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Influence of orientation errors on quasi-homogeneity of composite laminates
Composites Science and Technology 63 (2003) 739–749 www.elsevier.com/locate/compscitech
Influence of orientation errors on quasi-homogeneity of composite laminates A. Vincenti, P. Vannucci*, G. Verchery L.R.M.A., Laboratoire de Recherche en Me´canique et Acoustique, I.S.A.T., Institut Supe´rieur de l’Automobile et des Transports, Universite´ de Bourgogne, 49, rue Mademoiselle Bourgeois, BP 31, 58027 Nevers Cedex, France Received 28 December 2001; received in revised form 21 October 2002; accepted 28 October 2002
Abstract This paper presents a study on the effects of layer orientation defects on the property of quasi-homogeneity for composite laminates: a measure of the deviation from quasi-homogeneity, introducing the concept of degree of quasi-homogeneity, is proposed. Complete theoretical developments which lead to exact formulae in the case of a single orientation error on a layer of the laminate are showed and the results of a wide numerical analysis in the case of orientation errors randomly distributed on the stacking sequence are also presented. All the theoretical and numerical calculations are developed thanks to the polar method of representation of fourth order tensors introduced by Verchery. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: B. Defects; B. Thermomechanical properties; C. Laminates; Quasi-homogeneity
1. Introduction The general equations of the Classical Laminated Plate Theory (CLPT), describing the thermomechanical behaviour of a composite laminate, are [1]: N ¼ A"0 þ B 0 U
V; h
M ¼ B"0 þ D 0 V
W; h
A composite laminate is said to be quasi-homogeneous when it is uncoupled and it has the same behaviour in extension and in bending. Using the symbols of the CLPT, the property of quasi-homogeneity of a composite laminated plate can be expressed: B ¼ V ¼ O;
ð1Þ
where N and M are the tensors of in-plane forces and bending moments, "0 the tensor of in-plane strains in the middle plane, the tensor of curvatures, 0 the difference of temperature of the middle plane with respect to a nonstrain condition, the difference of temperature between the upper and lower face and h the thickness of the plate. A and D are the tensors describing the in- and out-ofplane stiffness behaviours of the plate, while B represents the coupling between these two behaviours. U, V and W have the same meaning as A, B and D, with respect to the forces and moments produced by thermal strains. * Corresponding author. Fax: +33-3-8671-5001. E-mail address: [email protected] (P. Vannucci).
C¼
A 12 D ¼ O; h h3
Z¼
U 12 W ¼ O: h h3
ð2Þ
Vannucci and Verchery [2,3] have shown the existence of a wide number of stacking sequences which give quasi-homogeneous laminates, called quasi-trivial solutions. These solutions have the form of a fixed sequence of different orientations, whose values are free; they are valid in the general case of laminates composed by identical plies. Moreover, other solutions concerning some particular cases are known: for instance, the whole set of quasi-homogeneous sequences for six-ply laminates with layers reinforced by balanced fabrics [4]. Only the exact matching of stacking sequences to the theoretical solutions assures the desired property for the
0266-3538/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00263-4
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laminate. Nevertheless, in the practice some defects may affect the production of a composite laminate, so that the real laminate has characteristics other than the designed one. This paper deals with orientation defects, which are common laminate imperfections, and investigates how they affect the property of quasi-homogeneity of a laminate. The most general case of laminates composed by identical plies is considered, and a measure of the deviation from quasi-homogeneity proposed by the introduction of the concept of degree of quasi-homogeneity. An exact formula for this quantity in the case of a single orientation error on a layer in the laminate is provided. Moreover, we develop a wide numerical analysis in the case of randomly distributed orientation errors in the stacking sequence; in this case, the influence of characteristic parameters (kind of material, number of plies, etc.) on the deviation from the designed property is studied, and an empirical function is proposed to describe the dependence of the degree of quasihomogeneity from these parameters. The theoretical study and the numerical investigation make use of the polar representation method of fourth order tensors proposed by Verchery [7,8]. A fore work already exists on the effect of orientation errors on uncoupling of composite laminates [5,6]. The present study completes this research considering the case of quasi-homogeneity and proposing a very extensive numerical investigation.
2. Recall of the polar method According to the polar method, the Cartesian components of a second order tensor L can be written as functions of three parameters T, R and F: L11 ¼ T þ Rcos2F; ð3Þ
L12 ¼ Rsin2F; L22 ¼ T Rcos2F:
fourth order tensor ) L ¼
If L is a fourth order tensor, the polar representation method expresses its Cartesian components as functions of 6 quantities T0 , T1 , R0 , R1 , F0 andF1 , called polar parameters: L1111 ¼ T0 L1112 ¼
þ 2T1 þ R0 cos4F0 þ 4R1 cos2F1 ; R0 sin4F0 þ 2R1 sin2F1 ;
L1122 ¼ T0 þ 2T1 R0 cos4F0 ; L1212 ¼ T0
R0 cos4F0 ;
L2212 ¼
R0 sin4F0 þ 2R1 sin2F1 ;
L2222 ¼ T0
In Eqs. (3) and (4), T, T0 and T1 are scalars, R, R0 and R1 are moduli and F, F0 and F1 are polar angles. T, T0 , T1 ,R, R0 , R1 and the difference F0 F1 are invariant under a rotation of the reference system through an angle . That is to say, in the new reference system we can still use Eqs. (3) and (4) to represent the Cartesian components of a second or fourth order tensor, respectively. It is sufficient, to this purpose, to change F in F in Eq. (3), and F0 , F1 in F0 , F1 in Eq. (4). The polar method is rather similar to the so-called technique of the lamination parameters, [9–11], but it is more effective, as it makes use of tensor invariants up to the third order. Instead, the lamination parameter technique is based upon the so-called invariants of Tsai and Pagano, only a part of which are first order invariants, the other being frame dependent quantities. The polar method is the most effective way to express material symmetries and to determine the position of symmetry axes, which are linked to the polar angles F0 and F1. Indeed, it is not a merely algebraic transformation of Cartesian tensor components: its origin must be found in a classical technique of mathematical physics, the use of complex variables, introduced firstly by Michell [12], and successively developed by Kolosov [13] and mainly by Muskhelishvili [14] and Green and Zerna [15]. The transformation proposed by Verchery is a modification of the Green and Zerna one which improves the effectiveness of the method, mainly in the case of anisotropic elasticity. For a complete and detailed explanation of the polar method, containing also a critical comparison with the lamination parameters technique, the reader is referred to [16]. In this study the quantity L, a quadratic combination of the polar parameters, has been used: it is a tensor invariant for L and has the same properties as the norm of a tensor [17]. We will refer to it as the norm of the tensor L: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi second order tensor ) L ¼ T 2 þ R 2 ;
þ 2T1 þ R0 cos4F0 þ 4R1 cos2F1 : ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T02 þ 2T12 þ R02 þ 4R12 : ð5Þ
3. Polar representation for quasi-homogeneity Tensors A, B and D, as well as U, V and W, can be represented by their polar components as shown in Eqs. (4) and (3). Starting from the composition laws for a laminated plate, they are expressed as functions of the polar components of the elementary layer. We still use T0 , T1 , etc. as symbols of the polar components for A, B, D, and T, R, F for U, V, W, and distinguish the inplane, coupling and bending components with superscripts -, ˆ and ˜ , respectively. Symbols T0 , T1 , etc. with
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no superscript represent the polar components of the elementary ply. For laminates composed by identical layers, it is (see [7,8]): p X
1 T 0 ; T^ 0 ; T~ 0 ¼ T0 zm zm k1 ; m k¼p k
bk ¼ 2k
p X 1 m e4iðF0 þk Þ zm R 0 e4iF 0 ; R^ 0 e4iF^ 0 ; R~ 0 e4iF~ 0 ¼ R0 k zk1 ; m k¼p p X 1 m e2iðF1 þk Þ zm R 1 e2iF 1 ; R^ 1 e2iF^ 1 ; R~ 1 e2iF~ 1 ¼ R1 k zk1 ; m k¼p
1 X m zk zm T ; T^ ; T~ ¼ T k1 ; m k¼p p
p
ð6Þ
where m=1, 2, 3 for extension, coupling and bending components respectively; n ¼ 2p or n ¼ 2p þ 1 is the number of layers of the laminate and zk represents the distance of the upper face of the k-th layer from the middle plane. Moreover, starting from Eq. (2) the polar components for tensor C can be expressed as functions of those of A and D, as well as for Z, using the polar components of U and W. For laminates composed by identical plies, it is: B and C ) T^ 0 ¼ T^ 1 ¼ 0; T0;C ¼ T1;C ¼ 0; V and Z ) T^ ¼ 0; TZ ¼ 0;
