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Micro-mechanics parameters controlling the strength of braided composites
ELSEVIER
MICRO-MECHANICS STRENGTH
PARAMETERS CONTROLLING OF BRAIDED COMPOSITES
THE
L. V. Smith & S. R. Swanson* Department
of Mechanical
Engineering.
University
(Received 25 July 1994: revised version received Abstract This study examines parameters which are believed to control the strength of triaxially braided carbon/epoxy composites under planar biaxial loading. Past studies have shown that the stiffness of these materials is readily determined and that consideration of a maximum fiber-direction strain has great utility in correlating failure. While the maximum fiber-direction strain is relatively constant over a wide range of biaxial stress states for these materials, it is lower than that found for similar laminated specimens. Many mechanisms, including textile processes and yarn undulation, undoubtedly contribute to the degraded properties. Their effects alone, however, do not adequately explain the low fiber-direction ultimate strains. Large planar variations in strain have experimentally been shown to exist. Local extreme fiber-direction failure strains may then be much closer to that found for laminates than was previously believed. One- and two-dimensional analytical models are used to study this behavior based on compliance changes resulting from discrete yarn bundles. It is hoped that an increased understanding of the complex strain response of braided composites will allow better failure prediction and an increased understanding of failure mechanisms. Keywords:
strength,
failure,
braid,
biaxial,
micro-
mechanics INTRODUCTION Textile composites are under consideration as a means of producing lightweight structural materials at reduced manufacturing costs. Although many types of textile composites are in use, the current study considers regular (two-step) triaxially braided carbon/epoxy composites. While research in previous studies has been successful at predicting the stiffness of these materials, until recently little has been known regarding their strength. Information regarding * To whom correspondence should be addressed.
of Utuh,
Salt Lake
14 February
City,
1995; accepted
Utuh X41 12. USA
8 March
1995)
strength will be required if design and optimization of these materials are to be accomplished. Previous studies have examined braided composites under in-plane biaxial loading conditions.‘-3 From this work it was found that consideration of a maximum fiber-direction strain from uniaxial tests was a good predictor of failure under biaxial loading environments. Similar results have been reported in the carbon/epoxy regarding laminated literature composites.“-’ The ability of the maximum fiberdirection strain failure criterion to correlate strength for both textile and laminated composites indicates that this failure criterion has broad application in predicting fiber-dominated failure for composite materials in general. The maximum fiber-direction strain failure criterion is applied by first determining the material strain state from externally applied loads. The strains are then transformed into each of the planar fiber directions of the composite. Fiber-direction strains are compared against ultimate strains determined from coupon tests for the material under consideration. Fibers oriented in the direction which reaches the ultimate strain first are predicted to fail. If the strength of the remaining fibers is insufficient to carry the applied load, which is usually the case, the part is predicted to fail. When applied to textile composites, the fiber-direction average strains are expected to differ from those actually experienced in the textile material. because of local strain concentrations associated with the microstructure. The intent of the current study is to examine the strength of braided composites and explain the experimentally observed trends from the braid architectures using micro-mechanics models. It is intended that this information may then be used in the design and optimization of these materials. While this study considers only one material system, the models are based on failures dominated by fiber-direction strain. It is reasonable to expect. therefore. that any material which obeys the maximum fiber-direction strain failure criterion would behave in a manner similar to the material under evaluation here.
178
L. V. Smith, S. R. Swanson
In the following, experimental procedures and results illustrating the utility of a maximum fiberdirection strain failure criterion are presented. Failure
600
modes and surfaces of representative specimens are examined. Comparison is made between the observed failure trends of braids and laminates. Two micromechanics models are presented which predict variations in strain resulting from changes in compliance in the transverse direction. Comparison of transverse strain variation is made with experimental data. The variation of strain in the braid direction is also reported to illustrate both its influence on strength and its sensitivity to fixed yarn spacing. EXPERIMENTAL
1
RESULTS
The experimental results, upon which this paper is by subjecting cylindrical based, were obtained specimens to axial loading and radial pressure in both tension and compression. The specimens were braided from AS4 carbon fiber and resin transfer molded with a Shell 1895 resin system. The inside diameter of the compression and tension specimens was 53 mm (2.2 in.) and 96 mm (3.8 in.), respectively. Both types of specimen had a wall thickness of 3.2 mm (0.13 in.). Even though only regular triaxially braided composites were studied, four different architectures (identified as A. B. C, and D) were examined to determine their effect on strength. A description of the architectures is presented in Table 1. Results of tests involving these materials have been reported in the literature, an example of which is given in Fig. 1. This figure illustrates the good agreement between the maximum fiber-direction strain failure criterion and the experiment in stress space for architecture B. Even though the range of in-plane biaxial stress varied widely between the specimens, two failure modes corresponding to the two fiber directions were observed. An axial failure mode, believed to be a result of fixed yarn failure, resulted in the creation of a failure surface normal to the fixed yarns. A braid
-200 -150 -100 -50 0 50 100 Hoop stress at failure (MPa) 0
Braid failure
0
Axial failure
Fig. 1. Biaxial failure
l
-
Biaxial failure Max. strain
envelope of the B architecture stress space.
Filament
count
code
A, BC CC DC A, B, C, Q
Braider
Fixed
12k 12k 6k 12k 12k 12k 6k 12k
6k 30k 30k 66k 9k 27k 33k 54k
in
failure mode, believed to be the result of braid yarn failure, produced a failure surface parallel to the fixed yarns, along paths of maximum curvature in the braid yarns. In both the axial and braid failure modes failure was sudden and catastrophic. Photographs of specimens illustrating these failure modes can be found in the literature.*‘” The failure surfaces were observed to be architecture and load direction (tension or compresArchitectures with braid yarns sion) dependent. oriented at rt45” (A and B) were observed to experience delamination near the failure surface. Figure 2 shows a photograph of a section taken parallel to the braid yarns of a failed *45” architecture. The delamination of this architecture was typical of both tensile and compressive failure.
Table 1. Description of the braid architectures. Subscript on braid code indicates specimen type, compression or tension
Braid
150 200
Fixed yarns (% of total)
Braid angle (degrees)
Number of fixed yarns
Number of plies
15 47 46 46 20 44 45 43
45 45 70 72 47 45 73 70
36 36 36 24 72 72 36 36
5 3 3 2 4 3 5 3
179
Parameters controlling the strength of braided composites
Fig. 2. Photomicrograph
of tensile braid failure of A architecture.
