epoxy laminates under in-plane biaxial compression

COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 65 (2005) 2105–2117 www.elsevier.com/locate/compscitech

Mechanisms-based failure laws for AS4/3502 graphite/epoxy laminates under in-plane biaxial compression Daniel Potter, Vijay Gupta *, Xu Chen, Jun Tian Department of Mechanical and Aerospace Engineering, University of California Los Angeles, 38-137E, Engg IV Buiding, Los Angeles CA 900951597, United States Received 5 October 2004; received in revised form 19 April 2005; accepted 4 May 2005 Available online 1 July 2005

Abstract Failure mechanisms and stress–strain behaviors have been investigated for [±30]12s and [±45]12s graphite-epoxy (AS4/3502) laminates under in-plane biaxial compression by using a cruciform biaxial test frame and microscopy of load-interrupted samples. The loading confinement ratio R was varied from 0.24 to nearly 1.0 to measure the sensitivity of sample failure mechanisms and stress– strain behavior to different stress states. Failure modes with an increasing loading confinement ratio for both fiber orientations transitioned from the uniaxial failure mode of in-plane shearing to out-of-plane shearing and massive delamination. The shear failures were on one or more planes that traversed the entire sample thickness and thus encompassed alternate plies in which fibers were aligned and misaligned with respect to the shear plane. The local failure was found to be triggered either by the matrix shear in fiber-aligned plies or fiber-shear in the fiber-misaligned plies. A standard classical laminate theory in combination with the standard rule of mixture theory for composites was used to calculate the maximum matrix and fiber shear stresses. The failure data conformed nicely with a Mohr–Coulomb-type shear failure law. Most interestingly, both types of failures correlated rather remarkably to a single shear failure law when the stresses were smeared and expressed on the scale of the laminae. This latter failure law should have widespread use because of its simplicity. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Biaxial testing; Failure laws; Failure mechanisms; Graphite/epoxy laminates

1. Introduction High specific strength and stiffness properties of fiberreinforced composite materials have resulted in their widespread use in load-bearing structures. This has motivated several theoretical and experimental studies for understanding their failure mechanisms and strengths under compression and tensile loading. Mechanical behavior under compression is important because the material can undergo a large reduction in its strength due to material defects in the form of fiber misalignment. Fiber microbuckling-induced failures in *

Corresponding author. Tel.: +1 310 825 0223; fax: +1 310 206 2302 E-mail address: [email protected] (V. Gupta).

0266-3538/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.05.007

composites under uniaxial loading have been exhaustively studied in the past three decades and a good understanding of the phenomenon has been largely accomplished (e.g., see the review in [1], and other related papers [2–5]). Subtleties aside, in the end, the original simple analysis of Argon [6] has essentially prevailed. The disproportionately large effort in dealing with the fiber microbuckling phenomenon has digressed the attention from other failure mechanisms such as interplay delaminations, fiber shear fracture, and matrix yielding. This is because these latter mechanisms become significant when the material is stressed under in-plane biaxial compression for which the experimental data has been relatively scarce. This is due to the few biaxial

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testing machines available and the difficulty of physically loading the specimen along the two axes. A lack of an understanding of the local failure mechanisms under biaxial loading has directly hindered the development of any reasonable failure law that can be conveniently used by the designers. Since material in most applications is loaded under multiaxial stress-state, this need has forced the development of semi-empirical failure laws, which has led to conservative designs. Furthermore, the absence of predictive capabilities in such models has precluded their widespread use in engines and other high-performance structural components, and has handicapped their further advance. Various approaches to develop failure criteria have been established, ranging from laminate level to micromechanical constituent models. However, the former approaches, such as the Tsai-Hill, Hoffman and TsiaWu criteria are empirical, and require experimentally determined laminate constants. Other methods model the laminae as plates subject to bending and buckling, as done by Wung and Chatterjee [7] and Shuart [8]. The effect of delamination among plies on bending and buckling loads was investigated by Kutla and Chang [9] and a constituent scale investigation was conducted by O
Brien and Hooper [10] to study the onset of matrix cracking and local delaminations between plies.

King et al. [11] used a finite element model to assess the strength of fiber and matrix interfaces. Furthermore, fracture mechanics models have also been developed to explain the nucleation and propagation of matrix cracks and subsequent stiffness degradation of composites [6,12,13]. The effects of fiber waviness and initial misalignment on micro-buckling of fibers, and the formation of kink bands in unidirectional laminates, has also been thoroughly studied [2,14–17], oftentimes using a beam on an elastic foundation model. It was shown previously that crack nucleation starts in the matrix when the local shear stress exceeds the shear strength of the epoxy in [±45]12s laminates [18,22]. This prompted a small decrement in the stress as the matrix sheared along planes aligned with the axial fiber directions (Fig. 1). The displacement of these shear planes appeared as transverse cracks on the specimen sides. Further strain was accommodated at a constant stress as additional shear planes were generated and specimens began to bulge in the lateral (z) direction. Transverse crack formation was accompanied by wing cracks extending into the interply interface due to the geometrical mismatch caused by shear sliding of two adjacent plies (Fig. 2). Once the sample was saturated with slip planes and transverse cracks, the interply wing cracks propagated to an extent to weaken the ply inter-

Fig. 1. A schematic showing the progression of cracking failure in a typical [±45]12s sample in association with the measured stress–strain characteristic.

