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Composites Science and Technology

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A unit circle failure criterion for carbon fiber reinforced polymer composites

Stephen W. Tsai a, Jose Daniel D. Melo b' *

a Department of Aeronautics & Astronautics, Stanford University, Durand Building, 496 Lomita Mall, Stanford, CA 94305-4035, USA b Department of Materials Engineering, Federal University of Rio Grande do Norte, Campus Central, 3000, Natal, RN, 59078-970, Brazil

ARTICLE INFO

ABSTRACT

Article history:

Received 19 July 2015

Received in revised form

26 November 2015

Accepted 16 December 2015

Available online 19 December 2015

Keywords:

Mechanical properties Failure criterion Laminate theory Unit circle

A novel invariant-based approach to describe elastic properties and failure of composite plies and laminates has been recently proposed in the literature. An omni strain failure envelope has been defined as the minimum inner failure envelope in strain space, which describes the failure of a given composite material for all ply orientations. In this work, a unit circle is proposed as a strain normalized failure envelope for any carbon fiber reinforced polymer laminate. Based on this unit circle, a failure envelope can be generated from the longitudinal tensile and compressive strains-to-failure of a unidirectional ply. The calculated failure envelope was found in good agreement with experimental data published in the well-known World Wide Failure Exercise.

© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND

license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Composite materials can offer a unique combination of properties that makes them very attractive to many applications. While in some applications they are a replacement for metallic materials, in others they can be the solution for delivering the required properties. The inherent anisotropy and complex failure mechanisms of these materials are fundamental to their superior properties and design flexibility, but make mechanical characterization rather complex, costly and time consuming. For unidirectional plies, there are four independent stiffness parameters to be measured; i.e., longitudinal, transverse and shear moduli and Poisson's ratio; and five strengths; i.e., longitudinal and transverse tensile and compressive, and shear. Thousands of coupons have been required to generate design allowable for aeronautic structures. Therefore, approaches leading to the characterization of composite materials a reduced number of tests are of significant practical interest, not only for the aeronautic industry, but also for the introduction of these materials in new applications.

Collaborative efforts such as the World Wide Failure Exercise [1] have demonstrated that failure criteria capable of predicting

* Corresponding author. E-mail addresses: stsai@stanford.edu (S.W. Tsai), jddmelo@gmail.com (J.D.D. Melo).

ultimate strength of composite laminates under biaxial loading remain a challenge. Typical failure criteria currently used for fiber-reinforced composites include Tsai-Wu [2], Hashin [3], Puck [4], Christensen [5,6], among others considered [7].

There are many physically based, multi-scale and fracture mechanics-based failure theories with claimed improvements in accuracy as compared to other traditional failure theories. Micro-mechanics relationships have been considered to predict lamina properties [8] with multi-scale approaches [9—11 ]. Physically based failure criteria consider specific equations to describe each failure mode: fiber failure and matrix failure [4]. Other criteria based on damage mechanics describe the initiation and evolution of damage leading up to failure [12,13] in some cases with remarkable agreement with experimental data [14]. However, while improvements in accuracy are possible with increasing complexity, these approaches can be complex to be implemented in design.

In a recent work published in the literature, an invariant-based approach to describe elastic properties and failure of composite plies and laminates was proposed [15]. Carbon fiber reinforced plastics were shown to share common stiffness properties if they are normalized by their respective trace of the plane stress stiffness matrix, Tr [Q]. Thus, a "master ply" was defined using universal trace-normalized stiffness components and the trace of the plane stress stiffness matrix becomes the only material property needed for the elastic characterization. In addition, an invariant "omni strain" failure envelope was proposed as the minimum inner

http://dx.doi.org/10.1016/j.compscitech.2015.12.011

0266-3538/© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Fig. 1. a) Superposition of LPF envelopes for T700/2510 using Tsai-Wu with Fxy* = -1/2 and Em* = 0.15. b) Omni strain LPF envelope for T700/2510.

