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Simulation of multiple delaminations in composite laminates under mixed-mode deformations
Simulation Modelling Practice and Theory 11 (2003) 483–500 www.elsevier.com/locate/simpat
Simulation of multiple delaminations in composite laminates under mixed-mode deformations q Paolo Lonetti, Raffaele Zinno
*
Department of Structural Engineering, University of Calabria, 87036 Cosenza, Italy Received 21 May 2002; received in revised form 16 October 2002; accepted 27 October 2002
Abstract A delamination analysis is developed in order to predict the defect evolution when a fiber composite laminate, with outer layers of equal or different thickness, is subjected to a compression load and multiple delaminations. In particular, the presence of two delaminated zones coaxially positioned on the two opposite sides of the laminate is analysed. In order to accurately predict the growth of delamination, when mode I and II are interacting, some fracture criteria have been proposed. Some results will be given to show the influence on the delamination buckling and on the growth phenomenon of the main geometrical and mechanical characteristics of the structure in order to show the pronounced role of the fracture modes on stable or unstable delamination growth. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Delamination buckling; Fiber composite laminate; Damage
1. Introduction Fibre reinforced laminate composites are widely used nowadays in load-bearing structures due to their light weight, high specific strength and stiffness, good corrosion resistance and superb fatigue strength limit. Many successful applications in q This is a revised/extended version of the paper ‘‘Numerical simulation of multiple delaminations in composite laminates under mixed-mode deformations’’ presented at the Eurosim 2001 International Congress, June 26–29, 2001, Delft, The Netherlands. * Corresponding author. Tel.: +39-0984-494020; fax: +39-0984-494045. E-mail address: [email protected] (R. Zinno).
1569-190X/03/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1569-190X(03)00054-6
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Nomenclature N Nc0i Nc l01 , l02 l1 , l2 t T w E B u n1 nlSD N Nc N 2;3 Nci N ð1Þ ð1Þ NLLB k q DU C GI GII GIC GIIC g
total compressive load buckling load of the i-layer in absence of buckling of the remaining layer buckling load upper and lower initial delamination length upper and lower post-buckling delamination length layer thickness total thickness of the laminate central transverse displacement Young modulus transverse length longitudinal displacement non-dimensional central transverse displacement of the ith layer non-dimensional central transverse displacement in the fictitious single delaminated body buckling load of the sublaminate composed by layer 2 and core compressive load of the layer 2 compressive load of the layer 2 and core buckling load of the i-layer post-buckling load of the layer 1 function of Nc2 adsorbed by buckled layer 1 non-dimensional initial layer 2/layer 2 delamination length ratio layer 1/layer 2 thickness ratio total potential energy increments density of the surface adhesion energy energy release rate for mode I energy release rate for mode II critical value of the energy release rate for mode I critical value of the energy release rate for mode II mode I/mode II critical energy release rate ratio
aerospace, mechanical, naval and, now, also in civil engineering rely on the exploitation of these properties as well as others. However, like metals, these materials lose much more of their structural integrity when damaged. In particular, laminated composites often present initial delaminations produced by several causes, such us manufacturing imperfections, stress concentrations, impacts and local or global buckling of laminae. In this paper a computer simulation of a characteristic failure of composite laminates, delamination, is presented. This phenomenon is one of the most dangerous kinds of damage in a laminate structure and, unfortunately, its modelling is very complicated without the help of purpose made computer program. Frequently, com-
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puter codes are developed by using a FEA computational procedures, or, as in this case, they are developed with the aim of analytical closed form solution. These closed form solutions are obtained by utilising the mechanical aspects of the delamination phenomenon and the linear elastic fracture mechanics theory. The occurrence of delaminations, in fact, may allow the laminate free surface to buckle locally (delamination buckling phenomenon) thereby creating conditions conducive to delamination growth. The delamination growth phenomenon depends upon the stress state at the crack tip and it is often governed by the mixed-mode stress intensity factor or by the mixed-mode strain energy release rate. This consideration is the main reason for several developed and published research works where interlaminar fracture toughness experimental tests about mode I and II are carried out or where mixed-mode delamination growth models for composite systems are developed. The present work falls within this context, concerning the behaviour of simple one-dimensional plates subject to compressive loads, in which the delamination is assumed to spread from initial interlaminar defects. In particular, attention is focused on locally buckled laminates containing multiple delaminations as depicted in Fig. 1, where the outer layers can have equal or different thickness. The present paper represents a collection of the analyses developed in [4–7] mainly devoted to similar cases. In order to accurately predict the growth of delamination, when mode I and II are interacting, some fracture criteria have been proposed taking into account experimental evidence. In the present paper, one of this mixed-mode fracture criteria [1,2] is adopted to obtain information on mixed-mode fracture behaviour of a laminated one-dimensional plates. Only in this last analysis the external layers have equal thickness.
B
1
N
N
2
L
1
w1 t T
N
N t 2
w2
L Fig. 1. Locally buckled laminated affected by symmetrically located delaminations.
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Some results show the influence on the delamination buckling and on the growth phenomenon of the main geometrical and mechanical characteristics of the structure in order to evidence the marked role of the fracture modes on the stable or unstable delamination growth.
2. Multiple delamination analysis for a narrow plate with outer layers of equal thickness With reference to Fig. 1, it is developed the analysis of the damage evolution produced by a compressive load on a narrow plate subjected to two coaxial delaminated zones. The thickness of the two external delaminated layers are equal. In particular, the relations governing the mechanical behaviour of the plate, after the second critical state are analysed. The hypothesis that the buckling of the second layer appears before delamination growth initiation in the first buckled layer, (Hyp. A), is supposed. The buckling load of the ith layer, in absence of the buckling of the remaining jth layer is 2 p 2 ti Nc0i ¼ EBT ; ð1Þ 3 l0i where E is the Young modulus of the homogeneous, linearly elastic, isotropic material constituting the layers, T is the total thickness of the laminate, and ti is the thickness of the layer. The first buckled layer, obviously, has length (l01 ) equal or greater than the second one (l02 ). The following non-dimensional parameters, which represent the central transverse displacements (w), are introduced in the computation in the model: w l21 w l22 n1 ¼ ; n2 ¼ : ð2a;bÞ l1 l2 The post-buckling relations which are valid for the first layer only have the following expressions [3]: 1 N ð1Þ ¼ Ncð1Þ þ Ncð1Þ n21 ; 8
ul1 N ð1Þ p2 2 þ n1 : ¼ l1 EBt 4
ð3a;bÞ
Because of the hypothesis that layer 2 buckles before the delamination growth initiation of layer 1, and by considering the compressive load N of the sublaminate composed by layer 2 and core 3, as the sum of the N ð2Þ and N ð3Þ relative to layer 2 and to core 3 respectively (see Fig. 2), it is possible to write: " # 2 N 1 3 l2 t 2 ¼1þ 1 ð4aÞ þ n2 ; 8 4 t T t Nc ul1 N t p2 l2 2 þ ¼ n: l1 EBðT tÞ T t 4 l1 2
ð4bÞ
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1
w1 t T
N
1
N
(1)
*
t
(2)
N =N +N
3
T*
(3)
2
2
w2
1
w1LLB t T
N
(1)
NLLB
1
(3) C2
NLLB
3
(2)
t
NC
2 2
Fig. 2. Layer compressive load distribution.