D. Potter et al. / Composites Science and Technology 65 (2005) 2105–2117

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Fig. 2. A scanning electron micrograph showing the edges of the transverse crack and its associated wing (interply) cracks on one of the lateral faces of a [±45]12s sample.

faces to allow delamination in the sample midsection and ultimate laminate failure. From a micro-mechanism viewpoint, matrix properties are primarily responsible for the laminate behavior. This subtle, yet significant influence, primarily dictates the more conspicuous final failure modes of ply bending and delamination. As the off-axis angle of the specimens was decreased, brittle failure increased and greater delamination accompanied laminate failure. [±30]12s uniaxially loaded samples with aspect ratios close to unity failed by inplane matrix shearing along two fiber-aligned planes with no delamination [18,22]. Unlike the [±45]12s samples, this orientation generated no transverse or wing cracks prior to ultimate failure. This catastrophic shearing failure mode indicates that upon reaching a critical matrix shear stress along a fiber direction, enough shear stress had been generated in fibers in the opposing direction to allow their instantaneous deformation and prompt ultimate laminate failure. The [±15]12s samples failed catastrophically by a combination of in-plane matrix shearing along the fiber direction and delamination [18,22]. Instead of a simple and clean slipping like the [±30]12s samples, the [±15]12s samples actually separated from one another with considerable fiber damage at the interface. [±10]12s samples failed by prevalent fiber bending and kinking that promoted extensive delamination, in addition to some in-plane matrix shearing. Uniaxially aligned samples failed by delamination, in addition to brooming and matrix crushing on the sample ends. In contrast, the relatively compliant [90]48 samples failed by matrix crushing due to the absence of fibers aligned with the loading axis. In general, increased alignment of the fibers with the loading axis prompted a transition from a dominant shearing failure mode to mechanisms of shearing and interply delamination.

2. Biaxial test configuration and sample preparation Testing was conducted with an Instron cruciform testing machine, consisting of two sets of 56 kip servohydraulic actuators with axially aligned load cells and hardened tool steel spacing shafts by following the procedures that are already described previously for testing under uniaxial compression [19–21]. All four actuators were used to apply end loading to samples as seen in Fig. 3. Great care was taken to insure that the platens along the machine
s two axes, which were designated as primary and secondary, were perpendicular to one another and that all eight edges of the platens met uniformly when brought together and provided uniform loading. The platens widths were approximately 92% the length of a test specimen edge to avoid a geometry constraint when compressing all four actuators on the

Fig. 3. Biaxial compression test setup.

D. Potter et al. / Composites Science and Technology 65 (2005) 2105–2117

compliant specimens. Consequently, relatively small areas of the test specimen corners were unloaded. Position compensation was used between opposing actuators along each axis so that the center of a biaxially loaded specimen remained stationary relative to the entire testing machine. Consequently, as one platen moved, its axis complement would move the same distance. This position compensated configuration was used so that side loading or shear traction would be minimized by the secondary axis platens as the sample deformed along the primary axis, an inherent concern in biaxial mechanical testing. Tests were conducted by applying a displacement rate into one primary axis actuator. As subsequent loading occurred along the primary axis, load was applied to the secondary axis, dictated with feedback control by the programmed confinement ratio R, the ratio of the load applied on the secondary axis to the primary axis. By varying the confinement ratio, distinct stress states were presented to the specimens. [±30]12s and [±45]12s graphite-epoxy (AS4/3502) samples with dimensions of approximately 25 mm2 and 6 mm thickness were cut from cured laminate plates and polished. In addition, angle ply laminates were also tested under uniaxial configuration to better understand the local failure mechanisms in biaxially loaded samples and also to determine the limiting properties of the constituents for use in the failure model.

