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Towards a bridge between the micro- and mesomechanics of delamination for laminated composites
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 66 (2006) 698–712 www.elsevier.com/locate/compscitech
Towards a bridge between the micro- and mesomechanics of delamination for laminated composites P. Ladeve`ze *, G. Lubineau, D. Marsal LMT-Cachan (E.N.S. de Cachan/Universite´ Paris 6/C.N.R.S.), 61 Avenue du Pre´sident Wilson, 94235 Cachan Cedex, France Available online 19 January 2005
Abstract We present a relatively complete bridge between the descriptions on the micro- and mesoscales of damaged laminated composites. The description on the microscale derives from the numerous theoretical and experimental works carried out in micromechanics. On the mesoscale, the plies and the interfaces are homogenized to arrive at a continuum damage mechanics approach. While previous works dealt mainly with the in-plane behavior of the laminate, here we introduce the out-of-plane part of the behavior, which is essential for the simulation of delamination. This results in a better understanding, consistent with micromechanics, of the mesomechanics of laminated composites introduced at Cachan and developed particularly there for more than 15 years. 2005 Elsevier Ltd. All rights reserved. Keywords: Composite; C. Laminates; C. Delamination
1. Introduction The analysis of composite structures may require the construction of damage models capable of predicting the intensities of the damage mechanisms and their evolution until final fracture. In addition, these models should be applicable to industrial structures subjected to complex loading situations, in order to use simulations to replace the numerous tests used today for designing high-gradient zones. Different tentative approaches have been followed. Micromechanics models, defined on the fiberÕs scale, are linked directly to experimental observations. These have led to numerous theoretical as well as experimental works, most of which focus on the study of the matrix microcracking phenomenon [1–6]. These models, while providing a sound understanding of the damage mechanisms, are still far from capable of predicting the evolu*
Corresponding author. Tel.: +33 1 47 40 24 02/22 53; fax: +33 1 47 40 27 85. E-mail address: [email protected] (P. Ladeve`ze). 0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2004.12.026
tion of damage until complete fracture of the ply. Furthermore, taking into account matrix microcracking alone does not enable one to explain certain kinds of behavior, such as the significant loss of stiffness of shear-loaded laminates [7]. Our approach for laminated composites is to use what we call a damage mesomodel for laminates (DML) [8–10], whose main characteristic is the use of an intermediate scale, which we call the mesoscale, between the scale of the laminate (macro) and the scale of the mechanisms (micro). This mesoscale enables one to take into account the mechanisms of damage easily as well as to simulate structures of industrial complexity. However, the missing link to the microdescription of damage is a weakness. The mesomodel was recently reconsidered in the light of the works done in micromechanics, which enabled us to refine the description of the single layer under inplane macroloading in the cases of matrix microcracking [11,12] and local interlaminar delamination [13]. Our main conclusion was that the mesomodel can be interpreted as the homogenized result of micromodels. We
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
proved that the damage mesoquantities (damage variables and associated forces) can be viewed as homogenized quantities which are independent of the stacking sequence. Thus, we derived micro–meso relations for the damage variables and the forces. The aim of this paper is to go one step further by extending this approach to out-of-plane macroloading in order to deal with delamination. With out-of-plane loading, the link between the micro- and mesoscales requires the resolution of two basic problems. The first problem, which concerns the homogenized single layer, is an extension of the former 2D-ply problem to nonplane stresses. The second problem, which concerns the homogenized interface, is a new 3D problem. A fundamental link between the micro- and mesoscales exists for both problems: two mesoquantities (the plane part of the mesostrains and the out-of-plane part of the mesostress) can be interpreted as mean values of the corresponding microquantities [14]. The first major characteristic of out-of-plane homogenization is that the relationships between micro- and mesoquantities still have intrinsic properties. The outof-plane damage variables of the single layer and of the interface will be detailed and a numerical proof based on many virtual tests will be given. The second major characteristic is that the homogenization method enables us to quantify the influence of intralaminar
699
cracks on damage at the interface. It appears that when the loads are purely normal to the laminateÕs plane the microcracking within the plies has no influence on the interface. This is a first step towards a better understanding of the coupling between intra- and interlaminar damage.
2. The damage mechanisms on the microscale The current micro–meso relations are restricted to four mechanisms of degradation which are clearly identified on the microscale (Fig. 1). The first two mechanisms are usual and have been studied by the community of micromechanics for laminated composites in numerous works [15–18], etc., detailed reviews of micromechanics can be found in [5,19,20]. The last two mechanisms have been included in the DML for a long time, but not in micromechanics. • Scenario 1: matrix microcracking. The transverse cracks are assumed to span the entire thickness of the ply. Moreover, one assumes that the microcracking pattern is periodic, at least locally, which is consistent with most practical situations. Thus, the level of microcracking is quantified by a cracking rate q defined by: q1 ¼ HL .
Fig. 1. The mechanisms of degradation on the microscale.
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P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
• Scenario 2: local delamination. This scenario deals with the interfaceÕs degradation initiated at the tip of the transverse microcrack. This phenomenon, whose pattern is assumed to be periodic, is quantified at each transverse crack tip by a local delamination ratio s ¼ He . • Scenario 3: diffuse damage. This scenario is associated mainly with fiber–matrix debonding. This is a fundamental mechanism in order to explain the behavior of the ply, especially under shear loading. This debonding is assumed to be homogeneous within the ply. Consequently, the transverse cracks (Scenario 1) ~ which is differoccur in a damaged ply of stiffness K, ent from the initial stiffness K0 of the plyÕs healthy material. • Scenario 4: diffuse delamination. This scenario deals with a diffuse degradation of the interface (microvoids, microdecohesions) on a smaller scale than that of Scenario 2. This degradation is also assumed to be homogeneous within the interface. Thus, the local delamination cracks (Scenario 2) occur in a damaged interface of stiffness ~k, which is different from the initial stiffness k0 of the interfaceÕs healthy material. Remark 1. Contrary to usual micromechanics analysis, the interface is introduced on the microscale and, as on the mesoscale, is a surface entity whose stiffness is related to that of a thin layer of matrix. In most practical cases, damage is initiated by Scenarios 3 and 4. Then, the accumulation of fiber–matrix debonding instances followed by their coalescence leads to transverse microcracking (Scenario 1). The competition between transverse microcracking and local delamination ends with the saturation of Scenario 1 and is relayed by the catastrophic development of Scenario 2. Finally, this last scenario, just as Scenario 4, can lead to macroscopic interlaminar debonding due to coalescence under out-of-plane macroloading (for example, close to geometric accidents, such as holes). We hope that the present work will give the beginning of an answer to these difficult localization/rupture problems. In summary, the microdescription of the damage state introduces two types of internal damage microvar-
iables: the cracking rates of the plies q 2 [0.;0.7] and the delamination ratios of the interfaces s 2 [0.;0.4] associated with each transverse crackÕs tip. The evolution of these microvariables is governed by energy release rates in the framework of fracture mechanics or finite fracture mechanics.
3. The damage mesomodel for laminates The DML describes the behavior of the laminate on the mesoscale through two mesoconstituents which are continuous media, the single layer and the interface (Fig. 2). The interface is a surface entity representing the thin layer of matrix which exists between two adjacent plies and is characterized by their relative orientation. Its stiffness is equal to the shear modulus of the matrix divided by its thickness which is taken, by assumption, as the thickness of an elementary ply divided by 5. The damage state of each mesoconstituent is quantified by damage variables linked to the mesoconstituentÕs loss of stiffness. Traditionally, the evolution of these variables depends on damage forces which quantify the variation of the mesoconstituentÕs energy with respect to damage. The DML is based on two hypotheses. The first assumption is that once the behavior of the mesoconstituents has been identified the DML can predict the behavior of any stacking sequence subjected to any loading. Consequently, the damage evolution laws of the ply and of the interface must be valid regardless of the stacking sequence or the loading. The second assumption is that the damage variables are constant throughout the thickness of each single layer, but can change from one layer to the next. Remark 2. We will come back on the first assumption by following the obtained micro–meso relations, see Remark 6. The single-layer takes into account diffuse debonding of fibers/matrix interfaces, transverse cracking, local delamination and fiber breakage [9]. The interface, which transfers displacements and stresses from one ply to the next, takes into account interlaminar delami-
Fig. 2. Decomposition of a laminated composite into mesoconstituents.
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nation [21]. Preliminary identifications were carried out in previous works for the single layer [22,23] and for the interface [24,25]. Reviews of the latest works dealing with delamination can be found in [26,27].
4. Equivalence principle and fundamental micro–meso link 4.1. The micro–meso equivalence principle The objective is to build a continuum damage mechanics model which is quasi-equivalent, from an energy standpoint, to the damaged laminate micromodel defined in Section 2. Indeed, one of the aims of deriving micro–meso relations is to study the evolution of damage, and the main fracture criteria themselves are based on energy concepts. Consequently, in this multiscale approach, the potential energy stored in the plies and in the interfaces must be the same on the micro- and mesoscales, as illustrated in Fig. 3. The equivalent mesomodel is considered to be completely achieved if two basic situations of equivalence are established, one for the ply and one for the interface, leading to a homogenization procedure which will be detailed in Section 5 for the ply and Section 6 for the interface. The homogenization must be carried out for any stacking sequence defined by the thicknesses and orientations of the plies, but also for any kind of microdamage state. Practically one reduces the domain of investigation of microdamage state using q 2 [0.;0.7] and s 2 [0.;0.4]. From experiments, the material could be considered as fully damaged for higher level of degradations. 4.2. Fundamental relationships between the micro and meso solutions Let us consider the most general case of a damaged laminate subjected to uniform macroloading. The associated micromechanics problem Pm is illustrated in Fig. 4(a). Each ply and each interface may be affected
701
by the microphenomenology adopted in Section 2. The macroloading is described by the plane part of the strain pep and the out-of-plane part of the stress rN3. p is the projection operator associated with the mean plane (P) of the laminate and N3 is the normal vector of (P). The objective is to replace the discretely damaged medium by an equivalent continuous and homogeneous medium. Let us denote sm (sm = [Um,em,rm]) the exact displacement, strain and stress fields on the microscale under such loading. With the assumption of linear elasticity, it is possible to decompose sm using a crack closure method such as: sm ¼ ~s þ s;
ð1Þ
~ in which all where ~s is the solution of a first problem P cracks and local delaminations are removed from the laminate (the ‘‘effective solution’’). Consequently, for this problem, the laminate is a stack of homogeneous plies and interfaces (possibly homogeneously damaged according to Scenarios 3 and 4) which yields the following usual solution at any point M of the laminate: p~epðMÞ ¼ pep;
~N 3 ðMÞ ¼ rN 3 : r
ð2Þ
In order to ensure static admissibility in the cracked zones of problem Pm , ~s must be corrected by solving a where each cracked area, whose residual problem P, normal vector is denoted n, is loaded by the residual stress ~ rn. The corresponding solution is denoted s. Now, let us focus on the vicinity of an interface Cj between two plies Si and Si + 1. Due to the localization of the residual problems around each cracked area, one can consider that the restriction of the solution s to the vicinity of the interface Cj depends only on the states of damage of this interface and of the neighboring plies with good approximation. Finally, in the vicinity of Cj, a good approximation of s is ob tained by solving the approximate problem Pj Cj defined on the cell (Fig. 5) subjected to the residual stress on the cracked areas of [Cj;Si,Si + 1], all other cracked areas being removed.
