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A unified fatigue failure criterion for unidirectional laminates
Composites: Part A 32 (2001) 107–118 www.elsevier.com/locate/compositesa
A unified fatigue failure criterion for unidirectional laminates J. Petermann a,*, A. Plumtree b a
b
Polymer Composites, Technical University Hamburg-Harburg, 21071 Hamburg, Germany Department of Mechanical Engineering, University of Waterloo, Waterloo, Canada N2L 3G1 Received 15 December 1999; revised 11 April 2000; accepted 19 May 2000
Abstract A unified fatigue failure criterion using micromechanics related to the fracture plane has been developed to predict fatigue lives of unidirectional fibre reinforced polymer composites subjected to cyclic off-axis tension–tension loading. Since the failure criterion incorporates both stresses and strains it may be characterized as energy based. Accounting for the fibre load angle as well as the stress ratio is the novelty of this fatigue failure criterion. The criterion only requires the stress ratio to be known from the experimental procedure. The relation between the applied load and the micro-stress and micro-strain field can be determined from a numerical method. The fatigue failure criterion has been verified by applying it to different sets of experimental data. Several fibre load angles, different fibre/matrix combinations as well as stress ratios are covered. The predicted fatigue lives are in good agreement with the experimental results for both different fibre load angles and stress ratios. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keyword: FRP
1. Introduction A fundamental problem in fatigue design is to predict the fatigue life of a structure under given load conditions. Since many composite components are subjected to dynamic loading in their applications, composite fatigue analysis has been the subject of many investigations during the past two decades. Some composite fatigue characteristics, for example a flat S–N curve and the large scatter of experimental data have been well recognized. Hence, a reliable fatigue curve obtained form a large sample population is the prerequisite to fatigue design. The complexity of the multi-damage mechanisms, damage related stress redistributions and their interactions in a heterogenous composite structure together with the inherent difficulties in fatigue have delayed the establishment of a general fatigue failure criterion [1]. Due to the large variety of laminates resulting from numerous materials, lay-ups and stacking sequences, it is uneconomic to determine fatigue life curves for any degree of generality by experiments only [2]. Obviously, the determination of a unified fatigue parameter and establishment of a life prediction model, both structure and loading mode independent, become important steps in composite fatigue design. This study presents an approach to predict fatigue * Corresponding author. Tel.: ⫹49-40-42878-8253; fax: ⫹49-40-428788239. E-mail address: [email protected] (J. Petermann).
lives of unidirectionally reinforced polymers under cyclic off-axis tensile loading.
2. Off-axis fatigue loading When a unidirectionally reinforced specimen is loaded off-axis in tension–tension fatigue, experimental observations [2–4] have shown that matrix cracking occurs preferentially. The potential fracture path coincides with the fibre direction as Fig. 1 shows. Since there are many stress concentration sites in the matrix immediately next to the fibres, matrix damage is a process of multiple initiation and coalescence of microcracks rather than 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 macrocrack stage is relatively short because of crack coalescence. Occasionally, it is not detected before failure occurs, presenting the so-called sudden-death behaviour [5]. In addition, matrix cracks are constrained by the fibres and confined to a direction parallel to the fibres.
3. Failure criteria 3.1. Limit and interactive criteria There are several ways to define failure. The most
1359-835X/01/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S1359-835 X( 00)00 099-3
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Fig. 1. Off-axis tensile coupons after fatigue failure [4].
obvious is when complete separation or fracture has occurred. More generally, failure would be when the component can no longer fulfil the function for which it was designed [6]. This definition not only includes total fracture but also large deformations like buckling or delamination. Limit criteria are based on maximum stress or maximum strain. Each consists of subcriteria corresponding to the strengths or ultimate strains of the fundamental modes of failure. According to the limit criteria, failure occurs when either one of the subcriteria is fulfilled. The maximum stress criterion does not account for possible interactions between stress components. The maximum strain criterion allows for some interaction due to Poisson’s effects. Although easy to use, limit criteria do not agree well with experimental data unless the fibre load angle is close to 0 or 90⬚, indicating failure dominated by only one normal stress. Interactive criteria describe the failure envelope using one single equation and, as the name suggests, are formulated in a way that stress interactions are taken into account. Based upon the von Mises yield criterion for predicting the onset of yielding in isotropic materials several criteria [7–9] of varying complexity have been proposed to include effects of anisotropy and to incorporate the basic strength parameters. However, one drawback of interactive criteria is
that, unlike with the limit criteria, the failure mode is not indicated. 3.2. Energy based criteria Energy-based criteria incorporate both stresses and strains. The multiplication of these components represents energy. This approach can be taken by measuring or calculating the hysteresis loop. When a material is subjected to an external load, the supplied energy is dissipated by friction and internal damping and some part is elastically stored in the material. Damage is caused by the irrecoverable part of the stored energy and may depend on the rate at which the strain energy is applied. Each material has a certain capability to absorb damage and failure is the result of damage accumulation. Because of the difficulty to measure the heat loss, it is generally assumed that damage development is proportional to the supplied energy. The damage caused in the material can therefore be related to the input energy through a functional relationship [10]. Since such a criterion does not rely on the different failure modes obtained in composites, it can be used regardless of the failure mechanism. Garud [11] related the plastic work done per load cycle to the fatigue life in a way that it provides the means to account
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Fig. 2. Deformations in the fracture plane.
