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A unified fatigue life model based on energy method
Composite Structures 75 (2006) 444–450 www.elsevier.com/locate/compstruct
A unified fatigue life model based on energy method M.M. Shokrieh *, F. Taheri-Behrooz Composites Research Laboratory, Mechanical Engineering Department, Iran University of Science and Technology, Tehran 16844, Iran Available online 12 June 2006
Abstract A new unified fatigue life model based on the energy method is developed for unidirectional polymer composite laminates subjected to constant amplitude, tension–tension or compression–compression fatigue loading. This new fatigue model is based on static failure criterion presented by Sandhu and substantially is normalized to static strength in fiber, matrix and shear directions. The proposed model is capable of predicting fatigue life of unidirectional composite laminates over the range of positive stress ratios in various fiber orientation angles. By using this new model all data points obtained from various stress ratios and fiber orientation angles are collapsed into a single curve. The new fatigue model is verified by applying it to different experimental data provided by other researchers. The obtained results by the new fatigue model are in good agreements with the experimental data of carbon/epoxy and E-glass/epoxy of unidirectional plies. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Polymer composites; Fatigue modeling; Energy method
1. Introduction Polymer based composites due to their specific strength and stiffness, resistance to environmental effects; low weight and high fatigue life are widely used by various industries. Usage of these materials as load bearing primary structures and exposing of them into cyclic loading, necessitates the fatigue life study as an important issue. Therefore, in spite of considerable research in this field, many researches are devoting their efforts to understand this phenomenon in more detail. The fatigue life of polymer composites strongly depends on the fiber directions, type of fibers and resins, number of layers, stacking sequence, type of loading, stress ratio, environmental conditions, and frequency of loading. These materials, before and after damage, are anisotropic and therefore the fatigue behavior of them is more complicated than traditional materials. In metals a dominant crack starts and propagates till final failure occurs. The crack propagation in composites is different and should be con*
Corresponding author. Tel./fax: +98 21 749 1206. E-mail address: [email protected] (M.M. Shokrieh).
0263-8223/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.04.041
sidered as a very complicated phenomenon. In these materials a large number of matrix micro-cracks initiates in the early stage of the fatigue loading and then various damage mechanisms like crazing, debonding between fibers and matrix, fiber breakage and delamination will happen. Therefore, the damage is not concentrated on the tip of a specific crack and is widely dispread [1]. Also, in the process of damage propagation in composites, the mechanical properties of unidirectional layers are decreased, the stresses are redistributed and the stress concentrations are decreased. By considering all theses complexities, understanding the step by step behavior of materials during the fatigue process is needed in order to predict the fatigue life of composites. In order to understand fatigue as a complicated phenomenon in composites, different methods are established by various authors. A comprehensive review on this topic is available in the literature [2–4]. Although fatigue behavior of composites is different from those of metals, however, many of the fatigue models are developed based on the well known S–N curves method. In this method a large number of experimental data is needed and the real damage mechanisms such as fiber and resin failure are not considered.
M.M. Shokrieh, F. Taheri-Behrooz / Composite Structures 75 (2006) 444–450
Hashin and Rotem [5] presented one of the earliest fatigue model based on S–N curve method. They presented the S–N curve by testing off-axis glass/epoxy specimens with different angles under a specific stress ratio. Then by replacing the fatigue strength as a function of the fatigue life instead of the static strength in a failure criterion, they predict the fatigue life of a unidirectional composite under a uni-axial loading condition in different directions. A model presented by Sims and Brogdon [6] is also similar to the model of Hashin and Rotem. Ellyin and El Kadi [7] showed that the strain energy density can be used as a fatigue criterion. In their model the total elastic strain energy density applied to the material in the first cycle is correlated to the fatigue life. Later, Ellyin and Fawaz [8] proposed a semi-log linear relationship between applied cyclic stress and number of cycles to failure. By having the S–N curve for a defined lay up as a primary block, their model is able to predict the fatigue life of the same lay up at different directions. Although Philippidis and Vassilopoulos [9] showed that the model of Ellyin and Fawaz [8] is very sensitive to the reference line and the predicted results of this model are not suitable for cylindrical specimens. Recently, Kawai [10] by normalization of the applied stress with respect to the static strength and defining the fatigue strength ratio presented a model for predicting the fatigue life of an orthotropic layer under positive and negative stress rations and arbitrary directions. Plumtree and Cheng [11] by defining the critical planes concept for unidirectional composites in planes parallel to the fiber expressed that the damage occurs due to the normal and shear components of stress in these planes. They used Smith–Watson–Topper (SWT) [12] parameter in their model to developed a new fatigue model for composites that is independent of stress ratio. Later, Plumtree and Petermann [13] employed a strain energy density concept in critical planes and presented another criterion for fatigue life simulation of composites. Both models of Plumtree and his colleagues [11,13] were not able to predict the fatigue life of the unidirectional composites loaded in the fiber direction. It is well known that the mode of failure changes from fiber mode to matrix mode by changing the direction of fibers [14]. However, their models [11,13] are established only for the matrix failure mode and for the angles lower than the transient angle the predicted results by their models expected to be in less agreement with the experimental data. In the present research a new fatigue failure criterion based on the static failure criterion of Sandhu and his colleagues [15] is established. This criterion considers the matrix and fiber modes of failure in one equation and consequently is capable of predicting the fatigue life of off-axis composites at angles less than the transient angle. The concept of the transient angle is explained in [14]. The proposed equation is in a normalized form and uses static failure value of Sandhu (sum of normalized strain energy density components) instead of the stress components as
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a new model of fatigue failure. This new fatigue failure criterion provides the capability of prediction of fatigue life for an orthotropic layer under tension–tension and compression–compression fatigue loading with arbitrary fiber angles. 2. Theoretical background Cumulative fatigue damage for composite materials is studied by various authors [16,17]. Initiation of crack depends on various parameters such as state of stress, stress ratio and fiber direction. In general, crack initiation occurs on the first cycles in the matrix, perpendicular to the fiber direction and develops as loading continues. Hahn [18] investigated the fatigue of unidirectional carbon/epoxy composites in directions of (h = 0°, 10°, 20°, 30°, 45°, 60°, and 90°) by well known S–N curves. Transition angle in tension for this material is obtained equal to 9.3° by using Eq. (1), hence, he only observed matrix failure mode according to modified Hashin criterion [14] in his test results. rffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffi Xt 56:9 htransition ¼ arctan ¼ arctan 1836 Yt ¼ 9:98
ð1Þ
where Xt and Yt are the tensile strength in fiber and matrix directions, respectively. For the matrix failure mode (fiber angles between transient angle and 90°) of a unidirectional off-axis lamia, matrix cracking occurs in the early stage of the loading and the matrix damage is a process of multiple initiation and coalescence of micro-cracks rather than the propagation of one single crack. Also, matrix cracking serves as a source of other damage due to stress singularities at the crack tips. The following macro-crack stage is relatively short because of crack coalescence. Occasionally, it is not detected before failure occurs, presenting the so-called sudden-death behavior [5]. In addition, matrix cracks are constrained by the fibers and confined to a direction parallel to the fibers. The fiber failure mode (fiber angles between 0° and transient angle) is different from the matrix failure mode as explained above. The matrix cracks in early stage of loading are integrated together as loading continues until they encounter with fibers and leads to fiber–matrix interface. By continuing loading, damage progress and cause fiber matrix debonding and leads to local fiber failure. This stage expected to follow a gradually material degradation regime, finally catastrophic fiber failure occurs. Plumtree and his colleagues [11,13] have developed their criterion based on the critical plane concept and presented their new fatigue parameter W* as Eq. (2): max max þ rmax W ¼ kðrmax 2 e2 6 e6 Þ 1R2
ð2Þ
where k ¼ 2 , r2, r6 and e2, e6 are stress and strain components in material directions and R is the stress ratio.
