7. FATIGUE OF COMPOSITE MATERIALS This section is aimed at introducing the basics of fatigue testing nomenclature and procedures. It is not by any means intended to cover all the basics of fatigue in general or for composite either for that matter, but rather to serve as an introduction to the concept and some initial thoughts about fatigue in composite laminates. The basics of fatigue theories and procedures are described in textbooks such as Fuchs and Stephens (1980), Suresh (1991) and Reifsnider (1991). First of all, the fatigue life of classical engineering materials are difficult to predict. For composites it is then not surprisingly even more difficult. There are several reasons for this. A composite lamina or laminate has many failure modes and failure mechanisms. These mechanisms will respond differently to fatigue loading. The stress distribution in a composite lamina may be several orders of magnitude different in different directions even tough the strains may be of the same order, due to a strong anisotropy. This in turn highlights the problem of using some generalised stress measure, like the von Mises stress commonly used for metals, in predicting the fatigue life.

7.1 The Stress Life Approach

The basic fatigue characteristics for an emerging material or material combination are essential to gain confidence in the durability of the material and its structural application. The stress life approach to fatigue was first introduced in the 1860s by Wöhler. Out of his work evolved the concept of an “endurance” limit, which characterises the applied stress amplitude below which a material is expected to have an infinite fatigue life. This empirical method has found widespread use in fatigue analysis although it does not account for plastic deformation during the cyclic loading. The basis for the creation of Wöhler curves or S/N curves is constant amplitude testing of smooth specimens, i.e. the test specimens are cyclically loaded between a maximum and minimum stress (or strain) level, Smax and Smin until failure occurs. The notation is illustrated in Fig.7.1, where the mean stress and the stress amplitude are defined as

S S Samp =

−max min

2 (7.1)

S S Smean =

+max min

2 (7.2)

7.1

Foundations of Fibre Composites

Smax

Smin

load cycle0

Smean

S

time

Samp

Figure 7.1 Nomenclature for constant stress amplitude loading

The S/N diagrams, where S is the applied stress and N is the number of load cycles can be plotted in a logarithmic or in a semi-logarithmic diagram (generally is S on a linear scale and N on a logarithmic) as shown in Fig.7.2. In a S/N diagram the total specimen life is plotted, where the total life implies the number of load cycles necessary to initiate fatigue cracks in the smooth specimens plus the number of cycles to propagate the dominant fatigue crack to failure. The stress, S, in eqs.(7.1-2) may be replaced with strain or even a stress intensity factor.

N

Region I Low cycle fatigue

Region II High cycle fatigue

Region III Endurance limit

S

Figure 7.2 Typical S/N diagram with the line showing a piece-wise linear representation of the fatigue function.

Under constant amplitude loading many engineering materials exhibit a plateau in the stress-life plot typically beyond about 106 fatigue cycles, which also seems to be valid for the sandwich core materials investigated herein. This load level below which the specimen may be cycled an infinite number of times without showing any crack initiation, or propagation of an existing crack, is called the endurance limit or threshold level. Tests performed below this level are generally interrupted and their corresponding result representation in the S/N diagram is accompanied by an arrow indicating a non-failed test, as illustrated in Fig.7.2. One usually divides the S/N into three regimes. The first regime is commonly denoted the Low cycle fatigue regime which is indicated by a high maximum stress level in the load cycle and low number of cycles to failure. The maximum stress is usually near or above the plastic yield stress of the material. The second regime is usually called the High cycle fatigue regime. In this regime the log(stress) vs. log(cycles) plot commonly lies on a straight line as indicated in Fig.7.2. This regime is commonly valid for in the regime up to 106 or 107 load cycles to failure. The maximum stress in the load cycle is now well within the elastic regime. The final regime is the so called Endurance limit. At fatigue stress levels below a certain value no

7.2

Fatigue of Composite Materials

failure can obtained whatever number of load cycles are applied. Typically, the number of load cycles applied in testing to obtain this limit is in the order of 106 to 109 depending on application. One can debate whether there is such a thing as an endurance limit and for some materials it is argued that such a limit does not really exist. Many components are actually designed based on the stress level near the endurance limit, particularly components that are subjected to many load cycles in its predicted life-life, one examples is rail-way wheel axles. The fatigue life of a material or a component may differ dramatically under constant amplitude loads with a maintained maximum applied load but changed minimum load or rather, changed amplitude. Therefore is the load ratio, R, introduced as

R SS

= min

max

(7.3)

where R<0 corresponds to a load cycle with both compression and tension loading or, as in the majority of the investigations in this thesis, positive and negative shear. The interval 0<R<1 represent tests under tension/tension loads and R>1 corresponds to compression/compression loading. The characterisation of a new material generally involves test series at different load ratios. The fatigue life of most materials will decrease with increasing mean stress level and thus increasing R-value. The schematic effect of increased load ratio is illustrated in Fig.7.3.

N

σ

R

σcr

Figure 7.3 S/N diagram for differnt R-values

The time to failure may be divided into two phases, the nucleation and formation of small cracks and then when a one or several dominating macroscopical cracks have formed, the crack propagation phase. The major part of the fatigue life is often the damage nucleation. Since this nucleation phase can vary significantly between, not only material types and qualities, but even between specimens from the same material batch. Therefore the data scatter must be considered when fatigue data is interpreted. A significant parameter in fatigue is the statistical distribution of fatigue data.