for the normal and the shear response, but does not consider mean stress effects. El Kadi and Ellyin [3] proposed a fatigue failure criterion for unidirectional composites which also allows for mean stress effects. Applying the maximum monotonic strain energy density to the criterion the effect of fibre load angle is taken into account indirectly. However, the fracture plane is not considered. 3.3. Critical plane concepts Critical plane concepts are based upon the observation that fatigue cracks initiate and propagate along planes, known as critical planes. Findley [12] proposed that a fatigue parameter should be expressed by addition of the alternating shear strain and some part of the normal strain in the critical plane. Brown and Miller [13] further developed this approach based upon mechanisms of fatigue crack growth as expressed in Eq. (1), where the nomenclature refers to Fig. 2 and K is a material constant. Dgⴱ Dg12 ⫹ KDe22
1
Good correlation with multi-axial fatigue data has been obtained under a variety of loading conditions and geometrical configurations using this approach for isotropic materials [14]. The multi-axial fatigue parameter has been also found to be successful in correlating fatigue data of composite materials [15]. However, the parameter Dg ⴱ given in Eq. (1) does not account for mean stress effects and can be criticized for the lack of formal correctness from the continuum mechanics viewpoint. To overcome this drawback, Glinka et al. [16] proposed a strain energy criterion related to the critical plane concept given by Eq. (2). DW ⴱ
Dg12 Dt12 De22 Ds 22 ⫹ 4 4
2
Since energy components are scalars, they can be added algebraically. The fatigue parameter DW ⴱ represents a sum
of strain energy densities contributed by components acting in the critical plane. 3.4. Extension of critical plane concepts to unidirecional laminates Multi-damage mechanisms and fibre/matrix interaction must be accounted for when developing a generalized fatigue parameter that indicates both the critical plane and strain energy density. In terms of an off-axis loaded unidirectional laminate, where the critical plane — the fracture plane — is parallel to the fibres, it accounts for fibre orientation effects. Fig. 2 shows the deformation in a unidirectional laminate under cyclic loading. Since inter-fibre fracture is caused by normal and shear components only, considering these two terms is sufficient. The amplitudes of normal and shear stresses/strains existing in the fracture plane are indicated and can be written as the difference between maximum and minimum value. Once a crack has formed, its tip is subjected to two displacement components: an opening mode normal to the fibres in 2-direction and a sliding mode parallel to the fibres in 1direction leading to a mixed mode crack growth. For an elemental cube subjected to cyclic tensile stresses only and behaving essentially linear-elastic, the stress ratio, R, is described by the ratio of minimum over maximum applied stress or strain. The assumption that the stress–strain response is of linear-elastic character and remains unchanged during the off-axis fatigue is validated by results reported in Ref. [3]. The strain energy density contributed by the normal components may be expressed for a given volume by Eq. (3) representing the area under the stress– strain curve. W1ⴱ
1 2
max max min min s 22 e22 ⫺ s 22 e22
1 2
max max 1 ⫺ R2 s 22 e22 3
Similarly, the strain energy density for pure shear is given
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Fig. 4. Symmetry of the periodic composite structure. Fig. 3. Flow-chart for predicting fatigue lives.
4. Numerical analysis
by Eq. (4). W2ⴱ
1 2
max min min tmax 12 g12 ⫺ t12 g12
1 2
max 1 ⫺ R2 tmax 12 g12
4
The proposed fatigue parameter combines both, normal and shear contributions. The unified fatigue parameter may then be expressed as max max max e22 ⫹ tmax W ⴱ W1ⴱ ⫹ W2ⴱ l s 22 12 g12
5
where
l
1 ⫺ R2 : 2
6
At this point it should be remembered, that Eqs. (5) and (6) were derived for tension–tension fatigue, i.e. R ⱖ 0: Here and in the following it is assumed that the stress ratio based on the local stresses does not differ from that globally prescribed. The parameter W ⴱ is calculated at a point and is assumed to control the material behaviour in that immediate neighbourhood. The extent of the region controlled by W ⴱ depends on the microstructure and the stress/strain gradient. The choice of this unified fatigue parameter has some obvious advantages: • addition of energy rather than stress or strain components, • accounts for the mean stress, • incorporates the stress ratio through a scale factor of l. When dealing with one particular set of experimental data obtained from only one stress ratio, the scale factor becomes secondary. However, when dealing with different stress ratios the scale factor must be included since a base for comparison is needed.
It is worth pointing out that Eq. (5) only requires the stress ratio to be known from the experimental procedure. Knowledge of the microstress and microstrain field can be determined from a numerical method based on the applied load. Fig. 3 shows the flow-chart for predicting the fatigue lives. To apply a numerical method an idealization of the laminate structure is necessary (compare Fig. 4). For convenience the analysis is always carried out using the same co-ordinate system, with the 1-direction parallel to the fibres. Due do its repeating structure a unidirectional laminate can be modelled as a periodic packing array. The stress/ strain field in each unit block must be similar because of the symmetric and periodic structure. Edge effects are neglected and the fibre–matrix interface is perfectly bonded. Therefore, it is sufficient to analyse one unit block only. Utilizing the four-fold symmetry of the unit block the numerical method can further be reduced to one quarter of the unit block. A finite element code was chosen to solve the equations relating the applied load and the fibre load angle to the microstresses. The material properties of fibre and matrix as the inputs to the numerical analysis are taken from literature. The state of plane strain is assumed to govern the numerical model involving a mesh of 124 nodes and 207 constant strain triangles. The code has been described in detail by Shen [1]. The calculations show that the highest combinations of both stress components, normal and shear, occur in the matrix adjacent to the fibre/matrix interface. The strains at highest stress concentration sites are calculated from Eqs. (7) and (8) using linear elastic mechanics assuming plane strain conditions.
e22
1 s ⫺ nm s 11 ⫹ s 33 Em 22
7
g12
1 t Gm 12
8
Having determined the microstresses with the finite element
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Fig. 5. Ratio of normal stress to shear stress with respect to the fibre load angle.
code and the microstrains using Eqs. (7) and (8), Eq. (5) can be evaluated since the stress ratio is known. 4.1. Fatigue life prediction for different fibre load angles For a constant stress ratio the scale factor need not be considered, e.g. l 1: When the unified fatigue parameter W ⴱ is plotted versus the experimentally observed fatigue lives in a log–log co-ordinate system, all data fall on a single straight line (see Section 5). In general terms such a curve is empirically expressed by log W ⴱ alog Nf ⫹ b:
9
Mathematically b is the intercept at Nf 1 and a is the slope of the curve. Nf is the number of cycles to failure. Both parameters, a and b are material constants and can be determined from experimental data. Eq. (9) may be written more conveniently in power law form as W ⴱ 10b Nf a K Nf a :
10
Here, the constant K represents the maximum strain energy density for cyclic failure Nf ⬎ 1: The unified fatigue curve W ⴱ versus Nf is applicable to all fibre load angles tested at a given stress ratio. Conversely, it can be used to predict the fatigue lives for any other fibre load angle at the same stress ratio. 4.2. Fatigue life prediction for different stress ratios Generally, the analysis follows the fatigue life prediction described in the previous section. Additionally, the scale factor l must be included. Then the unified fatigue parameter conveniently normalizes both fibre load angle and stress ratio. The influence of the stress ratio R on the scale factor l and thus on the unified fatigue parameter W ⴱ and
the fatigue life is given by Eqs. (6), (5) and (9). Assuming a constant maximum stress they indicate that as the stress ratio increases the fatigue life will increase. In fact this is covered by experimental observations [3]. Fatigue failure will not occur for l 0: Since the respective stress ratio is 1, indicating a constant loading, the scale factor l is consistent. 5. Experimental verification In the previous sections a model to describe composites whose fatigue behaviour is matrix controlled was presented. The unified fatigue parameter W ⴱ is now applied to demonstrate its capability of predicting the fatigue lives. Three different sets of data [2–4] are examined. The last one used unidirectional carbon/epoxy laminates to study the off-axis fatigue behaviour. The other studies carried out uniaxially loaded fatigue tests on unidirectional E-glass/ epoxy laminates for different fibre load angles. However, only El Kadi and Ellyin [3] examined different stress ratios. 5.1. Fatigue life predictions for different fibre load angles Following the flow-chart in Fig. 3, two of the available sets of data [2,4] are examined with respect to different fibre load angles only. The stress ratio was constant at R 0:1 for both studies. Applying the finite element analysis to the different fibre load angles, the ratio of normal stress s 22 to shear stress t 12 changes significantly as shown in Fig. 5 for the E-glass/epoxy data. As expected from the limit criteria, failure is dominated by shear at small fibre load angles and transverse normal stresses at higher fibre angles. This fact may cause difficulties when attempting to normalize fatigue life data using the classical theories mentioned in Section 3.1.