3. Failure mechanisms and stress–strain behavior Figs. 4 and 7 summarize the typical stress–strain behavior of the [±45]12s and [±30]12s layups, respectively, tested with different confinement ratios R. A summary of biaxial results is tabulated in Table 1. Typical uniaxial results (R = 0) are included for comparison. 3.1. [±45]12s orientation Fig. 5 shows typical biaxial [±45]12s samples after failure. It is evident that increasing the confinement ratio prompts greater delamination throughout the entire laminate, not solely midsection bulging as in the uniaxial

500

R=0.88 400

Compressive Stress (MPa)

2108

R=0.48 300

R=0.24

200

100

R=0

0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Compressive Strain

Fig. 4. Average stress–strain behavior of [±45]12s samples under inplane biaxial loading.

case. Since uniaxial samples tended to bulge in the lateral (y) direction also, increasing confinement ratios inhibited this deformation and thus significantly altered the failure mechanism to massive delamination. Not surprisingly, more ply bending and delamination were associated with the secondary axis sides due to greater loads in the primary direction. Compared to uniaxial compression (R = 0), increasing the confinement ratio to 0.24 reduced the number of transverse cracks and the extent of lateral bulging in the midsection. However, angled shear cracks developed in regions of delamination (Fig. 6). The angled cracks present similar features as the transverse cracks, including accompanying wing cracks, but are inclined at an angle close to 45° to the out-of-plane direction. This angle remained constant on the entire sample surfaces. The transition from transverse cracks to angled cracks is due to the increasing out-of-plane stress with increasing confinement ratio. This stress increases within the laminae due to initial fiber misalignment and tilts the maximum matrix shear plane in the out-of-plane direction as the secondary axis stress increases. It is interesting to note that the angled cracks developed at the same angle in plies primarily of the same fiber angle as

Table 1 Summary of biaxial test results Prepreg layup

Confinement ratio R

Ultimate stress ru (MPa)

[±45]12s [±45]12s [±45]12s [±45]12s [±30]12s [±30]12s [±30]12s [±30]12s

0.887 0.475 0.237 0 0.985 0.501 0.243 0


488
363
239
182
409
516
302
244

Ultimate strain eu

Elastic modulus (GPa)

LVDT

Strain gage

LVDT

Strain gage


0.0165
0.0233
0.0226
0.0233
0.00686
0.01631
0.0167
0.00972


0.00559
0.0134
0.0178

34.4 17.7 12.4 10.9 80.5 37.8 26.5 27.9

70.7 18.8 13.4


0.00342
0.00678
0.00639
0.00718

75.5 86.6 53.8 36.6

D. Potter et al. / Composites Science and Technology 65 (2005) 2105–2117

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Fig. 5. Global failure mechanisms observed in biaxial loaded [±45]12s samples at failure. The confinement ratio corresponding to the loading is indicated directly underneath each picture.

Fig. 6. Edges of angled shear cracks as observed on the sample plane containing its thickness.

evidenced in the figure by their appearance in every other ply. Increasing the confinement ratio to 0.48 resulted in substantial delamination of groups of plies throughout the entire length of the sample and the transition in the number of transverse cracks to angled cracks continued. A very few number of shear faults, a multiple number of laminae shearing along a plane in the thickness direction, were also observed. Shear faults, revealed on the samples
secondary axis sides, extended across about 5 plies, similar to kink band formations in unidirectional laminates. A confinement ratio of 0.88 caused extensive delamination at most of the ply interfaces (Fig. 5). However, less transverse and angled cracks developed com-

pared to smaller confinement ratios; instead, shear faults, 5–10 plies wide, ran across laminae sides. At this highly confined loading, angled cracks tended to propagate into adjacent plies to form shear faults rather than transition to wing cracks at ply interfaces. 3.2. [±30]12s orientation Fig. 8 shows typical biaxial [±30]12s samples after failure, again including a uniaxial sample for comparison. The effect of confinement ratio on the [±30]12s samples is similar to that of the [±45]12s specimens; increasing the biaxial load increased the amount of

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500

R=0.98

Compressive Stress (MPa)

400 R=0.50

300

R=0.24

200

100 R=0

0 0

0.005

0.01

0.015

0.02

With an increasing confinement ratio, failure mechanisms for both fiber orientations transitioned from the uniaxial failure modes to extensive delamination. However, the amount of delamination was greater in the [±45]12s samples for a specific confinement ratio, certainly because these samples
uniaxial ultimate failure mechanism was delamination. Thus, these samples were a step ahead in the relevant failure mechanism process. Furthermore, increasing the confinement ratio increased out-of-plane stresses that subsequently shifted transverse cracks to angled cracks, especially in regions of delamination where out-of-plane stresses were greatest. This transition of stress state in the [±45]12s specimens resulted in a shift from purely shear planes to partial shear planes and delamination, to extensive delamination and angled shear faults.

Compressive Strain Fig. 7. Average stress–strain behavior of biaxially-compressed [±30]12s laminates.

delamination. Two shear planes, the same uniaxial failure mechanism, were produced with a confinement ratio of 0.24 with the addition of moderate delamination at the intersection of the two planes. A confinement ratio of 0.50 produced the typical pair of shear planes but they did not extend through the entire specimen thickness. Angled interply cracks, as discussed for the [±45]12s biaxial samples, were also generated in the bent plies that had delaminated. These features indicate, at this confinement state, critical stresses to prompt shear slipping and interply delamination occurred simultaneously. For the near uniform biaxial compression state (R = 0.98), the prominent failure mechanism became solely delamination. Shear faults, 2–6 plies wide, were also discovered, in addition to angled cracks within plies that had experienced moderate bending and delamination.

