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Effects of morphology on properties of graphite composites
@X8-6223t89 $3X@+.tXt 8 1989Pergamon Press plc
CarbonVol. 27,No. 5, pp. 663-678.1989 Printed in GreatBritain.
EFFECTS
OF MORPHOLOGY ON PROPERTIES GRAPHITE COMPOSITES CARL
T.
OF
HERAKOVICH
Applied Mechanics Program, School of Engineering and Applied Sciences, Unive~i?y of Virginia, Charlottesville, VA 22903, U.S.A. Abstrart-The relationship between the morphology of graphite fiber composites and their effective mechanical properties, thermal properties, and failure characteristics is demonstrated. Typical results are presented for studies at the fiber/matrix, lamina, and laminate levels of analysis. It is shown that the range of property variations in nonhomogeneous, laminated graphite materials can be very large, in particular in comparison to monolithic materials. Further, this dependence of material response on morphology generally can be predicted using fundamental principles of mechanics.
Key Words Graphite composites, morphology, mechanical properties, thermal properties, failure.
transverse tensile strength being greater than 25.0. The coefficient of thermal expansion in the fiber direction is near zero compared to a very large value (similar to that of aluminum) in the transverse direction. Actual material properties are provided in the references cited. These very large variations in properties with orientation are the fundamental reason that laminated composites exhibit large internal stresses, unusual elastic and thermal properties, edge effects, and non self-similar crack growth.
1. INTRODUCTION
MorphoIogy is a term used frequentIy by materiah scientists, however, it is not used very often by mechanicians. One of the definitions of morphology as given by Webster[l] is “. . . any scientific study of form and structure.” With this definition of morphology in mind, it is possible to use fundamental principles of mechanics to study the influence of morphology on the response properties and failure characteristics of fibrous composite materials. Fibrous composites may be thought of as structures composed of fibers and matrix on one level and oriented layers on a second ievel. Accurate prediction of the response characteristics of these structures requires appropriate analysis of the interaction between the individual phases of the unidirectional composite or the individual layers of the laminate. Proper consideration of these interactions provides a clear understanding of the, otherwise, sometimes surprising properties of fibrous composites. A full understanding of the relationship between morphology and properties is invaluable to the materials scientist in the development of new materials, to the mechanician in understanding the results of an analysis, and to the designer in the development of structures. This article considers composites at the micro level, the lamina level, and the laminate level. Results are presented for the influence of fiber morphology on effective composite properties, and the influence of fiber orientation on inplane and out-ofplane Poisson’s ratios, inplane and out-of-plane coefficients of thermal expansion, and inplane coefficients of mutual influence. Failure and fracture are discussed as functions of fiber orientation and laminate stacking sequence. The results presented in this article are for typical graphite epoxies, primarily T300/5208 and AS41 3501-6. These materials are highly orthotropic with the ratio of axiat to transverse moduli being greater than 12.0 and the ratio of axial tensile strength to
2. EPHXTIVE PROPERTIES 2.1 Fiber allotropy Scanning electron micrographs of the fracture surfaces of pitch-based and pan-based graphite fibers (Fig. 1) clearly show that the two types of fibers have distinctly different morphologies[2]. Other fibers exhibit morphologies different from either of those shown in Fig. 1. For example, a boron fiber has a clearly identifiable tungsten core surrounded by an annulus of boron. In view of the different types of fiber morphology which may be present in a composite, it is necessary to evaluate the influence of fiber morphology on the composite properties in order to fully understand measured response and to optimize the morphoIogy of new composites. Different types of fiber microstructure were studied recently by Knott and Herakovich[3]. They considered circumferentially orthotropic, radially orthotropic, transversely isotropic, and combinations of these fiber morphologies (Fig. 2) in an isotropic resin matrix to study the influence of fiber morphology on the effective elastic and thermal properties of unidirectional composites. The designations radial and circumferential as used here refer to the preferred directions of the basai planes of the graphite crystals. The analysis was based upon the concentric cylinder assembiedge model of Hashin[4]. Results were pre663
C. T. HERAKOVICH
664
PITCH FIBER
PAN FIBER
Fig. 1. Fracture surfaces of pitch and pan graphite fibers.
sented for axial modulus, axial Poisson’s ratio, transverse bulk modulus, axial shear modulus, as well as axial and transverse coefficients of thermal expansion. Closed form expressions were obtained for the composite properties as functions of fiber and matrix
a) Clmumfemntially
Orthotmplc
properties and fiber that the expressions viously by Hashin[4] As these expressions repeated here. The
volume fraction. It was shown reduce to those presented prefor the case of isotropic fibers. are quite lengthy, they are not interested reader is referred to
t) Radially Otthotmplc
c) Tnnsvenely Isotropic
d) Clrcumhnntlally Orthotmplc with Tmnrwnely Isotmplc Con
a) Radially OfthotmPlc with Transversely Isotmplc Corn
Fig. 2. Idealized fiber morphologies.
665
Effects of morphology on properties of graphite composites
-
-x-*-*-K_______.
0.0
0.1
0.2
a-wwuvoanionww -yw BL*-.
0.3
0.U
0.5
0.6
0.7
0.9
0.9
FIBER ‘KILUME FRACRON
Fig. 3. Axial modulus as a function of fiber morphology.
0.31 -BY-
-x-*-*-w _-____. -----
---
_Y QrmrmopE I#wI)yLRoELlWllWC* v
G!%Ee
w
0.2.
g d 8 H
0.1.
o.oJ4 0.0
0.1
0.2
0.3
0.U
0.5
FIBER m.uw
0.6
0.7
0.6
0.9
1.0
FFAcnoN
Fig. 4. Axial Poisson’s ratio as a function of fiber morphology.
C.
666
T.
HERAKOVKH
Ok.,.,.,.,.,.,.,.,‘,. 0.0
0.1
0.2
0.3
0.Y
0.s
0.6
0.7
0.9
0.9
I.0
FIBERVOLWE fRNllON
Fig. 5. Transverse bulk modulus as a function of fiber morphology.
.__-. _....._..._.._ _______. -Y-A
---
-----
llwmasa TiuHRlERpL
EEiEZ:
FWER WLUUE FRACTION
Fig. 6. Axial shear modulus as a function of fiber morphology.
667
Effects of morphology on properties of graphite composites 200
600
150
V+= 0.623
-\
100
I
IA
0 50
\ ::
0
-50
r (pm1 Fig. 7. Thermal stresses in radially orthotropic fibers.
0.0
0.1
0.2
0.3
0.4
Knott and Herakovich[3] where they are given in detail. Typical results for effective composite properties as a function of fiber volume fraction are presented graphically in Figs. 3-6. These figures show that radially orthotropic and circumferentially orthotropic fibers give rise to identical effective composite properties. It was shown analytically in Knott and Herakovich[3] that this correspondence is exact. The figures also show that, as expected, if different properties (designated A, B & C) are used for a transversely isotropic fiber, the composite properties are different. The fact that the radially orthotropic and circumferentially orthotropic fibers give rise to identical effective composite properties is not too surprising if consideration is given to the fact that the basal planes are present in equal percentages in all directions for both fiber types. The morphology of the fiber can have a dramatic effect on the stress distribution within the fiber for some loading configurations. Figure 7 shows the distributions of the three non-zero components of stress in a radially orthotropic fiber when the composite is subjected to uniform thermal loading. As is evident in the figure, all three components of stress exhibit singular behavior at the center of the fiber. As shown
0.5 R/R0
0.6
0.7
0.6
0.6
1.0
t llhr-nulrlr -
Fig. 8. Radial loading stresses in radially orthotropic
fibers.