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Fig. 6. Original S–N data for various fibre load angles [2].
5.1.1. E-glass/epoxy The experimental data of Ref. [2], corresponding to six different fibre load angles u , namely 5, 10, 15, 20, 30 and 60⬚ are first examined. E-glass fibres Gevetex WS 13-320X1K921 in epoxy resin Bakelite ERL 2256 containing a fibre volume fraction of V f 0:6 have been used. The macrostress-fatigue life data s max versus Nf were obtained from rectangular specimens tested with R 0:1: The original S–N data indicated that the fatigue life was strongly dependent upon the fibre load angle as shown in Fig. 6. For all angles failure occurred by a crack through the specimen parallel to the fibre direction. These fracture planes represent critical planes as introduced. Hence, the
numerical method may be applied. In Fig. 7 the unified fatigue parameter is plotted against the number of reversals to failure using a log–log co-ordinate system. The scale factor l determined from Eq. (6) is included in the correlation. No obvious fibre load angle dependence is seen despite the existence of some scatter. The unified fatigue parameter collapses all data points on a single straight line determined by the least square method as log W ⴱ ⫺0:137 log Nf ⫺ 0:339 W ⴱ 0:458 Nf ⫺0:137 :
11
Since the unified fatigue parameter has not been derived
Fig. 7. Correlated fatigue lives for various fibre load angles.
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Fig. 8. Comparison of experimental data with predicted fatigue lives using W ⴱ for u 20⬚:
with respect to a particular fibre load angle, an arbitrary one can be chosen to predict the fatigue lives of specimens tested at the same stress ratio but different fibre load angles. To show the universal applicability, data obtained for u 20⬚ are used to determine the relationship between W ⴱ and Nf, which is log W ⴱ ⫺0:129 log Nf ⫺ 0:390 ⴱ
W 0:407 Nf
⫺0:129
:
12
In its turn Eq. (12) may be used to predict fatigue lives of the other specimens. A comparison of the predicted fatigue lives and the experimental results is given in Fig. 8. Predic-
tions and experimental data show very good agreement. However, the fatigue lives for u 5⬚ are slightly overestimated. This may be because of a degradation in fibre strength with a number of cycles when the fibre load angle is small [1], which is not taken into account in this matrix controlled fatigue model. 5.1.2. Carbon/epoxy To confirm the capability of the unified fatigue parameter to predict fatigue lives for different fibre load angles a second set of experimental data is analyzed. Awerbuch and Hahn [4] studied the off-axis fatigue behaviour of unidirectional AS4 Carbon in 3501-5A epoxy resin. The
Fig. 9. Original S–N data for various fibre load angles [4] and comparison with predicted fatigue lives using W ⴱ for u 20⬚:
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Fig. 10. Correlated fatigue lives for various fibre load angles.
nominal fibre content was 0.7 by volume. Five different offaxis angles were tested: 10, 20, 30, 45 and 60⬚. Constant amplitude loading tests at R 0:1 were performed at about 18 Hz until fracture or 10 6 cycles, whichever occurred first. Essentially, Awerbuch and Hahn found that no strength degradation was observed in the specimens surviving fatigue loading of 10 6 cycles regardless of the fibre load angle or fatigue stress level. For the other specimens the effect of the off-axis angle on the S–N relationship can be easily identified (Fig. 9) although large scatter were observed. The fatigue lives for the 30⬚-specimens even tend to increase as the maximum applied load increases. The authors concluded that a specimen being fatigue cycled at a certain dynamic stress level may fail randomly along the fibre directions (Fig. 1) at any cycle number without any warning or visible damage. Following the procedure described above, the unified fatigue parameter is applied to correlate the fatigue data. Fig. 10 shows that all data points fall on one single line as for Eglass/epoxy. The best fit relationship is log W ⴱ ⫺0:0741 log Nf ⫺ 0:438
13
W ⴱ 0:365 Nf ⫺0:074 :
It is interesting to note that the normalized fatigue data for Table 1 Correlated fatigue data 2Nf [1]
2
10 10 6
W ⴱ (MPa) E-Glass/Epoxy
Carbon/Epoxy
0.27 0.08
0.27 0.14
E-glass/epoxy and carbon/epoxy have the same order of magnitude as Table 1 indicates. For low cycle fatigue, the superior properties of carbon fibres are only of marginal influence. Both composites require the same energy input to failure. However, for high cycle fatigue the advantage of using carbon fibres instead of E-glass fibres is apparent due to the much shallower curve. Nearly twice as much energy must be supplied to cause failure in the carbon/epoxy composite when compared with E-glass/epoxy. A similar trend was observed by Curtis [17] where only the degradation of strength was considered. To be consistent with the analysis of the E-glass/epoxy data the fibre load angle of u 20⬚ is chosen to predict the fatigue lives. The relationship between W ⴱ and Nf is expressed in Eq. (14), which may now be applied to predict the fatigue lives for all other fibre load angles. Good agreement is seen for u 10; 45 and 60⬚ as Fig. 9 shows. The fatigue lives for u 30⬚ are overestimated. This may be explained with the atypical shape of the original S–N curve where the fatigue life increased as the maximum stress increased. Data from a larger sample population will probably show better agreement with the predictions. log W ⴱ ⫺0:085 log Nf ⫺ 0:321 W ⴱ 0:478 Nf ⫺0:085 :
14
It is worth pointing out that the unified fatigue parameter is sensitive to the number of cycles to failure. Rearranging Eq. (14) with respect to Nf shows that an increase of the unified fatigue parameter by 10% decreases the fatigue life by 68%. Despite this sensitivity good agreement between experimental data and predicted fatigue lives has been found. Therefore, one can conclude that the proposed model and the finite element analysis describe the physical
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Fig. 11. Original S–N data for various stress ratios and u 19⬚ [3].
reality sufficiently well to be extended to different stress ratios. 5.2. Fatigue life prediction for different stress ratios To assess the capability of the unified fatigue parameter correlating fatigue data obtained from different stress ratios, another set of experimental data is required. El Kadi and Ellyin [3] performed tests on 3M type 1003, a unidirectional E-glass fibre reinforced epoxy resin with a fibre volume fraction of 50%. The specimens were tested at room temperature under various stress ratios R 0:5; 0 and ⫺1) and off-axis angles u 19; 45 and 71⬚). It should be noted that these
specimen geometries differed considerably from those used by Hashin and co-workers [2,4] which is a direct consequence of the compression loading excursion included in the load cycle when applying R ⫺1: The cross-section area was approximately three times as large, whereas the gauge length was less than half as long. For the off-axis cyclic tests all specimens had an identical brittle failure mode with matrix failure parallel to the fibres. The original data indicated the slope of the S–N curve to be dependent on both the fibre load angle and the stress ratio. Fig. 11 shows the S–N curves for u 19⬚ tested under three different stress ratios. Here, the data for R ⫺1 are given for completeness since tension and compression do not contribute equal parts to the damage development. Plots
Fig. 12. Correlated fatigue data for stress ratios of R 0:5 and 0 and fibre load angles of u 19; 45 and 71⬚.