In relations (4a,b) Nc indicates the value of N capable of buckling layer 2: 2 p2 t Nc ¼ EBðT tÞ: ð5Þ 3 l2 In order to calculate the value of Nc2 which is the buckling load of the second layer, it is worth noting the l1 P l2 , which expresses the fact that the delamination fronts have different positions in the laminates. Moreover, it is possible to write the buckling load of the layer 2 as ð1Þ
Nc2 ¼ NLLB þ Nc ;
ð6Þ
ð1Þ NLLB
where represents the fraction of Nc2 adsorbed by buckled layer 1. Writing the compatibility equations of the axial displacements of the delaminated block, and introducing the Ncð2Þ layer 2 buckling load variable, the following relationship holds: 2 1 þ 6 lt2 ð1Þ ð2Þ ð7Þ NLLB ¼ 2 N c : 1 þ 6 lt1 By substituting Eqs. (5)–(7), it is possible to write: 2 # 2 " Nc2 t l1 T t 1 þ 6 lt2 ¼ Zðl1 ; l2 Þ ¼ þ 2 ; T l2 t Nc1 1 þ 6 l1 t
ð8Þ
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where Nc1 is the buckling load of layer 1, relative to the delamination length l1 , and it is equal to: 2 p2 t Nc1 ¼ EBT : ð9Þ 3 l1 By introducing the following non-dimensional ratios between the delamination lengths and the defects lengths: w1 ¼
l1 ; l01
w2 ¼
l2 ; l02
k¼
l02 ; l01
k 6 1;
we can rewrite Eqs. (8) and (9) as follows: # " 2 2 1 t 1 þ 6k 2 l01t w2 T t w21 ; Zðw1 ; w2 Þ ¼ 2 þ 2 k T t w22 1 þ l01t w21 " # 2 p2 t 1 1 Nc1 ¼ EBT ¼ 2 Nc01 : 2 3 l01 w1 w1
ð10a;b;c;dÞ
ð11a;bÞ
In order to obtain the force–displacement relations the compatibility between the axial displacements of the block ends must be recalled. In this way by equalizing Eqs. (3b) and (4b) it is possible to find the following equations: N ¼
T t ð1Þ p2 p2 l2 N þ EBðT tÞn21 EBt n21 : t 4 4 l1
ð12Þ
The total compressive load is N ¼ N ð1Þ þ N , and by using the expressions in Eqs. (3a) and (12) can be written as " # 2 2 N 1 3 l1 t 2 3 t l2 l1 ¼1þ 1 n22 : ð13Þ þ n1 Nc1 8 4 t T 4 T l1 t This expression substitutes Eq. (4a) when the buckling load appears in the second layer. By introducing: 2 2 1 3 l1 t 3 t l2 l2 W ¼ W ðl1 Þ ¼ þ 1 ; ; H2 ¼ H2 ðl1 ; l2 Þ ¼ 8 4 t T 4 T l1 t ð14a;bÞ it is possible rewrite (13) as N ¼ 1 þ W ðl1 Þn21 H2 ðl1 ; l2 Þn22 : ð15Þ Nc1 Moreover, by defining the ‘‘fictitious single delaminated body (SD)’’, obtained from the original laminate by suppressing the second delamination, we can write Eq. (13) as follows: " # 2 N 1 3 l1 t 2 ¼1þ 1 ð16Þ þ n1SD ¼ 1 þ W ðl1 Þn21SD : Nc1 8 4 t T
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By imposing, as previously, the compatibility conditions, by using Eqs. (3a), (4a) and (5), we reach: n22 ¼ Rðl1 ; l2 Þn21 Sðl1 ; l2 Þ;
ð17Þ
where 1 8
Rðl1 ; l2 Þ ¼ 2
2 t l1
þ 34 t
; t l2 þ 34 1 T t þ 34 T t l1 2 2 t þ lt1 l2 Sðl1 ; l2 Þ ¼ 2 3 t l ; 1 t 3 t þ 1 þ 4 T t l21 8 l2 4 T t 1 8
t l2
or in the other form: N ¼ 1 þ X ðl1 ; l2 Þ þ Y ðl1 ; l2 Þn21 ; Nc1
ð18Þ
with X ¼ X ðl1 ; l2 Þ ¼ H2 ðl1 ; l2 ÞSðl1 ; l2 Þ;
Y ¼ Y ðl1 ; l2 Þ ¼ W ðl1 Þ H2 ðl1 ; l2 ÞRðl1 ; l2 Þ:
It is also possible, by using the previous relation, to achieve the following expression for uL :
ξ1 2
2
W ξ1
precritical path
A
S
Z
X
2
ξ 1 SD
B
NC2 NC1
2
Y ξ1
N NC1
th l pa tica tcri s o tp firs
ond
sec
ath
al p
ritic
tc pos
1 (W-Y) 2 ξ R 2
ξ
2
O
1
2
ξ2 S R 2
ξ 2=R ξ 1-S 2
ξ
2 2
Fig. 3. Pre- and post-critical paths: displacement interpretation.
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N-NC2
N-NC1
2 t π 2 ξ2 t π ξ2 T 4 1 1+ T 4 2 2 t π 2 ξ2 T 4 1 1SD
NC2
second post-critical path
∆Φ2
B
NC1
A
∆Φ
1 first post-critical path
pre-critical path
u
L
Fig. 4. Pre- and post-critical paths: potential energy graphical interpretation.
uL ¼
NL t p2 2 þ l1 n1SD : EBT T 4
ð19Þ
In Figs. 3 and 4 are depicted the graphical interpretation of Eqs. (18) and (19). 3. Narrow plate with outer layers of different thickness affected by multiple delamination In this section the analysis of the damage evolution, due to a compressive load, is extended to the case of a laminated narrow plate with the different thickness outer layers, subjected to two coaxial delaminated zones. The previous hyp. A is again made. Fig. 1 is still valid but the thicknesses are: t1 and t2 , for the upper and bottom layer respectively. The buckling load of the ith layer, in absence of buckling of the remaining jth layer can still obtained by the previous equation (1). The same for Eqs. (2a,b) regarding the non-dimensional parameters representing the central transverse displacements (w). The ratios between the delamination lengths and the defects lengths (10a)–(10c) are also still valid, but the thickness ratio variable must be introduced in the formulation: t2 q¼ : ð20Þ t1 From Eq. (1) and Eqs. (10a)–(10c) and (20), the hypothesis that the ‘‘layer 1’’ lamina is the first to buckle, can be written equivalently as l01 t1 P ; l02 t2
or
k 61 q
ð21a;bÞ
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by the definition of Nc1 and Nc2 as the buckling loads of layers 1 and 2, relative to the delamination lengths l1 and l2 . In particular the value Nc1 , it is easy to obtain by substituting the value l1 to the value l0i in Eq. (1). In order to evaluate Nc2 variable the following straight lines r and s on the plane W1 and W2 are introduced in the formulation: r:
w1 ¼ ðk=qÞw2
s:
w1 ¼ kw2 :
ð22a;bÞ
In particular, the straight line r is relative to delamination lengths producing the delamination buckling phenomenon with an equal value of the external load parameter, i.e.: Nc1 ¼ Nc2 , while on s line the relation l1 ¼ l2 is valid, which expresses equal delamination lengths growth. So, by using these straight lines it is possible to recognise four cases: 1 q case AA: w2 6 w1 ; w2 6 w1 ; k k 1 q case AB: w2 6 w1 ; w2 P w1 ; ð23a;bÞ k k 1 q w1 ; w2 6 w1 ; k k 1 q case BB: w2 P w1 ; w2 P w1 : ð23c;dÞ k k From the geometrical and physical point of view the first relation in the four cases indicates: l1 P l2 (A) or l1 6 l2 (B), while the second relation indicates: first delamination on layer 1 (A) or first delamination on layer 2 (B). Following the same analytical procedure previously developed in the case of outer layers of equal thickness, it is possible to achieve the following equations that connect the mechanical parameters previously introduced: case BA:
w2 P
n22 ¼ Rðl1 ; l2 Þn21 Sðl1 ; l2 Þ;
uL ¼ uL ðN ; l1 ; l2 Þ ¼
N ¼ 1 þ X ðl1 ; l2 Þ þ Y ðl1 ; l2 Þn21 ; Nc1
N t1 p2 2 t2 p2 2 þ l1 n1 þ l2 n2 : EBT T 4 T 4
Nc2 ¼ Zðl1 ; l2 Þ; Nc1 ð24a;b;cÞ ð25Þ
Figs. 3 and 4 represent Eqs. (24a)–(24c) and (25), but now the expression of variables R, S, X , Y and Z depends on the region of plane W1 ; W2 , so depending on one of the four cases previously listed. When the first delaminated layer is upper lamina (‘‘layer 1’’) it is possible to write: G1 ¼
o DU ¼ CB; ol1
G2 ¼
o DU2 ¼ CB; ol2
ð26a;bÞ
while if the first delaminated layer is the lower lamina (‘‘layer 2’’), the expressions become: G1 ¼
o DU1 ¼ CB; ol1
G2 ¼
o DU ¼ CB: ol2
ð27a;bÞ
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Relations (26) and (27) are the particularised expressions of the limit equilibrium conditions in the well known Griffith criterion. The explicit and simple expressions for DU1 , DU2 and the sum DU can be obtained from a direct observation of Fig. 3. Obviously, in the case of Eq. (27) it is necessary change the position of DU1 , DU2 . For the sake of brevity, now we will explain only one of the four cases listed in Eqs. (23a)–(23d), and we will only summarize the main results for the remaining three cases. Case AA The post-buckling relations (3a,b) are valid for the first layer [3]. Eqs. (4a,b) must be modified to take into account the different thickness (see also Fig. 5 for the correct indexes): " 2 # N 1 3 l2 t2 ¼1þ 1 ð28aÞ þ n22 ; 8 4 t2 Nc T t1 ul1 N t2 p2 l2 2 þ ¼ n: l1 EBðT t1 Þ T t1 4 l1 2
ð28bÞ
In relations (28a,b), as in Eqs. (4a,b), Nc indicates the value of N capable of buckling layer 2: 2 p2 t2 Nc ¼ EBðT t1 Þ: ð29Þ 3 l2
1
w1 t1 T
N
1
N
(1)
*
T
t2
2
2
w2
1
w1LLB t1 T
(1)
NLLB
1
(3)
NC2
NLLB
3
(2)
t2
(2)
N =N +N
3
*
NC
2 2
Fig. 5. Layer load distribution.