Table 1

Intact and degraded properties of CFRPs (Em* = 0.15) [17].

Material Ex Ey Vx Intact (Degraded) Es X X ' Y Y ' S

IM7/977-3 191 9.94 (2.09) 0.35 (0.053) 7.79(1.23) 3250 1600 62 98 75

T300/5208 181 10.30 (2.51) 0.28 (0.042) 7.17 (1.48) 1500 1500 40 246 68

IM7/MTM45 175 8.20(1.77) 0.33 (0.050) 5.50 (1.00) 2500 1700 69 169 43

IM7/8552 159 8.96(1.72) 0.32 (0.048) 5.50 (0.95) 2501 1700 64 286 120

T300/F934 148 9.65 (1.89) 0.30 (0.045) 4.55 (1.00) 1314 1220 43 168 48

AS4/PEEK 134 8.90 (2.16) 0.28 (0.042) 5.10(1.21) 2130 1100 80 200 160

T700/2510 126 8.40 (1.79) 0.31 (0.046) 4.23 (0.96) 2172 1450 49 199 155

Note: Ex, Ey, Es, and vx are the longitudinal, transverse and shear moduli and major Poisson's ratio, respectively; X, X', Y, Y' and S are the longitudinal and transverse tensile and compressive and shear strengths, respectively. Elastic moduli in (GPa) and strengths in (MPa). Fxy* = -0.5 for intact plies and Fxy* = -0.075 for degraded plies. Em = 3.40 GPa

for all materials. Degraded moduli are calculated using micromechanics relations.

Fig. 2. Omni strain LPF envelopes for two CFRPs based on Tsai-Wu (solid line) and maximum strain (dashed line). a) IM7/977-3; b) T700/2510.

envelope in strain space, which defines the first-ply-failure (FPF) of a given composite material for all ply orientations [15].

The first-ply-failure (FPF) "omni strain" represents the most conservative design where all plies are intact, without any micro cracks. However, laminates can continue to carry load beyond FPF.

In fact, first-ply-failure is difficult to be clearly defined in coupon tests. For CFRP hard laminates under uniaxial tension, the stressstrain curve is linear up to failure with no easily observable kink or deviation from linearity to indicate FPF. It is therefore important to define and predict the continued load-carrying capability of

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 E,

Fig. 3. Unit circle failure envelope.

laminates beyond FPF. This approach would be identical to that of metals.

The present work extends the concept of "omni strain" FPF envelopes to the ultimate envelope. A unit circle is proposed as a failure envelope in strain space to all carbon fiber reinforced composites. Based on the unit circle, an envelope is generated from the longitudinal tensile and compressive strains-to-failure of a unidirectional ply, only. For other fiber-reinforced materials, the omni strain last-ply failure envelope would be equally valid and easy to use for the prediction of failure.

2. Omni strain last-ply failure envelope

The construction of the omni strain last-ply failure (LPF) envelope follows the same procedure described for the omni strain FPF [15]. Based on a given failure criterion, such as Tsai-Wu or maximum strain, failure envelopes are generated in strain space for a laminate with all ply orientations, from 0 to 90°. While for the omni strain FPF failure envelopes are obtained using intact ply properties, omni strain LPF envelopes are defined using degraded ply properties.

Degraded ply properties is a widely acceptable approach to account for the decreased matrix dominated stiffness components and increased failure strains, due to the presence of micro-cracks. Continuum damage mechanics has been proven effective in predicting stiffness degradation of composite laminates due to presence of transverse cracks [16]. Considering a matrix degradation factor (Em*), micromechanics is normally used to determine the loss of transverse and shear moduli, Poisson's ratio and the interaction term in the failure criterion (Fxy*). Strengths remain intact

In a previous study [17], the influence of matrix degradation factor on longitudinal tensile and compressive ultimate strains was investigated for various CFRPs and based on various failure criteria. It was observed that all criteria considered result in nearly the same predicted failure strains as degradation advances and the degradation factor approaches zero. A degradation factor of 0.15 has been recommended for the LPF prediction of CFRPs.