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Fig. 13. Comparison of experimental data for R 0 and u 45⬚ with predicted fatigue lives determined using W ⴱ for R 0:5 and u 45⬚:
for the other angles show the same effect on lower stress levels. For a given fatigue life and fibre load angle the failure stresses for R 0:5 are the highest whereas those for R ⫺1 are the lowest. For non-negative stress ratios this observation agrees with the proposed scale factor l . If a laminate is not only exposed to tensile but compressive loads, the modelling of damage development is much more difficult. A proper degradation model has to incorporate fibre dominated failure mechanisms such as fibre buckling and the development of kink bands. Those mechanisms are not covered by the proposed matrix-controlled fatigue model which was derived for tension–tension fatigue, i.e. non-negative stress ratios. Tests performed at those stress ratios are governed by similar damage mechanisms and their
results are suitable for comparison purposes. Hence, only data obtained from non-negative stress ratios are subsequently considered. Fig. 12 shows the correlated fatigue data for R 0:5 and R 0 and fibre load angles of u 19; 45 and 71⬚. The data points for R 0:5 and R 0 are no longer segregated as in the original S–N plots, but have been combined by the unified fatigue parameter. Compared with the previous analysis these data are located at a higher W ⴱ level. This may be due to the difference in specimen geometries. If the unified failure criterion were universally applicable for predicting fatigue lives for different stress ratios and different fibre load angles, any set of data obtained from an arbitrary combination of fibre load angle and stress
Fig. 14. Comparison of experimental data for R 0; u 19 and 71⬚ with predicted fatigue lives determined using W ⴱ for R 0:5 and u 45⬚:
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ratio must be capable of predicting fatigue lives for any other set. Analysis of the data sets [2,4] has already shown the applicability to different fibre load angles. Therefore, the unified failure criterion is now applied to different stress ratios. The best fit curve for R 0:5 and u 45⬚ is given in Eq. (15). The stress ratio of R 0:5 is chosen to eliminate the possibility of having a compressive load acting on the specimen when using R 0: The choice for u 45⬚ is based on having the most data available for determining the regression curve. In its turn, Eq. (15) is now applied to predict fatigue lives for other combinations of stress ratio and fibre load angle. log W ⴱ ⫺0:080 log Nf ⫺ 0:139 0:726 Nf ⫺0:080 : 15 Fig. 13 compares experimental data obtained for R 0 and u 45⬚ with predictions using Eq. (15). The fibre load angle remained unchanged whereas the stress has changed. A very good agreement is found. The most general case displays Fig. 14. Here, Eq. (15) is applied to predict fatigue lives for another set of stress ratio and fibre load angle, in particular R 0 with u 19 and 71⬚. Although the experimental data for u 19⬚ are underpredicted and those for 71⬚ slightly overpredicted, good agreemeent is still observed. Thus, the unified fatigue parameter is capable to predict fatigue lives of unidirectional laminates under cyclic tension–tension loading considering various combinations of fibre load angle and stress ratio. 6. Conclusions An approach to determine the fatigue properties of unidirectional laminates from limited experimental data under simple loading has been developed. The properties are written in a generalized form independent of the loading mode to subsequently apply them for predicting fatigue lives of laminae subjected to other loading modes. The proposed unified fatigue failure criterion is based on the actual microstresses and microstrains instead of imaginary laminate values. Since failure is related to micromechanics the generalization of fatigue data for different loading modes is more easily performed using the present technique than with traditional macromechanical approaches. Together with the proposed fatigue parameter a method of fatigue life prediction has been developed. Being based on strain energy densities the proposed unified fatigue parameter satisfies the conditions of continuum mechanics. Both, normal and shear contributions acting in the fracture plane are considered. The main advantages of the unified fatigue parameter are accounting for mean stress effects and being load path independent since only peak values instead of values throughout the cycle are needed. This methodology considerably reduces the number of parameters to be determined experimentally. The unified fatigue parameter has been successfully
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applied to predict fatigue lives of different unidirectional laminates under cyclic tension considering various combinations of fibre load angle and stress ratio. All data points are collapsed on a single straight line and expressed by an equation determined by the least square method. The predicted results and the experimental fatigue data are in good agreement. The successful correlation of the fatigue data using the fracture plane based unified fatigue criterion validated the assumption of matrix controlled fatigue. Using limited experimental data the unified fatigue parameter enables a generalized strain energy–life relationship to be determined. Once such a relationship is established, the fatigue life of unidirectional laminates cycled under any other combination of fibre load angle and stress ratio can be predicted. Therefore, the unified fatigue failure criterion is a potentially valuable tool for fatigue design.
7. Recommendations Further experimental verification is necessary to extend the unified fatigue parameter to loading other than unidirectional tension. The form of the proposed parameter suggests that multi-axial fatigue data can be successfully normalized as well. If extended to multi-directional laminates, the proposed model may also be used to describe delamination failure by defining the examined planes parallel to the layers. As an initial step only simple unidirectional laminates loaded in cyclic tension were considered in the present study. Incorporation of compressive failure mechanisms should be the challenge since tension and compression are governed by different failure mechanisms and do not contribute equal parts to damage development. Temperature should be considered as a variable, allowing the effect of temperature on fatigue life to be assessed. If such an approach is found to be satisfactory it may be extended to creep studies and environmental degradation.
References [1] ShenG. Fatigue life prediction of composites based on microstress analysis. PhD thesis, University of Waterloo, 1993. [2] Hashin Z, Rotem A. A fatigue failure criterion for fibre reinforced materials. Journal of Composite Materials 1973;7:448–64. [3] El Kadi H, Ellyin F. Effect of stress ratio on the fatigue of unidirectional glass/epoxy composite laminates. Composites 1994;10:1–8. [4] Awerbuch J, Hahn HT. Off-axis fatigue of graphite/epoxy composite. In: Fatigue of fibrous composite materials, ASTM STP 723. American Society for Testing and Materials, 1981. p. 243–73. [5] Chou PC, Croman R. Degradation and sudden-death models of fatigue of graphite/epoxy composites. In: Tsai SW, editor. Composite materials: testing and design, ASTM STP 674. American Society for Testing and Materials, 1979. p. 431–54. [6] Matthews FL, Rawlings RD. Composite materials: engineering and science. 1st ed. London: Chapman & Hall, 1984. [7] Hill R. A theory of the yielding and plastic flow of anistropic metals. Proceedings of the Royal Society, Series A 1948:193.