Delamination was not observed in the &70” architectures (C and D). Tensile loading typically created a failure surface normal to the composite surface, as shown in Fig. 3, while the failure surface from compressive loading usually followed the crimp angle, as shown in Fig. 4. A weak plane exists at this location in the resin-rich area between the braid and axial yarns. In both cases, however, failure of the braid yarns normally occurred in areas of maximum braid yarn curvature. While subtle differences in delamination and failure surfaces described above undoubtedly play a role in the failure process, the ability of fiber-direction strain to correlate failure over a wide range of stress states implies that for these materials, fiber-direction strain is the dominant parameter controlling failure. FAILURE TRENDS Using fiber-direction strain to determine the strength of a composite requires an understanding of the composite stiffness and fiber-direction maximum strain. Predicting the stiffness of braids is somewhat more involved than for laminates, but is readily achieved using various approaches. This study will therefore focus on mechanisms affecting the fiberdirection strain. Since the braid yarns are symmetric with respect to the fixed (axial) yarns, two fiber directions need to be considered: fixed and braid. Past studies have shown that failure in the axial direction is similar in many ways to laminated composites, with slightly lower ultimate values (on average near 1% strain for AS4/189Q8 Both compressive and tensile failure properties of fixed yarns follow the general trends of laminated composites. One example is a sensitivity observed in the compressive axial failure strain to the amount of adjacent bias material. As the relative amount of bias material (braid yarns for braided composites) is
Fig. 4. Photomicrograph of compressive braid failure of C
architecture. increased, the compressive failure strain of both laminated and braided composites has been observed to increase.’ Although this phenomenon has not been extensively studied, an increased amount of bias material is believed to add support to the critical fibers (fixed yarns for braided composites) which increase their critical load, or failure strain. Another similarity between the axial strength of braided composites and laminates has been a sensitivity in compression to fixed yarn straightness, while tensile strengths are relatively insensitive to fixed yarn waviness. The sensitivity to waviness in compression is again attributed to what is often termed as fiber micro-buckling, but more precisely is bending deformation of initially wavy fibers. It is generally believed that compression failure is expected to occur at lower loads for wavy fibers. In tension, however, fiber buckling is not a concern, and, therefore, waviness has a lesser effect. Strength in the braid direction is quite different from the axial direction and laminates. Three observations are worthy of note. First, the overall magnitude of the braid-direction failure strain was observed to be relatively low, on average near 0.4%. This is much lower than that observed for laminates containing fibers with similar crimp.“’ Second, the tensile and compressive braid-direction ultimate strains were similar in magnitude.’ This suggests that fiber buckling has a lesser effect in the braid yarns than in the fixed yarns. Third, the braid-direction failure strain is largely independent of braid crimp, while the failure strain of laminates with wavy fibers was directly related to crimp.“’ These observations indicate that a mechanism may be controlling failure which is significant in its magnitude and lowers the braid yarn strength below its critical load, reducing the effect of crimp. The following section investigates two models which have been developed to study such a mechanism. MICRO-MECHANICS
Fig. 3. Photomicrograph
of tensile braid failure of C architecture.
ANALYSIS
Textile materials are made from discrete yarns. Sroducing a preform in this way invariably results in resin-rich areas between the yarns. Compliance then varies not only through the thickness (as with laminates) but also becomes a function of surface position. Changes in compliance can be large, given
180
L. V. Smith, S. R. Swanson
the relative difference in stiffness between the fiber and matrix. This has significance in determining strength for materials which obey the maximum fiber-direction strain failure criterion, since compliance is used to obtain strength through Hooke’s law. Two micro-mechanics models have been employed to examine how changes in compliance affect local strain response. The first model considers a single braided layer, is analytical in nature, and is based on relatively simple strength of materials principles. The second model is numerical in nature, incorporating finite elements to approximate more closely the actual braided architecture. Experimental elastic-strain measurements on the surface of braided composites verify that strain is not uniform but is a function of position within the repeating unit cell. This has been found in strain gage studies in which the effect of strain gage size and position on the measured strain have been examined. In one study, strain varied by a factor of 2.5 over the surface of the specimen.’ A more precise indicator of surface strain gradients may be performed by examination of Moire fringe patterns. Moire plots have been reported for these materials, and the results support those found previously in the strain gage studies.” The largest strain gradients were observed in transverse loading, where the surface strains were found to be largely independent of interior braided layers. This is surprising since adjacent layers are often nested which encourages interaction between layers. The com-
pliance of the matrix interface between the layers is apparently sufficient, however, to allow independent strain response. As will be explained further below, it is believed that this variation in strain may explain in part the lower strength of the braided composites. If it is then assumed that layers act independently in transverse loading, a simplified analytical micromechanics model can be constructed to approximate the strain field. A single layer in the braid, shown schematically in Fig. 5, could be modeled onedimensionally in the transverse direction as two repeating regions in series. The bias yarns, continuous throughout the braid, would need to be represented in each region. The fixed yarns, with high compliance in the transverse direction, are not continuous in this direction and would only be represented in one region. Referring to the region with both fixed and braid yarns as compliant and the region with only braid yarns as stiff, in terms of laminated composites they would be represented as [O/k 01 and [& 01, respectively. A schematic of this material representation is depicted in Fig. 6. Whereas Figs 5 and 6 illustrate surface representations of the braid and analytical model, Fig. 7 contains through-the-thickness representations. From this view additional assumptions made in the analytical model are illustrated. The thickness of the braided layer, for example, is assumed to be constant and the shape of the axial yarn is not considered. Although a more complex representation of the actual geometry may more accurately predict strains, this simplified be used as a first approach can nevertheless approximation to the actual geometry.
/
Triaxial material
/
Biaxial material
Transverse
& F’ig. 5. Schematic of single braided layer, fixed yarns are shown extending past the braid yarns for clarity.
Axial
Fig. 6. Surface fiber orientation assumed by the analytical model.
