Matrix cracking in laminated composites: A review

Composites Engineering. Printed in Great Britain.

Vol.

MATRIX

I, No. 6. pp. 337-353.

1991

CRACKING

0961-9526/91 $3.00+ .Otl ‘C 1991 Pergamon Press plc

IN LAMINATED A REVIEW

COMPOSITES:

SERGE ABRATE Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, Rolla, MO 65401, U.S.A. (Received

4 June

1991; final

version

accepted

4 September

1991)

Abstract-This article presents an overview of the problem of matrix cracking in laminated composite materials. The parameters governing the onset and accumulation of internal damage under several types of loadings are reviewed. The analysis of laminates with matrix cracks can be performed using a shear-lag model, variational, elasticity, finite element or finite difference approaches. Prediction of further cracking is based on either a fracture mechanics approach or a probabilistic failure strength theory. Continuum damage mechanics are used by some investigators to model the behavior of the cracked laminate. Recent developments in this area are summarized here, and suggestions for future research are presented.

INTRODUCTION

Laminated composite materials are used extensively in aerospace and other applications, and extensive research has been directed toward understanding their basic failure modes. The stiffness of the reinforcing fibers being significantly larger than that of the matrix material, substantial strain magnification occurs in the matrix when a ply is loaded in the transverse direction. The strain at which failure occurs in off-axis plies is lower than that of plies aligned in the loading direction. In a laminate under tension, cracks develop in the off-axis plies as illustrated in Fig. la. Transverse matrix cracking can be followed by delamination of the 0,90interfaces starting at the top of the matrix cracks (Fig. lb). Because of a mismatch in the in-plane Poisson’s ratios of the longitudinal and transverse plies, matrix cracks can also appear in the longitudinal plies (Fig. lc). A similar type of damage also occurs in laminates subjected to fatigue, thermal cycling and other types of loading.

(cl Fig. 1. Failure modes in cross-ply laminates under tension. (a) Transverse matrix cracking; (b) transverse matrix crack and interface delamination; (c) matrix splitting of 0” plies. 337

S. AERATE

338

A clear understanding of this type of damage is necessary to develop advanced materials as well as more efficient design methods. In general, composite structures are designed using the First Ply Failure (FPF) criterion according to which a laminate is considered to have failed once a single ply has failed. When matrix cracking is the failure mode for FPF, the laminate can usually withstand higher loads and cannot be assumed to have failed, even if some properties such as the Young’s modulus and the coefficient of thermal expansion have been significantly altered. Allowing for some level of damage in a laminate would lead to more efficient designs but requires knowledge of damage development mechanisms and the effect of damage on the mechanical behavior of the laminate. Multiple cracking can have a number of adverse effects: (1) it reduces the effective stiffness of the cracked plies, causing a stress redistribution in the structure; (2) it alters the thermal expansion properties of the laminate; (3) it may lead to delamination between plies or fiber failure in the 0” plies; (4) it allows for fluid weepage in pressure vessels and for ingress of moisture and other detrimental effects. In this article, we will examine the characteristics of damage accumulation in a laminate subjected to tensile, fatigue, bending and thermal loadings. The prediction of the onset of cracking and damage accumulation under increased loading can be accomplished using a number of methods to perform a detailed stress analysis of the laminate and predict failure. The behavior of the damaged laminate can be predicted either from a micromechanics analysis of the damaged laminate or from a continuum damage mechanics approach. MATRIX

CRACKING

UNDER

TENSILE

LOADING

Many experimental investigations on damage growth in laminated composites have been conducted to identify various types of damage and determine possible causes. Most studies consider only symmetric laminates, and many are restricted to cross-ply laminates. Damage development in composite laminates containing 90” plies is discussed by Bader and Boniface (1985) and Morley (1987). Damage development during monotonic tension on a cross-ply laminate follows a well-defined pattern. There is a threshold value of strain, Ed,,, at which cracking begins in the transverse ply. Ply cracking is a matrix failure mode, not a fiber-matrix interface failure, and depends directly on the failure strain of the resin (Garret and Bailey, 1977b). Matrix cracking is the result of random microcracks that formed and coalesced under load (Wang, 1986). Unlike uniaxial specimens under transverse tension for which strain to failure is independent of thickness (Kistner et al., 1985), et, for a 90” ply in a laminate varies with thickness. The influence of ply thickness and the constraining effect of adjacent plies on the onset of matrix cracking are discussed by several investigators (i.e. Stinchcomb et al., 1981). Parvizi et al. (1978) consider threelayer glass-epoxy cross-ply laminates in which the outside 0” plies have a fixed thickness of 0.5 mm, while the thickness of the center ply was varied from 0.1 to 4 mm. For inner ply thicknesses above 0.4 mm, Ed,,does not depend on thickness, and the crack propagates instantaneously across the inner ply. Notice that in this case, the 90” ply cracks for a value of et, that is larger than the transverse failure strain of a unidirectional laminate. Below 0.4 mm, the nature of the cracking phenomena changes, and et, increases significantly. Small edge cracks that multiply and grow very slowly appear at the edges of the inner ply for thicknesses of 0.25 mm. When the inner-ply thickness is reduced to 0.15 mm, only very short edge cracks (1 mm in length) are observed prior to laminate failure; for thicknesses of 0.1 mm, no cracking is observed. The transverse strain to failure, et,, increases significantly as the ply thickness decreases, as shown in Flaggs and Kural (1982). Small delaminations appear at the ends of the cracks only for thick inner plies (4 mm). Because of the constraining effect of the adjacent layers, the strength of the transverse layer appears to be higher than that measured on a uniaxial specimen. For this reason, some authors (Kistner et al., 1985; Flaggs and Kural, 1982; Whitney and Browning, 1984) have studied what is called the “in situ” strength. Flaggs and Kural (1982) determine the in situ transverse lamina strength for (0,/90,),, (&30/90,), and (+60/90,), carbon-epoxy laminates and show that it depends on the thickness of the 90” layer as well as on the

Review Article

0

2

4

339

8

n

8

I 10

Fig. 2. In situ strength of 90” plies for carbon-epoxy laminates under tension. o[O,, 90,],, +[+30,90,],, *[+60,90,], laminates (from Flaggs and Kural, 1982).

rigidity of the constraining plies (Fig. 2). Kim and Hahn (1979) study laminates in which the 90” layers are relatively thick, so that the in situ transverse strength is close to that determined for a unidirectional coupon. First-ply failure stresses and laminate strength are found to follow a Weibull distribution and to be strongly affected by moisture and curing stresses. The models used by Parvizi et al. (1978), Flaggs and Kural (1982) or Aveston and Kelly (1973) assume that the strength of the 90” ply is uniform. So, while the first crack may be arbitrarily located, the ratio of maximum to minimum crack spacing cannot exceed 2. Experiments conducted by Manders et al. (1983) showed that this ratio is often largely exceeded, particularly for low strains. This implies that the uniform strength assumption is inadequate and statistical variability must be considered. The strength of the lamina is assumed to follow a Weibull cumulative distribution function (cdf)

where ago and a are the scale and shape parameters, respectively, and V is the volume of the 90” layer. The statistical variation of matrix fracture strain in 90” plies has been the object of a number of publications by Peters and coworkers (Peters, 1984, 1986, 1989; Peters and Meuseman, 1987; Peters and Chou, 1987; Peters and Andersen, 1989). Sheard and Jones (1987) showed that reinforcing fibers are generally not uniformly distributed in the matrix, creating resin-rich and resin-poor zones which focus the apparition of matrix cracks. Since transverse matrix cracking is a matrix failure mode, it is then logical to inquire which matrix properties govern the failure process. Lee and Schile (1982) showed that obtaining low residual stresses in the cured laminate and high fracture toughness of the resin were essential for improving the cracking resistance of the material. Another study by Penn et al. (1989) confirmed this finding by showing that using an additive to reduce residual stresses did not result in improved cracking resistance because the matrix toughness was also lowered in this case. First-ply failure, in itself, is not important, except that it indicates the beginning of the damage process (Nuismer and Tan, 1982). At first-ply failure, the load being carried by the transverse plies prior to crack formation is thrown to the adjacent plies which experience an increase in stress. During this process, the laminate undergoes additional