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[8] Azzi VD, Tsai SW. Anisotropic strength of composites. Experimental Mechanics 1965;5(9):283–8. [9] Tsai SW, Wu EM. A general theory of strength for anisotropic materials. Journal of Composites Materials 1971;5:58–80. [10] Ellyin F, El Kadi H. Predicting crack growth direction in unidirectional laminae. Engineering Fracture Mechanics 1990;36:27–37. [11] Garud YS. A new approach to the evaluation of fatigue under multiaxial loadings. Journal of Engineering Materials and Technology 1981;103:1118–215. [12] Findley WN. A theory for the effect of mean stress on fatigue of metals under combined torsional and axial load or bending. Journal of Engineering for Industry 1959;81:301–6. [13] Brown MW, Miller KJ. A theory for fatigue under multiaxial stress–
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A unified fatigue failure criterion for unidirectional laminates J. Petermann a,*, A. Plumtree b a
b
Polymer Composites, Technical University Hamburg-Harburg, 21071 Hamburg, Germany Department of Mechanical Engineering, University of Waterloo, Waterloo, Canada N2L 3G1 Received 15 December 1999; revised 11 April 2000; accepted 19 May 2000
Abstract A unified fatigue failure criterion using micromechanics related to the fracture plane has been developed to predict fatigue lives of unidirectional fibre reinforced polymer composites subjected to cyclic off-axis tension–tension loading. Since the failure criterion incorporates both stresses and strains it may be characterized as energy based. Accounting for the fibre load angle as well as the stress ratio is the novelty of this fatigue failure criterion. The criterion only requires the stress ratio to be known from the experimental procedure. The relation between the applied load and the micro-stress and micro-strain field can be determined from a numerical method. The fatigue failure criterion has been verified by applying it to different sets of experimental data. Several fibre load angles, different fibre/matrix combinations as well as stress ratios are covered. The predicted fatigue lives are in good agreement with the experimental results for both different fibre load angles and stress ratios. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keyword: FRP
1. Introduction A fundamental problem in fatigue design is to predict the fatigue life of a structure under given load conditions. Since many composite components are subjected to dynamic loading in their applications, composite fatigue analysis has been the subject of many investigations during the past two decades. Some composite fatigue characteristics, for example a flat S–N curve and the large scatter of experimental data have been well recognized. Hence, a reliable fatigue curve obtained form a large sample population is the prerequisite to fatigue design. The complexity of the multi-damage mechanisms, damage related stress redistributions and their interactions in a heterogenous composite structure together with the inherent difficulties in fatigue have delayed the establishment of a general fatigue failure criterion [1]. Due to the large variety of laminates resulting from numerous materials, lay-ups and stacking sequences, it is uneconomic to determine fatigue life curves for any degree of generality by experiments only [2]. Obviously, the determination of a unified fatigue parameter and establishment of a life prediction model, both structure and loading mode independent, become important steps in composite fatigue design. This study presents an approach to predict fatigue * Corresponding author. Tel.: ⫹49-40-42878-8253; fax: ⫹49-40-428788239. E-mail address: [email protected] (J. Petermann).
lives of unidirectionally reinforced polymers under cyclic off-axis tensile loading.
2. Off-axis fatigue loading When a unidirectionally reinforced specimen is loaded off-axis in tension–tension fatigue, experimental observations [2–4] have shown that matrix cracking occurs preferentially. The potential fracture path coincides with the fibre direction as Fig. 1 shows. Since there are many stress concentration sites in the matrix immediately next to the fibres, matrix damage is a process of multiple initiation and coalescence of microcracks rather than 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 macrocrack stage is relatively short because of crack coalescence. Occasionally, it is not detected before failure occurs, presenting the so-called sudden-death behaviour [5]. In addition, matrix cracks are constrained by the fibres and confined to a direction parallel to the fibres.
3. Failure criteria 3.1. Limit and interactive criteria There are several ways to define failure. The most
1359-835X/01/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S1359-835 X( 00)00 099-3
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Fig. 1. Off-axis tensile coupons after fatigue failure [4].
obvious is when complete separation or fracture has occurred. More generally, failure would be when the component can no longer fulfil the function for which it was designed [6]. This definition not only includes total fracture but also large deformations like buckling or delamination. Limit criteria are based on maximum stress or maximum strain. Each consists of subcriteria corresponding to the strengths or ultimate strains of the fundamental modes of failure. According to the limit criteria, failure occurs when either one of the subcriteria is fulfilled. The maximum stress criterion does not account for possible interactions between stress components. The maximum strain criterion allows for some interaction due to Poisson’s effects. Although easy to use, limit criteria do not agree well with experimental data unless the fibre load angle is close to 0 or 90⬚, indicating failure dominated by only one normal stress. Interactive criteria describe the failure envelope using one single equation and, as the name suggests, are formulated in a way that stress interactions are taken into account. Based upon the von Mises yield criterion for predicting the onset of yielding in isotropic materials several criteria [7–9] of varying complexity have been proposed to include effects of anisotropy and to incorporate the basic strength parameters. However, one drawback of interactive criteria is
that, unlike with the limit criteria, the failure mode is not indicated. 3.2. Energy based criteria Energy-based criteria incorporate both stresses and strains. The multiplication of these components represents energy. This approach can be taken by measuring or calculating the hysteresis loop. When a material is subjected to an external load, the supplied energy is dissipated by friction and internal damping and some part is elastically stored in the material. Damage is caused by the irrecoverable part of the stored energy and may depend on the rate at which the strain energy is applied. Each material has a certain capability to absorb damage and failure is the result of damage accumulation. Because of the difficulty to measure the heat loss, it is generally assumed that damage development is proportional to the supplied energy. The damage caused in the material can therefore be related to the input energy through a functional relationship [10]. Since such a criterion does not rely on the different failure modes obtained in composites, it can be used regardless of the failure mechanism. Garud [11] related the plastic work done per load cycle to the fatigue life in a way that it provides the means to account
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Fig. 2. Deformations in the fracture plane.
