Thermally induced damage in composite laminates: Predictive methodology and experimental investigation

Composites

Scimce and Technology

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THERMALLY INDUCED DAMAGE IN COMPOSITE LAMINATES: PREDICTIVE METHODOLOGY AND EXPERIMENTAL INVESTIGATION Cecelia

H. Park & Hugh

L. McManus*

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology,

(Received 22 December

1995; revised 2.5 March 1996; accepted 13 June 1996)

Abstract A general analysis method is presented to predict matrix cracks in a composite laminate, together with the resulting degradations of laminate properties, as functions of temperature or thermal cycles. All plies except the surface plies are considered. A shear-lag solution for the stresses in the vicinity of cracks and a fracture mechanics crack formation criterion are used to predict cracks. Damage is modeled incrementally, which allows the inclusion of the effects of temperaturedependent material properties and softening of the laminate due to previous cracking. The analysis is incorporated into an easy-to-use computer program. Experimentally, crack densities are measured in a variety of laminates exposed to decreasing temperatures. Crack densities are measured at the edges of specimens by microscopic inspection, and throughout the specimen volumes by X-radiography and sanding down of the edges. In specimens with thick ply groups (several plies of the same angle stacked together), cracks behave ‘classically’, running the width of the specimens. In specimens with single ply groups, smaller, discontinuous cracks developed. Correlation between the analytical results and the edge crack densities were reasonable for those specimens which behaved classically. Crack densities measured at specimen edges do not agree with internal crack densities (or analyses) in specimens with non-classical cracks. A free-edge stress analysis clarifies the reasons for these discrepancies. 0 I996 Elsevier Science Limited Keywords: composites, mal stresses, damage

analysis,

microcracking,

orbiting vehicle moves in and out of the Earth’s shadow can cause cracks in these materials. The cracks, referred to as matrix cracks or microcracks, are found in the individual plies, parallel to the fibers. They are caused by mismatch of the coefficients of thermal expansion (CTE) between plies of different orientations. Microcracks alter the thermal and mechanical properties of the laminate; in particular, they have unfavorable effects on dimensional stability. The present work is motivated by space applications, but the problem is observed in many applications. Microcracking due to mechanical loading and mechanical fatigue has been extensively studied. Early work used classical laminated plate theory (CLPT) to calculate ply stresses, and in situ transverse strength failure criteria.“* Flaggs and Kural’ found that the in situ strengths are not ply properties, but are dependent on ply thickness and laminate geometry. Attempts at improved strength-based methods included the shear-lag analysis of Lee and Daniel3 and the use by Peters of probabilistic strength distributions.4 Fracture-mechanics-based models, using shear-lag stress analyses and energy criteria to predict crack growth from assumed initial flaws, have been used by many authors.“-” Alternatives to shear-lag methods for calculating stress distributions include variational approaches,“.” 2D elasticity I4 and finite element methods.” The present work draws on this rich background; in particular, the crack formation model used here is based on the work of Laws and Dvorak.7 A more extensive review can be found elsewhere.” Many mechanical loading studies have included the effects of residual thermal stresses, but only a limited amount of work has been done to address the cracking of thermally loaded composites directly. Adams et a1.17 used CLPT and in situ transverse strength to predict the onset of microcracking. The ply discount method was used by Bowles” to predict reduction of properties with little success. These analyses emphasize damage at the ply level. Micromechanics methods approach the problem at the scale of individual fibers and the surrounding matrix. For

ther-

1BACKGROUND Advanced composite materials are commonly used in dimensionally critical space structures. However, the wide variations in temperature that occur as an * To whom correspondence

Cambridge, Massachusetts 02139, USA

should be addressed. 1209

C. H. Park, H. L. McManus

1210

example, Bowles and Griffin’” derived thermally induced stresses by using a finite element analysis. They then predicted fiber/matrix debonding initiation and location by comparing the maximum radial stress at the interface to the interfacial bond strength. Most of the work on the thermal loading problem has been experimental. Knouff et ~1.~” found fiber type and properties affected microcracking under thermal and numerous experiments have been cycling, conducted*‘-24 to measure properties such as CTE, stiffness, strain and crack density after thermal cycling. Manders and Maas’” tested laminates made from very thin plies to see if cracking and property degradation could be prevented. Bowles and She?’ tested fabric laminates for the same purposes. an analytical method McManus et a1.27 developed for thermally loaded laminates. Crack density in one ply group, and reduced laminate properties resulting from these cracks, were predicted as functions of monotonically decreasing temperature. The effects of thermal cycling were included in the analysis by using a material degradation fatigue model. Experimentally, specimens were monotonically cooled and inspected at the edges for cracks. Thermal cycling tests were also performed. The analyses correlated very well with the experiments. However, the method was severely limited in its practical use by the ability to predict crack density in only one ply group.