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r1 e1 r2 e2 r6 e6 þ þ ¼1 X e1u Y e2u Se6u
Fig. 1. Material (on-axis) 1–2 and loading (off-axis) x–y directions on an orthotropic layer.
Based on Eq. (2) only transverse normal stress and inplane shear stress in fracture plane are responsible for fatigue failure in all fiber load directions between 0° and 90°, while modified Hashin failure criterion has shown that in angles less than transition angle stress components in fiber direction and in-plane shear stress components are responsible for failure. Fig. 1 shows the material (on-axis) and loading (off-axis) directions on an orthotropic layer. Predicted results by different static failure criteria, especially modified Hashin criterion [14], also the present authors experiences in static and fatigue testing of glass/ epoxy and carbon/epoxy unidirectional composites identified that failure planes are not similar to that as proposed by Hahn [18] in all load fiber angles. Therefore in the following section a new energy based fatigue criterion is developed to consider all failure modes of a unidirectional laminate as matrix cracks, fiber–matrix interface debonding and fiber failure in a unique fatigue criterion. This criterion is capable of predicting the fatigue life of a unidirectional lamina at various fiber load angles and stress ratios. 3. Normalized fatigue failure criterion When S–N curves are employed to predict the fatigue life, maximum or amplitude of stresses are selected as fatigue measure and plotted vs. fatigue life. It has been shown by many researchers [5,18] that selecting a suitable fatigue model can strongly decrease the number of tests that are needed to forecast the fatigue life. In this research it has been shown that selecting a fatigue model based on elastic energy density components in material directions, is a good way of showing fatigue test results. To the best of our knowledge, the first static failure energy criterion based on strain energy was developed by Sandhu [15] for composite materials. This criterion, without considering the modes of failure, employs all stress and strain components in a normalized interactive equation to predict the failure. Sandhu expressed this criterion for an orthotropic laminate under in-plane loading as following:
ð3Þ
where r1, r2, r6 and e1, e2, e6 are stress and strain components in material directions, X,Y and S are maximum static strengths and e1u, e2u and e6u are maximum static strains in material directions. The advantage of this criterion in comparison with the stress based criteria is that this criterion uses both stress and strain to predict the failure. In this research, the magnitude of static failure of Sandhu criterion (sum of normalized strain energy density components) in each state of stress is introduced as a unified fatigue model for various fiber orientation angles and different stress ratios. The new fatigue model in the on-axis coordinates system can be written as follows: DW ¼ DW I þ DW II þ DW III ¼
Dr1 De1 Dr2 De2 Dr6 De6 þ þ X eu1 Y eu2 Seu6
ð4Þ
where D before a symbol indicates its range. The new fatigue model DW* represents a sum of strain energy densities contributed by all stress components in material directions. It should be mentioned here that in tension–tension loading X = Xt, Y = Yt and in compression–compression loading X = Xc, Y = Yc. Under cyclic fatigue loading, stress amplitude (state of stress) is always less than the maximum static strength, thus, the right side of Eq. (1) never reaches the unity except in tension-zero loading condition. The new fatigue criterion takes the value of static failure criterion in each state of stress as a new fatigue model. Total strain energy under cyclic loading can be shown by Eq. (5): 1 DW ¼ ðDr1 De1 þ Dr2 De2 þ Dr6 De6 Þ 2
ð5Þ
Normalized strain energy density contributed by the stress component along the fiber direction may be written as Eq. (6): DW I ¼
1 ðr1 max e1 max r1 min e1 min Þ X e1u
ð6Þ
Nonlinear response of unidirectional composites in matrix and shear directions is ignored in this model. Therefore the stress–strain response is assumed to be linear in material directions. X ¼ E1 e1u ! e1u ¼
X E1
and
r1 max ¼ E1 e1 max ! e1 max ¼
r1 max E1
ð7Þ
By substituting Eq. (7) into Eq. (6) it is concluded: DW I ¼
1 2 1 ð1 þ RÞ 2 ðDrÞ ðr r21 min Þ ¼ 2 X 2 1 max X ð1 RÞ
where R = rmin/rmax is the stress ratio.
ð8Þ
M.M. Shokrieh, F. Taheri-Behrooz / Composite Structures 75 (2006) 444–450
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The on-axis stresses are found by transforming of the off axis stresses to the on axis coordinates using a suitable transformation rule [19]. For the uniaxial case rx 5 0 and ry = rs = 0, stress components are obtained as follows:
where k and a are material constants and are independent of the stress ratio and fiber orientation. These two constants are obtained for each material by using one set of fatigue test data in an arbitrary stress ratio and a fiber orientation.
