed at the end of each cycle. 7.1. Influence of the environmental loading In order to illustrate the influence of cyclic environmental conditions on the evolution of the transient hygroscopic stresses, we have first considered the unidirectional laminates, then we have analysed the effect of these boundary conditions and the engendered hygroscopic loading on the studied parameters in the case of laminates having different in-plane degrees of anisotropy ðeA Þ. 7.1.1. Case of unidirectional laminates We have done the calculations of the residual stresses rrx and rry engendered by each humid environment (type A or B) for a 6-day period (Figs. 6–9). Some important points emerge from these results. Both stresses are
Fig. 6. Residual stresses rrx through the thickness of the unidirectional laminate with the period of 6 days (middle A).
Fig. 7. Residual stresses rrx through the thickness of the unidirectional laminate with the period of 6 days (middle B).
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Fig. 8. Residual stresses rry through the thickness of the unidirectional laminate with the period of 6 days (middle A).
Fig. 9. Residual stresses rry through the thickness of the unidirectional laminate with the period of 6 days (middle B).
compressive ones for the external plies and tensile ones for the internal plies during the absorption process. The order of magnitude of rrx is negligible compared to both longitudinal tensile and compressive strengths X and X 0 given in Table 3. On the other hand, the transversal residual stresses rry reach important values compared to both transverse tensile and compressive strengths Y and Y 0 . In fact, the maximum value of the transverse stress rry reaches about 45.27% of the tensile transverse strength Y in the case of the humid environment (B) and 24.44% in the case of humid environment (A). We notice that the compressive value is maximum at the beginning of the absorption. It has been observed that the compressive stress rry at the end of the first cycle is about 30.47% of the compressive strength Y 0 in the case of the humid environment (A) and about 30% in the case of the humid environment (B).
Secondly, we have calculated the stresses rry assuming that the two faces of the plate are submitted to the environmental conditions of type B and for periods of 6 h and 6 weeks (Figs. 10 and 11). The maximum value of the transverse stress rry reaches about 46.14% of the tensile strength Y for a period of 6 h and 45% for a period of 6 weeks. The maximum of the compressive stress is obtained at the beginning of the absorption. In fact, the maximum value reaches about 30.75% of the compressive strength Y 0 for a period of 6 h and 28.50% for a period of 6 weeks. Due to the symmetry of the present problem, the shear residual stresses rrS are always zero in the ply frame. This can also be explained by the fact that the degrees of anisotropy (in-plane and flexural degree of anisotropy) of unidirectional laminates of the type ½hrT stay constant whatever the value of the
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
399
Fig. 10. Residual stresses rry through the thickness of the unidirectional laminate with the period of 6 h (middle B).
Fig. 11. Residual stresses rry through the thickness of the unidirectional laminate with the period of six weeks (middle B).
angle h (fibres orientation) and whatever the number of plies. Also, we note that the values of the other stresses rrx and rry do not vary with the fibres orientation and the number of plies in the case of the unidirectional laminates. At last, and through these results, we notice that the duration of the period does not disturb too much the distribution of the transient residual stresses within the laminates, but its lightly influences the time of the appearance of maximum stresses. On the other hand, the environmental conditions, which the plate is submitted to, can clearly modify the values of these stresses in internal plies (tensile plies). For the external plies (compressive plies), we notice that the stresses keep almost the same values in the two middles (A and B). This is explained by the fact that the boundary conditions at the end of each cycle are identical in all cases studied (HR ¼ 82% and h ¼ 27 °C) and consequently are going to engender constant free expansions. The smaller dif-
ferences that appear are due only to the non-mechanical strains. We notice also the appearance of the edge effects which represent the fluctuating part of the transient residual stresses in the skin of the plate, particularly in the case of the periods of 6 weeks and 6 days. 7.1.2. Case of laminates with different in-plane degrees of anisotropy To assess the influence of boundary conditions and hygroscopic loading on the analysed parameters inside the laminates with in-plane degrees of anisotropy ðeA Þ, we have considered the two cases of symmetric boundary conditions (middles B and A) with the period of 6 days (the other periods are not tested because we have observed in the case of unidirectional laminates that the period influences very slightly the chosen quantities). In these tests, we have assumed that DT ¼ 0 is only analysis the hygroscopic stresses rrig . The considered laminates in this study are: ½0r (73%); ½20; 204S (59.9%); ½30; 304S (47.1%); ½0=45=90S (0%).
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A. Tounsi et al. / Composite Structures 55 (2002) 393–405
The obtained results (Tables 4–7) show well that the parameters analysed have the same tendencies in the two cases (middles A and B). The transversal stresses rryg keep the same variations at the end of different chosen cycles. For the other stresses rrxg and rrSg we have noticed that they keep the same tendencies except for the first cycles. Consequently, we have kept for the following, the boundary conditions defined in the middle of B and the hygroscopic loading engendered at 117 cycles. 7.2. Influence of anisotropy on the residual stresses Some configurations of materials were analysed aiming to study the influence of anisotropy on the hygrothermal behaviour of laminated composite plates. First, we have analysed the behaviour of hygroscopic stresses and thermal stresses inside the plate. The hyg-
roscopic stresses rrig are calculated assuming that the temperature difference DT is zero. On the other hand, the thermal stresses rrit are calculated assuming a null distribution of moisture concentration within the plate. In the Figs. 12 and 13, we present the evolution of the stresses rryg and rryt inside the considered laminates. A very interesting point which has been observed in this study is that for laminates with a lower degree of anisotropy, the transverse transient stress rryg decreases in the internal plies (tensile plies) and increases in the external plies (compressive plies) in absolute value (Fig. 12). The state of the transverse hygroscopic stress is oriented towards a compressive state in the laminates with lower degree of anisotropy and towards a tensile state in the laminates with higher degree of anisotropy. In the case of the thermal stress rryt (Fig. 13) the behaviour of laminates is reversed; the state of stresses leads towards a tensile state for laminates with lower
Table 4 Stresses (MPa) computed in the mid-plane of laminates (middle B) Degree of anisotropy (%)
1 Cycle rrxg
rryg
rrSg
21 Cycles rrxg
rryg
rrSg
117 Cycles rrxg
rryg
rrSg
3500 Cycles rrxg
rryg
rrSg
73 59.9 47.1 0
0.67 2.78 3.75 2.87
2.40 1.58 0.81 0.21
0 )1.04 )0.95 )0.23
3.02 8.53 12.28 15.03
10.80 7.28 3.84 0.84
0 )4.30 )4.07 )0.84
5.07 14.94 23.26 34.47
18.10 9.97 1.94 )5.30
0 )9.78 )9.38 )0.79
0.27 20.06 39.34 55.64
0.97 )18.32 )37.50 )54.39
0 )23 )22.23 )0.09
Table 5 Stresses (MPa) computed in the external faces of laminates (middle B) Degree of anisotropy (%)
1 Cycle rrxg
rryg
rrSg
21 Cycles rrxg
rryg
rrSg
117 Cycles rrxg
rryg
rrSg
3500 Cycles rrxg
rryg
rrSg
73 59.9 47.1 0
)20.7 )21.28 )20.73 )21.39
)73.95 )74.66 )75.41 )76.03
0 0.78 0.81 0
)18.30 )17 )13.90 )19.