Fig. 3. Energy equivalence between the micro and meso interpretations of damage.
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Fig. 4. The fundamental micro–meso link.
: approximate solution of s around an interface Cj. Fig. 5. Problem Pj Cj
Because the degradation pattern is periodic within Pj Pj Cj solutions of Pj [Cj;Si,Si + 1], the fields u Cj and r Cj verify periodicity and anti-periodicity boundary conditions, which leads to the following property: Pj
Pj
Property 1. hpe Cj pijCj ¼ 0 and h r Cj N 3 ijCj ¼ 0 8Cj , where hijCj denotes the mean value over Cj. . It is This property is verified exactly for problem Pj Cj not exactly true, but verified with very good approximation on the completely degraded geometry, which finally yields the following property:
Property 2. hpem pijCj #p~epjCj ¼ pep and hrm N 3 ijC # ~N 3 ¼ rN 3 8Cj . r Regarding this property, a strong link appears to exist between the micro and meso solutions. The plane part of the mesostrain and the out-of-plane part of the mesostress are mean values of the microfields. Thus, if one knows the solution on one scale, it is easy to obtain the solution on the other scale, which is the basis of an efficient multiscale method to study damage in laminates.
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
5. The basic ply problem 5.1. Traditional problem and potential energy of the equivalent ply Let us get back to the most general case of a damaged laminate (Fig. 4). Since the residual problems are localized around each ply, the first step consists in homogenizing the degradation associated with each ply separately. This simplified problem is a direct adaptation of the in-plane problem [11]. As illustrated in Fig. 6(a), only one ply S is degraded by Scenario 1 (microcracking rate q) and Scenario 2 (local delamination ratios ½s1 ; s2 ; s1þ ; s2þ ). The fibers of ply S are oriented along Direction 1. S 0 and S 0þ , whose fibers are oriented along h0 and h0þ , are the plies adjacent to ply S and represent the remaining structure. These plies are assumed to be homogeneous and possibly damaged according to Scenario 3. The procedure for skin plies, which is an extension of the in-plane procedure introduced in [12], is not detailed in this paper. As discussed in Section 4.2, the exact microsolution is split into an effective part ~s and a residual periodic corply rection sP , solution of the traditional basic ply problem ply P (Fig. 6(b)). The cell is subjected to periodicity boundary conditions corresponding to the periodicity
703
of damage in the ply S. The cracked area, whose normal rn. vector is denoted n, is loaded by the residual stress ~ Because of the linear elasticity assumption, the soluply tion sP can be written as the superposition of five elementary solutions (Fig. 7) associated with the five possible elementary loads on the cracked area. P r
ply
ply
ply
ply
ply
ply
P 12 þ r P 22 þ r P 13 þ r P 23 þ r P 33 : ¼r
ð3Þ
Result 1. Ply S can be homogenized and its potential energy can be expressed as: 2ESp t 2 ðI 22 ; I 12 Þ~ ~33 ½M ¼ ½p~ep ½M ep þ r r33 1 ðI 22 ; I 12 Þ½p~ jSj r223 3 ðI 22 ; I 12 Þ½p~ep ðI 23 þ 1Þ~ ~33 ½M þr ~ 23 G
2 r33 iþ ðI 13 þ 1Þ~ r213 I 33 h~ : ~ 13 ~3 E G
ð4Þ
2 and ½M 3 depend on the material 1 ; ½M Operators ½M properties of ply S. Consequently, on the microscale, the potential energy stored in ply S is completely determined through five damage indicators, I 22 ; I 12 ; I 23 ; I 13 and I 33 , which are
Fig. 6. Problem Pply: (a) geometry, (b) microdecomposition and (c) damaged mesoconstituents.
Fig. 7. Decomposition of problem Pply in terms of elementary loads.
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
7 6 5 4 3 2 1
-2 3
-1 3
33
33
-2 3 13
-3 3 12
-2 3 12
-1 3
12
-3 3 22
22
-2 3
0 -1 3
Remark 3. The derivation of expression (4) is ‘‘quasiexact’’ because it requires the approximation of negligible coupling effects among all the elementary residual problems of Fig. 7. This hypothesis is justified if the following relative energy eij kl between the diagonal solutions and the coupled solutions verifies: cijkl 2 eijkl ¼ pffiffiffiffiffiffiffiffiffiffi 1; 8ðij; klÞ 2 ½22; 12; 13; 23; 33 ; ij 6¼ kl; aij akl
8
-1 2
On the mesoscale, expression (4) completely defines the damaged homogeneous ply S (Fig. 6(c)), whose damage variables, associated with stiffness reductions, can be easily deduced from I 22 ; I 12 ; I 23 ; I 13 and I 33 .
-3
22
ð5Þ
9 x 10
Relative coupling term
defined as the integral of the strain energy for each basic residual problem. Denoting eij(M) the strain energy at any point M of the cell (X) in the basic ply problem P ply ij , one defines: Z Z ~ ~ I ii ¼ Ei I ij ¼ Gij e ðMÞ dS; eij ðMÞ dS: ii ~2ii S ~2ij S jSjr jSjr
22
704
considered coupling Fig. 8. Relative values of the coupling eij kl for the range of the 0 parameters: h0 2 ½90; 60; 45; 0; h0þ 2 ½90; 45; 45; 60; HH ¼ 2; 1 1 2 1 2 ¼ 0:5; s ¼ s ¼ s ¼ s ¼ 0:5. þ þ q
ð6Þ where Z ply ply aij ¼ Tr½ rP ij eP ij dS;
cijkl ¼
S
Z
ply
ply
Tr½ rP ij eP kl dS: S
ð7Þ A numerical proof of this approximation is given in Fig. 8, where this term is plotted for numerous geometric configurations of the stacking sequence. 5.2. The extended ply problem As previously stated, the traditional basic ply problem defines only the state of damage of ply S. This procedure
gives excellent results for in-plane stresses because the energy is really confined to the ply S being considered for such stresses. In the case of out-of-plane stresses, however, the damage state of ply S induces some nonnegligible energy in plies S 0 and S 0þ . Consequently, Pply defines the part of states of damage of S 0 and S 0þ on the mesoscale due to the microdegradations in S. In other words, under out-of-plane macroloading, the damage state of the ply being considered depends on the state of damage of the adjacent plies. To handle these considerations, one defines an extended ply problem, illustrated in Fig. 9. A detailed example is proposed. Let us consider a stacking sequence with only two identically damaged plies S and S 0þ whose fiber
Fig. 9. Extended Pply: (a) the traditional ply problem, (b) S 0þ Õs lower local delamination, (c) S 0þ Õs lower local delamination and transverse cracks, (d) S 0þ Õs upper and lower local delamination and transverse cracks.
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
• with the traditional ply problem [a] alone, • with the traditional ply problem [a] plus the contribution [b] from the lower local delamination of S 0þ , both problems considered to be uncoupled, • with the traditional ply problem [a] plus the contribution [c] from the lower local delamination and transverse cracks of S 0þ , both problems being considered uncoupled, • with the traditional ply problem [a] plus the contribution [d] from the lower and upper local delamination and transverse cracks of S 0þ , both problems considered to be uncoupled. The cell is subjected to a uniform residual stress loading 13 (in the basis of ply S) and the results, Fig. 10, show the significant impact of the whole cracked area of S 0þ on the energy stored in ply S and, consequently, on its damage state, which the extended ply problem enables one to simulate. The discrepancy between the 3D reference and the [a] + [d] solution comes from the uncoupled treatment of problems [a] and [b]. The results obtained for a uniform residual stress loading 33, Fig. 11, enable one to identify the negligible influence of the upper local delamination and transverse cracks of S 0þ on the damage state of ply S under 33 solicitation. In other words, the damage mesostate of a ply does not depend on the microcracking of the adjacent plies ~33 stress loading. The discrepancy between the 3D under r reference and the [a] + [d] solution still comes from the uncoupled treatment of problems [a] and [b] and a more
(Strain energy density) / (micro reference)
θ’−= -45 , θ’+=+45 , ρ|S=ρ|S’ =0.5 , τ|S=τ|S’ =0.3 +
0.8
θ’ = -45 , θ’ =+45 , ρ| =ρ| −
+
S
S’
=0.5 , τ| =τ|
+
S
S’
=0.3
+
1
0.8
0.6
0.4
0.2
0 0
Fig. 11. Extended problem Pply: influence of damage in ply S Õs on the energy of ply S under 33 loading.
accurate procedure may be required; nevertheless, the previously underlined point is not questioned. 5.3. Micro–meso relations for the ply The results of this section are associated with the traditional basic ply problem; the treatment of the extended ply problem is in progress. Result 2. The relations ðI 22 ; I 12 ; I 23 ; I 13 ; I 33 Þ $ ðq; s1 ; s2 ; s1þ ; s2þ Þ are approximately ply-material relations, i.e., the state of damage on the mesoscale is approximately independent of the configurations of the peripheral parts S 0 and S 0þ . An illustration of this result out-of-plane indicators. Instead tors I 13 ; I 23 and I 33 , we chose D13, D23 and D33 defined by: I 13 I 23 D13 ¼ ; D23 ¼ ; 1 þ I 13 1 þ I 23
is given in Fig. 12 for of the damage indicato plot the quantities
D33 ¼
I 33 1 þ I 33
ð8Þ
which are directly associated with the stiffness reductions commonly used in damage mechanics. Fig. 12(a) shows D13, D23 and D33 calculated for several states of microdegradation and for numerous geometric configurations (h0 2 ½90; 60; 45; h0þ 2 ½90; 45; 45; 60, 0 H 2 [helem;2helem;4helem], H 2 [helem;2helem;4helem], where helem is the thickness of the elementary material ply). It appears that for a given state of microdegradation, quantified by dimensionless microvariables ðq; s1 ; s2 ; s1þ ; s2þ Þ, the equivalent state of damage on the mesoscale is approximately intrinsic. This property is verified even better if one considers stacking sequences 0 with (H P H) (Fig. 12(b)).