340

S. AERATE

elongation at the same stress level which produces a “knee” in the stress-strain curve corresponding to a notable increase in strain for a constant stress level at the initial failure point (Ohira and Uda, 1985). The magnitude of this “knee” depends on the level of constraint provided by the 0” plies, as discussed by Taljera (1987). For laminates in which transverse plies are highly constrained, this “knee” is very small and can easily go unnoticed. For very low constraints, the adjacent plies cannot withstand the additional stresses, and laminate failure occurs immediately. For intermediate cases, after some additional elongation, the specimen will withstand increased stresses until further cracks appear. Garret and Bailey (1977a) demonstrate that, for cross-ply laminates, crack spacing in the transverse plies decreases with increasing stress and, in general, is not accompanied by delamination between plies. Similar observations have been reported by many investigators (Masters and Reifsnider, 1982). Eventually, an apparent saturation of crack density is reached; this crack pattern is called the Characteristic Damage State (CDS). Parvizi and Bailey (1978) present additional experimental data for glass-epoxy composites, explain the apparent discrepancy in the results of Garret and Bailey (1977a) and show very good agreement between theoretical predictions and experimental results. The evolution of crack density with increased loading generally follows a curve similar to Fig. 3. Once the mechanisms for First Ply Failure (FPF) and the development of a CDS are understood, the change in laminate properties as a function of damage must be determined. Experimental data are provided by Whitney and Browning (1984), who plot longitudinal and transverse crack density as a function of applied stress for a number of graphite/ epoxy laminates. Similar data on glass-epoxy laminates are presented by Highsmith and Reifsnider (1982). Talreja (1987) analyzes both sets of data and shows that the secant modulus of the laminate decreases monotonically after first-ply failure. Continuous reduction cannot be predicted by using the simple ply discount method in which the transverse modulus and the shear modulus for that ply are set to zero after the first crack appears. Several investigators (Harris et al., 1988; Kamimura, 1985; Groves et al. 1987; Allen et al., 1987b, 1988) report that the laminate modulus decreases almost linearly with transverse crack density. 0.6

0.5

0

50

100

150

200

250

Applied Stress @Pa) Fig. 3. Evolution of crack density with applied load for [0,90,], glass-epoxy laminates (from Kamimura, 1985).

Review

Article

341

The experimental study of Groves et al. (1987) identifies two types of matrix cracks. Curved cracks are formed after straight cracks and follow a repeatable pattern of location and orientation relative to the straight cracks. Angled cracks start either at the top or bottom of a transverse ply and progress at an angle toward the center of the ply in the direction of an existing straight crack. Partially-angled cracks are observed, and when two cracks emanating from the top and bottom of a ply join, a curved crack is formed. The number of curved cracks in cross-ply laminates is found to increase with the thickness of the 90” plies. MATRIX

CRACKING

UNDER

FATIGUE

LOADING

Generally, damage developed during the first phase of cyclic fatigue testing is similar to that produced by monotonic loading. A review of fatigue damage mechanisms in composite laminates is given by Stinchcomb and Reifsnider (1979), and Highsmith et al. (1984) discuss damage development under tension-tension, tension-compression and compression-compression cyclic loading. Jamison et al. (1984) describe the fatigue damage process under tension-tension fatigue as consisting of three phases. Phase I is similar to the damage process just described for monotonic loading. In phase II, saturation cracking is achieved in the transverse plies, and longitudinal cracks appear in the 0” plies as these plies are subjected to tension in the transverse direction. Interply delaminations occur at the intersection of longitudinal and transverse cracks. Another type of delamination initiates at the free edges of the specimen, although, usually, the laminate fails before edge delaminations can grow across the entire width. Stiffness losses during phase II are small (Allen et al., 1988; Jamison et al., 1984). Phase III is associated with rapid damage growth and localization leading to failure. Similar observations by Daniel et al. (1987) indicated that a sharp drop in laminate modulus during the first loading cycle (phase I) can be attributed to the introduction of transverse matrix cracks. The magnitude of the initial drop depends on the cyclic stress level. Phase II consists of the first 80% of the fatigue life of the specimen (log n/log N = 0.80). For the remaining 20% (phase III) of the fatigue life, stiffness losses are independent of the cyclic stress levels. A similar drop during the early stages of flexural fatigue testing of a unidirectional glass-fiber epoxy was reported by Agarwal and Joneja (1980). The flexural modulus had dropped to 30% of its initial value. Significant amounts of fiber fractures are observed in longitudinal plies during phase II at locations where the adjacent transverse plies are cracked. This can be explained by the fact that when transverse cracks are present, additional stresses are thrown upon the longitudinal plies, and the fiber stress is maximum at the crack locations (Jamison, 1985). A similar situation is observed for static loading even though fewer fiber failures are observed. Laminate stiffness reduction is thought to provide a reproducible correlation with per cent of life expanded (Highsmith and Reifsnider, 1982; Jamison et al., 1984). Lee et al. (1989) describe a procedure for predicting the fatigue life of cross-ply laminates based on the static and fatigue behavior of the unidirectional lamina. The fatigue life consists of a first part which corresponds to the development of a multiplication of damage in the 90” layers. The second part of the fatigue life after a CDS is reached for the 90” plies depends on the state of stress in the 0” plies. It is assumed that a 0” unidirectional lamina within the damaged laminate behaves like a 0” unidirectional lamina. The total fatigue life is taken as the sum of the number of cycles required to reach the CDS and the residual life after the CDS. Charewicz and Daniel (1985/1986) show a clear correspondence between the decrease in residual strength and remaining fatigue life. The normalized residual strength decreased in the same way the normalized residual modulus does as the number of cycles increases. THERMALLY-INDUCED

DAMAGE

Thermal expansion mismatch between the various plies in a laminate has been shown to cause microcracks in glass-fiber composites (Turner, 1946) and other composite materials. Spain (197 1) discussed thermally-induced microcracking of carbon-fiber

342

S. AEIRATE

composite jet engine fan blades. The core of the blades is composed of axially-aligned fibers with angled plies near the surface. The outer plies provide transverse rigidity to the blade and increase torsional rigidity. Cracking problems are encountered in this type of construction. Composite materials are currently used in space applications such as space telescopes and antennas and will be used in the future for large space structures. These materials are selected because of their low coefficient of thermal expansion (CTE) which will confer high-dimensional stability to the structure. For graphite-epoxy, the CTE in the fiber direction is slightly negative, while in the transverse direction larger positive values comparable to that of aluminum are observed. A laminate can then be tailored to obtain the desired CTE. Space structures are subjected to severe cyclic temperature changes (&250”F), and transverse matrix cracking can be induced by large thermal stresses. The presence of these cracks will affect the elastic and strength properties of the laminate as discussed earlier, but they will also reduce the coefficient of thermal expansion in a substantial manner. Adams et al. (1986) investigated thermally-induced microcracking in graphite-epoxy cross-ply laminates. Six configurations were studied: [0,9O,]z, [O,, 90,], , [O, , go],, [90/O,], , [go,, O,], and [90,/O],. The thermal load required to initiate transverse cracking and the accumulation of transverse cracks under cyclic loading between -250” and 250°F for up to 500 thermal cycles were studied experimentally. The crack density y (number of cracks per centimeter) in [O,, 90,], and [go,, O,], laminates was found to increase with the number of thermal cycles x according to y = A(1 - ehr),