for the normal and the shear response, but does not consider mean stress effects. El Kadi and Ellyin [3] proposed a fatigue failure criterion for unidirectional composites which also allows for mean stress effects. Applying the maximum monotonic strain energy density to the criterion the effect of fibre load angle is taken into account indirectly. However, the fracture plane is not considered. 3.3. Critical plane concepts Critical plane concepts are based upon the observation that fatigue cracks initiate and propagate along planes, known as critical planes. Findley [12] proposed that a fatigue parameter should be expressed by addition of the alternating shear strain and some part of the normal strain in the critical plane. Brown and Miller [13] further developed this approach based upon mechanisms of fatigue crack growth as expressed in Eq. (1), where the nomenclature refers to Fig. 2 and K is a material constant. Dgⴱ Dg12 ⫹ KDe22
1
Good correlation with multi-axial fatigue data has been obtained under a variety of loading conditions and geometrical configurations using this approach for isotropic materials [14]. The multi-axial fatigue parameter has been also found to be successful in correlating fatigue data of composite materials [15]. However, the parameter Dg ⴱ given in Eq. (1) does not account for mean stress effects and can be criticized for the lack of formal correctness from the continuum mechanics viewpoint. To overcome this drawback, Glinka et al. [16] proposed a strain energy criterion related to the critical plane concept given by Eq. (2). DW ⴱ
Dg12 Dt12 De22 Ds 22 ⫹ 4 4
2
Since energy components are scalars, they can be added algebraically. The fatigue parameter DW ⴱ represents a sum
of strain energy densities contributed by components acting in the critical plane. 3.4. Extension of critical plane concepts to unidirecional laminates Multi-damage mechanisms and fibre/matrix interaction must be accounted for when developing a generalized fatigue parameter that indicates both the critical plane and strain energy density. In terms of an off-axis loaded unidirectional laminate, where the critical plane — the fracture plane — is parallel to the fibres, it accounts for fibre orientation effects. Fig. 2 shows the deformation in a unidirectional laminate under cyclic loading. Since inter-fibre fracture is caused by normal and shear components only, considering these two terms is sufficient. The amplitudes of normal and shear stresses/strains existing in the fracture plane are indicated and can be written as the difference between maximum and minimum value. Once a crack has formed, its tip is subjected to two displacement components: an opening mode normal to the fibres in 2-direction and a sliding mode parallel to the fibres in 1direction leading to a mixed mode crack growth. For an elemental cube subjected to cyclic tensile stresses only and behaving essentially linear-elastic, the stress ratio, R, is described by the ratio of minimum over maximum applied stress or strain. The assumption that the stress–strain response is of linear-elastic character and remains unchanged during the off-axis fatigue is validated by results reported in Ref. [3]. The strain energy density contributed by the normal components may be expressed for a given volume by Eq. (3) representing the area under the stress– strain curve. W1ⴱ
1 2
max max min min s 22 e22 ⫺ s 22 e22
1 2
max max 1 ⫺ R2 s 22 e22 3
Similarly, the strain energy density for pure shear is given
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Fig. 4. Symmetry of the periodic composite structure. Fig. 3. Flow-chart for predicting fatigue lives.
4. Numerical analysis
by Eq. (4). W2ⴱ
1 2
max min min tmax 12 g12 ⫺ t12 g12
1 2
max 1 ⫺ R2 tmax 12 g12
4
The proposed fatigue parameter combines both, normal and shear contributions. The unified fatigue parameter may then be expressed as max max max e22 ⫹ tmax W ⴱ W1ⴱ ⫹ W2ⴱ l s 22 12 g12
5
where
l
1 ⫺ R2 : 2
6
At this point it should be remembered, that Eqs. (5) and (6) were derived for tension–tension fatigue, i.e. R ⱖ 0: Here and in the following it is assumed that the stress ratio based on the local stresses does not differ from that globally prescribed. The parameter W ⴱ is calculated at a point and is assumed to control the material behaviour in that immediate neighbourhood. The extent of the region controlled by W ⴱ depends on the microstructure and the stress/strain gradient. The choice of this unified fatigue parameter has some obvious advantages: • addition of energy rather than stress or strain components, • accounts for the mean stress, • incorporates the stress ratio through a scale factor of l. When dealing with one particular set of experimental data obtained from only one stress ratio, the scale factor becomes secondary. However, when dealing with different stress ratios the scale factor must be included since a base for comparison is needed.
It is worth pointing out that Eq. (5) only requires the stress ratio to be known from the experimental procedure. Knowledge of the microstress and microstrain field can be determined from a numerical method based on the applied load. Fig. 3 shows the flow-chart for predicting the fatigue lives. To apply a numerical method an idealization of the laminate structure is necessary (compare Fig. 4). For convenience the analysis is always carried out using the same co-ordinate system, with the 1-direction parallel to the fibres. Due do its repeating structure a unidirectional laminate can be modelled as a periodic packing array. The stress/ strain field in each unit block must be similar because of the symmetric and periodic structure. Edge effects are neglected and the fibre–matrix interface is perfectly bonded. Therefore, it is sufficient to analyse one unit block only. Utilizing the four-fold symmetry of the unit block the numerical method can further be reduced to one quarter of the unit block. A finite element code was chosen to solve the equations relating the applied load and the fibre load angle to the microstresses. The material properties of fibre and matrix as the inputs to the numerical analysis are taken from literature. The state of plane strain is assumed to govern the numerical model involving a mesh of 124 nodes and 207 constant strain triangles. The code has been described in detail by Shen [1]. The calculations show that the highest combinations of both stress components, normal and shear, occur in the matrix adjacent to the fibre/matrix interface. The strains at highest stress concentration sites are calculated from Eqs. (7) and (8) using linear elastic mechanics assuming plane strain conditions.
e22
1 s ⫺ nm s 11 ⫹ s 33 Em 22
7
g12
1 t Gm 12
8
Having determined the microstresses with the finite element
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Fig. 5. Ratio of normal stress to shear stress with respect to the fibre load angle.
code and the microstrains using Eqs. (7) and (8), Eq. (5) can be evaluated since the stress ratio is known. 4.1. Fatigue life prediction for different fibre load angles For a constant stress ratio the scale factor need not be considered, e.g. l 1: When the unified fatigue parameter W ⴱ is plotted versus the experimentally observed fatigue lives in a log–log co-ordinate system, all data fall on a single straight line (see Section 5). In general terms such a curve is empirically expressed by log W ⴱ alog Nf ⫹ b:
9
Mathematically b is the intercept at Nf 1 and a is the slope of the curve. Nf is the number of cycles to failure. Both parameters, a and b are material constants and can be determined from experimental data. Eq. (9) may be written more conveniently in power law form as W ⴱ 10b Nf a K Nf a :
10
Here, the constant K represents the maximum strain energy density for cyclic failure Nf ⬎ 1: The unified fatigue curve W ⴱ versus Nf is applicable to all fibre load angles tested at a given stress ratio. Conversely, it can be used to predict the fatigue lives for any other fibre load angle at the same stress ratio. 4.2. Fatigue life prediction for different stress ratios Generally, the analysis follows the fatigue life prediction described in the previous section. Additionally, the scale factor l must be included. Then the unified fatigue parameter conveniently normalizes both fibre load angle and stress ratio. The influence of the stress ratio R on the scale factor l and thus on the unified fatigue parameter W ⴱ and
the fatigue life is given by Eqs. (6), (5) and (9). Assuming a constant maximum stress they indicate that as the stress ratio increases the fatigue life will increase. In fact this is covered by experimental observations [3]. Fatigue failure will not occur for l 0: Since the respective stress ratio is 1, indicating a constant loading, the scale factor l is consistent. 5. Experimental verification In the previous sections a model to describe composites whose fatigue behaviour is matrix controlled was presented. The unified fatigue parameter W ⴱ is now applied to demonstrate its capability of predicting the fatigue lives. Three different sets of data [2–4] are examined. The last one used unidirectional carbon/epoxy laminates to study the off-axis fatigue behaviour. The other studies carried out uniaxially loaded fatigue tests on unidirectional E-glass/ epoxy laminates for different fibre load angles. However, only El Kadi and Ellyin [3] examined different stress ratios. 5.1. Fatigue life predictions for different fibre load angles Following the flow-chart in Fig. 3, two of the available sets of data [2,4] are examined with respect to different fibre load angles only. The stress ratio was constant at R 0:1 for both studies. Applying the finite element analysis to the different fibre load angles, the ratio of normal stress s 22 to shear stress t 12 changes significantly as shown in Fig. 5 for the E-glass/epoxy data. As expected from the limit criteria, failure is dominated by shear at small fibre load angles and transverse normal stresses at higher fibre angles. This fact may cause difficulties when attempting to normalize fatigue life data using the classical theories mentioned in Section 3.1.