bk ¼ 2k; ck ¼ 4 p2 þ p 3k2 ;
p X m 1 zk zm T 1 ; T^ 1 ; T~ 1 ¼ T1 k1 ; m k¼p
1 X 2iðFþk Þ m e zk zm R e2iF ; R^ e2iF^ ; R~ e2iF~ ¼ R k1 ; m k¼p
where hL ¼ h=n and coefficients bk and ck are:
ð7Þ
if n ¼ 2p þ 1; ð10Þ
k ; b0 ¼ 0; jkj
ck ¼ 4 p2 3k2 þ 3jkj 1 ; c0 ¼ 0;
if n ¼ 2p: ð11Þ
As is apparent from Eqs. (10) and (11), coefficients bk have a linear dependence on the layer index k. They have negative values for k < 0 and positive ones for k > 0: in fact, they have an odd variation with k, that is to say bk ¼ bk . Eq. (11) shows that coefficients ck have a quadratic dependence on the index k. They have negative values for p 4 k < k and k < k 4 p and non-negative values for k 4 k 4 k (the value of k depends on the number of layers n): their variation is symmetric with k, that is to say ck ¼ ck . Moreover, a property of coefficients ck is: k X
" ck ¼
ðX k þ1Þ
k¼k
k¼p
ck þ
p X
# ck :
ð12Þ
k¼k þ1
From Eqs. (10)–(12) we deduce the expression for the k P quantity ck : k¼k
and B ) R^ 0 e
4iF ^0
p X 1 ¼ R0 e4iF0 h2L bk e4ik ; 2 k¼p
k X
p X 1 R^ 1 e2iF^ 1 ¼ R1 e2iF1 h2L bk e2ik ; 2 k¼p
k¼k
1 R0 e4iF0 ck e4ik ; n3 k¼p
R1;C e2iF1;C ¼
p X 1 2iF1 R e ck e2ik ; 1 n3 k¼p
p X 1 2iF Re ck e2ik ; n3 k¼p
for n ¼ 2p:
ð8Þ
ð13Þ
ð9Þ
The quantity of Eq. (13) is used further in this paper, see Eqs. (24)–(30); it is worth noting that it can be expressed simply as a function of the number of layers n composing the laminate, without the need to know k*. Eq. (2) shows the necessary and sufficient conditions of quasi-homogeneity: all the polar components of B and C and those of V and Z must be equal to zero. It is equivalent to say that the norms B, C, V and Z of these tensors must be equal to zero. In terms of polar components, when Eq. (8) is injected into Eq. (5) B and C, as well as V and Z, become:
p X 1 V ) R^ e2iF^ ¼ Re2iF h2L bk e2ik ; 2 k¼p
Z ) R0;Z e2iF0;Z ¼
> p > P > > : n2 12k2 þ 12k 4 k¼1
p X
C ) R0;C e4iF0;C ¼
ck ¼
8 p P > 2 n2 12k2 1 for n ¼ 2p þ 1; > 1 þ n > > < k¼1
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vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p P u 2 u R0 þ 4R21 b2k u k¼p 2 u h B¼ Lu " #; 2 u p p u R20 cos4 k j t þ2 P P b b k j þ4R21 cos2 k j k¼p j¼kþ1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p P u 2 u R0 þ 4R12 c2k u k¼p u " C¼u #; u 2 p p u t þ2 P P ck cj R0 cos4 k j þ4R12 cos2 k j k¼p j¼kþ1
ð14Þ
8 > > > > ¼ 0 ) uncoupling between in-and > > > > > > out-of-plane behaviours; > > <
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX p p p X X 1 2u V ¼ RhL t b2k þ 2 bk bj cos2 k j ; 2 k¼p k¼p j¼kþ1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX p p X X u p 2 Z ¼ Rt ck þ 2 ck cj cos2 k j : k¼p
ð15Þ
k¼p j¼kþ1
From Eq. (14) it can be noticed that norms B and C only depend on the number and the orientations of layers in the laminate, and on the polar components R0 and R1 of the elementary ply, while the other polar components T0 , T1 , F0 and F1 do not appear in these formulas. It is the same for norms V and Z, which depend on the stacking sequence and on the thermal polar component R only.
4. Degree of quasi-homogeneity for a laminate The condition for elastic quasi-homogeneity is to have B and C equal to zero, that is equivalent to have their norms B and C equal to zero. On the contrary, when B and C are non-zero, the deviation from quasi-homogeneity can be assessed by the measure of the pair (B, C). In fact, B and C being norms, when these quantities are compared for two different laminates, it is possible to say which one is more uncoupled and which one has more similar behaviour in bending and in extension. Nevertheless, B and C are not homogeneous quantities: their values cannot be directly compared to say that the deviation from quasi-homogeneity for a laminate depends more on its uncoupling than on the difference between the in-plane and bending behaviours, or conversely. For this reason the pair (, ) is used in place of (B, C), where and are the ratios (see [5,6]): ¼
B C ; ¼ : Bmax Cmax
Bmax and Cmax are the maximum values of the norms B and C for a given number of plies and for a given material of the elementary ply. Quantities and are non-dimensional, so that pairs of (, ) can be compared, even for laminates with different numbers of layers and composed by different materials. and are also homogeneous and we can compare their values for a single laminate to establish the prevailing influence of uncoupling or of the difference between in- and out-of-plane behaviours on the deviation from quasi-homogeneity. Moreover, and are normalized quantities and their values belong to the range [0, 1]:
ð16Þ
> > > > ¼ 1 ) max: coupling between in-and > > > > > > out-of-plane behaviours; > > : ð17Þ 8 ¼ 0 ) same in-and out-of-plane behaviours; > > > < ¼ 1 ) max: difference between in-and > > > : out-of-plane behaviours: Hence, laminates can be classified on a scale of quasihomogeneity with the variation of and , that can be considered to be a degree of coupling and of extensionbending difference respectively. If the pair (, ) is considered as the representation of a vector in a plane, we can use the norm and the orientation tan of this vector as a measure of the deviation from quasi-homogeneity: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ 2 þ 2 ; tan ¼ : ð18Þ All the considerations on B and C are also valid for V and Z. Hence we introduce also the quantities and : ¼
V ; Vmax
¼
Z ; Zmax
ð19Þ
where Vmax and Zmax are the maximum values of the norms V and Z for a given number of plies and for a given material of the elementary ply. and have the same properties as and . So, the vector ( , ) can be used to measure the deviation from thermal quasi-homogeneity, that is to say its norm and its orientation tg!: ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2 ;
tg! ¼ :
ð20Þ
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5. Stacking sequences giving Bmax , Cmax and Vmax , Zmax It follows from Eq. (14) that the norms B and C are the sum of two terms, both depending on the number of layers and on the material of elementary plies. Nevertheless, one term does not depend on the orientation of the layers, while the second one changes with the stacking sequence: p p X X
B ) R12
k¼p j¼kþ1
C ) R12
Cmax
k 4 1 þ 2 X ¼ 3 R1 ck ; n 2 k¼k
if > 1:
ð25Þ
In the same way, from Eq. (15) we see that the stacking sequences giving Vmax and Zmax are those which maximize the sums:
bk bj ; k j ;
p p X X
1 þ 2 X 1 1 þ 2 2 n nmod2 ; bk ¼ R1 h2L 2 2 k¼1 2 p
Bmax ¼ 2R1 h2L
V )
ck cj ; k j ;
ð21Þ
p X p X
bk bj cos2 k j ;
k¼pj¼kþ1
k¼p j¼kþ1
Z ) ; k j ¼ 2 cos4 k j þ 4cos2 k j :
ð22Þ
Here we look for the stacking sequences corresponding to Bmax and Cmax , for a given number of plies and a given elementary material: they are the stacking sequences which maximize sums in Eq. (16). Taking into account the variation of coefficients bk and ck with the layer index k, see Eqs. (10) and (11), and that of the function ; k j , it can be easily shown that: 1. the condition to get Bmax [5] is that all the layers on each side of the middle plane have the same orientation. The difference between these two orientations must be equal to =2 if the elementary material has 4 1, but if > 1, the difference must be equal to : 1 ¼ arccos 1=2 ð23Þ 2 2. the condition for having Cmax is that all the layers with k 4 k 4 k have the same orientation (we recall that ck 0 for k 4 k 4 k ). All the other layers must have a different orientation. As in the preceding case, the difference between the two orientations must be equal to =2 if the elementary material has 4 1and to if > 1. When Eq. (14) is written for these stacking sequences, the expressions of Bmax and Cmax are obtained: p X 1 bk ¼ R1 h2L n2 nmod2 ; 2 k¼1
ð24Þ Cmax
k X 4 ¼ 3 R1 ck ; n k¼k
ck cj cos2 k j :
ð26Þ
k¼pj¼kþ1
where ¼ R0 =R1 and:
Bmax ¼ 2R1 h2L
p X p X
if 4 1
From the variation of bk and ck , see Eqs. (10) and (11), it can be easily seen that: 1. the condition for having Vmax is that all the layers on one side of the middle plane have the same orientation. The difference between these two orientations must be equal to =2 (see [5]); 2. the condition to get Zmax is that all the layers with k 4 k 4 k have the same orientation. All the other layers must have a different orientation. As in the preceding case, the difference between the two orientations must be equal to =2. It is worth noting that the stacking sequence for Vmax and the one for Zmax do not depend upon the elementary material composing the laminate. When Eq. (15) is written for these stacking sequences, we obtain the expressions of Vmax and Zmax : Vmax ¼
Rh2L
p k X 2 X bk ; Zmax ¼ 3 R ck : n k¼k k¼1
In Eqs. (24), (25) and (27) the quantity one of the expressions given in Eq. (13).