1.2
1
micro a a+b a+c a+d
1.2
(Strain energy density) / (micro reference)
directions are orthogonal. The local delamination ratios of both damaged plies verify sjS ¼ s1 jS ; s2 jS ; s1þ jS ; s2þ jS and sjS 0 ¼ s1 jS 0 ; s2 jS 0 ; s1þ jS 0 ; s2þ jS 0 . The strain energy density in S is first computed taking into account the complete 3D microdescription, which constitutes the energy microreference. Then, four calculations of the strain energy density in S are carried out:
705
+
micro a a+b a+c a+d
0.6
0.4
0.2
0 0
Fig. 10. Extended problem Pply: influence of damage in ply S on the energy of ply S under Loading 13 in the basis of ply S.
Remark 4. The out-of-plane damage indicators are independent of the initial level of diffuse damage within ply S or its adjacent interfaces. Fig. 13 shows the values of D33, D13 and D23 in the cases of initially healthy plies
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
706
0.5 0 0.5 0.4
0.3 0.2
1 2 τ+=τ+
0.1
0
0
0.5 0.3 0.4 0.1 0.2
τ1=τ2 -
1 0.5 0 0.5 0.4
0.3 0.2
τ1=τ2 +
0.1
0
0
0.5 0.3 0.4 0.1 0.2
1
1 0.5 0 0.5 0.4
0.3 0.2
τ1+=τ2+
0.1
0
0
0.5 0.3 0.4 0.1 0.2
τ1=τ2 -
1 0.5 0 0.5 0.4
-
0.3 0.2
1 2 τ+=τ+
0.1
0
0
0.5 0.3 0.4 0.1 0.2
τ1=τ2 -
-
1 0.5 0 0.5 0.4
0.3 0.2
τ1=τ2
2
τ- =τ-
+
(B)-Damage D13
1
(B)-Damage D33
(b)
(B)-Damage D23
(A)-Damage D23
(A)-Damage D13
(A)-Damage D33
(a)
+
0.1
0
0
0.5 0.3 0.4 0.1 0.2
0
0
0.5 0.3 0.4 0.1 0.2
+
1
2
τ- =τ-
1 0.5 0 0.5 0.4
0.3 0.2
τ1+=τ2+
0.1
τ1=τ2 -
-
Fig. 12. D13, D23 and D33 vs. the microstate of degradation and geometric configuration of the stacking sequence.
and interfaces, initially damaged ply (50% diffuse damage) and initially damaged interface (50% diffuse damage).
1
Healthy ply and interface Ply with diffuse damage Interface with diffuse damage
0.9 0.8
Damage level
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
D33
D13
D23
Fig. 13. Evolution of damage indicators D13, D23 and D33 with the state of diffuse damage of the mesoconstituents (microdegradation state: q1 ¼ 0:5; s1 ¼ s2 ¼ s1þ ¼ s2þ ¼ 0:3).
6. The basic interface problem 6.1. Potential energy of the equivalent interface Let us again consider the most general case of a damaged laminate subjected to uniform macroloading (Fig. 4). In order to homogenize the microdegradations associated with an interface Cj between two damaged plies Si and Si + 1 whose fiber directions are, respectively, Ni and Ni + 1, a good approximation of the solution in the vicinity of this interface can be obtained by considering the simplified 3D structure of Fig. 14. The angle between Ni and Ni + 1 is denoted h. One can introduce a local reference frame (N1,N2,N3) of orthotropic directions for the healthy interface; N3 is normal to the interface while N1 and N2 are the bisector directions associated with angle h. The external plies S 00 and S 00þ representing the remaining structure are homogenized and possibly damaged according to Scenario 3. With the exception of transverse cracks, each pair of vertical faces of the 3D cell must satisfy periodic conditions corresponding to Scenarios 1 and 2. The description of the state of microdegradation of the cell requires the microcracking rates and the local delamination ratios of plies Si and Si + 1: ðq; s1 ; s2 ; s1þ ; s2þ ÞjS i ; ðq; s1 ; s2 ; s1þ ; s2þ ÞjS iþ1 . The geometric parame-
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707
Fig. 14. The basic interface problem Pint.
ters of the interface are the angle h and the dimensionless quantities ðHH ji ; HHiþ1j Þ which quantify the unit thickness of the interface. The homogenization method is similar to that of the ply: the solution is written as an effective part ~s plus a int periodic correction sP using the crack closure method. The difficulty concerns this residual part, whose associated problem is called the basic interface problem Pint, in which the cracked area, whose normal vector is denoted n, is loaded by the residual stresses ~ rn. According to the linear elasticity assumption, the residual stresses are decomposed into three elementary solicitations in the orthotropic basis of the interface. Consequently, int P as the sum of one can express the residual solution r the three elementary solutions: P r
int
int
int
int
P 13 þ r P 23 þ r P 33 : ¼r
ð9Þ
Result 3. The interface Cj can be homogenized and its potential energy expressed as: 2Ejp ð1 þ I 1 Þ~ r213 ð1 þ I 2 Þ~ r223 ð1 þ I 3 Þ~ r233 ¼ ; ~k 1 ~k 2 ~k 3 jCj j
ð10Þ
where ~k 1 ; ~k 2 and ~k 3 are the elastic stiffness coefficients of interface Cj (possibly including initial diffuse damage). The potential energy is completely defined by three damage indicators I 1 ; I 2 and I 3 , which are defined as the integral of the strain energy over the interface for each elementary problem:
I i ¼
~k i Z int int Tr½ rP i3 eP i3 dC; 2 jCj j~ ri3 C
8i 2 ½1; 2; 3:
The derivation of Eq. (10) implies that the coupling effects between the residual problems are negligible. This approximation is justified if the following condition is verified: cpq ð12Þ epq ¼ pffiffiffiffiffiffiffiffiffi 1; 8ðp; qÞ 2 ½1; 2; 32 ; p 6¼ q; ap aq where Z int int ap ¼ Tr½ rP p3 eP p3 dC; C
cpq ¼
Z
int
int
Tr½ rP p3 eP q3 dC:
ð13Þ
C
In order to illustrate this condition, let us consider the damaged cell (Fig. 14) under the four states of degradation defined by Table 1. The local delamination ratios of both damaged plies verify sjS i ¼ s1 jS i ; s2 jS i ; s1þ jS i ; s2þ jS i and sjS iþ1 ¼ s1 jS iþ1 ; s2 jS iþ1 ; s1þ jS iþ1 ; s2þ jS iþ1 . The orientations of the fibers refer to direction N1 of the interface and the degradation of ply Si can take four configurations. First, this cell is loaded by a residual stress composed ~13 ; r ~23 and r ~33 applied of the three elementary loads r simultaneously. The strain energy stored in the interface under this solicitation is taken as the energy reference. Then, one solves six complementary problems on the same cell by considering each elementary loading and the corresponding coupling effects separately. Finally, the energy ratio of each complementary problem is defined by the ratio of the corresponding strain energy of the interface to the energy reference. The results are
Table 1 The four states of microdegradation associated with Fig. 15 Ply
Microcracking rate q
Local delamination ratio s
Orientation
Thickness
S 00þ
0. 0.4 0.4 or 0.5 0.0
0. 0.15 0.15 or 0.3 0.
30 +30 30 +30
1helem 1helem 1helem 1helem
Si + 1 Si S 00
ð11Þ
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708
ρ|S =0.40 τ|S =0.15 e12=3.5E 04
ρ|S =0.40 τ|S =0.30 e12=0.09
i
50
i
Energy ratio (%)
Energy ratio (%)
i
40 30 20 10 0
23 − 13 − 33 23 − 13 − 33 ρ|S =0.50 τ|S =0.15 e12=0.12 i
50 40 30 20 10 0
23 − 13 − 33
23 − 13 − 33
50 40 30 20 10 0
23 − 13 − 33 23 − 13 − 33 ρ|S =0.50 τ|S =0.30 e12=0.33 i
Energy ratio (%)
Energy ratio (%)
i
i
i
50 40 30 20 10 0
23 − 13 − 33
23 − 13 − 33
Fig. 15. Distribution of energy among the diagonal and coupling terms.
given in Fig. 15, which shows the distribution of the strain energy among the diagonal and coupling terms. The values of e12 are specified in the four cases. In the particular case of an identical microdegradation state in plies (Si) and (Si + 1), Condition (12) is verified, but as soon as the states of microdegradation in the plies differ the symmetry of the interface is broken and coupling between Residuals 23 and 13 appears in the behavior of the damaged interface. Nevertheless, this coupling never becomes the prominent factor in our domain of investigation and it will be neglected for this paper. The consequences of this approximation are currently being studied.
The potential energy equivalence principle between the micro- and mesoscales enables one to build the operators linking the damage indicators of the interface ðI 1 ; I 2 ; I 3 Þ to the microdegradation variables of the cell ðq; s1 ; s2 ; s1þ ; s2þ ÞjS i and ðq; s1 ; s2 ; s1þ ; s2þ ÞjS iþ1 numerically . To verify the consistency of these operators with the hypotheses of the mesomodel, one must evaluate their intrinsic properties with respect to the interface. For convenience, the results are presented in terms of losses of stiffness, which are linked to the damage indicators by the following relation: I i Di ¼ : ð14Þ 1 þ I i
6.2. Micro–meso relations for the interface
Let us consider an interface defined by Table 2. Once again, the local delamination ratios of both damaged plies verify sjS i ¼ s1 jS i ; s2 jS i ; s1þ jS i ; s2þ jS i and sjS iþ1 ¼ s1 jS iþ1 ; s2 jS iþ1 ; s1þ jS iþ1 ; s2þ jS iþ1 and the orientations of the fibers refer to the direction N1 of the interface. By taking several configurations of external plies S 00þ and S 00 for different orientations ðh00 ; h00þ Þ (Fig. 16)
Result 4. The relations ðI 1 ; I 2 ; I 3 Þ $ ððq; s1 ; s2 ; s1þ ; s2þ Þ jS i ; ðq; s1 ; s2 ; s1þ ; s2þ ÞjS iþ1 Þ are, with a very good approximation, ply-material properties, i.e. the micro–meso relations of the interface are independent of the stacking sequence of the laminate.