181
Parameters controlling the strength of braided composites
9
ot
Schematicof one layer in a braided composite
Biaxial material
This produces two equations (compliant and stiff regions) with three unknowns: E,, a”,, and o”,. A third equation may be obtained by assuming that the stresses in the axial direction sum to zero. To do this, the respective lengths of the compliant and stiff regions in the transverse direction must be known. This is readily measured, and can be expressed in dimensionless form as the ratio of the length of the stiff region to the combined length of both regions, r,, and is reported in Table 2. Summing forces in the axial direction, and assuming constant thickness u”,rb + u’,( 1 - rb) = 0
(2)
Combining these three expressions the axial stress in the compliant region can be found as Geometry assumd by the analyticalmodel
Fig. 7. Through-the-thickness representations braided layer.
of a single
a’, = Ot E”,(l - rb)/rb - E”,
(3)
and similarly for the stiff region as can be used to predict the stiffness of the compliant and stiff regions, including classical lamination theory if crimp in the braid yarns is neglected. In the current analysis a threedimensional lamination theory, which performs in-and out-of-plane transformations, was used to account for reduced properties resulting from braid crimp.’ Properties of both regions for each architecture are reported in Table 2. Note the large differences in moduli between the two regions for each architecture. Boundary conditions may be applied to these two regions by considering a uniform stress in the transverse direction, a,. and uniform strain in the axial direction, E,. Using the subscripts a and t to denote the axial and transverse directions, respectively, and the superscripts c, s, and i to denote the compliant, stiff, or either region, respectively, the strain in the axial direction is found from Hooke’s law to be Numerous
v”,,E’, - vC,,ES,
models
u”, = ut E’, - E&,/(rb - 1)
With the stresses in the axial and transverse direction for both regions known, Hooke’s law can again be applied and the strains in each of these regions found. From these, the ratio of the maximum to nominal strain in the transverse direction can be determined. The ratio for the A and B architectures was near 1.1 and the ratio for the C and D architectures fell between 1-2 and 1-3. This ratio of the maximum to nominal transverse strain has been reported in the literature for two of the architectures studied.” These were experimentally found to be 1.3 and 1.9 for the C and D architectures, respectively. Although this ratio was not reported for architectures A and B, Moire plots were provided where the ratio appeared to decrease in comparison with the C and D architectures, a trend that is consistent with the model results. Many geometrical characteristics of the braided
Table 2. Description of material properties used in the analytical model. Subscript on braid code indicates specimen type, compression or tension Braid code
Stiff region
Compliant
region
rh (average)
%
A, B, C, Q A, B, C, D,
(&a)
(&?a)
13.85 13.13 7.30 7.10 11.35 1290 6.98 7.16
13.85 13.13 62.98 67.14 13.95 1290 74.77 65.25
0.755 0.748 0.127 0.104 0.678 0.746 0.103 0.135
(4)
(&!a)
(G?a)
vat
49.03 63.70 63.57 68.75 45.47 67.67 58.59 65.39
18.40 15.05 34.07 30.24 18.38 13.92 40.87 33.40
0.699 0.642 0.146 0.131 0.633 0.620 0.121 0.155
0.62 0.30 0.29 0.30 0.48 0.22 0.40 0.32
182
L. V. Smith, S. R. Swanson
architecture that have been neglected in the analytical model can be accounted for using a numerical approach such as finite elements. While this method should improve the accuracy of the analysis, the braided geometry must still be approximated. In constructing the geometry for this numerical approach, a section normal to the fixed yarns was investigated, as shown in Fig. 8. For comparison, a photomicrograph of this architecture is given in Fig. 9. An investigation was first carried out to study how the shape of the fixed yarn affected the strain field. Two triangular and one elliptical fixed yarn crosssections were examined. The angle of the edges of one of the triangular sections was controlled, while the width of the other triangular section and the elliptical section were controlled. For each section, the area was determined from the given percentage of fixed yarn found in Table 1. The width of the triangular section with controlled crimp was larger than measured values. This is due in part to the sharp triangular corners in the model. The ratio of the maximum to minimum transverse strain was used to compare the effect of section shape. In comparing the two triangular sections, the ratio was 12% lower for the section with controlled crimp. In comparing sections with controlled width, the ratio was 5% lower for the ellipse. While the difference between these sections is not large, it does indicate a sensitivity to yarn spacing. It was decided, therefore, that the proper section should control width. Since the elliptical shape was able to model a wider variety of architectures and fixed yarn contents, it was used for the comparisons with the analytical model. The distance from the composite edge to the fixed yarn was measured for each architecture and used in positioning the yarns nearest the edges. Interior fixed yarns were assumed to be equally spaced through the thickness. The braid yarns and resin pockets were smeared into one material and surrounded the fixed yarns in the model. The vertical edges of the model
Fig. 9. Micro-photograph
of section normal to the fixed yarns of the compression B architecture.
were assumed to be lines of symmetry and the right edge was given a constant displacement. The model was run using generalized plane-strain in the axial direction. The material properties of the fixed and braid yarns were found using a fiber-inclination model. ’ The results of this numerical approach were similar to the analytical model. The ratio of the maximum to nominal transverse strain fell between 1.1 and 1.2 for the A and B architectures, and 1.2 and 1.4 for the C and D architectures. The maximum strains from the numerical solution were slightly higher than the analytical model, as might be expected since the numerical model does not average the extreme strains. Although correlation between these models and the experiment for the C architecture is quite good, predictions for the D architecture are relatively poor. The D architecture had the largest yarn sizes and therefore the coarsest microstructure of the four architectures studied. Differences may exist between the geometry of the architectures used in the models and that used to generate the experimental data from the Moire fringe patterns. While the relative fiber directions and yarn sizes were similar, the geometry for the models was taken from 3.2 mm (O-13 in.) thick cylindrical specimens and the Moire fringe patterns were taken from 6.4 mm (O-25 in.) thick flat coupons. As will be discussed in more detail in the next section, differences in yarn spacing and width, which undoubtedly exist between these types of specimens, directly affect the results. These models have also assumed that the fixed yarns are perfectly nested. If the fixed yarns are stacked, the strain variation would be expected to increase. Differences in both yarn spacing within layers and yarn alignment between layers may be causes for the poor correlation with the D architecture. DISCUSSION
Fig.
