A multiscale progressive failure approach for composite laminates based on thermodynamical viscoelastic and damage models

Composites: Part A 38 (2007) 198–209 www.elsevier.com/locate/compositesa

A multiscale progressive failure approach for composite laminates based on thermodynamical viscoelastic and damage models F. Laurin *, N. Carre`re, J.-F. Maire Department of Damage and Solid Mechanics, ONERA/DMSE, BP 72, 29 Avenue de la Division Leclerc, F-92322 Chaˆtillon Cedex, France Received 16 August 2005; received in revised form 3 January 2006; accepted 3 January 2006

Abstract This paper presents a multiscale progressive failure approach for composite materials which allows predicting the failure of a laminate from the thermo-mechanical properties (behaviour and strength) of the individual unidirectional plies. This kind of approach is predictive for different laminate layups and takes also into account the effects of ply failures on the macroscopic behaviour and the final laminate failure. For the first time to our knowledge, the viscosity of the matrix was introduced into the model for a more realistic description of the mesoscopic behaviour, especially for the non-linearity under shear loading. The mesoscopic failure criterion is based on physical principles and makes a distinction between the fibre failure and the interfibre failure modes. In order to describe the progressive failure of a laminate (ply after ply), a progressive degradation model, based on thermodynamical damage models, was developed. The definition of the final failure is also improved in order to predict the final failure more accurately, especially for [±h]s laminates. Finally, some results (stress vs. strain laminate responses and macroscopic failure envelopes) are presented and compared with test data found in the literature.  2006 Elsevier Ltd. All rights reserved. Keywords: A. Laminates; A. Fibres; B. Mechanical properties; B. Strength

1. Introduction Prediction of mechanical response and failure of composite laminates is essential for the design of engineering composite structures. Many different approaches exist in the literature, written at different scales (mesoscopic and microscopic scales), exhibiting a different complexity and leading to more or less successful results. Consequently, important safety factors have to be applied in aeronautic industry modelling because of the lack of confidence into existing failure criteria. In order to assess the prediction capabilities of the existing failure criteria, Hinton and Soden [1] have launched a state-of-the-art of the major multiscale failure approaches on different test cases performed on different composite

*

Corresponding author. Tel.: +33 1 46 73 46 92; fax: +33 1 46 73 48 91. E-mail address: [email protected] (F. Laurin).

1359-835X/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2006.01.018

materials, with representative layups, and multiaxial loading conditions. This investigation, entitled the ‘‘World Wide Failure Exercise’’ (WWFE), was divided into two parts: (i) the authors had to make ‘‘blind’’ theoretical predictions of the laminate failure, and compare the results with each other, and (ii) they realized an important collect of test data in order to analyse the validity of the different approaches. The conclusions of this exercise were: (i) ‘‘best’’ approaches predicted well the final failure of classical laminates under complex tensile loadings, but (ii) the theoretical macroscopic behaviours and the final failures for [±h]s laminates are in relatively poor agreement with test data and (iii) the predictions of failure in compression for every kind of layup overestimate the experimental data (because these material approaches neglect buckling of the tested tubes). The aim of the present study is to develop a multiscale progressive failure approach which improves the determination of the final failure of the laminate from the strength of the individual unidirectional ply under the plane stress

F. Laurin et al. / Composites: Part A 38 (2007) 198–209

hypothesis and then to validate it on the test data found in the literature. In order to be predictive for different laminate layups, we have chosen to develop a mesoscopic failure approach which takes into account the progressive aspect of a laminate failure. It means that after a first ply failure, the mechanical properties of the failed ply are degraded progressively, which reduces the effective rigidity of the laminate, and leads to overloading of the other plies. Failure approaches which neglect these phenomena thus overestimate, often dramatically, the final failure of the laminate. First, the principle of the progressive approach is described; the details of the failure model are presented in Section 2. The last section is devoted to the comparison between theory and test results on the macroscopic behaviour to failure. Failure envelopes of laminates are presented in Section 3. Finally, a comparison with the other approaches found in the literature (mainly from the WWFE) is presented.

199

a fibre failure mode (FF) and an interfibre failure mode (IFF). For each failure mode, a distinction is made between failures in tension and in compression due to their different failure mechanisms. The interfibre failure mode, tension failure (f2þ in Eq. (1)) and compression failure (f2 in Eq. (2)) are characterised separately, each one being modelled by the failure criterion !2  2 r22 s12 þ þ f2 ðr22 ; s12 ; d f Þ ¼ e Ye t S c ð1  pr22 Þ ¼ 1; if r22 P 0 !2  2 r22 s12  f2 ðr22 ; s12 ; d f Þ ¼ þ e Ye c S c ð1  pr22 Þ ¼ 1; with Ye t ¼ ð1  d f ÞY t ;

The proposed multiscale progressive failure approach (Fig. 1) can be divided into five steps: (i) the change of scale method, (ii) the choice of a mesoscopic behaviour, (iii) the definition of a mesoscopic failure criterion, (iv) degradation of the mechanical properties of the failed plies, and finally (v) definition of the final failure (laminate failure). The change of scale method used, is based on the classical laminate theory (CLT) extended to non-linear behaviour and written in an implicit way in order to ensure the equilibrium conditions at each scale, and to decrease the computational time. In the following, the other four considerations are discussed in more details. 2.1. Mesoscopic failure criterion

Laminate

The proposed failure criterion is based on the Hashin’s assumptions [2]; two failure modes are taken into account:

Unidirectional ply

Final failure of laminate

1

Determination of σ / ε in each ply Mesoscopic behaviour

e S c ¼ ð1  d f ÞS c

where Yt, Yc and Sc are respectively the transverse tensile and compressive strengths and the in-plane shear strength. Compared with the Hashin failure criterion, two improvements in the interfibre failure criterion have been carried out: (i) a better description of the strength of the UD ply under shear and transverse loadings (p coefficient in Eqs. (1) and (2)) and (ii) the introduction of the degradation of the interfibre strengths (df) due to premature single fibre failure. The reinforcement in high shear and weak compression in the interfibre failure mode, represented by the (1  pr22) term, could be explained at a micro-mechanical point of view: the transverse compressive loading increases the load transfer capability of the fibre/matrix interface and delays the percolation of micro-damages which are due to shear loading. On the contrary, the transverse tension furthers the percolation of micro-damages and reduces the interfibre strengths. The orthotropic character of the unidirec-

Macroscopic loading

2

and

ð3Þ

2. Multiscale progressive failure approach

Methods of change of scale

ð2Þ

if r22 < 0

Ye c ¼ ð1  d f ÞY c

ð1Þ

5

Definition of final failure

Ply failure in a laminate

3

Mesoscopic failure criterion

Degradation of failed ply

4

Progressive degradation model

Fig. 1. Principle of the multiscale progressive failure approach.

200

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tional ply is ensured. In fact, there is no interaction between the stress components r22 and s12, but between the shear strength Sc and the transverse stress r22. The independence of the sign of the shear stress s12 is also ensured (quadratic term). The biaxial failure envelopes, under combined r22 and s12 stresses, are reported in Fig. 2 for two different unidirectional plies: T300/LY556 (carbon/epoxy) and Eglass/ LY556 (Eglass/epoxy) plies [3]. The shapes of the envelopes are very similar (the maximum shear strength of the UD ply is reached for r22 = 0.5Yc), even for these two different materials; therefore, the identification of the p coefficient doesn’t need an additional test, it is defined as the solution of Eq. (4) of2 ðr22 ; s12 Þ ¼ 0; or22

for

r22 ¼

Yc 2

ð4Þ

The results presented in Fig. 2 are in very good agreement with test data on these two different materials. This failure criterion necessitates only the identification of the three interfibre uniaxial strengths (Yt, Yc, Sc) as the maximum stress criterion. Due to a statistical effect on the strength of the fibres (Weibull distribution), some single fibre failures occurred under the r11 uniaxial tension stress long before many fibre failures lead to the ultimate failure of the UD ply. In the vicinity of the fibre breaks, these premature failures lead to interfacial debondings and to the formation of microcracks in the matrix. Hoecker et al. [4] demonstrated experimentally that the quality of the fibre/matrix interface has a strong effect on the transverse tensile and in-plane shear strengths. Hence, one could deduce that these micro-damages (mainly fibre/matrix debondings), due to premature single fibre failures, facilitate the coalescence of microcracks, thus leading to a decrease in the interfibre failure strengths. 120

100

τ

12

MPa

80

60

40 Test data T300/LY556 Test data Eglass/LY556 Failure envelope Carbon Failure envelope Eglass

20

0

-200

-150

-100

σ22 MPa

-50

0

50

Fig. 2. Biaxial failure envelopes of unidirectional plies under combined r22 and s12 stresses.

This mechanism can be modelled using the degradation variable df, based on classical damage models [5], to represent the degradation of the interfibre strengths. This degradation is assumed to have the same effect on the whole interfibre strengths (Yt, Yc, Sc) d f ¼ af hsup½f1þ   f0 iþ

ð5Þ

where Æ æ+ are the Macauley brackets and where the sup function permits to store the maximum value reached by the fibre failure criterion f1þ (defined in Eq. (6)) during loading. The degradation variable df is increased when the tension fibre failure criterion attains the damage threshold f0. The af coefficient permits to describe the kinetics of degradation of the interfibre failure strengths: Yt, Yc and Sc (the kinetics of degradation being correlated to the kinetics of the premature single fibre failures). The degradation variable (df) is totally defined by the two ‘‘shape’’ parameters (Fig. 3): f1!2 which facilitates 2 the identification of the degradation threshold ðf0 ¼ f1!2 Þ and f2!1 which permits to determine easily the kinetics of degradation of the interfibre failure strengths with the rela2 tion: af ¼ ð1  f2!1 Þ=ð1  f1!2 Þ. These coefficients, which vary between 0 and 1 (excluded for f2!1), could be identified through multiaxial tests, but it was experimentally evidenced that the default values f1!2 = f2!1 = 0.5 lead to a good description of the failure envelopes for different materials (carbon/epoxy and Eglass/epoxy). The biaxial failure envelope, under combined r11 and r22 stresses, are reported in Fig. 3 (for f1!2 = f2!1 = 0.5). The failure envelope of the Eglass/MY750 unidirectional ply is in good agreement with the test data [6] which confirms the advantages of introducing coupling between the interfibre failure mode and the fibre failure mode. A distinction is also being made between tension ðf1þ Þ and compression ðf1 Þ failures in the fibre failure mode !2 r11 þ f1 ¼ ¼ 1; if r11 P 0 ð6Þ e t ðd 2 Þ X  2 r11 f2 ¼ ¼ 1; if r11 < 0 ð7Þ Xc e t , Xc are respectively the efficient longitudinal tenwhere X sile and compressive strengths of the unidirectional ply. The final failure of a quasi-isotropic laminate, due to fibre failure of the 0 plies, is usually considered, in a first approximation, as independent of the history of the thermo-mechanical loading. Nevertheless, Leroy [7] demonstrated experimentally that the longitudinal tensile strength of a T800/5250 unidirectional ply is 50% lower when the polymer matrix is removed (tests on dry fibre bundle). Leveque et al. [8] demonstrated also that the tensile strength of a T800H/F655-2 unidirectional ply is reduced significantly (about 40%) after thermal ageing, even if this ageing treatment affects mainly the mechanical properties of the polymer matrix but not those of the fibres.