)65.5 )68.9 )72.3 )75
0 3.90 3.85 )0.17
)14.29 )8.05 )0.31 )1.03
)51.03 )59.01 )67.02 )73.77
0 9.42 9.19 )0.50
)4.43 14.30 33.4 49.8
)15.80 )35.05 )54.23 )71.12
0 22.90 22.17 0
Table 6 Stresses (MPa) computed in the mid-plane of laminates (middle A) Degree of anisotropy (%)
1 Cycle rrxg
rryg
rrSg
21 Cycles rrxg
rryg
rrSg
117 Cycles rrxg
rryg
rrSg
3500 Cycles rrxg
rryg
rrSg
73 59.9 47.1 0
0.39 1.62 2.19 1.67
1.4 0.92 0.47 0.12
0 )0.60 )0.55 )0.13
1.61 4.59 6.6 7.98
5.76 3.88 2.05 0.45
0 )2.29 )2.17 )0.45
2.74 7.88 12.14 17.86
9.77 5.63 1.53 )2.16
0 )5 )4.78 )0.44
0.72 12.88 23.8 31.59
2.57 )8.22 )18.92 )28.28
0 )12.93 )12.45 )0.24
Table 7 Stresses (MPa) computed in the external faces of laminates (middle A) Degree of anisotropy (%)
1 Cycle rrxg
rryg
rrSg
21 Cycles rrxg
rryg
rrSg
117 Cycles rrxg
rryg
rrSg
3500 Cycles rrxg
rryg
rrSg
73 59.9 47.1 0
)20.98 )21.32 )21 )21.38
)74.95 )75.37 )75.80 )76.16
0 0.46 0.47 0
)19.76 )19.08 )17.43 )20.17
)70.59 )72.38 )74.2 )75.62
0 2.07 2.05 )0.09
)17.76 )14.70 )10.77 )11.43
)63.45 )67.51 )71.6 )75.02
0 4.80 4.67 )0.27
)11.93 )2.62 7.84 15.88
)42.61 )53.29 )63.97 )73.34
0 12.66 12.30 )0.002
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
401
Fig. 12. Influence of the degree of anisotropy eA on the transversal stresses rryg within the plate.
Fig. 13. Influence of the degree of anisotropy eA on the transversal stresses rryt within the plate.
Fig. 14. Influence of the degree of anisotropy eA on the longitudinal stresses rrxg within the plate.
degree of anisotropy contrarily to those having a higher degree of anisotropy where we notice that these stresses decrease as the degree of anisotropy takes higher values until the stresses become zero in the case of unidirectional laminates. We also note that the stresses rryg and
rryt calculated inside the laminates ½þ45; 459S and ½0=902S are nearly equal to those of the laminate ½þ60= 60=04 . Concerning the longitudinal transient stresses rrxg (Fig. 14), the laminates with lower degree of anisotropy
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A. Tounsi et al. / Composite Structures 55 (2002) 393–405
have the tendency of being in tension where we notice that these stresses increase gradually when the degree of anisotropy decreases, which is the opposite of what was noticed in the case of the thermal stresses rrxt Fig. 15 shows that laminates with lower degree of anisotropy
have the tendency of being in compression and where the stresses increase gradually in absolute value when the degree of anisotropy decreases. For the shear stresses (Figs. 16 and 17) we notice that the stresses rrSg and rrSt have different signs in the same
Fig. 15. Influence of the degree of anisotropy eA on the longitudinal stresses rrxt within the plate.
Fig. 16. Influence of the degree of anisotropy eA on the shear stresses rrSg within the plate.
Fig. 17. Influence of the degree of anisotropy eA on the shear stresses rrSt within the plate.
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
ply (one is positive, the other negative in the same ply and vice versa). In addition, in the quasi-isotropic laminate such as the laminate ½þ60= 60=04 and laminates with lower degree of anisotropy such as ½þ45; 459S and ½0=902S , the shear stresses tend towards zero. The same remark is made for the case of higher anisotropic laminates such as unidirectional laminates. The maximum of the resulting stress rrS is obtained for the case of the laminate ½þ20; 20rs which represents a degree of anisotropy of 59.9%. Once away from this degree of anisotropy, these stresses begin to decrease and tend towards zero for laminates with high degree of anisotropy (like the case of the unidirectional laminates), laminates with lower degree of anisotropy and quasiisotropic laminates. We also notice that these stresses are constant in each ply (at any time). This property is observed for all the laminates studied except for the case of the unsymmetric laminate ½þ60= 60=0r . This is due to the fact that the residual strains are defined by the difference between non-mechanical strains and the free expansions (Eq. (12)). The non-mechanical strains are constant within each ply for symmetrical boundary conditions and for uncoupled laminates. Because of the no symmetry of laminate ½þ60= 60=0r , the coupling matrix ½B will not be zero. This makes that the nonmechanical strains are not constant within each ply of this laminate. Secondly, we have studied the influence of the repetition index on the evolution of hygroscopic stresses rrig in the external faces and the mid-plane of laminates ½20; 20rs ; ½þ60= 60=0r and ½0=45=90rs (Figs. 18–23). All the obtained results have shown that the influence of the repetition index on the transversal hygroscopic stresses rryg , whether they were on external faces or in the mid-plane, is not considerable.In fact, we notice that the stresses rryg stay almost constant for any value of the repetition index ðrÞ. We have undertaken the same study for the longitudinal stresses rrxg where we can see that the influence of the repetition index is observed only in the first values after we have remarked that these stresses are not sen-
403
Fig. 19. Influence of the repetition of stacking sequences on the stresses computed in the external faces of laminate ½þ20; 20rs .