r1 max ¼ rx max cos2 h
4. Experimental verification ð9Þ
By substituting Eq. (9) into Eq. (8) one may be obtained: DW I ¼
1 ð1 þ RÞ 2 ðDrx Þ ðcos4 hÞ X 2 ð1 RÞ
ð10Þ
where h is the fiber orientation angle. Similarly the strain energy densities in matrix and shear directions are: 1 ð1 þ RÞ ðDrx Þ2 ðsin4 hÞ and Y 2 ð1 RÞ 1 ð1 þ RÞ ðDrx Þ2 ðsin2 h cos2 hÞ ¼ 2 S ð1 RÞ
DW II ¼ DW III
ð11Þ
By substituting Eqs. (10) and (11) into Eq. (4), a complete form of new fatigue model is obtained as: DW ¼ DW I þ DW I þ DW III 4 ð1 þ RÞ sin4 h sin2 h cos2 h 2 cos h ðDrx Þ ¼ þ þ ð1 RÞ X2 Y2 S2 ð12Þ At this point it should be mentioned that Eq. (12) is developed for different fiber load angles under positive stress ratios (R P 0) for both tension–tension and compression–compression cyclic loading. Whenever fiber and loading directions are coinciding together the first term of Eq. (12) is responsible for failure and is used as a new fatigue model. If the fiber orientation angle is perpendicular to the loading direction third term of the Eq. (12) is responsible for fatigue failure and used as a new fatigue model. Similarly in degrees less than transition angle failure is controlled by first and third terms and in degrees upper than transition angle second and third terms of Eq. (12) are responsible for failure. Therefore this new fatigue model is capable to take into account the effects of all different modes of failure of an orthotropic lamina. Ellyin and El Kadi [7] stated that fatigue life is related to the total input energy through a functional relationship, DWt = g(Nf). They have suggested a power law type equation of the form Eq. (13) to predict the fatigue life. DW t ¼ kN af þ C
ð13Þ
where k, a and C are material constants. k and a are functions of stress ratio and fiber direction. Here we rewrite Eq. (13) by substituting our proposed new fatigue model in the left side and setting the parameter C = 0 as: DW ¼ kN af
ð14Þ
In the previous section a new fatigue model is proposed to simulate the fatigue life of unidirectional composite materials for various fiber angles under different state of stresses and stress ratios. The inputs for the proposed model are ultimate strengths in material directions under tension and compression monotonic loadings and one set of fatigue test data to prescribe the material constants in Eq. (14). In order to verify the new fatigue model, it is applied to the available experimental data to show its capability in predicting of the fatigue life. Four different sets of experimental data available in the literature [5,10,18,20] are selected and examined to evaluate the capability of the new fatigue model. 4.1. Fatigue life prediction for unidirectional carbon/epoxy composites Two sets of experimental data of unidirectional carbon/ epoxy composites are used to verify the new fatigue model. The first set is the experimental data provided by Kawai [10] for different values of fiber angles and different positive stress ratios. The original fatigue data of a unidirectional carbon/ epoxy (T800H/2500) composite under the stress ratio of R = 0.1 and load frequency of 10 Hz is presented in Fig. 2. Fig. 2 clearly shows that the fiber orientation is an important parameter in fatigue life of a unidirectional ply. The new fatigue model data plotted in Fig. 3 using a log-log coordinate system. As shown all data points are distributed in a relatively narrow band and are nearly independent of fiber orientation. The least square method is
10000 Experimental data (Kawai and Suda) Carbon/Epoxy (T800H/2500) f=10Hz, R=0.1
Maximum stress (MPa)
r2 max ¼ rx max sin2 h r6 max ¼ rx max cos h sin h
0 15 45
10 30 90
1000
100
10
1
10
100
1000
10000 100000 1000000 10000000 1E+08
Number of reversals (2Nf)
Fig. 2. Comparison of predicted values with experimental results of Ref. [10], R = 0.1.
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10 0 30
10 45
Experimental (Awerbuch-Hahn), Carbon/Epoxy, AS/3501-5A, R=0.1, f=18Hz
15 90
ΔW*
ΔW*
Experimental (Kawai and Suda) Carbon/Epoxy (T800H/2500) f=10Hz, R=0.5
0.1
1
100
10000
1000000
30
1000
10000
100000
1000000
Fig. 5. Correlated fatigue lives for various fiber orientations [18].
with respect to the number of reversals to failure R = 0.1 [10].
applied to data points to find the best fit relationship to all experimental data in Fig. 3. All data points are collapsed to a master straight line expressed by the Eq. (15): DW ¼ 0:8928ð2N f Þ0:09717
ð15Þ
The predicted results using Eq. (15) are shown in Fig. 2. Solid lines show predicted fatigue life by using Eq. (15) and points show experimental data. As shown in this figure the predicted results are in good agreements with the experimental data. To demonstrate the capability of new fatigue model in predicting fatigue life for different stress ratios, Eq. (15) is employed to predict the fatigue life for stress ratio of R = 0.5 and various fiber orientations. Fig. 4 shows experimental data are in a good agreement with the predicted results by the Eq. (15). The second set of experimental data is selected to evaluate the new fatigue model, is provided by Awerbuch and Hahn [18] on unidirectional carbon/epoxy composites with fiber orientation angles (h = 0°, 10°, 20°, 30°, 45°, 60° and 90°) and stress ratio of R = 0.1. Comparison between predicted results by new fatigue parameter and experimental date are shown in Figs. 5 and 6. The best straight line fitted to these experimental data by least squares method is to be: DW ¼ 1:028ð2N f Þ0:1211
ð16Þ
100000 Experimental data (Kawai and Suda) Carbon/Epoxy (T800H/2500) f=10Hz, R=0.5
Maximum stress (MPa)
90
1
100000000
0
10
15
30
45
90
10000
1000
100
10
100
1000
10000
100000 1000000 10000000 100000000
Number of reversals (2Nf)
Fig. 4. Comparison of predicted values with experimental results R = 0.5 [10].
100000
Maximum stress (MPa)
DW*
1
20
60
Number of cycles (Nf)
Number of reversals (2Nf)
10
10
45
1
0 100
Fig. 3.
0
10000
Experimental data (Awerbuch-Hahn), Carbon/Epoxy, AS/3501-5A, R=0.1, f=18Hz
0
10
20
45
60
90
30
1000
100
10
1 100
1000
10000
100000
1000000
Number of cycles (Nf)
Fig. 6. Comparison of predicted values, Eq. (16), with experimental results of [18].
Figs. 2–6 confirm that the fiber orientation dependence of the off-axis fatigue data as well as the stress ratio dependence can be eliminated using the new fatigue model Eq. (12). As it is mentioned, the inputs of this new fatigue model are static strengths in material directions and one set of fatigue data in an arbitrary stress ratio and fiber load angle. 4.2. Fatigue life prediction for unidirectional E-glass/epoxy composites Two sets of experimental data of unidirectional E-glass/ epoxy composites are used to verify the new fatigue model and show the capability of the model in predicting fatigue life of different materials. The first set of experimental data is performed by El Kadi and Ellyin [20] on 3M type 1003 unidirectional E-glass/epoxy composites for various fiber orientations (h = 0°, 19°, 45°, 71° and 90°), different stress ratios (R = 0 and 0.5) and a loading frequency of 3.3 Hz. Fig. 7 shows the original experimental data of R = 0 and predicted results by the new fatigue model (Eq. (17)). New fatigue model (Eq. (12)) is applied to the experimental data in Fig. 7 and all data points are collapsed to a single curve as depicted in Fig. 8.
M.M. Shokrieh, F. Taheri-Behrooz / Composite Structures 75 (2006) 444–450 1
10000 0
Maximum stress (MPa)
Experimental data (Ellyin-ElKadi) E-glass/Epoxy f=3.3Hz, R=0.5
19
71
45
Experimental data (Hashin and Rotem) E-glass/Epoxy, R=0.1
90
100
1
100
10000
5
10
15
30
60
ΔW*
1000
10
449
1000000
0.1
100000000
Number of reversals (2Nf)
0.01 100
Fig. 7. Comparison of predicted values with experimental results of [20], R = 0.5.