ð27Þ k P
ck takes
k¼k
6. Effects of orientation errors on quasi-homogeneity: theoretical study In this section, we want to evaluate the effects of orientation defects upon quasi-homogeneity in the case of an ideally quasi-homogeneous laminate where a single orientation error "m affects the m-th ply; its real orientation angle becomes: "m ¼ m þ "m , where m is the theoretical direction.
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The deviation of the m-th ply from its ideal position causes the laminate to be coupled and to have a difference between the in- and out-of-plane behaviours. So, the norms of B, C, V and Z are not zero as for the ideal laminate. Starting from Eq. (14) with some simple algebraic transformations, the expressions for B and C due to an error "m are obtained: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 B" ¼ pffiffiffi R1 h2L jbm j 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ; 2 C" ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2R1 jcm j 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ:
ð28Þ
When Eqs. (24), (25) and (28) are injected in Eq. (16), the expressions for and become: 8 > > > > > > l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > pffiffiffi 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ if 4 1; > > 2 > > > < ¼ l 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > > > pffiffiffi 1 þ 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ > 2 > > > > > if > 1; > > > > > : 8 ð29Þ > > > > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > p ffiffiffi 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ if 4 1; > > > 2 > > > < ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > pffiffiffi 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ > > 2 1 þ > 2 > > > > > if > 1; > > > > :
¼
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l 1 cos2"m ;
¼
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 cos2"m :
ð32Þ
From Eqs. (28) and (31) it is easy to see that, for a given error "m , the maximum deviation from quasihomogeneity is when bk and ck are maximum, that is to say when k ¼ p (subscripts p correspond to the outer plies of the laminate). Hence, for a laminate composed by n layers of a given material, the effect of an orientation error is more important when it affects an external ply, that is to say when m ¼ p; in this case l¼2
n1 ; n2 nmod2
8 ðn 2Þðn 1Þ > > > p > P > > n2 1 þ n2 12k2 1 > > <
ð33Þ if n ¼ 2p þ 1;
k¼1
¼
> > ðn 2Þðn 1Þ > > > p > P > > : n2 12k2 þ 12k 4
if n ¼ 2p:
k¼1
Fig. 1 shows the function ð"; Þ=l, which, as apparent form Eqs. (29) and (30), is equal to ð"; Þ=. In Fig. 2 the function ð"Þ=l is given, which is again equal
where pffiffiffi 2j bm j 1 j bm j l¼ p ; ¼ 2 2P ðn nmod2Þ bk k¼1
¼
1 jcm j : k 2 P ck
ð30Þ
k¼k
Fig. 1. The functions
ð"; Þ ð"; Þ and . l
Fig. 2. The functions
ð"; Þ
ð"; Þ and pffiffiffi . l 2
In the same way V and Z for the laminate affected by the error "m can be found from Eq. (15), 1 h2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V" ¼ pffiffiffi R 2 bm 1 cos2"m ; 2 n Z" ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2R1 jcm j 1 cos2"m ;
ð31Þ
finally, from Eqs. (19) and (27) the expressions for and can be obtained:
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to ð"Þ=l. They illustrate the influence of the error angle " and of the material ratio on the deviation from quasihomogeneity. It can be noticed from Fig. 1 that for a given value of ", laminates composed by materials with lower are less sensitive to an orientation defect; on the contrary, if > 1 and in particular for tending towards infinity, the sensitivity to an orientation error increases: it is the case of plies reinforced by balanced fabrics. Nevertheless, the maximum value for the functions ð"; Þ=l and ð"; Þ= does not depend upon the variation of : 8 < " ¼ ; if 4 1; ð"; Þ ð"; Þ 2 ¼ ¼ 2 for : l max max " ¼ ; if > 1: ð34Þ It can be noticed in Fig. 2 that the deviation from thermal quasi-homogeneity depends only upon the error " and not on the elementary material. The function reaches its maximum value always for " ¼ =2: ð "Þ
ð"Þ ¼ ¼ 2: l max max
ð35Þ
It is important to remark once more that , , and depend only on the ratio , as the absolute quantities B, Bmax , C, Cmax , etc. depend only on the material parameters R0 and R1 .
7. Effects of orientation errors on quasi-homogeneity: a wide numerical investigation In the theoretical case of a single orientation error on a layer of a laminate exact formulas for , , and have been found. But, in reality, we can suppose that an orientation error affects each ply of a laminate and that these errors are randomly distributed among all the layers. In this case, it does not seem possible to find an exact expression of the degree of quasi-homogeneity. So, we have made a wide numerical investigation in order to study the variation of these quantities with orientation defects and to assess the influence of different parameters: number of plies of the laminate, material of the elementary layer, orientations of the layers, magnitude of the errors. Each calculation has been performed considering a vector E of angular defects: its generic component "k represents the error on the orientation of the k-th ply of the laminate. All the components of E are considered to be statistically independent. The error "k is assumed to be normally distributed around the theoretical angle for each ply. The characteristic angle is used in place of the standard deviation " to describe the distribution:
each "k belongs to range ½ ; with a probability of 95%. Hence it is " ¼ 1:96 . As E is a random error vector, each value of , , and has been computed as a mean on a population of np tests: to have a good stability of results, np has been fixed to 10 000. We have made tests on quasi-homogeneous sequences belonging to the set of quasi-trivial solutions found by Vannucci and Verchery [2,3]. To have the most general results, both non-symmetrical and symmetrical sequences have been used. We considered the case of laminates with two theoretical orientation angles, 0 and ; the number of plies for each angle is generally not the same. The influence on this phenomenon of the parameters variation is studied; these parameters are the orientation angle , the characteristic error angle , the number of layers n and the ratio for the elementary layer. In addition, a six-ply quasi-homogeneous laminates composed by balanced fabrics [4] has also been considered, as it is not of the quasi-trivial type. In this case the only free parameters are the characteristic angle and the stacking sequence, that is to say the layer orientation angles of the laminate, while the other parameters are fixed: n=6, ¼ 1. 7.1. Influence of the characteristic angle c We studied the influence of the characteristic angle in the case of various quasi-homogeneous stacking sequences, chosen in the set of quasi-trivial solutions, with different number n of plies and composed by elementary materials with different characteristic ratios . Tests with various values of the orientation angle have been made too. We found that the dependence of , and is linear with , while the slope of curves decreases with n. It is the same for , and . The variation with of , ,
and tan is shown in Figs. 3 and 4, for the cases of 8and 20-ply laminates with ¼ 30 and ¼ 0:92. For the same case, in Figs. 5 and 6 we show the variation of , , and tan!.
Fig. 3. Variation of and with
(=30 , =0.92).
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7.2. Influence of the orientation angle a We studied the influence of the orientation angle in the case of laminates with different number n of layers and for elementary materials with different . We fixed ¼ 5 , while the variation of is between 0 and 90 . The result is that , , and tan , as well as , , and tan!, are completely independent of . For this reason, we made all successive tests with a fixed value for the orientation angle and we chose a non-standard orientation, ¼ 30 . The results for , , and tan in the case of laminates with n=8 and n=20 are shown in Figs. 7 and 8, and for ¼ 0:92. For the same cases, in Figs. 9 and 10 we show the variation of , , and tan!.
Fig. 4. Variation of and tan with
Fig. 5. Variation of and with
Fig. 6. Variation of and tan! with
(=30 , =0.92).