(3)
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To obtain the force–displacement relations we have to impose the compatibility between the axial displacements of the block ends, so by equalising Eqs. (3b) and (28b) we obtain: N ¼
T t1 ð1Þ p2 p2 l2 N þ EBðT t1 Þn21 EBt2 n22 : t1 4 4 l1
ð30Þ
The total compressive load is N ¼ N ð1Þ þ N , and using the expressions in Eqs. (3a) and (30): " # 2 2 N 1 3 l1 t1 2 3 t2 l2 l1 ¼1þ 1 n22 : ð31Þ þ n2 Nc1 8 4 t1 4 T l1 t1 T This expression substitutes Eq. (28a) when the buckling in the second layer appears. By introducing: 2 2 1 3 l1 t1 3 t 2 l 2 l1 W ¼ W ðl1 Þ ¼ þ 1 ; ; H2 ¼ H2 ðl1 ; l2 Þ ¼ 8 4 t1 4 T l1 t1 T ð32a;bÞ it is possible rewrite (31) as N ¼ 1 þ W ðl1 Þn21 H2 ðl1 ; l2 Þn22 : Nc1
ð33Þ
Moreover, by defining the ‘‘fictitious single delaminated body (SD)’’, obtained from the original laminate, by suppressing the second delamination, we can write Eqs. (31) and (25) as follows: N ¼ 1 þ W n21SD ; Nc1
uL ¼
NL p2 t1 2 þ l1 n1SD : EBT 4 T
ð34a;bÞ
By imposing, as previously defined, the compatibility equations, by using Eqs. (3a), (28a) and (29), we finally reach Eqs. (17) or (24a), where 2 1 t1 3 þ 8 l1 4 R ¼ Rðl1 ; l2 Þ ¼ 2 ; ð35aÞ 1 t2 3 t2 3 t2 l2 1 þ þ 8 l2 4 4 T t1 l1 T t1
2 2 t2 t1 l2 l1 : S ¼ Sðl1 ; l2 Þ ¼ 2 1 t2 3 t2 3 t2 l2 1 þ þ 8 l2 4 4 T t 1 l1 T t1
ð35bÞ
By using the equivalent form of Eq. (34a) represented by Eq. (24a) we need: X ¼ X ðl1 ; l2 Þ ¼ H2 ðl1 ; l2 ÞSðl1 ; l2 Þ; Y ¼ Y ðl1 ; l2 Þ ¼ W ðl1 Þ H2 ðl1 ; l2 ÞRðl1 ; l2 Þ;
ð36a;bÞ
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while Eq. (24c) give us the expression of Z which explicitly becomes (" ) 2 #," 2 # t2 l1 l2 T t2 1 : 1þ6 Z ¼ Zðl1 ; l2 Þ ¼ 1þ6 þ L T t1 t2 t2
ð37Þ
In an analogous way we can achieve the relations for R, S, X , Y , Z in the other three cases: Case BA 2 1 t1 3 t1 l1 3 t1 1 þ þ 8 l1 4 T t2 l2 4 T t2 R ¼ Rðl1 ; l2 Þ ¼ ; 2 1 t2 3 þ 8 l2 4 2 2 t2 t1 l2 l1 ; ð38a;bÞ S ¼ Sðl1 ; l2 Þ ¼ 2 1 t2 3 þ 8 l2 4 X ¼ X ðl1 ; l2 Þ ¼ L ðH2 S þ 1Þ; Y ¼ Y ðl1 ; l2 Þ ¼ H2 R H1 ; " #," # 2 2 t2 3 t 1 l1 t1 3 t1 l1 Z ¼ Zðl1 ; l2 Þ ¼ W þ W þ ; 4 T l2 4 T l2 l2 l1
ð39a;bÞ ð40Þ
where 2 3 t1 l1 l1 H1 ¼ H1 ðl1 ; l2 Þ ¼ ; 4 T l2 t1 2 2 t2 l1 : Lðl1 ; l2 Þ ¼ t1 l2
1 3 H2 ¼ H2 ðl1 ; l2 Þ ¼ L þ 8 4
l1 t1
2 t2 1 ; T ð41a;b;cÞ
Case AB R and S are equal to Eqs. (35a,b), X and Y are equal to Eqs. (36a,b), while Z becomes Zy 2 t1 3 t 2 l2 Wy þ Nc1 4 T l1 l Zy ¼ Zy ðl1 ; l2 Þ ¼ ¼ 1 2 ; Nc2 t2 3 t 2 l2 Wy þ 4 T l1 l2 2 1 3 l2 t2 Wy ¼ Wy ðl2 Þ ¼ þ 1 ð42a;bÞ 8 4 l1 T in expression (33) the quantity Nc1 must be substituted by Nc1SD defined as 2 T ð1Þ p2 t1 EBT : ð43a;bÞ Nc1SD ¼ Nc ¼ t1 3 l1
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Case BB In this case we need to use the following equation, similar to Eq. (31): N ¼ Lðl1 ; l2 Þ H1 ðl1 ; l2 Þn21 þ H2 ðl1 ; l2 Þn22 ; Nc1SD
ð44Þ
where H1 , H2 and L are obtained from Eqs. ((41a)–(c)), and R, S are equals to Eqs. (38a,b), while for Zy the expression is equal to Eq. (42a) but with the layer indexes interchanged.
4. Limit equilibrium equations By remembering the Griffith criterion, the limit equilibrium conditions can be written by Eqs. (26a,b) for cases AA and BA, and by Eqs. (27a,b) for cases AB and BB. Here only the first two cases are discussed, being the analysis of the other two analogous cases. The cases of equal thickness produce the same following equations, only by substituting t1 for t in the expressions. Eqs. (26a,b), using relations (10a,b,c) and (20) are equivalent to the following:
1 o DU1 ¼ CB; l01 oW1
1 1 o DU2 ¼ CB: k l01 oW2
ð45a;bÞ
In the previous relations DU1 , DU2 are the total potential energy increments and C is the density of the surface adhesion energy. Observing Fig. 3, calculating from it, the simple expressions of DU1 and DU2 , so their sum DU, and by using the expressions of X , Y , R, S, W , Z previously introduced, we can obtain for Eq. (26a), equivalent to Eq. (45a): ð1Þ
P1 k2 þ P2 k þ P3 ¼ 2a0 ; where ð1Þ a0
4CB ¼ 2 t1 ; p T Nc01
o P1 ¼ P1 ðw1 ; w2 Þ ¼ ow1
ð46Þ
1 2 ðw þ kqRw2 Þw1 ; Y 1
P2 ¼ P2 ðw1 ; w2 Þ o 1 1 R ðX þ Z þ 1Þ ðZ 1Þ w1 þ kq ðX þ Z þ 1Þ þ S w2 ; ¼ ow1 Y W Y P3 ¼ P3 ðw1 ; w2 Þ ( ) o 1 1 1 R 1 ðX þ 1Þ þ ðX þ 1Þ þ S w2 Z ¼ 1 w1 þ kq ow1 Y W Z Y w21 and for Eq. (26b), equivalent to Eq. (45b): ð1Þ
Q1 k2 þ Q2 k þ Q3 ¼ 2a0 ;
ð47Þ
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where Q1 ¼ Q1 ðw1 ; w2 Þ ¼
o ow2
1 Y
Y 1 w1 þ kqRw w21 ; W
Q2 ¼ Q2 ðw1 ; w2 Þ o 1 1 R ðX þ Z þ 1Þ ðZ þ 1Þ w1 þ kq ðX þ Z þ 1Þ þ S w2 ; ¼ ow2 Y W Y Q3 ¼ Q3 ðw1 ; w2 Þ ( ) o 1 1 R 1 ðX þ 1Þ ðX þ 1Þ þ S w2 ¼ Z w1 þ kq : ow2 Y W Y w21 The solution of the last two equations (46) and (47) gives us: ð48Þ k^1 ¼ k^1 ðW1 ; W2 Þ; k^2 ¼ k^2 ðW1 ; W2 Þ; ^ that in the space W1 ; W2 ; k represent the limit load surfaces of the two delaminated layers.
