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Failure mechanisms in thick composites under compressive loading
Composites Part B 27B (1996) 543-552
Copyright 0 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/96/$15.00
ELSEVIER
Failure mechanisms in thick composites under compressive loading
Isaac M. Daniel Northwestern
and Hao-Ming
Hsiao
University, Evanston, IL 60208, USA
and Shi-Chang
Wooh
Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Received 27 January 1995; accepted 15 May 1995)
Failure mechanisms were studied in a unidirectional carbon/epoxy composite under uniform and linearly varying longitudinal compression. The first failure mechanism is shear yielding or shear failure in the matrix precipitated by initial fiber misalignment. It was shown how an initial fiber misalignment of 1S” can produce the measured compressive strength of 1725 MPa (250 ksi). Matrix failure is followed by fiber buckling and fracture, resulting in the formation of a kink band. The kink band orientation is constant in the range of /3 = 20-30”, whereas the kink angle (Yvaries from a small initial value to a maximum value of 2,& Kink band widths varied between 4 and 20 fiber diameters. Kink bands can occur on different planes which can rotate along the band. Kink band multiplication or broadening with increasing stress was observed at points where the maximum kink angle o was reached. Copyright 0 1996 Elsevier Science Limited (Keywords: failure mechanism;thick composites; compression testing; fiber misalignment;fiber microbuckling;kink bands, compressivestrength)
INTRODUCTION The design of some new composite structures requires advanced composite materials in sections up to 30cm (12 in.) in thickness. Many such applications of thick composites involve high compressive loading. Test results of thick composite cylinders loaded under hydrostatic pressure show premature failures at stresses below the compressive strengths measured in thin composite laminates’. These results show that compressive behavior of thick composites is not well understood. Early failures have been attributed primarily to fiber misalignment/waviness and nonlinear behavior of the matrix. Compressive failure mechanisms of composite materials have been studied by many investigators. Failure modes proposed for longitudinal compression include global Euler buckling, fiber microbuckling24, kinking14-16, matrix banding7-13, plastic kink yielding’71’8, shearing through both the fibers and the matrix” and buckling/kinking2s22.
Address for correspondence: Northwestern University, Center for Quality Engineering & Failure Prevention, 2137 North Sheridan Rd, Evanston, IL 60208-3020, USA
Greszczuk23 investigated the effects of matrix material, fiber diameter, fiber volume ratio, specimen manufacturing defects and specimen geometry, boundary conditions on longitudinal compressive strength. The influence of defects, such as fiber waviness and nonlinear matrix behavior, on compressive failure mechanisms and strength were studied by Davis24, Wang4, Hahn and Williams’, Suarez et a1.25 and Yeh and Teply26. Several investigators have addressed the phenomenon of kink band formation due to microbuckling. Berg and Salama’ were the first to observe kink banding in graphite/epoxy composites under longitudinal compression. Weaver and Williams’ suggested that kink banding is initiated by the transverse fracture in the region of buckled fibers and proceeds by progressive buckling and fracture of adjacent fibers and matrix. In this manner the kink band broadens and propagates across the composite section. Evans and Adler” presented a more detailed analysis of the various modes of kink formation, correlating some of the key microstructural parameters that govern kinking, under conditions of both axial compression and applied shear. Budiansky14 predicted the kink band angle for long-wave and short-wave imperfections. Experimental results of
543
Failure mechanisms
in thick composites:
I. M. Daniel
et al.
Hahn and Williams’ provided valuable information on the role of material properties (i.e. the matrix, interface) in causing different types of failure (e.g. microbuckling, kinking, etc.) at the microlevel. However, the influence of defects such as fiber waviness, etc., on the kink band geometry and the precise mechanism of the role of kink banding in reduction of the compressive strength have not been addressed in analyses to date. Compressive failure is often a near instantaneous and catastrophic phenomenon and for this reason it is difficult to monitor experimentally the nature and sequence of failure mechanisms. The latter are greatly influenced by the presence of defects such as fiber misalignment/waviness and by the local state of stress. This paper describes an investigation of failure mechanisms observed in unidirectional composite laminates under compressive loading.
with the fibers. The stress components referred to the principal material axes (fiber direction) obtained by stress transformation are 01
=
ax cos20P
+ 712)
o2
=
0,
sin’(cp + y12)
712 =
5
sin(cp
+ ~12)
(1)
cos(cp
+ ~12)
Assuming both the initial misalignment cp and the induced shear strain y12 to be small, equation (1) can be approximated as
Thus, the in-plane shear stress r12 is a nonlinear function of ylz, since o, is also a function of y12. From the last of equation (2) we obtain PREDICTION
OF COMPOSITE
FAILURE
Most theoretical models attempting to predict the longitudinal compressive strength of composites assume the existence of some fiber misalignment. They differ in the way they treat the effects of this misalignment and in the assumed response of the material, see, for example, Wang4, Hahn and Sohi13 and Haberle and Matthewsz7. The assumed initial fiber misalignment ‘p in a unidirectional composite, increases by the value of the induced in-plane shear strain y12 (or ye) under the axial load a, (Figure I). This situation then, is that of an offaxis lamina under uniaxial stress (T, at an angle of cp+ y12
The limiting value of (T, 2 g1 is the longitudinal compressive strength Fi,. The necessary condition for a maximum ox is da “=O %I2
or, from equation (3),
[
$(v+Y1Z) -T12]l
=o
(‘p + Y1212
or, (a)
Cp=
initial fiber misalignment
a-12 -=-
712
ay12
cp + 712
(4)
Given the shear stress versus shear strain curve obtained experimentally, the solution of equation (4) corresponds to the values 712 = T* and y12 = y*, where the tangent to the curve equals the slope T*/((P + y*). A measured shear stress-strain curve for carbon/epoxy (IM6G/3501-6) is shown in Figure 2 with an illustration of a graphical determination of T*. The longitudinal compressive strength is then Flc
=
(4nax
=
&
If the induced shear strain yi2 is much smaller than the initial fiber misalignment cp, which is still assumed to be small, equation (5) reduces to YIZ= in-plane shear strain = additional
fiber rotation
Figure 1 Unidirectional lamina with initial fiber misalignment and after axial compressive loading
544
before
F~,
N
(T12)max = F12 cp
cp
which is identical to Argon’s prediction7 and where Fi2 is the in-plane shear strength of the material. The same result is obtained by applying the maximum shear stress
Failure mechanisms
in thick composites:
I. M. Daniel et al.
/e
Compressive Suengthz
Figure 2 Graphical determination strength. (9 = initial misalignment)
criterion Then2*,
of
longitudinal
compressive
to the initially misaligned (off-axis) lamina. F12 FI,
=
ES
sincpcoscp-
cp
(7)
If the initial misalignment is negligible compared to the induced shear strain, i.e. if up< y12, then Rosen’s result2 FI,
g
(52
(8)
is recovered. With a further assumption of high shear stiffness fibers and application of a rule of mixtures series model, equation (8) can be written in the form F,, E
Gm
1 - v,
where G, is the matrix shear modulus and V, is the fiber volume ratio. Equation (9) is another form of the Rosen model prediction2. The above discussion clearly suggests that longitudinal compressive failure is intimately related to and governed by the in-plane shear response of the composite, even in the presence of the slightest initial fiber misalignment. Any compressive failure mechanisms observed must follow some form of initial shear failure of the composite, which is a matrix dominated property.