2 STATEMENT

OF THE

PROBLEM

Given laminate geometry, lay-up, material property information and thermal loading history, crack density and degraded laminate properties are predicted. The method can analyse all plies (except surface plies) in an arbitrary lay-up under monotonically increasing or cyclic thermal loading. Progressive material softening effects and temperature-dependent material properties are included. In the present work, this capability is pursued by: (1) developing a generalized analytical tool; (2) correlating the analysis with experiments; and (3) studying the validity of the assumptions underlying the analytical method.

3 ANALYTICAL

MODEL

3.1 Geometry Figure 1 shows the laminate geometry; the lay-up shown is simplified for clarity. In general, a thin, symmetric laminate aligned with a global coordinate system xyz is considered. The laminate is made up of unidirectional plies. Stacked plies with the same ply angle are assumed to act as a single thick ply, referred to as a ply group. Cracks are assumed to span the ply group thickness and propagate parallel to the fibers through the width of the composite laminate. Figure 1 shows cracks of this type in three ply groups. A local coordinate system X’Y’Z is defined for each crack. The y’ axis is aligned with the crack, parallel to the fiber direction of the ply group, the x’ axis is aligned with the transverse direction of the ply group, and the origin is at the center of the crack. The properties of this laminate in the global x direction will be determined. 3.2 Crack formation To predict cracking in any one ply group, the laminate is modeled as being made up of two components: the cracking ply group and the rest of the laminate. The cracking ply group is restrained by the rest of the laminate. The properties of the rest of the laminate are smeared, which implies that the analysis results will depend on lay-up, but not stacking sequence. Figure 2 shows a cross-section of the laminate, aligned with the local coordinate system of a hypothetical new crack which is postulated between two existing cracks. The laminate is subjected to a thermal load, AT, which is the difference between the current temperature and some stress free temperature, which is usually taken to be the cure temperature. The thermal load, AT, is assumed to be uniform throughout the laminate, and it is assumed that there are no applied mechanical loads or restraints. If the new crack forms, some of the strain energy stored in the laminate due to the thermal load will be released. A one-dimensional shear-lag model is used to estimate this energy. Laminate and cracking ply

Global Coordinate

Ply Group

45 90

b

4

2h Fig. 1. Laminate

geometry.

Fig. 2. Geometry

used in crack calculations.

1211

Thermally induced damage group properties are calculated in the x’ direction. The stress and displacement fields are found with and without the hypothetical new crack, and the strain energy in each case is then calculated. The difference between these strain energies is the energy that would be released by the appearance of the new crack. This is expressed, per unit area of new crack surface, as the energy release rate, G1. If G, is greater than the critical strain energy release rate of the material, GI,, the hypothetical crack will in fact form. This basic energy criteria is expressed as a function of applied thermal loading. The development is lengthy. An abbreviated development is given in the Appendix; extensive development is more presented elsewhere.‘hT28 The final expression is: G, =

a,aE,E,E,(cr, - CX,)~AT~ 25(a,E, + ac&) X [2tanh($i

- tanh(F)l

(1)

where the subscripts c and r represent properties of the cracking ply group and the rest of the laminate, respectively. Thus, (Y, is the CTE of the cracking ply group, CY,is the smeared CTE of the rest of the laminate, E, is the untracked stiffness of the cracking ply group, and E,. is the smeared stiffness of the remainder of the laminate, all in the x’ direction. The thickness of the cracking ply group is ac, and a, is the thickness of the rest of the laminate. The geometric fitting factor, [, is fixed at a value used in prior studies.27 Assuming an existing uniform crack spacing, 2h, a corresponding AT is found which satisfies eqn (1) with G, set equal to G,,. At any thermal load greater than this AT, new cracks will form midway between the existing cracks. Therefore, the minimum possible crack density at this AT is defined as p, where p = 1/(2h). Note that eqn (1) can be solved explicitly for AT given h, but if h (or p) for a given AT is required, it must be solved for graphically or numerically. If there are no pre-existing cracks in the cracking ply group, h is set to infinity, and eqn (1) simplifies to: GI =