(=30 , =0.92).
(=30 , =0.92).
Fig. 7. Variation of and with ( =5 ; =0.92).
Fig. 8. Variation of and tan with ( =5 ; =0.92).
Fig. 9. Variation of and with ( =5 ; =0.92).
Fig. 10. Variation of and tan! with ( =5 ; =0.92).
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7.3. Influence of the number of layers n The influence of the number of layers n in the case of quasi-homogeneous laminates composed by materials with different has been considered. We fixed in this case ¼ 5 and ¼ 30 . Figs. 11 and 12 show the results for , , and tan in the case of laminates with ¼ 0:92. For the same cases, Figs. 13 and 14 present the results for , , and tan!. We found that the dependence of , , and are described by two different curves for even or odd n; it is the same for , , and . It can be noticed that on logarithmic axes , and , as well as , and , are linear with n, for both even and odd n.
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7.4. Influence of the characteristic ratio r of the elementary material The influence of the material characteristic ratio ¼ R0 =R1 for laminates with different number of layers, n has been studied too; in this case, we fixed ¼ 5 and ¼ 30 . Figs. 15 and 16 show the results for , , and tan . It can be noticed that on logarithmic axes the curves representing , and have a step-like variation, the upper part of the curve being about twice the lower. Hence, the influence of orientation errors on the
Fig. 11. Variation of ln and ln with ln n ( =5 ; =0.92, =30 ).
Fig. 14. Variation of ln and ln tan ! with ln n ( =5 ; =0.92, =30 ).
Fig. 12. Variation of ln and ln tan with ln n ( =5 ; =0.92, =30 ).
Fig. 15. Variation of ln and ln with ln ( =5 , =30 ).
Fig. 13. Variation of ln and ln with ln n ( =5 ; =0.92, =30 ).
Fig. 16. Variation of ln and ln with ln ( =5 , =30 ).
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Fig. 17. Variation of ln and ln with ln ( =5 , =30 ).
Fig. 18. Variation of ln with ln and ln n ( =5 , even n).
deviation from quasi-homogeneity is much more important for laminates composed by materials with > 1. It is maximum when is toward infinity, that is to say in the case of plies reinforced by balanced fabrics. On the contrary, we found that the variation of has no influence on , and (see Fig. 17) and this confirms the result of Eq. (32) in the case of only one angular defect. 7.5. Tests on six-ply quasi-homogeneous laminates composed by balanced fabrics layers For six-ply laminates composed by balanced fabrics layers the whole set of quasi-homogeneous stacking sequences is known. These solutions are given in the form of a function of the orientation of a layer in the laminate and they are widely described in [4]. Hence, for this set of laminates the parameters that can influence the deviation from quasi-homogeneity are only the orientation angle and the characteristic angle . We found that , and are completely independent on the orientation angle , according to the results shown in the preceding paragraphs for stacking sequences belonging to the set of quasi-trivial solutions. So we can presume that the independence of the results from the theoretical orientations of the layers is a quite general result.
Fig. 19. Surface of f(n,) for ( =5 , even n).
Table 1 Coefficients ai of the empirical function lnf(n,) for
Even n Odd n
1
2
3
4
5
0.38 0.34
0.06 0.07
0.49 0.50
0.85 0.78
0.10 0.11
Curves in Figs. 11 and 12 show that , and have a linear dependence from . Hence, we can represent the variation of these three quantities with all the three parameters , , n by the empirical function:
7.6. Overall description of results gð ; ; nÞ ¼ Since the deviation from quasi-homogeneity does not depend upon the theoretical orientation angles of the stacking sequence, we suggest to represent the variation of , and with and n by the function fð; nÞ for a given value of : a2 ln f ð; nÞ ¼ a1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a5 lnn; ð þ a3 Þ2 þa4
ð36Þ
f ð; nÞ:
ð37Þ
The coefficients ai of Eq. (35) have been determined numerically, in the cases of , and , and they are shown for in Table 1, in both the cases of even and odd n. Figs. 18 and 19 represent the variation of
and of the empirical function with n and , in the case of even n and ¼ 5 ; the correspondence of the empirical function with the actual distribution is very accurate.
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8. Conclusions
References
This paper is a study about the influence of orientation errors on composite laminates designed to be quasi-homogeneous. In a first phase, the concept of degrees of quasihomogeneity, for both thermal and elastic behaviours, is introduced as a function of the norms of tensors B, C and V, Z, respectively. They measure the deviation from quasi-homogeneity for a laminate and their values can be compared in order to class several laminates on a scale of quasi-homogeneity. Further, we express these quantities by exact formulas in the case of a quasi-homogeneous laminate with one orientation error on a single ply. The formulas show that thermal quasi-homogeneity is only sensitive to the error angle , while the elastic quasi-homogeneity is also sensitive to the material ratio of the elementary layer. In particular, laminates composed by materials with higher result more influenced by orientation errors. Moreover, the paper shows the results of a wide numerical study in the case of orientation errors randomly distributed among all the layers of a laminate. We notice that the theoretical stacking sequence is not a relevant parameter for the deviation from quasi-homogeneity. On the contrary, other parameters affect the deviation from quasi-homogeneity. Actually, there is a linear dependence on the amplitude of orientation errors, described by the characteristic angle . The dependence on the number of layers n is linear in a logarithmic scale. Ratio does not affect the deviation from thermal quasi-homogeneity, according with the theoretical formulas written for a single error, but, in a logarithmic reference system, there is a step-like variation of the degree of elastic quasi-homogeneity with : materials with higher values of , such as plies reinforced by balanced fabrics, are more sensitive to orientation errors. Finally, we have proposed to describe these results by an empirical function, which expresses the variation of the degree of quasi-homogeneity with the parameters , , n.
[1] Jones RM. Mechanics of composite materials. USA: Taylor & Francis; 1975. [2] Vannucci P, Verchery G. A special class of uncoupled and quasihomogeneous laminates. Comp Sci Technol 2001;61(10):1465–73. [3] Vannucci P, Verchery G. Stiffness design of laminates using the polar method. Int J Solids Struct 2001;38(50-51):9281–94. [4] Vincenti A, Vannucci P, Verchery G. Anisotropy and symmetry for elastic properties of laminates reinforced by balanced fabrics. Composites Part A: Applied Science and Manufacturing 2001; 32(10):1525–32. [5] Vannucci P. On bending-tension coupling of laminates. J Elasticity 2001;64:13–28. [6] Belaı¨d F, Vannucci P, Verchery G. Numerical investigation of the influence of orientation defects on bending-tension coupling of laminates. In: Proceedings of ICCM 13, Beijing, China, 2001 [paper no. 1406]. [7] Verchery G. Les invariants des tenseurs d’ordre quatre du type de l’e´lasticite´. In: Proceedings of Euromech Colloquium 115, Villardde Lans, 1979. Paris: CNRS Editions 1982. p. 93–104 (in French). [8] Verchery G. Designing with anisotropy. Part 1: methods and general results for laminates. In: Proceedings of ICCM 12, Paris, France, 1999 [paper no. 734]. [9] Tsai SW, Pagano NJ. Invariant properties of composite materials. Composite Materials Workshop. Technomic; 1968. p. 233–53. [10] Miki M. Material design of composite laminates with required inplane elastic properties. In: Proceedings of ICCM 4, vol. 2, Tokyo, Japan, 1982. p. 1725–31. [11] Fukunaga H, Vanderplaats GN. Stiffness optimisation of orthotropic laminated composites using lamination parameters. AIAA Journal 1991;29(4):641–6. [12] Michell JH. The inversion of plane stress. Proc London Math Soc 1902;34:134–42. [13] Kolosov GV. On an application of complex function theory to a plane problem of the mathematical theory of elasticity. Russia: Yuriev; 1909. [14] Muskhelishvili NJ. Some basic problems of the mathematical theory of elasticity. 1933. Gro¨ningen (NL): P. Noordhoff; 1933. [Engl. trans. JRM Radoch, 1953]. [15] Green A, Zerna W. Theoretical elasticity. Oxford, UK: Clarendon Press; 1954. [16] Vannucci P. The`se pour l’Habilitation a` Diriger des Recherches. HDR dissertation, Universite´ de Bourgogne, France, 2002 [in French]. [17] Kandil N, Verchery G. New methods of design for stacking sequences of laminates. Brebbia CA, De Wilde WP, Blain WR, editors. Computational Mechanics Publications and Springer Verlag; 1988, p. 243–57.