C.
668
0.1
0.2
0.3
0.4
0.5
0.6
T. HERAKOVICH
426
:
-0.30
,I.
, . , 0.1
0.7
0.2
, . , . , 0.3
0.4
0.5
, . 0.6
I
0.7
f=n
Fig. 9. Shear stresses in radially orthotropic
by Avery and Herakovich[5], for thermal loading this singular stress distribution is unique to radially orthotropic fibers. However, Knott and Herakovich[3] showed that singular stress distributions also are present for radial loading of radially orthotropic fibers and axial shear displacement loading of radially orthotropic fibers (Figs. 8, 9). These sing&r stress dist~butions have the potential to initiate fiber failure in the form of fiber splitting. The presence of singular stresses in these idealized fibers is tempered by the fact that it is not physically possible to make a fiber which is orthotropic at its core. There must be a finite core size which is at most transversely isotropic. This is true because at r = 0, all directions in the transverse plane are equivalent. A transversely isotropic core has the effect of eliminating the singularity, but it does not eliminate the stress concentration as the core is approached. This
fibers.
point is discussed further in Knott and Herakovich[3]. An example of fiber splitting due to thermal loading is given in Herakovich and Hyer[6]. It is interesting to note that stress distributions in the fiber can be singular for some morphologies and loading conditions, but that these singular stress distributions do not carry over to singular effective properties. This is true because the effective properties of the composite are determined from average stresses and integration of the local stresses to determine average stresses eliminates the singularity. 2.2 ~~~~re~~~~ aide The morphology of an off-axis unidirectional or laminated composite may be classified in terms of
4’. 6
1
/’
-0.2 -
q;;
= f 2.166 Q il2’
’
-0.4 -
YF
= 0.37Qi
i t
-0.6
-
-aa
-
23*
1’
\ I I
/Ly,x 1
: -‘,?$Jo
I -60I
I
-3(-J I
I
0
/ \J
30 I
f
60I
,
0 (deg)
Fig. 11. Inplane Poisson’sratios for angle-ply laminates Fig. 10. Inplane properties of unidirectional composites.
and laminae.
Effects of morphology on properties of graphite composites
669
of the induced shear strain to applied axial strain
05 0.4
Poisson’s ratio represents a coupling between axial and transverse response and the coefficient of mutual influence represents a coupling between axial and shear strain. The variation in u,,. is symmetric about 0 = 0 whereas the variation in %?.I is antisymmetric. Poisson’s ratio varies over a relatively small range (0.06 to 0.37) as compared to the large variation in nX,.,X( +2.166 to -2.166). Because of the large variation possible in q__, it can be a very important parameter. It is a measure of the coupling which is present in off-axis testing of unidirectional composites[7] and, as will be discussed later, it is directly related to the development of interlaminar shear stresses near free edges.
0.3 0.2 01 ux2. 00 -0.1 -0.2 -0 3
00
30.0
15.0
45.0
60.0
7 5.0
90.0
THETA
Fig. 12. Through-the-thickness Poisson’s ratios for laminates.
the fiber orientation 8 of the individual layers. Varying the fiber orientation can have a dramatic influence on some mechanical and thermal properties of composites. Figure 10 shows the variations of inplane Poisson’s ratio u,,. and the coefficient of mutual influence IJ_ for unidirectional off-axis laminae. Poisson’s ratio is the ratio of induced lateral strain to applied axial strain
and the coefficient of mutual influence
0.0
10.0
20.0
30.0
40.0
50.0
60.0
is the ratio
70.0
BOO
90.0
e Fig. 13. Coefficients
of thermal expansion for angle-ply laminates.
2.3 Angle-ply laminates The influence of fiber orientation on several properties of angle-ply laminates is shown in Figs. ll13. The effective inplane laminate Poisson’s ratio for [( a6)]> laminates is shown in Fig. 11, along with the Poisson ratio for the unidirectional lamina. Clearly, the laminate exhibits much higher Poisson’s ratios than the lamina. The maximum u,,, for the laminate is in excess of 1.25. Thus the maximum Poisson’s ratio of the laminate is more than three times the maximum Poisson’s ratio of the lamina. This result is associated directly with the morphology (or structure) of the laminate. The high Poisson’s ratio of the laminate is the result of the internal stresses which must be present in the individual layers in order to satisfy equilibrium requirements. It can be shown from laminate analysis that this result is independent of the actual stacking sequence of the individual layers as long as the laminate is symmetric about the midplane. Possibly the most surprising aspect of the results in Fig. 11 is that the Poisson’s ratio of the laminate is greater than 1.0 for some fiber orientations. This means that the lateral contraction of the laminate is actually greater in magnitude than the applied axial strain. A most intriguing result indeed! In contrast, typical engineering metals exhibit a maximum Poisson’s ratio of 0.5, the value that corresponds to plastic flow. Since the coefficient of mutual influence of the unidirectional lamina is an odd function of the fiber orientation 0, the coefficient of mutual influence of a balanced laminate such as the [ (2 O)], is identically zero. The through the thickness Poisson’s ratio, u,, = - E,/E,, of a symmetric laminate can be estimated using laminate analysis and three dimensional constitutive equations(81. Typical results of such an analysis are shown in Fig. 12 as a function of the fiber orientation. This figure shows the variation in u,, for unidirectional, [(+e)]*, [O/*0],, and [0,/*8], lami-
670
C. T. HERAKOVICH
WOI,
clo,/-lo,1,
3,
c 30,/-309,
C(?30),1,
Fig. 14. Failed angle-ply laminates.
a) Mismatch
8 (deg) in 7?xy,r
8 (degI b) Maximum
0
I
0 cl Maximum
I
I
30
I
60
Tzx
I
90
(deg) Tensor Polynomial
Fig. 15. Correlation of delamination parameters for angle-ply laminates.
Effects of morphology on properties of graphite composites
671
b) [(SO)2], &’ = 1.4%
E = I 95 %
d = 0.9 %
e=0.9% c) [(f45)21, Ed’ = 1.95%
d) [(k60)2], &’ = 0.9%
Fig. 16. Edge damage in angle-ply laminates. nates. Again, a somewhat surprising result is predicted. The through-the-thickness Poisson’s ratio can vary from a positive value to a negative value of approximately the same magnitude, to a positive value of about twice the original value. Thus, the through-the-thickness Poisson’s ratio can vary by as much as 300% over a 90” range of fiber orientations. Further, positive and negative as well as zero values are obtainable when laminates are considered. The fiber angles corresponding to zero u;, are a function of the percentage of 0” plies in the laminate as indicated in Fig. 12.