0: = e, + 90” - 8,

(4)

Equation (4) effectively rotates the laminate into the coordinate system of the cracking ply x’y’z’. The necessary laminate properties for computing crack density and property degradation can be calculated using familiar CLPT relationships. The laminate stiffness in the x’y’z’ coordinate system is given by: A = i eiti i=l The rotated system are:

reduced

(5)

ply stiffnesses

el=

a,aCE,EC(aC - IX,)~AT~ 25(@,

in the x’y’z’

~-‘Qil;-T

(6)

+ a,&)

From eqn (2), the temperature change AT, at which cracking will start in an initially untracked ply group is given by: Ar,=

cycle number. For each ply, i, of thickness ti, these are Eli (longitudinal stiffness), E,, (transverse stiffness), u, (major Poisson’s ratio), G, (shear stiffness), cyli (longitudinal CTE) and atj (transverse CTE). Typical material properties are given in Table 1. Most of these properties are easily obtained for other materials. The exception to the rule is G,,, which is the effective critical strain energy release rate for the composite in the transverse direction. It is difficult to measure directly. Typically, measured delamination G,, values are used, as it can be argued that in both transverse cracking and delamination the crack front is advancing through matrix and/or along fiber/matrix interfaces, parallel to the fibers. Unfortunately, reported values for delamination, Gr,, for a given material vary widely.28 The values in Table 1 are mostly unpublished NASA Langley data; the GI, value for P75/934 and the 5 values for both materials are taken from fits to transverse cracking of cross-ply laminates.28 The values used were not altered to fit the data collected during this study. The crack formation analysis is carried out for each ply group in turn. The fibers of each ply are aligned at an angle @,to the global coordinate system. The ply group under consideration is referred to as the cracking ply group. The subscript c denotes this group, hence it has orientation 8,. The analysis is carried out in the local coordinate system of the cracking ply group x’y’z’ defined above. In this coordinate system, the angles of all plies are redefined:

%%a,& + a&) G - ay,)’ I’ J a,a,E,E,(a,

(3)

3.3 Laminate property constants The constants used in the above analysis are calculated by CLPT.29 First, appropriate material properties are obtained as functions of temperature or

Table 1. Material properties

Property E, (GPa) E, (GPa) L (GPa) G,c (J/m’) @IWV

P75/934 236.00 6.20 0.29 4.80 40.00 - 1.22 28.80 0.65

I’X/ERL1962 236.00 6.60 0.29 4.80 104.00 - o-95 39.60 0.65

C. H. Park, H. L. McManus

1212

where

Ti=

2sinB~cosf$

sin%!I

cos2 e;

-2sinB,‘cosf$

cos%;

sin20/ [ -sin@cosf9,!

c02e;

sinO;cosO;

-sin2&

(7) 1

After all the ply groups have been analysed in this manner, the degraded effective laminate properties in the global x direction (stiffness, Eeff, and CTE, creff) are calculated:

and Qi

=

[

EizIi

Elj:i

C8)

,11

Eeff = &

;

AiIW

=

A-‘;

11 0

&“ff

=

a7

(17)

where czy is the first element of ae = A-l 2 @(yiti i=l

and the reduced ply stiffnesses are

(18)

eqns (17) and (18) A, e and G are calculated from (5)-(11) with &’ = 0,. Note no bending is incorporated into the analysis, so it is strictly valid only for symmetric laminates.

III

eqns

Here ~~is a knockdown factor which accounts for the effects of pre-existing cracks; initially it has the value 1. The CTEs of each ply are:

3.4 Progressive

damage

ali (Yi =

(10)

ffti

11 0 In the ~‘y’z’ system the ply CTEs are: (Yi= T*~ffi

(11)

The laminate constants required in eqn (1) are now calculated. The total laminate thickness, ao, is: ao=2

ti

(12)

i=l

The necessary are:

constants

&=$;

for the cracking ply group cYc=cYtc; a, = t,

c

and the smeared properties are:

=A1t-Ecac.