(2)

where J = 3.41 x 10m3cm-’ and A = 7.3 cracks cm-‘. The temperature of transverse crack initiation showed considerable variability. Both Adams and Herakovich (1984) and Bowles (1984) used a quasi-3D finite element model of a quarter cell of a repetitive unit to determine the change in laminate CTE with crack spacing. Herakovich and Hyer (1986) used the same approach to study thermal cracking of flat laminates and tubes. Tests showed that the torsional stiffness of [90, O,, 901 graphite-epoxy tubes was significantly reduced by the introduction of these cracks. The effect of matrix cracks on the coefficient of thermal expansion of the laminate was predicted using a shear-lag approach by Lim and Hong (1989). Cracking of PEEKbased laminates designed for space applications is discussed in Barnes and Cogswell (1989). INFLUENCE

OF MATRIX

CRACKING

ON MATERIAL

DAMPING

While most investigators recognize stiffness changes as indicative of the state of damage, these variations are relatively small. The damping characteristics of the laminate will certainly be affected by the introduction of matrix cracks and other forms of damage. DiBenedetto et al. (1972) subject quasi-isotropic specimens to oscillatory loads of non-zero mean value and monitor the damping capacity at different mean stress levels. Significant changes in the damping capacity of the laminate are observed as cracks appear in certain layers. Smith et al. (1988) conducted a number of experiments on damaged cross-ply laminated beams in which transverse matrix cracking was induced by applying a tensile load in an MTS machine. For specimens with 90” plies on both the outside and inside of the laminate, the outside transverse plies accumulate damage much faster at the beginning of the loading process. The situation is then reversed at saturation levels where all the 90” plies have the same crack spacing. Each specimen is then placed on a special apparatus designed to measure the first-mode response of a cantilever beam. Damping ratios are calculated by the logarithmic decrement method. Six different laminates were tested at different levels of matrix crack damage and the effect of stacking sequence was investigated. Damage significantly increases damping in all cases. Kalyanasundaram and Allen (1988) present a model for predicting damping in laminated composites with damage due to transverse ply cracking. This work concentrates on cross-ply laminated beams using the Bernoulli-Euler model and a uniaxial constitutive equation with damage (from Allen et al., 1987a). The model assumes the damage level to be fixed and accounts

Review

Article

343

for both the damping increase and stiffness loss due to damage. The validity of the model is verified by comparison with the experimental data of Smith et al. (1988). In some instances, damping in the presence of damage has been increased 350%, while the corresponding reduction in natural frequencies is only 2%. This suggests that damping might be a more sensitive indicator of damage, as reported by Adams et al. (1973), Chandra et al. (1982) and DiBenedetto et al. (1972). FLEXURAL

DAMAGE

Few references deal with flexural damage of laminates. Rosensaft and Marom (1985) studied the three- and four-point static bending tests to composites laminates using three types of reinforcing fibers: glass, carbon and kevlar (shear force and moment diagrams, failure modes, interlaminar shear, tension compression, local buckling damage due to rollers). Schultz and Warwick (1971) impose oscillatory flexural loads of zero mean value on cantilever beam specimens. During this fatiguing, the real and imaginary parts of the complex modulus are measured periodically to detect fatigue crack damage. Crack damage is shown to induce changes in damping ratios, but no attempt is made to quantify the amount of cracking or to propose a model to predict the observed trends. Agarwal and Joneja (1979) study the progressive damage of a glass-epoxy laminate under constant deflection flexural fatigue. Progressive stiffness loss is observed until a crack develops at the root of the cantilevered specimen causing a very large drop in stiffness. Shih and Ebert (1987) study the load decay of unidirectional composite beams under four-point bending static and fatigue loadings. Extensive fractography and a discussion on the failure mechanisms in flexural fatigue are presented. PREDICTION

OF CRACK

SPACING

AND

ELASTIC

LAMINATE

PROPERTIES

Analyses of damaged composite laminates generally attempt to predict the onset of damage and its evolution as well as the influence of damage on laminate properties. Stress analyses of laminates with matrix cracks have been performed with shear-lag, elasticity, finite element, variational and finite difference methods. Failure of the off-axis layers has been examined using Griffith-type energy-balance methods, the concept of “effective flaw” distribution or a method based on a probabilistic strength distribution. Shear-lag methods (Garret and Bailey, 1977a; Parvizi and Bailey, 1978; Highsmith and Reifsnider, 1982; Lim and Hong, 1989; Reifsnider and Talug, 1980; Reifsnider and Jamison, 1982; Ogin et al., 1985) are used to analyze cross-ply laminates in which only the 90” plies are cracked, and most papers consider (O/90), laminates. Laws and Dvorak (1988) consider that the displacements are constant in each layer but vary from layer to layer, as have Masters and Reifsnider (1982) previsously. Garret and Bailey (1977a) assume a linear displacement profile in the transverse ply, while Steif is reported (Ogin et al., 1985) to have used a parabolic variation in an unpublished study. Lim and Hong (1989) also assume that displacements are constant within each layer but account for thermal loading and are able to predict the degradation of the coefficient of thermal expansion. The expression obtained by Steif for the longitudinal modulus of a [0,90], laminate with matrix cracks (Fig. 1) is

E,Eoc[l +5&LE!)!y-‘,

(3)

with I2 = 3G(b + d)Eo/(d2bE1E2),

(4)

where E, Eo, E1E2 are the elastic moduli of the cracked composite, untracked composite, longitudinal plies and transverse plies, respectively. G is the shear modulus of the transverse ply in the longitudinal direction, b and d are the thicknesses of individual longitudinal and transverse plies, respectively, and 2s is the crack spacing. For large crack spacings

344

S. ABRATE

tan As = 1 and, using a one-term Taylor series expansion, eqn (3) can be written as E=E,

l-c.;, c

I (6)

This shows that the laminate modulus decreases linearly with 1/2s, the crack density. For small crack spacings, tanh As t Ls and eqn (3) gives E = E,b/(b

+ d)

(7)

which is the result obtained by discounting the transverse ply properties. Identical results were obtained by Laws and Dvorak (1988), Han and Hahn (1989) and Kamimura (1985). For most cases, we can then assume that the laminate modulus will decrease linearly [eqn (5)] until it reaches its lower limit predicted by the ply discount method. Figure 4 shows that the experimental values of the laminate modulus obtained by Groves et al. (1987) decrease linearly with crack density, as predicted using eqn (5). The in-plane shear modulus of the cracked laminate can be predicted using eqn (5) in which E is replaced by GW (Han and Hahn, 1989). Flaggs (1985) presented a two-dimensional shear-lag model for [*0, go,], laminates subjected to general in-plane loading. The advantage of shear-lag methods is that closed-form solutions for stresses or strains can be obtained.

Number of Cracks/mm Fig. 4. Evolution

of laminate

modulus

with crack

density

for various

carbon-epoxy

laminates.