F. Laurin et al. / Composites: Part A 38 (2007) 198–209

201

σ11 = f1→ 2 Xt

50

σ 22 = f2→1Yt

σ 22 MPa

0 Hashin failure criterion Present failure criterion Test results -50

σ 22 = f2 →1Yc -100

-150 -1000

-500

0

σ11 MPa

500 0

1000

Fig. 3. Biaxial failure envelopes of the Eglass/MY750 unidirectional ply under combined r11 and r22 stresses.

This phenomenon could be explained through a statistical approach: after a premature single fibre failure (due to dispersion on fibre properties), the degraded matrix (by ageing or damage) leads to overloading of the other fibres in a larger domain than a sound matrix, so that the probability to break another ‘‘weakest link’’ is increased. The conclusion of these experiments is that the degradation of the matrix reduces the longitudinal tensile strength of the unidirectional ply because the load transfer between the fibres is distributed on a larger number of fibres. In order to describe the coupling between the interfibre degradation (i.e. matrix damage) and the longitudinal tene t Þ, sile strength, the effective longitudinal tensile strength ð X defined by Eq. (8), was introduced e t ¼ KX UD þ X dry ð1  KÞ X t t X UD t

with

K ¼ ehd 2

ð8Þ

where is the longitudinal tensile strength of a sound UD ply and X dry the strength of a dry fibre bundle. t This effective strength is based again on damage models, and is connected to the interfibre failure degradation variable d2 (see Section 2.3). The h coefficient represents the effect of degradation on the load transfer capability of the matrix and is in relation with the quality of the matrix and of the fibre/matrix bonding and needs to be identified through a laminate test where the same ply fails first in interfibre failure mode and then in fibre failure mode (which leads to the final failure). Eq. (8) may be interpreted as follows: the effective longie t Þ decreases progrestudinal tensile strength of a UD ply ð X sively from the initial strength X UD for a sound UD ply t (perfect fibre/matrix load transfer) to the dry fibre bundle strength ðX dry t Þ, which is equivalent to the strength of a UD ply with a fully degraded matrix. To conclude, the fibre failure criterion is similar to a maximum stress criterion; nevertheless, the longitudinal tensile strength depends on the degradation level of the matrix.

2.2. Mesoscopic behaviour In order to predict correctly a ply failure, it is necessary to have a good estimate of the mesoscopic stresses, the choice of the behaviour at the ply scale is thus essential. Eq. (9) presents the thermo–viscoelastic behaviour which is used to simulate the behaviour of carbon/epoxy and Eglass/epoxy unidirectional plies r ¼ C 0 : ðeT  eth  eve Þ;

with

eth ¼ aðT  T 0 Þ

ð9Þ

where is the stress, C0 the initial elastic stiffness, eT the total strain, eth the thermal strain and eve the viscous strain, expressed in the material axes. The thermal strain eth is defined classically in Eq. (9), a being the thermal tensor, T the test temperature and T0 the stress free temperature. The thermal aspect was introduced in order to take into account the thermal residual stresses (TRS) which could have a strong influence on the prediction of ply failures and especially on the first ply failure (FPF). Besides, the non-linear aspect of the behaviour under shear loading is addressed. While most of the authors agree that this non-linearity is mainly due to the viscosity of the matrix, non-linear elastic behaviours are widely used. Firstly, the non-linear behaviours (polynomial form [9], or Ramberg–Osgood equation [10]) are not able to describe creep and relaxation tests contrary to the viscoelastic model. Secondly, Fig. 4 illustrates the (important) effect of the loading rate not only on the macroscopic behaviour but also on the prediction of the final failure stress of a [±55]s Eglass/ MY750 laminate [11] subjected to a uniaxial tensile loading. The viscoelastic strain eve is calculated using the non-linear viscoelastic spectral model developed at Onera [12,13]. The main idea of this approach consists in expanding the viscosity into a series of elementary viscous mechanisms X 1 n_ i with n_ i ¼ ðli gðrÞS R : r  ni Þ ð10Þ e_ ve ¼ gðrÞ si i

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F. Laurin et al. / Composites: Part A 38 (2007) 198–209

where S0 is the elastic (initial) compliance, and where d1H1 and d2H2 are tensors which represent respectively the effect of fibre failure and interfibre failure on the compliance of the failed ply. For each degradation model, a distinction is made between the kinetics of degradation with the scalar variables di and the effect of failure on the mesoscopic behaviour with the effect tensors H  i . The degradation kinetics represents the evolution of the crack density in each ply during loading. The degradation variables d1 for fibre failure and d2 for interfibre failure are given by qffiffiffiffiffiffi þ  FF : d 1 ¼ a f1  1 and d_ 1 P 0 ð14Þ qffiffiffiffiffiffi þ IFF : d 2 ¼ b f  1 and d_ 2 P 0 2

Fig. 4. Influence of the loading rate on the macroscopic behaviour and prediction of the macroscopic failure stress for the [±55]s Eglass/MY750 laminate under uniaxial tension.

