A damage meso-mechanical approach to fatigue failure prediction of cross-ply laminate composites

International Journal of Fatigue 24 (2002) 429–435 www.elsevier.com/locate/ijfatigue

A damage meso-mechanical approach to fatigue failure prediction of cross-ply laminate composites Chingshen Li *, F. Ellyin, A. Wharmby

1

Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2G8

Abstract This paper focuses on the description and prediction of local fatigue damage development in fiber-reinforced cross-ply laminate composites. The transverse laminae and longitudinal laminae are homogenized and simulated by subcritical and critical effective laminae; and for each of them a meso-damage factor is introduced to represent the change of its in-situ elastic modulus. Based on the meso-damage theory, the redistributed local stresses in each lamina at the moment of crack saturation, or at the characteristic damage stage (CDS) are evaluated with measurable parameters of the laminate. This theory is therefore able to predict the CDS life and the stage II fatigue life of a cross-ply laminate. Experimental results of glass fiber/epoxy resin composite laminates are presented and found to be in agreement with the theoretical prediction.  2002 Elsevier Science Ltd. All rights reserved. Keywords: Composite; Effective lamina; Laminate; Local damage; Meso scale

1. Introduction High strength fiber-reinforced polymer composites are used or being considered for use in aircraft structures, vehicles and pressure vessels, etc. Consequently, increased attention is being devoted to describe and to predict the damage development in composites under monotonic and/or cyclic loading. A continuum damage mechanics framework was developed to characterize damage in the composite laminates by using the displacements arising from damage in the laminates as internal variables [1]. The overall response of the laminates to the induced damage is manifested in stiffness change, and this is expressed by a set of phenomenological constants, which can be determined experimentally. Prediction of fatigue failure of laminate composites has been an area of active investigation using damage theories, in the past two decades. The residual strength degradation [2], stiffness-degradation [3], and damage tolerance approaches are the main methods to study fatigue damage and to predict the fatigue life of composites. These global approaches are based on macro parameters such as residual strength,

* Corresponding author. 1 Deceased.

applied strain and/or modulus of laminates. Unfortunately, they are incapable of predicting a local fatigue failure in composite laminates with a specified laminate lay-up and stacking sequence, which play an important role in composite design. Inhomogeneity and anisotropy are two major structural characteristics of laminate composites, which affect damage development and fatigue life period. The inhomogeneity exists at micro-scale, such as fibers, and at meso-scale, such as laminae in the laminate composites [4]. The meso-scale inhomogeneity results from the layup of the anisotropic or orthotropic laminae. The inhomogeneity has a much greater influence on the laminate behavior under cyclic loading than under monotonic loading [5]. Local stress, damage states and the local in-situ “material properties” in each constituent member (lamina) may differ considerably from each other. This, then, poses a challenge for using the global homogenization concept of continuum damage mechanics, and for using the global approaches to predict the fatigue lives of the laminate composites. Using a local homogenization procedure and introducing a local damage factor for clusters undergoing localized damage, Li and Ellyin [6,7] proposed a meso-damage theory describing the influence of local damage development on the local stress, and local strain energy in the composites. The capability of meso-damage mechanics to predict local

0142-1123/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 1 ) 0 0 0 9 8 - 6

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Nomenclature Dg global damage factor for laminate ms , D Dms sc cr local damage factor for subcritical and critical effective lamina, respectively; Esc, Ecr in-situ elastic modulus of subcritical effective lamia and critical effective lamina, respectively; global nominal elastic modulus of laminate along x axis; Ex thickness of laminate and subcritical lamina, respectively; t, tsc s, ρ average crack distance and crack density, respectively; applied stress on laminate; σa,, σsc, σcr local stress in effective subcritical and critical effective lamina, respectively; N the number of loading cycles.

damage life in the clustering particulate composite has been demonstrated. This investigation aims at developing a meso-damage theory to illustrate the coupling effect between the local damage process and the local stress redistribution in the laminate components. The main objective of this study is to describe and to predict local fatigue damage development law in cross-ply laminate composites.