Im (1990) presented an elasticity solution for determining stresses in [0, 902, 0] and [90, 02, 901 laminates with matrix cracks under plane-strain extension. The finite element method is used by several authors (Adams and Herakovich, 1984; Herakovich and Hyer, 1986; Korczynskyj and Morley, 1981; Ohira, 1985; Sun and Jen, 1987; Allen et al., 1988) to obtain a detailed description of the stress fields. In a series of articles, Hashin (1985, 1986, 1987) presents a variational approach to this problem. In Hashin (1986), the evolution of the longitudinal modulus of a laminate subjected to in-plane normal loading and the evolution of the shear modulus under shear loading are predicted as a function of crack spacing in the 90” layer of a [90,0], laminate. This approach is also used (Hashin, 1987) to analyze orthogonally-cracked [O”,, 90”,], laminates, a typical damage state in phase II of cyclic loading. The laminate elastic

Review Article

345

modulus is obtained as a function of crack spacing, and expressions for Poisson’s ratio and the internal stresses are also obtained. The variation of laminate modulus with crack density given in Hashin (1987) indicates that, initially, losses are directly proportional to crack density and then tend to the limit predicted by the ply discount method. The damaged modulus reaches the limiting value calculated here only for very small crack spacings which may not be attainable, because laminate failure occurs shortly after crack saturation. Similar analyses were presented by Nairn (1988, 1989), and this problem is also studied by Aboudi et al. (1988) using a slightly different approach. Altus and Ishai (1986) apply the method of finite differences for their stress analysis of [90,0], laminates with both matrix cracking and delamination. Nuismer and Tan (1988) use an approximate elasticity solution to analyze the behavior of a laminate with transverse matrix cracks subjected to a general in-plane loading. These authors also provided closed-form solutions for the laminate axial modulus E, , shear modulus GU and Poisson’s ratio vxy based on the same approach (Tan and Nuismer, 1989), and also provide expressions for calculating the first-ply failure strains of laminates under tension or shear loading. Similar results were also presented by Han and Hahn (1989). The evolution of matrix cracking can be predicted using either a fracture mechanics approach or a probabilistic strength distribution approach. Parvizi et al. (1978) and Aveston and Kelly (1973) used a Griffith-type energy-balance method which states that to create a new crack, the release of strain energy must be equal to or larger than the energy formation of new crack surface. Cracks are assumed to span the entire thickness of the 90” plies, and crack growth mechanisms are not considered. This approach was also followed by several investigators (Flaggs and Kiral, 1982; Whitney and Browning, 1984; Nuismer and Tan, 1988; Tan and Nuismer, 1989; Han et al., 1988). Han et al. (1988) used a superposition approach to account for the influence of cracks and thermal stresses on the stress distribution in symmetric cross-ply laminates in tension. The perturbed stress field due to matrix cracks is obtained from equilibrium equations assuming a secondorder polynomial variation for the crack-opening displacement. The energy-balance method was then used to predict the increase in crack density. The strain for first cracking of the 90” ply in a [0,90], laminate is given by &FPC

=

1 Ez

l/4

(8)

where YF is the surface fracture energy density and G is the shear modulus in a plane normal to the fiber directions. The other parameters are defined as in eqn (3). Comparisons with experimental results published previously show reasonable agreement. Carbon-epoxy laminates are generally subjected to significant residual thermal stresses (Flaggs and Kural, 1982; Bailey, 1979) due to large differences in the coefficient of thermal expansion between longitudinal and transverse plies. As the laminate cools down from 340°F to room temperature (Flaggs and Kural, 1982) during curing, the transverse plies experience high tensile strains which must be added to the mechanically-induced strains generated during subsequent tensile testing. Similar results were reported by Bailey et al. (1979), who also show that thermal residual strains are lower for glass-epoxy laminates. Wang and coworkers (Wang et al., 1984, 1985; Wang, 1984) considered an “effective” flaw distribution in the 90” layers, The size of these flaws and their spacing both follow a normal probability distribution. The propagation of such a flaw will produce a transverse crack when the available energy release rate G becomes equal to the material’s critical energy release rate G, . The calculation of G for a specific laminate is performed numerically using a finite element model (Wang et al., 1985), and a MonteCarlo simulation procedure is used to determine the transverse crack density as a function of applied laminate stress. With the second approach, initiated by Manders et al. (1983), the strain to failure in the 90” plies is assumed to follow a two-parameter Weibull strength distribution. Fukunaga et al. (1984) used this approach in conjunction with a shear-lag model for determining the stress distribution in a cracked 90” layer and obtained expressions relating

346

S. ABRATE

the crack spacing to the applied stress and also a relationship axial strain and crack spacing. PREDICTIONS

OF LAMINATE

between the axial stress,

STRENGTH

Laminate failure, which generally is associated with the failure of the 0” plies, will be affected by the presence of transverse matrix cracks which will introduce stress concentrations in the adjoining 0” plies and reduce their strength. This effect was quantified by Sun and Jen (1987) using a two-dimensional finite element analysis and by Fukunaga et al. (1984) who used a shear-lag model accounting for the effect of thermal stresses. Using a two-parameter Weibull cumulative strength distribution, a simple expression was derived to predict the laminate strength, accounting for statistical variability and stress redistribution (Fukunaga et al., 1984). For [O,, go,], laminates, the strength of the laminate decreases as n increases since more load initially carried by the transverse plies is transferred to the 0” plies. Analyses and experiments (Fukunaga et al., 1984) indicate that the first cracking strain is very sensitive to the thickness of the 90” layer while the ultimate failure strain is not so sensitive to that parameter for [O,, go,], laminates. For [90, O,], laminates, the ultimate failure strength increases with n, but the rate of increase becomes smaller since the strength of the 0” ply decreases with thickness. This was verified by a series of tests on unidirectional specimens with increasing thickness (Sun and Jen, 1987). Results indicate significant scatter in the ultimate strength and a marked reduction in strength as the number of plies increases (Fig. 5). Inserting a thin adhesive film between each 0,90 interface helps to reduce the number of matrix cracks and the stress concentrations in the 0” plies as well as suppress delaminations resulting from cracks (Sun and Jen, 1987). 400,

1

0

1

2

3

4

Number of 0’ Plies

1

5

a

7

n

Fig. 5. Ultimate tensile strength of unidirectional [0,] carbon-epoxy laminates as a function of the number of plies (from Sun and Jen, 1987). CONTINUUM

DAMAGE

MECHANICS

The analyses reviewed in the previous section attempt to account for the details of damage present in the laminate. However, due to the complexity of the problem, only approximate solutions to very simple cases could be obtained. When dealing with the overall behavior of the laminate, detailed, accurate stress distributions are not necessary, and a damaged ply could be replaced by an equivalent layer having the same overall

Review Article

341

properties. Continuum Damage Mechanics (CDM) have been used to study the behavior of many different materials, such as rocks, metals, concrete and composites. The development of the general theory and its applications to practical problems has progressed rapidly in recent years due to the efforts of many investigators, such as Kachanov (1980, 1986), Lemaitre (1984, 1985a, b), Lemaitre and Chaboche (1978), Chaboche (1982, 1988a, b) Cordebois and Sidoroff (1982a, b) Krajcinovic (1979, 1983, 1984, 1985), Krajcinovic and Silva (1982), Krajcinovic and Fonseka (1981), Fonseka and Krajcinovic (1981), Fanella and Krajcinovic (1985), Murakami (1988), Chow and Wang (1987) and Karr (1985). These continuum damage mechanics concepts were applied by Sidoroff and Subagio (1987) to analyze the bending fatigue of glass-epoxy laminates. Peng et al. (1986) studied the stresses in a unidirectional panel with a hole in the center. Frantziskonis (1988) presented a theory to account for microdamage in composites. The basic notion is to quantify damage by appropriate variables and then to develop equations describing its evolution and influence on the mechanical behavior of the material. The first task is to define damage and establish a proper measure of it. The type of defect that can be handled by this approach is usually small compared to a typical microscopic scale of the material, and the material is assumed to be homogeneous on a macroscopic scale. Damage measures can be based on remaining life in fatigue studies; one can define damage as D= 1 -N/N,, (9) where N is the number of cycles already applied, and Nr is the total number of cycles to failure. Another approach is to characterize the microstructure (surface microcracks in fatigue, for example). Defects reduce S, the effective area of the specimen, and damage can be defined as D= 1 -S/S,,, (10) where S, is the initial area. Damage can also be measured through physical parameters. Density changes, decreases in the elastic modulus and acoustic emission (Karr, 1985) can be used. A discussion of the choice of a proper measure of damage is given by Chaboche (1988a). Depending on the problem, damage is characterized by either a scalar, a vector or, in general, a tensor. The use of a damage tensor, D, is required to account for the oriented nature of the cracks. A fourth-order tensor was used by Cordebois and Sidoroff (1982), but a second-order tensor can also be used. When cracks are uniformly distributed in all directions, damage is isotropic and is represented by a scalar. Next, the behavior of the material is related to the current state of damage, and a law for damage growth is introduced. In the most general case, the behavior of the undamaged material is anisotropic, as is damage. Several authors have developed models especially for laminated composites with transverse matrix cracks: Kamimura (1985), Talreja (1987), Nuismer and Tan (1982) and Allen et al. (1987a, b), for example. Kamimura (1985) describes damage with three parameters, D1, D2 and D,, so that the engineering elastic constants of the damaged ply are now given by P12 = vl,U - 01) v21