Variations in coefficients of thermal expansion (CTE) with fiber orientation also exhibit a wide range of values. As depicted in Fig. 13, the inplane CTE, (Y,, for [(tt3)], laminates ranges from a very small, near zero, value for 8 = o”, to a maximum negative value near 8 = 30”, and back through zero to a maximum positive value for 0 = 90”. The through-the-thickness CTE, c1,, varies symmetrically about 8 = 45”, which is the maximum value. The through-the-thickness CTE exhibits a 50% increase in magnitude as 0 increases from 0 to 45”. The variation of these properties with fiber ori-
672
C.
T.
HERAKOVICH
Fig. 17. Failed notched graphite epoxy tensile coupons.
Fig. 18. Normal stress ratio parameters.
Fig. 19. Crack growth in a [0,/902], laminate.
Effects of morphology on properties of graphite composites
673
-2; 0
I
2 a)
3
4
I
0
2
b) 8 =
8 = O”
0
I
3
4
3
4
15~
2
dl 8 = 60’
X Coordlnat~
e) 8
=
, mm
75’
X
Coordinate ,
Fig. 20. Crack growth in slotted laminae-Tensile entation can be very important to designers. All three properties-Poisson’s ratio, coefficient of mutual influence, and coefficient of thermal expansion-can lead to internal stresses which may be desirable depending upon the application. These properties also result in dimensional changes whichmay or may not be desirable. For example, frictional problems and buckling of roadways associated with thermal expansion are well known and usually undesirable; however, dimensional changes are desirable for thermostatic control. Awareness of the relationships between morphology and composite properties as indicated in Figs. 11-13 provides the designer with additional flexibility in tailoring a structure for optimum performance. 3. DELAMINATION
Laminates with free edges are known to exhibit interlaminar stresses[9]. The interlaminar stresses can have a strong influence on both the mode of failure and the ultimate load. Figure 14 shows several failed angle-ply laminates. Two basic types of laminates were tested, alternating [(+-t~),]~and clustered [&J - O,],[lO]. It is clear from the figure that the failure modes are entirely different depending upon the
mm
f)8=90° loading.
stacking sequence (morphology). The alternating configuration fails along a single fracture surface which is parallel to the fibers in half the layers and breaks fibers in the other half of the layers. In contrast, the clustered configuration exhibits several fracture surfaces and no fiber breakage. The clustered laminate fails entirely due to matrix or fiber/ matrix failure. In each layer there is a crack parallel to the fiber direction and, in addition, delamination has occurred at the + / - 0 interfaces. The ultimate stress of the alternating configuration was approximately 30% larger than the ultimate stress of the clustered configuration. It may be said that the morphology of the clustered configuration causes higher interlaminar stresses which result in early failure due to delamination. The interlaminar stresses are an edge effect. They are present in a small boundary layer region along the free edge of the laminate. They are the direct result of the mismatch in properties such as Poisson’s ratio and coefficient of mutual influence shown in Fig. 10. The morphology of the laminate (fiber orientation, stacking sequence and layer thickness) controls the magnitude of the interlaminar stresses. If these interlaminar stresses are sufficiently large,, delamination will occur prior to inplane failure. The
C. T.
HERAKOVICH
0
I
2 b) 8 = 30’
3
4
2 c)8=6oo
3
4
3
4
4
-3
Crack dlrsctlon -
j
I
0 X
2
3
Coordinate,
4
mm
a) e=oo 0t 0
3
I
8
2 I
0
I
2
3
4
, mm
X Coordinatr
e) in laminae
2
,
Fiber orientation
8
E I_ 0 f 5= 0 $ * -I l
-2 j -4
Crack dlrrctlon c
1
-3
I
-2
1
-I
0
a)
I
2
3
4
I , mm
2
3
4
8=O”
/ -4
-3
-2
-I
0
X Coordinatr
I
2
X Coordinate
d)8=75O
Fig. 21. Crack growth
0 II 0
, mm
8 = so0
with holes-Tensile
loading.
correlation between mismatch in coefficient of mutual influence, interlaminar shear stress, and tensor polynomial failure criterion for angle-ply laminates is demonstrated in Fig. 15. This figure shows that the fiber angle corresponding to the largest mismatch in coefficient of mutual influence also corresponds to the largest interlaminar shear stress and the maximum value of the failure criterion. Thus, the critical fiber orientation for delamination due to interlaminar shear stress in angle-ply laminates can be deduced directly from the mismatch in all?.*. The influence of fiber orientation on edge effects is demonstrated clearly in Fig. 16. This figure shows edge replicas of angle-ply laminates for fiber orientations of lo”, 30”, 45”, and 60”. The mode of failure is predominately delamination for low fiber orientations (high mismatch in coefficient of mutual influence), a combination of delamination and transverse cracking for intermediate angles (i.e. 45”) and only transverse cracking for the 60” fiber angle (low mismatch in coefficient of mutual influence).
4.
FRAmURE
b) 8=90°
Fig. 22. Crack growth in slotted laminae-Shear
loading.
The morphology of continuous fiber composite materials results in a material which is heteroge-
675
Effects of morphology on properties of graphite composites
-3: I
0
2
3
C)
(+I
8 = 30’
0
4
, mm
X Coordinate
I
rhear
2
3
4
Coordlnote , mm
X
d) 8 =
300(-1
I
2
*hear
3
2
I ’ _ 0 f c0 0 :, 0 0 -I l
-2
-3 0
I
2
3
X Coordinate
, mm
0) 13= 4!5’(+)
ahear
4
G
0
X Coordinate
f) 8=
Fig. 23. Crack growth in laminae with holes-Positive
neous with orthotropic or anisotropic response characteristics. The strength characteristics of graphite composites are highly orthotropic primarily because of the high tensile strength in the fiber direction and the very low tensile strength transverse to the fibers. The low transverse strength is generally associated with the low fiber/matrix bond strength as opposed to low tensile strength of the matrix itself. In fact, it is generally believed that there is no chemical bond between epoxy resins and graphite fibers. What bond there is is associated with mechanical interaction due to surface roughness, friction, and residual stresses from the curing process. The highly orthotropic strength characteristic of graphite composites results in crack growth characteristics which are completely different from the more commonly known characteristics of homogeneous, isotropic materials. Crack growth in isotropic materials is self similar. That is to say, if a crack is loaded perpendicular to the plane of the crack, the direction of crack extension is coincident with the original crack direction. In contrast to this self-similar crack growth of isotropic materials, unidirectional graphite epoxy exhibits crack growth which is
45’(-)
3
4
, mm shear
and negative shear.