FAILURE MECHANISMS UNDER UNIFORM COMPRESSIVE LOADING The material investigated in this study was carbon/epoxy (IM6G/3501-6, Hercules, Inc.). Unidirectional plates of various thicknesses were fabricated from the prepreg material, following previously developed fabrication
Figure 3 Schematic of NU compression adaptor a = 25 mm, base b = 38 mm)
text fixture.
techniques
described by Daniel
for thick composites
(Dimensions:
et a1.29 and Hsiao et al.30
Longitudinal compressive tests were conducted with a recently developed compression test method/fixture (NU fixture) incorporating both shear loading and end loading concepts, see Daniel et al.29 and Hsiao et al.30 (Figure 3). The specimens were 12.7 cm (5.00 in.) long, 1.27 cm (0.50 in.) wide, with thicknesses ranging between 2.03 mm (0.080in.) and 19.05 mm (0.75 in.). The specimens were tabbed with 5.40cm (2.125 in.) long glass/ epoxy tabs, leaving a 1.91 cm (0.75in.) long gage length. After tabbing and machining to final tolerances, steel end caps were bonded to the ends of the specimen to prevent premature crushing at the ends. A typical specimen configuration is shown in Figure 4. Strain gages were mounted on both sides and on the edges of the specimen in the gage section. All tests were carried out in a servohydraulic testing machine while recording the strains and failure mechanisms by video photography. A typical stress-strain curve to failure for a 48-ply specimen is shown in Figure 5. Using the experimentally obtained shear stress, 712, versus shear strain, 712, curve for the material in Figure 2 and equation (5) we obtain the variation of the predicted compressive strength, Flc, with initial fiber misalignment, cp, as depicted in Figure 6. The measured
545
Failure mechanisms
in thick composites:
i. M. Daniel
et al
i
‘0
E
R
I $1
OS
1.0
1.5
Fiber Misalignment,
rp,
LO 2.0
(deg)
Figure 6 Variation of predicted longitudinal with initial fiber misalignment for IM6G/3501-6
L. mm (ill.) 114 (4.50) 127 (5.00)
h, mm (1n.1
48.ply 72.~1~
133 (5.25)
25 (1.M))
16-ply
Figure 4
Schematic
of specimen
??
compressive strength carbon/epox:G
13 (0.50) 19 (0.75)
configuration
El = 168.9 GPa (24.5 Msi) Fi, = 1738 MPa (252 ksi) E’;, = 1.22 % 15w
-
050 0
02
04
06
08
10
12
1.4
Strain, E , (%)
Figure 5. Stress-strain curve of unidirectional 48-ply carbon/epoxy under longitudinal compressive loading (IM6G/3501-6)
compressive strength of 1725 MPa (250 ksi) corresponds to an initial fiber misalignment of cp g l.S’, or approximately one-fourth of the initial modulus Gt2 = 7.0 GPa (l.OMsi). The initiation and development of failure mechanisms cannot be observed easily, since the 3501-6 matrix is relatively brittle and fracture is near instantaneous and catastrophic. Failure mechanisms were recorded only after the test (post mortem) by means of optical as well as scanning electron photomicrography. Failure mechanisms in the gage sections are very difficult to observe
546
Figure 7 Illustration of narrow carbon/epoxy under longitudinal
and wide kink bands in IM6G/3501-6 compression
because of the catastrophic nature of failure. The easiest ones to observe are those occurring in constrained regions, such as the tabbed region of the specimen following premature failure initiation by end crushing. Early failures and discernible failure patterns occur in
Failure mechanisms in thick composites: I. M. Daniel et al.
Figure 9 Multiple kink bands of various orientations carbon/epoxy under longitudinal compression
Figure 8
Kink band geometry
regions where an additional shear stress component is added (through the tabs) to the direct compressive normal stress component. According to the previous discussion, adding shear is tantamount to increasing the fiber misalignment and causes early failure. Failure patterns appeared in the form of kink bands as shown in Figures 7 and 8. The kink band geometry is usually characterized by the kink band width, the inclination angle of the kinked fibers, CY,with respect to the loading axis, and the orientation angle of the kink band boundary, ,9, with respect to the transverse to the loading direction (Figure 8). In the upper photograph of Figure 7 the kink band width is equal to seven fiber diameters and the kink inclination and kink band boundary orientation angles are equal, cx = ,0 = 30”. In the lower photograph of the same figure, the kink band width is sixteen fiber diameters and the kink inclination and kink band boundary angles are also equal,
and widths
in
(Y= ,l?= 20”. Figure 9 shows multiple kink bands branching from a source region. Within each branch there are multiple kink bands of various orientations and widths. In the lower photograph of Figure 9, for example, one can identify a kink band that has a constant width of twenty fiber diameters with angles a g p % 23”, over some short length. However, these parameters may change along the kink band. A great deal of work has been reported on analysis and prediction of compressive failure mechanisms and, in particular, kink formation. In some early works a limiting relationship between the kink inclination and kink band angle was obtained as Q = 2/3. Chaplin” obtained this relation by assuming zero volumetric strain in the kinked zone, but did not justify that assumption. Similarly, Evans and Adler” arrived at the same conclusion by minimizing the elastic strain energy, which was tantamount to assuming zero axial strain in the kinked zone. In reality, the state of stress and deformation in the kinked zone is very complex and varies during the loading process. Experimental results do not show any fixed relationship between Q and p.