2. Effective lamina simulation 2.1. Stiffness reduction and damage stages Stiffness reduction of engineering materials, an indication of material deterioration, has been defined as a measure of damage in the continuum damage mechanics, e.g. see [8]. Obviously, a change in the stiffness reduction rate (the slope of elastic modulus versus fatigue cycles) frequently indicates a change of undergoing damage mechanisms. Fig. 1 displays the variation of

elastic modulus of a cross-ply laminate with the normalized fatigue life, N/Nf. The curve consists of two approximately linear parts: the relative time for the first one is much shorter, but the elastic modulus drops considerably in this period. The second part occupies a much greater proportion of the fatigue life, but the decrease of the modulus is much less. These two portions represent two damage stages of laminate composites, termed stage I and II, respectively. The magnitude of the elastic modulus drop in stage I and stage II for different laminates may vary slightly depending on the lay-up of the laminate. Fatigue damage mechanisms in cross-ply laminates generally consist of matrix cracking, longitudinal cracking, delamination and fiber fracture. The matrix cracking is a localized damage in the transverse ply at the first stage. The matrix cracking causes a steep drop of the elastic modulus of laminates. On the other hand, delamination, longitudinal cracking and fiber fracture mostly occur after CDS, i.e., in the second stage and are associated with the longitudinal lamina. The damage stages, which are of fundamental importance to engineering applications, are related to the two plies in cross-ply laminates, and have been termed as subcritical and critical elements [9]. 2.2. Effective lamina

Fig. 1. A schematic diagram indicating the variation of the elastic modulus of a cross-ply laminate during fatigue life

The material homogenization of laminate composites plays a fundamental role in the classical laminate theory, which assumes that each lamina is homogeneous and orthotropic [10]. Through the laminate theory, the local stress and strain in each lamina due to applied mechanical and thermal loads can be evaluated. This evaluation is only valid until local damage initiates in the laminate. A local homogenization method has been shown to be effective in considering the interaction among the local damage, the local microstructure and the local stressstrain [6]. In this study, the laminae having the same orientation and undergoing the same localized damage are simulated as a homogeneous and orthotropic medium with a local

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damage factor. This medium is termed an ‘effective lamina’. The meso-damage factor is a measure of the local damage, or the change of in-situ elastic modulus of the corresponding lamina or laminae. For example, the local damage factor of the effective lamina corresponding to transverse laminae in a cross-ply laminate, represents the influence of matrix cracking. Fig. 2 schematically displays the transformation of the transverse laminae of a cross-ply laminate undergoing matrix cracking into an effective lamina. The longitudinal laminae in the cross-ply laminates can also be simulated as an effective lamina, especially when local damage starts to develop within it. The insitu modulus change of the longitudinal laminae is also described by a meso-damage factor. Further description of these meso-damage factors will be given in the following sections. Based on a similar concept to that of Reifsnider’s [9], the effective lamina representing the longitudinal plies is termed a ‘critical effective lamina’, while the ‘subcritical effective lamina’ corresponds to the transverse laminae.

tion, two meso-damage factors are defined for the subcritical and critical effective laminae. A vectorial representation for the damage in the laminate was used by Talreja [1] to formulate a continuum damage mechanics framework for laminate composites. Due to the orthotropic nature of the lamina structure, the meso-damage factor is expressed as a first order-tensor, Dms(i,j,k). The three components of Dms(i,j,k) represent the damage due to fiber fracture, matrix cracking and interface damage in a lamina for which the x1 and x2 axes are along and perpendicular to the fiber direction, respectively. However, for a cross-ply laminate under uniaxial loading, the dominant components of damage factors for the subcritical and critical effective laminae are Dmssc(0, 1, 0) and Dmscr(1, 0, 0), respectively. To simplify the notation, these components of damage factors for the subcritical and critical effective laminae, will ms be denoted as Dms sc and Dcr . The meso-damage factor of the subcritical effective lamina is defined as the in-situ modulus change of the corresponding transverse ply in a laminate:

3. Theory

ms Esc(0)−Esc (r) Dms sc ⫽ Esc(0)

3.1. Meso-damage factors It was stated earlier that the fatigue damage in a crossply laminate occurs in two stages. The first one takes place in the transverse lamina, and the second is mostly restricted in the longitudinal lamina. To describe their lamina structure-dependency and to reveal their interac-