The use of strain space to generate omni strain failure envelopes

is essential to allow superposition of the individual failure envelopes of the constituent plies since their shapes are independent regardless of the presence of other plies. Thus, the omni strain failure envelope for the laminate is just the inner envelope defined by all individual envelopes superposed considering the all ply orientations in the laminate, from 0 to 90°.

Fig. 1 shows the LPF omni strain envelope for T700/2510. In this case, the failure envelopes for the individual plies were generated using Tsai-Wu failure criterion with normalized interaction term Fxy* = -1/2 and matrix degradation factor Em* = 0.15. Simultaneous ply degradation was considered with degraded material properties calculated according to relations presented in a previous work [17]. Intact and degraded properties for this and other CFRPs are shown in Table 1.

The individual envelopes in Fig. 1a) are defined in normal strain space while the omni envelope in Fig. 1b) is given in principal strain space. Since all ply angles are covered, the omni envelope is invariant, i.e., it remains the same for all angles of transformation. Then, the normal strain coordinates e1° and e2° can be replaced by their principal strain components ei° and en°. For a state of strain with non-zero shear, the principal strains can be determined by rotation of the reference coordinate system. The use of omni strain makes this conversion to principal strain simple and the failure envelope a material property.

The inner LPF envelope shown in Fig. 1 is controlled by [0] and [90] plies. This is observed for all CFRPs [17]. Shear strength would have to be significantly smaller for other plies to become controlling and no CFRP has been found with properties to meet this condition. Thus, CFRP laminates that contain these two plies will have the omni strain failure envelope controlled by them. If the laminate does not contain any of these two plies, the LPF omni envelope will be conservative. For other materials with a lesser degree of anisotropy such as glass fiber reinforced polymers, the controlling plies may be different, in particular for the positive strain to failure. Nevertheless, the omni strain envelope can still be used since it does not assume failure controlled by the [0] and [90] plies.

3. Unit circle failure envelope

Because the omni LPF envelope for CFRPs is controlled by [0] and [90] plies, the failure envelope for this class of materials can be further simplified. Omni LPF strain envelopes for two CFRPs - IM7/ 977-3 and T700/2510 - are shown in Fig. 2 with anchor points based on an interactive failure criterion (Tsai-Wu) and a non-interactive failure criterion (maximum strain). The properties for these materials are provided in Table 1.

The envelopes shown in Fig. 2 were calculated using a matrix degradation factor (Em*) of 0.15. The omni LFP envelopes for interactive and non-interactive failure criteria become coincident when the matrix degradation factor approaches zero. Although the envelopes shown for interactive and non-interactive failure criteria are different, the same anchors apply to both: one tensile and one compressive failure strain. The shear strain-to-failure of the laminate can be found by a loading vector along the q-axis, the bisector of the coordinate axes in the 2nd and 4th quadrant. They are also shown in Fig. 2 as solid squares based on Tsai-Wu, a more conservative criterion than maximum strain.

In principle, the tensile and compressive anchor points shown in Fig. 2 cannot be determined directly from experiments since [0] coupons have Poisson's ratio higher than zero. However, considering degraded ply properties, Poisson's ratios are very small (approaching zero) and thus, failure strains determined from uniaxial tests are essentially the same as the anchor points in Fig. 2. Thus, a very simple envelope for CFRPs can be defined based on the

equation of a unit circle if the anchor points are defined as shown in Fig. 3.

Based on the unit circle (Fig. 3), failure envelopes can be derived for any CFRP if the anchor points are multiplied by the respective strains-to-failure ex and eX'. In Fig. 4, failure envelopes for six CRPFs based on the unit circle and on last-ply failure omni envelopes are shown for purposes of comparison.