7.2 Fatigue Life Representation

For emerging material systems and combinations, applications normally precede and drive the development of life prediction methods. As a result, many empirical models have been used and are still used to characterise the fatigue life of sandwich structures and the materials of the core and faces. Some of these are Suresh (1991)

(7.4) ( )S N S Ncra=

7.3

Foundations of Fibre Composites

(7.5) ( )S N S b Ncr= − log

where S is the applied maximum cyclic load, N is the number of cycles, Scr is the static strength and a and b are material parameters. Eq.(7.4) is the classical power-law fatigue criterion producing a linear S/N curve in a logarithmic plot while Eq.(7.5) is a linear representation of the fatigue data in a semi-logarithmic plot. A more sophisticated model including the fatigue threshold, Sth, is the following smooth fatigue data prediction curve

(7.6) S N S S S eth cr thN a b

( ) ( ) log( / )= + − −

This equation was proposed by Weibull (1951, 1961) to describe statistical variations in e.g. mechanical testing of materials. The purpose of a curve fit is to obtain a simple representation of the stress life behaviour, which can be used in the design process. The material parameters are found as a best curve fit by i.e. minimising the quadratic error between the test data and the theoretical value of eq.(7.6). Mean Stress Effect It is possible to plot the test results with their curve fits (or prediction curves) for different load ratios, R, in one S/N diagram. But more generally is the effect on the fatigue life as a function of the mean stress or the stress amplitude of interest. The mean stress effects in fatigue can also be represented in terms of constant-life diagrams, as shown in Fig.7.4. In these models different combinations of the stress amplitude and mean stress are plotted to provide a constant life. Most well known among these models are those due to Goodman (1899)

SS

SS

amp

fs

mean= −1$

, (7.7)

Gerber (1874),

SS

SS

amp

fs

mean= −

12

$ (7.8)

and Soderberg (1939)

SS

SS

amp

fs

mean

yield

= −

1 (7.9)

where Sfs is the fatigue strength (for a fixed life) for fully reversed loading (Smean=0 and R=−1) and Syield is the yield strength of the material. The constant amplitude diagram is usually used when only the static strength and the fatigue strength have been evaluated or are accessible.

7.4

Fatigue of Composite Materials

Samp S fs

Smean

GerberGoodman

Syield

Soderberg

1

1 Scr

Figure 7.4 Constant life diagram

If a sufficient number of tests are conducted, a Haigh diagram as shown in Fig.7.4, can be constructed for constant lifetime curves. Again the mean stress is represented on the x-axis and the stress amplitude is given on the y-axis. If this type of diagram is to be fully constructed numerous test results for different combinations of mean stresses and amplitudes are required. When a Haigh diagram has been used in the preceding chapters a short description of its construction is of interest. The mean stress is a linear function of the stress amplitude for a fixed R and these ratios are plotted in Fig.7.5 as solid lines. Actual test results may be included in a Haigh diagram but the interpretation is made easier by using data from the curve fits (Eq.(7.6)) based on the test results. Data points for a fixed interval, one per decade, of load cycles are then plotted in the diagram. These data points are not actual test results but are based on the best curve fit and are thus a good representation of the actual test results. By connecting these points with a curve or a line a visual representation of the relation between the mean stress and the stress amplitude is possible.

R = 0

R = 0.5

R = -0.5

106

102

smean

s amp

scrit

High cycle fatigue

Low cycle fatigue

Figure 7.5 Principle of Haigh diagram

The prediction curves in Figs.7.3-4 can be extrapolated to the left of the ordinate axis to represent the effect of compressive mean stresses.

7.3 Material degradation due to Fatigue Loading

When performing fatigue tests at constant stress amplitudes the cyclic softening or hardening of the material is accompanied by an increase or a decrease in the strain amplitude. The cyclic softening behaviour is shown schematically in Fig.7.6. Consequently, if a constant strain amplitude is applied, an increase or a decrease of the stress level in the material will take place.

7.5

Foundations of Fibre Composites

σ ε

(a) (b)

t t

Figure 7.6 (a) Stress controlled loading and (b) the strain response due to cyclic softening.

The cyclic softening during the fatigue life can be expressed graphically in stress-strain loops, as shown in Fig.7.7.

ε

σ

ε

σ

E

Cyclic stress-strain curve

ε∆

εp εe

σ∆

(a) (b)

Figure 7.7 Schematic representation of (a) stable stress-strain hysteresis and (b) cyclic stress-strain curve drawn through the tips of the stable loops.

Stress-strain hysteresis is particularly important for polymeric materials, the matrix material, and less important for the fibres.

7.4 Crack Growth Approach

A material that has some sort of sharp notch or crack will behave differently under fatigue loading. An un-notched specimen of part of a structure can usually be subjected to a certain number of load cycles without any significant change in the behaviour. The stiffness and strength remains unchanged. However, after a critical number of load cycles, called the initiation time, or number of cycles to initiation, defects or cracks will form which will grow for each load cycle. The time or number of load cycles it takes for these to grow so large that the structure will fail is usually much shorter than the initiation time, it may only be a handful of load cycles. A structure containing a defect or crack with some macroscopic dimension will respond to fatigue in a quite different manner. The defect will now rather grow, a small distance for each load cycle. The crack propagation rate, da/dN, is usually plotted against the stress intensity range, ∆K, where the crack propagation is divided into three phases: initiation, stable crack

7.6

Fatigue of Composite Materials

growth and unstable crack growth, as illustrated in Fig.7.8. The first part deals primarily with non continuum failure processes, where the increment of average crack growth is less than 10-