4. Modulus and strength results Fig. 9 plots the moduli derived from the actuator displacement transducers versus the confinement ratio on the [±30]12s and [±45]12s samples. The machine and fixture-related compliances were removed from the raw data to calculate the moduli presented in Fig. 9. Additionally, the sample
s entire length and width were taken to calculate the overall strains. Thus, the plotted values are volume average values and therefore expected to be lower than the highest modulus values that can be measured locally using strain gauges mounted directly on the specimens. Physically, the values plotted in Fig. 9 are more representative of material behavior as there are always stress and strain variations across the sample volume even in controlled testing that was used in this work. This is also representative of large composite structural applications where such variations are likely to be present in a similar-sized volume element that is often utilized by designers to model the structure. It is

Fig. 8. Global failure mechanisms observed in biaxially-compressed [±30]12s samples at failure. The confinement ratio for each test is written directly underneath the picture.

D. Potter et al. / Composites Science and Technology 65 (2005) 2105–2117

Moduluslvdt (GPa)

30

2111

[ 30]12s

25 20 15

[ 45]12s

10 5 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Confinement Ratio R Fig. 9. The effect of confinement ratio R on the elastic modulus of the laminates.

apparent from the linear trend lines that the stiffness of the samples increases with biaxial load. Increased suppression of sample lateral expansion by confinement loading leads to greater primary stiffness due to Poisson effects and the difference in specimen stiffness between the two loading axes. Since the curves in Fig. 9 are approximately parallel and the [±45]12s moduli values are lower, the percentage of modulus change versus confinement ratio is greater in the [±45]12s samples compared to the [±30]12s specimens. Fig. 10 plots the ultimate stress of the biaxial samples versus the confinement ratio R. The ultimate stress curves show a similar trend as the modulus curves in Fig. 9, except a more linear relationship exists for the former case. The [±30]12s data point at the 0.98 confinement ratio is not included in the plot since feedback control instabilities were generated. This was caused by the combination of the high confinement ratio and the large Poisson ratios of the samples and led to premature sample failures. Also, ultimate strain trends were not as consistent as the stress trends, indicating failure is more influenced by critical stress levels rather than strain limits. The ultimate strength increases with greater confinement ratios because the shear stress in the matrix is re-

Ultimate Stress (MPa)

-600

[±30]12

-500 -400

[±45]12s

-300 -200 -100 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Confinement Ratio R

Fig. 10. The effect of confinement ratio R on the ultimate strength of the laminates.

duced and the shearing failure mode is suppressed. This shifts the primary failure modes from transverse crack development and shear slipping, as in the [±45]12s and [±30]12s samples, respectively, to interply shearing and delamination. Since increasing the confinement loading raises out-of-plane stresses while lowering in-plane matrix shear stresses, critical stresses to initiate failure via the latter failure mechanisms occur before the epoxy shear strength is reached.

5. Analytical modeling and results The uniaxial and biaxial results were analyzed to investigate the link between failure mechanisms and stress–strain behavior, including laminate strengths. A constituent level model with stresses generated from laminate theory was developed to investigate fiber and matrix properties and their influence on laminate strengths in relation to varying failure mechanisms. An understanding of these aids development of laminate failure criteria based on constituent properties that can provide not only quantitative strengths, but also qualitative explanations of failure modes. Thus, a more complete understanding of composite behavior can be achieved compared to phenomenological results from empirical criteria. Classical lamination theory (CLT) [23,24] was used to analytically derive the stresses in plies of laminates with different fiber orientations. Although CLT assumes two-dimensional material behavior and does not include thickness direction effects, it provides a simple and convenient method to provide the general stress state within a lamina away from edges. In this sense, the analysis is unable to determine the influence of stress concentrations at the specimen edges, which is typical of direct loading as utilized in this work. These effects were however kept to a minimum in the present experiments as the failure was observed within the central region of the specimens away from