As shown in Fig. 4, the unit circle envelope is basically inscribed in the omni last-ply failure envelope. However, while the determination of the omni last-ply failure envelope requires the complete characterization of the ply properties - four stiffness parameters and five strengths — the failure envelope in strain space based on the unit circle only requires the strains-to-failure of a [0] coupon measured in tension and compression. This is a great

Fig. 4. Failure envelopes in strain space for CFRPs based on unit circle and omni strain LPF. a) IM7/977-3; b) T300/5208; c) IM7/MTM45; d) IM7/8552; e) T300/F934; f) AS4/PEEK.

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

Fig. 5. a) Unit Tresca-like envelope; b) Omni LPF (solid line) and Tresca-like (dashed line) envelopes for T700/2510.

advantage in terms of specimen preparation, time and cost.

Tresca-like envelopes could also be generated using the normalized strains to failure as anchor points (Fig. 5). However, as shown in Fig. 5, a Tresca-like envelope proved unsafe for most CFRPs in the first quadrant of the strain space (tension—tension). In addition, a circle is single-valued and a smooth function with continuous derivative. It is therefore a simple equation, which can be easily implemented in algorithms for design. Although other options of having smooth functions exist such as power law, the unit circle is also justified because it is limited by the same strain-to-failure of a [0] ply in all directions of the strain space. Therefore, it is invariant with respect to coordinate system rotation.

The unit circle is also applicable for cold-dry and hot-wet conditions. In this case, the [0] specimens must be tested after they have been conditioned. Hot-wet conditions will affect mainly the compressive strain to failure of [0] unidirectional specimens. Thus, the additional compressive tests of these environmentally conditioned specimens may be sufficient in many situations.

4. Envelopes in stress space

Failure envelopes are most commonly presented in stress space. The omni strain LPF and the unit circle are both invariant in strain space for a given material. However, they can represent multiple envelopes in stress space, depending on the laminate. The envelopes in stress space can be obtained using the stress-strain relations for laminate (Equation (1)).

fsig = h[Aj {j

for ij = 1, 2, 6. [Aj] is the laminate stiffness matrix and h is the laminate thickness.

In this case, degraded elastic properties for the ply are used for the calculation of [Aj] to account for matrix cracking prior to failure. Thus, each laminate composition represents one unique envelope in stress space while there is only one unit circle in strain space. In Fig. 6, the unit circle failure envelope for T700/2510 in strain space and failure envelopes various laminates in stress space are presented.

5. Comparison with experimental data

In order to evaluate the accuracy of the proposed envelopes for predicting failure in composite laminates a comparison was made

with published data. First, the failure envelope for quasi-isotropic [0/±45/90]s AS4/3501-6 based on the unit circle was compared with experimental data. Biaxial failure stresses for this laminate under a variety of stress ratios were published in the literature as part of the World Wide Failure Exercise [18]. The ply data used to generate the failure envelope was published in another paper, which is also part of the same study [19]. The mechanical properties for this material needed to generate the unit circle envelope are given in Table 2, which also shows properties for two E-glass/epoxy plies. The unit circle failure envelope for AS4/3501-6 is shown in Fig. 7.

Based on this unit circle shown in Fig. 7, the envelope can be generated in stress space for the give laminate. Considering that the only elastic modulus used as material data was in the fiber orientation (Ex), a simplified procedure was applied for the determination of degraded ply elastic properties. Master or universal ply properties were first used for the calculation of the ply elastic properties based on the elastic modulus (Ex), as described in Ref. [15]. Based on the master ply properties normalized by the trace of the plane stress stiffness matrix, the other elastic constants Ey, vx and Es were determined [15]. Then, a matrix degradation factor (Em*) of 0.15 was multiplied by these three constants while Ex was kept intact. This procedure produces elastic moduli Ey, and Es, which are smaller than those predicted using micromechanics equations [17], but the effect on the failure envelope was found insignificant. In addition, the degradation procedure based on micromechanics equations includes empirical factors, which are not necessarily a better estimation than the procedure used herein. Thus, this procedure can generate a failure envelope in stress space based on the unit circle in strain space using only the elastic modulus in the fiber orientation (Ex). In case all ply elastic constants, fiber volume fraction and matrix modulus are provided, the degraded properties can be calculated using micromechanics equations [17].