Each elementary viscous mechanism (ni) is defined by its relaxation time (si) and its weight (li). These two quantities define the temporal spectrum which is supposed to have a Gaussian form (Eq. (11)). In order to facilitate the identification procedure, the viscous compliance of a UD ply, SR, (Eq. (11)) is expressed as a function of the elastic compliance S0  2 ! 1 i  n0 i si ¼ e ; li ¼ pffiffiffi exp  nc n0 p 2 3 0 0 0 6 7 0 R 0 ð11Þ S ¼ 4 0 b22 S 22 5 0 0 0 b66 S 66 In order to describe the non-linear aspect of viscosity in a creep test [12], a non-linear function, depending only on stress, was introduced into the model pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin tr : SR : r gðrÞ ¼ 1 þ c ð12Þ Only two tests are necessary to identify the six coefficients of the viscoelastic model: a (multiple) creep test on a [±45]s laminate and a creep test on a [90]s laminate. To conclude, introducing viscosity is of absolute necessity in order to predict accurately the behaviour and the final failure of [±h]s laminates (see for instance Fig. 4), but also to predict the sequence of ply failures even in classical laminates (for instance quasi-isotropic [0/±45/90]s laminates). 2.3. Progressive degradation model

e S ¼ S þ d 1H 1 þ d 2H 2

( with g1 ¼ 2

The progressive degradation model is based on damage models which were developed at Onera [14]. After failure of a ply in a laminate, the effective elastic compliance ð e S Þ of the failed ply is increased 0

When the failure criterion value is higher than one, the ply is broken and its mechanical properties are progressively degraded (due to the load transfer from the other sound plies in the laminate). The evolution of the degradation variable is positive in order to ensure the Clausius–Duhem inequality (this model is thermodynamically consistent). This condition permits also to describe a realistic behaviour even under complex loading/unloading paths. On the one hand, the fibre failure is usually catastrophic for simple specimens (tube, plate, . . .), leading to the final failure. In a practical manner, the calculation is stopped, consequently the coefficient a has not to be identified. On the other hand, interfibre failure is not necessarily catastrophic for a laminate, thus the coefficient b, which represents the kinetics of interfibre degradation of a failed ply in a laminate, has to be identified through a laminate test. The effect tensors represent the effects of a crack on the mesoscopic behaviour of a failed ply; these effects are very different if the failure is a fibre failure (catastrophic) or an interfibre failure. Moreover, the effect tensors are different if the ply is under tensile or compressive loadings, the cracks being opened or closed. The effect tensors for fibre failure (FF) and interfibre failure (IFF) in tension and compression are given by Eqs. (15) and (16). The gi activation index allows to distinguish the opened cracks (gi = 1) from the closed ones (gi = 0) 2 3  0 0 0 ðg1 hþ 11 þ ð1  g1 Þh11 ÞS 11 6 7 FF : H 1 ¼ 4 0 0 0 5;

ð13Þ

0 (

0

1;

if r11 P 0;

0;

if r11 < 0

0 hFF 66 S 66

ð15Þ

0

0

3

 0 ðg2 hþ 22 þ ð1  g2 Þh22 ÞS 22

0

7 5;

0

0 hIFF 66 S 66

0

6 IFF : H 2 ¼ 4 0

with g2 ¼

0

1;

if r22 P 0;

0;

if r22 < 0

ð16Þ

F. Laurin et al. / Composites: Part A 38 (2007) 198–209  FF IFF The coefficient h 11 , h22 , h66 and h66 represent the effect of cracks on the different components of the mesoscopic compliance and are determined through a micro-mechanical approach [15] (based on failure mechanics [16]) by energy equivalence. Consequently, no additional test is necessary for identification of these effect tensors. It should be noted that this progressive degradation model takes into account the unilateral aspect of damage, which is essential in order to predict failure not only for non-proportional tests, but also for some proportional tests subjected to multiaxial loadings.

2.4. Definition of final failure The definition of laminate failure depends on the industrial application. Only tubes are addressed in this study. Obviously, fibre failures (tension and compression) and transverse compression failure (inducing an explosive wedge effect) are considered as ultimate failure for laminates. However, for [±h]s laminate tubes, failure could occur without the previously described catastrophic ply failures. Moreover, the commonly used last-ply failure approach [17] underestimates dramatically the experimental final failure of [±h]s laminates for some loading conditions (for example [±55]s laminates under Rxx/Ryy = 1/2 loading where the first ply failure is observed (through oil leaking) at 40% of the final failure load [18]). In order to predict the final failure of this kind of laminate, a maximum macroscopic strain criterion is widely used [9,19]. Nevertheless, this criterion being not connected with the state of degradation of the laminate, it could not predict accurately the final failure of [±h]s laminates subjected to creep tests (where there would be important strains without any ply failure). It is thus necessary to propose a criterion which permits to quantify the effect of the different failure modes (fibre

Indicator of loss of macroscopic rigidity : Es

1

203

and interfibre failure) on the macroscopic behaviour, in order to determine which ply failure mechanisms are catastrophic for the laminate. We proposed to use a normalized and directional indicator (Es) which permits to quantify by a scalar value the loss of macroscopic rigidity in the loading directions (even for multiaxial tests), defined by R: S macro :R 0 t R: e macro S :R t