6 mm (cycle)-1. In this regime, the stress intensity factor range approaches the fatigue crack growth threshold, Kth. The intermediate part is also called the Paris regime and is generally the most interesting since there is a linear relation in the log-log plot between the crack propagation rate and the stress intensification range due to the notch. This corresponds to a stable crack growth and this relation is written as

dadN

c K m= ∆ (7.10)

where m is the slope of the curve and c is the point where an extension of the curve will intersect with ∆K=1 MPa m. Regime III, at very high ∆K values, the fatigue crack growth rates are significantly higher than those observed in regime II, Paris regime. A higher sensitivity of crack growth to microstructure, load ratios and stress ratios is also noticed in regime III. Additionally in Fig.7.8 the effect of load ratio is shown. The enhanced influence of load ratio is a consequence of the critical condition that the maximum stress intensity factor value for the fatigue cycle, Kmax, approaches the fracture toughness of the material, Kc. Since the ∆K values at which Kmax begins to approach Kc are lower for high R ratios, catastrophic fatigue failure occurs at lower ∆K values with increased load ratio.

log dadN

log ∆K

I II III

m

K c∆ thK

low Rhigh R

Figure 7.8 Fatigue crack growth da/dN versus stress intensity amplitude ∆K.

Pre-cracked specimens are primarily used to study the crack propagation rate in the considered materials. These types of specimen have one or two initial cracks from which propagation will occur. Two standard types of such specimens are shown in Fig.7.9: the single edge notched bending specimen, SENB, and the compact tension specimen, CT. Both are used for Mode I crack growth investigations and the test methods and the geometry of these specimens are thoroughly described for example in ASTM standards.

7.7

Foundations of Fibre Composites

P

W

4W

aa

P

P

W

1.2W

(a) (b)

Figure 7.9 (a) The single edge notched bend (SENB) and (b) the compact tension (CT) specimens for measuring the Mode I crack propagation rate (the dashed lines indicate the crack growth direction).

For metals, the compact tension (CT) specimen, shown in Fig.7.8(b) is commonly used to measure the fatigue crack growth in the Paris’ regime (regime II) and the test procedure specifications are found in ASTM specification E647-93. The stress intensity factor for this configuration, as function of crack length, is given by the relation

∆∆K P

BWaW

= ⋅

1 2/ φ (7.11)

where φ is a finite width correction factor, approximately given by the relation

φaw

=2

1

0886 4 64 1332 14 72 56032

2 3+

−

+

−

+

−

aW

aW

aW

aW

aW

aW

. . . . .4

(7.12)

For composite laminates, crack propagation is common in practice as the growth of delaminations, which may occur in both Mode I and Mode II. The specimens used to test this are the same as for static fracture mechanics testing described in Chapter 9.9. A typical Paris’ law relation is shown in Fig.7.10. This one is, however, not obtained for a classical engineering material, but for a closed cell polymer foam. The data is given for crack propagation in both mode I (opening) and mode II (shear). As seen, the both the crack propagation rate and slope of the Paris’ law curve is higher in mode I than in mode II.

7.8

Fatigue of Composite Materials

10-7

10-6

10-5

0.0001

0.001

0.01

0.1

0.02 0.06 0.1log ∆K, MPa √m

log

(da/

dN),

mm

/cyc

le

H100 (mode I)C=3995m=7.59

H100 (mode II)C=0.17m=5.68

(a)

Figure 7.10 Paris’ law curves for crack propagation in mode I and mode II for a closed cell polymer foam (Divinycell H100)

7.5 Fatigue of Solid Polymers The characterization of the fatigue performance of polymers is performed in many ways done as for metallic materials. Typical Wöhler-curves are based on constant strain or stress fatigue testing. As seen in Fig.7.2 the curve consists of three distinct regions. The existence of a region I is dependent on whether crazes are formed and whether these crazes causes micro cracks to nucleate under the high loading or not. If the maximum tensile stress in the very first cycles is not sufficiently large to form crazes, this region I will not exist. The region I will mere be an extrapolation of the slope of the S/N curve in region II. At the high end of region II the dominating damage mechanisms is the slow growth of crazes and their transformation into cracks. Region III forms the endurance limit for the polymer, the fatigue life in this region is controlled by the incubation time for the nucleation of microscopic flaws. Three different, irreversible damage mechanisms can occur in polymeric solids; crazing, slip-band formation, and thermal degradation. If the damage is introduced by crazing, slip-band formation can accumulate under either monotonic or cyclic loading. Polymeric materials undergo cyclic softening when subjected to fatigue loads as illustrated in Fig.7.7. This is most pronounced for ductile polymers, but amorphous, semi-crystalline and polymer-matrix composites also exhibit cyclic softening. Changes in the extent of cristallinity mainly affect the degree and the rate of the cyclic softening. Crazes Crazes occur in glassy polymers that are subjected to tensile stress at low temperatures. Above a certain stress level, striations appear in planes perpendicular to the direction of the loading. Glassy polymers are e.g. polystyren (PS), polymethylmetacrylate (PMMA),

7.9

Foundations of Fibre Composites

polypropylen (PP). There is a continuity of material across a craze whereas the faces of a Griffith crack in a in a brittle solid are fully separated. A craze contains fibrils of highly orientated molecules separated by porous regions. The density of the material in the craze is some 40-60% of that of the polymer itself. Cyclic deformation, fatigue, in many polymers is dictated by the nucleation, growth and breakdown of crazes. Crazes will only form under local tensile loading, cyclic softening is only observed in the tensile portion of fatigue, and hysteresis loop remains stable in the compression portion. Craze zones ahead of a crack in a polymer is somewhat analogous to the plastic zone ahead of a crack in a brittle solid. Slipband At stress levels lower than the strength of the glassy polymer, ‘plastic’ deformation can be initiated by the formation of shear bands, or slip-bands. Heating A major aspect in fatigue of polymers is the question of adiabatic heating, which can lead to failure due to thermal heating. Energy is absorbed under cyclic loading, either by a high surrounding temperature or by internal micro-crack friction. This heating will weaken the polymeric chains and thus reduce the resistance to deformation. At high strain rates, i.e. high frequencies the fatigue resistance will decrease due to the internal heating as illustrated in Fig.7.11. The fatigue resistance is further dependent on the volume to surface ratio of the specimen, the heat loss is through the external surface of the specimen. A larger specimen will suffer from increased internal heating and temperature raise compared with a smaller specimen with the same cross section geometry and stress. Not only will the geometry affect the heating, different polymeric materials have different visco-elastic damping. The fatigue strength of a material is increased with increased visco-elastic damping.

f

stre

ss

log N

Figure 7.11 Effect of cyclic frequency, f, on the fatigue strength.