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platen edges. This is probably because fiber kinking was not observed as the key failure mode, which in turn, is sensitive to imperfections and edge stresses. This investigation further applied these results to lamina constituents to investigate how constituent stresses invoke failure mechanisms. Hygrothermal effects were also included in the analysis to account for residual stresses generated after curing. An outline of the analytical procedure is described below. By using the properties of the AS4/3502 prepreg (Table 2), reduced constitutive relationships in terms of the compliance matrix for plane stress and plane strain were calculated for the behavior of a lamina relative to the fiber direction [24]. The x–y–z coordinate system was aligned with the loading axes such that the x and y axes were aligned with the primary and confinement loading directions, respectively, while the 1– 2–3 local coordinate system was oriented in an individual ply with the fiber axes aligned with the 1-direction. Since applied moments were negligible during testing and no bending-stretching coupling existed because the laminates had a symmetric layup, the stress–strain relation is simply
 N þ N H xy ¼ ½Afe gxy ; ð1Þ where [A] is the in-plane laminate stiffness matrix and e0 is the total laminate in-plane strain, assumed to be consistent across all plies. The in-plane forces per unit width {N}xy were obtained by summing the transformed reduced stiffness matrices of each ply. The hygrothermal loads per unit width fN gH xy were obtained in a similar manner with the transformed thermal and moisture expansion coefficient vectors. The ultimate laminate forces {N}xy, measured from testing in the laminate coordinate system, and the H hygrothermal forces fN gxy , both adjusted with laminate thicknesses, were then used in to obtain the total laminate strains. The mechanical (mech) stresses that cause

Table 2 Prepreg and constituent material properties mf

0.552

E1 (GPa) E2 (GPa) G12 (GPa) m12 m23 a1 a2 b1 b2 Efa (GPa) Ema (GPa)

124 7.64 GPa 3.85 GPa 0.351 0.520
0.9 Æ 10
6/°C 27 Æ 10
6/°C 0.01/moisture content 0.2/moisture content 221 4.8

a

Fiber (f) and matrix (m) properties.

failure were obtained by subtracting the hygrothermal stresses from the total laminate stresses from n oo   n
0  ¼ Q e
a DT þ b Dc ð2Þ rmech f g f g xy xy xy k xy xy k

with ½Qxy being the stiffness of each ply (k) in the laminate coordinate system. The temperature change during curing (DT) and the moisture content (Dc) of each lamina were estimated at
121 °C and 0.005, respectively [23,25,26]. The resulting stresses within each ply were then transformed back into the laminae coordinate systems with the two-dimensional transformation matrix [T] and the ply stiffness matrix [Q]: mech

frg12k ¼ ½Q½T k fegxy k .

ð3Þ

Ply stresses from Eq. (3) were used to calculate stress states within the fiber and matrix at ultimate stress levels determined from testing. These values were then applied to a standard rule-of-mixture-type model to determine the local stresses in the fibers and the matrix and expressed in the fiber-aligned 1–2–3 coordinate system. The constituent properties needed for these calculations are provided in Table 2. Although polymer matrix behavior is typically nonlinear and is affected by the local geometry with respect to the fibers [24,27], the response was assumed linear to facilitate a simple model and analysis. In these calculations, it was also assumed that the fiber and matrix deformed uniformly. In addition, the normal (r22) and shear (s12) stresses were each assumed to act evenly on both the matrix and fiber [24]. The maximum shear anywhere in the sample, and the accompanying normal stress across the maximum shear plane were also determined. For the matrix, the hydrostatic stress across the maximum shear plane was also calculated. The out-of-plane stress r33, as needed for determining the maximum shear stress, was determined by setting the out-of-plane displacement to zero with the plane strain relations. Except for the [±45]12s samples under R = 0 and 0.24 confinements, samples in other orientations exhibited purely brittle elastic failure with very little inelasticity. Calculations pertaining to these [±45]12s samples using an elastic analysis is justified as the peak sample loads, which the model presented here calculates, occurs at the onset of yield in the matrix. The model essentially takes the peak sample loads measured experimentally and apportions them to fiber-aligned and misaligned plies, and then eventually to individual constituents to setup the local failure conditions. For this purpose an elastic analysis should be sufficient. However, the analysis in the paper cannot explain the continuous evolution of stress beyond this peak point as the matrix continues to yield and leads to stress redistributions within the constituents. Several recent papers [28–30] provide the evolution of stress for composites subjected to biaxial loading.