Using the laminate stress-strain relations (Equation (1)), the envelope in stress space can be obtained from the unit circle. In Fig. 8, biaxial failure stress envelope for a AS4/3501-6 [0/±45/90]s laminate is shown and compared to experimental data [18].

The correlation between the prediction using the unit circle failure envelope and the experimental data in Fig. 8 is good in the tension-tension quadrant and, to a lesser extent, in the tensioncompression quadrant. For the compression-compression quadrant, a poor correlation between test results and theoretical prediction was observed. However, a large data scatter is also observed

2000 1600 1200 800

co 400

e. o o"

-400 -800 -1200

-1200-800 -400 0 400 800 12001600 2000

cr^MPa)

2000 1600 1200 800 "co 400

-400 -800 -1200

-1200-800 -400 0 400 800 12001600 2000

OilMPa)

Fig. 6. Unit circle failure envelope for T700/2510 in strain space and the envelopes in stress space considering various laminates. a) Unit Circle in strain space; b) Quasi-isotropic [0/ ±45/90]s; c) Hard [07/±45/90]s; Soft [0/±454/90]s; Cross-ply [0/90]s.

in this quadrant. This can be related to buckling which affected the experimental data reducing ultimate strengths. This discrepancy between the theoretical predictions and the experimental data in the compression—compression quadrant was also observed for all current theories evaluated in Ref. [7] and it was attributed to

buckling. Nevertheless, compared to all failure theories considered in Ref. [7], the unit circle gives the smallest difference between the prediction and the experimental data in the compression—compression quadrant.

For E-Glass/epoxy, omni strain LPF failure envelopes were also

Table 2

Mechanical properties of unidirectional laminae [19].

Material Ex Ey Vx Intact (Degraded) Es X X' Y Y' S

AS4/3501-6 126 1950 1480

LY556/HT907/DY063 epoxy 53.48 17.7 (2.16) 0.278 (0.042) 5.83 (1.10) 1140 570 35 114 72

MY750/HY917/DY063 epoxy 45.6 16.2 (2.02) 0.278 (0.042) 5.83 (1.03) 1280 800 40 145 73

Note: Ex, Ey, Es, and vx are the longitudinal, transverse and shear moduli and major Poisson's ratio, respectively; X, X', Y, Y' and S are the longitudinal and transverse tensile and compressive and shear strengths, respectively. Elastic moduli in (GPa) and strengths in (MPa). Fxy* = -0.5 for intact plies and Fxy* = -0.075 for degraded plies. Em = 3.40 GPa for AS4/3501-6 and Em = 3.35 GPa for both E-glass/epoxy materials. Degraded moduli are calculated using micromechanics relations.

-0.02 -0.01 0.00 0.01 0.02

Fig. 7. Unit circle failure envelope for AS4/3501—6.

compared to experimental data. For these materials, unit circle envelope is not used since the controlling plies may vary according to the material properties and are not necessarily [0] as for CFRPs. The E-Glass/epoxy ply data used to generate the failure envelopes

are shown in Table 2. The degraded properties of the two E-glass/ epoxy plies shown in Table 2 were not provided in Ref. [19] and were determined using micromechanics equations for Em* = 0.15 [17].

Two E-glass/epoxy laminates were considered: [±30/90]s and [±55]. The failure data of both laminates under biaxial stress loading are presented in Ref. [18]. The data was obtained using tubes under combined pressure and axial load and combined torsion and axial load. All [±30/90]s specimens tested under internal or external pressure were lined with a flexible adhesive liner and all the data are for final rupture. For the [±55] specimens, some tests were carried out without a flexible plastic liner. These specimens without liner failed typically be weeping and were thus not considered for comparison with the envelope.