Table 2 States of microdegradation associated with Figs. 16 and 17 Ply S 00þ Si + 1 Si S 00
Microcracking rate q 0.0 0.5 0.35 0.0
Local delamination ratio s
Orientation
Thickness
0.0 0.25 0.15 0.0
h00þ
H 00þ 1helem 1helem H 00
22.5 22.5 h00
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
D
D3
0.7
1
0.7
D2
0.7
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0 0
50
100
150
θ’’
0
50
100
0
150
0
θ’’
50
100
150
θ’’
−
+
709
0
50
100
150
0
θ’’
100
150
θ’’
−
+
50
0
50
100
150
θ’’
−
+
Fig. 16. Influence of the orientations of the external plies.
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
D
D3
0.7
1
0.7
D2
0.7
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0 0
0.5 0.5 0
H’’+
H’’
0 0
0.5 0.5 0
H’’+
0
H’’
Fig. 17. Influence of the thicknesses of the external plies.
0.5 0.5 0
H’’+
H’’
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710
0.7
0.4
healthy ply and interface interface loss of stiffness 50% ply loss of stiffness 50%
All cracks taken into account without transverse microcracking and neighboring delaminations 0.35
ρ|
S
=0.4 , τ|
i+1
S
0.6
=0.15 , θ=60
ρ| =0.5 , ρ|
i+1
S
0.3
Level of mesodamage
Mesodamage D
3
0.5
0.25
0.2
0.15
i
S
=0.4 , τ| =0.25 , τ|
i+1
S
S
i
=0.2 , θ=60
i+1
0.4
0.3
0.2
0.1
0.1
0.05
0
0
(ρ|S ,τ|S ) : (0.4,0.15) – (0.5,0.15) – (0.5,0.225) – (0.5,0.3) i
Mesodamage : D2– D 1– D
i
3
Fig. 18. Influence of intralaminar cracks on delamination under r ~33 stress loading.
Fig. 19. Influence of Scenarios 3 and 4.
and thicknesses ðH 00 ; H 00þ Þ (Fig. 17), one can note that the intrinsic properties of the three micro–meso operators are verified with very good accuracy (65%).
• with healthy materials, • with plies affected by Scenario 3 leading to a 50% loss of stiffness, • with interfaces affected by Scenario 4 leading to a 50% loss of stiffness.
Remark 5. The intralaminar cracks have no influence ~33 stress loading. The local on the delamination under r delaminations ðs1þ jS i ; s2þ jS i ; s1 jS iþ1 ; s2 jS iþ1 Þ of the interfaces adjacent to an interface Cj are called ‘‘neighboring delaminations’’. The identification of the operator associated with Damage 33, illustrated by Fig. 18, was carried out with and without transverse cracking and ‘‘neighboring delaminations’’ during a physically reasonable evolution of the degradation state of ply Si. The local delamination ratios of the two damaged plies verified sjS i ¼ s1 jS i ; s2 jS i ; s1þ jS i ; s2þ jS i and sjS iþ1 ¼ s1 jS iþ1 ; s2 jS iþ1 ; s1þ jS iþ1 ; s2þ jS iþ1 . It appears that intralaminar cracks and neighboring delaminations have a negligible influence on ~33 stress loading. Conversely, interface damage under r these cracks play a major role in interface damage under ~13 and r ~23 stress loading, which makes us hope that a r more specific link between local delamination and transverse cracking will be derived in the near future. Remark 6. The damage state of the interface could depend on the state of the adjacent layers which contradicts the first main hypothesis of our standard mesomodel, (see Remark 2). However, it is relatively easy to extend it through a non-local mesomodel, which will be done in a companion paper. Remark 7. Our method enables one to study the influence of Scenarios 3 and 4 on the micro–meso relations of the interface. Let us consider a particular stacking sequence (h = 60, Hi = Hi + 1 = helem) and a combination of Scenarios 1 and 2 in plies Si and Si + 1 described by ðqjS i ¼ 0:5; sjS i ¼ 0:25; qjS iþ1 ¼ 0:4; sjS iþ1 ¼ 0:2Þ. Fig. 19 shows the three homogenized damage states corresponding to the three cases:
Despite the catastrophic level of diffuse damage, the changes of homogenized mesodamage are relatively insignificant. For usual levels of diffuse damage 3 and 4, the influence of such damage on the homogenized damage of the interface is negligible. 7. Outlook Up to now, we carried out the homogenization procedures for any type of microdamage state defined by microcracking rates q 2 [0.;0.7] and delamination ratios s 2 [0.;0.4], and for any stacking sequence defined by the thicknesses and orientations of the plies. In order to homogenize a ply, the adjacent plies must be at least as thick as the ply being considered. This is the only condition on the stacking sequence required to obtain sufficiently accurate intrinsic properties. One can use the results of such homogenization and the ‘‘virtual testing’’ procedure for modeling. This opens new possibilities for the choice of internal variables: the description of damage can be fully meso or fully micro, but many hybrid micro–meso descriptions are also allowable (Table 3). Then, the thermodynamic forces, constructed according to the description chosen, lead to equivalent constitutive laws. In particular, these laws can still be written in terms of traditional mesodamage variables and associated mesodamage forces. A typical treatment would be the following: Scenarios 3 and 4 initiate damage according to:
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711
Table 3 Allowable choices of internal variables Description
Ply (i)
Hybrid ply and interface Meso ply and interface Hybrid interface
~ i; D ~ i; D ~ i; D
Interface (j)
~ 0i ; qi D ~ 0 ; Di ; D0 D i i ~ 0 ; Di ; D0 D i i
~ 1; D ~ 1; D ~ 1; D
~ 2; D ~ 2; D ~ 2; D
~ 3 ; s1þ ; s2þ ; s1 ; s2 D ~ 3 ; D1 ; D2 ; D 3 D ~ 3 ; s1 ; s2 ; s1 ; s2 D þ þ
Table 4 Unilateral effect: behavior on the mesoscale Ply (i)
rmeso 22 rmeso 22 rmeso 33 rmeso 33
Interface (j)
>0 rmeso 33 rmeso 60 33
>0 60 >0 60
Open transverse cracks Closed transverse cracks (no microcracks) Open delamination cracks Closed delamination cracks (no delamination cracks) Open delamination cracks Closed delamination cracks
h
i ~_ 22 ; D~_ 0 12 ji;t ¼ f ð½Y~ 22 ; Y~ 0 ji;t ; s 6 tÞ; D 12
ð15Þ
h
i ~_ 1 ; D ~_ 1 ; D ~_ 2 ; D ~_ 3 ji;t ¼ gð½Y~ 1 ; Y~ 0 ; Y~ 0 ji;t ; s 6 tÞ; D 2 3
ð16Þ
where f and g are functions of the thermodynamic forces and of the history of the mesoconstituents. These functions can be defined by phenomenological or micromechanical results, whichever are available. Then, the percolation of fiber–matrix debonding leads to Scenario 1, whose evolution is governed by Gqi 6 Gqc (where Gqi is the energy release rate associated with the birth of a new microcrack in the finite fracture mechanics sense, and Gqc is the critical energy release rate, a property of the material). Local delamination is considered inactive until the saturation of Scenario 1. Then, the level of Scenario 2 increases with respect to Gsi 6 Gsc (where Gsi is the energy release rate associated with a small propagation of local delamination, and Gsc the corresponding critical rate, a property of the material). Once this last scenario has been initiated, it propagates catastrophically and leads to macroscopic delamination, which can be described only from a macro point of view. Remark 8. Microcracks and local delamination cracks can be open or closed, which leads to different types of behavior (Ladeve`ze, 02 [28]). A simple way to model these phenomena is to make the mesobehavior dependent on the sign of the stress, according to Table 4.
8. Conclusion A virtual testing approach for the derivation of complete relationships between micro- and mesomechanics under out-of-plane loading has been presented. Consequently, the damage mesomodel is now being reconsidered as a systematic homogenization of traditional micromechanics models. Pragmatic rules will be given
in a companion paper. Moreover, it is possible to describe the actual microdamage mechanisms and their intensities using mesoquantities. Some interesting properties have been highlighted; for example, we showed that microcracking does not affect the behavior of the interface in pure 33 stress loading. There is work in progress to deal with the coupling of damage and (visco)plasticity, which was not considered here.
References [1] Flaggs D, Kural M. Experimental determination of the in situ transverse lamina strength in graphite/epoxy laminates. J Compos Mater 1982;16:103–15. [2] Hashin Z. Analysis of cracked laminates: a variational approach. Mech Mater 1985;4:121–36. [3] Boniface L, Smith P, Ogin S, Bader M. Observations on tranverse ply crack growth in a [0/902]s CFRP laminate under monotonic and cyclic loading. In: Proceedings of the 6th international conference on composite materials, vol. 3; 1987. p. 156–65. [4] Tan S, Nuismer R. A theory for progressive matrix cracking in composites laminates. J Comput Mater 1989;23:1029–47. [5] Nairn J, Hu S. Matrix microcracking. In: Taljera, editor. Damage mechanics of composite materials, Elsevier; 1994. p. 187–243. [6] Joffe R, Varna J. Analytical modeling of stiffness reduction in symmetric and balanced laminates due to cracks in 90 layer. Compos Sci Technol 1999;59:1641–52. [7] Lagattu F, Lafarie-Fre´not M. Variation of PEEK matrix crystallinity in APC-2 composite subjected to large shearing deformation. Compos Sci Technol 2000;60:605–12. [8] Ladeve`ze P. About the damage mechanic of composites. In: Bathias C, Menke`s D, editors. Comptes-rendus des JNC5. Paris: Pluralis Publication; 1986. p. 667–83 [in French]. [9] Ladeve`ze P. About a damage mechanics approach. In: Baptiste D, editor. Mechanics and mechanisms of damage in composite and multimaterials. MEP; 1989. p. 119–42. [10] Herakovich C. Mechanics of fibrous composites. New York: Wiley; 1998. [11] Ladeve`ze P, Lubineau G. On a damage mesomodel for laminates: micro–meso relationships, possibilities and limits. Compos Sci Technol 2001;61(15):2149–58. [12] Ladeve`ze P, Lubineau G. On a damage mesomodel for laminates: micromechanics basis and improvement. Mech Mater 2002;35: 763–75.