8. Finite
element
mesh of architecture.
the
compression
B
The micro-mechanics models presented above allow strain variations caused by changes in compliance to be quantified. Obtaining a ratio of the maximum to average strain in the fiber direction will indicate local fiber-strain extremes from average strain measurements. This ratio was determined using both the
Parameters
controlling
the strength of braided
analytical and numerical models presented above. The fiber-direction strains were found by transforming the axial and transverse strains into the braid direction. Figure 10 compares the results of these models for the tension A architecture which is representative of the trends observed in the remaining architectures. Results of the numerical model were taken at two locations through the thickness. The first location, marked A in Fig. 8, was on the composite surface. which initially indicates an increase in strain variation with rb. As noted previously, rb is the ratio of the length of the stiff region to the combined length of both regions. As r, increases, however, the effect of the reduced fixed yarn width on the composite surface decreases. Results were taken from a second location in the numerical model at the interface between the fixed and braid yarn nearest the surface, marked B in Fig. 8. Results at this location show a continued increase in strain variation with rh similar to that predicted by the analytical model. This indicates that at locations of high rh surface strain measurements may be a poor indicator of interior strain levels. Perhaps most important about the relationship between the strain variation and r, is the larger variations in strain which occur at higher r,. Given architectures with similar variation in r,,, those with a higher average r, are predicted ,to exhibit lower strength. Fixed yarn widths and spacing were measured for each of the architectures presented in Table 1. Results of these measurements and the micro-mechanics models may explain some of the observed braid failure trends. The architecture with the most uniform yarn widths and spacing was observed to exhibit the highest braid-direction failure strain. while the architecture with the least uniform fixed yam width 7
0 Analytical model 0 FEM at surface
composites
183
and spacing exhibited the lowest braid-direction failure strain. While it is believed that this variation is caused by debulking during resin transfer molding. it suggests that a bulky preform may have lower strength not only from wrinkled yarns but from increased strain variations as the yarn spacing becomes less uniform. The strain concentration factors presented here are not expected to explain fully the large decrease in the observed braid strength. The intent of this study has been to assess the influence of discrete yarn bundles on strain variation, and not to predict the failure strain of a braid yarn in general. Stress concentrations from yarn curvature. for example, have not been addressed in this study, but are likely to affect strength. The parameters of the C and D architectures were chosen to investigate the effect of yarn size. The experimental evidence shows that architectures with larger yarns exhibit higher strain variations. The micro-mechanics models presented here complement this finding by defining mechanisms which may explain how this occurs. Although this study neglects some of the details of the three-dimensional geometry found in braided composites, it nevertheless aids in understanding mechanisms which may determine strength. It suggests, for example, that although a braid yarn may have identical undulation to a fixed yarn. because compliance varies more with surface position in the braid direction, the braid yarn will have a lower ultimate strength. While numerous parameters and mechanisms play a role in determining the strength of braided composites. strain variations from discrete yarn bundles explain both the low average ultimate strains of the braid yarns and the trends between the architectures. It is believed, therefore. that compliance variation is a governing parameter affecting the fiber-dominated ultimate strength of braid yarns, and likely contributes to the decreased strength of the fixed yarns. but to a lesser degree.
A FEM at axial yarn P
0.4
0.6
CONCLUSIONS
0.8
‘b Fig. 10. Comparison of the predicted strain variation for the analytical and numerical micro-mechanics models.
The failure of regular triaxially braided composites under in-plane loading has been examined. The good correlation between a maximum fiber-direction strain failure criterion and experiment has prompted a more detailed study of the fiber-direction failure process from a micro-mechanics point of view. Photomicrographs showing fiber-direction failure modes have been presented. Failure in the axial direction was observed to follow trends similar to laminated composites when exposed to both compressive and tensile loading, with slightly lower ultimate values. Failure in the braid direction resulted in braid-direction ultimate values significantly lower than that found in the fixed yarns
184
L. V. Smith,
or even in fibers with similar waviness for compressive and tensile loading. Consideration variations in strain caused by changes in
both to
local compliance was given both experimentally and using micro-mechanics models. General correlation between the models and experiment was achieved. Spacing of the fixed yarns was observed to affect the local compliance directly. Since strength is controlled by ultimate fiber-direction strain, strain variation caused by changes in compliance provide explanation for the lower average failure strains observed in braided composites. ACKNOWLEDGEMENT This research was performed under the sponsorship of the NASA Langley Research Center under grant no. NAG-I-1379. This support is gratefully acknowledged.
S. R. Swanson J. Comp. Mater. 4. Swanson, S. R. & Nelson,
carbon/epoxy biaxial stress. Conference
Smith, L. V. & Swanson, S. R., Response of braided composites under compressive loading. Comp. Engng, 3 (1993) 1165-84. Smith, L. V. & Swanson, S. R., Failure of braided carbon/epoxy composites under biaxial compression. J. Comp. Mater., 28, (1994) 1158-78.
Smith, L. V. & Swanson, S. R., Failure of braided composite cylinders under biaxial tension. To appear in
Proceedings Composite
M., Failure properties of under tension-compression of the Third Japan-U.S. Materials, Tokyo, (1986)
pp. 279-86. S. R. & Qian, Y., Multiaxial characterization of T800/3900-1 carbon/epoxy composites. Comp. Sci.
5. Swanson,
Technol., 43 (1991) 197-203. S. R. & Trask, 6. Swanson,
quasi-isotropic
laminates
B. C., Strength of under off-axis loading. Camp.
Sci. Technol. J., 34 (1989) 19-34. 7. Calvin, G. E. & Swanson, S. R., Characterization
of the failure properties of IM7/8551-7 carbon/epoxy under multiaxial stress. J. Engng Mater. Technol., 112(1990)
61-7. 8. Smith, L. V. & Swanson, S. R., Effect of architecture
on the strength of braided tubes under biaxial tension and compression. Submitted to J. Engng Mater. Technol.
(1994). 9. Calvin,
strength
REFERENCES
on
laminates
G. E. & Swanson, of carbonlepoxy
S. R., In-situ compressive AS4/3501-6 laminates. J.
Engng Mater. Technol., 115(1993) 122-8. 10. Adams, D. 0. & Hyer, M. W., Fabrication
and compression testing of layer waviness in thermoplastic composite laminates. Proc. 36th Int. SAMPE Symp. and
Exhibition, 36 (1991) 1094- 108. 11. Naik, R. A., Ifju, P. G. & Masters. J. E., Effect of fiber
architecture parameters on mechanical performance of braided composites. Fourth NASA/DOD Advanced Composite Technical Cant, 1 (1993) 525-54, Salt Lake City, Utah, 7-11 June 1993.