Influence of orientation errors on quasi-homogeneity of composite laminates A. Vincenti, P. Vannucci*, G. Verchery L.R.M.A., Laboratoire de Recherche en Me´canique et Acoustique, I.S.A.T., Institut Supe´rieur de l’Automobile et des Transports, Universite´ de Bourgogne, 49, rue Mademoiselle Bourgeois, BP 31, 58027 Nevers Cedex, France Received 28 December 2001; received in revised form 21 October 2002; accepted 28 October 2002
Abstract This paper presents a study on the effects of layer orientation defects on the property of quasi-homogeneity for composite laminates: a measure of the deviation from quasi-homogeneity, introducing the concept of degree of quasi-homogeneity, is proposed. Complete theoretical developments which lead to exact formulae in the case of a single orientation error on a layer of the laminate are showed and the results of a wide numerical analysis in the case of orientation errors randomly distributed on the stacking sequence are also presented. All the theoretical and numerical calculations are developed thanks to the polar method of representation of fourth order tensors introduced by Verchery. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: B. Defects; B. Thermomechanical properties; C. Laminates; Quasi-homogeneity
1. Introduction The general equations of the Classical Laminated Plate Theory (CLPT), describing the thermomechanical behaviour of a composite laminate, are [1]: N ¼ A"0 þ B 0 U
V; h
M ¼ B"0 þ D 0 V
W; h
A composite laminate is said to be quasi-homogeneous when it is uncoupled and it has the same behaviour in extension and in bending. Using the symbols of the CLPT, the property of quasi-homogeneity of a composite laminated plate can be expressed: B ¼ V ¼ O;
ð1Þ
where N and M are the tensors of in-plane forces and bending moments, "0 the tensor of in-plane strains in the middle plane, the tensor of curvatures, 0 the difference of temperature of the middle plane with respect to a nonstrain condition, the difference of temperature between the upper and lower face and h the thickness of the plate. A and D are the tensors describing the in- and out-ofplane stiffness behaviours of the plate, while B represents the coupling between these two behaviours. U, V and W have the same meaning as A, B and D, with respect to the forces and moments produced by thermal strains. * Corresponding author. Fax: +33-3-8671-5001. E-mail address: [email protected] (P. Vannucci).
C¼
A 12 D ¼ O; h h3
Z¼
U 12 W ¼ O: h h3
ð2Þ
Vannucci and Verchery [2,3] have shown the existence of a wide number of stacking sequences which give quasi-homogeneous laminates, called quasi-trivial solutions. These solutions have the form of a fixed sequence of different orientations, whose values are free; they are valid in the general case of laminates composed by identical plies. Moreover, other solutions concerning some particular cases are known: for instance, the whole set of quasi-homogeneous sequences for six-ply laminates with layers reinforced by balanced fabrics [4]. Only the exact matching of stacking sequences to the theoretical solutions assures the desired property for the
0266-3538/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00263-4
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laminate. Nevertheless, in the practice some defects may affect the production of a composite laminate, so that the real laminate has characteristics other than the designed one. This paper deals with orientation defects, which are common laminate imperfections, and investigates how they affect the property of quasi-homogeneity of a laminate. The most general case of laminates composed by identical plies is considered, and a measure of the deviation from quasi-homogeneity proposed by the introduction of the concept of degree of quasi-homogeneity. An exact formula for this quantity in the case of a single orientation error on a layer in the laminate is provided. Moreover, we develop a wide numerical analysis in the case of randomly distributed orientation errors in the stacking sequence; in this case, the influence of characteristic parameters (kind of material, number of plies, etc.) on the deviation from the designed property is studied, and an empirical function is proposed to describe the dependence of the degree of quasihomogeneity from these parameters. The theoretical study and the numerical investigation make use of the polar representation method of fourth order tensors proposed by Verchery [7,8]. A fore work already exists on the effect of orientation errors on uncoupling of composite laminates [5,6]. The present study completes this research considering the case of quasi-homogeneity and proposing a very extensive numerical investigation.
2. Recall of the polar method According to the polar method, the Cartesian components of a second order tensor L can be written as functions of three parameters T, R and F: L11 ¼ T þ Rcos2F; ð3Þ
L12 ¼ Rsin2F; L22 ¼ T Rcos2F:
fourth order tensor ) L ¼
If L is a fourth order tensor, the polar representation method expresses its Cartesian components as functions of 6 quantities T0 , T1 , R0 , R1 , F0 andF1 , called polar parameters: L1111 ¼ T0 L1112 ¼
þ 2T1 þ R0 cos4F0 þ 4R1 cos2F1 ; R0 sin4F0 þ 2R1 sin2F1 ;
L1122 ¼ T0 þ 2T1 R0 cos4F0 ; L1212 ¼ T0
R0 cos4F0 ;
L2212 ¼
R0 sin4F0 þ 2R1 sin2F1 ;
L2222 ¼ T0
In Eqs. (3) and (4), T, T0 and T1 are scalars, R, R0 and R1 are moduli and F, F0 and F1 are polar angles. T, T0 , T1 ,R, R0 , R1 and the difference F0 F1 are invariant under a rotation of the reference system through an angle . That is to say, in the new reference system we can still use Eqs. (3) and (4) to represent the Cartesian components of a second or fourth order tensor, respectively. It is sufficient, to this purpose, to change F in F in Eq. (3), and F0 , F1 in F0 , F1 in Eq. (4). The polar method is rather similar to the so-called technique of the lamination parameters, [9–11], but it is more effective, as it makes use of tensor invariants up to the third order. Instead, the lamination parameter technique is based upon the so-called invariants of Tsai and Pagano, only a part of which are first order invariants, the other being frame dependent quantities. The polar method is the most effective way to express material symmetries and to determine the position of symmetry axes, which are linked to the polar angles F0 and F1. Indeed, it is not a merely algebraic transformation of Cartesian tensor components: its origin must be found in a classical technique of mathematical physics, the use of complex variables, introduced firstly by Michell [12], and successively developed by Kolosov [13] and mainly by Muskhelishvili [14] and Green and Zerna [15]. The transformation proposed by Verchery is a modification of the Green and Zerna one which improves the effectiveness of the method, mainly in the case of anisotropic elasticity. For a complete and detailed explanation of the polar method, containing also a critical comparison with the lamination parameters technique, the reader is referred to [16]. In this study the quantity L, a quadratic combination of the polar parameters, has been used: it is a tensor invariant for L and has the same properties as the norm of a tensor [17]. We will refer to it as the norm of the tensor L: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi second order tensor ) L ¼ T 2 þ R 2 ;
þ 2T1 þ R0 cos4F0 þ 4R1 cos2F1 : ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T02 þ 2T12 þ R02 þ 4R12 : ð5Þ
3. Polar representation for quasi-homogeneity Tensors A, B and D, as well as U, V and W, can be represented by their polar components as shown in Eqs. (4) and (3). Starting from the composition laws for a laminated plate, they are expressed as functions of the polar components of the elementary layer. We still use T0 , T1 , etc. as symbols of the polar components for A, B, D, and T, R, F for U, V, W, and distinguish the inplane, coupling and bending components with superscripts -, ˆ and ˜ , respectively. Symbols T0 , T1 , etc. with
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no superscript represent the polar components of the elementary ply. For laminates composed by identical layers, it is (see [7,8]): p X
1 T 0 ; T^ 0 ; T~ 0 ¼ T0 zm zm k1 ; m k¼p k
bk ¼ 2k
p X 1 m e4iðF0 þk Þ zm R 0 e4iF 0 ; R^ 0 e4iF^ 0 ; R~ 0 e4iF~ 0 ¼ R0 k zk1 ; m k¼p p X 1 m e2iðF1 þk Þ zm R 1 e2iF 1 ; R^ 1 e2iF^ 1 ; R~ 1 e2iF~ 1 ¼ R1 k zk1 ; m k¼p
1 X m zk zm T ; T^ ; T~ ¼ T k1 ; m k¼p p
p
ð6Þ
where m=1, 2, 3 for extension, coupling and bending components respectively; n ¼ 2p or n ¼ 2p þ 1 is the number of layers of the laminate and zk represents the distance of the upper face of the k-th layer from the middle plane. Moreover, starting from Eq. (2) the polar components for tensor C can be expressed as functions of those of A and D, as well as for Z, using the polar components of U and W. For laminates composed by identical plies, it is: B and C ) T^ 0 ¼ T^ 1 ¼ 0; T0;C ¼ T1;C ¼ 0; V and Z ) T^ ¼ 0; TZ ¼ 0;