547
Failure
mechanisms
Figure 10 sive stress
Pure bending
in thick
composites.
test for producing
linearly
i. M. Daniel
varying
et al
compres-
at the points of maximum curvature. ‘ihc 1110 krnh boundaries are of unequal length. The kink propagatcb farther along the points of the fiber where the high tensile strain (convexity) in the fiber is on the side of highcl global compressive stress. Attempts have been made to predict the kmh hand width. Results of various analytical approaches predict kink band widths of the order of lo-15 fiber diameters” A rough estimate can be obtained by using the simple Euler buckling formula for an unsupported fiber. Assuming that compressive failure is initiated by fiber buckling, following shear yielding or excessive shear deformation of the matrix. the critical fiber length at buckling into the slgmoidal shape of I-‘i,~ul-~~1.3. is given bq Figure 11 field
FAILURE VARYING
.4rrested
kink band in linearly
MECHANISMS COMPRESSIVE
varying
compressive
stress
UNDER LINEARLY STRESS
where df = fiber diameter
In order to study the initiation and progression of compressive failure, a pure bending test was designed to produce a linearly varying compressive stress in a unidirectional composite (Figure 10). This type of loading produces compressive failure at the top of the beam which propagates down to a point where the compressive stress is not high enough to initiate failure. Thus, an arrested kink band is obtained with features corresponding to the varying compressive stress along its length. Figure II shows such a kink band whose width is approximately 4-5 fiber diameters and whose inclination angle ,B is 30”. The kink angle N seems to increase from a very small initial value to approximately 50”. An enlargement of the region at the end of the kink band, or the beginning of failure, is shown in Figure 12. It is seen that fiber fracture is preceded by fiber buckling. The buckled fiber has a sigmoidal shape and fractures occur
548
Ef = fiber modulus
07t:cr= critical fiber stress at buckling For the material
under
investigation,
E,- = 276 GPa
and
where (a, )pL is the proportional limit in the compressive stress-strain curve of Figure 5 and Vf is the fiber volume ratio. The proportional limit was used because it corresponds to the initiation of local compressive failure and kink band formation. Therefore, the fiber stress corresponding to this proportional limit stress is taken equal to the critical buckling stress. Substituting the above values into equation (10) we obtain L,, ” I I .9 d,
Failure mechanisms
1 P
7 --
-
-
in thick composites:
I. M. Daniel
et al.
= 2p = 60”
a-
Figure 14 Illustration
of kink
band
broadening
with
increasing
compression
Substituting numerical values for the material under investigation, F12 = 73 MPa, cff.cr= 1212 MPa and Vf = 0.66, we obtain Lug= 5.3”
-
Figure 13 Geometry
of buckled
which is very close to what has been observed. As the compressive stress increases the kink angle increases from this initial value to a maximum value which should be Q = 2/3 according to Weaver and Williams’ and
-
fiber
Chaplin
lo.
Another
phenomenon observed and illustrated in is kink band broadening with increasing compression. At two locations in the figure it is seen that the local kink angle reaches a maximum value, which was measured to be Q = 60” = 2,0. As the compressive stress increases, this angle Q does not increase beyond this maximum value but a new kink band is generated at that point, adjacent to the band with the maximum Q angle. Thereafter the kink angle (Yfalls below its maximum value as the broader kink band propagates. All measurements of kink parameters were made after the specimen was unloaded. Figure
In the buckling mode observed in Figure 12 and shown in Figure 13 there is one inflection point at the midpoint between the assumed fixed ends of the fiber. Therefore, the length L in Figure 13 is equivalent to the length of a pin-ended column for which the critical value is given by equation (10). The distance between the two fiber failures within the length L, which is the kink band width, is approximately equal to L,,/2. The predicted band width would be higher if the lateral constraint of the matrix were taken into account and if the ends of the sigmoid-shaped segment of the fiber were not assumed fixed. If fiber buckling is assumed to follow immediately after shear yielding or shear failure of the matrix, then, the critical fiber stress at buckling/shear failure can be related to the initial kink angle a0 as follows 2F12 gf.cr
=
Vf
sin 2~0
=
sin 2~~
or 2F12 vpf,.,r
(11)
14
FAILURE MECHANISMS GRADIENT STRESSES
UNDER HIGH
To further investigate the initiation and progression of compressive failure, axial compressive tests were conducted on unidirectional specimens with semi-circular edge notches (Figure 15). This type of loading produces a high compressive stress at the center of the notch. A stress concentration of approximately 6.8 exists at the center of the notch. However, a relatively high shear stress, 712, exists at approximately 10” off the center of the notch, whose value is 712 = 0.65 ao, where a0 is the far field
549
Failure
mechanisms
in thick
composites:
I. M. Daniel
et al
Shear failure
Hole boundary Figure 16
Initial shear fGlure on notch boundary axial compression
1
Compression
Kink Bads
01 speumen
Second major kink band (out-of-plane)
Shear failure (initiated from hole)
I
4
urldcl
and Vertical Shear Splitting
P Figure 15 Unidirectional
notched
specimen
under axial compression
applied uniaxial stress. For the carbon/epoxy material investigated, compressive failure at the center of the notch would be initiated at an applied nominal stress of
Flc “O = 6.8 =
1725
~
6.8
= 254 MPa First major kink band
whereas location
shear failure at an applied F12 c~=~===
would start at the 10” off center stress of 73
Figure 17 Major kink bands starting
from shear failure
112MPa
Indeed, as seen in Figure 16, the initial failure around the notch is axial shear failure, propagating along the vertical (loading) direction for some distance. Major kink bands are initiated from this shear failure because the fibers adjacent to the shear failure band lose their lateral support and start failing. In-plane and out-ofplane major kink bands starting from shear failure are shown in Figure 17. Major kink bands branch off or are
connected with narrower secondary kink bands and additional axial shear splitting as shown in Figure 18. Kink bands can start on one plane and rotate along their length (Figure 19). In many cases where a kink band is arrested, it is clearly seen that the kinking mode of compressive failure is transformed into an axial shear failure at the end of the kink band as shown in Figure 20. This again highlights the significance of matrix shear failure as a precursor to kink banding.
Failure mechanisms
in thick composites:
Second major kink band (out-of-plane)
1. M. Daniel
et al.
Shear failure (initiated from kink band)
\
Figure 20
Illustration
of shear failure at the end of a kink band
First major kink band Figure 18 Connection and shear splitting
of major kink bands through
Figure 19 Illustration
of kink band with changing
SUMMARY
AND
secondary
bands
plane
CONCLUSIONS
Failure mechanisms were investigated in a unidirectional carbon/epoxy composite under longitudinal compression. The material investigated was IM6G/3501-6 carbon/epoxy in thicknesses up to 72 plies (9.1 mm; 0.36in.). Tests were conducted under uniform and linearly varying compressive stress. The first failure mechanism appears to be shear yielding or shear failure in the matrix precipitated by pre-existing fiber misalignment. It was shown that the longitudinal compressive strength of the material can be predicted on the basis of the shear stress-strain response
of the material (7i2 versus y12) and the initial fiber misalignment. It was calculated that it takes an initial fiber misalignment of approximately cp E 1.5” to predict the measured compressive strength of Fi, = 1725 MPa (250 ksi). Matrix shear yielding is followed by fiber buckling and fracture. Fracture of a fiber increases shear in its vicinity, thereby precipitating further matrix yielding which in turn causes further fiber buckling and fracture. The buckled fiber has a sigmoidal shape and results in two fractures on the convex sides at the points of maximum curvature. A simple buckling calculation predicts a distance between fiber fractures equal to at least six fiber diameters. The resulting failure pattern takes the form of a kink band. The kink orientation angle at initiation of kinking is very small, but it increases with compressive stress. For the material investigated this initial angle was estimated to be o. = 5.3” which is close to experimental observations. This angle o: can increase under increasing stress up to a maximum value amax = 2p, where p is the kink band orientation. When this condition is reached, further increase in stress causes kink band multiplication or broadening, starting at the point where CX~,,is reached. Thereafter, (Y decreases with the propagation of the multiple or broadened kink band. Kink angles czbetween very small values and nearly 60” were observed. The kink band angle p seems to depend on the local and global state of stress. In the cases studied it was found to vary between 20 and 30”. The kink band width varied between 4 and 20 fiber diameters. No significant effect of thickness was noticed on the overall compressive strength. However, the kink band geometry was more complex for thicker specimens. Kink bands occurred on different planes which, in some cases, rotated along the same band.