Fig. 20. Influence of the repetition of stacking sequences on the stresses computed in the mid-plane of laminate ½þ60= 60=0r .
Fig. 21. Influence of the repetition of stacking sequences on the stresses computed in the external faces of laminate ½þ60= 60=0r .
Fig. 22. Influence of the repetition of stacking sequences on the stresses computed in the mid-plane of laminate ½0=45=90rs .
Fig. 18. Influence of the repetition of stacking sequences on the stresses computed in the mid-plane of laminate ½þ20; 20rs .
sitive to the repetition of stacking sequences. In fact, their stabilisation ðrrxg Þ is observed in the studied laminates ð½þ20; 20rs ; ½þ60= 60=0r and ½0=45=90rs Þ as
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Fig. 23. Influence of the repetition of stacking sequences on the stresses computed in the external faces of laminate ½0=45=90rs .
soon as the repetition index takes the values 6, 4 and 9, respectively. For the shear stresses rrSg , we notice that these stresses stabilise for the first values of the repetition index. At last, few laminates with random sequences are analysed. Figs. 24–27 illustrate the variation of the residual stresses rrig and rrit in the external faces and the mid-plane of laminates versus the in-plane degree of anisotropy. The obtained results confirm almost all those obtained in the previous case for laminates with controlled stacking sequences except for a few values of longitudinal stresses rrxg and rrxt which have not respected the evolution observed for the case of controlled sequences. The explanation that we can give to this is
Fig. 26. Influence of the degree of anisotropy on the hygroscopic stresses rrig computed in the external faces of the laminates (when DT ¼ 0).
Fig. 27. Influence of the degree of anisotropy on the thermal stresses rrit computed in the external faces of the laminates (when C ¼ 0).
Fig. 24. Influence of the degree of anisotropy on the hygroscopic stresses rrig computed in the mid-plane of the laminates (when DT ¼ 0).
tied to the repetition index which is in this cases equal to 1 for all laminates analysed with random stacking sequences; this means that the calculated stresses are not yet stabilised and consequently at this stage, these stresses are not only dependent on the in-plane degree of anisotropy. The transversal stresses rryg and rryt vary lightly with the degree of anisotropy. This is due to the fact that all the laminates analysed have a lower degree of anisotropy. For the same reason, we have noticed that the shear stresses rrSg and rrSt are almost zero.
8. Conclusions
Fig. 25. Influence of the degree of anisotropy on the thermal stresses rrit computed in the mid-plane of the laminates (when C ¼ 0).
This work represents an original attempt to analyses the influence of anisotropy on the hygrothermal behaviour of laminated composite plates. The non-mechanical residual stresses across the plate were studied as a function of the degree of anisotropy for different controlled and random stacking sequences. Such a study has permitted us to explain the partial compensation of the thermal and hygroscopic residual stresses. Indeed,
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
this is due to the different variations of the two stresses with the degree of anisotropy of laminates: if one decreases with the degree of anisotropy, the other increases, or in the case of shear stresses, if one is positive, the other is negative and vice versa. Also, this study has permitted us to predict the state of the two residual stresses (thermal and hygroscopic stresses) inside the laminated plates and therefore, to find more resistant stacking sequences to a moist environment. References [1] Verchery G. Designing with anisotropy. In: Hamelin P, Verchery G, editors. Textile composite in building construction, Part 3 Mechanical behaviour, design and application. Paris: Pluralis; 1990. p. 29–42. [2] Kandil N, Verchery G. New methods of design for stacking sequence of laminates. In: Brebbia CA, de Wilde WP, Blain WR, editors. Computer aided design in composite material technology. Berlin: Computational Mechanics Publications/Springer; 1988. p. 243–57. [3] Kandil N, Verchery G. Some new developments in the design of stacking sequences of laminates. In: Yunshu W, Zhenlong G, Renjie W, editors. Proceeding of the seventh International Conference on Composite Materials, vol. 3. Oxford: International Academic Publishers/Pergamon Press; 1989. p. 358–63. [4] Verchery G, Vong TS. Une methode d’aide graphique a la conception des sequences d’empilement dans les stratifies. In: Bathias C, Menkes D, editors. Proceedings of the 5th French Conference on Composite Materials. Paris: Pluralis; 1986. p. 267–80. [5] Cheikh Saad Bouh AB, Aivazzadeh S, El Sheikh MS, Verchery G. A quantitative study of the influence of anisotropy on the bending
[6]
[7] [8] [9] [10]
[11]
[12]
[13]
[14]
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deformation of laminates. In: 8th International Conference on Composite Materials, Honolulu (USA), Section 26-J, 1991. p. 1– 12. Benkeddad A, Grediac M, Vautrin A. On the transient hygroscopic stresses in laminated composite plates. Compos Struct 1995;30:201–15. Hahn HT, Kim RY. Swelling of composite laminates. Adv Compos Mater – Environ Eff, ASTM STP 1978;658:98–120. Tsai SW. In: Composite design. 4th ed. Dayton, USA: Think Composites; 1988. Springer G. Moisture content of composites under transient condition. J Compos Mater 1977;11:107–22. Springer G. Numerical procedures for the solutions of onedimensional Fickian diffusion problems. In: Environmental effects on composite materials. Westport, USA: Technomic Publishing Co; 1981. p. 166–99. Springer G. Environmental effects. In: Tsai SW, editor. Composites design, Section 16. 4th ed. Dayton, USA: Think Composites; 1988. Adda-Bedia EA, Han WS, Verchery G. An asymptotic characterisation of the moisture diffusion in polymer matrix composites with cyclic environmental conditions. Compos Struct 2000;49(3):269–74. Adda-Bedia EA, Han WS, Verchery G. Edge effects in resin matrix composites with cyclic environmental conditions. In: Miravet A, editor. Proceeding of the 9th International Conference on Composite Materials (ICCM/9), vol. V. University of Zaragoza and Woodhead Publishing; 1993. p. 617–24. Vechery G. Moisture diffusion in polymer matrix composites with cyclic environmental conditions. In: Bunsell AR, Jamet JF, Massiah A, editors. Developments in The Sciences and Technology of Composite Materials, Proceeding of the 5th European Conference on Composite (ECCM5), European Association for Composite Materials (EACM), Bordeaux, France, 1992. p. 505–10.