MICRO-MECHANICS STRENGTH
PARAMETERS CONTROLLING OF BRAIDED COMPOSITES
THE
L. V. Smith & S. R. Swanson* Department
of Mechanical
Engineering.
University
(Received 25 July 1994: revised version received Abstract This study examines parameters which are believed to control the strength of triaxially braided carbon/epoxy composites under planar biaxial loading. Past studies have shown that the stiffness of these materials is readily determined and that consideration of a maximum fiber-direction strain has great utility in correlating failure. While the maximum fiber-direction strain is relatively constant over a wide range of biaxial stress states for these materials, it is lower than that found for similar laminated specimens. Many mechanisms, including textile processes and yarn undulation, undoubtedly contribute to the degraded properties. Their effects alone, however, do not adequately explain the low fiber-direction ultimate strains. Large planar variations in strain have experimentally been shown to exist. Local extreme fiber-direction failure strains may then be much closer to that found for laminates than was previously believed. One- and two-dimensional analytical models are used to study this behavior based on compliance changes resulting from discrete yarn bundles. It is hoped that an increased understanding of the complex strain response of braided composites will allow better failure prediction and an increased understanding of failure mechanisms. Keywords:
strength,
failure,
braid,
biaxial,
micro-
mechanics INTRODUCTION Textile composites are under consideration as a means of producing lightweight structural materials at reduced manufacturing costs. Although many types of textile composites are in use, the current study considers regular (two-step) triaxially braided carbon/epoxy composites. While research in previous studies has been successful at predicting the stiffness of these materials, until recently little has been known regarding their strength. Information regarding * To whom correspondence should be addressed.
of Utuh,
Salt Lake
14 February
City,
1995; accepted
Utuh X41 12. USA
8 March
1995)
strength will be required if design and optimization of these materials are to be accomplished. Previous studies have examined braided composites under in-plane biaxial loading conditions.‘-3 From this work it was found that consideration of a maximum fiber-direction strain from uniaxial tests was a good predictor of failure under biaxial loading environments. Similar results have been reported in the carbon/epoxy regarding laminated literature composites.“-’ The ability of the maximum fiberdirection strain failure criterion to correlate strength for both textile and laminated composites indicates that this failure criterion has broad application in predicting fiber-dominated failure for composite materials in general. The maximum fiber-direction strain failure criterion is applied by first determining the material strain state from externally applied loads. The strains are then transformed into each of the planar fiber directions of the composite. Fiber-direction strains are compared against ultimate strains determined from coupon tests for the material under consideration. Fibers oriented in the direction which reaches the ultimate strain first are predicted to fail. If the strength of the remaining fibers is insufficient to carry the applied load, which is usually the case, the part is predicted to fail. When applied to textile composites, the fiber-direction average strains are expected to differ from those actually experienced in the textile material. because of local strain concentrations associated with the microstructure. The intent of the current study is to examine the strength of braided composites and explain the experimentally observed trends from the braid architectures using micro-mechanics models. It is intended that this information may then be used in the design and optimization of these materials. While this study considers only one material system, the models are based on failures dominated by fiber-direction strain. It is reasonable to expect. therefore. that any material which obeys the maximum fiber-direction strain failure criterion would behave in a manner similar to the material under evaluation here.
178
L. V. Smith, S. R. Swanson
In the following, experimental procedures and results illustrating the utility of a maximum fiberdirection strain failure criterion are presented. Failure
600
modes and surfaces of representative specimens are examined. Comparison is made between the observed failure trends of braids and laminates. Two micromechanics models are presented which predict variations in strain resulting from changes in compliance in the transverse direction. Comparison of transverse strain variation is made with experimental data. The variation of strain in the braid direction is also reported to illustrate both its influence on strength and its sensitivity to fixed yarn spacing. EXPERIMENTAL
1
RESULTS
The experimental results, upon which this paper is by subjecting cylindrical based, were obtained specimens to axial loading and radial pressure in both tension and compression. The specimens were braided from AS4 carbon fiber and resin transfer molded with a Shell 1895 resin system. The inside diameter of the compression and tension specimens was 53 mm (2.2 in.) and 96 mm (3.8 in.), respectively. Both types of specimen had a wall thickness of 3.2 mm (0.13 in.). Even though only regular triaxially braided composites were studied, four different architectures (identified as A. B. C, and D) were examined to determine their effect on strength. A description of the architectures is presented in Table 1. Results of tests involving these materials have been reported in the literature, an example of which is given in Fig. 1. This figure illustrates the good agreement between the maximum fiber-direction strain failure criterion and the experiment in stress space for architecture B. Even though the range of in-plane biaxial stress varied widely between the specimens, two failure modes corresponding to the two fiber directions were observed. An axial failure mode, believed to be a result of fixed yarn failure, resulted in the creation of a failure surface normal to the fixed yarns. A braid
-200 -150 -100 -50 0 50 100 Hoop stress at failure (MPa) 0
Braid failure
0
Axial failure
Fig. 1. Biaxial failure
l
-
Biaxial failure Max. strain
envelope of the B architecture stress space.
Filament
count
code
A, BC CC DC A, B, C, Q
Braider
Fixed
12k 12k 6k 12k 12k 12k 6k 12k
6k 30k 30k 66k 9k 27k 33k 54k
in
failure mode, believed to be the result of braid yarn failure, produced a failure surface parallel to the fixed yarns, along paths of maximum curvature in the braid yarns. In both the axial and braid failure modes failure was sudden and catastrophic. Photographs of specimens illustrating these failure modes can be found in the literature.*‘” The failure surfaces were observed to be architecture and load direction (tension or compresArchitectures with braid yarns sion) dependent. oriented at rt45” (A and B) were observed to experience delamination near the failure surface. Figure 2 shows a photograph of a section taken parallel to the braid yarns of a failed *45” architecture. The delamination of this architecture was typical of both tensile and compressive failure.