(1)

in which Esc(0) is the elastic modulus of the corresponding transverse ply without cracks, and Escms(ρ) is the elastic modulus of the subcritical effective lamina corresponding to the transverse ply with a crack density ρ. The Escms(ρ) indicates the remaining load-carrying capability of the cracked transverse lamina. Similarly, the meso-damage factor for the critical effective lamina is correlated with the in-situ elastic modulus change in the corresponding longitudinal lamina, i.e. ms Ecr(0)−Ecr Dms cr ⫽ Ecr(0)

(2)

in which Ecrms is the in-situ elastic modulus of the longitudinal lamina, due to delamination, longitudinal cracking, fiber fracture, etc. within the lamina. Based on an approach similar to that of Talreja [1] for a cross-ply laminate, we assume Dms cr (r)⫽

atsc ⫽artsc s

(3)

in which α is a constant, s, ρ and t are the average crack distance, crack density and the thickness of the effective transverse ply, respectively. Note that in the above the damage factor is only a function of the crack density and the thickness of the subcritical effective lamina. Substituting Eq. (1) into the above yields Ems sc (r)⫽Esc(0)(1⫺artsc) Fig. 2. Simulation of a transverse lamina undergoing progressive matrix cracking via an effective subcritical lamina with a local damage factor

(4)

The dependence of in-situ elastic modulus of the transverse ply on crack density, and the ply thickness indi-

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cated in the above equation, has been experimentally confirmed by Ryder and Crossman [11] and Lee, Daniel and Yahiv [12]. For a given laminate lay-up [0m/90n]s, the crack density at the characteristic damage state (CDS) is known or can be measured. So is the Escms(CDS). Hence, the Escms (CDS) can be viewed as an important variable indicating the damage stage transition. The relationship between the elastic modulus of a laminate, Ex and the laminate’s constituent moduli is determined by the lay-up of the laminate, and is given by

冋

t−tsc tsc Ecr⫹ E (r) Ex(r)⫽ t t−tsc sc

册

(5)

in which, t is the laminate thickness. The overall response of a structural body to the local damage of its constituents can be obtained through a spatial average of the local damage over the whole body [13]. In the first damage stage of a laminate, the global damage factor of the laminate, Dg , therefore, can be expressed as: tsc Dg⫽Dms sc t

(6)

The above equation shows that at the first damage stage, the global damage factor is proportional to the local damage and its relative thickness. At the second damage stage, the global damage factor can be obtained as a summation of the local damage in each lamina weighted by their volume fraction, that is, Dg⫽[tscD ⫹(t⫺tsc)D ]/t ms sc

ms cr

(7)

evaluated by using the above. This stress will be used to predict the CDS life. The average stress in the critical effective lamina can be expressed as sa[tEx−tscEsc(r)] scr⫽ Ex(r)(t−tsc)

(9)

Substitution of Eqs. (1) and (5) into the above yields Ecr sa scr⫽ tcr Ex0 1− Dms t sc

冉

冊

(10)

The above equations indicate that the local average stress in the critical lamina increases while the stress in the subcritical lamina decreases as damage develops in a laminate under a constant amplitude cyclic loading. The local stress redistribution from the transverse ply to the longitudinal ply depends on the damage development in the transverse ply and the relative thickness (tcr/t) of the ply in the laminate. The ratio of fatigue cycles of the CDS over fatigue life for [90n/0m] cross-ply laminates was found to be less than 20% [15]. This suggests that a drastic stressredistribution occur in a relative short period in early life. After the CDS, the local stresses in each lamina are stabilized. The redistributed stress in the critical lamina at the CDS is: scr,CDS⫽

tEcr s (t−tsc)Ecr+tscEsc,CDS a

(11)

The redistributed stress in the critical lamina at the CDS will be applied to predict the stage II fatigue life of laminate composites.

3.2. Local stress redistribution and damage globalization 4. Applications The local stress redistribution occurs with the local damage development, as well as local geometry change [14]. In this study, the local geometry change is negligible, and the dependency of local stress variation with the damage development for each lamina of the crossply laminate under a constant amplitude cyclic loading will be investigated. From the classical laminate theory, the average stress in the subcritical effective lamina can be obtained from Esc(r) s ssc(r)⫽ Ex(r) a

(8)

in which Ex(ρ) is the elastic modulus of the laminate along the x-direction with transverse crack density ρ, and σa is the applied stress. At the CDS, the saturated crack density can be measured. The Esc,CDS(ρ) and Ex,CDS(ρ) at that stage can be determined by using Eqs. (4) and (5), respectively. Therefore, the redistributed stress in the subcritical effective lamina at this stage, σsc,CDS, can be