Influence of anisotropy on the transient hygroscopic stresses in polymer matrix composites with cyclic environmental conditions A. Tounsi a, E. Adda Bedia a
a,*
, G. Verchery
b
Laboratoire de M ecanique et Mat eriaux, Universit e de Sidi Bel Abbes, BP 89 Cit e Ben M’hidi, 22000 Sidi Bel Abbes, Alg erie b Institut Sup erieur de l’Automobile et des Transports, BP 31, 58027 Nevers Cedex, France
Abstract The transient residual stresses in polymer matrix composites induced by cyclic variations in temperature and moisture are very important and must be taken into account in the design of composite materials, particularly aerospace structures e.g. aircraft. Aiming to reduce these stresses, we have studied the influence of anisotropy on the hygrothermal behaviour of the laminated plates. The objective, here, is to contribute to the optimal design of composite structures. The anisotropy is evaluated using the degree of anisotropy introduced from polar representation of tensors [1–5]. Various laminates, with controlled and random stacking sequences, were analysed under typical boundary conditions and with the same hygrothermal loading. The results show that the thermal and hygroscopic stresses vary differently with the degree of anisotropy in such a way that their superposition does not lead to a growth of the intern resulting stresses. The other obtained results show that the knowledge of the degree of anisotropy of the laminate permits us to predict the state of different stresses in the ply frame. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction One of the most important issues in the use of composite materials in the aeronautical structures is the degradation of the material due to moisture absorption through long years of service under cyclic environmental conditions. This problem has been extensively investigated to calculate the moisture content of composite materials exposed to air by varying temperature and relative humidity simulating long-term service. On the other hand, the residual stresses within the laminated structures induced by environmental conditions in terms of temperature and moisture were the object of rare studies [6,7]. These stresses are investigated in this work with some simplifying assumptions. First, the environmental loading is supposed to be cyclic and the basic cycle is described by varying the temperature and relative humidity. Secondly, the residual stresses due to the distribution of temperature across the plate and those due to the moisture concentration within the same plate are supposed to be uncoupled. This is to be permitted if moisture sorption and heat diffusion are not coupled. However, two circumstances generally authorise us to
*
Corresponding author. E-mail address: [email protected] (E.A. Bedia).
uncouple the analyses of moisture concentration and heat diffusion. First, in polymeric materials, these phenomena have very different characteristic times: their ratio of the diffusivities can be as high as six orders of magnitude. Secondly, for thicknesses of some millimetres, the variations of external temperature and moisture are slow compared to the time necessary to reach thermal equilibrium or steady state. Consequently, the stress components due to the change of temperature can be reasonably computed using the classical laminated plate theory [8] due to the uniform distribution of temperature across the plate at the thermal equilibrium. On the other hand, particular attention must be paid to the calculation of the transient stress components [6] due to the non-uniform moisture concentration. We note that all these hypotheses constitute the basis of many studies such as those presented by Springer [9–11], for the prediction of moisture diffusion with cyclic environmental conditions. Furthermore, the purpose of this work is to contribute to the design methodology by investigating and assessing the influence of anisotropy on the hygrothermal behaviour of the laminated plates. The polar representation of anisotropic properties is selected as a convenient tool for the study and can serve to explain the partial compensation of the two stresses which are originally different (stresses due to the moisture
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 6 1 - 1
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concentration and those due to distribution of the temperature within the plate). Finally, this contribution to the optimal design of materials composites permits us to find laminates with more resistant stacking sequences to the environmental condition.
2. Calculation of the moisture concentration The widespread and simple moisture diffusion model for the computation of the kinetic water absorption is Fick’s law. The analysis of the moisture content Cðz; tÞ in the composite material can be done with the hypothesis of an instantaneous thermal equilibrium, i.e. the temperature through the thickness of the material is equal to the given external time-dependent temperature. Then, the solution for a plate with thickness equal to h, exposed to the same moisture distribution on both faces, when described as one-dimensional Fickian problem with diffusivity D, satisfies the following partial differential equation and boundary conditions: oCðz; tÞ o2 Cðz; tÞ ¼ DðtÞ ; ot oz2 h ; t ¼ C0 ðtÞ and C 2 Cðz; oÞ ¼ 0:
ð1Þ
h ;t C 2
Fig. 1. Definition of the different strains involved in the calculation of the residual stresses in the ‘‘i ðx or yÞ’’ direction: (a) initial state; (b) free expansions; (c) actual final state.
puted at each point zk of the thickness where the moisture concentration C is given by the computer code ‘‘W8GAIN’’. The value of k varies from 0 to N , where N is the number of plies and subplies (in counting, the subplies are considered as a ply). In all considered laminates we have taken N ¼ 144. The free expansions are given by the following equation: eðKÞ ¼ ax DT þ bx cðzK ; tÞ; x ¼ ay DT þ by cðzK ; tÞ; eðKÞ y
¼ C0 ðtÞ;
ð4Þ
ð2Þ
ðKÞ eS
ð3Þ
The non-mechanical stresses are then easily computed in the ply frame
When the environmental conditions are cyclic, the functions DðtÞ and C0 ðtÞ are cyclic functions, with the period T . The solution of Eq. (1) with the identical boundary conditions on both faces of the plate specified in Eq. (2) has been obtained by using the program ‘‘W8GAIN’’, based on the step by step integration method, developed and published by Springer [10] which is programmed in FORTRAN 77 on a workstation ‘‘Apollo DN 4500’’ (processor 68030 with a Floating Point Accelerator). This program has been used in order to validate the analytical method suggested by Adda Bedia EA and Verchery G [12–14] which also permits the solution of such a type of problem.
nðKÞ
ri
¼ 0:
ðKÞ
¼ Qij ej ;
i; j ¼ x; y; s:
ð5Þ
In the laminate frame defined in Fig. 2, these stresses become rnðkÞ a , a ¼ 1; 2; 6, and the non-mechanical inplane stress components Na and moment components Ma are Z h=2 Z h=2 Nan ¼ rna dz and Man ¼ zrna dz; h=2
a ¼ 1; 2; 6:
h=2
ð6Þ
3. Calculation of the non-mechanical stresses due to the moisture concentration distribution The moisture concentration distribution inside the plate gives rise to residual stresses. Such stresses can be computed using the method presented by Benkeddada [6], which is an extension of the method given by Tsai [8] for uniform moisture concentration distributions. Time t being given, the moisture concentration distribution is given by the computer code ‘‘W8GAIN’’. The first step is to compute the on-axis free expansions ex and ey defined in Fig. 1. These expansions are com-
Fig. 2. Laminate frame ð1; 2Þ and local frame ðx; yÞ of one of the plies.