E

r

a,

(13)

of the rest of the laminate

CY, =

a; ;

’

a, = a, -a,

(14)

The smeared CTE of the remaining plies, Q$, is the first element of the vector of CTEs which are calculated thus: (Y’= [A - &-‘( Equation (1) can now density in the cracking factor for this group relationship developed E

a

c

c

E$, &@j

- e,&t,)

(15)

be solved to predict the crack ply group, pc. The knockdown is also recalculated, using the previously:27

@!&tan&

K, = +Ea

25

Pcac >

=tanhz

(16)

Edge effects

The analytical procedure developed predicts stresses in each ply of a laminate using the assumptions of CLPT. However, at the free edge of a laminate, the assumptions of CLPT are no longer valid. Given that most experimental results involve observation of microcracks at specimen edges, the free-edge stress state is clearly important. The analysis of Bhat is used to explore analytically the three-dimensional free-edge stress state. A modification of the computer code bY Bhat computes the in-plane stress distribution in each PlY group near a free edge parallel to the global x direction. This analysis is used for comparison of expefimental results, but it is not incorporated into the crack prediction calculations. It will be fully described in a later publication.

Thermally induced damage

/ Repeat until target temperature c reached Decrease Temperature One Increment I

1213

-Done I

t

Compute Material 1 Proper&s at Temperature

--i

1

Reueat for

I

Calculate New Laminate Properties Update Knockdown Factors Fig. 3. Algorithm flowchart.

4 COMPUTER

PROGRAM

The computer code CRACKOMATIC II takes material properties, laminate geometries, and thermal loading history and predicts crack density and corresponding degraded laminate properties. The two types of calculations are: (1) tabulation of crack densities and degraded material properties as functions of monotonically decreasing temperature; and (2) tabulation of crack densities and degraded material properties as functions of thermal cycles. The resulting output is a table with columns listing thermal load (AT or N), corresponding crack density of any selected plies, and corresponding reduced laminate properties. These can be used to generate plots of cracking and changing laminate properties as the laminate is thermally cycled or progressively cooled. The program output reports crack densities as they could be observed on the edge of a specimen. This requires that the calculated results, which are expressed in ply coordinates, be adjusted to account for a geometric effect. Figure 4 illustrates this effect. The figure shows a laminate with cracks in the 45” and 90” direction. The crack spacing, h, is the same, but due to the geometry of the laminate, the edge crack count is higher in the 90” (9 cracks) than in the 45” ply (6 cracks). The reported crack density, pEdge, is the calculated density, p. multiplied by a geometric

Crack density measured on this side

P 'P 90” 45”

I

I

I

I

Fig. 4. Apparent crack densities. factor: pEdge= pcsin( 0,) The CRACKOMATIC request from the authors.

II program

(19) is available on

5 EXPERIMENTS

Experiments were conducted to correlate with analytical predictions and to check the assumptions on which the analysis was based. P75/934 and P75/ERL1962 graphite/epoxy laminates listed in Table 2 were fabricated at NASA Langley Research Center using O-127 mm (0.005”) thick plies and vendor

1214

C. H. Park, H. L. McManus Table 2. Laminates tested

Lay-up &/%I, ;;//‘“/;Y$% [oj45;9Oj- 45]$ [O/90/ f 4515 [O/ f 45/901, W45,/9W - 4521,