highly dependent on the direction of minimum strength, irrespective of the orientation of the crack relative to the fiber direction and irrespective of the direction of loading relative to the plane of the crack[ 111. Figure 17 shows failed graphite epoxy tensile specimens with different fiber orientations. In all cases the loading was tension perpendicular to the plane of the original slot-type notch. As the results clearly show, crack growth from the existing notch is always parallel to the fiber direction. Crack growth in the unidirectional graphite epoxy is self similar only for the very special case of loading perpendicular to the fiber direction. The results and conclusions discussed in the preceding paragraph are limited to the case of unidirectional materials, that is, laminae. Laminates are an entirely different situation. Consider, for example, the failure surfaces of the angle-ply laminates in Fig. 14. Clearly, for the alternating [(t-@)?I, laminate there is fiber breakage in half the layers. Thus, crack growth in laminates is not necessarily parallel to the fiber direction in all layers. This indicates that crack growth in a laminate is an even more complicated problem, strongly infuenced by the morphol-
676
C. T. HERAKOVICH
b) 1901LAMINA Fig. 24. Actual crack growth in notched ogy of the laminate as well as the morphology of the individual layers. The fact that cracks do not grow in a self-similar manner in fibrous composites means that classical fracture mechanics is not applicable. It also means that the direction of crack growth is a fundamental unknown of the problem. The problem of crack growth in unidirectional composites was addressed recently in a general way by Beuth and Herakovich[ 111, where additional references on the subject are given. If the existing notch is not sharp, an additional unknown is the location of the crack initiation site from the existing notch. For isotropic materials, the crack initiation site corresponds to the point of maximum stress. This will not necessarily be the case for anisotropic materials because the
[0] and [W] tensile
coupons.
crack initiation site is a function of both the stress magnitude and the directionally dependent strength. The problem of crack initiation site as a function of notch geometry was studied by Gurdal and Herakovich[l2]. Crack growth in unidirectional fibrous composites is thus a many faceted problem. Critical parameters which must be determined and modeled include: a) the site of crack initiation, b) the direction of crack growth, c) the load for crack initiation, and d) the stability of crack growth. The extent to which all of these parameters are a function of far-field loading, fiber orientation, notch orientation, notch geometry, and environmental conditions must be included in a comprehensive fracture model. A theory for predicting fracture in fibrous com-
Effects
of morphology
on properties
that has shown promise is the normal stress ratio (NSR) theory originally proposed by Buczek and Herakovich[l3]. This theory simply states that crack growth is controlled by the normal stress at a point normalized with respect to the tensile strength, both of which are functions of the plane under consideration. The theory states that crack growth will initiate at the site and in the direction where this ratio first attains a value of 1.0 as the load is increased. Mathematically. posites
NSR = am,,,/Tmm where T is defined T = Xrsin’B
+ Yrcos’8
with I3 being the angle between the plan in question and the fiber direction (Fig. 18). This theory is based upon the physical argument that crack growth is controlled by normal stress only, but that the directional dependent strength must also be included in a general theory. Because it has not been possible to measure the strength of a unidirectional composite on predetermined planes, it has been necessary to define Tbmas given above. This definition satisfies the minimum requirements that if the material is isotropic, T,, is independent of B; if the crack grows parallel to the fibers, T,, equals Yr, the transverse strength of the composite; and if the crack grows perpendicular to the fibers, T,, equals X,, the axial strength of the composite. This theory has been applied with reasonably good success in a number of applications. Figure 19 shows
Fig. 25.
Actual
of graphite
composites
671
one of the first applications, a sequence of deformed finite element grids depicting crack growth in a [9OJ Or],laminate using the NSR theory[l3]. As is evident from the figure, the theory predicts that a transverse crack in a 90” layer changes direction and becomes a delamination between layers when the crack reaches the O/90 interface. Figures 20-23 show the results of using the theory to predict both the location and direction of crack growth from notched laminae with different notch geometry, fiber orientation and far-field loading. As indicated in these figures, the site of crack initiation varies considerable with the problem parameters, but the direction of growth is always parallel to the fibers. Typical results from an extensive experimental investigation are detailed in Figs. 24 and 25 and Table 1. The photographs in the figures show that the location of the crack initiation site is variable and that the crack always grows parallel to the fibers (as predicted by the normal stress ratio theory). Table 1 includes comparisons between theory and experiment for the critical far-field stress corresponding to crack initiation. The critical stresses in the table were based upon linear elastic stress distributions obtained using anisotropic elasticity theory (Lekhnitskii[l4], Erdogan and Sih[lS]). Theoretical predictions were obtained at a critical distance from the tip of a sharp crack and on the surface of an elliptical notch. The results of the 90” test were used to determine the critical distance from the sharp crack. Experimental results are presented for the far-field stress at crack initiation and at failure. Since these two stress values are not always identical (Table l), it is apparent that stable crack growth was observed
crack growth in notched
[30] lamina
under shear loading.
C. T.
678
HERAKOVICH
Table 1. Theoretical and experimental critical siresses tension and shear tests (ksi/MPa) Experimental
Theoretical
e
Initiation
Failure
Sharp
Elliptical
0 45 90”
38.21263 4.12128.4 2.81/19.4
58.8/405 41.2128.4 2.81119.4
20.9/144 5.02134.6 -
22.81157 48.3133.3 -
mechanician, and the structural designer. The influence of morphology generally can be predicted using fundamental principles of mechanics.
REFERENCES
Iosipescu shear tests 0” 15” 30” 45
4.03127.8 3.50/24.1 4.13128.5 4.08128.1
12.1j83.6 9.85167.9 7.52151.8 4.08128.1
3.10121.4 2.88119.9 3.01/20.8 3.80126.2
1. WebsterS New World Dictionary of the American Language, College Edition, The
2. 6.51144.9 5.12135.3 4.56131.4 4.70132.4
3.
4. 5.
in some cases. The comparison between theory and experiment in Table 1 is quite good. This is particularly true when it is recalled that the critical load for crack initiation under far-field shear loading is predicted quite well with a theory which is based upon the normal stress at the crack tip and is independent of local shear stresses. 5.
SUMMARY
It has been shown that the morphology (structure) of graphite composites can have a strong influence on the elastic, thermal and fracture characteristics of the material. Many properties of graphite composites exhibit characteristics which may appear surprising until the effects of the morphology are considered. Knowledge of the influence of morphology is important to the materials scientist, the
6.
World Publishing Co, New
York (1953). R. Bacon, ‘Phil. Trans. R. Sot. Lond. A 294. 437 (1979). T. W. Knott and C. T. Herakovich, Final Report, NSF Grant MSM-8613090, University of Virginia (May 1988). Z. Hashin, NASA CR-1974 (March 1974). W. B. Avery and C. T. Herakovich, J. Appl. Mech. 53, p. 751 (1986). C. T. Herakovich and M. W. Hyer, Engineering Fracture Mechanics 25, 779 (1986).
7. M. J. Pindera and C. T. Herakovich, Journal of Composite Materials 21, 1164 (1987). 8. C. T. Herakovich, Journal of Comoosite Materials 18. 447 (1984). 9. R. B. Pipes and N. J. Pagano, Journal of Composite Materials 4, 538 (1970). 10. C. T. Herakovicd, Journal of Composite Materials 16, I
.
216 (1982).
11. J. L. Beuth Jr., and C. T. Herakovich, Theoretical and Applied Fracture Mechanics, 11, 27-46 (1989). 12. Z. Gurdal and C. T. Herakovich, Theoretical and Applied Fracture Mechanics 8, 59 (1987).
13. M. B. Buczek and C. T. Herakovich, Journal of Composite Materials 19, 544 (1985).
14. S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body, English Translation by Holden-Day, Inc., SanFranscisco (1963). 15. F. Erdogan and G. C: Sih, Journal of Basic Engineering 85, 519 (1963).