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
Here Nan and Man are computed assuming a linear distribution of the non-mechanical stresses rnðkÞ between a the points zk and zkþ1 ; k ¼ 0; . . . ; N 1: Nan ¼
N 1 X
aðkÞ a
ðz2kþ1 z2k Þ þ bðkÞ a ðzkþ1 zk Þ; 2
aðkÞ a
ðz3kþ1 z3k Þ ðz2kþ1 z2k Þ þ bðkÞ a 3 2
k¼0
Man
¼
N 1 X k¼0
ð7Þ
with aðkÞ a ¼
ranðkþ1Þ rnðkÞ a ; zkþ1 zk
bka ¼ rnðkÞ aðkÞ a a zk ;
5. Validation of the program ð9Þ
The non-mechanical strains are deduced using the usual strain–stress relation in laminated plates [8]. 0n ¼ ½afN n g þ ½bfM n g; e ð10Þ t fk n g ¼ ½b fN n g þ ½dfM n g: are then computed at The non-mechanical strains enðkÞ a each point zk n enðkÞ ¼ e0n a a þ zk ka :
ð11Þ
In the ply frame, these non-mechanical strains become nðkÞ ei . The residual strains are then defined by the difference between the non-mechanical strains and the free expansions: rðkÞ
ei
ðkÞ
nðkÞ
¼ ei
ei :
ð12Þ rðkÞ
Then we can obtain the residual stresses ri frame as follows: rðkÞ
ri
rðkÞ
¼ Qij ej :
in the ply ð13Þ
rðkÞ
Each stress ri can be expressed by the summation of two stresses which are originally different rðkÞ
ri
rðkÞ
rðkÞ
¼ rig þ rit ;
convenient for many applications. Its special application is the definition of the degree of anisotropy of a material. It is defined as the relative deviation in stiffness between the material and its isotropic part (the isotropic part of a material should be defined from polar components, but for a laminate, it can more simply be seen as the homogeneous material with properties identical to the in-plane properties of a quasi-isotropic laminate). The degree of anisotropy so defined is positive and is always less than 1.
ð8Þ
ðkÞ rnðkÞ ¼ aðkÞ a a z þ ba :
zk < z < zkþ1 ;
395
ð14Þ
rðkÞ rig
is the hygroscopic stress calculated with a where null distribution of temperature within the plate (DT ¼ 0). rðkÞ rit is the thermal stress calculated with a null distribution of moisture concentration within the plate ðCðzK ; tÞ ¼ 0Þ.
4. Polar representation method This method was developed by Verchery and his coworkers [1–5]. It uses the representation of tensors by parameters which are the so-called polar components. For plane elastic stiffness, six parameters are required. These polar components give the key to compare various anisotropic materials together, using a relative deviation defined from the polar components of the materials to be compared. This relative deviation is
The validation of this program has been achieved in three steps. The first consists in testing the computer code ‘‘W8GAIN’’ developed and published by Springer [10], used for the calculation of the moisture concentration inside the plate which is submitted to the cyclic environmental conditions, wherein we have checked the same results obtained by Adda Bedia EA and Verchery G [12–14] who have used the same computer code. The second step shows that the evolution of the transverse transient stress rrðkÞ obtained by a part of the program in y charge of calculating stresses and strains has the same tendencies with those of Hahn and Kim [7]. The calculations have been made for three different graphite/epoxy laminates exposed to a constant moist environment: ½08T ; ½0; 903 s and ½04; 904 T . The appeared discrepancies are very probably due to the analytical model for the swelling strains which takes into account a threshold value of the concentration below which the swelling strains are zero [7]. The same results compared to those of Benkeddad A [6] using the same laminates confirm the validation of this part of the program which nearly has given identical results. This is due to the used method which follows the same step described by Benkeddad A [6]. The third step consists in validating a part of the program which is in charge of the calculation of the degree of anisotropy where we have checked with those of laminates which we already know.
6. Material and laminated plates used The material used for the present studies is a graphite/ epoxy, in which properties are summarised in Tables 1–3 and the thickness of the laminated plate is 12.7 mm. Two categories of multilayer composite plates are used in this work: Laminates with controlled and random stacking sequences. 6.1. Controlled stacking sequences laminated plates Three types of laminates with controlled stacking sequences are used in this work.
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Table 1 Properties of material [11] Diffusivity, mm2 /s
D ¼ 0:57 expð4993=T Þ, T : temperature (K)
Moisture concentration at the surface as a function of relative humidity in atmosphere, %
C0 ¼ 0:015H, H: relative humidity (%)
Table 2 Mechanical properties of material [8] Ex (GPa)
Ey (GPa)
mx
Gxy (GPa)
ax ð106 K1 Þ
ay ð106 K1 Þ
bx
by
181
10.3
0.28
7.17
0.02
22.5
0
0.6
Table 3 Strength data of material [8] Material
X (MPa) X 0 (MPa)
Y (MPa) Y 0 (MPa) S (MPa)
Graphite/ epoxy
1500
40
1500
246
temperature [8] and room temperature is DT ¼ 100 °C. The variations in temperature and relative humidity retained are shown in Figs. 3–5 as a function of the reduced time, tr ¼ t/period. These environmental conditions are similar to those described by Springer [9] to simulate the 20-yr service life of a military aircraft. The temperature and humidity were assumed to vary in a cyclical way. For the 6-day period (144 h), normal atmospheric conditions are active for 123 h, followed by
68
6.1.1. Unidirectional laminates Since all unidirectional laminates have the same hygrothermal behaviour, we have considered only the laminates of type ½0rT , where r is a repetition index, equal to 36. 6.1.2. Cross-ply laminates The selected symmetric stacking sequences are: ð½0=90=0=90=0=90=0=90=0rÞs, noted under condensed shape ½0=90rs and ½þh; hps; h ¼ 20°; 30°; 45°. With: r ¼ 2; p ¼ 1; 4; 6; 9; 18 and 36.
Fig. 3. Ambient relative humidity before and after the flight versus the reduced time: tr ¼ t=ð6 hÞ, t=ð6 daysÞ, t=ð6 weeksÞ.