D. Potter et al. / Composites Science and Technology 65 (2005) 2105–2117

For the [±45]12s samples loaded with confinement ratios of 0, 0.25 and 0.48, the ultimate in-plane matrix shear stress was calculated to be between 89 and 92 MPa, which agrees well with the shear strength of the epoxy measured independently by Shuart [8]. In [±45]12s samples, this critical stress is reached and starts matrix crack nucleation in all confinement ratio samples except those that approach a confinement ratio of one. At this latter point, the in-plane shear stress in the sample is minimized so that out-of-plane stresses become dominant and invoke fiber buckling and shear fault generation. These failure mechanisms are of a more brittle nature and are directed out-of-plane causing massive delamination, as in Fig. 5 (R = 0.88). In addition, the normal stress on the matrix and fiber is actually tensile for the angle cross-plies until approximately ±45°, an effect of the high in-plane Poisson ratios of these configurations. The direction of the normal stress, either compressive or tensile, is indicative of the failure mechanism too. Laminates with tensile normal stresses all failed catastrophically without cracking and stiffness degradation prior to ultimate failure. Tensile stresses tend to facilitate matrix crack nucleation and propagation in a mode I manner throughout the laminae at the point of initial matrix cracking, thus leading to catastrophic failure. Increasing the confinement ratio from zero in the [±45]12s samples also led to the generation of angled cracks (Fig. 6) in replacement of straight transverse cracks. The crack or shear plane angle, where matrix cracks coalesce and become visible on the sample sides, is evidence of the changing stress field within the plies. The increased confinement loading rotates the shear plane approximately 45° since more out-of-plane stresses are generated. However, angle crack development does not increase beyond a confinement ratio of about 0.5 since out-of-plane stresses are of magnitudes to cause massive interply shearing once one shear plane develops in a sample. In general, biaxial loading on [±45]12s samples involves a relatively broad range of failure mechanisms and demonstrates the increasing influence of out-of-plane stresses with confinement ratio. Transverse crack development and large strain accommodation with uniaxial loading transitions to angled crack development and finally to extremely brittle behavior with greater biaxial loads. For smaller fiber orientation angles, namely [±30]12s, [±15]12s and [±10]12s, uniaxial loading caused brittle behavior. These fiber orientations have large tensile stresses across the uniaxial shear plane (rnormal) that facilitate matrix crack nucleation and propagation, leading to solely brittle behavior. An analyses similar to that discussed above resulted in calculating tensile r22 stress

of 21.3 MPa and 38.0, respectively, for the [±10]12s and [±15]12s laminate configurations. It can be confidently predicted that angle cross-ply samples with fiber orientations below about ± 30° under biaxial loading will fail to brittle extents exceeding their respective uniaxial brittle behaviors. As the sample fiber angle was reduced, failure transitioned from shearing along one major fiber-aligned plane to dominant interply shearing or delamination and less fiber-aligned shear slipping. Decreasing the fiber angle with uniaxial loading is analogous to increasing the biaxial confinement ratio on samples of a consistent fiber orientation angle. Either of these conditions increase the magnitudes of out-of-plane stresses in relation to in-plane stresses and prompt greater sample brittleness. Even in [±30]12s samples with a confinement ratio of 0.5, the large compressive normal stress across the shear plane could not suppress the large out-of-plane shear stress from eliciting failure. However, Fig. 8 (R = 0.50) shows that this configuration failed in segmented delaminations in addition to in-plane shear planes, indicating that the large normal compressive stress may have minimized massive interply shearing. Angle cross-ply laminates have relatively large inplane Poisson ratios mxy compared to angle-plies, exceeding unity for many orientations. This is illustrated in Fig. 11 with angle-ply data. This figure, derived from plane stress and plane strain classical lamination theory (CLT) calculations, shows the large differences of mxy versus the fiber orientation angle for both layup configurations. The assumption that the modulus behavior versus confinement loading is governed solely by the in-plane Poisson ratio, or the amount of lateral expansion restrained, is not valid. This conclusion is drawn since [±45]12s specimens were found to be more sensitive to confinement loading than [±30]12s samples, whereas the latter configuration has a larger Poisson ratio. CLT results from biaxial testing provide new insight into the fiber scissoring phenomenon in the [±45]12s

1.6 ν xy [± ]12s - Angle Cross-plies

1.4

In-plane Poisson Rati

6. Discussion of failure mechanisms in relation to calculated stresses

2113

1.2

ν xy Measured Experimentally

1 0.8

——— Plane Stress from CLT --------- Plane Strain from CLT

0.6 0.4 ν xy [+ ]48 - Angle-plies

0.2 0 0

10

20

30

40

50

60

70

80

90

Fiber Orientation Angle (degrees)

Fig. 11. In-plane Poisson ratio mxy as a function of the fiber orientation.

D. Potter et al. / Composites Science and Technology 65 (2005) 2105–2117

specimens previously investigated for solely uniaxial compression. The combination of large matrix shear stresses, large interlaminar stresses and relatively low fiber stresses, produces a unique mechanism; specifically, it allows fibers within both ply orientations to slip along their respective directions while not allowing generation of enough shear stress in the fibers to prompt catastrophic failure. On the other hand, the lack of crack development prior to ultimate failure and subsequent lateral expansion in the [±30]12s samples indicates that fibers along one orientation reach a critical stress or strain limit to promote instantaneous laminate failure along shear planes that extend through the entire laminate thickness. This is supported by the local stress analysis results, which showed that the axial fiber stress increases rapidly for fiber orientations below about ±35°. This result, in addition to Pagano
s result [31] that the interlaminar shear stress sxz decreases with angle cross-ply orientations below ±35°, shows that another stress becomes influential in the shift toward delaminating failure modes. This is presumably the out-of-plane normal stress rz which is generated by micro-bending of the fibers and becomes significant at small fiber angles where the axial fiber stress is large compared to the matrix stress. In either failure mode however, shearing or delamination, the initial micro-failure mechanism was cleaving of the matrix.