The constitutive

=

v2lU

-

D2)

I?, = E,(l

- Dl)

E2 = E,(l

- D2)

&

- 06).

= E&l

equations for the damaged ply are given by

(11)

348

S. ABRATE

where S,, = I/[,??,(1 - Di)], S,, = l/&(1 - Q)], S,, = l/[&(1 - De)] and SIz = S2i = v,,/E, = v,,/E, . In this formulation, the damage parameters represent the change in modulus. For transverse matrix cracking, Dz is determined from a shear-lag analysis of one basic repetitive unit, and when the laminate is subjected to uniaxial tension, D, and D6 are not required. This model predicts the onset of cracking as a function of ply thickness and other laminate parameters and also determines the variation of crack density with laminate strain. Good agreement with experimental results is obtained (Kamimura, 1985). Nuismer and Tan (1982) study the same problem but define a different damage parameter. The damaged ply behavior is given by E = D&T,

(13)

where S is the compliance of the undamaged ply, and the damage state variable, DN, is equal to 1 for an undamaged ply and greater than 1 for a damaged ply. DN is related to the damage parameter, D, , of Kamimura (1985) by l/D, = 1 - D, . Starting with a given crack spacing, the energy stored in a typical cell is proportional to the compliance of that cell. When a new crack surface is created, the decrease in energy stored is equal to the energy needed to create a new crack. The stiffness of the cell is taken from the approximate elasticity, which is more accurate than that obtained from the shear-lag theory. Relatively good predictions of first-ply failure were obtained for a number of laminates. Talreja (1987) presents a more general approach based on a scalar field characterization of damage. For one orientation of cracks, the damage parameter is defined by D, = n.1.

w-f,

(14)

where n is the crack density, I and w are the average crack length and width and f is a factor between 0 and 1 depending on the crack shape and the constraint to crack opening provided by the surrounding plies. Damage is then characterized by a vector normal to the plane of the cracks with magnitude D. Constitutive equations are derived considering damage as an internal state variable. The stiffness of a damaged ply is given by cpq = c,q + c;q,

(15)

where Cjq is the stiffness of the undamaged ply. Cdq is a function of D and the orientation of the damage. The form of that matrix is given in Talreja (1987) for an orthotropic lamina with different types of matrix cracks, a number of constants having to be determined experimentally. Once these constants are determined, very good agreement with experimental results is obtained. Allen et al. (1987a) describe damage by the second-order tensor “ij = uinj, where the UiS are the components of the crack-opening displacements and the njS are the components of the normal to the crack surface. Constitutive equations for a damaged ply under isothermal conditions derived from thermodynamic consideration, with the oij as an internal state variable tensor, are given by Gij = 0: + Cijk/&k/ + Iij,,Olk/ 3

(16)

where a,Tis the residual stress, Cij,/ is the undamaged elastic modulus tensor and Zijk/ is the damage modulus tensor. This theory is applied to laminates with matrix cracks, the damage growth law being determined experimentally. Allen et al. (1988) extend their model to deal with laminates containing both transverse matrix cracking and interply delaminations. However, since interply delamination is not distributed throughout a ply, it cannot be included in the continuum model. Its effect is accounted for by a special kinematical constraint. Results are presented concerning stiffness degradation due to both types of damage. The influence of interply delaminations on the stiffness loss of the laminate is small. Further results on the applicability of this model are given in Harris and Allen (1988). In spite of their varying degrees of complexity, these four approaches (Nuismer and Tan, 1982; Talreja, 1987; Kamimura, 1985; Allen et al., 1987a), basically consist of modifying the constitutive equations as certain measures of damage change and have been applied to symmetric laminates under tension. The damage parameter is based on a

Review Article

349

measure of the crack geometry. However, this approach may not be convenient, since determining the required information about the crack distribution is very time consuming and requires interruption of the testing. In another approach used for the analysis of fatigue data, the loss in elastic modulus, monitored during testing, is taken as the damage parameter and is related to the number of fatigue cycles and some measure of the cyclic loading. Sidoroff and Subagio (1987) used a unidirectional model of elastic damage in which the damage variable is defined from the elastic modulus. The stress-strain relation is given by 0 = E(D) * E, (17) where the elastic modulus E is related to the damage variable D by E(D)

= E,,(l

- D),

(18)

E. being the initial modulus. The fatigue evolution law relating the damage increment per cycle is taken as

dD -= dN

A(Ac)‘/(a L0

- D)b

in tension in compression,

(19)

where AE is the strain amplitude and A, b and c are constants to be determined experimentally. This model has been applied successfully to the analysis of results from bending fatigue tests. The evolution of damage during fatigue life in composite materials was also discussed by Poursatrip et al. (1982) for a tension-compression specimen. Ogin et al. (1985) assumed that the rate of increase of damage is given by (20)

where A and n are constants. Excellent agreement with experimental results is reported. The applications of continuum damage mechanics to composite materials has seen some theoretical development and limited but successful practical applications. CONCLUSIONS

In laminated composites several types of internal damage develop. Matrix cracking is observed in the off-axis plies of laminates under tension, since, due to strain magnification, their strain to failure is lower than that of the 0” plies. As loads are increased, delamination may appear at the crack tip. The presence of matrix cracks in the 90” plies has the effect of increasing the load carried by the 0” plies and unloading the 90” plies in the immediate vicinity of the crack. In addition, a stress concentration is also introduced at the crack tip. This eventually causes fiber failures in the 0” plies and leads to total laminate failure. Another type of damage, encountered particularly with glass-epoxy laminates, is matrix splitting in the 0” layers which is caused by a Poisson’s ratio mismatch between 0” and 90” plies. Matrix cracking and delamination also occur in laminates subjected to fatigue loading, thermal cycling, flexural and complex loadings. The appearance of this type of damage generally occurs well before complete laminate failure, and it is thought that if the damage initiation and development process is better understood, the first-ply failure criterion can be replaced by a less conservative approach. The initiation and development of matrix cracks in cross-ply laminates under tension has been the object of several experimental investigations and is well documented. These studies show that the strength of a 90” ply depends on the constraining effect of the adjacent plies so that the in situ strength of that ply is higher than that of a 90” lamina of equal thickness. As loading is increased, further cracking develops, and the crack density increases until saturation will correspond to a Characteristic Damage State (CDS) with arrays of regularly spaced cracks. The introduction of internal damage and matrix cracks, in particular, affects the overall behavior of the laminate. The laminate modulus can decrease significantly, depending on the lay-up, and this change is sometimes taken as a measure of damage.