6.1.3. Quasi-isotropic laminates The selected stacking sequences are: ½60= 60=0= 60= 0=60=0=60= 60r and ð½0= þ 45=90= 45pÞs, noted, respectively, under condensed shape ½þ60= 60=0r and ½0=45=90ps with r ¼ 1;2;4;8 and 16; p ¼ 1;2;3;6;9 and 18. 6.2. Random stacking sequences laminated plates Various 18-ply symmetric laminates were selected at random with a uniform distribution of the angles. A sample of 20 random sequences was studied. It was found that the in-plane degree of anisotropy varies from 5.99% to 42.22%. The extreme cases are shown below:
Fig. 4. Constant relative humidity before and after the flight versus the reduced time: tr ¼ t=ð6 hÞ, t=ð6 daysÞ, t=ð6 weeksÞ.
½35=8=65=140=118=128=75=126=86=132=74=15=7=2= 75=141=41=27=S ;
eA ¼ 5:99%;
½23=26=170=13=142=1=21=167=51=33=112=71=6=18= 81=31=148=28=S ;
eA ¼ 42:22%:
7. Results The results below are given under the following assumptions. The difference DT between the stress-free
Fig. 5. Ambient temperature before and after the flight versus the reduced time: tr ¼ t=ð6 hÞ, t=ð6 daysÞ, t=ð6 weeksÞ.
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
a flight for 11 h, and normal atmospheric conditions once more for 10 h. In Fig. 3, the relative humidity of the environment was assumed to be reduced as the temperature increases, except during flight, when the relative humidity decreases to zero. In Fig. 4, the relative humidity of the environment was assumed to be constant (82%), except during flight when the relative humidity decreases to zero. In order to facilitate the interpretation of results we have considered two moist middles. Humid environment of type A or B. These two middles are defined as follows: Environment of the type A: In this middle the plate has been submitted of these two faces in cyclic environmental conditions illustrated in Figs. 3 and 5. Environment of the type B: In this middle the plate has been submitted of these two faces to the cyclic environmental conditions represented in Figs. 4 and 5. We present some examples with different cyclic environmental conditions (only on the relative humidity).
397
The transient hygroscopic stresses are computed at the end of each cycle. 7.1. Influence of the environmental loading In order to illustrate the influence of cyclic environmental conditions on the evolution of the transient hygroscopic stresses, we have first considered the unidirectional laminates, then we have analysed the effect of these boundary conditions and the engendered hygroscopic loading on the studied parameters in the case of laminates having different in-plane degrees of anisotropy ðeA Þ. 7.1.1. Case of unidirectional laminates We have done the calculations of the residual stresses rrx and rry engendered by each humid environment (type A or B) for a 6-day period (Figs. 6–9). Some important points emerge from these results. Both stresses are
Fig. 6. Residual stresses rrx through the thickness of the unidirectional laminate with the period of 6 days (middle A).
Fig. 7. Residual stresses rrx through the thickness of the unidirectional laminate with the period of 6 days (middle B).
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A. Tounsi et al. / Composite Structures 55 (2002) 393–405
Fig. 8. Residual stresses rry through the thickness of the unidirectional laminate with the period of 6 days (middle A).
Fig. 9. Residual stresses rry through the thickness of the unidirectional laminate with the period of 6 days (middle B).
compressive ones for the external plies and tensile ones for the internal plies during the absorption process. The order of magnitude of rrx is negligible compared to both longitudinal tensile and compressive strengths X and X 0 given in Table 3. On the other hand, the transversal residual stresses rry reach important values compared to both transverse tensile and compressive strengths Y and Y 0 . In fact, the maximum value of the transverse stress rry reaches about 45.27% of the tensile transverse strength Y in the case of the humid environment (B) and 24.44% in the case of humid environment (A). We notice that the compressive value is maximum at the beginning of the absorption. It has been observed that the compressive stress rry at the end of the first cycle is about 30.47% of the compressive strength Y 0 in the case of the humid environment (A) and about 30% in the case of the humid environment (B).
Secondly, we have calculated the stresses rry assuming that the two faces of the plate are submitted to the environmental conditions of type B and for periods of 6 h and 6 weeks (Figs. 10 and 11). The maximum value of the transverse stress rry reaches about 46.14% of the tensile strength Y for a period of 6 h and 45% for a period of 6 weeks. The maximum of the compressive stress is obtained at the beginning of the absorption. In fact, the maximum value reaches about 30.75% of the compressive strength Y 0 for a period of 6 h and 28.50% for a period of 6 weeks. Due to the symmetry of the present problem, the shear residual stresses rrS are always zero in the ply frame. This can also be explained by the fact that the degrees of anisotropy (in-plane and flexural degree of anisotropy) of unidirectional laminates of the type ½hrT stay constant whatever the value of the
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
399
Fig. 10. Residual stresses rry through the thickness of the unidirectional laminate with the period of 6 h (middle B).
Fig. 11. Residual stresses rry through the thickness of the unidirectional laminate with the period of six weeks (middle B).
angle h (fibres orientation) and whatever the number of plies. Also, we note that the values of the other stresses rrx and rry do not vary with the fibres orientation and the number of plies in the case of the unidirectional laminates. At last, and through these results, we notice that the duration of the period does not disturb too much the distribution of the transient residual stresses within the laminates, but its lightly influences the time of the appearance of maximum stresses. On the other hand, the environmental conditions, which the plate is submitted to, can clearly modify the values of these stresses in internal plies (tensile plies). For the external plies (compressive plies), we notice that the stresses keep almost the same values in the two middles (A and B). This is explained by the fact that the boundary conditions at the end of each cycle are identical in all cases studied (HR ¼ 82% and h ¼ 27 °C) and consequently are going to engender constant free expansions. The smaller dif-
ferences that appear are due only to the non-mechanical strains. We notice also the appearance of the edge effects which represent the fluctuating part of the transient residual stresses in the skin of the plate, particularly in the case of the periods of 6 weeks and 6 days. 7.1.2. Case of laminates with different in-plane degrees of anisotropy To assess the influence of boundary conditions and hygroscopic loading on the analysed parameters inside the laminates with in-plane degrees of anisotropy ðeA Þ, we have considered the two cases of symmetric boundary conditions (middles B and A) with the period of 6 days (the other periods are not tested because we have observed in the case of unidirectional laminates that the period influences very slightly the chosen quantities). In these tests, we have assumed that DT ¼ 0 is only analysis the hygroscopic stresses rrig . The considered laminates in this study are: ½0r (73%); ½20; 204S (59.9%); ½30; 304S (47.1%); ½0=45=90S (0%).