Es ¼

ð17Þ

where S macro and e S macro are respectively the initial compli0 ance of the laminate and the macroscopic degraded compliance due to the ply failures in the laminate and where R is the macroscopic stress in the loading axes. If the laminate is undamaged, the indicator remains equal to 1. When some cracks appear in plies, the macroscopic compliance is progressively increased due to the degraded mechanical properties of the failed plies; the indicator decreases progressively and tends asymptotically to 0. Fig. 5 presents the evolution of the indicator of loss of macroscopic rigidity in a [±55]s Eglass/MY750 laminate for different loading ratios under biaxial macroscopic stresses (Rxx/Ryy). For the loading ratios (1/0) and (1/1), after the first ply failure (transverse tension failure), the rigidity of the laminate decreases significantly (under 50%), this transverse interfibre failure is thus catastrophic for the laminate. Nevertheless, for the loading ratios (1/2) and (1/3), the rigidity progressively decreases as a function of loading. These transverse tension failures are not catastrophic for the laminate, contrary to fibre failures which lead to a sudden and important loss of rigidity of the laminate. In the following, a laminate, which has lost more than 50% of its macroscopic rigidity, is considered as broken. The loss of macroscopic rigidity criterion is thus defined as Es 6 0.5.

First ply failure : Transverse tension

0.9

Fibre tension failure Fibre failure in tension

0.8

0.7

Σ xx /Σ yy=1 / 0 Σ xx /Σ yy=1 / 1 Σ xx / Σ yy=1 / 2 Σ xx / Σ yy=1 / 3

0.6

0.5

0.4 0

100

200

300

400 500 600 Macroscopic Stresses : Σ MPa

700

800

900

Fig. 5. Evolution of macroscopic rigidity for different loading ratios on the [±55]s Eglass/MY750 laminate.

204

F. Laurin et al. / Composites: Part A 38 (2007) 198–209

3. Test results and discussion

800

In this section, the predictions of the present failure approach are compared with some test data found in the literature. Three different kinds of laminates are investigated under uniaxial or multiaxial loadings: (i) ‘‘classical’’ laminates ([0/90]s and [0/±45/90]s), (ii) [±h]s laminates ([±45]s and [±55]s) and (iii) other laminates such as [90/ ±30]s which fail through different mechanisms. In the following, these test data are divided into two main classes: (i) stress vs. strain curves to failure for laminates under biaxial and uniaxial loading and (ii) biaxial failure envelopes of laminates. In the following, because of a lack of information on the manufacturing process of the laminates, the stress free temperature T0 is assumed to be reduced by 50% to take into account the effect of moisture absorption after cooling (due to atmosphere humidity) on the residual thermal stresses. The non-linear viscoelastic model has been identified from the stress/strain curves of the unidirectional plies [20]. The value of the different parameters of the multiscale failure approach for the Eglass/epoxy [20,21] and carbon/ epoxy [22,23] materials, studied in the present paper, are reported in Table 1.

700

2.

3.1. Macroscopic behaviour to final failure The macroscopic behaviour of a [0/±60/0]s T300/ 5208 laminate [23] under uniaxial macroscopic loading (Rxx/Ryy = 1/0) with a stress loading rate R_ xx ¼ 1:7 MPa/s is given in Fig. 6. This laminate has a linear behaviour to final failure (the laminate behaviour is mainly governed

Experimental failure : Σxx=788 MPa Experimental : Exx xx Experimental : Eyy yy Simulation Numerical : EEyy yy xx xx Simulation Numerical : EExx xx yy yy

600 500

Σxx MPa

2.

400

1.

1.

300 200 Failure modes 1. Trans. tension ±60˚ ±60° plies 2. Longi. tension 0 0˚° plies

100 0 -4

-2

E Eyyyy

0

2

4

6 Exx xx

8

10 --3

x 10

Fig. 6. Macroscopic behaviour of the [0/±60/0]s T300/5208 laminate under Rxx/Ryy = 1/0 uniaxial loading.

by the elastic behaviour of the 0 plies). The prediction of the final failure of the laminate under longitudinal tension (i.e. fibre failure) is not modified by the presence of viscosity (the longitudinal behaviour is dominated by fibre linear elastic behaviour) and by the thermal residual stresses (the TRS have negligible values as compared with longitudinal strengths), whereas they have a strong effect on the prediction of the first ply failure (FPF). Nevertheless, it is necessary to take into account the effect of the ply failures (transverse tension of ±60 plies) in order to estimate (in a conservative way) the final failure of this laminate. Finally, the prediction of the present approach is in good

Table 1 Mechanical and thermal properties of the different materials AS4/3501-6

T300/5208

Eglass/LY556

Eglass/MY750

E1 E2 m12 G12

126,000 MPa 11,000 MPa 0.28 6600 MPa

160,000 MPa 11,500 MPa 0.3 6500 MPa

53,480 MPa 17,700 MPa 0.278 5830 MPa

45,600 MPa 16,200 MPa 0.278 5830 MPa

aL aT T0

1 · 106 C1 26 · 106 C1 60 C

1 · 106 C1 26 · 106 C1 60 C

8.6 · 106 C1 26.4 · 106 C1 60 C

8.6 · 106 C1 26.4 · 106 C1 60 C

n0 nc c n b22 b66

8 5 32 2 0.001 0.1

8 5 1.9 3 0.1 0.9

7 6.5 5.8 3 0.35 0.9

7 6.5 3.3 2 0.1 0.8

Xt Xc Yt Yc Sc

1950 MPa 1000 MPa 48 MPa 165 MPa 79 MPa

1619 MPa 900 MPa 49 MPa 140 MPa 76 MPa

1140 MPa 570 MPa 40 MPa 135 MPa 61 MPa

1280 MPa 800 MPa 40 MPa 145 MPa 73 MPa

a b h

10 1.5 0.01

10 1.5 0.05

10 7 0.1

10 7 0.11

F. Laurin et al. / Composites: Part A 38 (2007) 198–209

700 R

3.