Material P?5/934 P75/934 P75/934 P751934 P75/934 P75/934 P75/ERL1962

Total specimens 3 4 3 6 4 4 3

recommended cure procedures. Panels 30-5 cm X 30.5 cm (12” X 12”) were cut into 76 cm X 1.2 cm (3.0” x 0.5”) specimens. All the specimens were dried to a constant weight to eliminate moisture effects. Specimens were monotonically cooled to progressively lower temperatures in a thermal environmental chamber. Starting at room temperature, the specimens were cooled to target temperatures as low as -157°C ( - 25O’F) at a rate of 14”C/min. The cooling and heating rates were low enough to avoid significant thermal gradients in the specimens. Specimens were soaked at the target temperature for 5 min and returned to room temperature. The edges of the specimens were then inspected for cracks under an optical microscope at 200 X magnification. An observed crack extending more than half the thickness of a ply or ply group was counted as a crack. The crack configuration throughout the volume of the laminate was determined using two methods. The first was dye penetrant enhanced X-radiography. Specimens were soaked in dye penetrant, and X-ray photographs were then taken. These photographs showed only cracks in surface plies and those cracks in inner layers that ran continuously from an edge (and hence were accessible to the dye penetrant). The second method was a direct inspection of the volume of the specimen. This was accomplished via a series of edge inspections done after sanding down the edges of the specimens. The sanding was continued until the entire specimen volume had been inspected (i.e. the specimen was completely destroyed). This method was found to work better than alternative methods such as cutting up the specimen, as the cutting itself created additional cracks. Control experiments were performed to ensure that the sanding procedure did not create additional cracks; details are found elsewhere.16

Experimental results shown are average values with one standard deviation error bars. Edge inspection data is collected from two or more specimens, while the interior data is collected from eight or more locations in a single specimen by sanding down the specimen edges. 6.1 Thick ply groups The [OJ45J902/ - 45& specimens were produced specifically because they have thick ply groups (two or more plies oriented in the same direction stacked together) that produce large cracks that can be easily observed. Qualitatively, the cracks in these specimens are ‘classic’ cracks which continued through the laminate, as shown in Fig. 1. The crack distributions measured by edge inspection and X-radiography are identical, and the X-ray pictures show that the cracks run across the full width of the specimens. The -452 ply group in the [O/45/90/ - 451s laminate and the 904 ply group in the [OJ90& laminate also show ‘classic’ behavior. Quantitatively, the analytical predictions of crack density at low temperatures ( - 157°C) are reasonably accurate in most cases (see Fig. 5). The temperature below which cracking is first observed is predicted reasonably well in the 904, -454, -452 and 452 groups ( - 45, shown in Fig. 5, additional data from previous resultsr6). The numbers of cracks developed at a given temperature are well predicted in the 90, group,” and underpredicted in the 45, groups. In these groups, the crack density increases gradually, rather than jumping up at the temperature of first cracking as predicted. The 902 ply groups have large numbers of cracks at room temperature (Fig. 5). It is suspected that these

-I”

_

: W5lEZU1962 [02/452/902/-4521, 25 -

-

20 -

- - - Analysis-454 A Experiment 902 0 Experiment -454

Analysis902

6 RESULTS The experimental results are compared with analytical predictions made using the material properties in Table 1. The material properties are assumed to be temperature independent. Selected results from a previous studyI are presented here. Crack densities are plotted as functions of decreasing temperature.

-200

-150

-100

-50

0

Temperature (“C) Fig. 5. Cracking in thicker ply groups.

50

Thermally induced damage

are due to damage in specimen preparation.16 The data for these ply groups also shows a great deal of scatter. In all cases, experimental results approach analytical predictions at the lowest temperature studied. 6.2 Single plies Single plies do not crack in the classical way shown in Fig. 1. The cracks do go from the top to the bottom of each ply (so cross-sections such as shown in Fig. 2 have the expected appearance), but do not go straight through the width of the specimen. Without continuous cracks to follow, the X-ray dye cannot penetrate into the specimen, so no cracks are visible in these plies in the X-ray photographs. Sanding down specimen edges reveals large variations in crack densities within each specimen. The 30” plies in the [O/ & 301s laminates are predicted to remain untracked during testing. A few cracks are observed at specimen edges, but none in the interior of specimens. Conversely, there are many more cracks in the single 45” plies of [O/45/90/ - 4.51slaminates away from the edges than near edges. There is a great deal of variability from one location to another within these plies. The correlation between the analytical predictions and the crack densities measured at the specimen free edges is poor in all cases. In the 45” plies, few cracks are observed at the edges. However, the correlation between the analytical prediction and the average internal crack density is quite good (Fig. 6). Cracks are observed at the edges of 90” plies at higher temperatures than predicted (Fig. 7). Again, damage during specimen manufacturing is suspected. The

1215

Analysis

-

16

l

EdgeData

0

Interior Data

[0/45/90/45]s 90” Ply i

Fig. 7.