5. Modal interaction in symmetrically located delamination growth Delamination growth has a constrained path (interlaminar). In this way, the total potential energy is function of the fracture modes involved in the phenomenon. Because of the plane deformation characteristic of the phenomenon, the tearing fracture mode (Mode III) is practically absent. In this section we confine the analysis to a narrow plate with outer layers of equal thickness. For the two layers (1 and 2) the following relations remain still valid: G1 ðk; W1 ; W2 Þ ¼ G1I þ G1II ¼
p2 t Nc01 ðP1 k2 þ P2 k þ P3 Þ; 8 T
ð49Þ
p2 t 1 Nc01 ðQ1 k2 þ Q2 k þ Q3 Þ ; ð50Þ k 8 T where the adopted symbols are those previously introduced. With reference to Fig. 6, the following quantities are introduced: G2 ðk; W1 ; W2 Þ ¼ G2I þ G2II ¼
e¼
N ; EBT
e1 ¼
N ð1Þ ; EBt
e2 ¼
N ð2Þ ; EBt
e3 ¼
N ð3Þ ; EBðT 2tÞ
e ¼
N ðÞ : EBðT tÞ
The extensional potential energy of the system is E ¼ EðN ; l1 ; l2 Þ 1 1 1 1 1 ¼ N eðL l1 Þ N ð1Þ e1 l1 N ð2Þ e2 l2 N ð3Þ e3 l2 N ðÞ e ðl1 l2 Þ 2 2 2 2 2 i Bh 2 2 2 2 2 ¼ e kðL l1 Þ þ e1 k1 l1 þ e2 k2 l2 þ e3 k3 l2 þ e k ðl1 l2 Þ ; ð51Þ 2
P. Lonetti, R. Zinno / Simulation Modelling Practice and Theory 11 (2003) 483–500 L-
L- 1 2
T
ε1
2
2
N
ε
ε* ε3
t 2
1
2
1 1
t
497
1
2
2
ε*
N
ε
ε2
L Fig. 6. Extensional deformations.
where the following extensional stiffness parameters are adopted: k1 ¼ Et;
k2 ¼ Et;
k3 ¼ EðT 2tÞ;
k ¼ EðT tÞ;
k ¼ ET :
The unitary energy release rates at the delamination front are G1II ¼
oE B B k1 k 2 2 ¼ ðe22 k þ e21 k1 þ e k Þ ¼ ðeð1Þ e Þ ; ol1 2 2 k1 þ k
ð52Þ
G2II ¼
oE B 2 B k2 k3 2 ¼ ðe k þ e23 k3 e k Þ ¼ ðeð2Þ e3 Þ2 : ol2 2 2 2 k2 þ k3
ð53Þ
From expressions (52) and (53) it is possible to reach the explicit expressions of G1II and G2II qualitatively similar to the expressions (49) and (50). Moreover, from relations equations (49) and (50), by subtracting G1II and G2II , represented by Eqs. (52) and (53), it is possible to obtain the expressions for G1I and G2I . The delamination growth condition, or the limit equilibrium in mixed mode, can be obtained from the expression proposed in [1]: e ¼ GI þ gGII ¼ CB; G
g¼
GIC ; GIIC
ð54Þ
where GIC and GIIC are the interlaminar strength for mode I and II, respectively. For layer 1 it is possible to write: e 1 ðk; W1 ; W2 Þ ¼ G1I ðk; W1 ; W2 Þ þ gG1II ðk; W1 ; W2 Þ ¼ CB G
ð55Þ
and from here a relation analogous to Eq. (46): ð1Þ Pe1 k^21 þ Pe2 k^1 þ Pe3 ¼ 2a0 ;
ð56Þ
where Pi ¼ PiI þ gPiII (i ¼ 1; 2; 3). The explicit expressions can be obtained in the same way as for P1 , P2 , P3 in Eq. (46). Analogously for the second layer: e 2 ðk; W1 ; W2 Þ ¼ G2I ðk; W1 ; W2 Þ þ gG2II ðk; W1 ; W2 Þ ¼ CB G
ð57Þ
and from it the equation: e 2 k^1 þ Q e 3 ¼ 2kað1Þ ; e 1 k^2 þ Q Q 0 1
Qi ¼ QiI þ gQiII ; ði ¼ 1; 2; 3Þ
ð58Þ
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in Eqs. (56) and (58) the coefficient a0 is equal to .h t i ð1Þ a0 ¼ ½4CB p Nc01 : T
ð59Þ
6. Numerical results and conclusions When the hypothesis that the buckling of layer 2, relative to the initial delamination lengths, precedes the delamination growth onset of layer 1, four situations appear possible: (a) stable path for the delamination of the first layer (in the upper layer), while the second one (the bottom layer) is in post-buckling conditions; (b) stable growth for the delamination of the upper layer and no-stable growth for the bottom layer, but stable growth in the ‘‘Joined Limiting Equilibrium Load Path (JLEP)’’ which is the representation of the joined limit equilibrium path in plane (W1 , W2 ) in function of the abscissa s where s : W1 ¼ kW2 . (c) no-stable growth for delamination of layer 1, but stable growth in JLEP; (d) stable growth of delamination in both the layers. k=0.35 3
5 4
12
1
7 6.5 6 λ 5.5 5
4
Ψ2 5
k=0.35
1
2
3
4
Ψ2 3
2 1
7 6.5 6 5.5
2
3
4
λ
5
Ψ1
Fig. 7. Limit surfaces as functions of dimensionless delamination lengths.
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In particular, Fig. 7, depict the first situation. The light grey surface is relative to the upper layer, while the dark grey is relative to the bottom one. To draw this picture the following values are fixed: k¼
l02 ¼ 0; 35; l01
t ¼ 0; 02; l01
t ¼ 0; 01; T
ð1Þ
a0 ¼
4CB ¼ 0; 01: p2 Tt Nc01
Fig. 8a and b show the limit load surfaces of the two delaminated layers. In particular the following values are fixed: k¼
l02 ¼ 0; 35; l01
t l01
¼ 0; 02;
t ¼ 0; 01; T
ð1Þ
a0 ¼
4CB ¼ 0; 01: Nc01
p2 Tt
Finally, in order to show the results for laminates with outer layers of different thickness the following parameters have been used: t=l01 ¼ 0; 01; t=T ¼ ð1Þ 0; 01; a0 ¼ 0; 01; k ¼ 0; 5 and q ¼ t2 =t1 ¼ 0; 5 (in Fig. 9a) and q ¼ t2 =t1 ¼ 0; 9 (in Fig. 9b).
Fig. 8. (a) Limit surfaces g ¼ GIC =GIIC ¼ 1. (b) Limit surfaces g ¼ GIC =GIIC ¼ 0; 15.
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λ 16
5
14 4
12 1
3 2
3 4
1
(a)
2
2 5 5
5
5
2 4
1.5 3
1
2
2 2
3 4
(b)
1
5 1
Fig. 9. (a) Limit surfaces: K ¼ 0; 5, Q ¼ 0; 5. (b) Limit surfaces: K ¼ 0; 5, Q ¼ 0; 9.
References [1] C. Amarg Garg, Delamination––a damage mode in composite structures, Engineering Fracture Mechanics 29 (5) (1988) 557–584. [2] E. Altus, D. Ishai, Delamination buckling criterion for composite laminates: a macro approach, Engineering Fracture Mechanics 41 (5) (1992) 737–751. [3] D. Bruno, Delamination buckling in composite laminates with interlaminar defects, Theoretical and Applied Fracture Mechanics 9 (2) (1988) 145–179. [4] D. Bruno, G. Porco, Delaminazione in buckling di piastre laminate, in: Proceedings of the XXIII AIAS National Conference, Rende, Cosenza, Italy, 21–24 September 1994. [5] F. Greco, R. Zinno An analysis of multiple interlaminar crack propagation under mixed-mode deformations, in: Proceedings of XV National Conference of the IGF ItalianGroup of Fracture, Bari, Italy, 3–5 Maggio 2000. [6] P. Lonetti, R. Zinno, Numerical simulation of multiple delaminations in composite laminates under mixed-mode deformations, in: Proceedings of EUROSIM 2001 Congress on Shape the Future with Simulation, Delft, Netherlands, 2001. [7] G. Porco, R. Zinno, Modal interaction in the debonding problems in laminates, in Proceedings of the Fifth Conference of Unilateral problems in Structural Analysis, Ferrara, Italy, 12–14 June 1997.