1000
10000
100000
1000000
Number of cycles to failure (Nf)
Fig. 10. Correlated fatigue lives for various fiber orientations [5].
10
1000 0
19
71
90
1
0.1
Experimental data (Hashin and Rotem) E-glass/Epoxy, R=0.1
45
Maximum stress (MPa)
ΔW*
Experimental data (Ellyin-ElKadi) E-glass/Epoxy f=3.3Hz, R=0.5
10
100
1000
10000
100000 100000
1E+07
1E+08
10 60
15
100
10 100 1
5 30
1000
10000
100000
1000000
Number of cycles to failure (Nf)
Number of reversals (2Nf)
Fig. 8. Correlated fatigue lives for various fiber orientations Ref. [20], R = 0.5.
The best fit to the experimental data is obtained by using least square method is as follows: DW ¼ 0:7583ð2N f Þ
0:07762
ð17Þ
Eq. (17) is also applied to the existing data [20] and other stress ratio (R = 0), shown in Fig. 9. The obtained results are in good agreement with the experimental data and show that the model is capable of predicting life for various stress ratios and fiber orientations.
Maximum stress (MPa)
10000 Experimental data (Ellyin-ElKadi) E-glass/Epoxy f=3.3Hz ,R=0
0
19
71
90
45
1000
100
10 1
100
10000
1000000
100000000
Number of reversals (2Nf)
Fig. 9. Comparison of predicted values with experimental results of Ref. [20], R = 0.
Fig. 11. Comparison of predicted values, Eq. (18), with experimental results of [5].
The second set of experimental data is performed by Hashin and Rotem [5] on the off-axis E-glass/epoxy unidirectional composites with fiber directions of (h = 0°, 5°, 10°, 15°, 20°, 30° and 60°) and stress ratio of R = 0.1. Predicated results by using the Eq. (18) along with experimental data are shown in Figs. 10 and 11. Results show good correlation between experimental data and the model (Eq. (18)). The best fit to the experimental dada is obtained by the following equation: 0:1402
DW ¼ 0:3675ð2N f Þ
ð18Þ
As Figs. 10 and 11 illustrate predicted fatigue life by new fatigue model are in good agreement over the range of (0°– 90°) with experimental results. While criteria based on the critical planes concept [11,13] are not capable to predict life for 0 and 90 degrees fiber directions. Also the predicted life by these criteria [11,13] for angles less than the transition angle (i.e., 5°) are not matched with the experimental data. This problem comes from there that the models based on critical planes concept [11,13] does not take into account the effect of transition angle; consequently the part of input energy which is absorbed by the fibers is omitted. Obtained results confirm that the new fatigue model Eq. (12) becomes a unified strength model to eliminate the effects of positive stress ratios as well as the fiber
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orientation on the off-axis fatigue behavior of unidirectional composite materials subjected to constant amplitude cyclic loading. 5. Conclusion The new fatigue model proposed in this research is capable of predicting the fatigue life for unidirectional composites with arbitrary fiber orientations under various states of stress and stress ratios. In the new fatigue model all stress components are responsible for fatigue failure. Another capability of this model is predicting fatigue life for positive stress ratios for the compression–compression as well as the tension–tension fatigue loading condition. It means, new fatigue model is capable of distinguishing between stress ratios 0.5 and 0.5 by using different magnitudes for maximum static strength in material direction under tension and compression loading. References [1] Harris B. Fatigue and accumulation of damage in reinforced plastics, SEE fatigue group conference on fatigue of FRP, London, June 1977. [2] Degreck J, Van Paepegem W. Fatigue damage modeling of fibre-reinforced composite materials review. Appl Mech Rev 2001;54(4):279–300. [3] Stinchcomb WW, Reifsnider KL. Fatigue damage mechanism in composite materials: a review, fatigue mechanisms. In: Fong JT, editor. Proceeding of an ASTM–NBS-NSF symposium, Kansas City, Mo., May 1978. ASTM STP 675. American Society for Testing and Materials; 1979. p. 762–87. [4] Sendeckyj GP. Life prediction for resin–matrix composite materials. In: Reifsnider KL, editor. Fatigue of composite materials. Elsevier Science Publishers; 1990. p. 431–83. [5] Hashin Z, Rotem R. A fatigue failure criterion for fiber reinforced materials. J Comp Mater 1973;7:448–64.
[6] Sims DF, Brogdon VH. Fatigue behavior of composites under different loading modes. In: Reifsnider KL, Lauraitis KN, editors. Fatigue of filamentary composite materials. ASTM STP 636 1977, pp. 185–205. [7] Ellyin F, EL-Kadi H. A fatigue failure criterion for fiber reinforced composite lamina. Compos Struct 1990;15:61–74. [8] Fawaz Z, Ellyin F. Fatigue failure model for fiber-reinforced materials under general loading conditions. J Compos Mater 1994;28(15):1432–51. [9] Philippidis TP, Vassilopoulos AP. Fatigue strength prediction under multiaxial stress. J Compos Mater 1999;33(17):1578–99. [10] Kawai M. A phenomenological model for off-axis fatigue behavior of unidirectional polymer matrix composites under different stress ratios. Compos Part A 2004;35:955–63. [11] Plumtree A, Cheng GX. A fatigue damage parameter for offaxis unidirectional fiber reinforced composites. Int J Fatigue 1999;21:849–56. [12] Smith KN, Watson P, Topper TH. A stress strain function for the fatigue of metals. J Mater, JMLSA 1970;5:767–78. [13] Petermann J, Plumtree A. A unified fatigue failure criterion for unidirectional laminates. Compos Part A 2001;32:107–18. [14] Shokrieh MM. Progressive fatigue damage modeling of composite materials, Ph.D. Thesis, Department of Mechanical Engineering, McGill University, Montreal, Canada, February 1996. [15] Sandhu RS, Gallo RL, Sendeckyj GP. Initiation and accumulation of damage in composite laminates. Compos Mater 1982:163–82. [16] Talreja R. Damage characterization by internal variables. In: Talreja R, editor. Damage mechanics of composite materials. Elsevier Science B.V.; 1994. p. 53–78. [17] Reifsnider KL. Damage and damage mechanics. In: Reifsnider KL, editor. Fatigue of composite materials. Elsevier Publishers B.V.; 1990. p. 11–77. [18] Awerbuch J, Hahn HT. Off-axis fatigue of graphite/epoxy composites. Fatigue of fibrous composite materials. ASTM STP 1981;723:243–47. [19] Tsai SW. Introduction to composite materials. Westport, Connecticut: Technomoic Publication Company; 1980. [20] El Kadi H, Ellyin F. Effect of stress ratio on the fatigue of unidirectional glass/epoxy composite laminates. Composites 1994; 10:1–8.