Temperature (“C) Cracking in thin 90” plies.

average internal crack density correlation Fig. 7 is inconclusive.

shown in

6.3 Property changes

The change of laminate properties due to cracking was studied analytically. The laminate stiffness and CTE changes in a [O/45/90/ - 451s laminate due to predicted microcracking are shown in Figs 8 and 9, respectively. In both cases, the effect of cracking in the central -452 ply group is seen first, followed by the effects of cracking in the 45” and 90” plies. Figure 8 shows normalized laminate longitudinal stiffness

20 [O/45/90/-451s 45” Ply

P75/934 [O/45/90/-451s

-

Analysis

all plies

97

-200

-150

-100

-50

0

50

-200

Temperature (“C) Fig. 6.

Cracking of thin 45” plies.

Fig. 8.

I . . . . I . , . . I . . , . , . . . .

-150

-100

-50

0

50

Temperature (“C) Effect of cracking on laminate longitudinal stiffness.

C. H. Park, H. L. M&anus

1216

the fiber direction in each ply group of a thermally loaded [O/45/90/ - 453, laminate is shown through the first 2 mm from the free edge. The stresses reach expected (CLPT) values by 1 mm into the laminate. The transverse stress in the 45” ply groups drops to zero near the free edge. Experimentally, few cracks were seen at the edges of these ply groups, while internal crack densities reached predicted values within l-2 mm of the edge. 7 DISCUSSION

-0.6 -200

-150

-100

-50

0

Temperature (“C) Fig. 9. Effect of cracking on laminate longitudinal

CTE.

versus decreasing temperature and illustrates the small change in stiffness seen in all laminates considered. Figure 9 shows that the CTE is greatly affected by thermal loading and cracking. The values change by about 300% in the 25 - 157°C temperature range. The figures do not show temperature dependent material properties; the results reflect permanent property changes due to cracking damage. 6.4 Edge effects The free edge stress solution is shown in Fig. 10 for the [O/45/90/ - 453, laminate. The stress transverse to

AND

CONCLUSIONS

A method for predicting thermally induced cracking in composite laminates, and its effects on laminate properties, has been developed. In contrast with previous work, which concentrated on single transverse ply groups in simple laminates, cracking is predicted in all but the surface plies of any symmetric laminate. Experimental correlations show the method works well for predicting volume average crack densities in laminates exposed to low temperatures. The method does not correlate well with edge crack counts in some cases. Two effects, not accounted for by the assumptions used to develop the analysis, explain these discrepancies: the configuration of the cracks, and the effects of the specimen free edges. Cracks in groups of two or more 0.128 mm plies with the same orientation behave in the assumed way, with cracks continuous through the width of the laminate. Single plies do not crack in this fashion. Cracks are short, and crack density varies widely through the width of the laminate. Near edges, angle plies (such as 45” plies) show much lower levels of cracking than in the interior. This is due to the low transverse stresses in these plies near free edges. Conversely, 90” plies show more cracks near edges, and have cracks near edges even at room temperature. The reason for this is less clear, but it has been noted that these plies are sensitive to handlingI and the exact condition of the edge during loading.16,30 This work shows that edge crack counts, which have been accepted as a useful metric of damage, are not necessarily good indicators of the internal state of the laminate. This problem does not appear in thick, multi-ply groups, typical of the specimens used in many previous studies, but it is acute for thin individual plies typical of spacecraft structure. For these structures, both new screening test techniques (to replace edge crack counting) and improved analytical techniques (to incorporate the effects of free edges and to account for the non-ideal nature of cracks in these plies) are needed.

ACKNOWLEDGEMENT 0

0.5

1

1.5

DistanceFrom Free Edge (mm) Fig. 10. Stresses due to thermal loading near the free edge.

2

This work was supported by NASA Grant NAG-l1463, supervised by Steve Tompkins and Howard Maahs.

1217

Thermally induced damage

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4. Peters, P., The strength distribution of 90” plies in O/90/0 graphite-epoxy laminates. J. Cornp. Mater., 18 (1984) 545-5.55. 5. Highsmith, A. & Reifsnider, K., Stiffness-reduction mechanisms in composite laminates. Damage in Composite Materials, ASTM STP 775, 1982, pp. 102-117 of tensile matrix failure in 6. Flaggs, D., Prediction composite laminates. J. Comp. Muter., 19 (1985) 29-50. 7. Laws, N. & Dvorak, G., Progressive transverse cracking in composite laminates. J. Comp. Muter., 22 (1988) 900-916.