Simulation of multiple delaminations in composite laminates under mixed-mode deformations q Paolo Lonetti, Raffaele Zinno
*
Department of Structural Engineering, University of Calabria, 87036 Cosenza, Italy Received 21 May 2002; received in revised form 16 October 2002; accepted 27 October 2002
Abstract A delamination analysis is developed in order to predict the defect evolution when a fiber composite laminate, with outer layers of equal or different thickness, is subjected to a compression load and multiple delaminations. In particular, the presence of two delaminated zones coaxially positioned on the two opposite sides of the laminate is analysed. In order to accurately predict the growth of delamination, when mode I and II are interacting, some fracture criteria have been proposed. Some results will be given to show the influence on the delamination buckling and on the growth phenomenon of the main geometrical and mechanical characteristics of the structure in order to show the pronounced role of the fracture modes on stable or unstable delamination growth. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Delamination buckling; Fiber composite laminate; Damage
1. Introduction Fibre reinforced laminate composites are widely used nowadays in load-bearing structures due to their light weight, high specific strength and stiffness, good corrosion resistance and superb fatigue strength limit. Many successful applications in q This is a revised/extended version of the paper ‘‘Numerical simulation of multiple delaminations in composite laminates under mixed-mode deformations’’ presented at the Eurosim 2001 International Congress, June 26–29, 2001, Delft, The Netherlands. * Corresponding author. Tel.: +39-0984-494020; fax: +39-0984-494045. E-mail address: [email protected] (R. Zinno).
1569-190X/03/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1569-190X(03)00054-6
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Nomenclature N Nc0i Nc l01 , l02 l1 , l2 t T w E B u n1 nlSD N Nc N 2;3 Nci N ð1Þ ð1Þ NLLB k q DU C GI GII GIC GIIC g
total compressive load buckling load of the i-layer in absence of buckling of the remaining layer buckling load upper and lower initial delamination length upper and lower post-buckling delamination length layer thickness total thickness of the laminate central transverse displacement Young modulus transverse length longitudinal displacement non-dimensional central transverse displacement of the ith layer non-dimensional central transverse displacement in the fictitious single delaminated body buckling load of the sublaminate composed by layer 2 and core compressive load of the layer 2 compressive load of the layer 2 and core buckling load of the i-layer post-buckling load of the layer 1 function of Nc2 adsorbed by buckled layer 1 non-dimensional initial layer 2/layer 2 delamination length ratio layer 1/layer 2 thickness ratio total potential energy increments density of the surface adhesion energy energy release rate for mode I energy release rate for mode II critical value of the energy release rate for mode I critical value of the energy release rate for mode II mode I/mode II critical energy release rate ratio
aerospace, mechanical, naval and, now, also in civil engineering rely on the exploitation of these properties as well as others. However, like metals, these materials lose much more of their structural integrity when damaged. In particular, laminated composites often present initial delaminations produced by several causes, such us manufacturing imperfections, stress concentrations, impacts and local or global buckling of laminae. In this paper a computer simulation of a characteristic failure of composite laminates, delamination, is presented. This phenomenon is one of the most dangerous kinds of damage in a laminate structure and, unfortunately, its modelling is very complicated without the help of purpose made computer program. Frequently, com-
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485
puter codes are developed by using a FEA computational procedures, or, as in this case, they are developed with the aim of analytical closed form solution. These closed form solutions are obtained by utilising the mechanical aspects of the delamination phenomenon and the linear elastic fracture mechanics theory. The occurrence of delaminations, in fact, may allow the laminate free surface to buckle locally (delamination buckling phenomenon) thereby creating conditions conducive to delamination growth. The delamination growth phenomenon depends upon the stress state at the crack tip and it is often governed by the mixed-mode stress intensity factor or by the mixed-mode strain energy release rate. This consideration is the main reason for several developed and published research works where interlaminar fracture toughness experimental tests about mode I and II are carried out or where mixed-mode delamination growth models for composite systems are developed. The present work falls within this context, concerning the behaviour of simple one-dimensional plates subject to compressive loads, in which the delamination is assumed to spread from initial interlaminar defects. In particular, attention is focused on locally buckled laminates containing multiple delaminations as depicted in Fig. 1, where the outer layers can have equal or different thickness. The present paper represents a collection of the analyses developed in [4–7] mainly devoted to similar cases. In order to accurately predict the growth of delamination, when mode I and II are interacting, some fracture criteria have been proposed taking into account experimental evidence. In the present paper, one of this mixed-mode fracture criteria [1,2] is adopted to obtain information on mixed-mode fracture behaviour of a laminated one-dimensional plates. Only in this last analysis the external layers have equal thickness.
B
1
N
N
2
L
1
w1 t T
N
N t 2
w2
L Fig. 1. Locally buckled laminated affected by symmetrically located delaminations.
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Some results show the influence on the delamination buckling and on the growth phenomenon of the main geometrical and mechanical characteristics of the structure in order to evidence the marked role of the fracture modes on the stable or unstable delamination growth.
2. Multiple delamination analysis for a narrow plate with outer layers of equal thickness With reference to Fig. 1, it is developed the analysis of the damage evolution produced by a compressive load on a narrow plate subjected to two coaxial delaminated zones. The thickness of the two external delaminated layers are equal. In particular, the relations governing the mechanical behaviour of the plate, after the second critical state are analysed. The hypothesis that the buckling of the second layer appears before delamination growth initiation in the first buckled layer, (Hyp. A), is supposed. The buckling load of the ith layer, in absence of the buckling of the remaining jth layer is 2 p 2 ti Nc0i ¼ EBT ; ð1Þ 3 l0i where E is the Young modulus of the homogeneous, linearly elastic, isotropic material constituting the layers, T is the total thickness of the laminate, and ti is the thickness of the layer. The first buckled layer, obviously, has length (l01 ) equal or greater than the second one (l02 ). The following non-dimensional parameters, which represent the central transverse displacements (w), are introduced in the computation in the model: w l21 w l22 n1 ¼ ; n2 ¼ : ð2a;bÞ l1 l2 The post-buckling relations which are valid for the first layer only have the following expressions [3]: 1 N ð1Þ ¼ Ncð1Þ þ Ncð1Þ n21 ; 8
ul1 N ð1Þ p2 2 þ n1 : ¼ l1 EBt 4
ð3a;bÞ
Because of the hypothesis that layer 2 buckles before the delamination growth initiation of layer 1, and by considering the compressive load N of the sublaminate composed by layer 2 and core 3, as the sum of the N ð2Þ and N ð3Þ relative to layer 2 and to core 3 respectively (see Fig. 2), it is possible to write: " # 2 N 1 3 l2 t 2 ¼1þ 1 ð4aÞ þ n2 ; 8 4 t T t Nc ul1 N t p2 l2 2 þ ¼ n: l1 EBðT tÞ T t 4 l1 2
ð4bÞ
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487
1
w1 t T
N
1
N
(1)
*
t
(2)
N =N +N
3
T*
(3)
2
2
w2
1
w1LLB t T
N
(1)
NLLB
1
(3) C2
NLLB
3
(2)
t
NC
2 2
Fig. 2. Layer compressive load distribution.