A unified fatigue life model based on energy method M.M. Shokrieh *, F. Taheri-Behrooz Composites Research Laboratory, Mechanical Engineering Department, Iran University of Science and Technology, Tehran 16844, Iran Available online 12 June 2006
Abstract A new unified fatigue life model based on the energy method is developed for unidirectional polymer composite laminates subjected to constant amplitude, tension–tension or compression–compression fatigue loading. This new fatigue model is based on static failure criterion presented by Sandhu and substantially is normalized to static strength in fiber, matrix and shear directions. The proposed model is capable of predicting fatigue life of unidirectional composite laminates over the range of positive stress ratios in various fiber orientation angles. By using this new model all data points obtained from various stress ratios and fiber orientation angles are collapsed into a single curve. The new fatigue model is verified by applying it to different experimental data provided by other researchers. The obtained results by the new fatigue model are in good agreements with the experimental data of carbon/epoxy and E-glass/epoxy of unidirectional plies. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Polymer composites; Fatigue modeling; Energy method
1. Introduction Polymer based composites due to their specific strength and stiffness, resistance to environmental effects; low weight and high fatigue life are widely used by various industries. Usage of these materials as load bearing primary structures and exposing of them into cyclic loading, necessitates the fatigue life study as an important issue. Therefore, in spite of considerable research in this field, many researches are devoting their efforts to understand this phenomenon in more detail. The fatigue life of polymer composites strongly depends on the fiber directions, type of fibers and resins, number of layers, stacking sequence, type of loading, stress ratio, environmental conditions, and frequency of loading. These materials, before and after damage, are anisotropic and therefore the fatigue behavior of them is more complicated than traditional materials. In metals a dominant crack starts and propagates till final failure occurs. The crack propagation in composites is different and should be con*
Corresponding author. Tel./fax: +98 21 749 1206. E-mail address: [email protected] (M.M. Shokrieh).
0263-8223/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.04.041
sidered as a very complicated phenomenon. In these materials a large number of matrix micro-cracks initiates in the early stage of the fatigue loading and then various damage mechanisms like crazing, debonding between fibers and matrix, fiber breakage and delamination will happen. Therefore, the damage is not concentrated on the tip of a specific crack and is widely dispread [1]. Also, in the process of damage propagation in composites, the mechanical properties of unidirectional layers are decreased, the stresses are redistributed and the stress concentrations are decreased. By considering all theses complexities, understanding the step by step behavior of materials during the fatigue process is needed in order to predict the fatigue life of composites. In order to understand fatigue as a complicated phenomenon in composites, different methods are established by various authors. A comprehensive review on this topic is available in the literature [2–4]. Although fatigue behavior of composites is different from those of metals, however, many of the fatigue models are developed based on the well known S–N curves method. In this method a large number of experimental data is needed and the real damage mechanisms such as fiber and resin failure are not considered.
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Hashin and Rotem [5] presented one of the earliest fatigue model based on S–N curve method. They presented the S–N curve by testing off-axis glass/epoxy specimens with different angles under a specific stress ratio. Then by replacing the fatigue strength as a function of the fatigue life instead of the static strength in a failure criterion, they predict the fatigue life of a unidirectional composite under a uni-axial loading condition in different directions. A model presented by Sims and Brogdon [6] is also similar to the model of Hashin and Rotem. Ellyin and El Kadi [7] showed that the strain energy density can be used as a fatigue criterion. In their model the total elastic strain energy density applied to the material in the first cycle is correlated to the fatigue life. Later, Ellyin and Fawaz [8] proposed a semi-log linear relationship between applied cyclic stress and number of cycles to failure. By having the S–N curve for a defined lay up as a primary block, their model is able to predict the fatigue life of the same lay up at different directions. Although Philippidis and Vassilopoulos [9] showed that the model of Ellyin and Fawaz [8] is very sensitive to the reference line and the predicted results of this model are not suitable for cylindrical specimens. Recently, Kawai [10] by normalization of the applied stress with respect to the static strength and defining the fatigue strength ratio presented a model for predicting the fatigue life of an orthotropic layer under positive and negative stress rations and arbitrary directions. Plumtree and Cheng [11] by defining the critical planes concept for unidirectional composites in planes parallel to the fiber expressed that the damage occurs due to the normal and shear components of stress in these planes. They used Smith–Watson–Topper (SWT) [12] parameter in their model to developed a new fatigue model for composites that is independent of stress ratio. Later, Plumtree and Petermann [13] employed a strain energy density concept in critical planes and presented another criterion for fatigue life simulation of composites. Both models of Plumtree and his colleagues [11,13] were not able to predict the fatigue life of the unidirectional composites loaded in the fiber direction. It is well known that the mode of failure changes from fiber mode to matrix mode by changing the direction of fibers [14]. However, their models [11,13] are established only for the matrix failure mode and for the angles lower than the transient angle the predicted results by their models expected to be in less agreement with the experimental data. In the present research a new fatigue failure criterion based on the static failure criterion of Sandhu and his colleagues [15] is established. This criterion considers the matrix and fiber modes of failure in one equation and consequently is capable of predicting the fatigue life of off-axis composites at angles less than the transient angle. The concept of the transient angle is explained in [14]. The proposed equation is in a normalized form and uses static failure value of Sandhu (sum of normalized strain energy density components) instead of the stress components as
445
a new model of fatigue failure. This new fatigue failure criterion provides the capability of prediction of fatigue life for an orthotropic layer under tension–tension and compression–compression fatigue loading with arbitrary fiber angles. 2. Theoretical background Cumulative fatigue damage for composite materials is studied by various authors [16,17]. Initiation of crack depends on various parameters such as state of stress, stress ratio and fiber direction. In general, crack initiation occurs on the first cycles in the matrix, perpendicular to the fiber direction and develops as loading continues. Hahn [18] investigated the fatigue of unidirectional carbon/epoxy composites in directions of (h = 0°, 10°, 20°, 30°, 45°, 60°, and 90°) by well known S–N curves. Transition angle in tension for this material is obtained equal to 9.3° by using Eq. (1), hence, he only observed matrix failure mode according to modified Hashin criterion [14] in his test results. rffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffi Xt 56:9 htransition ¼ arctan ¼ arctan 1836 Yt ¼ 9:98
ð1Þ
where Xt and Yt are the tensile strength in fiber and matrix directions, respectively. For the matrix failure mode (fiber angles between transient angle and 90°) of a unidirectional off-axis lamia, matrix cracking occurs in the early stage of the loading and the matrix damage is a process of multiple initiation and coalescence of micro-cracks rather than the propagation of one single crack. Also, matrix cracking serves as a source of other damage due to stress singularities at the crack tips. The following macro-crack stage is relatively short because of crack coalescence. Occasionally, it is not detected before failure occurs, presenting the so-called sudden-death behavior [5]. In addition, matrix cracks are constrained by the fibers and confined to a direction parallel to the fibers. The fiber failure mode (fiber angles between 0° and transient angle) is different from the matrix failure mode as explained above. The matrix cracks in early stage of loading are integrated together as loading continues until they encounter with fibers and leads to fiber–matrix interface. By continuing loading, damage progress and cause fiber matrix debonding and leads to local fiber failure. This stage expected to follow a gradually material degradation regime, finally catastrophic fiber failure occurs. Plumtree and his colleagues [11,13] have developed their criterion based on the critical plane concept and presented their new fatigue parameter W* as Eq. (2): max max þ rmax W ¼ kðrmax 2 e2 6 e6 Þ 1R2
ð2Þ
where k ¼ 2 , r2, r6 and e2, e6 are stress and strain components in material directions and R is the stress ratio.