CarbonVol. 27,No. 5, pp. 663-678.1989 Printed in GreatBritain.
EFFECTS
OF MORPHOLOGY ON PROPERTIES GRAPHITE COMPOSITES CARL
T.
OF
HERAKOVICH
Applied Mechanics Program, School of Engineering and Applied Sciences, Unive~i?y of Virginia, Charlottesville, VA 22903, U.S.A. Abstrart-The relationship between the morphology of graphite fiber composites and their effective mechanical properties, thermal properties, and failure characteristics is demonstrated. Typical results are presented for studies at the fiber/matrix, lamina, and laminate levels of analysis. It is shown that the range of property variations in nonhomogeneous, laminated graphite materials can be very large, in particular in comparison to monolithic materials. Further, this dependence of material response on morphology generally can be predicted using fundamental principles of mechanics.
Key Words Graphite composites, morphology, mechanical properties, thermal properties, failure.
transverse tensile strength being greater than 25.0. The coefficient of thermal expansion in the fiber direction is near zero compared to a very large value (similar to that of aluminum) in the transverse direction. Actual material properties are provided in the references cited. These very large variations in properties with orientation are the fundamental reason that laminated composites exhibit large internal stresses, unusual elastic and thermal properties, edge effects, and non self-similar crack growth.
1. INTRODUCTION
MorphoIogy is a term used frequentIy by materiah scientists, however, it is not used very often by mechanicians. One of the definitions of morphology as given by Webster[l] is “. . . any scientific study of form and structure.” With this definition of morphology in mind, it is possible to use fundamental principles of mechanics to study the influence of morphology on the response properties and failure characteristics of fibrous composite materials. Fibrous composites may be thought of as structures composed of fibers and matrix on one level and oriented layers on a second ievel. Accurate prediction of the response characteristics of these structures requires appropriate analysis of the interaction between the individual phases of the unidirectional composite or the individual layers of the laminate. Proper consideration of these interactions provides a clear understanding of the, otherwise, sometimes surprising properties of fibrous composites. A full understanding of the relationship between morphology and properties is invaluable to the materials scientist in the development of new materials, to the mechanician in understanding the results of an analysis, and to the designer in the development of structures. This article considers composites at the micro level, the lamina level, and the laminate level. Results are presented for the influence of fiber morphology on effective composite properties, and the influence of fiber orientation on inplane and out-ofplane Poisson’s ratios, inplane and out-of-plane coefficients of thermal expansion, and inplane coefficients of mutual influence. Failure and fracture are discussed as functions of fiber orientation and laminate stacking sequence. The results presented in this article are for typical graphite epoxies, primarily T300/5208 and AS41 3501-6. These materials are highly orthotropic with the ratio of axiat to transverse moduli being greater than 12.0 and the ratio of axial tensile strength to
2. EPHXTIVE PROPERTIES 2.1 Fiber allotropy Scanning electron micrographs of the fracture surfaces of pitch-based and pan-based graphite fibers (Fig. 1) clearly show that the two types of fibers have distinctly different morphologies[2]. Other fibers exhibit morphologies different from either of those shown in Fig. 1. For example, a boron fiber has a clearly identifiable tungsten core surrounded by an annulus of boron. In view of the different types of fiber morphology which may be present in a composite, it is necessary to evaluate the influence of fiber morphology on the composite properties in order to fully understand measured response and to optimize the morphoIogy of new composites. Different types of fiber microstructure were studied recently by Knott and Herakovich[3]. They considered circumferentially orthotropic, radially orthotropic, transversely isotropic, and combinations of these fiber morphologies (Fig. 2) in an isotropic resin matrix to study the influence of fiber morphology on the effective elastic and thermal properties of unidirectional composites. The designations radial and circumferential as used here refer to the preferred directions of the basai planes of the graphite crystals. The analysis was based upon the concentric cylinder assembiedge model of Hashin[4]. Results were pre663
C. T. HERAKOVICH
664
PITCH FIBER
PAN FIBER
Fig. 1. Fracture surfaces of pitch and pan graphite fibers.
sented for axial modulus, axial Poisson’s ratio, transverse bulk modulus, axial shear modulus, as well as axial and transverse coefficients of thermal expansion. Closed form expressions were obtained for the composite properties as functions of fiber and matrix
a) Clmumfemntially
Orthotmplc
properties and fiber that the expressions viously by Hashin[4] As these expressions repeated here. The
volume fraction. It was shown reduce to those presented prefor the case of isotropic fibers. are quite lengthy, they are not interested reader is referred to
t) Radially Otthotmplc
c) Tnnsvenely Isotropic
d) Clrcumhnntlally Orthotmplc with Tmnrwnely Isotmplc Con
a) Radially OfthotmPlc with Transversely Isotmplc Corn
Fig. 2. Idealized fiber morphologies.
665
Effects of morphology on properties of graphite composites
-
-x-*-*-K_______.
0.0
0.1
0.2
a-wwuvoanionww -yw BL*-.
0.3
0.U
0.5
0.6
0.7
0.9
0.9
FIBER ‘KILUME FRACRON
Fig. 3. Axial modulus as a function of fiber morphology.
0.31 -BY-
-x-*-*-w _-____. -----
---
_Y QrmrmopE I#wI)yLRoELlWllWC* v
G!%Ee
w
0.2.
g d 8 H
0.1.
o.oJ4 0.0
0.1
0.2
0.3
0.U
0.5
FIBER m.uw
0.6
0.7
0.6
0.9
1.0
FFAcnoN
Fig. 4. Axial Poisson’s ratio as a function of fiber morphology.
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T.
HERAKOVKH
Ok.,.,.,.,.,.,.,.,‘,. 0.0
0.1
0.2
0.3
0.Y
0.s
0.6
0.7
0.9
0.9
I.0
FIBERVOLWE fRNllON
Fig. 5. Transverse bulk modulus as a function of fiber morphology.
.__-. _....._..._.._ _______. -Y-A
---
-----
llwmasa TiuHRlERpL
EEiEZ:
FWER WLUUE FRACTION
Fig. 6. Axial shear modulus as a function of fiber morphology.
667
Effects of morphology on properties of graphite composites 200
600
150
V+= 0.623
-\
100
I
IA
0 50
\ ::
0
-50
r (pm1 Fig. 7. Thermal stresses in radially orthotropic fibers.
0.0
0.1
0.2
0.3
0.4
Knott and Herakovich[3] where they are given in detail. Typical results for effective composite properties as a function of fiber volume fraction are presented graphically in Figs. 3-6. These figures show that radially orthotropic and circumferentially orthotropic fibers give rise to identical effective composite properties. It was shown analytically in Knott and Herakovich[3] that this correspondence is exact. The figures also show that, as expected, if different properties (designated A, B & C) are used for a transversely isotropic fiber, the composite properties are different. The fact that the radially orthotropic and circumferentially orthotropic fibers give rise to identical effective composite properties is not too surprising if consideration is given to the fact that the basal planes are present in equal percentages in all directions for both fiber types. The morphology of the fiber can have a dramatic effect on the stress distribution within the fiber for some loading configurations. Figure 7 shows the distributions of the three non-zero components of stress in a radially orthotropic fiber when the composite is subjected to uniform thermal loading. As is evident in the figure, all three components of stress exhibit singular behavior at the center of the fiber. As shown
0.5 R/R0
0.6
0.7
0.6
0.6
1.0
t llhr-nulrlr -
Fig. 8. Radial loading stresses in radially orthotropic
fibers.