4.1. Stage I (CDS) life prediction The CDS is a well-defined damage state in laminate composites indicating matrix crack density saturation in the transverse laminae, or the termination of localized damage in the laminae. There are several investigations on the crack saturation in different laminate lay-ups showing that the crack density at the CDS is a laminate property, or it is dependent only on the laminate lay-up and the material properties of the composite constituents, e.g. see ref. [3]. Although stage I fatigue life occupies a relatively short proportion of fatigue life of a transverse laminate, the prediction of CDS life of transverse laminate may shed light on the development of a general methodology for the cross-ply laminates or for composite structures [16]. Moreover, for some composite structures subjected to extreme low frequency-cyclic loading, such as press-

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ure vessel, the CDS life could be the only useful design life, because of possibility of fluid leakage. Hence, the prediction of CDS life in such cases is critical for a safety design. The localized damage development corresponding to the matrix cracking is governed by the local stress, σsc, in the transverse effective lamina, i.e. dDms sc ⫽␬tsscn dN

(12)

in which ␬t and n are constants for the laminate. As the increase of matrix cracks causes the decrease of σsc, then according to (12) the damage growth rate would tend to decrease. The in-situ damage process of the transverse laminae has been assumed to be the same as the unidirectional transverse laminae [12]. Based on this assumption, the CDS life of the cross ply laminates is predicted by using the S–N curve of the transverse unidirectional laminate, which can be approximately expressed as a power function or a polynominal on a log-log scale [12,17]. Notable differences between the in-situ damage process of unidirectional transverse laminates and the transverse laminae in the cross-ply laminates exist in the following two aspects. First, the damage process in the former one is controlled by a constant applied maximum cyclic stress (nominal) for load control tests or a constant applied strain for strain control tests. In contrast, the local maximum cyclic stress in the transverse laminae of cross-ply laminates decreases considerably during the load cycles before the CDS. Secondly, a single crack causes failure of a unidirectional transverse ply, and the number of cracks in the unidirectional laminae at failure is generally less than that in the cross-ply laminates’ 90° laminae due to the constraint of the 0° laminae. To include the influence of these two factors, the N(CDS) is predicted by using a modified S–N curve of unidirectional laminae[9020] given in ref. [17], N(CDS)⫽bt(s¯ sc)m⫹Ct

(13)

in which the exponent m, coefficient βt and constant Ct are the same as those of the unidirectional transverse laminae. s¯ sc in the above is the average maximum local cyclic stress in the 90° laminae of the cross-ply laminate, 1 and is approximately given by ssc= [ssc(0)+ssc,CDS]. 2 4.2. Stage II fatigue life prediction It was mentioned earlier that the second damage stage occupies a major proportion of the fatigue life of a laminate, and is controlled by the damage process around and within the critical lamina. In the second damage stage following the CDS, the damage in the sub-critical lamina is fully developed and the redistributed stress is stabil-

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ized. This represents a marked difference from the stage I damage during which the local stresses change continuously and in a significant amount. The second stage fatigue damage development in laminates is governed by the redistributed maximum cyclic stress in the critical effective lamina at the CDS, and can be expressed as, dDcr ⫽␬l(scr,CDS)n⬘ dN

(14)

in which ␬l and n⬘ are constants for the laminate. This damage process has been assumed to be the same as the fatigue process of the corresponding unidirectional longitudinal laminae, and has been predicted by using the S–N curve of the latter. Experimental results have shown that the S–N curve of the corresponding unidirectional longitudinal laminae [18] can be written as Nf⫽bl(s1max)m⬘⫹Cl

(15)