bk ¼ 2k; ck ¼ 4 p2 þ p 3k2 ;
p X m 1 zk zm T 1 ; T^ 1 ; T~ 1 ¼ T1 k1 ; m k¼p
1 X 2iðFþk Þ m e zk zm R e2iF ; R^ e2iF^ ; R~ e2iF~ ¼ R k1 ; m k¼p
where hL ¼ h=n and coefficients bk and ck are:
ð7Þ
if n ¼ 2p þ 1; ð10Þ
k ; b0 ¼ 0; jkj
ck ¼ 4 p2 3k2 þ 3jkj 1 ; c0 ¼ 0;
if n ¼ 2p: ð11Þ
As is apparent from Eqs. (10) and (11), coefficients bk have a linear dependence on the layer index k. They have negative values for k < 0 and positive ones for k > 0: in fact, they have an odd variation with k, that is to say bk ¼ bk . Eq. (11) shows that coefficients ck have a quadratic dependence on the index k. They have negative values for p 4 k < k and k < k 4 p and non-negative values for k 4 k 4 k (the value of k depends on the number of layers n): their variation is symmetric with k, that is to say ck ¼ ck . Moreover, a property of coefficients ck is: k X
" ck ¼
ðX k þ1Þ
k¼k
k¼p
ck þ
p X
# ck :
ð12Þ
k¼k þ1
From Eqs. (10)–(12) we deduce the expression for the k P quantity ck : k¼k
and B ) R^ 0 e
4iF ^0
p X 1 ¼ R0 e4iF0 h2L bk e4ik ; 2 k¼p
k X
p X 1 R^ 1 e2iF^ 1 ¼ R1 e2iF1 h2L bk e2ik ; 2 k¼p
k¼k
1 R0 e4iF0 ck e4ik ; n3 k¼p
R1;C e2iF1;C ¼
p X 1 2iF1 R e ck e2ik ; 1 n3 k¼p
p X 1 2iF Re ck e2ik ; n3 k¼p
for n ¼ 2p:
ð8Þ
ð13Þ
ð9Þ
The quantity of Eq. (13) is used further in this paper, see Eqs. (24)–(30); it is worth noting that it can be expressed simply as a function of the number of layers n composing the laminate, without the need to know k*. Eq. (2) shows the necessary and sufficient conditions of quasi-homogeneity: all the polar components of B and C and those of V and Z must be equal to zero. It is equivalent to say that the norms B, C, V and Z of these tensors must be equal to zero. In terms of polar components, when Eq. (8) is injected into Eq. (5) B and C, as well as V and Z, become:
p X 1 V ) R^ e2iF^ ¼ Re2iF h2L bk e2ik ; 2 k¼p
Z ) R0;Z e2iF0;Z ¼
> p > P > > : n2 12k2 þ 12k 4 k¼1
p X
C ) R0;C e4iF0;C ¼
ck ¼
8 p P > 2 n2 12k2 1 for n ¼ 2p þ 1; > 1 þ n > > < k¼1
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vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p P u 2 u R0 þ 4R21 b2k u k¼p 2 u h B¼ Lu " #; 2 u p p u R20 cos4 k j t þ2 P P b b k j þ4R21 cos2 k j k¼p j¼kþ1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p P u 2 u R0 þ 4R12 c2k u k¼p u " C¼u #; u 2 p p u t þ2 P P ck cj R0 cos4 k j þ4R12 cos2 k j k¼p j¼kþ1
ð14Þ
8 > > > > ¼ 0 ) uncoupling between in-and > > > > > > out-of-plane behaviours; > > <
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX p p p X X 1 2u V ¼ RhL t b2k þ 2 bk bj cos2 k j ; 2 k¼p k¼p j¼kþ1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX p p X X u p 2 Z ¼ Rt ck þ 2 ck cj cos2 k j : k¼p
ð15Þ
k¼p j¼kþ1
From Eq. (14) it can be noticed that norms B and C only depend on the number and the orientations of layers in the laminate, and on the polar components R0 and R1 of the elementary ply, while the other polar components T0 , T1 , F0 and F1 do not appear in these formulas. It is the same for norms V and Z, which depend on the stacking sequence and on the thermal polar component R only.
4. Degree of quasi-homogeneity for a laminate The condition for elastic quasi-homogeneity is to have B and C equal to zero, that is equivalent to have their norms B and C equal to zero. On the contrary, when B and C are non-zero, the deviation from quasi-homogeneity can be assessed by the measure of the pair (B, C). In fact, B and C being norms, when these quantities are compared for two different laminates, it is possible to say which one is more uncoupled and which one has more similar behaviour in bending and in extension. Nevertheless, B and C are not homogeneous quantities: their values cannot be directly compared to say that the deviation from quasi-homogeneity for a laminate depends more on its uncoupling than on the difference between the in-plane and bending behaviours, or conversely. For this reason the pair (, ) is used in place of (B, C), where and are the ratios (see [5,6]): ¼
B C ; ¼ : Bmax Cmax
Bmax and Cmax are the maximum values of the norms B and C for a given number of plies and for a given material of the elementary ply. Quantities and are non-dimensional, so that pairs of (, ) can be compared, even for laminates with different numbers of layers and composed by different materials. and are also homogeneous and we can compare their values for a single laminate to establish the prevailing influence of uncoupling or of the difference between in- and out-of-plane behaviours on the deviation from quasi-homogeneity. Moreover, and are normalized quantities and their values belong to the range [0, 1]:
ð16Þ
> > > > ¼ 1 ) max: coupling between in-and > > > > > > out-of-plane behaviours; > > : ð17Þ 8 ¼ 0 ) same in-and out-of-plane behaviours; > > > < ¼ 1 ) max: difference between in-and > > > : out-of-plane behaviours: Hence, laminates can be classified on a scale of quasihomogeneity with the variation of and , that can be considered to be a degree of coupling and of extensionbending difference respectively. If the pair (, ) is considered as the representation of a vector in a plane, we can use the norm and the orientation tan of this vector as a measure of the deviation from quasi-homogeneity: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ 2 þ 2 ; tan ¼ : ð18Þ All the considerations on B and C are also valid for V and Z. Hence we introduce also the quantities and : ¼
V ; Vmax
¼
Z ; Zmax
ð19Þ
where Vmax and Zmax are the maximum values of the norms V and Z for a given number of plies and for a given material of the elementary ply. and have the same properties as and . So, the vector ( , ) can be used to measure the deviation from thermal quasi-homogeneity, that is to say its norm and its orientation tg!: ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2 ;
tg! ¼ :
ð20Þ
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5. Stacking sequences giving Bmax , Cmax and Vmax , Zmax It follows from Eq. (14) that the norms B and C are the sum of two terms, both depending on the number of layers and on the material of elementary plies. Nevertheless, one term does not depend on the orientation of the layers, while the second one changes with the stacking sequence: p p X X
B ) R12
k¼p j¼kþ1
C ) R12
Cmax
k 4 1 þ 2 X ¼ 3 R1 ck ; n 2 k¼k
if > 1:
ð25Þ
In the same way, from Eq. (15) we see that the stacking sequences giving Vmax and Zmax are those which maximize the sums:
bk bj ; k j ;
p p X X
1 þ 2 X 1 1 þ 2 2 n nmod2 ; bk ¼ R1 h2L 2 2 k¼1 2 p
Bmax ¼ 2R1 h2L
V )
ck cj ; k j ;
ð21Þ
p X p X
bk bj cos2 k j ;
k¼pj¼kþ1
k¼p j¼kþ1
Z ) ; k j ¼ 2 cos4 k j þ 4cos2 k j :
ð22Þ
Here we look for the stacking sequences corresponding to Bmax and Cmax , for a given number of plies and a given elementary material: they are the stacking sequences which maximize sums in Eq. (16). Taking into account the variation of coefficients bk and ck with the layer index k, see Eqs. (10) and (11), and that of the function ; k j , it can be easily shown that: 1. the condition to get Bmax [5] is that all the layers on each side of the middle plane have the same orientation. The difference between these two orientations must be equal to =2 if the elementary material has 4 1, but if > 1, the difference must be equal to : 1 ¼ arccos 1=2 ð23Þ 2 2. the condition for having Cmax is that all the layers with k 4 k 4 k have the same orientation (we recall that ck 0 for k 4 k 4 k ). All the other layers must have a different orientation. As in the preceding case, the difference between the two orientations must be equal to =2 if the elementary material has 4 1and to if > 1. When Eq. (14) is written for these stacking sequences, the expressions of Bmax and Cmax are obtained: p X 1 bk ¼ R1 h2L n2 nmod2 ; 2 k¼1
ð24Þ Cmax
k X 4 ¼ 3 R1 ck ; n k¼k
ck cj cos2 k j :
ð26Þ
k¼pj¼kþ1
where ¼ R0 =R1 and:
Bmax ¼ 2R1 h2L
p X p X
if 4 1
From the variation of bk and ck , see Eqs. (10) and (11), it can be easily seen that: 1. the condition for having Vmax is that all the layers on one side of the middle plane have the same orientation. The difference between these two orientations must be equal to =2 (see [5]); 2. the condition to get Zmax is that all the layers with k 4 k 4 k have the same orientation. All the other layers must have a different orientation. As in the preceding case, the difference between the two orientations must be equal to =2. It is worth noting that the stacking sequence for Vmax and the one for Zmax do not depend upon the elementary material composing the laminate. When Eq. (15) is written for these stacking sequences, we obtain the expressions of Vmax and Zmax : Vmax ¼
Rh2L
p k X 2 X bk ; Zmax ¼ 3 R ck : n k¼k k¼1
In Eqs. (24), (25) and (27) the quantity one of the expressions given in Eq. (13).