7.6 The Fatigue Process in Composite Lamina – Loading Parallel to the Fibres

The fatigue mechanisms in composites are similar to that for metals in that they both involve a fatigue crack initiation phase and crack propagation phase. In another sense it is very dissimilar, containing a complicated micro-mechanical process. The fatigue process described in this chapter is more thoroughly described by Talreja (1987). Firstly, we should consider plotting the standard stress-life (S-N) diagram in a somewhat different form. It appears to be much more logical to use strain versus number of cycles than stress. Now, why is that so? There are two ways to reason; firstly, imagine two unidirectional lamina of identical material constituents but with different fibre volume fractions. The one

7.10

Fatigue of Composite Materials

with high fibre volume fraction will have a much higher strength, but the strain to failure is governed by fibre properties and will be (usually!) the same. The fatigue life should also then be same in terms of strain, but not stress. If we then make a laminate with layers other than zero plies, the strength will decrease compared to the unidirectional laminate, but since final failure is governed by fibre fractures the strength will be different, but not the strain to failure. A similar argumentation can be applied to fatigue. Before actually looking at fatigue data to see what happens, let’s make a phenomenological discussion about fatigue. The discussion is basically taken from the book by Talreja (1987) and most of the figures and graphs are from the same book (reproduced with permission). Fatigue at high loads, near the static strength We all know that material fracture to some extent is statistical, i.e., the fracture stress is not a sole given number, but varies slightly between two identical specimens. Our composite laminates consists of a large number of fibres, each fibre having slightly different strength and the strength may even vary somewhat along the fibre itself. At high load levels, near the static strength of the laminate, some fibres will fail during the first load cycle, providing the stress is beyond the strength of weakest fibre in the laminate. This will happen everywhere in the laminate and independently of each other. At the next load application (second load cycle), the stress state will be somewhat different due to the already failed fibres, and some more fibres will fail, still independently of each other. The same thing will happen at the third load cycle, and so on. One can argue that this process is unaffected by any other type of failures occurring, like matrix cracking and that the fatigue life is totally governed by this random fibre fracture process. When the number of fibre failure becomes large enough in a local area, the load redistribution in such an area implies that more fibre fractures will occur due to the stress concentration created and a crack will be formed that will grow very fast during the next few load cycles, leading to final failure. This is schematically illustrated in Fig.7.12.

Figure 7.12 Fibre breakage in unidirectional composites under loading parallell to fibres.(reproduced with kind permission of Prof. R. Talreja)

According to this discussion, the fatigue life of the composite can vary between a few load cycles only (actually the specimen can fail already in the first load cycle) or at basically any number of cycles, since the process of fibre fracture chaotic in nature and does not involve any growth mechanism until the very last few load cycles. One can then argue that the fatigue

7.11

Foundations of Fibre Composites

life according to this mechanism short produce a horizontal scatter band in the fatigue life diagram, as shown in Fig.7.13. The strain at this level is a scatter band around the strength of composite, denoted εc.

Figure 7.13 Fatigue life diagram for uni-directional composites under loading parallell to fibres.(reproduced with kind permission of Prof. R. Talreja)

The process of fibre breakage is non-progressive since there is no damage zone growing from the early stages of fatigue loading, i.e., the damage progression cannot be traced. Due to this, the scatter in the fibre breakage is not dependent on the fatigue life, but the apparent fatigue life dependency in this band comes from the fact that the probability of finding a cross-section with enough broken fibres to cause failure under the applied maximum strain increases with the number of cycles. Fatigue at low loads We can continue this discussion for the case of very low loads. Assume that the strain during fatigue is so that during the first load cycle no failures occur at all – no fibre break, there are no crack developing in the matrix. At the second load cycle, nothing has changed, there is no active failure mechanism, no energy dissipating mechanisms has occurred. Thus, the second load cycle is identical to the first and there will be no failures even in the second load cycle. For such loads, the composite will not suffer from fatigue, which is called the fatigue limit. Actually, this load level is found, in most cases, to be the same as the fatigue limit of the matrix material, denoted εm, and will appear in the fatigue life diagram as another horizontal line (see Fig.7.13). We can look at this in a slightly different manner too. The fatigue mechanism in a polymeric matrix is similar to that of metals, involving crack propagation normal to the tensile load. Under cyclic loading, the constrained matrix is subjected to strain controlled fatigue and if the strain is above the failure strain the matrix cracks. At high strains, these cracks could grow, while at low strains, they could stop at the fibre/matrix interfaces. If the load is low enough, matrix cracks could form, but they would all stop due to the inhomogeneous nature of the material and a fatigue limit is reached. Thus, although there may exist an energy dissipating mechanism at the early stages of fatigue loading, it will cease to act if all matrix cracks will