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[13] Ladeve`ze P, Lubineau G. An enhanced mesomodel for laminates based on micromechanics. Compos Sci Technol 2002;62: 533–41. [14] Ladeve`ze P, Lubineau G. Bridge between the ‘‘micro’’ and ‘‘meso’’ mechanics of laminated composites. CR Me´canique 2003;331:537–44 [in French]. [15] Garret K, Bailey J. Multiple transverse fracture in 909 cross-ply laminates of glass-fibre reinforced polyester. J Mater Sci 1977;12:157–68. [16] Crossman F, Wang A. The dependence of transverse cracking and delamination on ply thickness in graphite/epoxy laminates. In: Reifsnider K, editor. Damage in composite materials, ASTM-STP 775; 1982. p. 118–39. [17] Hashin Z. Analysis of cracked laminates: a variational approach. Mech Mater 1985;4:121–36. [18] Berthelot J, Corre JL. A model for transverse cracking and delamination in cross-ply laminates. Compos Sci Technol 2000;60: 1055–66. [19] Nairn J. Matrix microcracking in composites. In: Taljera-Manson, editor. Polymer matrix composites, comprehensive composite materials. Amsterdam: Elsevier; 2000. p. 403–32. [Chapter 13]. [20] Berthelot J. Transverse cracking and delamination in cross-ply laminates under static and fatigue loadings. Appl Mech Rev 2003;56:111–47. [21] Allix O, Ladeve`ze P. Interlaminar interface modelling for the prediction of laminate delamination. Compos Struct 1992;22: 235–42.
[22] Ladeve`ze P, LeDantec E. Damage modeling of the elementary ply for laminated composites. Compos Sci Technol 1992;43(3): 257–67. [23] Allix O, Bahlouli N, Cluzel C, Perret L. Modelling and identification of temperature-dependent mechanical behaviour of the elementary ply in carbon/epoxy laminates. Compos Sci Technol 1996;56:883–8. [24] Allix O, Le´veˆque D, Perret L. Identification dÕun mode`le dÕinterface interlaminaire pour la pre´vision du de´lamiange dans les composites stratifie´s. In: Baptiste D, Vautrin A, editors. Comptes Rendus des Dixie`mes Journe´es Nationales sur les Composites AMAC; 1996. p. 1041–52. [25] Allix O, Le´ve`que D, Perret L. Interlaminar interface model identification and forecast of delamination in composite laminates. Compos Sci Technol 1998;56:671–8. [26] Allix O. Interface damage mechanics: application to delamination. In: Allix O, Hild F, editors. Continuum damage mechanics of materials and structures. Amsterdam: Elsevier; 2002. p. 295–324. [27] De Borst R, Schipperen J. Computational methods for delamination and fracture in composites. In: Allix O, Hild F, editors. Continuum damage mechanics of materials and structures. Amsterdam: Elsevier; 2002. p. 325–52. [28] Ladeve`ze P. An anisotropic damage theory and unilateral effects: application to laminates, to three and four directional composites. In: Allix O, Hild F, editors. Continuum damage mechanics of materials and structures. Amsterdam: Elsevier; 2002. p. 205–33.
Towards a bridge between the micro- and mesomechanics of delamination for laminated composites P. Ladeve`ze *, G. Lubineau, D. Marsal LMT-Cachan (E.N.S. de Cachan/Universite´ Paris 6/C.N.R.S.), 61 Avenue du Pre´sident Wilson, 94235 Cachan Cedex, France Available online 19 January 2005
Abstract We present a relatively complete bridge between the descriptions on the micro- and mesoscales of damaged laminated composites. The description on the microscale derives from the numerous theoretical and experimental works carried out in micromechanics. On the mesoscale, the plies and the interfaces are homogenized to arrive at a continuum damage mechanics approach. While previous works dealt mainly with the in-plane behavior of the laminate, here we introduce the out-of-plane part of the behavior, which is essential for the simulation of delamination. This results in a better understanding, consistent with micromechanics, of the mesomechanics of laminated composites introduced at Cachan and developed particularly there for more than 15 years. 2005 Elsevier Ltd. All rights reserved. Keywords: Composite; C. Laminates; C. Delamination
1. Introduction The analysis of composite structures may require the construction of damage models capable of predicting the intensities of the damage mechanisms and their evolution until final fracture. In addition, these models should be applicable to industrial structures subjected to complex loading situations, in order to use simulations to replace the numerous tests used today for designing high-gradient zones. Different tentative approaches have been followed. Micromechanics models, defined on the fiberÕs scale, are linked directly to experimental observations. These have led to numerous theoretical as well as experimental works, most of which focus on the study of the matrix microcracking phenomenon [1–6]. These models, while providing a sound understanding of the damage mechanisms, are still far from capable of predicting the evolu*
Corresponding author. Tel.: +33 1 47 40 24 02/22 53; fax: +33 1 47 40 27 85. E-mail address: [email protected] (P. Ladeve`ze). 0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2004.12.026
tion of damage until complete fracture of the ply. Furthermore, taking into account matrix microcracking alone does not enable one to explain certain kinds of behavior, such as the significant loss of stiffness of shear-loaded laminates [7]. Our approach for laminated composites is to use what we call a damage mesomodel for laminates (DML) [8–10], whose main characteristic is the use of an intermediate scale, which we call the mesoscale, between the scale of the laminate (macro) and the scale of the mechanisms (micro). This mesoscale enables one to take into account the mechanisms of damage easily as well as to simulate structures of industrial complexity. However, the missing link to the microdescription of damage is a weakness. The mesomodel was recently reconsidered in the light of the works done in micromechanics, which enabled us to refine the description of the single layer under inplane macroloading in the cases of matrix microcracking [11,12] and local interlaminar delamination [13]. Our main conclusion was that the mesomodel can be interpreted as the homogenized result of micromodels. We
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
proved that the damage mesoquantities (damage variables and associated forces) can be viewed as homogenized quantities which are independent of the stacking sequence. Thus, we derived micro–meso relations for the damage variables and the forces. The aim of this paper is to go one step further by extending this approach to out-of-plane macroloading in order to deal with delamination. With out-of-plane loading, the link between the micro- and mesoscales requires the resolution of two basic problems. The first problem, which concerns the homogenized single layer, is an extension of the former 2D-ply problem to nonplane stresses. The second problem, which concerns the homogenized interface, is a new 3D problem. A fundamental link between the micro- and mesoscales exists for both problems: two mesoquantities (the plane part of the mesostrains and the out-of-plane part of the mesostress) can be interpreted as mean values of the corresponding microquantities [14]. The first major characteristic of out-of-plane homogenization is that the relationships between micro- and mesoquantities still have intrinsic properties. The outof-plane damage variables of the single layer and of the interface will be detailed and a numerical proof based on many virtual tests will be given. The second major characteristic is that the homogenization method enables us to quantify the influence of intralaminar
699
cracks on damage at the interface. It appears that when the loads are purely normal to the laminateÕs plane the microcracking within the plies has no influence on the interface. This is a first step towards a better understanding of the coupling between intra- and interlaminar damage.
2. The damage mechanisms on the microscale The current micro–meso relations are restricted to four mechanisms of degradation which are clearly identified on the microscale (Fig. 1). The first two mechanisms are usual and have been studied by the community of micromechanics for laminated composites in numerous works [15–18], etc., detailed reviews of micromechanics can be found in [5,19,20]. The last two mechanisms have been included in the DML for a long time, but not in micromechanics. • Scenario 1: matrix microcracking. The transverse cracks are assumed to span the entire thickness of the ply. Moreover, one assumes that the microcracking pattern is periodic, at least locally, which is consistent with most practical situations. Thus, the level of microcracking is quantified by a cracking rate q defined by: q1 ¼ HL .
Fig. 1. The mechanisms of degradation on the microscale.
700
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• Scenario 2: local delamination. This scenario deals with the interfaceÕs degradation initiated at the tip of the transverse microcrack. This phenomenon, whose pattern is assumed to be periodic, is quantified at each transverse crack tip by a local delamination ratio s ¼ He . • Scenario 3: diffuse damage. This scenario is associated mainly with fiber–matrix debonding. This is a fundamental mechanism in order to explain the behavior of the ply, especially under shear loading. This debonding is assumed to be homogeneous within the ply. Consequently, the transverse cracks (Scenario 1) ~ which is differoccur in a damaged ply of stiffness K, ent from the initial stiffness K0 of the plyÕs healthy material. • Scenario 4: diffuse delamination. This scenario deals with a diffuse degradation of the interface (microvoids, microdecohesions) on a smaller scale than that of Scenario 2. This degradation is also assumed to be homogeneous within the interface. Thus, the local delamination cracks (Scenario 2) occur in a damaged interface of stiffness ~k, which is different from the initial stiffness k0 of the interfaceÕs healthy material. Remark 1. Contrary to usual micromechanics analysis, the interface is introduced on the microscale and, as on the mesoscale, is a surface entity whose stiffness is related to that of a thin layer of matrix. In most practical cases, damage is initiated by Scenarios 3 and 4. Then, the accumulation of fiber–matrix debonding instances followed by their coalescence leads to transverse microcracking (Scenario 1). The competition between transverse microcracking and local delamination ends with the saturation of Scenario 1 and is relayed by the catastrophic development of Scenario 2. Finally, this last scenario, just as Scenario 4, can lead to macroscopic interlaminar debonding due to coalescence under out-of-plane macroloading (for example, close to geometric accidents, such as holes). We hope that the present work will give the beginning of an answer to these difficult localization/rupture problems. In summary, the microdescription of the damage state introduces two types of internal damage microvar-
iables: the cracking rates of the plies q 2 [0.;0.7] and the delamination ratios of the interfaces s 2 [0.;0.4] associated with each transverse crackÕs tip. The evolution of these microvariables is governed by energy release rates in the framework of fracture mechanics or finite fracture mechanics.
3. The damage mesomodel for laminates The DML describes the behavior of the laminate on the mesoscale through two mesoconstituents which are continuous media, the single layer and the interface (Fig. 2). The interface is a surface entity representing the thin layer of matrix which exists between two adjacent plies and is characterized by their relative orientation. Its stiffness is equal to the shear modulus of the matrix divided by its thickness which is taken, by assumption, as the thickness of an elementary ply divided by 5. The damage state of each mesoconstituent is quantified by damage variables linked to the mesoconstituentÕs loss of stiffness. Traditionally, the evolution of these variables depends on damage forces which quantify the variation of the mesoconstituentÕs energy with respect to damage. The DML is based on two hypotheses. The first assumption is that once the behavior of the mesoconstituents has been identified the DML can predict the behavior of any stacking sequence subjected to any loading. Consequently, the damage evolution laws of the ply and of the interface must be valid regardless of the stacking sequence or the loading. The second assumption is that the damage variables are constant throughout the thickness of each single layer, but can change from one layer to the next. Remark 2. We will come back on the first assumption by following the obtained micro–meso relations, see Remark 6. The single-layer takes into account diffuse debonding of fibers/matrix interfaces, transverse cracking, local delamination and fiber breakage [9]. The interface, which transfers displacements and stresses from one ply to the next, takes into account interlaminar delami-
Fig. 2. Decomposition of a laminated composite into mesoconstituents.