In relations (4a,b) Nc indicates the value of N capable of buckling layer 2: 2 p2 t Nc ¼ EBðT tÞ: ð5Þ 3 l2 In order to calculate the value of Nc2 which is the buckling load of the second layer, it is worth noting the l1 P l2 , which expresses the fact that the delamination fronts have different positions in the laminates. Moreover, it is possible to write the buckling load of the layer 2 as ð1Þ
Nc2 ¼ NLLB þ Nc ;
ð6Þ
ð1Þ NLLB
where represents the fraction of Nc2 adsorbed by buckled layer 1. Writing the compatibility equations of the axial displacements of the delaminated block, and introducing the Ncð2Þ layer 2 buckling load variable, the following relationship holds: 2 1 þ 6 lt2 ð1Þ ð2Þ ð7Þ NLLB ¼ 2 N c : 1 þ 6 lt1 By substituting Eqs. (5)–(7), it is possible to write: 2 # 2 " Nc2 t l1 T t 1 þ 6 lt2 ¼ Zðl1 ; l2 Þ ¼ þ 2 ; T l2 t Nc1 1 þ 6 l1 t
ð8Þ
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where Nc1 is the buckling load of layer 1, relative to the delamination length l1 , and it is equal to: 2 p2 t Nc1 ¼ EBT : ð9Þ 3 l1 By introducing the following non-dimensional ratios between the delamination lengths and the defects lengths: w1 ¼
l1 ; l01
w2 ¼
l2 ; l02
k¼
l02 ; l01
k 6 1;
we can rewrite Eqs. (8) and (9) as follows: # " 2 2 1 t 1 þ 6k 2 l01t w2 T t w21 ; Zðw1 ; w2 Þ ¼ 2 þ 2 k T t w22 1 þ l01t w21 " # 2 p2 t 1 1 Nc1 ¼ EBT ¼ 2 Nc01 : 2 3 l01 w1 w1
ð10a;b;c;dÞ
ð11a;bÞ
In order to obtain the force–displacement relations the compatibility between the axial displacements of the block ends must be recalled. In this way by equalizing Eqs. (3b) and (4b) it is possible to find the following equations: N ¼
T t ð1Þ p2 p2 l2 N þ EBðT tÞn21 EBt n21 : t 4 4 l1
ð12Þ
The total compressive load is N ¼ N ð1Þ þ N , and by using the expressions in Eqs. (3a) and (12) can be written as " # 2 2 N 1 3 l1 t 2 3 t l2 l1 ¼1þ 1 n22 : ð13Þ þ n1 Nc1 8 4 t T 4 T l1 t This expression substitutes Eq. (4a) when the buckling load appears in the second layer. By introducing: 2 2 1 3 l1 t 3 t l2 l2 W ¼ W ðl1 Þ ¼ þ 1 ; ; H2 ¼ H2 ðl1 ; l2 Þ ¼ 8 4 t T 4 T l1 t ð14a;bÞ it is possible rewrite (13) as N ¼ 1 þ W ðl1 Þn21 H2 ðl1 ; l2 Þn22 : ð15Þ Nc1 Moreover, by defining the ‘‘fictitious single delaminated body (SD)’’, obtained from the original laminate by suppressing the second delamination, we can write Eq. (13) as follows: " # 2 N 1 3 l1 t 2 ¼1þ 1 ð16Þ þ n1SD ¼ 1 þ W ðl1 Þn21SD : Nc1 8 4 t T
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489
By imposing, as previously, the compatibility conditions, by using Eqs. (3a), (4a) and (5), we reach: n22 ¼ Rðl1 ; l2 Þn21 Sðl1 ; l2 Þ;
ð17Þ
where 1 8
Rðl1 ; l2 Þ ¼ 2
2 t l1
þ 34 t
; t l2 þ 34 1 T t þ 34 T t l1 2 2 t þ lt1 l2 Sðl1 ; l2 Þ ¼ 2 3 t l ; 1 t 3 t þ 1 þ 4 T t l21 8 l2 4 T t 1 8
t l2
or in the other form: N ¼ 1 þ X ðl1 ; l2 Þ þ Y ðl1 ; l2 Þn21 ; Nc1
ð18Þ
with X ¼ X ðl1 ; l2 Þ ¼ H2 ðl1 ; l2 ÞSðl1 ; l2 Þ;
Y ¼ Y ðl1 ; l2 Þ ¼ W ðl1 Þ H2 ðl1 ; l2 ÞRðl1 ; l2 Þ:
It is also possible, by using the previous relation, to achieve the following expression for uL :
ξ1 2
2
W ξ1
precritical path
A
S
Z
X
2
ξ 1 SD
B
NC2 NC1
2
Y ξ1
N NC1
th l pa tica tcri s o tp firs
ond
sec
ath
al p
ritic
tc pos
1 (W-Y) 2 ξ R 2
ξ
2
O
1
2
ξ2 S R 2
ξ 2=R ξ 1-S 2
ξ
2 2
Fig. 3. Pre- and post-critical paths: displacement interpretation.
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N-NC2
N-NC1
2 t π 2 ξ2 t π ξ2 T 4 1 1+ T 4 2 2 t π 2 ξ2 T 4 1 1SD
NC2
second post-critical path
∆Φ2
B
NC1
A
∆Φ
1 first post-critical path
pre-critical path
u
L
Fig. 4. Pre- and post-critical paths: potential energy graphical interpretation.
uL ¼
NL t p2 2 þ l1 n1SD : EBT T 4
ð19Þ
In Figs. 3 and 4 are depicted the graphical interpretation of Eqs. (18) and (19). 3. Narrow plate with outer layers of different thickness affected by multiple delamination In this section the analysis of the damage evolution, due to a compressive load, is extended to the case of a laminated narrow plate with the different thickness outer layers, subjected to two coaxial delaminated zones. The previous hyp. A is again made. Fig. 1 is still valid but the thicknesses are: t1 and t2 , for the upper and bottom layer respectively. The buckling load of the ith layer, in absence of buckling of the remaining jth layer can still obtained by the previous equation (1). The same for Eqs. (2a,b) regarding the non-dimensional parameters representing the central transverse displacements (w). The ratios between the delamination lengths and the defects lengths (10a)–(10c) are also still valid, but the thickness ratio variable must be introduced in the formulation: t2 q¼ : ð20Þ t1 From Eq. (1) and Eqs. (10a)–(10c) and (20), the hypothesis that the ‘‘layer 1’’ lamina is the first to buckle, can be written equivalently as l01 t1 P ; l02 t2
or
k 61 q
ð21a;bÞ
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491
by the definition of Nc1 and Nc2 as the buckling loads of layers 1 and 2, relative to the delamination lengths l1 and l2 . In particular the value Nc1 , it is easy to obtain by substituting the value l1 to the value l0i in Eq. (1). In order to evaluate Nc2 variable the following straight lines r and s on the plane W1 and W2 are introduced in the formulation: r:
w1 ¼ ðk=qÞw2
s:
w1 ¼ kw2 :
ð22a;bÞ
In particular, the straight line r is relative to delamination lengths producing the delamination buckling phenomenon with an equal value of the external load parameter, i.e.: Nc1 ¼ Nc2 , while on s line the relation l1 ¼ l2 is valid, which expresses equal delamination lengths growth. So, by using these straight lines it is possible to recognise four cases: 1 q case AA: w2 6 w1 ; w2 6 w1 ; k k 1 q case AB: w2 6 w1 ; w2 P w1 ; ð23a;bÞ k k 1 q w1 ; w2 6 w1 ; k k 1 q case BB: w2 P w1 ; w2 P w1 : ð23c;dÞ k k From the geometrical and physical point of view the first relation in the four cases indicates: l1 P l2 (A) or l1 6 l2 (B), while the second relation indicates: first delamination on layer 1 (A) or first delamination on layer 2 (B). Following the same analytical procedure previously developed in the case of outer layers of equal thickness, it is possible to achieve the following equations that connect the mechanical parameters previously introduced: case BA:
w2 P
n22 ¼ Rðl1 ; l2 Þn21 Sðl1 ; l2 Þ;
uL ¼ uL ðN ; l1 ; l2 Þ ¼
N ¼ 1 þ X ðl1 ; l2 Þ þ Y ðl1 ; l2 Þn21 ; Nc1
N t1 p2 2 t2 p2 2 þ l1 n1 þ l2 n2 : EBT T 4 T 4
Nc2 ¼ Zðl1 ; l2 Þ; Nc1 ð24a;b;cÞ ð25Þ
Figs. 3 and 4 represent Eqs. (24a)–(24c) and (25), but now the expression of variables R, S, X , Y and Z depends on the region of plane W1 ; W2 , so depending on one of the four cases previously listed. When the first delaminated layer is upper lamina (‘‘layer 1’’) it is possible to write: G1 ¼
o DU ¼ CB; ol1
G2 ¼
o DU2 ¼ CB; ol2
ð26a;bÞ
while if the first delaminated layer is the lower lamina (‘‘layer 2’’), the expressions become: G1 ¼
o DU1 ¼ CB; ol1
G2 ¼
o DU ¼ CB: ol2
ð27a;bÞ
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Relations (26) and (27) are the particularised expressions of the limit equilibrium conditions in the well known Griffith criterion. The explicit and simple expressions for DU1 , DU2 and the sum DU can be obtained from a direct observation of Fig. 3. Obviously, in the case of Eq. (27) it is necessary change the position of DU1 , DU2 . For the sake of brevity, now we will explain only one of the four cases listed in Eqs. (23a)–(23d), and we will only summarize the main results for the remaining three cases. Case AA The post-buckling relations (3a,b) are valid for the first layer [3]. Eqs. (4a,b) must be modified to take into account the different thickness (see also Fig. 5 for the correct indexes): " 2 # N 1 3 l2 t2 ¼1þ 1 ð28aÞ þ n22 ; 8 4 t2 Nc T t1 ul1 N t2 p2 l2 2 þ ¼ n: l1 EBðT t1 Þ T t1 4 l1 2
ð28bÞ
In relations (28a,b), as in Eqs. (4a,b), Nc indicates the value of N capable of buckling layer 2: 2 p2 t2 Nc ¼ EBðT t1 Þ: ð29Þ 3 l2
1
w1 t1 T
N
1
N
(1)
*
T
t2
2
2
w2
1
w1LLB t1 T
(1)
NLLB
1
(3)
NC2
NLLB
3
(2)
t2
(2)
N =N +N
3
*
NC
2 2
Fig. 5. Layer load distribution.