7.12

Fatigue of Composite Materials

stop propagating. Then, from load cycle n to cycle n+1 there will be no more fracture surfaces created and we are below the fatigue limit. Fatigue at intermediate loads Firstly we must recognise that for cracks or damage to grow during fatigue loading, there must exist some kind of irreversible mechanism. In the case of metals, this energy dissipating mechanism is plastic yielding, e.g. in front of a growing crack tip. Thus, for cracks and damage to grow, the state of stress and strain in the material must change from one load cycle to the next. If there are no such changes, no energy is dissipated and there will be no growth. Under cyclic loading the matrix is subjected to strain controlled fatigue since the matrix is constrained between the load bearing fibres, (remember the assumptions made when the deriving E1 using the rule of mixtures in chapter 2.1). If this strain exceeds the fatigue (strain) limit for the matrix (not the composite!) matrix cracks will be induced. If the strain level is low, these matrix cracks will stop at a fibre interface, and we have reached the fatigue as discussed above. If, however, the strain level is high these crack may continue to grow during the next load cycle – there is damage growth, energy dissipated and the material will continue to degrade. In this stage, matrix cracks will form in the early stages of fatigue loading. If, however, the strain level is high the stress intensity at the crack tip might exceed the fracture stress of the fibre and hence this might lead to fibre failure. Now, a macro crack is formed, which will propagate in an opening mode to the next interface. Either this fibre also will fail and the crack will continue to propagate in an opening mode, or the shear stress at the crack tip will cause the crack to propagate in a shearing mode in the interface between the fibre and the surrounding matrix (see Fig.7.14). The debonding length depends on the shear strength of the interface and is usually small, of the order of a few fibre diameters. Again, the load will be redistributed to other fibres, which may again break, and more interfacial cracking will appear. The material will in this manner degrade until there are so many fibre failures that the entire laminate will fracture. Since there is slow degradation mechanism, the fatigue life of the material will depend on the load and the number of load cycles. In this regime there will thus be a dependence between the applied load (or strain) and the number of cycles creating an inclined relation in the fatigue life diagram, as illustrated in Fig.7.13.

Figure 7.14 Fatigue damage mechanisms in unidirectional composite laminate (a) fibre breakage with interfacial debonding, (b) matrix cracking and (c) interfacial shear failure.

Of course, all three types of damage mechanisms, matrix cracking, fibre fracture and interface cracking, may occur simultaneously. However, the predominant mechanism leading to failure would be effective in a limited range of the applied strain, i.e. the fatigue life at a specific

7.13

Foundations of Fibre Composites

strain level will be dependent of one of the three damage mechanisms. Based on this assumption it would be expected that the fatigue-life diagram would consist of different components, each corresponding to the underlying damage mechanism. Fig.7.15 shows an idealised schematic of fatigue damage mechanisms in unidirectional composites under tensile loading parallel to the fibres. The resulting fatigue-life diagram is shown in Fig.13 where the maximum applied strain is plotted against the logarithmic of the number of cycles to failure.

Figure 7.15 Fatigue damage mechanisms in unidirectional composite under loading parallal to fibers (reproduced with kind permission of Prof. R. Talreja)

The damage mechanisms in tensile fatigue may be divided into three types. The scatter band centred around the composite failure strain corresponds to fibre breakage. The sloping scatter extending from this band to the horizontal line representing the fatigue limit corresponds to matrix cracking and interfacial shear failure. The fatigue limit is defined as the strain corresponding to the boundary between the non-propagating matrix cracks and the propagating matrix cracks at 106 cycles. Effect of composite stiffness on fatigue performance Recall the strength prediction of composites discussed in chapter 4.3 and especially Fig.4.6. The strength of the composite is related to the lowest of either the matrix strain at failure or the fibre strain at failure. In most practical cases the fibre strain at failure is lower than the matrix strain at failure. For high modulus fibres this is most certainly the case. The fatigue life of unidirectional composites loaded in the fibre direction is also dependent on the modulus of the fibres. Let’s study two specific cases, one having a very stiff fibre with long strain to failure and the other being a less stiff fibre with higher strain to failure but still lower than the matrix strain to failure. Schematic stress-strain relations of these are shown in Fig.716.

7.14

Fatigue of Composite Materials

σ

ε ε

σ

εm εc εcεm

fibre fibre

matrix matrix

composite composite

Fig.7.16 Stress-strain relation for (a) low stiffness fibres and ( b) high stiffness fibres

We can now start to discuss the effect of these assumptions made. By plotting the strain versus the number of cycles in the S-N diagram (S is here interpreted as the strain, rather than stress), the effect of the fibre volume fraction should disappear, although the laminates will have very different strengths depending on the fibre volume fraction. This is indeed seen from Fig.7.17 where a unidirectional glass/epoxy laminate of various fibre volume fractions are shown.

Figure 7.17 Fatigue life diagram for glass-epoxy unidirectional composite.(reproduced with kind permission of Prof. R. Talreja)

The next issue is to discuss the effect of the different failure mechanisms. Assume first that we have a unidirectional laminate subjected to fatigue loading parallel to the fibres and that the fibres are very stiff and have a low strain to failure. If the fibre failure strain is lower than the strain at which matrix cracks develop, or at least so low that matrix cracks cannot grow, then the fatigue life is governed by the process of fibre breakage, and all data should lie within the scatter band corresponding to this failure mode. Data for such a unidirectional laminate are shown in Fig.7.18. The data is obtained for a high-modulus carbon fibre with a very low strain to failure (0.5%). In this case, and the cases discussed following this, the fatigue limit for matrix material (epoxy) is taken as 0.6% strain.