Table 1. Description of the braid architectures. Subscript on braid code indicates specimen type, compression or tension
Braid
150 200
Fixed yarns (% of total)
Braid angle (degrees)
Number of fixed yarns
Number of plies
15 47 46 46 20 44 45 43
45 45 70 72 47 45 73 70
36 36 36 24 72 72 36 36
5 3 3 2 4 3 5 3
179
Parameters controlling the strength of braided composites
Fig. 2. Photomicrograph
of tensile braid failure of A architecture.
Delamination was not observed in the &70” architectures (C and D). Tensile loading typically created a failure surface normal to the composite surface, as shown in Fig. 3, while the failure surface from compressive loading usually followed the crimp angle, as shown in Fig. 4. A weak plane exists at this location in the resin-rich area between the braid and axial yarns. In both cases, however, failure of the braid yarns normally occurred in areas of maximum braid yarn curvature. While subtle differences in delamination and failure surfaces described above undoubtedly play a role in the failure process, the ability of fiber-direction strain to correlate failure over a wide range of stress states implies that for these materials, fiber-direction strain is the dominant parameter controlling failure. FAILURE TRENDS Using fiber-direction strain to determine the strength of a composite requires an understanding of the composite stiffness and fiber-direction maximum strain. Predicting the stiffness of braids is somewhat more involved than for laminates, but is readily achieved using various approaches. This study will therefore focus on mechanisms affecting the fiberdirection strain. Since the braid yarns are symmetric with respect to the fixed (axial) yarns, two fiber directions need to be considered: fixed and braid. Past studies have shown that failure in the axial direction is similar in many ways to laminated composites, with slightly lower ultimate values (on average near 1% strain for AS4/189Q8 Both compressive and tensile failure properties of fixed yarns follow the general trends of laminated composites. One example is a sensitivity observed in the compressive axial failure strain to the amount of adjacent bias material. As the relative amount of bias material (braid yarns for braided composites) is
Fig. 4. Photomicrograph of compressive braid failure of C
architecture. increased, the compressive failure strain of both laminated and braided composites has been observed to increase.’ Although this phenomenon has not been extensively studied, an increased amount of bias material is believed to add support to the critical fibers (fixed yarns for braided composites) which increase their critical load, or failure strain. Another similarity between the axial strength of braided composites and laminates has been a sensitivity in compression to fixed yarn straightness, while tensile strengths are relatively insensitive to fixed yarn waviness. The sensitivity to waviness in compression is again attributed to what is often termed as fiber micro-buckling, but more precisely is bending deformation of initially wavy fibers. It is generally believed that compression failure is expected to occur at lower loads for wavy fibers. In tension, however, fiber buckling is not a concern, and, therefore, waviness has a lesser effect. Strength in the braid direction is quite different from the axial direction and laminates. Three observations are worthy of note. First, the overall magnitude of the braid-direction failure strain was observed to be relatively low, on average near 0.4%. This is much lower than that observed for laminates containing fibers with similar crimp.“’ Second, the tensile and compressive braid-direction ultimate strains were similar in magnitude.’ This suggests that fiber buckling has a lesser effect in the braid yarns than in the fixed yarns. Third, the braid-direction failure strain is largely independent of braid crimp, while the failure strain of laminates with wavy fibers was directly related to crimp.“’ These observations indicate that a mechanism may be controlling failure which is significant in its magnitude and lowers the braid yarn strength below its critical load, reducing the effect of crimp. The following section investigates two models which have been developed to study such a mechanism. MICRO-MECHANICS
Fig. 3. Photomicrograph
of tensile braid failure of C architecture.
ANALYSIS
Textile materials are made from discrete yarns. Sroducing a preform in this way invariably results in resin-rich areas between the yarns. Compliance then varies not only through the thickness (as with laminates) but also becomes a function of surface position. Changes in compliance can be large, given
180
L. V. Smith, S. R. Swanson
the relative difference in stiffness between the fiber and matrix. This has significance in determining strength for materials which obey the maximum fiber-direction strain failure criterion, since compliance is used to obtain strength through Hooke’s law. Two micro-mechanics models have been employed to examine how changes in compliance affect local strain response. The first model considers a single braided layer, is analytical in nature, and is based on relatively simple strength of materials principles. The second model is numerical in nature, incorporating finite elements to approximate more closely the actual braided architecture. Experimental elastic-strain measurements on the surface of braided composites verify that strain is not uniform but is a function of position within the repeating unit cell. This has been found in strain gage studies in which the effect of strain gage size and position on the measured strain have been examined. In one study, strain varied by a factor of 2.5 over the surface of the specimen.’ A more precise indicator of surface strain gradients may be performed by examination of Moire fringe patterns. Moire plots have been reported for these materials, and the results support those found previously in the strain gage studies.” The largest strain gradients were observed in transverse loading, where the surface strains were found to be largely independent of interior braided layers. This is surprising since adjacent layers are often nested which encourages interaction between layers. The com-
pliance of the matrix interface between the layers is apparently sufficient, however, to allow independent strain response. As will be explained further below, it is believed that this variation in strain may explain in part the lower strength of the braided composites. If it is then assumed that layers act independently in transverse loading, a simplified analytical micromechanics model can be constructed to approximate the strain field. A single layer in the braid, shown schematically in Fig. 5, could be modeled onedimensionally in the transverse direction as two repeating regions in series. The bias yarns, continuous throughout the braid, would need to be represented in each region. The fixed yarns, with high compliance in the transverse direction, are not continuous in this direction and would only be represented in one region. Referring to the region with both fixed and braid yarns as compliant and the region with only braid yarns as stiff, in terms of laminated composites they would be represented as [O/k 01 and [& 01, respectively. A schematic of this material representation is depicted in Fig. 6. Whereas Figs 5 and 6 illustrate surface representations of the braid and analytical model, Fig. 7 contains through-the-thickness representations. From this view additional assumptions made in the analytical model are illustrated. The thickness of the braided layer, for example, is assumed to be constant and the shape of the axial yarn is not considered. Although a more complex representation of the actual geometry may more accurately predict strains, this simplified be used as a first approach can nevertheless approximation to the actual geometry.
/
Triaxial material
/
Biaxial material
Transverse
& F’ig. 5. Schematic of single braided layer, fixed yarns are shown extending past the braid yarns for clarity.
Axial
Fig. 6. Surface fiber orientation assumed by the analytical model.