C.
668
0.1
0.2
0.3
0.4
0.5
0.6
T. HERAKOVICH
426
:
-0.30
,I.
, . , 0.1
0.7
0.2
, . , . , 0.3
0.4
0.5
, . 0.6
I
0.7
f=n
Fig. 9. Shear stresses in radially orthotropic
by Avery and Herakovich[5], for thermal loading this singular stress distribution is unique to radially orthotropic fibers. However, Knott and Herakovich[3] showed that singular stress distributions also are present for radial loading of radially orthotropic fibers and axial shear displacement loading of radially orthotropic fibers (Figs. 8, 9). These sing&r stress dist~butions have the potential to initiate fiber failure in the form of fiber splitting. The presence of singular stresses in these idealized fibers is tempered by the fact that it is not physically possible to make a fiber which is orthotropic at its core. There must be a finite core size which is at most transversely isotropic. This is true because at r = 0, all directions in the transverse plane are equivalent. A transversely isotropic core has the effect of eliminating the singularity, but it does not eliminate the stress concentration as the core is approached. This
fibers.
point is discussed further in Knott and Herakovich[3]. An example of fiber splitting due to thermal loading is given in Herakovich and Hyer[6]. It is interesting to note that stress distributions in the fiber can be singular for some morphologies and loading conditions, but that these singular stress distributions do not carry over to singular effective properties. This is true because the effective properties of the composite are determined from average stresses and integration of the local stresses to determine average stresses eliminates the singularity. 2.2 ~~~~re~~~~ aide The morphology of an off-axis unidirectional or laminated composite may be classified in terms of
4’. 6
1
/’
-0.2 -
q;;
= f 2.166 Q il2’
’
-0.4 -
YF
= 0.37Qi
i t
-0.6
-
-aa
-
23*
1’
\ I I
/Ly,x 1
: -‘,?$Jo
I -60I
I
-3(-J I
I
0
/ \J
30 I
f
60I
,
0 (deg)
Fig. 11. Inplane Poisson’sratios for angle-ply laminates Fig. 10. Inplane properties of unidirectional composites.
and laminae.
Effects of morphology on properties of graphite composites
669
of the induced shear strain to applied axial strain
05 0.4
Poisson’s ratio represents a coupling between axial and transverse response and the coefficient of mutual influence represents a coupling between axial and shear strain. The variation in u,,. is symmetric about 0 = 0 whereas the variation in %?.I is antisymmetric. Poisson’s ratio varies over a relatively small range (0.06 to 0.37) as compared to the large variation in nX,.,X( +2.166 to -2.166). Because of the large variation possible in q__, it can be a very important parameter. It is a measure of the coupling which is present in off-axis testing of unidirectional composites[7] and, as will be discussed later, it is directly related to the development of interlaminar shear stresses near free edges.
0.3 0.2 01 ux2. 00 -0.1 -0.2 -0 3
00
30.0
15.0
45.0
60.0
7 5.0
90.0
THETA
Fig. 12. Through-the-thickness Poisson’s ratios for laminates.
the fiber orientation 8 of the individual layers. Varying the fiber orientation can have a dramatic influence on some mechanical and thermal properties of composites. Figure 10 shows the variations of inplane Poisson’s ratio u,,. and the coefficient of mutual influence IJ_ for unidirectional off-axis laminae. Poisson’s ratio is the ratio of induced lateral strain to applied axial strain
and the coefficient of mutual influence
0.0
10.0
20.0
30.0
40.0
50.0
60.0
is the ratio
70.0
BOO
90.0
e Fig. 13. Coefficients
of thermal expansion for angle-ply laminates.
2.3 Angle-ply laminates The influence of fiber orientation on several properties of angle-ply laminates is shown in Figs. ll13. The effective inplane laminate Poisson’s ratio for [( a6)]> laminates is shown in Fig. 11, along with the Poisson ratio for the unidirectional lamina. Clearly, the laminate exhibits much higher Poisson’s ratios than the lamina. The maximum u,,, for the laminate is in excess of 1.25. Thus the maximum Poisson’s ratio of the laminate is more than three times the maximum Poisson’s ratio of the lamina. This result is associated directly with the morphology (or structure) of the laminate. The high Poisson’s ratio of the laminate is the result of the internal stresses which must be present in the individual layers in order to satisfy equilibrium requirements. It can be shown from laminate analysis that this result is independent of the actual stacking sequence of the individual layers as long as the laminate is symmetric about the midplane. Possibly the most surprising aspect of the results in Fig. 11 is that the Poisson’s ratio of the laminate is greater than 1.0 for some fiber orientations. This means that the lateral contraction of the laminate is actually greater in magnitude than the applied axial strain. A most intriguing result indeed! In contrast, typical engineering metals exhibit a maximum Poisson’s ratio of 0.5, the value that corresponds to plastic flow. Since the coefficient of mutual influence of the unidirectional lamina is an odd function of the fiber orientation 0, the coefficient of mutual influence of a balanced laminate such as the [ (2 O)], is identically zero. The through the thickness Poisson’s ratio, u,, = - E,/E,, of a symmetric laminate can be estimated using laminate analysis and three dimensional constitutive equations(81. Typical results of such an analysis are shown in Fig. 12 as a function of the fiber orientation. This figure shows the variation in u,, for unidirectional, [(+e)]*, [O/*0],, and [0,/*8], lami-
670
C. T. HERAKOVICH
WOI,
clo,/-lo,1,
3,
c 30,/-309,
C(?30),1,
Fig. 14. Failed angle-ply laminates.
a) Mismatch
8 (deg) in 7?xy,r
8 (degI b) Maximum
0
I
0 cl Maximum
I
I
30
I
60
Tzx
I
90
(deg) Tensor Polynomial
Fig. 15. Correlation of delamination parameters for angle-ply laminates.
Effects of morphology on properties of graphite composites
671
b) [(SO)2], &’ = 1.4%
E = I 95 %
d = 0.9 %
e=0.9% c) [(f45)21, Ed’ = 1.95%
d) [(k60)2], &’ = 0.9%
Fig. 16. Edge damage in angle-ply laminates. nates. Again, a somewhat surprising result is predicted. The through-the-thickness Poisson’s ratio can vary from a positive value to a negative value of approximately the same magnitude, to a positive value of about twice the original value. Thus, the through-the-thickness Poisson’s ratio can vary by as much as 300% over a 90” range of fiber orientations. Further, positive and negative as well as zero values are obtainable when laminates are considered. The fiber angles corresponding to zero u;, are a function of the percentage of 0” plies in the laminate as indicated in Fig. 12.