7.15

Foundations of Fibre Composites

Figure 7.18 Fatigue life diagram for high modulus carbon-epoxy unidirectional composite.(reproduced with kind permission of Prof. R. Talreja)

Data like the one in Fig.7.18 came out rather early in the development, the data in Fig.7.18 is from 1973 (recall that carbon fibres first were produced in a lab in 1961). Indeed all fatigue data points appear to be on, or near the scatter band extending from quasi-static tensile failure. As a result of such data the general opinion was that composites do not suffer from fatigue degradation, a view sometimes still heard, and a quite misleading conclusion. As the development of carbon fibres continued, more ductile carbon fibres were produced with higher strain to failure. An example of a fatigue life diagram for an intermediate stiffness carbon fibre composite is shown in Fig.7.19, a fibre with a strain to failure closer to 1%. According to the discussion above, there should be a region of progressive damage growth and degradation in the strain regime between the fibre failure scatter band and the fatigue limit of the matrix. As seen in Fig.7.19 a small such region seems to exist.

Figure 7.19 Fatigue life diagram for intermediate modulus carbon-epoxy unidirectional composite.(reproduced with kind permission of Prof. R. Talreja)

As the stain to failure for the fibre increases, the gap between the fibre fracture scatter band and the matrix fatigue limit increases and there should be a more pronounced region of fatigue degradation. Fatigue data for a high strain to failure carbon fibre laminate is shown in

7.16

Fatigue of Composite Materials

Fig.7.20 that seems to verify this hypothesis. The data for the glass/epoxy laminate shown in Fig.7.17 again points to the same thing.

Figure 7.20 Fatigue life diagram for high strain to failure carbon-epoxy unidirectional composite.(reproduced with kind permission of Prof. R. Talreja)

7.7 The Fatigue Process in Composite Lamina – Loading Off-Axis

The strength of a unidirectional composite loaded at an off-axis angle to the fibre will be drastically reduced compared to the on axis strength. The predominate failure mode will be matrix shear or delamination between fibres, as shown in Fig.7.21. The crack initiation in a unidirectional laminate subjected to off-axis fatigue loading will most probably be in this region, the interface between the matrix and the fibre. Such a crack will be subjected to a mixed mode loading; an opening and an in-plane (parallel to the fibres) shear loading. The opening part of the mix-mode loading will increase with the off-axis angle. The opening mode is the more critical of the two and hence the limiting strain under which no crack propagation will occur will decrease with decreasing off-axis angle. There is one main difference between the off-axis case and the case of loading along the fibres; any matrix cracks forming in the case of loading parallel to the fibres could arrest when meeting a matrix-fibre interface. In the off-axis case there are no discontinuities in the material that a growing crack can intercept. Thus, in the off-axis case one can assume that the fatigue limit, still being a matrix controlled property, will be lower in the off-axis case, being without any arrest mechanisms.

7.17

Foundations of Fibre Composites

Figure 7.21 Matrix and interfacial cracking under off-axis of unidirectional composites: (a) mixed mode crack growth, 0 < θ < 90° (b) opening mode crack growth, θ = 90°. .(reproduced with kind

permission of Prof. R. Talreja)

In Fig.7.22 the fatigue life diagram for off-axis fatigue of unidirectional composites is shown. At an off-axis angle of 90°, i.e. when the applied loading is perpendicular to the fibres and hence the crack growth will be in the opening mode only. This will lead to debonding between the fibre and matrix at a low strain and with a low fatigue limit.

Figure 7.22 Fatigue-life diagram for off-axis fatigue of unidirectional composites. Dotted lines correspond to the fatigue-life diagram for on-axis fatigue.(reproduced with kind permission of Prof. R.

Talreja)

Once the off-axis angle decreases from 90 degrees, the matrix will grow in a mixed mode I-mode II state, which requires a higher strain and also gives a higher fatigue limit. At sufficiently low off-axis angle matrix cracks will have to grow almost entirely in mode II and the crack arresting mechanism may again be present. The fatigue limit as function of off-axis angle is shown in Fig.7.23.

7.18

Fatigue of Composite Materials

Figure 7.23 Fatigue limit as function of off-axis angle for unidirectional glass-epoxy composite. (reproduced with kind permission of Prof. R. Talreja)

The lower limit of the curve in Fig.7.23 is usually denoted εd – fatigue limit for transverse fibre debonding which is the limiting strain value for the formation of interface cracks in a 90-degree lamina.

7.8 The Fatigue Process in Composites Laminates

Angle-plied laminates Angle-plied laminates consist of multiple layers of off-axis layer and thus the fracture mechanism is the same as in off-axis laminates, i.e. fibre matrix delamination. A new mechanism is added for angle-plied laminates, delamination between layers. The delamination is caused by the interlaminar stresses acting between the layers. This region is matrix dominated and thus the failure is governed by the matrix properties. There is one other important point to consider as well; as a fibre-matrix crack in one layer has grown all across the width of the specimen, the specimen will NOT fail completely, as it would have done for a unidirectional laminate. The reason is simply that there is another layer in another direction that will cross-over the crack and carry the load. Thus, one crack is not sufficient to break the laminate. As one crack in one layer has grown completely across the laminate, other cracks may start growing. However many of these crack will grow completely across the laminate, in either ply, will still not break the laminate. For complete failure of the laminate, delamination between the layers must develop. The fibre-matrix cracks in one layer create stress concentrations that will promote delamination (compare with discussion in Chapter 6 in delamination stresses appearing at free edges). For a [+θ/−θ]s where θ is a small angle, the cracks will grow mainly in mode II (slow) and very large delaminations must develop prior to complete fracture. For large angles, the reversed can be argued. For small angles, the fatigue limit will then be close to that of the matrix itself, εm, whereas for large angles it will approach the fatigue limit for the formation of delaminations - εd,l. An angle-ply laminate cannot have a fatigue limit lower than that of a unidirectional off-axis laminate. The mechanism of delamination required for failure will make angle-ply laminates much better, at least for small angles. This is schematically shown in Fig.7.24.