350

S. ABRATE

Several factors influencing matrix cracking have been identified. Residual thermal stresses developed during curing can reach significant levels, particularly with carbon-epoxy laminates, and should be lowered. In addition, the fracture toughness of the laminate should be increased. A number of approaches have been presented to predict the onset of cracking and the evolution of crack density as loading increases. Micromechanics analyses of laminates with matrix cracks are often performed using shear-lag methods, while several other methods are also used. The prediction of further cracking is based either on a fracture mechanics approach or on a probabilistic failure strength approach. Comparison with experiments shows reasonable agreement, especially for the variation of in-plane laminate properties with crack density and the evolution of crack density with load. Several attempts have been made to model the behavior of a damaged laminate using a continuum damage approach. The effective properties of a damaged ply are defined in terms of a certain measure of damage, which is usually a tensor to account for anisotropy. To date, most of the research has been focused on understanding the behavior of cross-ply laminates under in-plane loading. The next step should be to extend this work to include more general lay-ups and study damage development under out-of-plane deformation and the evolution of bending rigidities with damage. REFERENCES Aboudi, J., Lee, S. W. and Herakovich, C. T. (1988). Three-dimensional analysis of laminates with cross cracks. J. Appl. Mech. 55, 389-397. Adams, D. S., Bowles, D. E. and Herakovich, C. T. (1986). Thermally induced transverse cracking in graphite-epoxy cross-ply laminates. J. Reinf. Plastics Compos. 5, 152-169. Adams, R. D., Flitcroft, J. E., Hancox, N. L. and Reynolds, W. N. (1973). Effects of shear damage on the torsional behavior of carbon reinforced plastics. J. Compos. Mater. 7, 68-75. Adams, D. S. and Herakovich, C. T. (1984). Influence of damage on the thermal response of graphite epoxy laminates. J. Thermal Stresses, 7, 91-103. Agarwal, B. D. and Joneja, S. K. (1979). Progressive damage in GFRP under constant deflection flexural fatigue. Fibre Sci. Technol. 12, 341. Agarwal, B. D. and Joneja, S. K. (1980). Flexural fatigue of a unidirectional composite in the longitudinal direction. Mat. Sci. Engng. 46, 63. Allen, D. H. Groves, S. E. and Harris, C. E. (1988). A cumulative damage model for continuous fiber composite laminates with matrix cracking and interply delaminations. ASTM STP 972, pp. 57-80, ASTM, Philadelphia, PA. Allen, D. H., Harris, C. E. and Groves, S. E. (1987a). A thermomechanical constitutive theory for elastic composites with distributed damage-Part I: Theoretical development. Int. J. Solids Structures, 23, 1301-1308. Allen, D. H., Harris, C. E. and Groves, S. E. (1987b). A thermomechanical constitutive theory for elastic composites with distributed damage-Part II: Application to matrix cracking in laminated composites. Int. J. Solids Structures, 23, 1319-1338. Allen, D. H., Harris, C. E., Groves, S. E. and Norwell, R. G. (1978). Characterization of stiffness loss in crossply laminates with curved cracks. J. Compos. Muter. 22(l), 71-80. Altus, E. and Ishai, 0. (1986). Transverse cracking and delamination interaction in the failure process of composite laminates. Camp. Sci. Technol. 26, 59-77. Aveston, J. and Kelly, A. (1973). Theory of multiple fracture of fibrous composites. J. Mater. Sci. 8, 352-362. Bader, M. G. and Boniface, L. (1985). Damage development during quasistatic and cyclic loading in GRP and CFRP laminates containing 90” plies. 5th Int. Conf. on Composite Materials, ICCM-V, San Diego (Edited by W. C. Harrigan, J. Strife and A. K. Dhingra), pp. 221-232. Metallurgical Society Inc., Warrendale, PA. Bailey, J. E., Curtis, P. T. and Parvizi, A. (1979). On transverse cracking and longitudinal splitting behaviour of glass and carbon fibre reinforced epoxy cross ply laminates and the effect of Poisson and thermally generated strain. Proc. R. Sot. London, Series A, 366, 599-623. Barnes, J. A. and Cogswell, F. N. (1989). Thermoplastics for space. SAMPE Q. 20(3), 22-27. Bowles, D. E. (1984). Effect of microcracks on the thermal expansion of composite laminates. J. Compos. Mater. 17, 173-186. Chaboche, J. L. (1982). Le concept de contrainte effective applique a l’elasticite et a la viscoelasticite en presence d’un endommagement anisotrope. Colloques Internationaux du CNRS, No. 295, Comportement m&unique des Solides Anisotropes (Edited by J. P. Boehler), pp. 737-760. Martinus Nijhoff, The Hague. Chaboche, J. L. (1988a). Continuum damage mechanics: Part I-General concepts. J. Appt. Mech. 55, 59-64. Chaboche, J. L. (1988b). Continuum damage mechanics: Part II-Damage growth, crack initiation, and crack growth. J. Appl. Mech. 55, 65-72. Chandra, R., Malik, A. K. and Prabhakaran, R. (1982). Damping as a measure of damage in composites. J. Tech. Councils ASCE, 108, 106-111. Charewicz, A. and Daniel, I. M. (1986). Fatigue damage mechanisms and residual properties of graphite/epoxy laminates. IUTAM Symp. on Mechanics of Damage and Fatipue, Haifa, Israel. and Emma. - - Fract. Mech. 25(5-6), 793-808. Chow, C. L. and Wang, J. (1987). Anisotropic theory of elasticity for continuum damage mechanics. Int. J. Fract. 33(3), 3-16.

Review Article

351

Cordebois, J. P. and Sidoroff, F. (1982a). Endommagement anisotrope en Clasticite et plasticite. J. Met. Theor. Appliq., Numero Special, 145-60. Cordebois, J. P. and Sidoroff, F. (1982b). Damage induced elastic anisotropy. Colloques Internationaux du CNRS, No. 295, Comportement Mecanique des Solides Anisotropes (Edited by J. P. Boehler), p. 761. Martinus Nijhoff, The Hague. Daniel, I. M., Lee, J. W. and Yaniv, G. (1987). Damage mechanisms and stiffness degradation in graphite/epoxy composites. 6th Int. Conf. on Composite Materials, ICCM-VI, combined with the 2nd European Conf. on Composite Materials, ECCM-II, London, Vol. 4 (Edited by F. L. Matthews, N. C. R. Buskell, J. M. Hodginson and J. Morton), pp. 4.129-4.138. Elsevier Applied Science, London. DiBenedetto, A. T., Gavehel, J. V., Thomas, R. L. and Barlow, J. W. (1972). Non-destructive determination of fatigue crack damage in composites using vibration tests. J. Materials 7, 21 l-215. Fanella, D. and Krajcinovic, D. (1985). Continuum damage mechanics of fiber reinforced concrete. J. Engng. Mech. 111, 995-1009. Flaggs, D. L. (1985). Prediction of tensile matrix failure in composite laminates. J. Compos. Mater. 19, 29-50. Flaggs, D. L. and Kural, M. H. (1982). Experimental determination of the in-situ transverse lamina strength in graphite/epoxy laminates. J. Compos. Mater. 16, 103-116. Fonseka, G. U. and Krajcinovic, D. (1981). The continuous damage theory of brittle materials Part 2: Uniaxial and plane response modes. J. Appl. Mech. 48, 816-824. Frantziskonis, G. (1988). Distributed damage in composites, theory and verification. Compos. Struct. 10, 165-184.