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
nation [21]. Preliminary identifications were carried out in previous works for the single layer [22,23] and for the interface [24,25]. Reviews of the latest works dealing with delamination can be found in [26,27].
4. Equivalence principle and fundamental micro–meso link 4.1. The micro–meso equivalence principle The objective is to build a continuum damage mechanics model which is quasi-equivalent, from an energy standpoint, to the damaged laminate micromodel defined in Section 2. Indeed, one of the aims of deriving micro–meso relations is to study the evolution of damage, and the main fracture criteria themselves are based on energy concepts. Consequently, in this multiscale approach, the potential energy stored in the plies and in the interfaces must be the same on the micro- and mesoscales, as illustrated in Fig. 3. The equivalent mesomodel is considered to be completely achieved if two basic situations of equivalence are established, one for the ply and one for the interface, leading to a homogenization procedure which will be detailed in Section 5 for the ply and Section 6 for the interface. The homogenization must be carried out for any stacking sequence defined by the thicknesses and orientations of the plies, but also for any kind of microdamage state. Practically one reduces the domain of investigation of microdamage state using q 2 [0.;0.7] and s 2 [0.;0.4]. From experiments, the material could be considered as fully damaged for higher level of degradations. 4.2. Fundamental relationships between the micro and meso solutions Let us consider the most general case of a damaged laminate subjected to uniform macroloading. The associated micromechanics problem Pm is illustrated in Fig. 4(a). Each ply and each interface may be affected
701
by the microphenomenology adopted in Section 2. The macroloading is described by the plane part of the strain pep and the out-of-plane part of the stress rN3. p is the projection operator associated with the mean plane (P) of the laminate and N3 is the normal vector of (P). The objective is to replace the discretely damaged medium by an equivalent continuous and homogeneous medium. Let us denote sm (sm = [Um,em,rm]) the exact displacement, strain and stress fields on the microscale under such loading. With the assumption of linear elasticity, it is possible to decompose sm using a crack closure method such as: sm ¼ ~s þ s;
ð1Þ
~ in which all where ~s is the solution of a first problem P cracks and local delaminations are removed from the laminate (the ‘‘effective solution’’). Consequently, for this problem, the laminate is a stack of homogeneous plies and interfaces (possibly homogeneously damaged according to Scenarios 3 and 4) which yields the following usual solution at any point M of the laminate: p~epðMÞ ¼ pep;
~N 3 ðMÞ ¼ rN 3 : r
ð2Þ
In order to ensure static admissibility in the cracked zones of problem Pm , ~s must be corrected by solving a where each cracked area, whose residual problem P, normal vector is denoted n, is loaded by the residual stress ~ rn. The corresponding solution is denoted s. Now, let us focus on the vicinity of an interface Cj between two plies Si and Si + 1. Due to the localization of the residual problems around each cracked area, one can consider that the restriction of the solution s to the vicinity of the interface Cj depends only on the states of damage of this interface and of the neighboring plies with good approximation. Finally, in the vicinity of Cj, a good approximation of s is ob tained by solving the approximate problem Pj Cj defined on the cell (Fig. 5) subjected to the residual stress on the cracked areas of [Cj;Si,Si + 1], all other cracked areas being removed.
Fig. 3. Energy equivalence between the micro and meso interpretations of damage.
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
702
Fig. 4. The fundamental micro–meso link.
: approximate solution of s around an interface Cj. Fig. 5. Problem Pj Cj
Because the degradation pattern is periodic within Pj Pj Cj solutions of Pj [Cj;Si,Si + 1], the fields u Cj and r Cj verify periodicity and anti-periodicity boundary conditions, which leads to the following property: Pj
Pj
Property 1. hpe Cj pijCj ¼ 0 and h r Cj N 3 ijCj ¼ 0 8Cj , where hijCj denotes the mean value over Cj. . It is This property is verified exactly for problem Pj Cj not exactly true, but verified with very good approximation on the completely degraded geometry, which finally yields the following property:
Property 2. hpem pijCj #p~epjCj ¼ pep and hrm N 3 ijC # ~N 3 ¼ rN 3 8Cj . r Regarding this property, a strong link appears to exist between the micro and meso solutions. The plane part of the mesostrain and the out-of-plane part of the mesostress are mean values of the microfields. Thus, if one knows the solution on one scale, it is easy to obtain the solution on the other scale, which is the basis of an efficient multiscale method to study damage in laminates.
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
5. The basic ply problem 5.1. Traditional problem and potential energy of the equivalent ply Let us get back to the most general case of a damaged laminate (Fig. 4). Since the residual problems are localized around each ply, the first step consists in homogenizing the degradation associated with each ply separately. This simplified problem is a direct adaptation of the in-plane problem [11]. As illustrated in Fig. 6(a), only one ply S is degraded by Scenario 1 (microcracking rate q) and Scenario 2 (local delamination ratios ½s1 ; s2 ; s1þ ; s2þ ). The fibers of ply S are oriented along Direction 1. S 0 and S 0þ , whose fibers are oriented along h0 and h0þ , are the plies adjacent to ply S and represent the remaining structure. These plies are assumed to be homogeneous and possibly damaged according to Scenario 3. The procedure for skin plies, which is an extension of the in-plane procedure introduced in [12], is not detailed in this paper. As discussed in Section 4.2, the exact microsolution is split into an effective part ~s and a residual periodic corply rection sP , solution of the traditional basic ply problem ply P (Fig. 6(b)). The cell is subjected to periodicity boundary conditions corresponding to the periodicity
703
of damage in the ply S. The cracked area, whose normal rn. vector is denoted n, is loaded by the residual stress ~ Because of the linear elasticity assumption, the soluply tion sP can be written as the superposition of five elementary solutions (Fig. 7) associated with the five possible elementary loads on the cracked area. P r
ply
ply
ply
ply
ply
ply
P 12 þ r P 22 þ r P 13 þ r P 23 þ r P 33 : ¼r
ð3Þ
Result 1. Ply S can be homogenized and its potential energy can be expressed as: 2ESp t 2 ðI 22 ; I 12 Þ~ ~33 ½M ¼ ½p~ep ½M ep þ r r33 1 ðI 22 ; I 12 Þ½p~ jSj r223 3 ðI 22 ; I 12 Þ½p~ep ðI 23 þ 1Þ~ ~33 ½M þr ~ 23 G
2 r33 iþ ðI 13 þ 1Þ~ r213 I 33 h~ : ~ 13 ~3 E G
ð4Þ
2 and ½M 3 depend on the material 1 ; ½M Operators ½M properties of ply S. Consequently, on the microscale, the potential energy stored in ply S is completely determined through five damage indicators, I 22 ; I 12 ; I 23 ; I 13 and I 33 , which are
Fig. 6. Problem Pply: (a) geometry, (b) microdecomposition and (c) damaged mesoconstituents.
Fig. 7. Decomposition of problem Pply in terms of elementary loads.
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
7 6 5 4 3 2 1
-2 3
-1 3
33
33
-2 3 13
-3 3 12
-2 3 12
-1 3
12
-3 3 22
22
-2 3
0 -1 3
Remark 3. The derivation of expression (4) is ‘‘quasiexact’’ because it requires the approximation of negligible coupling effects among all the elementary residual problems of Fig. 7. This hypothesis is justified if the following relative energy eij kl between the diagonal solutions and the coupled solutions verifies: cijkl 2 eijkl ¼ pffiffiffiffiffiffiffiffiffiffi 1; 8ðij; klÞ 2 ½22; 12; 13; 23; 33 ; ij 6¼ kl; aij akl
8
-1 2
On the mesoscale, expression (4) completely defines the damaged homogeneous ply S (Fig. 6(c)), whose damage variables, associated with stiffness reductions, can be easily deduced from I 22 ; I 12 ; I 23 ; I 13 and I 33 .
-3
22
ð5Þ
9 x 10
Relative coupling term
defined as the integral of the strain energy for each basic residual problem. Denoting eij(M) the strain energy at any point M of the cell (X) in the basic ply problem P ply ij , one defines: Z Z ~ ~ I ii ¼ Ei I ij ¼ Gij e ðMÞ dS; eij ðMÞ dS: ii ~2ii S ~2ij S jSjr jSjr
22
704
considered coupling Fig. 8. Relative values of the coupling eij kl for the range of the 0 parameters: h0 2 ½90; 60; 45; 0; h0þ 2 ½90; 45; 45; 60; HH ¼ 2; 1 1 2 1 2 ¼ 0:5; s ¼ s ¼ s ¼ s ¼ 0:5. þ þ q
ð6Þ where Z ply ply aij ¼ Tr½ rP ij eP ij dS;
cijkl ¼
S
Z
ply
ply
Tr½ rP ij eP kl dS: S
ð7Þ A numerical proof of this approximation is given in Fig. 8, where this term is plotted for numerous geometric configurations of the stacking sequence. 5.2. The extended ply problem As previously stated, the traditional basic ply problem defines only the state of damage of ply S. This procedure
gives excellent results for in-plane stresses because the energy is really confined to the ply S being considered for such stresses. In the case of out-of-plane stresses, however, the damage state of ply S induces some nonnegligible energy in plies S 0 and S 0þ . Consequently, Pply defines the part of states of damage of S 0 and S 0þ on the mesoscale due to the microdegradations in S. In other words, under out-of-plane macroloading, the damage state of the ply being considered depends on the state of damage of the adjacent plies. To handle these considerations, one defines an extended ply problem, illustrated in Fig. 9. A detailed example is proposed. Let us consider a stacking sequence with only two identically damaged plies S and S 0þ whose fiber
Fig. 9. Extended Pply: (a) the traditional ply problem, (b) S 0þ Õs lower local delamination, (c) S 0þ Õs lower local delamination and transverse cracks, (d) S 0þ Õs upper and lower local delamination and transverse cracks.