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15. Lim, S. & Hong, C., Effect of transverse cracks on the thermomechanical properties of cross-ply laminated composites. Comp. Sci. Technol., 34 (1989) 145-162. 16. Park, C., Analysis of thermally induced damage in composite space structures. MSc thesis, Massachusetts Institute of Technology, 1994. 17. Adams, D., Bowles, D. & Herakovich, C., Thermally induced transverse cracking in graphite/epoxy cross-ply laminates. J. Reinf Plust. Comp., 5 (1986) 152-169. 18. Bowles, D., Effect of microcracks on the thermal expansion of composite laminates. J. Comp. Mater., 18 (1984) 173-187. 19. Bowles, D. & Griffin, O., Micromechanics analysis of space simulated thermal stresses in composites: I. Theory and unidirectional laminates and II. Multidirectional laminates and failure predictions. J. Rein& Plast. Comp., 10 (1991) 505-539. 20. Knouff, B., Tompkins, S. & Jayaraman, N., The effect of graphite fiber properties on microcracking due to thermal cycling of epoxy-cyanate matrix laminates. ASTM 5th Symp. on Composite Materials: Fatigue and Fracture, Atlanta, GA, 1993, pp. 2-27.

21. Tompkins, S. & Williams, S., Effects of thermal cycling on mechanical properties of graphite polyimide. J. Spacecraft and Rockets, 21 (1984) 274-279. ** LL.

Tompkins, S., Bowles, D., Slemp, Response of composite materials orbit environment. AZAAINASA Williamsburg, VA, 1988. 23. Tompkins, S., Effects of thermal materials for space structures.

W. & Teichman, A., to the space station Space Station Symp.,

cycling on composite

NASAlSDIO Space Environmental Effects on Materials Workshop, 1989, pp.

447-470. 24. Tompkins, S., Funk, J., Bowles, D., Towell, T. & Connell, J., Composite materials for precision space reflector panels. SPIE Int. Symp. and Exhibition on Optical Engineering and Photonics, Orlando, FL, 1992. 25. Manders, P. & Maas, D., Thin carbon fiber prepregs for dimensionally critical structures. SPZE Advances in Optical Structure Systems, Orlando, FL, 1990, pp. 536-545. 26. Bowles, D. & Shen, J., Thermal cycling effects on the dimensional stability of P75 and P75-T300 (fabric) hybrid graphite/epoxy laminates. 33rd Int. SAMPE Symp. and Exhibition, Anaheim, CA, 1988. 27. McManus, H., Bowles, D. & Tompkins, S., Prediction of thermally induced matrix cracking. J. Reinf Plast. Comp. (in press). 28. Maddocks, J., Microcracking in composite laminates under thermal and mechanical loading. MSc thesis, Massachusetts Institute of Technology, 1995. 29. Jones, R., Mechanics of Composite Materials. Hemisphere, 1975. 30. Kitano, A., Yoshioka, K., Noguchi, K. & Matsui, J., Edge finishing effects on transverse cracking of cross-ply CFRP laminates. 9th Int. Conf on Composite Materials, Vol. 5, Madrid, 1993, pp. 169-176

APPENDIX The calculation of the strain energy released by a crack (eqn (1)) is derived using a shear-lag model. The model is illustrated in Fig. Al. The stress in the cracking layer, a,, the stress in the rest of the laminate, a,, and the shear stresses between the layers, q, are all taken to be functions only of x’. The coordinate system of Fig. 2 is used, centered on a virtual crack between two existing cracks at x’ = f h. A unit laminate width (in the y’ direction) is assumed for simplicity. In Fig. 1, equilibrium of each layer requires:

(AlI

AT

I

7

I

a,/2

-dx’

-I

Fig. Al. Shear-lag stress model.