7.19

Foundations of Fibre Composites

Figure 7.24 Variation of the fatigue limit with fibre angle in a symmetric angle-ply laminate of glass-epoxy (reproduced with kind permission of Prof. R. Talreja)

Cross-ply laminates ([0/90]-type laminates) In a cross-ply laminate ([0/90]-type laminates) the first event of failure is fibre matrix interface debonding in the 90-degree layers. These cracks then grow toward the layer interfaces and upon meeting them create stress concentration. These stress concentrations create delamination in a similar way as in the angle-ply laminates, essentially reducing the laminate to a unidirectional lamina. The difference between the cross-ply and the unidirectional laminate is that the fatigue limit for cross-ply laminates now being governed by the fatigue limit for delamination growth, rather than the matrix material fatigue limit. A fatigue life diagram for a cross-ply carbon-epoxy laminate is shown in Fig.7.25. The top scatter band corresponding to fibre breakage is essentially the same as for a unidirectional laminate. The fatigue limit is given by the strain under which no transverse debonding cracks are formed. The progressive damage in the 90-degree layers leading to delamination is primarily responsible for the sloping scatter band.

Figure 7.25 Fatigue life diagram for cross-ply carbon epoxy laminate (reproduced with kind permission of Prof. R. Talreja)

7.20

Fatigue of Composite Materials

Other laminates The most common type of laminates consists of 0, 90 and ±45-degree layers, where the 0-degree layers are in the main loading direction. The mechanism of damage if found to be failure of the 90-degree plies (transverse fibre debonding) as in the cross-ply laminates, which leads to delamination and overloading of the 0-degree fibres. The fatigue life diagram is thus anticipated to show features as for unidirectional laminates, angle-ply and cross-ply laminates. The upper part of the fatigue life diagram (high strains) will then show a fibre breakage scatter band. This is followed by a sloping scatter band corresponding to matrix cracking, interfacial fracture and delamination. The lower limit will again be governed by the minimum strain causing delamination due to debonding in the 90-degree layers. For the epoxy resin used in the tests presented in Figs.7.26 and 7.27, this strain was found to be 0.46%.

Figure 7.26 Fatigue life diagram for [0/±45/90]s glass epoxy laminate (reproduced with kind permission of Prof. R. Talreja)

Figure 7.27 Fatigue life diagram for [0/45/90/-452/90/45/0]s carbon epoxy laminate (reproduced with kind permission of Prof. R. Talreja)

Recapitulating and summarising the development of damage in composite laminates under fatigue: In the initial stage matrix cracks will form in the layers with an off-axis lay-up (not parallel to the principal loading). These cracks will increase with the number of load cycles

7.21

Foundations of Fibre Composites

and will eventually form macroscopic cracks. These macroscopic cracks will show a classical crack propagation pattern. This stage is called the Characteristic Damage State, CDS, and has found to be independent of loading history but determined by the laminate properties, i.e. stacking sequence and ply stiffness. The matrix cracks will generally run through the ply thickness and also the ply width. These cracks will initiate microcracks in adjacent plies. In the ply interfaces close to the macroscopic and microscopic cracks strong interlaminar stresses develop which lead to separation between the plies locally. At this stage the rate of damage progression increases rapidly and will soon lead to an area where the local stresses reaches a level above the critical and a fracture is initiated. This is illustrated in the damage progression-time diagram in Fig.7.28. Two dominating stages can be clearly identified: the formation of local, independent matrix micro cracks and a second stage where various types and orientations of cracks interact with increasing rates which finally will lead to failure.

Figure 7.28. Development of damage in composite laminates under fatigue (reproduced with kind permission of Prof. R. Talreja)

7.9 Fatigue Design Criteria

In the drive for more optimised structures and products, the development is towards lighter, stronger and more reliable structures. Some of these requirements are in contradiction with each other and hence there is a demand for good and reliable design criteria. Different fatigue design methods are used depending of the application as described in e.g. (Fuchs, 1980). Infinite-Life A part or product designed using this approach part should not break during its entire estimated life. The stress level applied should be so low that they are safely below the fatigue threshold. This approach is used on components difficult to inspect or where an over-sized design is not crucial, e.g. valves in engines. This is the oldest criteria.

7.22

Fatigue of Composite Materials

Safe-Life This part should withstand a certain number of load cycles at a specified loading. The safe-life design often uses safety factors based on stress-life or strain-life calculations, either the results are taken from S/N diagrams or are results from spectrum fatigue tests which correspond to the load conditions of the part. When a part has reached its service life, it is replaced, whether it has failed or not. The safe life of a part has many uncertainties, such as changes in load conditions, scatter in the test results, and variations in the materials. By selecting large margins of safety or “safety factor” a safe operating life can be guaranteed. In the strive for high performance and low cost this design procedure is very conservative and maybe not optimum. The part may still fail due to initial flaws producing cracks which will propagate under fatigue loading and invoke a premature failure. Fail-Safe Design The weight penalty for the high safety factors used in the safe life designs call upon a new design criteria. In the fail-safe criteria fatigue cracks are tolerated but should not cause failure before they are detected and can be repaired or kept under supervision. Another way of achieving structural integrity is the usage of multiple load paths, i.e. if a part fails other parts will carry the load. The implementation of crack stoppers is also common in parts where a fail-safe design strategy has been used. The approach is commonly used in e.g. the aircraft industry (wings and fuselages). Damage Tolerant Design The approach assumes that fatigue cracks are present and uses fracture mechanics analyses and tests to check whether such cracks will grow large enough to cause failure before they are detected during a periodical inspection. In order to use this very sophisticated method designers have to evaluate all possible locations for fatigue crack onset. This is generally performed using high accuracy finite element evaluation. Of course this procedure requires massive resources and hence this design approach is almost only used in the aerospace industry.