Variations in coefficients of thermal expansion (CTE) with fiber orientation also exhibit a wide range of values. As depicted in Fig. 13, the inplane CTE, (Y,, for [(tt3)], laminates ranges from a very small, near zero, value for 8 = o”, to a maximum negative value near 8 = 30”, and back through zero to a maximum positive value for 0 = 90”. The through-the-thickness CTE, c1,, varies symmetrically about 8 = 45”, which is the maximum value. The through-the-thickness CTE exhibits a 50% increase in magnitude as 0 increases from 0 to 45”. The variation of these properties with fiber ori-
672
C.
T.
HERAKOVICH
Fig. 17. Failed notched graphite epoxy tensile coupons.
Fig. 18. Normal stress ratio parameters.
Fig. 19. Crack growth in a [0,/902], laminate.
Effects of morphology on properties of graphite composites
673
-2; 0
I
2 a)
3
4
I
0
2
b) 8 =
8 = O”
0
I
3
4
3
4
15~
2
dl 8 = 60’
X Coordlnat~
e) 8
=
, mm
75’
X
Coordinate ,
Fig. 20. Crack growth in slotted laminae-Tensile entation can be very important to designers. All three properties-Poisson’s ratio, coefficient of mutual influence, and coefficient of thermal expansion-can lead to internal stresses which may be desirable depending upon the application. These properties also result in dimensional changes whichmay or may not be desirable. For example, frictional problems and buckling of roadways associated with thermal expansion are well known and usually undesirable; however, dimensional changes are desirable for thermostatic control. Awareness of the relationships between morphology and composite properties as indicated in Figs. 11-13 provides the designer with additional flexibility in tailoring a structure for optimum performance. 3. DELAMINATION
Laminates with free edges are known to exhibit interlaminar stresses[9]. The interlaminar stresses can have a strong influence on both the mode of failure and the ultimate load. Figure 14 shows several failed angle-ply laminates. Two basic types of laminates were tested, alternating [(+-t~),]~and clustered [&J - O,],[lO]. It is clear from the figure that the failure modes are entirely different depending upon the
mm
f)8=90° loading.
stacking sequence (morphology). The alternating configuration fails along a single fracture surface which is parallel to the fibers in half the layers and breaks fibers in the other half of the layers. In contrast, the clustered configuration exhibits several fracture surfaces and no fiber breakage. The clustered laminate fails entirely due to matrix or fiber/ matrix failure. In each layer there is a crack parallel to the fiber direction and, in addition, delamination has occurred at the + / - 0 interfaces. The ultimate stress of the alternating configuration was approximately 30% larger than the ultimate stress of the clustered configuration. It may be said that the morphology of the clustered configuration causes higher interlaminar stresses which result in early failure due to delamination. The interlaminar stresses are an edge effect. They are present in a small boundary layer region along the free edge of the laminate. They are the direct result of the mismatch in properties such as Poisson’s ratio and coefficient of mutual influence shown in Fig. 10. The morphology of the laminate (fiber orientation, stacking sequence and layer thickness) controls the magnitude of the interlaminar stresses. If these interlaminar stresses are sufficiently large,, delamination will occur prior to inplane failure. The
C. T.
HERAKOVICH
0
I
2 b) 8 = 30’
3
4
2 c)8=6oo
3
4
3
4
4
-3
Crack dlrsctlon -
j
I
0 X
2
3
Coordinate,
4
mm
a) e=oo 0t 0
3
I
8
2 I
0
I
2
3
4
, mm
X Coordinatr
e) in laminae
2
,
Fiber orientation
8
E I_ 0 f 5= 0 $ * -I l
-2 j -4
Crack dlrrctlon c
1
-3
I
-2
1
-I
0
a)
I
2
3
4
I , mm
2
3
4
8=O”
/ -4
-3
-2
-I
0
X Coordinatr
I
2
X Coordinate
d)8=75O
Fig. 21. Crack growth
0 II 0
, mm
8 = so0
with holes-Tensile
loading.
correlation between mismatch in coefficient of mutual influence, interlaminar shear stress, and tensor polynomial failure criterion for angle-ply laminates is demonstrated in Fig. 15. This figure shows that the fiber angle corresponding to the largest mismatch in coefficient of mutual influence also corresponds to the largest interlaminar shear stress and the maximum value of the failure criterion. Thus, the critical fiber orientation for delamination due to interlaminar shear stress in angle-ply laminates can be deduced directly from the mismatch in all?.*. The influence of fiber orientation on edge effects is demonstrated clearly in Fig. 16. This figure shows edge replicas of angle-ply laminates for fiber orientations of lo”, 30”, 45”, and 60”. The mode of failure is predominately delamination for low fiber orientations (high mismatch in coefficient of mutual influence), a combination of delamination and transverse cracking for intermediate angles (i.e. 45”) and only transverse cracking for the 60” fiber angle (low mismatch in coefficient of mutual influence).
4.
FRAmURE
b) 8=90°
Fig. 22. Crack growth in slotted laminae-Shear
loading.
The morphology of continuous fiber composite materials results in a material which is heteroge-
675
Effects of morphology on properties of graphite composites
-3: I
0
2
3
C)
(+I
8 = 30’
0
4
, mm
X Coordinate
I
rhear
2
3
4
Coordlnote , mm
X
d) 8 =
300(-1
I
2
*hear
3
2
I ’ _ 0 f c0 0 :, 0 0 -I l
-2
-3 0
I
2
3
X Coordinate
, mm
0) 13= 4!5’(+)
ahear
4
G
0
X Coordinate
f) 8=
Fig. 23. Crack growth in laminae with holes-Positive
neous with orthotropic or anisotropic response characteristics. The strength characteristics of graphite composites are highly orthotropic primarily because of the high tensile strength in the fiber direction and the very low tensile strength transverse to the fibers. The low transverse strength is generally associated with the low fiber/matrix bond strength as opposed to low tensile strength of the matrix itself. In fact, it is generally believed that there is no chemical bond between epoxy resins and graphite fibers. What bond there is is associated with mechanical interaction due to surface roughness, friction, and residual stresses from the curing process. The highly orthotropic strength characteristic of graphite composites results in crack growth characteristics which are completely different from the more commonly known characteristics of homogeneous, isotropic materials. Crack growth in isotropic materials is self similar. That is to say, if a crack is loaded perpendicular to the plane of the crack, the direction of crack extension is coincident with the original crack direction. In contrast to this self-similar crack growth of isotropic materials, unidirectional graphite epoxy exhibits crack growth which is
45’(-)
3
4
, mm shear
and negative shear.