7. Failure laws 7.1. Failure laws based on local failure criterion of matrix and fiber shearing When viewed globally, the failure in all material orientations and confinements was observed to be on one or more shear planes (or faults) that traversed the entire sample thickness. Thus, in the width direction, these failure planes encompassed alternate plies in which fibers were aligned and misaligned with respect to the shear plane. When examined microscopically, the failure planes showed evidence of shearing in both the fiberaligned and fiber-misaligned plies. Shear failure in fiber-aligned plies must be locally governed by the shear failure of the matrix as the fibers are essentially parallel to the failure plane and thus offer no resistance to shear. On the other hand, the failure in the neighboring plies where the fibers intersect the failure plane at a substantial angle must be governed by the shear failure of the fibers as the fibers provide the most resistance to shear in this configuration. Given the level of testing resolution and the catastrophic brittle nature of the failure (except for the [±45]12s samples), it could not be ascertained whether the formation of the terminal shear plane was triggered by the local failure of the matrix in the fiberaligned plies or by fiber shearing in the non-aligned

plies. We show in the following that the experimental data correlated remarkably well to a Mohr–Coulombtype criterion for matrix and fiber shear failure. A general pressure or normal stress–dependent Mohr–Coulomb criterion for either the matrix or the fiber failure can be expressed as s ¼ lðr
r0 Þ þ s0 s ¼ s0 if r 6 r0 ;

if r > r0 ;

ð4Þ

where s is the maximum shear stress in the constituent and r is the stress normal to the maximum shear plane. For polymeric matrix where the hydrostatic stress is known to influence its shear resistance [32,33], r can be also taken as the hydrostatic stress. Further, l, r0 and s0 are material constants. Particularly, s0 is the pure shear strength of the epoxy material, r0 is the critical pressure or the normal stress across the shear plane above which the shear strength will increase linearly with the pressure (or the normal stress), and l is a measure of internal friction. These parameters can be determined by carrying out experiments on pure epoxy or by carrying out compression experiments on angle ply laminates. The model proposed above stems directly from earlier works [34–36] where the compressive strengths of unidirectional carbon/epoxy and glass/resin composites were found to correlate linearly with an increase in the hydrostatic pressure. In these works, however, the failure was triggered by the fiber kinking mechanism. The maximum matrix shear stress for the AS4/3502 [±45°] cross-ply laminates at R = 0 and R = 0.24 and for the AS4/3502 [10°] [30°] [45°] [60°] angled-ply laminates is plotted as a function of the hydrostatic pressure and the stress normal to the shear plane in Figs. 12 and 13, respectively, by open legends. All these samples are believed to have failed by matrix shearing in the fiberaligned plies. The figures show that all data points except the one for the [60°] sample fall rather nicely on a

AS4/3502 Maximum shear stress in matrix (MPa)

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180 150 120 90 60 30 0

-10

10

30

50

70

90

Hydrostatic stress in matrix (MPa)

Fig. 12. Variation of maximum shear stress in the 3502 epoxy matrix corresponding to experimentally measured failure strength as a function of the hydrostatic stress for all samples tested in this work. Note the data depicted by filled legend does not follow the failure law.

D. Potter et al. / Composites Science and Technology 65 (2005) 2105–2117

2115

Maximum shear stress in matrix (MPa)

AS4/3502 180 150 120 90 60 30 0 -10

10

30

50

70

90

110

130

Normal stress across the maximum shear stress plane in matrix (MPa) Fig. 13. Variation of maximum shear stress in the 3502 epoxy matrix corresponding to experimentally measured failure strength as a function of the normal stress across the maximum shear stress plane for all samples tested in this work. Note the data depicted by filled legend does not follow the failure law.

the failure criterion of Eq. (3). Thus, it is clear that the failure in these samples was triggered in the fiber misaligned plies and then spread laterally to encompass the adjacent fiber-aligned plies. Together they formed the width of the fault plane. This failure is exactly like the widely observed and recognized phenomenon of fiber microbuckling in unidirectional composites except that the kink bands of finite widths cannot be formed here because the adjoining plies provide lateral support to the deforming fibers. Thus, the failure is forced on a single shear plane in the fiber-misaligned plies. Apparently as soon as this happens the failure spreads rather easily laterally into the neighboring plies where fibers provide no resistance to lateral shear crack growth. The three material constants should be obtained from independent compression experiments on individual fibers. Unfortunately, the individual fiber experiments are difficult to perform. The only parameter available is the s0 value for the AS4 fiber which is about 210 MPa [37]. In the absence of these tests, the user