551
Failure
mechanisms
in thick
composites:
I M. Daniel
In the case of’more complex states of stress. such as in the vicinity of notches, in-plane shear l‘ailure is a more likely initial failure mechanism. Compressive failure in the form of kink bands is initiated from the shear failure at a much lower compressive stress than would be required in the absence of the shear failure. Major kink bands are connected through narrower secondary ones and additional axial shear failures.
et at
4 i5
16
17 18
ACKNOWLEDGEMENTS The work described in this paper was sponsored by the Office of Naval Research. We are grateful to Dr Y. D. S. Rajapakse of ONR for his encouragement and cooperation and to Mrs Yolande Mallian for typing the manuscript.
2x37
19 20
21
REFERENCES 22 1
2
3 4
5
6
552
Garala. H.G. Experimental evaluation of graphiteeexpoxy cylinders subjected to external hydrostatic compressive loading. In ‘Proc. 1987 SEM Spring Conf. on Experimental Mechanics’, Society for Experimental Mechanics, Bethel, CL 1987, pp. 948-951 Rosen, B.W. Mechanics of composite strengthening. In ‘Fiber Composite Materials.’ American Society of Metals. Metals Park. OH, 1965. pp. 35.-75 Greszczuk, L.B. Microbuckling failure of circular fiber-reinforced composites. AIAA J. 1975. 13, 1311 1318 Wang, A.S.D. A non-linear microbuckling model predicting the compressive strength of unidirectional composites. ASME Paper 7%WA:Aero-1, 1978 Hahn, H.T. and Williams J.G. Compression failure mechanisms in unidirectional composites. In ‘Composite Materials: Testing and Design’, (Seventh Conference), ASTM STP 893, American Sot. For Testing and Materials, Philadelphia, 1986, pp. 115-139 Chaudhuri, R.A. Prediction of the compressive strength of thick-section advanced composite laminates. J. Compos. Mater. 1991, 25. 1244 Argon. A.S. Fracture of composites. In ‘Treatise on Materials Science and Technology’, Vol. 1, (Ed. H. Herman) Academic Press. New York. 1972, pp. 79 114 Berg, C.A. and Salama, M. Fatigue of graphite fiber-reinforced Fibre Sci. and Tech. 1973. 6, 79 epoxy in compression. Weaver. C.W. and Williams, J.G. Deformation of a carbon epoxy composite under hydrostatic pressure. J. Mater. Sci. 1975. IO, 1323 Chaplin, R. Compressive fracture in unidirectional glass-reinforced plastics. J. Mater. Sci. 1977. 12, 347 Evans, A.G. and Adler, W.G. Kinking as a mode of structural degradation in carbon fiber composites. Actu Metali. 1978, 26. 725 Parry. T.V. and Wronski, A.S. Kinking and compressive failure in uniaxially aligned carbon fibre composites tested under superposed hydrostatic pressure. J. Mater. Sci. 1982. 17, 893
Wahu. i-1.1 ‘md SohI. y1.M Bucklmg oi embedded m epoxy. Ffhrc ‘w/ li,c,ir. 1986. 27. Budianskv. B .Micromechamcs. (~~q~c/~~u\ )0X3, 16. : Budiansky. B. aud Fleck. N.A. Compressive taiiurc ,.r. Solids. 1993, 41 (I), 183 Jelf, P.M. and Fleck, N.A. Compression failure mechanisms iii unidirectional composites. J. Compos. Muter. 1992. 26 ( IX). 2706 Swift. D.G. Elastic moduli of fibrous composites contaming misaligned fibers. J. Phjx. D: Appl. Phys. 1975. 8. 223 Piggott. M.R. A theoretical framework for the compressive properties of aligned fiber composites. J. Muter. SC;. 198 I. 16.
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26 27
28 29
30
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Hancox. N.L The compression strength of unidirectional carbon fiber reinforced epoxy resin. J. Muter. Sci. 1975. 10, 234 Steif. P.S. A model for kinking in fiber composites: I. Fiber breakage via microbuckling: II. Kink band formatinn. hrr. .I Solid.7 Strwt. 1990. 26. 549 (‘hung, I. and Weitsman. Y.J. Model for the micro-huckhng micro-kinking compressive response of fiber-reinforced composites. In ‘Mechanics USA 1994, Proc. of the Twe!fth US Nat. Congress of iZpplied Mechanics’, (Ed. A.S. Kobayashi). ASME, 1994. S2566S261 Schapery. R.A. Prediction of compressive strength dnd kmk bands in composites using a work potential. Int. J. Solid.\.S/rrrc! 1995. 32.(6!7). 739 Greszczuk. I .R. On farlure modes of mudirectional composite\ under compressive loading. In ‘Fracture of Composite Materials: Proc 2nd [ISA USSR Symposium’. Martinus NijhotTPubhshers. 19X2, pp. 731 244 Davis. J.G Compression strength ot’fiber reinforced composite ASTM STP HO. materials. In ‘Composite Reliability.’ American Sot. for Testing and Materials, Philadelphia. !c)75. pp. 264 3?’ Suarez, J.A.. Whiteside. J.B. and Hadcock, R.N. The umurnce of local failure modes on the compressive strength of boron epoxy composite. In ‘Composite Materials: Testing and Design (Second Conference)‘. ASTM STP 497, American Sot. for Testing and Materials, Philadelphia. 1972, pp. 237 -256 Yeh, J.R. and Teply. J.L. Compressive response of kevlar:epoxy composites. J. Compos. Muter. 1988, 22. 245 Haberle, J.G. and Matthews, F.L. A micromechanics model for compressive failure of unidirectional fibre-reinforced plastics. J. Compos. Muter. 1994, 28 (17). 1618 Daniel, I.M. and Ishai. 0. ‘Engineering Mechanics of C‘omposite Materials’, Oxford University Press, New York. 1994 Daniel, I.M., Hsiao. H.M., Wooh, SC. and Vittoser. .I Processing and compressive behavior of thick composites. In ‘AMD-Vol. 162. Mechanics of Thick Composites’. (Ed. Y.D.S. Rajapakse). ASME, New York, 1993, pp. 107. 126 Hsiao, H.M . Wooh, S.C. and Daniel, I.M. Fabrication methods for unidirectional and crossply composites with fiber waviness. J. .4&a. Murer. 1995, 26 (I), 19 Hsiao, H.M.. Daniel. I.M. and Wooh, SC‘. A new compression test method for thick composites. J. Compo.y. MatcJr. 1995. 29 (13). 1789 Fleck. N.A . Deng, L. and Budiansky. B. Prediction of kink band width m fiber composites. Report MECH-203. Harvard Univcrsit\. 1907
Copyright 0 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/96/$15.00
ELSEVIER
Failure mechanisms in thick composites under compressive loading
Isaac M. Daniel Northwestern
and Hao-Ming
Hsiao
University, Evanston, IL 60208, USA
and Shi-Chang
Wooh
Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Received 27 January 1995; accepted 15 May 1995)
Failure mechanisms were studied in a unidirectional carbon/epoxy composite under uniform and linearly varying longitudinal compression. The first failure mechanism is shear yielding or shear failure in the matrix precipitated by initial fiber misalignment. It was shown how an initial fiber misalignment of 1S” can produce the measured compressive strength of 1725 MPa (250 ksi). Matrix failure is followed by fiber buckling and fracture, resulting in the formation of a kink band. The kink band orientation is constant in the range of /3 = 20-30”, whereas the kink angle (Yvaries from a small initial value to a maximum value of 2,& Kink band widths varied between 4 and 20 fiber diameters. Kink bands can occur on different planes which can rotate along the band. Kink band multiplication or broadening with increasing stress was observed at points where the maximum kink angle o was reached. Copyright 0 1996 Elsevier Science Limited (Keywords: failure mechanism;thick composites; compression testing; fiber misalignment;fiber microbuckling;kink bands, compressivestrength)