Experimental failure : Σ xx=610 MPa

3.

100

Experimental failure : Σ yy=95 MPa

2.

2.

90 80 1.

1.

70

Σyy MPa

agreement with experimental data for macroscopic behaviour and final failure. Consequently, for carbon/epoxy laminates composed of at least one layer in the loading direction (as [0/±60/0]s or quasi-isotropic [0/±45/ 90]s), the macroscopic behaviour is quasi-linear to the final failure which occurs in the fibre failure mode. The macroscopic behaviour of a [0/90]s Eglass/MY750 laminate [18] under macroscopic uniaxial loading (Rxx/ Ryy = 1/0) with a stress loading rate R_ xx ¼ 1:7 MPa/s is given in Fig. 7. The macroscopic behaviour becomes nonlinear after the first ply failure because the mechanical properties of the failed plies (90 plies in transverse tension) are progressively degraded thereby leading to a decrease in the laminate stiffness determined by homogenisation. This test case demonstrates the necessity to take into account the effect of plies failure on the prediction of the behaviour of the laminate and also on the prediction of the final failure. For the [0/90]s Eglass/epoxy laminate, due to constrained Poisson’s effect, the 0 plies fail first in transverse tension, the cracks (the so-called splitting cracks) are parallel to both the fibres and the loading direction. The matrix of the 0 plies is thus degraded and its longitudinal tensile strength decreases, which leads to the prediction of a final failure at RRxx ¼ 608 MPa (close to the experimental failure stresses at 609 MPa). Without this coupling, the final failure stress is predicted at Rxx = 675 MPa, a value which is close to predictions of other failure approaches in the WWFE [9,19]. We have demonstrated that, as the other ‘‘best’’ failure approaches found in the literature, the present approach describes very well the macroscopic behaviour and the final failure of classical laminates (in carbon/epoxy and Eglass/ epoxy) subjected to (multiaxial) tensile loadings. The macroscopic behaviour of a [±45]s Eglass/MY750 laminate subjected to a macroscopic biaxial loading (Rxx/ Ryy = 1/1) under a strain loading rate E_ xx ¼ 2:78

205

60 50

Failure modes 1. Shear ±45° plies 2. Loss of macro. rigidity

Experimental : E yy yy yy Experimental : E xx xx xx Simulation Numerical ::EEyy yy yy yy Simulation Numerical ::EExx xx xx xx

40 30 20 10 0

-0.1 .1

Exx xx

-0.05 .

0

0.05

Eyy yy

0.1

Fig. 8. Macroscopic behaviour of the [±45]s Eglass/MY750 laminate under Rxx/Ryy =  1/1 biaxial loading.

105 s1 [24] is given in Fig. 8. For this particular loading, the ±45 plies are mainly (there are TRS in each ply) subjected to plane shear loading. The introduction of viscosity, which describes the non-linearity of the shear behaviour at the mesoscopic scale, is essential to predict accurately the macroscopic behaviour and also the final failure. The prediction of the macroscopic behaviour until the final failure is in good agreement with the experimental data. The predicted final failure stress is Ryy = 96 MPa (instead of 95 MPa from test results) and the final failure strain is Eyy = 9% (instead of 10% for test data). It is worth mentioning that the final failure of this laminate is not due to the first ply failure (in-plane shear failure of ±45 plies), but to the catastrophic loss of macroscopic rigidity in the loading directions (Eq. (17)). The last-ply failure approach [17] underestimates dramatically the experimental final failure (point 1 in Fig. 8). Thus, this test case emphasises the necessity to define the final failure of a laminate, which is not reduced to the fibre mode, especially for [±h]s laminates where specimens could be broken without ‘‘catastrophic’’ failures.

600

Σxx MPa

400

300

3.2. Failure envelopes of different laminates

Experimental : E xx Experimental : E yy Simulations E Numerical : E:xx xx xx Simulations E Numerical : E:yy yy yy

500

2.

2.

200

100 1.

0 -0.005 E 0 Eyy yy

1.

0.005

Failure modes 1. Trans. Tension 90° plies 2. Trans. Tension 0° plies 3. Longi. Tension 0° plies

0.01 E 0.015 xx Exx

0.02

0.025

0.03

Fig. 7. Macroscopic behaviour of the [0/90]s Eglass/MY750 laminate under Rxx/Ryy = 1/0 uniaxial loading.

The biaxial failure envelope of a [0/±45/90]s AS4/ 3501 laminate, under combined Rxx and Ryy macroscopic stresses, and test data found in the literature [22,25–28] are reported in Fig. 9. First, Fig. 9 shows that the predicted failure envelope is in very good agreement with test data for multiaxial tensile loadings. The macroscopic envelope is very similar to a macroscopic maximum strain failure envelope, because the final failures for this laminate are attributed to fibre failures (in tension or compression). It is important to note that the first ply failure occurs long before the final failure in a tension fibre mode. Consequently, a first-ply failure design method for a quasi-isotropic laminate underestimates the strength of this laminate

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1000

Initial failure il Final i failure

Σyy MPa

800 600

[22] Swanson & Christoforou [22] Swanson & Trask [28] Swanson & Nelson [27] Colvin & Swanson [26] Swanson & Colvin [29]

400 200 0 -200 -400 -600 -800

-600

--400

-200

0

200 Σ xx MPa xx

400

600

800

1000

Fig. 9. Biaxial failure envelope of the [0/±45/90]s AS4/3501 laminate under combined Rxx and Ryy macroscopic stresses.