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The obtained results (Tables 4–7) show well that the parameters analysed have the same tendencies in the two cases (middles A and B). The transversal stresses rryg keep the same variations at the end of different chosen cycles. For the other stresses rrxg and rrSg we have noticed that they keep the same tendencies except for the first cycles. Consequently, we have kept for the following, the boundary conditions defined in the middle of B and the hygroscopic loading engendered at 117 cycles. 7.2. Influence of anisotropy on the residual stresses Some configurations of materials were analysed aiming to study the influence of anisotropy on the hygrothermal behaviour of laminated composite plates. First, we have analysed the behaviour of hygroscopic stresses and thermal stresses inside the plate. The hyg-
roscopic stresses rrig are calculated assuming that the temperature difference DT is zero. On the other hand, the thermal stresses rrit are calculated assuming a null distribution of moisture concentration within the plate. In the Figs. 12 and 13, we present the evolution of the stresses rryg and rryt inside the considered laminates. A very interesting point which has been observed in this study is that for laminates with a lower degree of anisotropy, the transverse transient stress rryg decreases in the internal plies (tensile plies) and increases in the external plies (compressive plies) in absolute value (Fig. 12). The state of the transverse hygroscopic stress is oriented towards a compressive state in the laminates with lower degree of anisotropy and towards a tensile state in the laminates with higher degree of anisotropy. In the case of the thermal stress rryt (Fig. 13) the behaviour of laminates is reversed; the state of stresses leads towards a tensile state for laminates with lower
Table 4 Stresses (MPa) computed in the mid-plane of laminates (middle B) Degree of anisotropy (%)
1 Cycle rrxg
rryg
rrSg
21 Cycles rrxg
rryg
rrSg
117 Cycles rrxg
rryg
rrSg
3500 Cycles rrxg
rryg
rrSg
73 59.9 47.1 0
0.67 2.78 3.75 2.87
2.40 1.58 0.81 0.21
0 )1.04 )0.95 )0.23
3.02 8.53 12.28 15.03
10.80 7.28 3.84 0.84
0 )4.30 )4.07 )0.84
5.07 14.94 23.26 34.47
18.10 9.97 1.94 )5.30
0 )9.78 )9.38 )0.79
0.27 20.06 39.34 55.64
0.97 )18.32 )37.50 )54.39
0 )23 )22.23 )0.09
Table 5 Stresses (MPa) computed in the external faces of laminates (middle B) Degree of anisotropy (%)
1 Cycle rrxg
rryg
rrSg
21 Cycles rrxg
rryg
rrSg
117 Cycles rrxg
rryg
rrSg
3500 Cycles rrxg
rryg
rrSg
73 59.9 47.1 0
)20.7 )21.28 )20.73 )21.39
)73.95 )74.66 )75.41 )76.03
0 0.78 0.81 0
)18.30 )17 )13.90 )19.
)65.5 )68.9 )72.3 )75
0 3.90 3.85 )0.17
)14.29 )8.05 )0.31 )1.03
)51.03 )59.01 )67.02 )73.77
0 9.42 9.19 )0.50
)4.43 14.30 33.4 49.8
)15.80 )35.05 )54.23 )71.12
0 22.90 22.17 0
Table 6 Stresses (MPa) computed in the mid-plane of laminates (middle A) Degree of anisotropy (%)
1 Cycle rrxg
rryg
rrSg
21 Cycles rrxg
rryg
rrSg
117 Cycles rrxg
rryg
rrSg
3500 Cycles rrxg
rryg
rrSg
73 59.9 47.1 0
0.39 1.62 2.19 1.67
1.4 0.92 0.47 0.12
0 )0.60 )0.55 )0.13
1.61 4.59 6.6 7.98
5.76 3.88 2.05 0.45
0 )2.29 )2.17 )0.45
2.74 7.88 12.14 17.86
9.77 5.63 1.53 )2.16
0 )5 )4.78 )0.44
0.72 12.88 23.8 31.59
2.57 )8.22 )18.92 )28.28
0 )12.93 )12.45 )0.24
Table 7 Stresses (MPa) computed in the external faces of laminates (middle A) Degree of anisotropy (%)
1 Cycle rrxg
rryg
rrSg
21 Cycles rrxg
rryg
rrSg
117 Cycles rrxg
rryg
rrSg
3500 Cycles rrxg
rryg
rrSg
73 59.9 47.1 0
)20.98 )21.32 )21 )21.38
)74.95 )75.37 )75.80 )76.16
0 0.46 0.47 0
)19.76 )19.08 )17.43 )20.17
)70.59 )72.38 )74.2 )75.62
0 2.07 2.05 )0.09
)17.76 )14.70 )10.77 )11.43
)63.45 )67.51 )71.6 )75.02
0 4.80 4.67 )0.27
)11.93 )2.62 7.84 15.88
)42.61 )53.29 )63.97 )73.34
0 12.66 12.30 )0.002
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Fig. 12. Influence of the degree of anisotropy eA on the transversal stresses rryg within the plate.
Fig. 13. Influence of the degree of anisotropy eA on the transversal stresses rryt within the plate.
Fig. 14. Influence of the degree of anisotropy eA on the longitudinal stresses rrxg within the plate.
degree of anisotropy contrarily to those having a higher degree of anisotropy where we notice that these stresses decrease as the degree of anisotropy takes higher values until the stresses become zero in the case of unidirectional laminates. We also note that the stresses rryg and
rryt calculated inside the laminates ½þ45; 459S and ½0=902S are nearly equal to those of the laminate ½þ60= 60=04 . Concerning the longitudinal transient stresses rrxg (Fig. 14), the laminates with lower degree of anisotropy
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have the tendency of being in tension where we notice that these stresses increase gradually when the degree of anisotropy decreases, which is the opposite of what was noticed in the case of the thermal stresses rrxt Fig. 15 shows that laminates with lower degree of anisotropy
have the tendency of being in compression and where the stresses increase gradually in absolute value when the degree of anisotropy decreases. For the shear stresses (Figs. 16 and 17) we notice that the stresses rrSg and rrSt have different signs in the same
Fig. 15. Influence of the degree of anisotropy eA on the longitudinal stresses rrxt within the plate.
Fig. 16. Influence of the degree of anisotropy eA on the shear stresses rrSg within the plate.
Fig. 17. Influence of the degree of anisotropy eA on the shear stresses rrSt within the plate.