ð27Þ k P
ck takes
k¼k
6. Effects of orientation errors on quasi-homogeneity: theoretical study In this section, we want to evaluate the effects of orientation defects upon quasi-homogeneity in the case of an ideally quasi-homogeneous laminate where a single orientation error "m affects the m-th ply; its real orientation angle becomes: "m ¼ m þ "m , where m is the theoretical direction.
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The deviation of the m-th ply from its ideal position causes the laminate to be coupled and to have a difference between the in- and out-of-plane behaviours. So, the norms of B, C, V and Z are not zero as for the ideal laminate. Starting from Eq. (14) with some simple algebraic transformations, the expressions for B and C due to an error "m are obtained: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 B" ¼ pffiffiffi R1 h2L jbm j 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ; 2 C" ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2R1 jcm j 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ:
ð28Þ
When Eqs. (24), (25) and (28) are injected in Eq. (16), the expressions for and become: 8 > > > > > > l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > pffiffiffi 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ if 4 1; > > 2 > > > < ¼ l 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > > > pffiffiffi 1 þ 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ > 2 > > > > > if > 1; > > > > > : 8 ð29Þ > > > > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > p ffiffiffi 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ if 4 1; > > > 2 > > > < ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > pffiffiffi 2 ð1 cos4"m Þ þ 4ð1 cos2"m Þ > > 2 1 þ > 2 > > > > > if > 1; > > > > :
¼
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l 1 cos2"m ;
¼
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 cos2"m :
ð32Þ
From Eqs. (28) and (31) it is easy to see that, for a given error "m , the maximum deviation from quasihomogeneity is when bk and ck are maximum, that is to say when k ¼ p (subscripts p correspond to the outer plies of the laminate). Hence, for a laminate composed by n layers of a given material, the effect of an orientation error is more important when it affects an external ply, that is to say when m ¼ p; in this case l¼2
n1 ; n2 nmod2
8 ðn 2Þðn 1Þ > > > p > P > > n2 1 þ n2 12k2 1 > > <
ð33Þ if n ¼ 2p þ 1;
k¼1
¼
> > ðn 2Þðn 1Þ > > > p > P > > : n2 12k2 þ 12k 4
if n ¼ 2p:
k¼1
Fig. 1 shows the function ð"; Þ=l, which, as apparent form Eqs. (29) and (30), is equal to ð"; Þ=. In Fig. 2 the function ð"Þ=l is given, which is again equal
where pffiffiffi 2j bm j 1 j bm j l¼ p ; ¼ 2 2P ðn nmod2Þ bk k¼1
¼
1 jcm j : k 2 P ck
ð30Þ
k¼k
Fig. 1. The functions
ð"; Þ ð"; Þ and . l
Fig. 2. The functions
ð"; Þ
ð"; Þ and pffiffiffi . l 2
In the same way V and Z for the laminate affected by the error "m can be found from Eq. (15), 1 h2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V" ¼ pffiffiffi R 2 bm 1 cos2"m ; 2 n Z" ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2R1 jcm j 1 cos2"m ;
ð31Þ
finally, from Eqs. (19) and (27) the expressions for and can be obtained:
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A. Vincenti et al. / Composites Science and Technology 63 (2003) 739–749
to ð"Þ=l. They illustrate the influence of the error angle " and of the material ratio on the deviation from quasihomogeneity. It can be noticed from Fig. 1 that for a given value of ", laminates composed by materials with lower are less sensitive to an orientation defect; on the contrary, if > 1 and in particular for tending towards infinity, the sensitivity to an orientation error increases: it is the case of plies reinforced by balanced fabrics. Nevertheless, the maximum value for the functions ð"; Þ=l and ð"; Þ= does not depend upon the variation of : 8 < " ¼ ; if 4 1; ð"; Þ ð"; Þ 2 ¼ ¼ 2 for : l max max " ¼ ; if > 1: ð34Þ It can be noticed in Fig. 2 that the deviation from thermal quasi-homogeneity depends only upon the error " and not on the elementary material. The function reaches its maximum value always for " ¼ =2: ð "Þ
ð"Þ ¼ ¼ 2: l max max
ð35Þ
It is important to remark once more that , , and depend only on the ratio , as the absolute quantities B, Bmax , C, Cmax , etc. depend only on the material parameters R0 and R1 .
7. Effects of orientation errors on quasi-homogeneity: a wide numerical investigation In the theoretical case of a single orientation error on a layer of a laminate exact formulas for , , and have been found. But, in reality, we can suppose that an orientation error affects each ply of a laminate and that these errors are randomly distributed among all the layers. In this case, it does not seem possible to find an exact expression of the degree of quasi-homogeneity. So, we have made a wide numerical investigation in order to study the variation of these quantities with orientation defects and to assess the influence of different parameters: number of plies of the laminate, material of the elementary layer, orientations of the layers, magnitude of the errors. Each calculation has been performed considering a vector E of angular defects: its generic component "k represents the error on the orientation of the k-th ply of the laminate. All the components of E are considered to be statistically independent. The error "k is assumed to be normally distributed around the theoretical angle for each ply. The characteristic angle is used in place of the standard deviation " to describe the distribution:
each "k belongs to range ½ ; with a probability of 95%. Hence it is " ¼ 1:96 . As E is a random error vector, each value of , , and has been computed as a mean on a population of np tests: to have a good stability of results, np has been fixed to 10 000. We have made tests on quasi-homogeneous sequences belonging to the set of quasi-trivial solutions found by Vannucci and Verchery [2,3]. To have the most general results, both non-symmetrical and symmetrical sequences have been used. We considered the case of laminates with two theoretical orientation angles, 0 and ; the number of plies for each angle is generally not the same. The influence on this phenomenon of the parameters variation is studied; these parameters are the orientation angle , the characteristic error angle , the number of layers n and the ratio for the elementary layer. In addition, a six-ply quasi-homogeneous laminates composed by balanced fabrics [4] has also been considered, as it is not of the quasi-trivial type. In this case the only free parameters are the characteristic angle and the stacking sequence, that is to say the layer orientation angles of the laminate, while the other parameters are fixed: n=6, ¼ 1. 7.1. Influence of the characteristic angle c We studied the influence of the characteristic angle in the case of various quasi-homogeneous stacking sequences, chosen in the set of quasi-trivial solutions, with different number n of plies and composed by elementary materials with different characteristic ratios . Tests with various values of the orientation angle have been made too. We found that the dependence of , and is linear with , while the slope of curves decreases with n. It is the same for , and . The variation with of , ,
and tan is shown in Figs. 3 and 4, for the cases of 8and 20-ply laminates with ¼ 30 and ¼ 0:92. For the same case, in Figs. 5 and 6 we show the variation of , , and tan!.
Fig. 3. Variation of and with
(=30 , =0.92).
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7.2. Influence of the orientation angle a We studied the influence of the orientation angle in the case of laminates with different number n of layers and for elementary materials with different . We fixed ¼ 5 , while the variation of is between 0 and 90 . The result is that , , and tan , as well as , , and tan!, are completely independent of . For this reason, we made all successive tests with a fixed value for the orientation angle and we chose a non-standard orientation, ¼ 30 . The results for , , and tan in the case of laminates with n=8 and n=20 are shown in Figs. 7 and 8, and for ¼ 0:92. For the same cases, in Figs. 9 and 10 we show the variation of , , and tan!.
Fig. 4. Variation of and tan with
Fig. 5. Variation of and with
Fig. 6. Variation of and tan! with
(=30 , =0.92).
(=30 , =0.92).