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Fig. 6. Original S–N data for various fibre load angles [2].
5.1.1. E-glass/epoxy The experimental data of Ref. [2], corresponding to six different fibre load angles u , namely 5, 10, 15, 20, 30 and 60⬚ are first examined. E-glass fibres Gevetex WS 13-320X1K921 in epoxy resin Bakelite ERL 2256 containing a fibre volume fraction of V f 0:6 have been used. The macrostress-fatigue life data s max versus Nf were obtained from rectangular specimens tested with R 0:1: The original S–N data indicated that the fatigue life was strongly dependent upon the fibre load angle as shown in Fig. 6. For all angles failure occurred by a crack through the specimen parallel to the fibre direction. These fracture planes represent critical planes as introduced. Hence, the
numerical method may be applied. In Fig. 7 the unified fatigue parameter is plotted against the number of reversals to failure using a log–log co-ordinate system. The scale factor l determined from Eq. (6) is included in the correlation. No obvious fibre load angle dependence is seen despite the existence of some scatter. The unified fatigue parameter collapses all data points on a single straight line determined by the least square method as log W ⴱ ⫺0:137 log Nf ⫺ 0:339 W ⴱ 0:458 Nf ⫺0:137 :
11
Since the unified fatigue parameter has not been derived
Fig. 7. Correlated fatigue lives for various fibre load angles.
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Fig. 8. Comparison of experimental data with predicted fatigue lives using W ⴱ for u 20⬚:
with respect to a particular fibre load angle, an arbitrary one can be chosen to predict the fatigue lives of specimens tested at the same stress ratio but different fibre load angles. To show the universal applicability, data obtained for u 20⬚ are used to determine the relationship between W ⴱ and Nf, which is log W ⴱ ⫺0:129 log Nf ⫺ 0:390 ⴱ
W 0:407 Nf
⫺0:129
:
12
In its turn Eq. (12) may be used to predict fatigue lives of the other specimens. A comparison of the predicted fatigue lives and the experimental results is given in Fig. 8. Predic-
tions and experimental data show very good agreement. However, the fatigue lives for u 5⬚ are slightly overestimated. This may be because of a degradation in fibre strength with a number of cycles when the fibre load angle is small [1], which is not taken into account in this matrix controlled fatigue model. 5.1.2. Carbon/epoxy To confirm the capability of the unified fatigue parameter to predict fatigue lives for different fibre load angles a second set of experimental data is analyzed. Awerbuch and Hahn [4] studied the off-axis fatigue behaviour of unidirectional AS4 Carbon in 3501-5A epoxy resin. The
Fig. 9. Original S–N data for various fibre load angles [4] and comparison with predicted fatigue lives using W ⴱ for u 20⬚:
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Fig. 10. Correlated fatigue lives for various fibre load angles.
nominal fibre content was 0.7 by volume. Five different offaxis angles were tested: 10, 20, 30, 45 and 60⬚. Constant amplitude loading tests at R 0:1 were performed at about 18 Hz until fracture or 10 6 cycles, whichever occurred first. Essentially, Awerbuch and Hahn found that no strength degradation was observed in the specimens surviving fatigue loading of 10 6 cycles regardless of the fibre load angle or fatigue stress level. For the other specimens the effect of the off-axis angle on the S–N relationship can be easily identified (Fig. 9) although large scatter were observed. The fatigue lives for the 30⬚-specimens even tend to increase as the maximum applied load increases. The authors concluded that a specimen being fatigue cycled at a certain dynamic stress level may fail randomly along the fibre directions (Fig. 1) at any cycle number without any warning or visible damage. Following the procedure described above, the unified fatigue parameter is applied to correlate the fatigue data. Fig. 10 shows that all data points fall on one single line as for Eglass/epoxy. The best fit relationship is log W ⴱ ⫺0:0741 log Nf ⫺ 0:438
13
W ⴱ 0:365 Nf ⫺0:074 :
It is interesting to note that the normalized fatigue data for Table 1 Correlated fatigue data 2Nf [1]
2
10 10 6
W ⴱ (MPa) E-Glass/Epoxy
Carbon/Epoxy
0.27 0.08
0.27 0.14
E-glass/epoxy and carbon/epoxy have the same order of magnitude as Table 1 indicates. For low cycle fatigue, the superior properties of carbon fibres are only of marginal influence. Both composites require the same energy input to failure. However, for high cycle fatigue the advantage of using carbon fibres instead of E-glass fibres is apparent due to the much shallower curve. Nearly twice as much energy must be supplied to cause failure in the carbon/epoxy composite when compared with E-glass/epoxy. A similar trend was observed by Curtis [17] where only the degradation of strength was considered. To be consistent with the analysis of the E-glass/epoxy data the fibre load angle of u 20⬚ is chosen to predict the fatigue lives. The relationship between W ⴱ and Nf is expressed in Eq. (14), which may now be applied to predict the fatigue lives for all other fibre load angles. Good agreement is seen for u 10; 45 and 60⬚ as Fig. 9 shows. The fatigue lives for u 30⬚ are overestimated. This may be explained with the atypical shape of the original S–N curve where the fatigue life increased as the maximum stress increased. Data from a larger sample population will probably show better agreement with the predictions. log W ⴱ ⫺0:085 log Nf ⫺ 0:321 W ⴱ 0:478 Nf ⫺0:085 :
14
It is worth pointing out that the unified fatigue parameter is sensitive to the number of cycles to failure. Rearranging Eq. (14) with respect to Nf shows that an increase of the unified fatigue parameter by 10% decreases the fatigue life by 68%. Despite this sensitivity good agreement between experimental data and predicted fatigue lives has been found. Therefore, one can conclude that the proposed model and the finite element analysis describe the physical
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Fig. 11. Original S–N data for various stress ratios and u 19⬚ [3].