INTRODUCTION The design of some new composite structures requires advanced composite materials in sections up to 30cm (12 in.) in thickness. Many such applications of thick composites involve high compressive loading. Test results of thick composite cylinders loaded under hydrostatic pressure show premature failures at stresses below the compressive strengths measured in thin composite laminates’. These results show that compressive behavior of thick composites is not well understood. Early failures have been attributed primarily to fiber misalignment/waviness and nonlinear behavior of the matrix. Compressive failure mechanisms of composite materials have been studied by many investigators. Failure modes proposed for longitudinal compression include global Euler buckling, fiber microbuckling24, kinking14-16, matrix banding7-13, plastic kink yielding’71’8, shearing through both the fibers and the matrix” and buckling/kinking2s22.
Address for correspondence: Northwestern University, Center for Quality Engineering & Failure Prevention, 2137 North Sheridan Rd, Evanston, IL 60208-3020, USA
Greszczuk23 investigated the effects of matrix material, fiber diameter, fiber volume ratio, specimen manufacturing defects and specimen geometry, boundary conditions on longitudinal compressive strength. The influence of defects, such as fiber waviness and nonlinear matrix behavior, on compressive failure mechanisms and strength were studied by Davis24, Wang4, Hahn and Williams’, Suarez et a1.25 and Yeh and Teply26. Several investigators have addressed the phenomenon of kink band formation due to microbuckling. Berg and Salama’ were the first to observe kink banding in graphite/epoxy composites under longitudinal compression. Weaver and Williams’ suggested that kink banding is initiated by the transverse fracture in the region of buckled fibers and proceeds by progressive buckling and fracture of adjacent fibers and matrix. In this manner the kink band broadens and propagates across the composite section. Evans and Adler” presented a more detailed analysis of the various modes of kink formation, correlating some of the key microstructural parameters that govern kinking, under conditions of both axial compression and applied shear. Budiansky14 predicted the kink band angle for long-wave and short-wave imperfections. Experimental results of
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Failure mechanisms
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et al.
Hahn and Williams’ provided valuable information on the role of material properties (i.e. the matrix, interface) in causing different types of failure (e.g. microbuckling, kinking, etc.) at the microlevel. However, the influence of defects such as fiber waviness, etc., on the kink band geometry and the precise mechanism of the role of kink banding in reduction of the compressive strength have not been addressed in analyses to date. Compressive failure is often a near instantaneous and catastrophic phenomenon and for this reason it is difficult to monitor experimentally the nature and sequence of failure mechanisms. The latter are greatly influenced by the presence of defects such as fiber misalignment/waviness and by the local state of stress. This paper describes an investigation of failure mechanisms observed in unidirectional composite laminates under compressive loading.
with the fibers. The stress components referred to the principal material axes (fiber direction) obtained by stress transformation are 01
=
ax cos20P
+ 712)
o2
=
0,
sin’(cp + y12)
712 =
5
sin(cp
+ ~12)
(1)
cos(cp
+ ~12)
Assuming both the initial misalignment cp and the induced shear strain y12 to be small, equation (1) can be approximated as
Thus, the in-plane shear stress r12 is a nonlinear function of ylz, since o, is also a function of y12. From the last of equation (2) we obtain PREDICTION
OF COMPOSITE
FAILURE
Most theoretical models attempting to predict the longitudinal compressive strength of composites assume the existence of some fiber misalignment. They differ in the way they treat the effects of this misalignment and in the assumed response of the material, see, for example, Wang4, Hahn and Sohi13 and Haberle and Matthewsz7. The assumed initial fiber misalignment ‘p in a unidirectional composite, increases by the value of the induced in-plane shear strain y12 (or ye) under the axial load a, (Figure I). This situation then, is that of an offaxis lamina under uniaxial stress (T, at an angle of cp+ y12
The limiting value of (T, 2 g1 is the longitudinal compressive strength Fi,. The necessary condition for a maximum ox is da “=O %I2
or, from equation (3),
[
$(v+Y1Z) -T12]l
=o
(‘p + Y1212
or, (a)
Cp=
initial fiber misalignment
a-12 -=-
712
ay12
cp + 712
(4)
Given the shear stress versus shear strain curve obtained experimentally, the solution of equation (4) corresponds to the values 712 = T* and y12 = y*, where the tangent to the curve equals the slope T*/((P + y*). A measured shear stress-strain curve for carbon/epoxy (IM6G/3501-6) is shown in Figure 2 with an illustration of a graphical determination of T*. The longitudinal compressive strength is then Flc
=
(4nax
=
&
If the induced shear strain yi2 is much smaller than the initial fiber misalignment cp, which is still assumed to be small, equation (5) reduces to YIZ= in-plane shear strain = additional
fiber rotation
Figure 1 Unidirectional lamina with initial fiber misalignment and after axial compressive loading
544
before
F~,
N
(T12)max = F12 cp
cp
which is identical to Argon’s prediction7 and where Fi2 is the in-plane shear strength of the material. The same result is obtained by applying the maximum shear stress
Failure mechanisms
in thick composites:
I. M. Daniel et al.
/e
Compressive Suengthz
Figure 2 Graphical determination strength. (9 = initial misalignment)
criterion Then2*,
of
longitudinal
compressive
to the initially misaligned (off-axis) lamina. F12 FI,
=
ES
sincpcoscp-
cp
(7)
If the initial misalignment is negligible compared to the induced shear strain, i.e. if up< y12, then Rosen’s result2 FI,
g
(52
(8)
is recovered. With a further assumption of high shear stiffness fibers and application of a rule of mixtures series model, equation (8) can be written in the form F,, E
Gm
1 - v,
where G, is the matrix shear modulus and V, is the fiber volume ratio. Equation (9) is another form of the Rosen model prediction2. The above discussion clearly suggests that longitudinal compressive failure is intimately related to and governed by the in-plane shear response of the composite, even in the presence of the slightest initial fiber misalignment. Any compressive failure mechanisms observed must follow some form of initial shear failure of the composite, which is a matrix dominated property.