(=30 , =0.92).
Fig. 7. Variation of and with ( =5 ; =0.92).
Fig. 8. Variation of and tan with ( =5 ; =0.92).
Fig. 9. Variation of and with ( =5 ; =0.92).
Fig. 10. Variation of and tan! with ( =5 ; =0.92).
A. Vincenti et al. / Composites Science and Technology 63 (2003) 739–749
7.3. Influence of the number of layers n The influence of the number of layers n in the case of quasi-homogeneous laminates composed by materials with different has been considered. We fixed in this case ¼ 5 and ¼ 30 . Figs. 11 and 12 show the results for , , and tan in the case of laminates with ¼ 0:92. For the same cases, Figs. 13 and 14 present the results for , , and tan!. We found that the dependence of , , and are described by two different curves for even or odd n; it is the same for , , and . It can be noticed that on logarithmic axes , and , as well as , and , are linear with n, for both even and odd n.
747
7.4. Influence of the characteristic ratio r of the elementary material The influence of the material characteristic ratio ¼ R0 =R1 for laminates with different number of layers, n has been studied too; in this case, we fixed ¼ 5 and ¼ 30 . Figs. 15 and 16 show the results for , , and tan . It can be noticed that on logarithmic axes the curves representing , and have a step-like variation, the upper part of the curve being about twice the lower. Hence, the influence of orientation errors on the
Fig. 11. Variation of ln and ln with ln n ( =5 ; =0.92, =30 ).
Fig. 14. Variation of ln and ln tan ! with ln n ( =5 ; =0.92, =30 ).
Fig. 12. Variation of ln and ln tan with ln n ( =5 ; =0.92, =30 ).
Fig. 15. Variation of ln and ln with ln ( =5 , =30 ).
Fig. 13. Variation of ln and ln with ln n ( =5 ; =0.92, =30 ).
Fig. 16. Variation of ln and ln with ln ( =5 , =30 ).
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Fig. 17. Variation of ln and ln with ln ( =5 , =30 ).
Fig. 18. Variation of ln with ln and ln n ( =5 , even n).
deviation from quasi-homogeneity is much more important for laminates composed by materials with > 1. It is maximum when is toward infinity, that is to say in the case of plies reinforced by balanced fabrics. On the contrary, we found that the variation of has no influence on , and (see Fig. 17) and this confirms the result of Eq. (32) in the case of only one angular defect. 7.5. Tests on six-ply quasi-homogeneous laminates composed by balanced fabrics layers For six-ply laminates composed by balanced fabrics layers the whole set of quasi-homogeneous stacking sequences is known. These solutions are given in the form of a function of the orientation of a layer in the laminate and they are widely described in [4]. Hence, for this set of laminates the parameters that can influence the deviation from quasi-homogeneity are only the orientation angle and the characteristic angle . We found that , and are completely independent on the orientation angle , according to the results shown in the preceding paragraphs for stacking sequences belonging to the set of quasi-trivial solutions. So we can presume that the independence of the results from the theoretical orientations of the layers is a quite general result.
Fig. 19. Surface of f(n,) for ( =5 , even n).
Table 1 Coefficients ai of the empirical function lnf(n,) for
Even n Odd n
1
2
3
4
5
0.38 0.34
0.06 0.07
0.49 0.50
0.85 0.78
0.10 0.11
Curves in Figs. 11 and 12 show that , and have a linear dependence from . Hence, we can represent the variation of these three quantities with all the three parameters , , n by the empirical function:
7.6. Overall description of results gð ; ; nÞ ¼ Since the deviation from quasi-homogeneity does not depend upon the theoretical orientation angles of the stacking sequence, we suggest to represent the variation of , and with and n by the function fð; nÞ for a given value of : a2 ln f ð; nÞ ¼ a1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a5 lnn; ð þ a3 Þ2 þa4
ð36Þ
f ð; nÞ:
ð37Þ
The coefficients ai of Eq. (35) have been determined numerically, in the cases of , and , and they are shown for in Table 1, in both the cases of even and odd n. Figs. 18 and 19 represent the variation of
and of the empirical function with n and , in the case of even n and ¼ 5 ; the correspondence of the empirical function with the actual distribution is very accurate.
A. Vincenti et al. / Composites Science and Technology 63 (2003) 739–749
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8. Conclusions
References
This paper is a study about the influence of orientation errors on composite laminates designed to be quasi-homogeneous. In a first phase, the concept of degrees of quasihomogeneity, for both thermal and elastic behaviours, is introduced as a function of the norms of tensors B, C and V, Z, respectively. They measure the deviation from quasi-homogeneity for a laminate and their values can be compared in order to class several laminates on a scale of quasi-homogeneity. Further, we express these quantities by exact formulas in the case of a quasi-homogeneous laminate with one orientation error on a single ply. The formulas show that thermal quasi-homogeneity is only sensitive to the error angle , while the elastic quasi-homogeneity is also sensitive to the material ratio of the elementary layer. In particular, laminates composed by materials with higher result more influenced by orientation errors. Moreover, the paper shows the results of a wide numerical study in the case of orientation errors randomly distributed among all the layers of a laminate. We notice that the theoretical stacking sequence is not a relevant parameter for the deviation from quasi-homogeneity. On the contrary, other parameters affect the deviation from quasi-homogeneity. Actually, there is a linear dependence on the amplitude of orientation errors, described by the characteristic angle . The dependence on the number of layers n is linear in a logarithmic scale. Ratio does not affect the deviation from thermal quasi-homogeneity, according with the theoretical formulas written for a single error, but, in a logarithmic reference system, there is a step-like variation of the degree of elastic quasi-homogeneity with : materials with higher values of , such as plies reinforced by balanced fabrics, are more sensitive to orientation errors. Finally, we have proposed to describe these results by an empirical function, which expresses the variation of the degree of quasi-homogeneity with the parameters , , n.
[1] Jones RM. Mechanics of composite materials. USA: Taylor & Francis; 1975. [2] Vannucci P, Verchery G. A special class of uncoupled and quasihomogeneous laminates. Comp Sci Technol 2001;61(10):1465–73. [3] Vannucci P, Verchery G. Stiffness design of laminates using the polar method. Int J Solids Struct 2001;38(50-51):9281–94. [4] Vincenti A, Vannucci P, Verchery G. Anisotropy and symmetry for elastic properties of laminates reinforced by balanced fabrics. Composites Part A: Applied Science and Manufacturing 2001; 32(10):1525–32. [5] Vannucci P. On bending-tension coupling of laminates. J Elasticity 2001;64:13–28. [6] Belaı¨d F, Vannucci P, Verchery G. Numerical investigation of the influence of orientation defects on bending-tension coupling of laminates. In: Proceedings of ICCM 13, Beijing, China, 2001 [paper no. 1406]. [7] Verchery G. Les invariants des tenseurs d’ordre quatre du type de l’e´lasticite´. In: Proceedings of Euromech Colloquium 115, Villardde Lans, 1979. Paris: CNRS Editions 1982. p. 93–104 (in French). [8] Verchery G. Designing with anisotropy. Part 1: methods and general results for laminates. In: Proceedings of ICCM 12, Paris, France, 1999 [paper no. 734]. [9] Tsai SW, Pagano NJ. Invariant properties of composite materials. Composite Materials Workshop. Technomic; 1968. p. 233–53. [10] Miki M. Material design of composite laminates with required inplane elastic properties. In: Proceedings of ICCM 4, vol. 2, Tokyo, Japan, 1982. p. 1725–31. [11] Fukunaga H, Vanderplaats GN. Stiffness optimisation of orthotropic laminated composites using lamination parameters. AIAA Journal 1991;29(4):641–6. [12] Michell JH. The inversion of plane stress. Proc London Math Soc 1902;34:134–42. [13] Kolosov GV. On an application of complex function theory to a plane problem of the mathematical theory of elasticity. Russia: Yuriev; 1909. [14] Muskhelishvili NJ. Some basic problems of the mathematical theory of elasticity. 1933. Gro¨ningen (NL): P. Noordhoff; 1933. [Engl. trans. JRM Radoch, 1953]. [15] Green A, Zerna W. Theoretical elasticity. Oxford, UK: Clarendon Press; 1954. [16] Vannucci P. The`se pour l’Habilitation a` Diriger des Recherches. HDR dissertation, Universite´ de Bourgogne, France, 2002 [in French]. [17] Kandil N, Verchery G. New methods of design for stacking sequences of laminates. Brebbia CA, De Wilde WP, Blain WR, editors. Computational Mechanics Publications and Springer Verlag; 1988, p. 243–57.