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
ply (one is positive, the other negative in the same ply and vice versa). In addition, in the quasi-isotropic laminate such as the laminate ½þ60= 60=04 and laminates with lower degree of anisotropy such as ½þ45; 459S and ½0=902S , the shear stresses tend towards zero. The same remark is made for the case of higher anisotropic laminates such as unidirectional laminates. The maximum of the resulting stress rrS is obtained for the case of the laminate ½þ20; 20rs which represents a degree of anisotropy of 59.9%. Once away from this degree of anisotropy, these stresses begin to decrease and tend towards zero for laminates with high degree of anisotropy (like the case of the unidirectional laminates), laminates with lower degree of anisotropy and quasiisotropic laminates. We also notice that these stresses are constant in each ply (at any time). This property is observed for all the laminates studied except for the case of the unsymmetric laminate ½þ60= 60=0r . This is due to the fact that the residual strains are defined by the difference between non-mechanical strains and the free expansions (Eq. (12)). The non-mechanical strains are constant within each ply for symmetrical boundary conditions and for uncoupled laminates. Because of the no symmetry of laminate ½þ60= 60=0r , the coupling matrix ½B will not be zero. This makes that the nonmechanical strains are not constant within each ply of this laminate. Secondly, we have studied the influence of the repetition index on the evolution of hygroscopic stresses rrig in the external faces and the mid-plane of laminates ½20; 20rs ; ½þ60= 60=0r and ½0=45=90rs (Figs. 18–23). All the obtained results have shown that the influence of the repetition index on the transversal hygroscopic stresses rryg , whether they were on external faces or in the mid-plane, is not considerable.In fact, we notice that the stresses rryg stay almost constant for any value of the repetition index ðrÞ. We have undertaken the same study for the longitudinal stresses rrxg where we can see that the influence of the repetition index is observed only in the first values after we have remarked that these stresses are not sen-
403
Fig. 19. Influence of the repetition of stacking sequences on the stresses computed in the external faces of laminate ½þ20; 20rs .
Fig. 20. Influence of the repetition of stacking sequences on the stresses computed in the mid-plane of laminate ½þ60= 60=0r .
Fig. 21. Influence of the repetition of stacking sequences on the stresses computed in the external faces of laminate ½þ60= 60=0r .
Fig. 22. Influence of the repetition of stacking sequences on the stresses computed in the mid-plane of laminate ½0=45=90rs .
Fig. 18. Influence of the repetition of stacking sequences on the stresses computed in the mid-plane of laminate ½þ20; 20rs .
sitive to the repetition of stacking sequences. In fact, their stabilisation ðrrxg Þ is observed in the studied laminates ð½þ20; 20rs ; ½þ60= 60=0r and ½0=45=90rs Þ as
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Fig. 23. Influence of the repetition of stacking sequences on the stresses computed in the external faces of laminate ½0=45=90rs .
soon as the repetition index takes the values 6, 4 and 9, respectively. For the shear stresses rrSg , we notice that these stresses stabilise for the first values of the repetition index. At last, few laminates with random sequences are analysed. Figs. 24–27 illustrate the variation of the residual stresses rrig and rrit in the external faces and the mid-plane of laminates versus the in-plane degree of anisotropy. The obtained results confirm almost all those obtained in the previous case for laminates with controlled stacking sequences except for a few values of longitudinal stresses rrxg and rrxt which have not respected the evolution observed for the case of controlled sequences. The explanation that we can give to this is
Fig. 26. Influence of the degree of anisotropy on the hygroscopic stresses rrig computed in the external faces of the laminates (when DT ¼ 0).
Fig. 27. Influence of the degree of anisotropy on the thermal stresses rrit computed in the external faces of the laminates (when C ¼ 0).
Fig. 24. Influence of the degree of anisotropy on the hygroscopic stresses rrig computed in the mid-plane of the laminates (when DT ¼ 0).
tied to the repetition index which is in this cases equal to 1 for all laminates analysed with random stacking sequences; this means that the calculated stresses are not yet stabilised and consequently at this stage, these stresses are not only dependent on the in-plane degree of anisotropy. The transversal stresses rryg and rryt vary lightly with the degree of anisotropy. This is due to the fact that all the laminates analysed have a lower degree of anisotropy. For the same reason, we have noticed that the shear stresses rrSg and rrSt are almost zero.
8. Conclusions
Fig. 25. Influence of the degree of anisotropy on the thermal stresses rrit computed in the mid-plane of the laminates (when C ¼ 0).
This work represents an original attempt to analyses the influence of anisotropy on the hygrothermal behaviour of laminated composite plates. The non-mechanical residual stresses across the plate were studied as a function of the degree of anisotropy for different controlled and random stacking sequences. Such a study has permitted us to explain the partial compensation of the thermal and hygroscopic residual stresses. Indeed,
A. Tounsi et al. / Composite Structures 55 (2002) 393–405
this is due to the different variations of the two stresses with the degree of anisotropy of laminates: if one decreases with the degree of anisotropy, the other increases, or in the case of shear stresses, if one is positive, the other is negative and vice versa. Also, this study has permitted us to predict the state of the two residual stresses (thermal and hygroscopic stresses) inside the laminated plates and therefore, to find more resistant stacking sequences to a moist environment. References [1] Verchery G. Designing with anisotropy. In: Hamelin P, Verchery G, editors. Textile composite in building construction, Part 3 Mechanical behaviour, design and application. Paris: Pluralis; 1990. p. 29–42. [2] Kandil N, Verchery G. New methods of design for stacking sequence of laminates. In: Brebbia CA, de Wilde WP, Blain WR, editors. Computer aided design in composite material technology. Berlin: Computational Mechanics Publications/Springer; 1988. p. 243–57. [3] Kandil N, Verchery G. Some new developments in the design of stacking sequences of laminates. In: Yunshu W, Zhenlong G, Renjie W, editors. Proceeding of the seventh International Conference on Composite Materials, vol. 3. Oxford: International Academic Publishers/Pergamon Press; 1989. p. 358–63. [4] Verchery G, Vong TS. Une methode d’aide graphique a la conception des sequences d’empilement dans les stratifies. In: Bathias C, Menkes D, editors. Proceedings of the 5th French Conference on Composite Materials. Paris: Pluralis; 1986. p. 267–80. [5] Cheikh Saad Bouh AB, Aivazzadeh S, El Sheikh MS, Verchery G. A quantitative study of the influence of anisotropy on the bending
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