in which βl, Cl and m⬘ are constants for the laminae. Hence, the second stage fatigue life of a laminate can be simply obtained by substituting the σ1max in the above with the redistributed maximum cyclic stress at CDS, σcr,CDS in the laminate. As the fatigue lives in the stage I and II of the laminate are determined separately from Eqs. (13) and (15), the total fatigue life of a laminate, obviously, will be a summation of them. For laminates under a high cyclic stress amplitude, the second stage of fatigue damage would likely start before the CDS. However, the percentage of the overlap of the first with the second damage stage over the entire fatigue life would be relatively small. Neglecting this portion will not appreciably influence the accuracy of the fatigue life prediction, but will considerably reduce the calculation time. The meso-damage theory emphasizes the roles of the transverse lamina and longitudinal lamina in the laminate damage process. In this regard, it is similar to the concept of the critical and subcritical elements of Reifsnider[9]. However, the use of local damage factors makes it possible to characterize the local damage processes, to identify the condition of the damage mode transition, and to predict the fatigue life. 5. Comparison with experimental results Two groups of cross-ply laminate coupon specimens of lay-up [02/903]s and [02/90]s were tested under cyclic loading. The specimens were made from 3M Scotchply type 1003 continuous E-glass-fiber/epoxy-resinreinforced plastic. The coupon specimens were cured at 1500C for 12 hrs. The final dimensions of the coupon specimens were 204x22x2.5mm. More details of the specimens and fabrication procedures can be found in ref. [19].

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Fatigue tests were conducted under load-control at room temperature. The test frequency was 3 Hz with load ratio, R=σmin/σmax=0.1. During fatigue tests the crack density was measured periodically either by an insitu imaging process or by a microscope observation on the edge of specimens. Both methods were used to complement each other. Each fatigue life datum was generally obtained by using one specimen. However, in case when a datum, or the failure appearance of a specimen looked abnormal, the datum was replaced by using a second specimen loaded at the same condition. It is found that the saturated crack density for [02/903]s specimens was 0.91/mm, while for the [02/90]s specimens was 1.6/mm. The effect of transverse ply thickness on the crack density in cross-ply laminates has been described by Crossman and Wang[20]. Eq. (4) is used to determine the in-situ elastic modulus of transverse ply at the CDS. The constant in the eqn., α= ⫺0.7143, was obtained from the best fit to the experimental data. The elastic moduli were used to determine the redistributed stresses from Eq. (11). The experimental fatigue life data of unidirectional transverse laminate [9020], was taken from ref. [20], and was used to predict the CDS life of the laminates. The experimental S–N curve of the corresponding unidirectional longitudinal glass-fiber/epoxy lamina [08] is shown in Fig. 3. Figs. 4 and 5 show the experimental data of stressfatigue lives of [02/903]s and [02/90]s specimens, respectively. The theoretical prediction is also shown in the figures. It can be seen that the predicted fatigue life is in good agreement with the experimental data for both groups of specimens. This indicates that the meso-damage theory properly describes the two dominant damage processes in the laminate composites.

Fig. 4. Predicted and measured fatigue life data of [02/903]s laminate specimens

Fig. 5. Predicted and measured fatigue life data of [02/90]s laminate specimens

6. Conclusions A damage meso-mechanics framework for composite laminates is presented to describe and to predict local fatigue damage development in cross-ply laminates. Experimental study of fatigue damage development in glass fiber-reinforced cross-ply laminates has also been conducted and the results have been compared with the theoretical predictions. The following conclusions are drawn:

Fig. 3. The S-N curve of a unidirectional longitudinal glass fiber/epoxy laminate[08]

1. The transverse plies undergoing progressive cracking in cross-ply laminates are homogenized and simulated with a subcritical effective lamina, for which a mesodamage factor is introduced representing the local damage state due to matrix cracking. A similar simulation is also performed for the longitudinal laminae. This simulation provides a logical approach to ana-

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lyze the local stress-strain, as well as the local damage development. 2. The meso-damage theory correlates the local stress redistribution with the local damage development, and the in-situ material’s properties of the laminate composites, so that it facilities the prediction of the local damage lives, and fatigue lives of the laminates. 3. Redistributed stresses at CDS play a key role in the prediction of both stage I(CDS life) and stage II fatigue lives of a laminate. Basic data required for this prediction include measurable crack density of the laminate at CDS, stress-fatigue life data (S–N) of unidirectional transverse and longitudinal laminae. However, a modification is to be incorporated for the laminate’s CDS life prediction to reflect the influence of the local stress variation and the longitudinal laminae’s constraint in the laminate. 4. The experimental results of fatigue lives of glass fiber/epoxy resin cross-ply laminates of different 90° laminae thickness are presented showing a good agreement with the theoretical prediction.

[5]

[6]

[7]

[8] [9] [10] [11]

[12]

[13] [14]

Acknowledgements The financial support provided by the Natural Sciences and Engineering Research Council of Canada is acknowledged.

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