highly dependent on the direction of minimum strength, irrespective of the orientation of the crack relative to the fiber direction and irrespective of the direction of loading relative to the plane of the crack[ 111. Figure 17 shows failed graphite epoxy tensile specimens with different fiber orientations. In all cases the loading was tension perpendicular to the plane of the original slot-type notch. As the results clearly show, crack growth from the existing notch is always parallel to the fiber direction. Crack growth in the unidirectional graphite epoxy is self similar only for the very special case of loading perpendicular to the fiber direction. The results and conclusions discussed in the preceding paragraph are limited to the case of unidirectional materials, that is, laminae. Laminates are an entirely different situation. Consider, for example, the failure surfaces of the angle-ply laminates in Fig. 14. Clearly, for the alternating [(t-@)?I, laminate there is fiber breakage in half the layers. Thus, crack growth in laminates is not necessarily parallel to the fiber direction in all layers. This indicates that crack growth in a laminate is an even more complicated problem, strongly infuenced by the morphol-
676
C. T. HERAKOVICH
b) 1901LAMINA Fig. 24. Actual crack growth in notched ogy of the laminate as well as the morphology of the individual layers. The fact that cracks do not grow in a self-similar manner in fibrous composites means that classical fracture mechanics is not applicable. It also means that the direction of crack growth is a fundamental unknown of the problem. The problem of crack growth in unidirectional composites was addressed recently in a general way by Beuth and Herakovich[ 111, where additional references on the subject are given. If the existing notch is not sharp, an additional unknown is the location of the crack initiation site from the existing notch. For isotropic materials, the crack initiation site corresponds to the point of maximum stress. This will not necessarily be the case for anisotropic materials because the
[0] and [W] tensile
coupons.
crack initiation site is a function of both the stress magnitude and the directionally dependent strength. The problem of crack initiation site as a function of notch geometry was studied by Gurdal and Herakovich[l2]. Crack growth in unidirectional fibrous composites is thus a many faceted problem. Critical parameters which must be determined and modeled include: a) the site of crack initiation, b) the direction of crack growth, c) the load for crack initiation, and d) the stability of crack growth. The extent to which all of these parameters are a function of far-field loading, fiber orientation, notch orientation, notch geometry, and environmental conditions must be included in a comprehensive fracture model. A theory for predicting fracture in fibrous com-
Effects
of morphology
on properties
that has shown promise is the normal stress ratio (NSR) theory originally proposed by Buczek and Herakovich[l3]. This theory simply states that crack growth is controlled by the normal stress at a point normalized with respect to the tensile strength, both of which are functions of the plane under consideration. The theory states that crack growth will initiate at the site and in the direction where this ratio first attains a value of 1.0 as the load is increased. Mathematically. posites
NSR = am,,,/Tmm where T is defined T = Xrsin’B
+ Yrcos’8
with I3 being the angle between the plan in question and the fiber direction (Fig. 18). This theory is based upon the physical argument that crack growth is controlled by normal stress only, but that the directional dependent strength must also be included in a general theory. Because it has not been possible to measure the strength of a unidirectional composite on predetermined planes, it has been necessary to define Tbmas given above. This definition satisfies the minimum requirements that if the material is isotropic, T,, is independent of B; if the crack grows parallel to the fibers, T,, equals Yr, the transverse strength of the composite; and if the crack grows perpendicular to the fibers, T,, equals X,, the axial strength of the composite. This theory has been applied with reasonably good success in a number of applications. Figure 19 shows
Fig. 25.
Actual
of graphite
composites
671
one of the first applications, a sequence of deformed finite element grids depicting crack growth in a [9OJ Or],laminate using the NSR theory[l3]. As is evident from the figure, the theory predicts that a transverse crack in a 90” layer changes direction and becomes a delamination between layers when the crack reaches the O/90 interface. Figures 20-23 show the results of using the theory to predict both the location and direction of crack growth from notched laminae with different notch geometry, fiber orientation and far-field loading. As indicated in these figures, the site of crack initiation varies considerable with the problem parameters, but the direction of growth is always parallel to the fibers. Typical results from an extensive experimental investigation are detailed in Figs. 24 and 25 and Table 1. The photographs in the figures show that the location of the crack initiation site is variable and that the crack always grows parallel to the fibers (as predicted by the normal stress ratio theory). Table 1 includes comparisons between theory and experiment for the critical far-field stress corresponding to crack initiation. The critical stresses in the table were based upon linear elastic stress distributions obtained using anisotropic elasticity theory (Lekhnitskii[l4], Erdogan and Sih[lS]). Theoretical predictions were obtained at a critical distance from the tip of a sharp crack and on the surface of an elliptical notch. The results of the 90” test were used to determine the critical distance from the sharp crack. Experimental results are presented for the far-field stress at crack initiation and at failure. Since these two stress values are not always identical (Table l), it is apparent that stable crack growth was observed
crack growth in notched
[30] lamina
under shear loading.
C. T.
678
HERAKOVICH
Table 1. Theoretical and experimental critical siresses tension and shear tests (ksi/MPa) Experimental
Theoretical
e
Initiation
Failure
Sharp
Elliptical
0 45 90”
38.21263 4.12128.4 2.81/19.4
58.8/405 41.2128.4 2.81119.4
20.9/144 5.02134.6 -
22.81157 48.3133.3 -
mechanician, and the structural designer. The influence of morphology generally can be predicted using fundamental principles of mechanics.
REFERENCES
Iosipescu shear tests 0” 15” 30” 45
4.03127.8 3.50/24.1 4.13128.5 4.08128.1
12.1j83.6 9.85167.9 7.52151.8 4.08128.1
3.10121.4 2.88119.9 3.01/20.8 3.80126.2
1. WebsterS New World Dictionary of the American Language, College Edition, The
2. 6.51144.9 5.12135.3 4.56131.4 4.70132.4
3.
4. 5.
in some cases. The comparison between theory and experiment in Table 1 is quite good. This is particularly true when it is recalled that the critical load for crack initiation under far-field shear loading is predicted quite well with a theory which is based upon the normal stress at the crack tip and is independent of local shear stresses. 5.
SUMMARY
It has been shown that the morphology (structure) of graphite composites can have a strong influence on the elastic, thermal and fracture characteristics of the material. Many properties of graphite composites exhibit characteristics which may appear surprising until the effects of the morphology are considered. Knowledge of the influence of morphology is important to the materials scientist, the
6.
World Publishing Co, New
York (1953). R. Bacon, ‘Phil. Trans. R. Sot. Lond. A 294. 437 (1979). T. W. Knott and C. T. Herakovich, Final Report, NSF Grant MSM-8613090, University of Virginia (May 1988). Z. Hashin, NASA CR-1974 (March 1974). W. B. Avery and C. T. Herakovich, J. Appl. Mech. 53, p. 751 (1986). C. T. Herakovich and M. W. Hyer, Engineering Fracture Mechanics 25, 779 (1986).
7. M. J. Pindera and C. T. Herakovich, Journal of Composite Materials 21, 1164 (1987). 8. C. T. Herakovich, Journal of Comoosite Materials 18. 447 (1984). 9. R. B. Pipes and N. J. Pagano, Journal of Composite Materials 4, 538 (1970). 10. C. T. Herakovicd, Journal of Composite Materials 16, I
.
216 (1982).
11. J. L. Beuth Jr., and C. T. Herakovich, Theoretical and Applied Fracture Mechanics, 11, 27-46 (1989). 12. Z. Gurdal and C. T. Herakovich, Theoretical and Applied Fracture Mechanics 8, 59 (1987).
13. M. B. Buczek and C. T. Herakovich, Journal of Composite Materials 19, 544 (1985).
14. S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body, English Translation by Holden-Day, Inc., SanFranscisco (1963). 15. F. Erdogan and G. C: Sih, Journal of Basic Engineering 85, 519 (1963).