FAILURE MECHANISMS UNDER UNIFORM COMPRESSIVE LOADING The material investigated in this study was carbon/epoxy (IM6G/3501-6, Hercules, Inc.). Unidirectional plates of various thicknesses were fabricated from the prepreg material, following previously developed fabrication
Figure 3 Schematic of NU compression adaptor a = 25 mm, base b = 38 mm)
text fixture.
techniques
described by Daniel
for thick composites
(Dimensions:
et a1.29 and Hsiao et al.30
Longitudinal compressive tests were conducted with a recently developed compression test method/fixture (NU fixture) incorporating both shear loading and end loading concepts, see Daniel et al.29 and Hsiao et al.30 (Figure 3). The specimens were 12.7 cm (5.00 in.) long, 1.27 cm (0.50 in.) wide, with thicknesses ranging between 2.03 mm (0.080in.) and 19.05 mm (0.75 in.). The specimens were tabbed with 5.40cm (2.125 in.) long glass/ epoxy tabs, leaving a 1.91 cm (0.75in.) long gage length. After tabbing and machining to final tolerances, steel end caps were bonded to the ends of the specimen to prevent premature crushing at the ends. A typical specimen configuration is shown in Figure 4. Strain gages were mounted on both sides and on the edges of the specimen in the gage section. All tests were carried out in a servohydraulic testing machine while recording the strains and failure mechanisms by video photography. A typical stress-strain curve to failure for a 48-ply specimen is shown in Figure 5. Using the experimentally obtained shear stress, 712, versus shear strain, 712, curve for the material in Figure 2 and equation (5) we obtain the variation of the predicted compressive strength, Flc, with initial fiber misalignment, cp, as depicted in Figure 6. The measured
545
Failure mechanisms
in thick composites:
i. M. Daniel
et al
i
‘0
E
R
I $1
OS
1.0
1.5
Fiber Misalignment,
rp,
LO 2.0
(deg)
Figure 6 Variation of predicted longitudinal with initial fiber misalignment for IM6G/3501-6
L. mm (ill.) 114 (4.50) 127 (5.00)
h, mm (1n.1
48.ply 72.~1~
133 (5.25)
25 (1.M))
16-ply
Figure 4
Schematic
of specimen
??
compressive strength carbon/epox:G
13 (0.50) 19 (0.75)
configuration
El = 168.9 GPa (24.5 Msi) Fi, = 1738 MPa (252 ksi) E’;, = 1.22 % 15w
-
050 0
02
04
06
08
10
12
1.4
Strain, E , (%)
Figure 5. Stress-strain curve of unidirectional 48-ply carbon/epoxy under longitudinal compressive loading (IM6G/3501-6)
compressive strength of 1725 MPa (250 ksi) corresponds to an initial fiber misalignment of cp g l.S’, or approximately one-fourth of the initial modulus Gt2 = 7.0 GPa (l.OMsi). The initiation and development of failure mechanisms cannot be observed easily, since the 3501-6 matrix is relatively brittle and fracture is near instantaneous and catastrophic. Failure mechanisms were recorded only after the test (post mortem) by means of optical as well as scanning electron photomicrography. Failure mechanisms in the gage sections are very difficult to observe
546
Figure 7 Illustration of narrow carbon/epoxy under longitudinal
and wide kink bands in IM6G/3501-6 compression
because of the catastrophic nature of failure. The easiest ones to observe are those occurring in constrained regions, such as the tabbed region of the specimen following premature failure initiation by end crushing. Early failures and discernible failure patterns occur in
Failure mechanisms in thick composites: I. M. Daniel et al.
Figure 9 Multiple kink bands of various orientations carbon/epoxy under longitudinal compression
Figure 8
Kink band geometry
regions where an additional shear stress component is added (through the tabs) to the direct compressive normal stress component. According to the previous discussion, adding shear is tantamount to increasing the fiber misalignment and causes early failure. Failure patterns appeared in the form of kink bands as shown in Figures 7 and 8. The kink band geometry is usually characterized by the kink band width, the inclination angle of the kinked fibers, CY,with respect to the loading axis, and the orientation angle of the kink band boundary, ,9, with respect to the transverse to the loading direction (Figure 8). In the upper photograph of Figure 7 the kink band width is equal to seven fiber diameters and the kink inclination and kink band boundary orientation angles are equal, cx = ,0 = 30”. In the lower photograph of the same figure, the kink band width is sixteen fiber diameters and the kink inclination and kink band boundary angles are also equal,
and widths
in
(Y= ,l?= 20”. Figure 9 shows multiple kink bands branching from a source region. Within each branch there are multiple kink bands of various orientations and widths. In the lower photograph of Figure 9, for example, one can identify a kink band that has a constant width of twenty fiber diameters with angles a g p % 23”, over some short length. However, these parameters may change along the kink band. A great deal of work has been reported on analysis and prediction of compressive failure mechanisms and, in particular, kink formation. In some early works a limiting relationship between the kink inclination and kink band angle was obtained as Q = 2/3. Chaplin” obtained this relation by assuming zero volumetric strain in the kinked zone, but did not justify that assumption. Similarly, Evans and Adler” arrived at the same conclusion by minimizing the elastic strain energy, which was tantamount to assuming zero axial strain in the kinked zone. In reality, the state of stress and deformation in the kinked zone is very complex and varies during the loading process. Experimental results do not show any fixed relationship between Q and p.