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r1 e1 r2 e2 r6 e6 þ þ ¼1 X e1u Y e2u Se6u
Fig. 1. Material (on-axis) 1–2 and loading (off-axis) x–y directions on an orthotropic layer.
Based on Eq. (2) only transverse normal stress and inplane shear stress in fracture plane are responsible for fatigue failure in all fiber load directions between 0° and 90°, while modified Hashin failure criterion has shown that in angles less than transition angle stress components in fiber direction and in-plane shear stress components are responsible for failure. Fig. 1 shows the material (on-axis) and loading (off-axis) directions on an orthotropic layer. Predicted results by different static failure criteria, especially modified Hashin criterion [14], also the present authors experiences in static and fatigue testing of glass/ epoxy and carbon/epoxy unidirectional composites identified that failure planes are not similar to that as proposed by Hahn [18] in all load fiber angles. Therefore in the following section a new energy based fatigue criterion is developed to consider all failure modes of a unidirectional laminate as matrix cracks, fiber–matrix interface debonding and fiber failure in a unique fatigue criterion. This criterion is capable of predicting the fatigue life of a unidirectional lamina at various fiber load angles and stress ratios. 3. Normalized fatigue failure criterion When S–N curves are employed to predict the fatigue life, maximum or amplitude of stresses are selected as fatigue measure and plotted vs. fatigue life. It has been shown by many researchers [5,18] that selecting a suitable fatigue model can strongly decrease the number of tests that are needed to forecast the fatigue life. In this research it has been shown that selecting a fatigue model based on elastic energy density components in material directions, is a good way of showing fatigue test results. To the best of our knowledge, the first static failure energy criterion based on strain energy was developed by Sandhu [15] for composite materials. This criterion, without considering the modes of failure, employs all stress and strain components in a normalized interactive equation to predict the failure. Sandhu expressed this criterion for an orthotropic laminate under in-plane loading as following:
ð3Þ
where r1, r2, r6 and e1, e2, e6 are stress and strain components in material directions, X,Y and S are maximum static strengths and e1u, e2u and e6u are maximum static strains in material directions. The advantage of this criterion in comparison with the stress based criteria is that this criterion uses both stress and strain to predict the failure. In this research, the magnitude of static failure of Sandhu criterion (sum of normalized strain energy density components) in each state of stress is introduced as a unified fatigue model for various fiber orientation angles and different stress ratios. The new fatigue model in the on-axis coordinates system can be written as follows: DW ¼ DW I þ DW II þ DW III ¼
Dr1 De1 Dr2 De2 Dr6 De6 þ þ X eu1 Y eu2 Seu6
ð4Þ
where D before a symbol indicates its range. The new fatigue model DW* represents a sum of strain energy densities contributed by all stress components in material directions. It should be mentioned here that in tension–tension loading X = Xt, Y = Yt and in compression–compression loading X = Xc, Y = Yc. Under cyclic fatigue loading, stress amplitude (state of stress) is always less than the maximum static strength, thus, the right side of Eq. (1) never reaches the unity except in tension-zero loading condition. The new fatigue criterion takes the value of static failure criterion in each state of stress as a new fatigue model. Total strain energy under cyclic loading can be shown by Eq. (5): 1 DW ¼ ðDr1 De1 þ Dr2 De2 þ Dr6 De6 Þ 2
ð5Þ
Normalized strain energy density contributed by the stress component along the fiber direction may be written as Eq. (6): DW I ¼
1 ðr1 max e1 max r1 min e1 min Þ X e1u
ð6Þ
Nonlinear response of unidirectional composites in matrix and shear directions is ignored in this model. Therefore the stress–strain response is assumed to be linear in material directions. X ¼ E1 e1u ! e1u ¼
X E1
and
r1 max ¼ E1 e1 max ! e1 max ¼
r1 max E1
ð7Þ
By substituting Eq. (7) into Eq. (6) it is concluded: DW I ¼
1 2 1 ð1 þ RÞ 2 ðDrÞ ðr r21 min Þ ¼ 2 X 2 1 max X ð1 RÞ
where R = rmin/rmax is the stress ratio.
ð8Þ
M.M. Shokrieh, F. Taheri-Behrooz / Composite Structures 75 (2006) 444–450
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The on-axis stresses are found by transforming of the off axis stresses to the on axis coordinates using a suitable transformation rule [19]. For the uniaxial case rx 5 0 and ry = rs = 0, stress components are obtained as follows:
where k and a are material constants and are independent of the stress ratio and fiber orientation. These two constants are obtained for each material by using one set of fatigue test data in an arbitrary stress ratio and a fiber orientation.