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
• with the traditional ply problem [a] alone, • with the traditional ply problem [a] plus the contribution [b] from the lower local delamination of S 0þ , both problems considered to be uncoupled, • with the traditional ply problem [a] plus the contribution [c] from the lower local delamination and transverse cracks of S 0þ , both problems being considered uncoupled, • with the traditional ply problem [a] plus the contribution [d] from the lower and upper local delamination and transverse cracks of S 0þ , both problems considered to be uncoupled. The cell is subjected to a uniform residual stress loading 13 (in the basis of ply S) and the results, Fig. 10, show the significant impact of the whole cracked area of S 0þ on the energy stored in ply S and, consequently, on its damage state, which the extended ply problem enables one to simulate. The discrepancy between the 3D reference and the [a] + [d] solution comes from the uncoupled treatment of problems [a] and [b]. The results obtained for a uniform residual stress loading 33, Fig. 11, enable one to identify the negligible influence of the upper local delamination and transverse cracks of S 0þ on the damage state of ply S under 33 solicitation. In other words, the damage mesostate of a ply does not depend on the microcracking of the adjacent plies ~33 stress loading. The discrepancy between the 3D under r reference and the [a] + [d] solution still comes from the uncoupled treatment of problems [a] and [b] and a more
(Strain energy density) / (micro reference)
θ’−= -45 , θ’+=+45 , ρ|S=ρ|S’ =0.5 , τ|S=τ|S’ =0.3 +
0.8
θ’ = -45 , θ’ =+45 , ρ| =ρ| −
+
S
S’
=0.5 , τ| =τ|
+
S
S’
=0.3
+
1
0.8
0.6
0.4
0.2
0 0
Fig. 11. Extended problem Pply: influence of damage in ply S Õs on the energy of ply S under 33 loading.
accurate procedure may be required; nevertheless, the previously underlined point is not questioned. 5.3. Micro–meso relations for the ply The results of this section are associated with the traditional basic ply problem; the treatment of the extended ply problem is in progress. Result 2. The relations ðI 22 ; I 12 ; I 23 ; I 13 ; I 33 Þ $ ðq; s1 ; s2 ; s1þ ; s2þ Þ are approximately ply-material relations, i.e., the state of damage on the mesoscale is approximately independent of the configurations of the peripheral parts S 0 and S 0þ . An illustration of this result out-of-plane indicators. Instead tors I 13 ; I 23 and I 33 , we chose D13, D23 and D33 defined by: I 13 I 23 D13 ¼ ; D23 ¼ ; 1 þ I 13 1 þ I 23
is given in Fig. 12 for of the damage indicato plot the quantities
D33 ¼
I 33 1 þ I 33
ð8Þ
which are directly associated with the stiffness reductions commonly used in damage mechanics. Fig. 12(a) shows D13, D23 and D33 calculated for several states of microdegradation and for numerous geometric configurations (h0 2 ½90; 60; 45; h0þ 2 ½90; 45; 45; 60, 0 H 2 [helem;2helem;4helem], H 2 [helem;2helem;4helem], where helem is the thickness of the elementary material ply). It appears that for a given state of microdegradation, quantified by dimensionless microvariables ðq; s1 ; s2 ; s1þ ; s2þ Þ, the equivalent state of damage on the mesoscale is approximately intrinsic. This property is verified even better if one considers stacking sequences 0 with (H P H) (Fig. 12(b)).
1.2
1
micro a a+b a+c a+d
1.2
(Strain energy density) / (micro reference)
directions are orthogonal. The local delamination ratios of both damaged plies verify sjS ¼ s1 jS ; s2 jS ; s1þ jS ; s2þ jS and sjS 0 ¼ s1 jS 0 ; s2 jS 0 ; s1þ jS 0 ; s2þ jS 0 . The strain energy density in S is first computed taking into account the complete 3D microdescription, which constitutes the energy microreference. Then, four calculations of the strain energy density in S are carried out:
705
+
micro a a+b a+c a+d
0.6
0.4
0.2
0 0
Fig. 10. Extended problem Pply: influence of damage in ply S on the energy of ply S under Loading 13 in the basis of ply S.
Remark 4. The out-of-plane damage indicators are independent of the initial level of diffuse damage within ply S or its adjacent interfaces. Fig. 13 shows the values of D33, D13 and D23 in the cases of initially healthy plies
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
706
0.5 0 0.5 0.4
0.3 0.2
1 2 τ+=τ+
0.1
0
0
0.5 0.3 0.4 0.1 0.2
τ1=τ2 -
1 0.5 0 0.5 0.4
0.3 0.2
τ1=τ2 +
0.1
0
0
0.5 0.3 0.4 0.1 0.2
1
1 0.5 0 0.5 0.4
0.3 0.2
τ1+=τ2+
0.1
0
0
0.5 0.3 0.4 0.1 0.2
τ1=τ2 -
1 0.5 0 0.5 0.4
-
0.3 0.2
1 2 τ+=τ+
0.1
0
0
0.5 0.3 0.4 0.1 0.2
τ1=τ2 -
-
1 0.5 0 0.5 0.4
0.3 0.2
τ1=τ2
2
τ- =τ-
+
(B)-Damage D13
1
(B)-Damage D33
(b)
(B)-Damage D23
(A)-Damage D23
(A)-Damage D13
(A)-Damage D33
(a)
+
0.1
0
0
0.5 0.3 0.4 0.1 0.2
0
0
0.5 0.3 0.4 0.1 0.2
+
1
2
τ- =τ-
1 0.5 0 0.5 0.4
0.3 0.2
τ1+=τ2+
0.1
τ1=τ2 -
-
Fig. 12. D13, D23 and D33 vs. the microstate of degradation and geometric configuration of the stacking sequence.
and interfaces, initially damaged ply (50% diffuse damage) and initially damaged interface (50% diffuse damage).
1
Healthy ply and interface Ply with diffuse damage Interface with diffuse damage
0.9 0.8
Damage level
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
D33
D13
D23
Fig. 13. Evolution of damage indicators D13, D23 and D33 with the state of diffuse damage of the mesoconstituents (microdegradation state: q1 ¼ 0:5; s1 ¼ s2 ¼ s1þ ¼ s2þ ¼ 0:3).
6. The basic interface problem 6.1. Potential energy of the equivalent interface Let us again consider the most general case of a damaged laminate subjected to uniform macroloading (Fig. 4). In order to homogenize the microdegradations associated with an interface Cj between two damaged plies Si and Si + 1 whose fiber directions are, respectively, Ni and Ni + 1, a good approximation of the solution in the vicinity of this interface can be obtained by considering the simplified 3D structure of Fig. 14. The angle between Ni and Ni + 1 is denoted h. One can introduce a local reference frame (N1,N2,N3) of orthotropic directions for the healthy interface; N3 is normal to the interface while N1 and N2 are the bisector directions associated with angle h. The external plies S 00 and S 00þ representing the remaining structure are homogenized and possibly damaged according to Scenario 3. With the exception of transverse cracks, each pair of vertical faces of the 3D cell must satisfy periodic conditions corresponding to Scenarios 1 and 2. The description of the state of microdegradation of the cell requires the microcracking rates and the local delamination ratios of plies Si and Si + 1: ðq; s1 ; s2 ; s1þ ; s2þ ÞjS i ; ðq; s1 ; s2 ; s1þ ; s2þ ÞjS iþ1 . The geometric parame-
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
707
Fig. 14. The basic interface problem Pint.
ters of the interface are the angle h and the dimensionless quantities ðHH ji ; HHiþ1j Þ which quantify the unit thickness of the interface. The homogenization method is similar to that of the ply: the solution is written as an effective part ~s plus a int periodic correction sP using the crack closure method. The difficulty concerns this residual part, whose associated problem is called the basic interface problem Pint, in which the cracked area, whose normal vector is denoted n, is loaded by the residual stresses ~ rn. According to the linear elasticity assumption, the residual stresses are decomposed into three elementary solicitations in the orthotropic basis of the interface. Consequently, int P as the sum of one can express the residual solution r the three elementary solutions: P r
int
int
int
int
P 13 þ r P 23 þ r P 33 : ¼r
ð9Þ
Result 3. The interface Cj can be homogenized and its potential energy expressed as: 2Ejp ð1 þ I 1 Þ~ r213 ð1 þ I 2 Þ~ r223 ð1 þ I 3 Þ~ r233 ¼ ; ~k 1 ~k 2 ~k 3 jCj j
ð10Þ
where ~k 1 ; ~k 2 and ~k 3 are the elastic stiffness coefficients of interface Cj (possibly including initial diffuse damage). The potential energy is completely defined by three damage indicators I 1 ; I 2 and I 3 , which are defined as the integral of the strain energy over the interface for each elementary problem:
I i ¼
~k i Z int int Tr½ rP i3 eP i3 dC; 2 jCj j~ ri3 C
8i 2 ½1; 2; 3:
The derivation of Eq. (10) implies that the coupling effects between the residual problems are negligible. This approximation is justified if the following condition is verified: cpq ð12Þ epq ¼ pffiffiffiffiffiffiffiffiffi 1; 8ðp; qÞ 2 ½1; 2; 32 ; p 6¼ q; ap aq where Z int int ap ¼ Tr½ rP p3 eP p3 dC; C
cpq ¼
Z
int
int
Tr½ rP p3 eP q3 dC:
ð13Þ
C
In order to illustrate this condition, let us consider the damaged cell (Fig. 14) under the four states of degradation defined by Table 1. The local delamination ratios of both damaged plies verify sjS i ¼ s1 jS i ; s2 jS i ; s1þ jS i ; s2þ jS i and sjS iþ1 ¼ s1 jS iþ1 ; s2 jS iþ1 ; s1þ jS iþ1 ; s2þ jS iþ1 . The orientations of the fibers refer to direction N1 of the interface and the degradation of ply Si can take four configurations. First, this cell is loaded by a residual stress composed ~13 ; r ~23 and r ~33 applied of the three elementary loads r simultaneously. The strain energy stored in the interface under this solicitation is taken as the energy reference. Then, one solves six complementary problems on the same cell by considering each elementary loading and the corresponding coupling effects separately. Finally, the energy ratio of each complementary problem is defined by the ratio of the corresponding strain energy of the interface to the energy reference. The results are
Table 1 The four states of microdegradation associated with Fig. 15 Ply
Microcracking rate q
Local delamination ratio s
Orientation
Thickness
S 00þ
0. 0.4 0.4 or 0.5 0.0
0. 0.15 0.15 or 0.3 0.
30 +30 30 +30
1helem 1helem 1helem 1helem
Si + 1 Si S 00
ð11Þ
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
708
ρ|S =0.40 τ|S =0.15 e12=3.5E 04
ρ|S =0.40 τ|S =0.30 e12=0.09
i
50
i
Energy ratio (%)
Energy ratio (%)
i
40 30 20 10 0
23 − 13 − 33 23 − 13 − 33 ρ|S =0.50 τ|S =0.15 e12=0.12 i
50 40 30 20 10 0
23 − 13 − 33
23 − 13 − 33
50 40 30 20 10 0
23 − 13 − 33 23 − 13 − 33 ρ|S =0.50 τ|S =0.30 e12=0.33 i
Energy ratio (%)
Energy ratio (%)
i
i
i
50 40 30 20 10 0
23 − 13 − 33
23 − 13 − 33
Fig. 15. Distribution of energy among the diagonal and coupling terms.
given in Fig. 15, which shows the distribution of the strain energy among the diagonal and coupling terms. The values of e12 are specified in the four cases. In the particular case of an identical microdegradation state in plies (Si) and (Si + 1), Condition (12) is verified, but as soon as the states of microdegradation in the plies differ the symmetry of the interface is broken and coupling between Residuals 23 and 13 appears in the behavior of the damaged interface. Nevertheless, this coupling never becomes the prominent factor in our domain of investigation and it will be neglected for this paper. The consequences of this approximation are currently being studied.