Fukunaga, H., Chou, T. W., Peters, P. W. M. and Schulte, K. (1984). Probabilistic failure strength analyses of graphite/epoxy cross-ply laminates. J. Compos. Mater. 18, 339-356. Garret, K. W. and Bailey, J. E. (1978). Multiple transverse fracture in 90” cross-ply laminates of glass fiberreinforced polyester. J. Mater. Sci. 12, 157. Garret, K. W. and Bailey, J. E. (1977b). The effect of resin failure strain in the tensile properties of glass fiber reinforced polyester cross-ply laminates. J. Mater. Sci. 12, 2189-2194. Groves, S. E., Harris, C. E., Highsmith, A. L., Allen, D. H. and Norwell, R. G. (1987). An experimental and analytical treatment of matrix cracking in cross-ply laminates. Exp. Mech. 27(l), 73-79. Han, Y. M. and Hahn, H. T. (1989). Ply cracking and property degradation of symmetric balanced laminates under general in-plane loading. Compos. Sci. Technol. 35, 317-397. Han, Y. M., Hahn, H. T. and Croman, R. B. (1988). A simplified analysis of transverse ply cracking in cross-ply laminates. Compos. Sci. Technol. 31, 165-177. Hashin, Z. (1985). Analysis of cracked laminates: a variational approach. Mech. Mater. 4, 121-136. Hashin, Z. (1986). Analysis of stiffness reduction of cracked cross-ply laminates. Engng. &act. Mech. 25(5-6), 771-778. Hashin, Z. (1987). Analysis of orthogonally cracked laminates under tension. J. Appl. Mech. 54, 872-879. Harris, C. E. and Allen, D. H. (1988). A continuum damaae model of fatiaue-induced damaee in laminated composites. SAMPE J. 24(4), 43-51. Harris, C. E., Allen, D. H. and Nottorf, E. W. (1988). Modelling stiffness loss in quasi-isotropic laminates due to microstructural damage. J. Engng. Mater. Technol. 110, 128-133. Herakovich, C. T. and Hyer, M. W. (1986). Damage induced property changes in composites subjected to cyclic thermal loading. Engng. Pratt. Mech. 25(5-6), 779-791. Highsmith, A. L. and Reifsnider, K. L. (1982). Stiffness-reduction mechanisms in composite laminates. ASTM STP 775, pp. 103-l 17, ASTM, Philadelphia, PA. Highsmith, A. L. Stinchcomb, W. W. and Reifsnider, K. L. (1984). Effect of fatigue induced defects on the residual response of composite laminates. ASTM STP 836, pp. 194-216, ASTM, Philadelphia, PA. Im, S. (1990). Asymptotic stress field around a crack normal to the ply interface of an anisotropic composite laminate. Int. J. Solids Structures 26(l), 11l-127. Jamison, R. D. (1985). The role of microdamage in tensile failure of graphite/epoxy laminates. Compos. Sci. Technol. 24, 83-99. Jamison, R. D., Schulte, K., Reifsnider, K. L. and Stinchcomb, W. W. (1984). Characterization and analysis of damage mechanisms in tension-tension fatigue of graphite/epoxy laminates. ASTM STP 836, pp. 21-55, ASTM, Philadelphia, PA. Kachanov, L. M. (1980). Continuum model of medium with cracks. J. Engng. Mech. 106(EM5), 1039-1051. Kachanov, L. M. (1986). Introduction to Continuum Damage Mechanics, Martinus Nijhoff, Dordrecht. Kalyanasundaram, S. and Allen, D. H. (1988). A model for predicting damage dependent damping in laminated composites. Comput. Struct. 30(1/2), 113-l 18. Kamimura, K. (1985). Modelisation theorique de la croissance d’endommagement appliquee a la theorie des plaques stratifiees. J. M&c. Theor. Appliq. 4(4), 537-553. Karr, D. G. (1985). A damage mechanics model for uniaxial deformation of ice. J. Energy Resources Techn. 107, 363-368.

Kim, R. Y. and Hahn, H. T. (1979). Effect of curing stresses on the first ply failure in composite laminates. J. Compos. Mater. 13, 2-16. Kistner, M., Whitney, J. M. and Browning, C. E. (1985). First-ply failure of graphite/epoxy laminates. ASTM STP 864, pp. 44-61, ASTM, Philadelphia, PA. Korczynskyj, Y. and Morley, J. G. (1981). Constrained cracking in cross-ply laminates. J. Muter. Sci. 16, 1785-1795.

Krajcinovic, Krajcinovic, Krajcinovic, Krajcinovic, 829-834. Krajcinovic, theory. J.

D. D. D. D.

(1979). Distributed (1983). Constitutive (1984). Continuum (1985). Continuum

damage theory of beams in pure bending. J. Appl. Mech. 46, 592-596. equations for damaging materials. J. Appl. Mech. 50, 355-360. damage mechanics. Appl. Mech. Rev. 37(l), l-6. damage revisited: basic concepts and definitions. J. Appl. Mech. 52,

D. and Fonseka, G. U. (1981). The continuous damage theory of brittle materials Part 1: General Appl. Mech. 48, 809-815.

S. ABRATE

352

Krajcinovic, D. and Silva, M. A. G. (1982). Statistical aspects of the continuous damage theory. Int. J. Solids Structures

18(7),

551-562.

Laws, N. and Dvorak, G. J. (1988). Progressive transverse cracking in composite laminates. J. Compos. Muter. 22, 900-916. Lee, J. W., Daniel, I. M. and Yaniv, G. (1989). Fatigue life prediction of cross-ply composite laminates. ASTM STP 1012, pp. 19-28, ASTM, Philadelphia, PA. Lee, S. M. and Schile, R. (1982). An investigation of material variables of epoxy resins controlled transverse cracking in composites. J. Muter. Sci. 17, 2095-2106. Lemaitre, J. (1984). How to use damage mechanics. Nuclear Engng. Design 80, 233-245. Lemaitre, J. (1985a). A continuous damage mechanics model for ductile fracture. J. Engng. Mater. Technol. 107, 83-89.

Lemaitre, J. (1985b). Coupled elasto-plasticity and damage constitutive equations. Camp. Meth. Appl. Mech. Engng. 51, 31-49. Lemaitre, J. and Chaboche, J. L. (1978). Aspect Phtnomenologique de la rupture par endommagement. J. Mac. Appliq.

2, 317-365.

Lim, S. G. and Hong, C. S. (1989). Effect of transverse cracks on the thermomechanical properties of cross-ply laminated composites. Compos. Sci. Technol. 34, 145-162. Manders, P. W., Chou, T. W., Jones, F. R. and Rock, J. W. (1983). Statistical analyses of multiple fracture in O/90/0 glass fibre/epoxy resin laminates. J. Muter. Sci. 18, 2876-2889. Masters, J. E. and Reifsnider, K. L. (1982). An investigation of cumulative damage development in quasiisotropic graphite/epoxy laminates. ASTM STP 775, pp. 40-62, ASTM, Philadelphia, PA. Morley, J. G. (1987). High Performance Fibre Composites. Academic Press, New York. Murakami, S. (1988). Mechanical modeling of material damage. J. Appl. Mech. 55, 280-286. Nairn, J. A. (1988). The initiation of microcracking in cross-ply laminates: a variational mechanics analysis. Proc. American Sot. for Composites 3rd Technical Conf., Seattle, WA, pp. 472-48 1. Technomic Publishing Co., Inc., Lancaster, PA. Nairn, J. A. (1989). The strain energy release rate of composite microcracking: a variational approach. J. Compos. Muter. 23(11), 1106-1129. Nuismer, R. J. and Tan, S. C. (1982). The role of matrix cracking in the continuum constitutive behavior of a damaged composite ply. In Mechanics of Composite Materials: Recent Advances, Proc. IUTAM Symp. on Mechanics of Composite Materials, VPI, Blacksburg, VA (Edited by Z. Hashin and C. T. Herakovich), pp. 437-448. Pergamon Press, New York. Niusmer, R. J. and Tan, S. C. (1988). Constitutive relations of a cracked composite lamina. J. Compos. Mater. 22, 306-321.