547
Failure
mechanisms
Figure 10 sive stress
Pure bending
in thick
composites.
test for producing
linearly
i. M. Daniel
varying
et al
compres-
at the points of maximum curvature. ‘ihc 1110 krnh boundaries are of unequal length. The kink propagatcb farther along the points of the fiber where the high tensile strain (convexity) in the fiber is on the side of highcl global compressive stress. Attempts have been made to predict the kmh hand width. Results of various analytical approaches predict kink band widths of the order of lo-15 fiber diameters” A rough estimate can be obtained by using the simple Euler buckling formula for an unsupported fiber. Assuming that compressive failure is initiated by fiber buckling, following shear yielding or excessive shear deformation of the matrix. the critical fiber length at buckling into the slgmoidal shape of I-‘i,~ul-~~1.3. is given bq Figure 11 field
FAILURE VARYING
.4rrested
kink band in linearly
MECHANISMS COMPRESSIVE
varying
compressive
stress
UNDER LINEARLY STRESS
where df = fiber diameter
In order to study the initiation and progression of compressive failure, a pure bending test was designed to produce a linearly varying compressive stress in a unidirectional composite (Figure 10). This type of loading produces compressive failure at the top of the beam which propagates down to a point where the compressive stress is not high enough to initiate failure. Thus, an arrested kink band is obtained with features corresponding to the varying compressive stress along its length. Figure II shows such a kink band whose width is approximately 4-5 fiber diameters and whose inclination angle ,B is 30”. The kink angle N seems to increase from a very small initial value to approximately 50”. An enlargement of the region at the end of the kink band, or the beginning of failure, is shown in Figure 12. It is seen that fiber fracture is preceded by fiber buckling. The buckled fiber has a sigmoidal shape and fractures occur
548
Ef = fiber modulus
07t:cr= critical fiber stress at buckling For the material
under
investigation,
E,- = 276 GPa
and
where (a, )pL is the proportional limit in the compressive stress-strain curve of Figure 5 and Vf is the fiber volume ratio. The proportional limit was used because it corresponds to the initiation of local compressive failure and kink band formation. Therefore, the fiber stress corresponding to this proportional limit stress is taken equal to the critical buckling stress. Substituting the above values into equation (10) we obtain L,, ” I I .9 d,
Failure mechanisms
1 P
7 --
-
-
in thick composites:
I. M. Daniel
et al.
= 2p = 60”
a-
Figure 14 Illustration
of kink
band
broadening
with
increasing
compression
Substituting numerical values for the material under investigation, F12 = 73 MPa, cff.cr= 1212 MPa and Vf = 0.66, we obtain Lug= 5.3”
-
Figure 13 Geometry
of buckled
which is very close to what has been observed. As the compressive stress increases the kink angle increases from this initial value to a maximum value which should be Q = 2/3 according to Weaver and Williams’ and
-
fiber
Chaplin
lo.
Another
phenomenon observed and illustrated in is kink band broadening with increasing compression. At two locations in the figure it is seen that the local kink angle reaches a maximum value, which was measured to be Q = 60” = 2,0. As the compressive stress increases, this angle Q does not increase beyond this maximum value but a new kink band is generated at that point, adjacent to the band with the maximum Q angle. Thereafter the kink angle (Yfalls below its maximum value as the broader kink band propagates. All measurements of kink parameters were made after the specimen was unloaded. Figure
In the buckling mode observed in Figure 12 and shown in Figure 13 there is one inflection point at the midpoint between the assumed fixed ends of the fiber. Therefore, the length L in Figure 13 is equivalent to the length of a pin-ended column for which the critical value is given by equation (10). The distance between the two fiber failures within the length L, which is the kink band width, is approximately equal to L,,/2. The predicted band width would be higher if the lateral constraint of the matrix were taken into account and if the ends of the sigmoid-shaped segment of the fiber were not assumed fixed. If fiber buckling is assumed to follow immediately after shear yielding or shear failure of the matrix, then, the critical fiber stress at buckling/shear failure can be related to the initial kink angle a0 as follows 2F12 gf.cr
=
Vf
sin 2~0
=
sin 2~~
or 2F12 vpf,.,r
(11)
14
FAILURE MECHANISMS GRADIENT STRESSES
UNDER HIGH
To further investigate the initiation and progression of compressive failure, axial compressive tests were conducted on unidirectional specimens with semi-circular edge notches (Figure 15). This type of loading produces a high compressive stress at the center of the notch. A stress concentration of approximately 6.8 exists at the center of the notch. However, a relatively high shear stress, 712, exists at approximately 10” off the center of the notch, whose value is 712 = 0.65 ao, where a0 is the far field
549
Failure
mechanisms
in thick
composites:
I. M. Daniel
et al
Shear failure
Hole boundary Figure 16
Initial shear fGlure on notch boundary axial compression
1
Compression
Kink Bads
01 speumen
Second major kink band (out-of-plane)
Shear failure (initiated from hole)
I
4
urldcl
and Vertical Shear Splitting
P Figure 15 Unidirectional
notched
specimen
under axial compression
applied uniaxial stress. For the carbon/epoxy material investigated, compressive failure at the center of the notch would be initiated at an applied nominal stress of
Flc “O = 6.8 =
1725
~
6.8
= 254 MPa First major kink band
whereas location
shear failure at an applied F12 c~=~===
would start at the 10” off center stress of 73
Figure 17 Major kink bands starting
from shear failure
112MPa
Indeed, as seen in Figure 16, the initial failure around the notch is axial shear failure, propagating along the vertical (loading) direction for some distance. Major kink bands are initiated from this shear failure because the fibers adjacent to the shear failure band lose their lateral support and start failing. In-plane and out-ofplane major kink bands starting from shear failure are shown in Figure 17. Major kink bands branch off or are
connected with narrower secondary kink bands and additional axial shear splitting as shown in Figure 18. Kink bands can start on one plane and rotate along their length (Figure 19). In many cases where a kink band is arrested, it is clearly seen that the kinking mode of compressive failure is transformed into an axial shear failure at the end of the kink band as shown in Figure 20. This again highlights the significance of matrix shear failure as a precursor to kink banding.
Failure mechanisms
in thick composites:
Second major kink band (out-of-plane)
1. M. Daniel
et al.
Shear failure (initiated from kink band)
\
Figure 20
Illustration
of shear failure at the end of a kink band
First major kink band Figure 18 Connection and shear splitting
of major kink bands through
Figure 19 Illustration
of kink band with changing
SUMMARY
AND
secondary
bands
plane
CONCLUSIONS
Failure mechanisms were investigated in a unidirectional carbon/epoxy composite under longitudinal compression. The material investigated was IM6G/3501-6 carbon/epoxy in thicknesses up to 72 plies (9.1 mm; 0.36in.). Tests were conducted under uniform and linearly varying compressive stress. The first failure mechanism appears to be shear yielding or shear failure in the matrix precipitated by pre-existing fiber misalignment. It was shown that the longitudinal compressive strength of the material can be predicted on the basis of the shear stress-strain response
of the material (7i2 versus y12) and the initial fiber misalignment. It was calculated that it takes an initial fiber misalignment of approximately cp E 1.5” to predict the measured compressive strength of Fi, = 1725 MPa (250 ksi). Matrix shear yielding is followed by fiber buckling and fracture. Fracture of a fiber increases shear in its vicinity, thereby precipitating further matrix yielding which in turn causes further fiber buckling and fracture. The buckled fiber has a sigmoidal shape and results in two fractures on the convex sides at the points of maximum curvature. A simple buckling calculation predicts a distance between fiber fractures equal to at least six fiber diameters. The resulting failure pattern takes the form of a kink band. The kink orientation angle at initiation of kinking is very small, but it increases with compressive stress. For the material investigated this initial angle was estimated to be o. = 5.3” which is close to experimental observations. This angle o: can increase under increasing stress up to a maximum value amax = 2p, where p is the kink band orientation. When this condition is reached, further increase in stress causes kink band multiplication or broadening, starting at the point where CX~,,is reached. Thereafter, (Y decreases with the propagation of the multiple or broadened kink band. Kink angles czbetween very small values and nearly 60” were observed. The kink band angle p seems to depend on the local and global state of stress. In the cases studied it was found to vary between 20 and 30”. The kink band width varied between 4 and 20 fiber diameters. No significant effect of thickness was noticed on the overall compressive strength. However, the kink band geometry was more complex for thicker specimens. Kink bands occurred on different planes which, in some cases, rotated along the same band.