(3)
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493
To obtain the force–displacement relations we have to impose the compatibility between the axial displacements of the block ends, so by equalising Eqs. (3b) and (28b) we obtain: N ¼
T t1 ð1Þ p2 p2 l2 N þ EBðT t1 Þn21 EBt2 n22 : t1 4 4 l1
ð30Þ
The total compressive load is N ¼ N ð1Þ þ N , and using the expressions in Eqs. (3a) and (30): " # 2 2 N 1 3 l1 t1 2 3 t2 l2 l1 ¼1þ 1 n22 : ð31Þ þ n2 Nc1 8 4 t1 4 T l1 t1 T This expression substitutes Eq. (28a) when the buckling in the second layer appears. By introducing: 2 2 1 3 l1 t1 3 t 2 l 2 l1 W ¼ W ðl1 Þ ¼ þ 1 ; ; H2 ¼ H2 ðl1 ; l2 Þ ¼ 8 4 t1 4 T l1 t1 T ð32a;bÞ it is possible rewrite (31) as N ¼ 1 þ W ðl1 Þn21 H2 ðl1 ; l2 Þn22 : Nc1
ð33Þ
Moreover, by defining the ‘‘fictitious single delaminated body (SD)’’, obtained from the original laminate, by suppressing the second delamination, we can write Eqs. (31) and (25) as follows: N ¼ 1 þ W n21SD ; Nc1
uL ¼
NL p2 t1 2 þ l1 n1SD : EBT 4 T
ð34a;bÞ
By imposing, as previously defined, the compatibility equations, by using Eqs. (3a), (28a) and (29), we finally reach Eqs. (17) or (24a), where 2 1 t1 3 þ 8 l1 4 R ¼ Rðl1 ; l2 Þ ¼ 2 ; ð35aÞ 1 t2 3 t2 3 t2 l2 1 þ þ 8 l2 4 4 T t1 l1 T t1
2 2 t2 t1 l2 l1 : S ¼ Sðl1 ; l2 Þ ¼ 2 1 t2 3 t2 3 t2 l2 1 þ þ 8 l2 4 4 T t 1 l1 T t1
ð35bÞ
By using the equivalent form of Eq. (34a) represented by Eq. (24a) we need: X ¼ X ðl1 ; l2 Þ ¼ H2 ðl1 ; l2 ÞSðl1 ; l2 Þ; Y ¼ Y ðl1 ; l2 Þ ¼ W ðl1 Þ H2 ðl1 ; l2 ÞRðl1 ; l2 Þ;
ð36a;bÞ
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while Eq. (24c) give us the expression of Z which explicitly becomes (" ) 2 #," 2 # t2 l1 l2 T t2 1 : 1þ6 Z ¼ Zðl1 ; l2 Þ ¼ 1þ6 þ L T t1 t2 t2
ð37Þ
In an analogous way we can achieve the relations for R, S, X , Y , Z in the other three cases: Case BA 2 1 t1 3 t1 l1 3 t1 1 þ þ 8 l1 4 T t2 l2 4 T t2 R ¼ Rðl1 ; l2 Þ ¼ ; 2 1 t2 3 þ 8 l2 4 2 2 t2 t1 l2 l1 ; ð38a;bÞ S ¼ Sðl1 ; l2 Þ ¼ 2 1 t2 3 þ 8 l2 4 X ¼ X ðl1 ; l2 Þ ¼ L ðH2 S þ 1Þ; Y ¼ Y ðl1 ; l2 Þ ¼ H2 R H1 ; " #," # 2 2 t2 3 t 1 l1 t1 3 t1 l1 Z ¼ Zðl1 ; l2 Þ ¼ W þ W þ ; 4 T l2 4 T l2 l2 l1
ð39a;bÞ ð40Þ
where 2 3 t1 l1 l1 H1 ¼ H1 ðl1 ; l2 Þ ¼ ; 4 T l2 t1 2 2 t2 l1 : Lðl1 ; l2 Þ ¼ t1 l2
1 3 H2 ¼ H2 ðl1 ; l2 Þ ¼ L þ 8 4
l1 t1
2 t2 1 ; T ð41a;b;cÞ
Case AB R and S are equal to Eqs. (35a,b), X and Y are equal to Eqs. (36a,b), while Z becomes Zy 2 t1 3 t 2 l2 Wy þ Nc1 4 T l1 l Zy ¼ Zy ðl1 ; l2 Þ ¼ ¼ 1 2 ; Nc2 t2 3 t 2 l2 Wy þ 4 T l1 l2 2 1 3 l2 t2 Wy ¼ Wy ðl2 Þ ¼ þ 1 ð42a;bÞ 8 4 l1 T in expression (33) the quantity Nc1 must be substituted by Nc1SD defined as 2 T ð1Þ p2 t1 EBT : ð43a;bÞ Nc1SD ¼ Nc ¼ t1 3 l1
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495
Case BB In this case we need to use the following equation, similar to Eq. (31): N ¼ Lðl1 ; l2 Þ H1 ðl1 ; l2 Þn21 þ H2 ðl1 ; l2 Þn22 ; Nc1SD
ð44Þ
where H1 , H2 and L are obtained from Eqs. ((41a)–(c)), and R, S are equals to Eqs. (38a,b), while for Zy the expression is equal to Eq. (42a) but with the layer indexes interchanged.
4. Limit equilibrium equations By remembering the Griffith criterion, the limit equilibrium conditions can be written by Eqs. (26a,b) for cases AA and BA, and by Eqs. (27a,b) for cases AB and BB. Here only the first two cases are discussed, being the analysis of the other two analogous cases. The cases of equal thickness produce the same following equations, only by substituting t1 for t in the expressions. Eqs. (26a,b), using relations (10a,b,c) and (20) are equivalent to the following:
1 o DU1 ¼ CB; l01 oW1
1 1 o DU2 ¼ CB: k l01 oW2
ð45a;bÞ
In the previous relations DU1 , DU2 are the total potential energy increments and C is the density of the surface adhesion energy. Observing Fig. 3, calculating from it, the simple expressions of DU1 and DU2 , so their sum DU, and by using the expressions of X , Y , R, S, W , Z previously introduced, we can obtain for Eq. (26a), equivalent to Eq. (45a): ð1Þ
P1 k2 þ P2 k þ P3 ¼ 2a0 ; where ð1Þ a0
4CB ¼ 2 t1 ; p T Nc01
o P1 ¼ P1 ðw1 ; w2 Þ ¼ ow1
ð46Þ
1 2 ðw þ kqRw2 Þw1 ; Y 1
P2 ¼ P2 ðw1 ; w2 Þ o 1 1 R ðX þ Z þ 1Þ ðZ 1Þ w1 þ kq ðX þ Z þ 1Þ þ S w2 ; ¼ ow1 Y W Y P3 ¼ P3 ðw1 ; w2 Þ ( ) o 1 1 1 R 1 ðX þ 1Þ þ ðX þ 1Þ þ S w2 Z ¼ 1 w1 þ kq ow1 Y W Z Y w21 and for Eq. (26b), equivalent to Eq. (45b): ð1Þ
Q1 k2 þ Q2 k þ Q3 ¼ 2a0 ;
ð47Þ
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P. Lonetti, R. Zinno / Simulation Modelling Practice and Theory 11 (2003) 483–500
where Q1 ¼ Q1 ðw1 ; w2 Þ ¼
o ow2
1 Y
Y 1 w1 þ kqRw w21 ; W
Q2 ¼ Q2 ðw1 ; w2 Þ o 1 1 R ðX þ Z þ 1Þ ðZ þ 1Þ w1 þ kq ðX þ Z þ 1Þ þ S w2 ; ¼ ow2 Y W Y Q3 ¼ Q3 ðw1 ; w2 Þ ( ) o 1 1 R 1 ðX þ 1Þ ðX þ 1Þ þ S w2 ¼ Z w1 þ kq : ow2 Y W Y w21 The solution of the last two equations (46) and (47) gives us: ð48Þ k^1 ¼ k^1 ðW1 ; W2 Þ; k^2 ¼ k^2 ðW1 ; W2 Þ; ^ that in the space W1 ; W2 ; k represent the limit load surfaces of the two delaminated layers.