Ogin, S. L., Smith, P. A. and Beaumont, P. W. R. (1985). Matrix cracking and stiffness reduction during the fatigue of a (O/90), GFRP laminate. Compos. Sci. Technol. 22, 23-3 1. Ohira, H. (1985). Analysis of the stress distributions in the cross-ply composite after transverse cracking. 5th Int. Conf. on Composite Materials, ICCM-V, San Diego (Edited by W. C. Harrigan, J. Strife and A. K. Dhingra), pp. 1115-l 124. Metallurgical Society Inc., Warrendale, PA. Ohira, H. and Uda, N. (1985). Knee point and post failure behavior of some laminated composites subjected to uniaxial tension. ASTM STP 864. .DD.. 110-127. ASTM. Philadelohia. PA. Parvizi, A. and Bailey, J. E. (1978). On multiple transversecracking ht glass fibre epoxy cross-ply laminates. .I. Mater. Sci. 13, 2131-2136. Parvizi, A., Garret, K. W. and Bailey, J. E. (1978). Constrained cracking in glass fiber reinforced epoxy cross-ply laminates. J. Mater. Sci. 13, 195-201. Peng, L., Yang, F., Xiao, Z. and Zhu, H. (1986). Anisotropic damage in orthotropic fiber-reinforced composite plate. Proc. Int. Symp. on Composite Materials and Structure, Beijing, China (Edited by T. T. Loo and C. T. Sun) pp. 530-536. Technomic Publishing Co., Inc., Lancaster, PA. Penn, L. S., Chou, R. C. T., Wang A. S. D. and Binienda, W. K. (1989). The effect of matrix shrinkage on damage accumulation in composites. J. Compos. Mater. 23, 570-586. Peters, P. W. M. (1984). The strength distribution of 90” plies in O/90/0 graphite-epoxy laminates. J. Compos. Mater.

18, 545-556.

Peters, P. W. M. (1986). Constrained 90-deg ply cracking in 0/90/O and f 45/90/ t- 45 CFRP laminates. ASTM STP 907, pp. 84-99, ASTM, Philadelphia, PA. Peters, P. W. M. (1989). The influence of fibre, matrix and interface on crossply cracking in carbon fibre reinforced plastics cross-ply laminates. ASTM, STP 1012, pp. 103-117, ASTM, Philadelphia, PA. Peters, P. W. M. and Andersen, S. I. (1989). The influence of matrix fracture strain and interface strength on cross-ply cracking in CFRP in the temperature range of - 100°C to + 100°C. J. Compos. Mater. 23,955-960. Peters, P. W. M. and Chou, T. W. (1987). On cross-ply cracking in glass- and carbon-fibre reinforced epoxy laminates. Composites 18(l), 40-46. Peters, P. W. M. and Meuseman, H. (1987). On cross ply cracking in graphite, glass and aramid fibre reinforced plastics. 6th Int. Conf. on Composite Materials, ICCM-VI, combined with the 2nd European Conf. on Composite Materials, ECCM-II, London (Edited by F. L. Matthews, N. C. R. Buskell, J. M. Hodginson and J. Morton), pp. 3.508-3.525. Elsevier Applied Science, London. Poursartip A., Ashby, M. F. and Beaumont, P. W. R. (1982). Damage accumulation during fatigue of composites. Scripta Metallurgica 16, 601-606. Reifsnider, K. L. and Jamison, K. (1982). Fracture of fatigue-loaded composite laminates. Int. J. Fatigue 4, 187-197.

Reifsnider, K. L. and Talug, A. (1980). Analysis of fatigue damage in composite laminates. Int. J. Fatigue 2(l), 3-11. Rosensaft, M. and Marom, G. (1985). Evaluation of bending test methods for composite materials. J. Camp. Technol. Res. 7(l), 12-16. Schultz, A. B. and Warwick, D. N. (1971). Vibration response: a nondestructive test for fatigue crack damage in filament reinforced composites. J. Comp. Muter. 5, 394-404.

Review Article

353

Sheard, P. A. and Jones, F. R. (1987). Computer simulation of the transverse cracking process in glass fibre composites. 6th Int. Conf. on Composite Materials, ICCM- VI, combined with the 2nd European Conf. on CompositeMaterials, ECCM-II, London (Edited by F. L. Matthews, N. C. R. Buskell, J. M. Hodginson and .I. Morton), Vol. 3, pp. 3.123-3.135. Elsevier Applied Science, London. Shih, G. C. and Ebert, L. J. (1987). The effect of the fiber/matrix interface on the flexural fatigue performance of unidirectional fiberglass composites. Compos. Sci. Technol. 28, 137-161. Sidoroff, F. and Subagio, B. (1987). Fatigue damage modelling of composite materials from bending tests. 6th Int. Conf. on Composite Materials, ICCM-VI, combined with the 2nd European Conf. on Composite Materials, ECCM-ZI, London (Edited by F. L. Matthews, N. C. R. Buskell, J. M. Hodginson and J. Morton), Vol. 4, pp. 4.32-4.39. Elsevier Applied Science, London. Smith, S. A. Allen, D. H. and Harris, C. E. (1988). An experimental investigation of damage-dependent material damping in laminated composites. Proc. AIAA 29th Structures, Structural Dynamics and Materials Conf., pp. 1501-1510. AIAA, New York. Spain, R. G. (1971). Thermal microcracking of carbon fibrejresin composites. Composites, 2, 33-37. Stinchcomb, W. W. and Reifsnider, K. L. (1979). Fatigue damage mechanisms in composite materials: a review. ASTM STP 679, pp. 762-787, ASTM, Philadelphia, PA. Stinchcomb, W. W., Reifsnider, K. L., Yeung, P. and Masters, J. (1981). Effect of ply constraint on fatigue damage development in composite material laminates. ASTM STP 723, pp. 65-84, ASTM, Philadelphia, PA. Sun, C. T. and Jen, K. C. (1987). On the effect of matrix cracks on laminate strength. J. Reinf. Plastics Compos. 6(3), 208-222. Talreja, R. (1987). Fatigue of Composite Materials. Technomic, Stamford, Connecticut. Tan, S. C. and Nuismer, R. J. (1989). A theory for progressive matrix cracking in composite laminates. J. Compos. Mater. 23, 1029-1047. Turner, P. S. (1946). Thermal-expansion stress in reinforced plastics. J. Res. National Bureau Stand. 37, 339-350. Wang, A. S. D. (1984). Fracture mechanics of sublaminate cracks in composite materials. Compos. Technol. Rev. 6(2), 45-62. Wang, A. S. D. (1986). On fracture mechanics of matrix cracking in composite laminates. Proc. Int. Symp. on Composite Materials and Structures, Beijing, China, lo-13 June (Edited by T. T. Loo and C. T. Sun), pp. 576-584. Technomic Publishing Co., Inc., Lancaster, PA. Wang, A. S. D., Chou, P. C. and Lei, S. C. (1984). A stochastic model for the growth of matrix cracks in composite laminates. J. Compos. Mater. 18, 239-254. Wang, A. S. D., Kishore, N. N. and Li, C. A. (1985). Crack development in graphite-epoxy cross-ply laminates under uniaxial tension. Compos. Sci. Technol. 24, l-31. Whitney, J. M. and Browning, C. E. (1984). Materials characterization for matrix dominated failure modes. ASTM STP 836, pp. 104-124, ASTM, Philadelphia, PA.