reality sufficiently well to be extended to different stress ratios. 5.2. Fatigue life prediction for different stress ratios To assess the capability of the unified fatigue parameter correlating fatigue data obtained from different stress ratios, another set of experimental data is required. El Kadi and Ellyin [3] performed tests on 3M type 1003, a unidirectional E-glass fibre reinforced epoxy resin with a fibre volume fraction of 50%. The specimens were tested at room temperature under various stress ratios R 0:5; 0 and ⫺1) and off-axis angles u 19; 45 and 71⬚). It should be noted that these
specimen geometries differed considerably from those used by Hashin and co-workers [2,4] which is a direct consequence of the compression loading excursion included in the load cycle when applying R ⫺1: The cross-section area was approximately three times as large, whereas the gauge length was less than half as long. For the off-axis cyclic tests all specimens had an identical brittle failure mode with matrix failure parallel to the fibres. The original data indicated the slope of the S–N curve to be dependent on both the fibre load angle and the stress ratio. Fig. 11 shows the S–N curves for u 19⬚ tested under three different stress ratios. Here, the data for R ⫺1 are given for completeness since tension and compression do not contribute equal parts to the damage development. Plots
Fig. 12. Correlated fatigue data for stress ratios of R 0:5 and 0 and fibre load angles of u 19; 45 and 71⬚.
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Fig. 13. Comparison of experimental data for R 0 and u 45⬚ with predicted fatigue lives determined using W ⴱ for R 0:5 and u 45⬚:
for the other angles show the same effect on lower stress levels. For a given fatigue life and fibre load angle the failure stresses for R 0:5 are the highest whereas those for R ⫺1 are the lowest. For non-negative stress ratios this observation agrees with the proposed scale factor l . If a laminate is not only exposed to tensile but compressive loads, the modelling of damage development is much more difficult. A proper degradation model has to incorporate fibre dominated failure mechanisms such as fibre buckling and the development of kink bands. Those mechanisms are not covered by the proposed matrix-controlled fatigue model which was derived for tension–tension fatigue, i.e. non-negative stress ratios. Tests performed at those stress ratios are governed by similar damage mechanisms and their
results are suitable for comparison purposes. Hence, only data obtained from non-negative stress ratios are subsequently considered. Fig. 12 shows the correlated fatigue data for R 0:5 and R 0 and fibre load angles of u 19; 45 and 71⬚. The data points for R 0:5 and R 0 are no longer segregated as in the original S–N plots, but have been combined by the unified fatigue parameter. Compared with the previous analysis these data are located at a higher W ⴱ level. This may be due to the difference in specimen geometries. If the unified failure criterion were universally applicable for predicting fatigue lives for different stress ratios and different fibre load angles, any set of data obtained from an arbitrary combination of fibre load angle and stress
Fig. 14. Comparison of experimental data for R 0; u 19 and 71⬚ with predicted fatigue lives determined using W ⴱ for R 0:5 and u 45⬚:
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ratio must be capable of predicting fatigue lives for any other set. Analysis of the data sets [2,4] has already shown the applicability to different fibre load angles. Therefore, the unified failure criterion is now applied to different stress ratios. The best fit curve for R 0:5 and u 45⬚ is given in Eq. (15). The stress ratio of R 0:5 is chosen to eliminate the possibility of having a compressive load acting on the specimen when using R 0: The choice for u 45⬚ is based on having the most data available for determining the regression curve. In its turn, Eq. (15) is now applied to predict fatigue lives for other combinations of stress ratio and fibre load angle. log W ⴱ ⫺0:080 log Nf ⫺ 0:139 0:726 Nf ⫺0:080 : 15 Fig. 13 compares experimental data obtained for R 0 and u 45⬚ with predictions using Eq. (15). The fibre load angle remained unchanged whereas the stress has changed. A very good agreement is found. The most general case displays Fig. 14. Here, Eq. (15) is applied to predict fatigue lives for another set of stress ratio and fibre load angle, in particular R 0 with u 19 and 71⬚. Although the experimental data for u 19⬚ are underpredicted and those for 71⬚ slightly overpredicted, good agreemeent is still observed. Thus, the unified fatigue parameter is capable to predict fatigue lives of unidirectional laminates under cyclic tension–tension loading considering various combinations of fibre load angle and stress ratio. 6. Conclusions An approach to determine the fatigue properties of unidirectional laminates from limited experimental data under simple loading has been developed. The properties are written in a generalized form independent of the loading mode to subsequently apply them for predicting fatigue lives of laminae subjected to other loading modes. The proposed unified fatigue failure criterion is based on the actual microstresses and microstrains instead of imaginary laminate values. Since failure is related to micromechanics the generalization of fatigue data for different loading modes is more easily performed using the present technique than with traditional macromechanical approaches. Together with the proposed fatigue parameter a method of fatigue life prediction has been developed. Being based on strain energy densities the proposed unified fatigue parameter satisfies the conditions of continuum mechanics. Both, normal and shear contributions acting in the fracture plane are considered. The main advantages of the unified fatigue parameter are accounting for mean stress effects and being load path independent since only peak values instead of values throughout the cycle are needed. This methodology considerably reduces the number of parameters to be determined experimentally. The unified fatigue parameter has been successfully
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applied to predict fatigue lives of different unidirectional laminates under cyclic tension considering various combinations of fibre load angle and stress ratio. All data points are collapsed on a single straight line and expressed by an equation determined by the least square method. The predicted results and the experimental fatigue data are in good agreement. The successful correlation of the fatigue data using the fracture plane based unified fatigue criterion validated the assumption of matrix controlled fatigue. Using limited experimental data the unified fatigue parameter enables a generalized strain energy–life relationship to be determined. Once such a relationship is established, the fatigue life of unidirectional laminates cycled under any other combination of fibre load angle and stress ratio can be predicted. Therefore, the unified fatigue failure criterion is a potentially valuable tool for fatigue design.
7. Recommendations Further experimental verification is necessary to extend the unified fatigue parameter to loading other than unidirectional tension. The form of the proposed parameter suggests that multi-axial fatigue data can be successfully normalized as well. If extended to multi-directional laminates, the proposed model may also be used to describe delamination failure by defining the examined planes parallel to the layers. As an initial step only simple unidirectional laminates loaded in cyclic tension were considered in the present study. Incorporation of compressive failure mechanisms should be the challenge since tension and compression are governed by different failure mechanisms and do not contribute equal parts to damage development. Temperature should be considered as a variable, allowing the effect of temperature on fatigue life to be assessed. If such an approach is found to be satisfactory it may be extended to creep studies and environmental degradation.
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