181
Parameters controlling the strength of braided composites
9
ot
Schematicof one layer in a braided composite
Biaxial material
This produces two equations (compliant and stiff regions) with three unknowns: E,, a”,, and o”,. A third equation may be obtained by assuming that the stresses in the axial direction sum to zero. To do this, the respective lengths of the compliant and stiff regions in the transverse direction must be known. This is readily measured, and can be expressed in dimensionless form as the ratio of the length of the stiff region to the combined length of both regions, r,, and is reported in Table 2. Summing forces in the axial direction, and assuming constant thickness u”,rb + u’,( 1 - rb) = 0
(2)
Combining these three expressions the axial stress in the compliant region can be found as Geometry assumd by the analyticalmodel
Fig. 7. Through-the-thickness representations braided layer.
of a single
a’, = Ot E”,(l - rb)/rb - E”,
(3)
and similarly for the stiff region as can be used to predict the stiffness of the compliant and stiff regions, including classical lamination theory if crimp in the braid yarns is neglected. In the current analysis a threedimensional lamination theory, which performs in-and out-of-plane transformations, was used to account for reduced properties resulting from braid crimp.’ Properties of both regions for each architecture are reported in Table 2. Note the large differences in moduli between the two regions for each architecture. Boundary conditions may be applied to these two regions by considering a uniform stress in the transverse direction, a,. and uniform strain in the axial direction, E,. Using the subscripts a and t to denote the axial and transverse directions, respectively, and the superscripts c, s, and i to denote the compliant, stiff, or either region, respectively, the strain in the axial direction is found from Hooke’s law to be Numerous
v”,,E’, - vC,,ES,
models
u”, = ut E’, - E&,/(rb - 1)
With the stresses in the axial and transverse direction for both regions known, Hooke’s law can again be applied and the strains in each of these regions found. From these, the ratio of the maximum to nominal strain in the transverse direction can be determined. The ratio for the A and B architectures was near 1.1 and the ratio for the C and D architectures fell between 1-2 and 1-3. This ratio of the maximum to nominal transverse strain has been reported in the literature for two of the architectures studied.” These were experimentally found to be 1.3 and 1.9 for the C and D architectures, respectively. Although this ratio was not reported for architectures A and B, Moire plots were provided where the ratio appeared to decrease in comparison with the C and D architectures, a trend that is consistent with the model results. Many geometrical characteristics of the braided
Table 2. Description of material properties used in the analytical model. Subscript on braid code indicates specimen type, compression or tension Braid code
Stiff region
Compliant
region
rh (average)
%
A, B, C, Q A, B, C, D,
(&a)
(&?a)
13.85 13.13 7.30 7.10 11.35 1290 6.98 7.16
13.85 13.13 62.98 67.14 13.95 1290 74.77 65.25
0.755 0.748 0.127 0.104 0.678 0.746 0.103 0.135
(4)
(&!a)
(G?a)
vat
49.03 63.70 63.57 68.75 45.47 67.67 58.59 65.39
18.40 15.05 34.07 30.24 18.38 13.92 40.87 33.40
0.699 0.642 0.146 0.131 0.633 0.620 0.121 0.155
0.62 0.30 0.29 0.30 0.48 0.22 0.40 0.32
182
L. V. Smith, S. R. Swanson
architecture that have been neglected in the analytical model can be accounted for using a numerical approach such as finite elements. While this method should improve the accuracy of the analysis, the braided geometry must still be approximated. In constructing the geometry for this numerical approach, a section normal to the fixed yarns was investigated, as shown in Fig. 8. For comparison, a photomicrograph of this architecture is given in Fig. 9. An investigation was first carried out to study how the shape of the fixed yarn affected the strain field. Two triangular and one elliptical fixed yarn crosssections were examined. The angle of the edges of one of the triangular sections was controlled, while the width of the other triangular section and the elliptical section were controlled. For each section, the area was determined from the given percentage of fixed yarn found in Table 1. The width of the triangular section with controlled crimp was larger than measured values. This is due in part to the sharp triangular corners in the model. The ratio of the maximum to minimum transverse strain was used to compare the effect of section shape. In comparing the two triangular sections, the ratio was 12% lower for the section with controlled crimp. In comparing sections with controlled width, the ratio was 5% lower for the ellipse. While the difference between these sections is not large, it does indicate a sensitivity to yarn spacing. It was decided, therefore, that the proper section should control width. Since the elliptical shape was able to model a wider variety of architectures and fixed yarn contents, it was used for the comparisons with the analytical model. The distance from the composite edge to the fixed yarn was measured for each architecture and used in positioning the yarns nearest the edges. Interior fixed yarns were assumed to be equally spaced through the thickness. The braid yarns and resin pockets were smeared into one material and surrounded the fixed yarns in the model. The vertical edges of the model
Fig. 9. Micro-photograph
of section normal to the fixed yarns of the compression B architecture.
were assumed to be lines of symmetry and the right edge was given a constant displacement. The model was run using generalized plane-strain in the axial direction. The material properties of the fixed and braid yarns were found using a fiber-inclination model. ’ The results of this numerical approach were similar to the analytical model. The ratio of the maximum to nominal transverse strain fell between 1.1 and 1.2 for the A and B architectures, and 1.2 and 1.4 for the C and D architectures. The maximum strains from the numerical solution were slightly higher than the analytical model, as might be expected since the numerical model does not average the extreme strains. Although correlation between these models and the experiment for the C architecture is quite good, predictions for the D architecture are relatively poor. The D architecture had the largest yarn sizes and therefore the coarsest microstructure of the four architectures studied. Differences may exist between the geometry of the architectures used in the models and that used to generate the experimental data from the Moire fringe patterns. While the relative fiber directions and yarn sizes were similar, the geometry for the models was taken from 3.2 mm (O-13 in.) thick cylindrical specimens and the Moire fringe patterns were taken from 6.4 mm (O-25 in.) thick flat coupons. As will be discussed in more detail in the next section, differences in yarn spacing and width, which undoubtedly exist between these types of specimens, directly affect the results. These models have also assumed that the fixed yarns are perfectly nested. If the fixed yarns are stacked, the strain variation would be expected to increase. Differences in both yarn spacing within layers and yarn alignment between layers may be causes for the poor correlation with the D architecture. DISCUSSION
Fig.