5. Modal interaction in symmetrically located delamination growth Delamination growth has a constrained path (interlaminar). In this way, the total potential energy is function of the fracture modes involved in the phenomenon. Because of the plane deformation characteristic of the phenomenon, the tearing fracture mode (Mode III) is practically absent. In this section we confine the analysis to a narrow plate with outer layers of equal thickness. For the two layers (1 and 2) the following relations remain still valid: G1 ðk; W1 ; W2 Þ ¼ G1I þ G1II ¼
p2 t Nc01 ðP1 k2 þ P2 k þ P3 Þ; 8 T
ð49Þ
p2 t 1 Nc01 ðQ1 k2 þ Q2 k þ Q3 Þ ; ð50Þ k 8 T where the adopted symbols are those previously introduced. With reference to Fig. 6, the following quantities are introduced: G2 ðk; W1 ; W2 Þ ¼ G2I þ G2II ¼
e¼
N ; EBT
e1 ¼
N ð1Þ ; EBt
e2 ¼
N ð2Þ ; EBt
e3 ¼
N ð3Þ ; EBðT 2tÞ
e ¼
N ðÞ : EBðT tÞ
The extensional potential energy of the system is E ¼ EðN ; l1 ; l2 Þ 1 1 1 1 1 ¼ N eðL l1 Þ N ð1Þ e1 l1 N ð2Þ e2 l2 N ð3Þ e3 l2 N ðÞ e ðl1 l2 Þ 2 2 2 2 2 i Bh 2 2 2 2 2 ¼ e kðL l1 Þ þ e1 k1 l1 þ e2 k2 l2 þ e3 k3 l2 þ e k ðl1 l2 Þ ; ð51Þ 2
P. Lonetti, R. Zinno / Simulation Modelling Practice and Theory 11 (2003) 483–500 L-
L- 1 2
T
ε1
2
2
N
ε
ε* ε3
t 2
1
2
1 1
t
497
1
2
2
ε*
N
ε
ε2
L Fig. 6. Extensional deformations.
where the following extensional stiffness parameters are adopted: k1 ¼ Et;
k2 ¼ Et;
k3 ¼ EðT 2tÞ;
k ¼ EðT tÞ;
k ¼ ET :
The unitary energy release rates at the delamination front are G1II ¼
oE B B k1 k 2 2 ¼ ðe22 k þ e21 k1 þ e k Þ ¼ ðeð1Þ e Þ ; ol1 2 2 k1 þ k
ð52Þ
G2II ¼
oE B 2 B k2 k3 2 ¼ ðe k þ e23 k3 e k Þ ¼ ðeð2Þ e3 Þ2 : ol2 2 2 2 k2 þ k3
ð53Þ
From expressions (52) and (53) it is possible to reach the explicit expressions of G1II and G2II qualitatively similar to the expressions (49) and (50). Moreover, from relations equations (49) and (50), by subtracting G1II and G2II , represented by Eqs. (52) and (53), it is possible to obtain the expressions for G1I and G2I . The delamination growth condition, or the limit equilibrium in mixed mode, can be obtained from the expression proposed in [1]: e ¼ GI þ gGII ¼ CB; G
g¼
GIC ; GIIC
ð54Þ
where GIC and GIIC are the interlaminar strength for mode I and II, respectively. For layer 1 it is possible to write: e 1 ðk; W1 ; W2 Þ ¼ G1I ðk; W1 ; W2 Þ þ gG1II ðk; W1 ; W2 Þ ¼ CB G
ð55Þ
and from here a relation analogous to Eq. (46): ð1Þ Pe1 k^21 þ Pe2 k^1 þ Pe3 ¼ 2a0 ;
ð56Þ
where Pi ¼ PiI þ gPiII (i ¼ 1; 2; 3). The explicit expressions can be obtained in the same way as for P1 , P2 , P3 in Eq. (46). Analogously for the second layer: e 2 ðk; W1 ; W2 Þ ¼ G2I ðk; W1 ; W2 Þ þ gG2II ðk; W1 ; W2 Þ ¼ CB G
ð57Þ
and from it the equation: e 2 k^1 þ Q e 3 ¼ 2kað1Þ ; e 1 k^2 þ Q Q 0 1
Qi ¼ QiI þ gQiII ; ði ¼ 1; 2; 3Þ
ð58Þ
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P. Lonetti, R. Zinno / Simulation Modelling Practice and Theory 11 (2003) 483–500 ð1Þ
in Eqs. (56) and (58) the coefficient a0 is equal to .h t i ð1Þ a0 ¼ ½4CB p Nc01 : T
ð59Þ
6. Numerical results and conclusions When the hypothesis that the buckling of layer 2, relative to the initial delamination lengths, precedes the delamination growth onset of layer 1, four situations appear possible: (a) stable path for the delamination of the first layer (in the upper layer), while the second one (the bottom layer) is in post-buckling conditions; (b) stable growth for the delamination of the upper layer and no-stable growth for the bottom layer, but stable growth in the ‘‘Joined Limiting Equilibrium Load Path (JLEP)’’ which is the representation of the joined limit equilibrium path in plane (W1 , W2 ) in function of the abscissa s where s : W1 ¼ kW2 . (c) no-stable growth for delamination of layer 1, but stable growth in JLEP; (d) stable growth of delamination in both the layers. k=0.35 3
5 4
12
1
7 6.5 6 λ 5.5 5
4
Ψ2 5
k=0.35
1
2
3
4
Ψ2 3
2 1
7 6.5 6 5.5
2
3
4
λ
5
Ψ1
Fig. 7. Limit surfaces as functions of dimensionless delamination lengths.
P. Lonetti, R. Zinno / Simulation Modelling Practice and Theory 11 (2003) 483–500
499
In particular, Fig. 7, depict the first situation. The light grey surface is relative to the upper layer, while the dark grey is relative to the bottom one. To draw this picture the following values are fixed: k¼
l02 ¼ 0; 35; l01
t ¼ 0; 02; l01
t ¼ 0; 01; T
ð1Þ
a0 ¼
4CB ¼ 0; 01: p2 Tt Nc01
Fig. 8a and b show the limit load surfaces of the two delaminated layers. In particular the following values are fixed: k¼
l02 ¼ 0; 35; l01
t l01
¼ 0; 02;
t ¼ 0; 01; T
ð1Þ
a0 ¼
4CB ¼ 0; 01: Nc01
p2 Tt
Finally, in order to show the results for laminates with outer layers of different thickness the following parameters have been used: t=l01 ¼ 0; 01; t=T ¼ ð1Þ 0; 01; a0 ¼ 0; 01; k ¼ 0; 5 and q ¼ t2 =t1 ¼ 0; 5 (in Fig. 9a) and q ¼ t2 =t1 ¼ 0; 9 (in Fig. 9b).
Fig. 8. (a) Limit surfaces g ¼ GIC =GIIC ¼ 1. (b) Limit surfaces g ¼ GIC =GIIC ¼ 0; 15.
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P. Lonetti, R. Zinno / Simulation Modelling Practice and Theory 11 (2003) 483–500
λ 16
5
14 4
12 1
3 2
3 4
1
(a)
2
2 5 5
5
5
2 4
1.5 3
1
2
2 2
3 4
(b)
1
5 1
Fig. 9. (a) Limit surfaces: K ¼ 0; 5, Q ¼ 0; 5. (b) Limit surfaces: K ¼ 0; 5, Q ¼ 0; 9.
References [1] C. Amarg Garg, Delamination––a damage mode in composite structures, Engineering Fracture Mechanics 29 (5) (1988) 557–584. [2] E. Altus, D. Ishai, Delamination buckling criterion for composite laminates: a macro approach, Engineering Fracture Mechanics 41 (5) (1992) 737–751. [3] D. Bruno, Delamination buckling in composite laminates with interlaminar defects, Theoretical and Applied Fracture Mechanics 9 (2) (1988) 145–179. [4] D. Bruno, G. Porco, Delaminazione in buckling di piastre laminate, in: Proceedings of the XXIII AIAS National Conference, Rende, Cosenza, Italy, 21–24 September 1994. [5] F. Greco, R. Zinno An analysis of multiple interlaminar crack propagation under mixed-mode deformations, in: Proceedings of XV National Conference of the IGF ItalianGroup of Fracture, Bari, Italy, 3–5 Maggio 2000. [6] P. Lonetti, R. Zinno, Numerical simulation of multiple delaminations in composite laminates under mixed-mode deformations, in: Proceedings of EUROSIM 2001 Congress on Shape the Future with Simulation, Delft, Netherlands, 2001. [7] G. Porco, R. Zinno, Modal interaction in the debonding problems in laminates, in Proceedings of the Fifth Conference of Unilateral problems in Structural Analysis, Ferrara, Italy, 12–14 June 1997.