The LPF omni envelope for the [±30/90]s laminate shown in Fig. 9 proved conservative in tension—tension and tensioncompression quadrants. For the compression—compression quadrant, one data point suggests unconservative prediction. The same was observed for all failure criteria considered in Ref. [7]. For the [±55] laminate, the shape of the envelope presented in Fig. 10 was found to describe well the experimental data. The envelope proved conservative in all quadrants.

With the exception of the compression—compression quadrant in Fig. 9, the omni failure envelopes in Fig. 9 and in Fig. 10 provided a conservative envelope in all quadrants, rather than an agreement with the measured shape of the failure envelope. The reason for the conservative prediction is the fact that the omni envelopes consider all ply orientations and these laminates have 3 and 2 ply

-1200 H—

-1200 -800 -400 0 400 800 1200

a2(MPa)

a2(MPa)

Fig. 8. Biaxial failure stress for [0/±45/90] s AS4/3501—6 laminate. Experimental data and unit circle failure envelope.

Fig. 9. Biaxial failure stress for [±30/90] (82.8%/17.2%) E-Glass/LY556/HT907/DY063 laminate. Experimental data vs. omni strain last ply failure envelope.

"to q.

I. ° 0

-1000 -500 0 500 1000

a2(MPa)

Fig. 10. Biaxial failure stress for [±55] E-Glass/MY750/HY917/DY063 laminate. Experimental data vs. omni strain last ply failure envelope.

orientations, respectively, and thus, fewer failure modes associated. The unconservative prediction in the compression-compression quadrant in Fig. 9 can be related to structural instability, as previously discussed.

The predictive capability of failure criteria for laminate strength based on ply properties is normally associated with some simplifying assumptions. In this case, the failure envelope in stress space considers a matrix degradation factor and the assumption that all plies degrade simultaneously. However, the failure process in composites is known to be more complicated with interaction between intra-lamina failure and interlamina failure. A more accurate prediction may be possible if a progressive damage scheme is implemented to generate failure envelopes in stress space.

6. Conclusions

In this work, invariant omni last-ply failure envelopes were described for fiber-reinforced composites. A unit circle was proposed as a failure envelope in strain space to all carbon fiber reinforced composites. The circle is based on the fact that, for laminates with all ply angles, last-ply failure envelopes in strain space are anchored by the uniaxial tensile and compressive strains-to-failure. The unit circles are obtained by using the uniaxial tensile and compressive failure strains as appropriate normalizing factors. These values can be directly measured from [0] coupon tests, since the Poisson's ratio of degraded plies approaches zero due to the presence of micro-cracks. Transverse tensile and compressive strengths and shear strength are not needed. Predictions using the failure envelopes were compared with experimental data a good agreement was obtained. The failure envelopes can be further extended for cold-dry and hot-wet conditions if the [0] specimens are tested after they have been conditioned. Ultimately, the unit circle makes failure analysis of carbon fiber reinforced composite laminates very simple to understand and implement. One criterion can be applied to all laminates regardless of ply orientation. With this failure envelope, significant savings in time and simulation cost as well as design and testing can be realized. All [0] panels are easy

to fabricate, produce minimum scrap, and are less prone to uncertainties/defects from the lamination process.

The unit circle failure criterion applies to materials to which the omni LPF strain envelopes are controlled by [0] and [90] plies. This is the case of carbon fiber reinforced composite laminates. For glass fiber reinforced composites, the controlling plies may vary according to the material properties and omni strain LPF failure envelopes are recommended.

Although there is no question about the importance of failure theories able to provide more accurate prediction of composites, criteria able to predict failure using a reduced number of tests are equally important, in particular to industry. Thus, the significance of the unit circle is not its improved accuracy as compared to existing failure theories or the ability to capture the detailed physics associated with the various failure mechanisms, but rather its improved simplicity - as measured by the number of tests required to define the failure envelope — and invariance with respect to ply orientation. In addition, the unit circle offers flexibility to accommodate the variations of environmental conditions corresponding to the state of pristine (trace-based), manufacturing defective, and/or usage damaged.

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