which is used only at 30% of its real strength potential. The experimental dispersion of the ultimate macroscopic failure stresses is very important for tests under compressive loadings (733 MPa < Rxx <  305 MPa), these tests are difficult to realize because of structural phenomena such as buckling. The present material failure approach overestimates the laminate strength for bi-compressive loadings. It is observed experimentally [27] that these premature failures are due to buckling of the tubes subjected to axial compression and external pressure. The biaxial failure envelope of a [±55]s Eglass/MY750 laminate, under combined Rxx and Ryy macroscopic stresses, and test data found in the literature [29] are given in Fig. 10. In the tension–tension quadrant, the predicted failure envelope is conservative. The failure of the laminate for loading ratios between (Rxx/Ryy = 1/0) and (Rxx/Ryy = 2/3)

is due to an important and sudden loss of macroscopic rigidity (Eq. (17)). For these loadings, the interfibre failure (shear and transverse tension) are catastrophic for the laminate. For loading ratios between (Rxx/Ryy = 2/3) and (Rxx/ Ryy = 1/4), the fibre failure leads to final failure. The coupling between the degradation of the matrix and the longitudinal tensile strength is very important in order to obtain conservative values, contrary to the other failure approaches reported in the literature. For ratios between (Rxx/Ryy = 1/4) and (Rxx/Ryy = 0/1), the model predicts that the transverse compression failure is catastrophic for tubes because this kind of ply failure induces an important delamination, leading to crack coalescence and to a loss of sealing for the tube. Nevertheless, for tubes with liner, the transverse compression failure is not catastrophic; the predicted macroscopic failure stresses for tubes with liner are thus too conservative for these experiments. In the tension–compression quadrant, the laminate fails in transverse compression failure and the prediction are in very good agreement with experimental data. Finally, in the compression–compression quadrant, the predictions are too conservative, but the test data are obtained from laminates with a fibre volume fraction Vf = 0.68 instead of Vf = 0.6 for the other tests (in tension), the material of each ply is thus not the same (different behaviour and failure properties). Anyway, the predicted shape seems realistic and conservative. It should be noted that the final failure modes are various and that there is an important variation in the strength of this laminate as a function of the applied loading. In fact, the macroscopic ultimate stress lies between RRxx ¼ 78 MPa for uniaxial tension and RRxx ¼ 810 MPa for tension/internal pressure for a loading ratio (Rxx/Ryy = 1/3). The last test case, presented in Fig. 11, is the biaxial failure envelope of a [90/±30]s Eglass/LY556 laminate,

400

Σ yy MPa

300 200 100 0 --100 IInitial Failure Final failure Test data

--200 --300 -400

-200

0

200

400

600

800

Σxx MPa

Fig. 10. Biaxial failure envelope of the [±55]s Eglass/MY750 laminate under combined Rxx and Ryy macroscopic stresses.

Fig. 11. Biaxial failure envelope of the [90/ 30]s Eglass/LY556 laminate under combined Rxx and Ryy macroscopic stresses.

F. Laurin et al. / Composites: Part A 38 (2007) 198–209

under combined Rxx and Ryy macroscopic stresses, and test data reported in [21]. In the half-plane (Ryy P 0), the failure stresses are in very good agreement with the test data, especially in the tension–tension quadrant, due to coupling between matrix degradation and fibre strength which permits to obtain conservative failure predictions. Nevertheless, the predictions of final failure in compression (Ryy < 0 external pressure) overestimate dramatically the experimental stresses (as all the failure approaches reported in the literature). The premature failures in the compression–compression quadrant are due to buckling [21] which can not be described by a material failure approach. A correct modelling of the failure of laminates subjected to multiaxial compressive loadings necessitates considering the laminate as a structure.

and the effect of failure (different for closed and opened cracks) on the mesoscopic behaviour. It is worth mentioning that the present progressive degradation model is able to take into account the unilateral aspect of damage to describe accurately complex loading/unloading paths. The present progressive multiscale failure approach is able to predict macroscopic stress/strain curves and failure envelopes of laminates from the thermo-mechanical properties of the elementary unidirectional ply. These predictions are in as good an agreement with the experimental test data found in the literature on classical ([0/90]s or [0/±45/90]s) laminates under (multiaxial) tensile loadings (see Fig. 12), as the ‘‘best’’ approaches presented in the ‘‘World Wide Failure Exercise’’ [30]. Moreover, con-

4. Conclusion and perspectives

a Experimental failure : Σyy=847 MPa

900 800 E xx 700

EEyyyy Σyy MPa

600 500 400 300

Present approach Tsai's approach Zinoviev's approach Puck's approach Test results

200 100 0

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Macroscopic strain

Fig. 12. Comparison of the previsions of macroscopic behaviour of the [0/±45/90]s AS4/3501-6 laminate under Rxx/Ryy = 1/2 biaxial loading with different failure approaches.