8. Finite
element
mesh of architecture.
the
compression
B
The micro-mechanics models presented above allow strain variations caused by changes in compliance to be quantified. Obtaining a ratio of the maximum to average strain in the fiber direction will indicate local fiber-strain extremes from average strain measurements. This ratio was determined using both the
Parameters
controlling
the strength of braided
analytical and numerical models presented above. The fiber-direction strains were found by transforming the axial and transverse strains into the braid direction. Figure 10 compares the results of these models for the tension A architecture which is representative of the trends observed in the remaining architectures. Results of the numerical model were taken at two locations through the thickness. The first location, marked A in Fig. 8, was on the composite surface. which initially indicates an increase in strain variation with rb. As noted previously, rb is the ratio of the length of the stiff region to the combined length of both regions. As r, increases, however, the effect of the reduced fixed yarn width on the composite surface decreases. Results were taken from a second location in the numerical model at the interface between the fixed and braid yarn nearest the surface, marked B in Fig. 8. Results at this location show a continued increase in strain variation with rh similar to that predicted by the analytical model. This indicates that at locations of high rh surface strain measurements may be a poor indicator of interior strain levels. Perhaps most important about the relationship between the strain variation and r, is the larger variations in strain which occur at higher r,. Given architectures with similar variation in r,,, those with a higher average r, are predicted ,to exhibit lower strength. Fixed yarn widths and spacing were measured for each of the architectures presented in Table 1. Results of these measurements and the micro-mechanics models may explain some of the observed braid failure trends. The architecture with the most uniform yarn widths and spacing was observed to exhibit the highest braid-direction failure strain. while the architecture with the least uniform fixed yam width 7
0 Analytical model 0 FEM at surface
composites
183
and spacing exhibited the lowest braid-direction failure strain. While it is believed that this variation is caused by debulking during resin transfer molding. it suggests that a bulky preform may have lower strength not only from wrinkled yarns but from increased strain variations as the yarn spacing becomes less uniform. The strain concentration factors presented here are not expected to explain fully the large decrease in the observed braid strength. The intent of this study has been to assess the influence of discrete yarn bundles on strain variation, and not to predict the failure strain of a braid yarn in general. Stress concentrations from yarn curvature. for example, have not been addressed in this study, but are likely to affect strength. The parameters of the C and D architectures were chosen to investigate the effect of yarn size. The experimental evidence shows that architectures with larger yarns exhibit higher strain variations. The micro-mechanics models presented here complement this finding by defining mechanisms which may explain how this occurs. Although this study neglects some of the details of the three-dimensional geometry found in braided composites, it nevertheless aids in understanding mechanisms which may determine strength. It suggests, for example, that although a braid yarn may have identical undulation to a fixed yarn. because compliance varies more with surface position in the braid direction, the braid yarn will have a lower ultimate strength. While numerous parameters and mechanisms play a role in determining the strength of braided composites. strain variations from discrete yarn bundles explain both the low average ultimate strains of the braid yarns and the trends between the architectures. It is believed, therefore. that compliance variation is a governing parameter affecting the fiber-dominated ultimate strength of braid yarns, and likely contributes to the decreased strength of the fixed yarns. but to a lesser degree.
A FEM at axial yarn P
0.4
0.6
CONCLUSIONS
0.8
‘b Fig. 10. Comparison of the predicted strain variation for the analytical and numerical micro-mechanics models.
The failure of regular triaxially braided composites under in-plane loading has been examined. The good correlation between a maximum fiber-direction strain failure criterion and experiment has prompted a more detailed study of the fiber-direction failure process from a micro-mechanics point of view. Photomicrographs showing fiber-direction failure modes have been presented. Failure in the axial direction was observed to follow trends similar to laminated composites when exposed to both compressive and tensile loading, with slightly lower ultimate values. Failure in the braid direction resulted in braid-direction ultimate values significantly lower than that found in the fixed yarns
184
L. V. Smith,
or even in fibers with similar waviness for compressive and tensile loading. Consideration variations in strain caused by changes in
both to
local compliance was given both experimentally and using micro-mechanics models. General correlation between the models and experiment was achieved. Spacing of the fixed yarns was observed to affect the local compliance directly. Since strength is controlled by ultimate fiber-direction strain, strain variation caused by changes in compliance provide explanation for the lower average failure strains observed in braided composites. ACKNOWLEDGEMENT This research was performed under the sponsorship of the NASA Langley Research Center under grant no. NAG-I-1379. This support is gratefully acknowledged.
S. R. Swanson J. Comp. Mater. 4. Swanson, S. R. & Nelson,
carbon/epoxy biaxial stress. Conference
Smith, L. V. & Swanson, S. R., Response of braided composites under compressive loading. Comp. Engng, 3 (1993) 1165-84. Smith, L. V. & Swanson, S. R., Failure of braided carbon/epoxy composites under biaxial compression. J. Comp. Mater., 28, (1994) 1158-78.
Smith, L. V. & Swanson, S. R., Failure of braided composite cylinders under biaxial tension. To appear in
Proceedings Composite
M., Failure properties of under tension-compression of the Third Japan-U.S. Materials, Tokyo, (1986)
pp. 279-86. S. R. & Qian, Y., Multiaxial characterization of T800/3900-1 carbon/epoxy composites. Comp. Sci.
5. Swanson,
Technol., 43 (1991) 197-203. S. R. & Trask, 6. Swanson,
quasi-isotropic
laminates
B. C., Strength of under off-axis loading. Camp.
Sci. Technol. J., 34 (1989) 19-34. 7. Calvin, G. E. & Swanson, S. R., Characterization
of the failure properties of IM7/8551-7 carbon/epoxy under multiaxial stress. J. Engng Mater. Technol., 112(1990)
61-7. 8. Smith, L. V. & Swanson, S. R., Effect of architecture
on the strength of braided tubes under biaxial tension and compression. Submitted to J. Engng Mater. Technol.
(1994). 9. Calvin,
strength
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and compression testing of layer waviness in thermoplastic composite laminates. Proc. 36th Int. SAMPE Symp. and
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architecture parameters on mechanical performance of braided composites. Fourth NASA/DOD Advanced Composite Technical Cant, 1 (1993) 525-54, Salt Lake City, Utah, 7-11 June 1993.