bilinear curve consistent with the Mohr–Coulomb criterion of Eq. (3). This indicates that the shear strength of the epoxy is linearly pressure-dependent. The three material constants needed to define the failure law can be obtained by performing three experiments on pure epoxy or on angle ply laminates. The maximum matrix shear stress for the remaining AS4/3502 samples, which are marked by filled legends in Figs. 12 and 13, does not follow the trend. Thus, for these samples, the matrix shear resistance was not reached at the time of sample failure. Since the failure was still seen in these samples, it was explored if the failure and the formation of the failure plane was triggered by shear failure of fibers in the fiber-misaligned plies and then spread laterally to shear the adjacent fiber-aligned plies. To this end, the local stresses in the fibers were determined from which the maximum shear stress and the stress normal to the maximum shear plane were calculated. These latter stresses are plotted in Fig. 14, which shows a remarkable correlation consistent with

AS4/3502

Maximum shear stress in fibers [MPa]

900

600

300

0 0

200

400

600

800

1000

Normal stress across the maximum shear stress plane [MPa]

Fig. 14. Variation of maximum shear stress in the fiber corresponding to experimentally measured failure strength as a function of the normal stress across the maximum shear stress plane.

D. Potter et al. / Composites Science and Technology 65 (2005) 2105–2117

Maximum Shear Stress in the Composite [MPa]

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AS4/3502

500 400 300 200 100 0 0

100

200

300

400

500

Normal stress across the maximum shear stress plane [MPa] Fig. 15. Variations of maximum shear stress calculated at the laminate level at sample failure as a function of normal stress across the maximum shear stress plane.

can still define the failure law by carrying out three experiments of his choice and then apply the law for more general loading. Thus, whether the failure is triggered by matrix failure in the fiber-aligned plies or fiber failure in the fiber-misaligned plies, the overall failure takes the form of similar-looking fault planes. Even though the experimental techniques were unable to resolve this fast-occurring phenomenon, the model appears to sort it out rather nicely. 7.2. Failure law based on overall laminate properties Rather interestingly, the failure data for all the above samples in Figs. 12–14 fell on the same curve when the laminate-level shear stress for each sample was plotted as a function of either the laminate-level stress normal to the shear plane (Fig. 15) or the laminate-level hydrostatic pressure (not shown). This remarkable correlation provides the designer a rather convenient failure law in which the overall compressive failure of the laminate under any material orientation and biaxiality ratio, whether triggered locally by the matrix or fiber failure, can be determined by knowing only three material constants, l, r0 and s0. These material constants are now defined at the laminate level and can be determined by carrying out three tests of user
s choice. This correlation at the laminate level is purely coincidental but it can be very useful to a designer as it is rather simple to implement. Even with its empirical flair the designer can be assured that when he uses this laminate level criterion then the actual failure is at the constituent level, which in turn, is governed by one of the applicable mechanism-based failure laws shown in Figs. 12, 13 or 14. That is, the design is not purely empirical. The question however arises why the laminate-level stresses correlate as they do in Fig. 15? The simple reason could be that the parameters plotted in Fig. 15 can be represented mechanistically in terms of the local fiber

and matrix shear stresses and the normal pressure on the maximum shear plane that are shown to correlate in Figs. 12–14.

8. Conclusions Failure mechanisms and stress–strain behaviors were investigated for [±30]12s and [±45]12s graphiteepoxy (AS4/3502) laminates under in-plane biaxial compression by using a cruciform biaxial test frame and microscopy of load-interrupted samples. The loading confinement ratio R was varied from 0.24 to nearly 1.0 to measure the sensitivity of sample failure mechanisms and stress–strain behavior to different stress states. Biaxial compression testing with increasing confinement ratios increased the specimen stiffnesses and strengths. The failure modes transitioned from the uniaxial failure mode of in-plane shearing to out-of-plane shearing and massive delamination with increasing R. The shear failures were on one or more planes that traversed the entire sample thickness and thus encompassed alternate plies in which fibers were aligned and misaligned with respect to the shear plane. The local failure was found to be triggered either by the matrix shear in fiber-aligned plies or fiber-shear in the fiber-misaligned plies. The standard classical laminate theory in combination with the standard rule of mixture theory for composites was used to calculate the maximum matrix and fiber shear stresses. The relationship between material constituent stresses and their corresponding failure mechanisms was discussed. The failure data conformed nicely with a Mohr–Coulomb-type shear failure law. Most interestingly, both types of failures correlated rather remarkably on a single shear failure law when the stresses were smeared and expressed on the scale of the laminate. This latter failure law should have widespread use because of its simplicity.

D. Potter et al. / Composites Science and Technology 65 (2005) 2105–2117

Acknowledgement This work was sponsored in part by the Air Force Office of Scientific Research, USAF, under grant number F49620-97-1-0450, and by the National Science Foundation under grant number CMS 9696096.

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