Experimental failure : Σ =95 MPa

100

yy

90 80 70

Σyy MPa

In order to predict the ply failure sequence in a laminate until the final failure, a multiscale progressive failure approach, defined at the scale of the elementary unidirectional ply, was developed. The present mesoscopic failure criterion, based on the Hashin’s assumptions, introduces a distinction between the fibre failure and the interfibre failure modes. Two main improvements have been included in the interfibre failure criterion: (i) a better description of the strength of the UD ply under shear and transverse loadings and (ii) the use of a degradation variable, based on classical damage models, to model the degradation of the interfibre strengths due to premature single fibre failures (statistical effect on fibre strength). Several authors have demonstrated experimentally that the level of the degradation of the matrix has a strong influence on the longitudinal tensile strength of the UD ply. Consequently, a coupling between the efficient longitudinal tensile strength and the interfibre degradation of the matrix has been introduced in the fibre failure criterion. The biaxial failure envelopes of UD plies are in good agreement with test data found in the literature. It has been evidenced that it is essential to describe accurately the mesoscopic behaviour in order to predict ply failures. Thermo–viscoelastic behaviour is used to model the behaviour of polymer matrix composites. In the present investigation, the thermal aspect was introduced in order to take into account the thermal residual stresses which could have a strong effect on the prediction of ply failures. Moreover, the viscosity of the matrix is addressed in order to describe creep and relaxation tests and take also into account the effect of the loading rate which could have a strong influence on the prediction of the macroscopic behaviour and of the final failure. The degradation model, based on thermo-dynamical damage models, decreases progressively the effective stiffness of the failed ply in the laminate. Degradation due to fibre failure and degradation due to interfibre failure are characterised separately. For each degradation model, a distinction is made between the kinetics of degradation (i.e. evolution of the crack density)

207

60 50 40 Present approach Tsai's approach Zinoviev's approach Puck's approach Test results

30 20 10 0

-0.1

-0.05

Exx xx

0

0.05

Eyy

0.1

Fig. 13. Comparison of the previsions of macroscopic behaviour of the [±45]s Eglass/MY750 laminate under Rxx/Ryy =  1/1 biaxial loading with different failure approaches.

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Fig. 14. Comparison between finite element results on a [±55]s Eglass/MY750 thick tube subjected to axial compressive loading and test data [31].

trary to other failure approaches, the predictions of the present model are also in good agreement with test data on [±h]s laminates (see Fig. 13), which is mainly due to the introduction of the viscosity of the matrix and to a better definition of the final failure of a laminate. Nevertheless, the present failure approach (as other material failure approaches) overestimates the ultimate failure of laminates subjected to (multiaxial) compressive loadings because the ‘‘premature’’ failures are due to buckling of composite structures. A correct modelling of the failure of laminates subjected to multiaxial compressive loadings necessitates considering the laminate as a structure. This is the reason for which the present failure approach has been implemented into a finite element (FE) code in order to perform calculations on composite structures. The predictions of the final failure of thin tubes subjected to compressive loadings, with FE calculations performed in order to take into account the effect of buckling, will be presented in a next paper. As an example of structural calculations, the comparison between experimental data [31], realised on a [±55]s Eglass/MY750 laminate thick tube (to avoid buckling) subjected to an axial compressive loading, and finite element results are given in Fig. 14. Due to viscosity of the matrix, the macroscopic behaviour is strongly non-linear. The transverse compression ply failure leads to the final failure of the laminate tube. The prediction of the behaviour (inside and outside the tube) and the final failure is in very good agreement with the experimental data. The present approach is written under the plane stress hypothesis; the extension of this model to 3-D problems is also under investigation. Acknowledgements This work was carried out under the AMERICO project (Multiscale Analyses: Innovating Research for COmpos-

ites) directed by ONERA (French Aeronautics and Space Research Centre) and funded by the DGA/STTC (French Ministry of Defence) which is gratefully acknowledged. The authors would like to express their sincere gratitude to Dr. R. Valle, for valuable and helpful discussions. References [1] Hinton MJ, Soden PD. Predicting failure in composite laminates: background to the exercise. Compos Sci Technol 1998;58:1001–10. [2] Hashin Z. Failure criteria for unidirectional fibre composites. J Appl Mech 1980;47:329–34. [3] Cuntze RG, Freund A. The predictive capability of failure mode concept-based strength criteria for multidirectional laminates. Compos Sci Technol 2004;64:343–77. [4] Hoecker F, Friedrich K, Blumberg H, Karger-Kocsis J. Effects of fibre/matrix adhesion on off-axis mechanical response in carbon– fibre/epoxy–resin composites. Compos Sci Technol 1995;54:317– 27. [5] Lemaitre J, Chaboche J-L. Me´canique des mate´riaux solides. DUNOD edition published in collaboration with GRECO, CNRS and ONERA; 1985. [6] Al-Khalil MFS, Soden PD, Kitching R, Hinton MJ. The effects of radial stresses on the strength of thin-walled filament wound GRP composite pressure cylinders. Int J Mech Sci 1996;38:97–120. [7] Leroy F-H. Rupture des composites unidirectionnels a` fibres de carbone et matrice thermodurcissable: approche micro–macro. Ph.D. Thesis, Universite´ de Bordeaux I (Bordeaux—France); 1996. [8] Leveque D, Schieffer A, Mavel A, Chemineau N, Maire J-F. Analyse multie´chelle des effets du vieillissement sur la tenue me´canique des composites a` matrice organique, In: Perreux D, Dubois C, editors. Revue des composites et des mate´riaux avance´s. Herme`s; 2002, vol. 12. p. 139–62. [9] Puck A, Schu¨rmann H. Failure analysis of FRP laminates by means of physically based phenomenological models: Part A. Compos Sci Technol 1998;58:1045–67. [10] Bogetti TA. Predicting the nonlinear response and progressive failure of composite laminates. Compos Sci Technol 2004;64:329–42. [11] Carroll M, Ellyin F, Kujawski D, Chiu AS. The rate dependent behaviour of ±55 filament wound glass–fibre/epoxy tubes under biaxial loading. Compos Sci Technol 1995;55:391–403. [12] Petipas C. Analyse et pre´vision du comportement a` long terme des composites fibres de carbone/matrice organique. Ph.D. Thesis, UFR

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