The potential energy equivalence principle between the micro- and mesoscales enables one to build the operators linking the damage indicators of the interface ðI 1 ; I 2 ; I 3 Þ to the microdegradation variables of the cell ðq; s1 ; s2 ; s1þ ; s2þ ÞjS i and ðq; s1 ; s2 ; s1þ ; s2þ ÞjS iþ1 numerically . To verify the consistency of these operators with the hypotheses of the mesomodel, one must evaluate their intrinsic properties with respect to the interface. For convenience, the results are presented in terms of losses of stiffness, which are linked to the damage indicators by the following relation: I i Di ¼ : ð14Þ 1 þ I i
6.2. Micro–meso relations for the interface
Let us consider an interface defined by Table 2. Once again, the local delamination ratios of both damaged plies verify sjS i ¼ s1 jS i ; s2 jS i ; s1þ jS i ; s2þ jS i and sjS iþ1 ¼ s1 jS iþ1 ; s2 jS iþ1 ; s1þ jS iþ1 ; s2þ jS iþ1 and the orientations of the fibers refer to the direction N1 of the interface. By taking several configurations of external plies S 00þ and S 00 for different orientations ðh00 ; h00þ Þ (Fig. 16)
Result 4. The relations ðI 1 ; I 2 ; I 3 Þ $ ððq; s1 ; s2 ; s1þ ; s2þ Þ jS i ; ðq; s1 ; s2 ; s1þ ; s2þ ÞjS iþ1 Þ are, with a very good approximation, ply-material properties, i.e. the micro–meso relations of the interface are independent of the stacking sequence of the laminate.
Table 2 States of microdegradation associated with Figs. 16 and 17 Ply S 00þ Si + 1 Si S 00
Microcracking rate q 0.0 0.5 0.35 0.0
Local delamination ratio s
Orientation
Thickness
0.0 0.25 0.15 0.0
h00þ
H 00þ 1helem 1helem H 00
22.5 22.5 h00
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
D
D3
0.7
1
0.7
D2
0.7
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0 0
50
100
150
θ’’
0
50
100
0
150
0
θ’’
50
100
150
θ’’
−
+
709
0
50
100
150
0
θ’’
100
150
θ’’
−
+
50
0
50
100
150
θ’’
−
+
Fig. 16. Influence of the orientations of the external plies.
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
D
D3
0.7
1
0.7
D2
0.7
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0 0
0.5 0.5 0
H’’+
H’’
0 0
0.5 0.5 0
H’’+
0
H’’
Fig. 17. Influence of the thicknesses of the external plies.
0.5 0.5 0
H’’+
H’’
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
710
0.7
0.4
healthy ply and interface interface loss of stiffness 50% ply loss of stiffness 50%
All cracks taken into account without transverse microcracking and neighboring delaminations 0.35
ρ|
S
=0.4 , τ|
i+1
S
0.6
=0.15 , θ=60
ρ| =0.5 , ρ|
i+1
S
0.3
Level of mesodamage
Mesodamage D
3
0.5
0.25
0.2
0.15
i
S
=0.4 , τ| =0.25 , τ|
i+1
S
S
i
=0.2 , θ=60
i+1
0.4
0.3
0.2
0.1
0.1
0.05
0
0
(ρ|S ,τ|S ) : (0.4,0.15) – (0.5,0.15) – (0.5,0.225) – (0.5,0.3) i
Mesodamage : D2– D 1– D
i
3
Fig. 18. Influence of intralaminar cracks on delamination under r ~33 stress loading.
Fig. 19. Influence of Scenarios 3 and 4.
and thicknesses ðH 00 ; H 00þ Þ (Fig. 17), one can note that the intrinsic properties of the three micro–meso operators are verified with very good accuracy (65%).
• with healthy materials, • with plies affected by Scenario 3 leading to a 50% loss of stiffness, • with interfaces affected by Scenario 4 leading to a 50% loss of stiffness.
Remark 5. The intralaminar cracks have no influence ~33 stress loading. The local on the delamination under r delaminations ðs1þ jS i ; s2þ jS i ; s1 jS iþ1 ; s2 jS iþ1 Þ of the interfaces adjacent to an interface Cj are called ‘‘neighboring delaminations’’. The identification of the operator associated with Damage 33, illustrated by Fig. 18, was carried out with and without transverse cracking and ‘‘neighboring delaminations’’ during a physically reasonable evolution of the degradation state of ply Si. The local delamination ratios of the two damaged plies verified sjS i ¼ s1 jS i ; s2 jS i ; s1þ jS i ; s2þ jS i and sjS iþ1 ¼ s1 jS iþ1 ; s2 jS iþ1 ; s1þ jS iþ1 ; s2þ jS iþ1 . It appears that intralaminar cracks and neighboring delaminations have a negligible influence on ~33 stress loading. Conversely, interface damage under r these cracks play a major role in interface damage under ~13 and r ~23 stress loading, which makes us hope that a r more specific link between local delamination and transverse cracking will be derived in the near future. Remark 6. The damage state of the interface could depend on the state of the adjacent layers which contradicts the first main hypothesis of our standard mesomodel, (see Remark 2). However, it is relatively easy to extend it through a non-local mesomodel, which will be done in a companion paper. Remark 7. Our method enables one to study the influence of Scenarios 3 and 4 on the micro–meso relations of the interface. Let us consider a particular stacking sequence (h = 60, Hi = Hi + 1 = helem) and a combination of Scenarios 1 and 2 in plies Si and Si + 1 described by ðqjS i ¼ 0:5; sjS i ¼ 0:25; qjS iþ1 ¼ 0:4; sjS iþ1 ¼ 0:2Þ. Fig. 19 shows the three homogenized damage states corresponding to the three cases:
Despite the catastrophic level of diffuse damage, the changes of homogenized mesodamage are relatively insignificant. For usual levels of diffuse damage 3 and 4, the influence of such damage on the homogenized damage of the interface is negligible. 7. Outlook Up to now, we carried out the homogenization procedures for any type of microdamage state defined by microcracking rates q 2 [0.;0.7] and delamination ratios s 2 [0.;0.4], and for any stacking sequence defined by the thicknesses and orientations of the plies. In order to homogenize a ply, the adjacent plies must be at least as thick as the ply being considered. This is the only condition on the stacking sequence required to obtain sufficiently accurate intrinsic properties. One can use the results of such homogenization and the ‘‘virtual testing’’ procedure for modeling. This opens new possibilities for the choice of internal variables: the description of damage can be fully meso or fully micro, but many hybrid micro–meso descriptions are also allowable (Table 3). Then, the thermodynamic forces, constructed according to the description chosen, lead to equivalent constitutive laws. In particular, these laws can still be written in terms of traditional mesodamage variables and associated mesodamage forces. A typical treatment would be the following: Scenarios 3 and 4 initiate damage according to:
P. Ladeve`ze et al. / Composites Science and Technology 66 (2006) 698–712
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Table 3 Allowable choices of internal variables Description
Ply (i)
Hybrid ply and interface Meso ply and interface Hybrid interface
~ i; D ~ i; D ~ i; D
Interface (j)
~ 0i ; qi D ~ 0 ; Di ; D0 D i i ~ 0 ; Di ; D0 D i i
~ 1; D ~ 1; D ~ 1; D
~ 2; D ~ 2; D ~ 2; D
~ 3 ; s1þ ; s2þ ; s1 ; s2 D ~ 3 ; D1 ; D2 ; D 3 D ~ 3 ; s1 ; s2 ; s1 ; s2 D þ þ
Table 4 Unilateral effect: behavior on the mesoscale Ply (i)
rmeso 22 rmeso 22 rmeso 33 rmeso 33
Interface (j)
>0 rmeso 33 rmeso 60 33
>0 60 >0 60
Open transverse cracks Closed transverse cracks (no microcracks) Open delamination cracks Closed delamination cracks (no delamination cracks) Open delamination cracks Closed delamination cracks
h
i ~_ 22 ; D~_ 0 12 ji;t ¼ f ð½Y~ 22 ; Y~ 0 ji;t ; s 6 tÞ; D 12
ð15Þ
h
i ~_ 1 ; D ~_ 1 ; D ~_ 2 ; D ~_ 3 ji;t ¼ gð½Y~ 1 ; Y~ 0 ; Y~ 0 ji;t ; s 6 tÞ; D 2 3
ð16Þ
where f and g are functions of the thermodynamic forces and of the history of the mesoconstituents. These functions can be defined by phenomenological or micromechanical results, whichever are available. Then, the percolation of fiber–matrix debonding leads to Scenario 1, whose evolution is governed by Gqi 6 Gqc (where Gqi is the energy release rate associated with the birth of a new microcrack in the finite fracture mechanics sense, and Gqc is the critical energy release rate, a property of the material). Local delamination is considered inactive until the saturation of Scenario 1. Then, the level of Scenario 2 increases with respect to Gsi 6 Gsc (where Gsi is the energy release rate associated with a small propagation of local delamination, and Gsc the corresponding critical rate, a property of the material). Once this last scenario has been initiated, it propagates catastrophically and leads to macroscopic delamination, which can be described only from a macro point of view. Remark 8. Microcracks and local delamination cracks can be open or closed, which leads to different types of behavior (Ladeve`ze, 02 [28]). A simple way to model these phenomena is to make the mesobehavior dependent on the sign of the stress, according to Table 4.
8. Conclusion A virtual testing approach for the derivation of complete relationships between micro- and mesomechanics under out-of-plane loading has been presented. Consequently, the damage mesomodel is now being reconsidered as a systematic homogenization of traditional micromechanics models. Pragmatic rules will be given
in a companion paper. Moreover, it is possible to describe the actual microdamage mechanisms and their intensities using mesoquantities. Some interesting properties have been highlighted; for example, we showed that microcracking does not affect the behavior of the interface in pure 33 stress loading. There is work in progress to deal with the coupling of damage and (visco)plasticity, which was not considered here.
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