551
Failure
mechanisms
in thick
composites:
I M. Daniel
In the case of’more complex states of stress. such as in the vicinity of notches, in-plane shear l‘ailure is a more likely initial failure mechanism. Compressive failure in the form of kink bands is initiated from the shear failure at a much lower compressive stress than would be required in the absence of the shear failure. Major kink bands are connected through narrower secondary ones and additional axial shear failures.
et at
4 i5
16
17 18
ACKNOWLEDGEMENTS The work described in this paper was sponsored by the Office of Naval Research. We are grateful to Dr Y. D. S. Rajapakse of ONR for his encouragement and cooperation and to Mrs Yolande Mallian for typing the manuscript.
2x37
19 20
21
REFERENCES 22 1
2
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Garala. H.G. Experimental evaluation of graphiteeexpoxy cylinders subjected to external hydrostatic compressive loading. In ‘Proc. 1987 SEM Spring Conf. on Experimental Mechanics’, Society for Experimental Mechanics, Bethel, CL 1987, pp. 948-951 Rosen, B.W. Mechanics of composite strengthening. In ‘Fiber Composite Materials.’ American Society of Metals. Metals Park. OH, 1965. pp. 35.-75 Greszczuk, L.B. Microbuckling failure of circular fiber-reinforced composites. AIAA J. 1975. 13, 1311 1318 Wang, A.S.D. A non-linear microbuckling model predicting the compressive strength of unidirectional composites. ASME Paper 7%WA:Aero-1, 1978 Hahn, H.T. and Williams J.G. Compression failure mechanisms in unidirectional composites. In ‘Composite Materials: Testing and Design’, (Seventh Conference), ASTM STP 893, American Sot. For Testing and Materials, Philadelphia, 1986, pp. 115-139 Chaudhuri, R.A. Prediction of the compressive strength of thick-section advanced composite laminates. J. Compos. Mater. 1991, 25. 1244 Argon. A.S. Fracture of composites. In ‘Treatise on Materials Science and Technology’, Vol. 1, (Ed. H. Herman) Academic Press. New York. 1972, pp. 79 114 Berg, C.A. and Salama, M. Fatigue of graphite fiber-reinforced Fibre Sci. and Tech. 1973. 6, 79 epoxy in compression. Weaver. C.W. and Williams, J.G. Deformation of a carbon epoxy composite under hydrostatic pressure. J. Mater. Sci. 1975. IO, 1323 Chaplin, R. Compressive fracture in unidirectional glass-reinforced plastics. J. Mater. Sci. 1977. 12, 347 Evans, A.G. and Adler, W.G. Kinking as a mode of structural degradation in carbon fiber composites. Actu Metali. 1978, 26. 725 Parry. T.V. and Wronski, A.S. Kinking and compressive failure in uniaxially aligned carbon fibre composites tested under superposed hydrostatic pressure. J. Mater. Sci. 1982. 17, 893
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Hancox. N.L The compression strength of unidirectional carbon fiber reinforced epoxy resin. J. Muter. Sci. 1975. 10, 234 Steif. P.S. A model for kinking in fiber composites: I. Fiber breakage via microbuckling: II. Kink band formatinn. hrr. .I Solid.7 Strwt. 1990. 26. 549 (‘hung, I. and Weitsman. Y.J. Model for the micro-huckhng micro-kinking compressive response of fiber-reinforced composites. In ‘Mechanics USA 1994, Proc. of the Twe!fth US Nat. Congress of iZpplied Mechanics’, (Ed. A.S. Kobayashi). ASME, 1994. S2566S261 Schapery. R.A. Prediction of compressive strength dnd kmk bands in composites using a work potential. Int. J. Solid.\.S/rrrc! 1995. 32.(6!7). 739 Greszczuk. I .R. On farlure modes of mudirectional composite\ under compressive loading. In ‘Fracture of Composite Materials: Proc 2nd [ISA USSR Symposium’. Martinus NijhotTPubhshers. 19X2, pp. 731 244 Davis. J.G Compression strength ot’fiber reinforced composite ASTM STP HO. materials. In ‘Composite Reliability.’ American Sot. for Testing and Materials, Philadelphia. !c)75. pp. 264 3?’ Suarez, J.A.. Whiteside. J.B. and Hadcock, R.N. The umurnce of local failure modes on the compressive strength of boron epoxy composite. In ‘Composite Materials: Testing and Design (Second Conference)‘. ASTM STP 497, American Sot. for Testing and Materials, Philadelphia. 1972, pp. 237 -256 Yeh, J.R. and Teply. J.L. Compressive response of kevlar:epoxy composites. J. Compos. Muter. 1988, 22. 245 Haberle, J.G. and Matthews, F.L. A micromechanics model for compressive failure of unidirectional fibre-reinforced plastics. J. Compos. Muter. 1994, 28 (17). 1618 Daniel, I.M. and Ishai. 0. ‘Engineering Mechanics of C‘omposite Materials’, Oxford University Press, New York. 1994 Daniel, I.M., Hsiao. H.M., Wooh, SC. and Vittoser. .I Processing and compressive behavior of thick composites. In ‘AMD-Vol. 162. Mechanics of Thick Composites’. (Ed. Y.D.S. Rajapakse). ASME, New York, 1993, pp. 107. 126 Hsiao, H.M . Wooh, S.C. and Daniel, I.M. Fabrication methods for unidirectional and crossply composites with fiber waviness. J. .4&a. Murer. 1995, 26 (I), 19 Hsiao, H.M.. Daniel. I.M. and Wooh, SC‘. A new compression test method for thick composites. J. Compo.y. MatcJr. 1995. 29 (13). 1789 Fleck. N.A . Deng, L. and Budiansky. B. Prediction of kink band width m fiber composites. Report MECH-203. Harvard Univcrsit\. 1907