C. H. Park, H. L. McManus

1218 and global requires:

equilibrium,

given

no

applied

grar + uCuC= 0

The stress-displacement

uc k

-=-E, dx’

!3=2- du E, dx’

a,A~

+ w%)

2@,E,

u, = F

c

(cxc- cx,)AT (A5)

; h = +=-

a,)AT

(A6)

c

eqn (A5) reduces to the form: & 452 dX,Z-,Za,=

GI =

where AW is the external work performed during the formation of the crack (here, as no mechanical load is being applied, AW = 0) and AU is the change in strain energy within the laminate. The strain energy in the initial state is calculated from the contributions of the normal stresses, U, and the shear stresses, U,: (A13)

u, = h 42&t

I

+Bcosh(F)

+$

-h

(A8)

1 _ cosh(25x’lac)

cosh(2,$hla,)

1

1 -

1 _ cosh(2&“lac)

x

cosh(Wa,) 1

[

(A9)

ur*=

Combining eqns (A2) and (A9) gives:

1

acErEc

x

1 _ cosh(2&“la,)

Note the terms in the braces are the far-field stresses

(away from any cracks) and the terms in square brackets show the altered distribution of stress due to

4*=5

VW

(a, - CYJAT]

arEr + w% [

(Al’3

(A14)

K

Equation (A14) lacks the factor l/2 because two interfaces must be considered (see Fig. Al). The change is between the stress state with crack spacing 2h, solved above, and the state with crack spacing h, (i.e. with the virtual crack in Fig. 2 existing). The stresses in the new state can be calculated by the same method used above, but considering two identical regions with cracks at x” = f h/2. The coordinates x” are defined for convenience to be centered between the new crack and an existing one, i.e. x” =x’ -h/2. The stresses in this case will be: ,*= c

[

(Al21

(A7)

Here, h is only a notation of convenience, but 6 captures in dimensionless form an important feature of the solution. Physically (from eqn (A6)) it is a ratio between the shear and axial stiffnesses of the shear lag problem; its effect on the solution can be seen in eqn (A8), where it and the cracking layer thickness, a,, control the length scale of the solution. First consider the stresses without the virtual crack. Applying the boundary conditions a, = 0 at x’ = f h to eqn (A8) and using eqns (A6) to simplify gives:

x

AW-AU

2 u, = 5

-A

(All)

Again, the term in square brackets controls the distribution of the stress. In this case, shear stress is maximized at the location of cracks, and reduces to zero in the far-field. The energy released per unit area by the appearance of a crack at x’ = 0 is expressed as:

which is an ODE with solutions of the form: a,=Asinh(F)

I

a,

+ a&

aracEr&

KG&

sinh(25x’/a,) cosh(2,$h/a,)

Using the definitions:

5=

1_

(A3) (A4)

a&

ar;:E;f;Ec (ac- u,)AT)

4=5 { -

are:

where u, and u, are the displacements in the cracking layer and the rest of the laminate, and K is an effective shear stiffness of the bonding layer between them. Placing eqn (A4) into the first of eqns (Al), taking the derivative of the result with respect to x’ and using eqns (A2) and (A3) to express the result in u, yields: d=u ---2K dxr2

the existing cracks at &h. Finally, the shear stress is from the above and eqn (Al):

(A2)

relationships

cu,AT;

forces,

t

$?:;IE

cash@ /a,) I

c

(‘416)

(a, - +T)

1 _ sinh(2@“/a,) cosh(Eh/a,)

1

(AI7)

Thermally induced damage

Now the energies can be calculated: h/2 U:

=a,

I -h/2

*2

h/2

“dx”+a, I -hl2

E, hi2

q

=

2 I -hi2

1219

Collecting terms and simplifying gives: *2

(7, &”

G = _ ar&Wc (AN

E,

K

2Wr

- aJ2AT2 + a&J

_ cosh(2&“IaJ

*2

4&”

I

(Al9)

cosh(&/a,) _

Equation (A13) can now be expressed as:

cosh(2&‘/a,) cosh(2[h/a,)

1 [cosh(@/a,) I) 1[ 2 + sinh(2&“la,)

2+

sinh(2&‘la,)

2 dx,, dx,

cosh(2,$h/a,)

(A21) GI=

-1

Ci

U:-UC+U;-Uq

(A20)

Equation (A20) can be evaluated by substitution of eqns (A9)-(All) into eqns (A13) and (A14), and eqns (A15)-(A17) into eqns (AH) and (A19).

The integrals of eqn (A21) can then be evaluated with the aid of standard identities for the hyperbolic functions, and the results rearranged into the form of eqn (1). A more detailed treatment of this development is given elsewhere.28