r1 max ¼ rx max cos2 h
4. Experimental verification ð9Þ
By substituting Eq. (9) into Eq. (8) one may be obtained: DW I ¼
1 ð1 þ RÞ 2 ðDrx Þ ðcos4 hÞ X 2 ð1 RÞ
ð10Þ
where h is the fiber orientation angle. Similarly the strain energy densities in matrix and shear directions are: 1 ð1 þ RÞ ðDrx Þ2 ðsin4 hÞ and Y 2 ð1 RÞ 1 ð1 þ RÞ ðDrx Þ2 ðsin2 h cos2 hÞ ¼ 2 S ð1 RÞ
DW II ¼ DW III
ð11Þ
By substituting Eqs. (10) and (11) into Eq. (4), a complete form of new fatigue model is obtained as: DW ¼ DW I þ DW I þ DW III 4 ð1 þ RÞ sin4 h sin2 h cos2 h 2 cos h ðDrx Þ ¼ þ þ ð1 RÞ X2 Y2 S2 ð12Þ At this point it should be mentioned that Eq. (12) is developed for different fiber load angles under positive stress ratios (R P 0) for both tension–tension and compression–compression cyclic loading. Whenever fiber and loading directions are coinciding together the first term of Eq. (12) is responsible for failure and is used as a new fatigue model. If the fiber orientation angle is perpendicular to the loading direction third term of the Eq. (12) is responsible for fatigue failure and used as a new fatigue model. Similarly in degrees less than transition angle failure is controlled by first and third terms and in degrees upper than transition angle second and third terms of Eq. (12) are responsible for failure. Therefore this new fatigue model is capable to take into account the effects of all different modes of failure of an orthotropic lamina. Ellyin and El Kadi [7] stated that fatigue life is related to the total input energy through a functional relationship, DWt = g(Nf). They have suggested a power law type equation of the form Eq. (13) to predict the fatigue life. DW t ¼ kN af þ C
ð13Þ
where k, a and C are material constants. k and a are functions of stress ratio and fiber direction. Here we rewrite Eq. (13) by substituting our proposed new fatigue model in the left side and setting the parameter C = 0 as: DW ¼ kN af
ð14Þ
In the previous section a new fatigue model is proposed to simulate the fatigue life of unidirectional composite materials for various fiber angles under different state of stresses and stress ratios. The inputs for the proposed model are ultimate strengths in material directions under tension and compression monotonic loadings and one set of fatigue test data to prescribe the material constants in Eq. (14). In order to verify the new fatigue model, it is applied to the available experimental data to show its capability in predicting of the fatigue life. Four different sets of experimental data available in the literature [5,10,18,20] are selected and examined to evaluate the capability of the new fatigue model. 4.1. Fatigue life prediction for unidirectional carbon/epoxy composites Two sets of experimental data of unidirectional carbon/ epoxy composites are used to verify the new fatigue model. The first set is the experimental data provided by Kawai [10] for different values of fiber angles and different positive stress ratios. The original fatigue data of a unidirectional carbon/ epoxy (T800H/2500) composite under the stress ratio of R = 0.1 and load frequency of 10 Hz is presented in Fig. 2. Fig. 2 clearly shows that the fiber orientation is an important parameter in fatigue life of a unidirectional ply. The new fatigue model data plotted in Fig. 3 using a log-log coordinate system. As shown all data points are distributed in a relatively narrow band and are nearly independent of fiber orientation. The least square method is
10000 Experimental data (Kawai and Suda) Carbon/Epoxy (T800H/2500) f=10Hz, R=0.1
Maximum stress (MPa)
r2 max ¼ rx max sin2 h r6 max ¼ rx max cos h sin h
0 15 45
10 30 90
1000
100
10
1
10
100
1000
10000 100000 1000000 10000000 1E+08
Number of reversals (2Nf)
Fig. 2. Comparison of predicted values with experimental results of Ref. [10], R = 0.1.
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10 0 30
10 45
Experimental (Awerbuch-Hahn), Carbon/Epoxy, AS/3501-5A, R=0.1, f=18Hz
15 90
ΔW*
ΔW*
Experimental (Kawai and Suda) Carbon/Epoxy (T800H/2500) f=10Hz, R=0.5
0.1
1
100
10000
1000000
30
1000
10000
100000
1000000
Fig. 5. Correlated fatigue lives for various fiber orientations [18].
with respect to the number of reversals to failure R = 0.1 [10].
applied to data points to find the best fit relationship to all experimental data in Fig. 3. All data points are collapsed to a master straight line expressed by the Eq. (15): DW ¼ 0:8928ð2N f Þ0:09717
ð15Þ
The predicted results using Eq. (15) are shown in Fig. 2. Solid lines show predicted fatigue life by using Eq. (15) and points show experimental data. As shown in this figure the predicted results are in good agreements with the experimental data. To demonstrate the capability of new fatigue model in predicting fatigue life for different stress ratios, Eq. (15) is employed to predict the fatigue life for stress ratio of R = 0.5 and various fiber orientations. Fig. 4 shows experimental data are in a good agreement with the predicted results by the Eq. (15). The second set of experimental data is selected to evaluate the new fatigue model, is provided by Awerbuch and Hahn [18] on unidirectional carbon/epoxy composites with fiber orientation angles (h = 0°, 10°, 20°, 30°, 45°, 60° and 90°) and stress ratio of R = 0.1. Comparison between predicted results by new fatigue parameter and experimental date are shown in Figs. 5 and 6. The best straight line fitted to these experimental data by least squares method is to be: DW ¼ 1:028ð2N f Þ0:1211
ð16Þ
100000 Experimental data (Kawai and Suda) Carbon/Epoxy (T800H/2500) f=10Hz, R=0.5
Maximum stress (MPa)
90
1
100000000
0
10
15
30
45
90
10000
1000
100
10
100
1000
10000
100000 1000000 10000000 100000000
Number of reversals (2Nf)
Fig. 4. Comparison of predicted values with experimental results R = 0.5 [10].
100000
Maximum stress (MPa)
DW*
1
20
60
Number of cycles (Nf)
Number of reversals (2Nf)
10
10
45
1
0 100
Fig. 3.
0
10000
Experimental data (Awerbuch-Hahn), Carbon/Epoxy, AS/3501-5A, R=0.1, f=18Hz
0
10
20
45
60
90
30
1000
100
10
1 100
1000
10000
100000
1000000
Number of cycles (Nf)
Fig. 6. Comparison of predicted values, Eq. (16), with experimental results of [18].
Figs. 2–6 confirm that the fiber orientation dependence of the off-axis fatigue data as well as the stress ratio dependence can be eliminated using the new fatigue model Eq. (12). As it is mentioned, the inputs of this new fatigue model are static strengths in material directions and one set of fatigue data in an arbitrary stress ratio and fiber load angle. 4.2. Fatigue life prediction for unidirectional E-glass/epoxy composites Two sets of experimental data of unidirectional E-glass/ epoxy composites are used to verify the new fatigue model and show the capability of the model in predicting fatigue life of different materials. The first set of experimental data is performed by El Kadi and Ellyin [20] on 3M type 1003 unidirectional E-glass/epoxy composites for various fiber orientations (h = 0°, 19°, 45°, 71° and 90°), different stress ratios (R = 0 and 0.5) and a loading frequency of 3.3 Hz. Fig. 7 shows the original experimental data of R = 0 and predicted results by the new fatigue model (Eq. (17)). New fatigue model (Eq. (12)) is applied to the experimental data in Fig. 7 and all data points are collapsed to a single curve as depicted in Fig. 8.
M.M. Shokrieh, F. Taheri-Behrooz / Composite Structures 75 (2006) 444–450 1
10000 0
Maximum stress (MPa)
Experimental data (Ellyin-ElKadi) E-glass/Epoxy f=3.3Hz, R=0.5
19
71
45
Experimental data (Hashin and Rotem) E-glass/Epoxy, R=0.1
90
100
1
100
10000
5
10
15
30
60
ΔW*
1000
10
449
1000000
0.1
100000000
Number of reversals (2Nf)
0.01 100
Fig. 7. Comparison of predicted values with experimental results of [20], R = 0.5.
1000
10000
100000
1000000
Number of cycles to failure (Nf)
Fig. 10. Correlated fatigue lives for various fiber orientations [5].
10
1000 0
19
71
90
1
0.1
Experimental data (Hashin and Rotem) E-glass/Epoxy, R=0.1
45
Maximum stress (MPa)
ΔW*
Experimental data (Ellyin-ElKadi) E-glass/Epoxy f=3.3Hz, R=0.5
10
100
1000
10000
100000 100000
1E+07
1E+08
10 60
15
100
10 100 1
5 30
1000
10000
100000
1000000
Number of cycles to failure (Nf)
Number of reversals (2Nf)
Fig. 8. Correlated fatigue lives for various fiber orientations Ref. [20], R = 0.5.
The best fit to the experimental data is obtained by using least square method is as follows: DW ¼ 0:7583ð2N f Þ
0:07762
ð17Þ
Eq. (17) is also applied to the existing data [20] and other stress ratio (R = 0), shown in Fig. 9. The obtained results are in good agreement with the experimental data and show that the model is capable of predicting life for various stress ratios and fiber orientations.
Maximum stress (MPa)
10000 Experimental data (Ellyin-ElKadi) E-glass/Epoxy f=3.3Hz ,R=0
0
19
71
90
45
1000
100
10 1
100
10000
1000000
100000000
Number of reversals (2Nf)
Fig. 9. Comparison of predicted values with experimental results of Ref. [20], R = 0.
Fig. 11. Comparison of predicted values, Eq. (18), with experimental results of [5].
The second set of experimental data is performed by Hashin and Rotem [5] on the off-axis E-glass/epoxy unidirectional composites with fiber directions of (h = 0°, 5°, 10°, 15°, 20°, 30° and 60°) and stress ratio of R = 0.1. Predicated results by using the Eq. (18) along with experimental data are shown in Figs. 10 and 11. Results show good correlation between experimental data and the model (Eq. (18)). The best fit to the experimental dada is obtained by the following equation: 0:1402
DW ¼ 0:3675ð2N f Þ
ð18Þ
As Figs. 10 and 11 illustrate predicted fatigue life by new fatigue model are in good agreement over the range of (0°– 90°) with experimental results. While criteria based on the critical planes concept [11,13] are not capable to predict life for 0 and 90 degrees fiber directions. Also the predicted life by these criteria [11,13] for angles less than the transition angle (i.e., 5°) are not matched with the experimental data. This problem comes from there that the models based on critical planes concept [11,13] does not take into account the effect of transition angle; consequently the part of input energy which is absorbed by the fibers is omitted. Obtained results confirm that the new fatigue model Eq. (12) becomes a unified strength model to eliminate the effects of positive stress ratios as well as the fiber
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orientation on the off-axis fatigue behavior of unidirectional composite materials subjected to constant amplitude cyclic loading. 5. Conclusion The new fatigue model proposed in this research is capable of predicting the fatigue life for unidirectional composites with arbitrary fiber orientations under various states of stress and stress ratios. In the new fatigue model all stress components are responsible for fatigue failure. Another capability of this model is predicting fatigue life for positive stress ratios for the compression–compression as well as the tension–tension fatigue loading condition. It means, new fatigue model is capable of distinguishing between stress ratios 0.5 and 0.5 by using different magnitudes for maximum static strength in material direction under tension and compression loading. References [1] Harris B. Fatigue and accumulation of damage in reinforced plastics, SEE fatigue group conference on fatigue of FRP, London, June 1977. [2] Degreck J, Van Paepegem W. Fatigue damage modeling of fibre-reinforced composite materials review. Appl Mech Rev 2001;54(4):279–300. [3] Stinchcomb WW, Reifsnider KL. Fatigue damage mechanism in composite materials: a review, fatigue mechanisms. In: Fong JT, editor. Proceeding of an ASTM–NBS-NSF symposium, Kansas City, Mo., May 1978. ASTM STP 675. American Society for Testing and Materials; 1979. p. 762–87. [4] Sendeckyj GP. Life prediction for resin–matrix composite materials. In: Reifsnider KL, editor. Fatigue of composite materials. Elsevier Science Publishers; 1990. p. 431–83. [5] Hashin Z, Rotem R. A fatigue failure criterion for fiber reinforced materials. J Comp Mater 1973;7:448–64.
[6] Sims DF, Brogdon VH. Fatigue behavior of composites under different loading modes. In: Reifsnider KL, Lauraitis KN, editors. Fatigue of filamentary composite materials. ASTM STP 636 1977, pp. 185–205. [7] Ellyin F, EL-Kadi H. A fatigue failure criterion for fiber reinforced composite lamina. Compos Struct 1990;15:61–74. [8] Fawaz Z, Ellyin F. Fatigue failure model for fiber-reinforced materials under general loading conditions. J Compos Mater 1994;28(15):1432–51. [9] Philippidis TP, Vassilopoulos AP. Fatigue strength prediction under multiaxial stress. J Compos Mater 1999;33(17):1578–99. [10] Kawai M. A phenomenological model for off-axis fatigue behavior of unidirectional polymer matrix composites under different stress ratios. Compos Part A 2004;35:955–63. [11] Plumtree A, Cheng GX. A fatigue damage parameter for offaxis unidirectional fiber reinforced composites. Int J Fatigue 1999;21:849–56. [12] Smith KN, Watson P, Topper TH. A stress strain function for the fatigue of metals. J Mater, JMLSA 1970;5:767–78. [13] Petermann J, Plumtree A. A unified fatigue failure criterion for unidirectional laminates. Compos Part A 2001;32:107–18. [14] Shokrieh MM. Progressive fatigue damage modeling of composite materials, Ph.D. Thesis, Department of Mechanical Engineering, McGill University, Montreal, Canada, February 1996. [15] Sandhu RS, Gallo RL, Sendeckyj GP. Initiation and accumulation of damage in composite laminates. Compos Mater 1982:163–82. [16] Talreja R. Damage characterization by internal variables. In: Talreja R, editor. Damage mechanics of composite materials. Elsevier Science B.V.; 1994. p. 53–78. [17] Reifsnider KL. Damage and damage mechanics. In: Reifsnider KL, editor. Fatigue of composite materials. Elsevier Publishers B.V.; 1990. p. 11–77. [18] Awerbuch J, Hahn HT. Off-axis fatigue of graphite/epoxy composites. Fatigue of fibrous composite materials. ASTM STP 1981;723:243–47. [19] Tsai SW. Introduction to composite materials. Westport, Connecticut: Technomoic Publication Company; 1980. [20] El Kadi H, Ellyin F. Effect of stress ratio on the fatigue of unidirectional glass/epoxy composite laminates. Composites 1994; 10:1–8.