7.10 Fatigue Testing

Some specimens used for fatigue testing are described in chapter 9.8 of this text. These specimens are unnotched and thus used for the extraction of S/N-curves. The specimen is usually subjected to a constant amplitude sinusoidal load variation with a given R-value and run until failure. One specimen will then provide one point for the S/N-curve. Testing at high load values, close to the static strength of the specimen, is difficult. Tests at low load values are very time consuming. The bulk specimens tested are usually at intermediate load values at fatigue lives in the range of 103-106 cycles to failure. Testing for crack growth data, or Paris’ law data, can be done using the same specimens for fracture toughness testing, as described in chapter 9.9. The loading is again commonly sinusoidal, with the load or the displacement controlled. The crack growth is monitored and from and knowledge of the load (or displacement) the stress intensity factor can be calculated and the Paris’ law curve created. In theory, the load or displacement can be varied so that the entire Paris’ law can be obtained with just one specimen, but usually at least a few specimens are required. Testing can be done in both mode I (e.g. with the DCB-specimen) or in mode II (e.g. the ENF specimen).

7.23

Foundations of Fibre Composites

7.11 Fatigue Life Estimation Theories

Fatigue prediction model by Hashin and Rotem Hashin and Rotem (1973) were one of the first to attempt to formulate a criterion for the prediction of the fatigue life of composite laminates. It is based on classical lamination theory and takes on a form similar to the Tsai-Hill criterion. They study a symmetric laminate (i.e. B = 0) subjected to uni-axial in-plane loading. The global co-ordinate system for the laminate is as usual denoted (x,y) and the local by (1,2). In a general state of plane stress, the global stresses transform into local stresses through the use of the rotation angle θ, which is the angle between the local (lamina fixed) and global (laminate fixed) co-ordinate systems. For a uni-axial loading, the global stress vector can be written as

σ = (σx, σy, τxy)t = (σ, 0, 0)t (7.13)

The stress in a lamina rotated an angle θ from the global co-ordinate system is then obtained through stress transformation to

σl = (σ cos2θ, σ sin2θ, σ cosθ sinθ)t (7.14)

From tests it has been observed that there are basically two different failure modes; for small θ (smaller than 1 to 2°) specimens fail by cumulative fibre failure. For larger angles the failure mode is a crack through the matrix, parallel to the fibres. By first studying the static case one can first assume that the observed failure modes are independent. Hashin and Rotem (1973) then suggest a failure criterion of the kind

11 σ̂σ = and (7.15)

1),( 122 =τσF

where F is function approximated by

1212122

22 =++ ττσσ CBA

Since the material should be insensitive to the direction and sign of the shear stress, the constant B must be taken as zero. Then, for the criterion to be applicable for stress in transverse loading alone (2-direction) and pure shear loading alone, the constants A and C must equal the square of the strengths under those conditions. Next, one must also realise that the failure stress may be different for tensile and compressive loading. By using this, the static criterion of eq.(7.15) may be written

t11 σ̂σ = when σ1 > 0 and c11 σ̂σ = when σ1 < 0 (7.16a)

1ˆˆ

2

12

12

2

2

2 =

+

ττ

σσ

t

when σ2 > 0 and 1ˆˆ

2

12

12

2

2

2 =

+

ττ

σσ

c

when σ2 < 0 (7.16b)

In the case of cyclic fatigue loading we first define

σmax – maximum applied stress during the load cycle

7.24

Fatigue of Composite Materials

σmin – minimum applied stress during the load cycle

N - number of load cycles to failure

minσ

σ mzxR = (7.17)

Hashin and Rotem (1973) then assumed that one could use the same criterion in fatigue loading. Thus,

(7.18a) N11 σσ =

12

12

12

2

2

2 =

+

NN τ

τσσ (7.18b)

If one then writes the fatigue strength (with superscript N) for the lamina principal axes as function of the static strength, the load ratio R and the number of cycles as

)

)

)

,(ˆ 111 NRfN σσ =

) (7.19) ,(ˆ 222 NRfN σσ =

,(ˆ 121212 NRfN ττ =

where f are material fatigue functions, which basically describes the function for the strength as function of number of load cycles, i.e., the Wöhler curves in the three principal stress components. In the case the maximum and minimum cyclic stresses have the same sign, R is positive. For R = 1, there is no cyclic loading and the static case is obtained for which

f1(1,N) = f2(1,N) = f12(1,N) = 1

We can use this in the static strength criterion to become

(7.20)

<>

=0 when ),(ˆ0 when ),(ˆ

111

1111 σσ

σσσ

NRgNRfN

<>

=0 when ),(ˆ0 when ),(ˆ

222

2222 σσ

σσσ

NRgNRfN

,(ˆ 121212 NRfN ττ =

If the stresses change sign in the cycles, i.e. for R < 0, then the choice of 1σ̂ and 2σ̂ becomes ambiguous since it is no longer clear whether the tensile or compressive stresses should be chosen. Still, these parameters can be regarded as stress parameters, which are determined by curve fitting to experimental results.

7.25

Foundations of Fibre Composites

Assume for simplicity that σmax and σmin have the same sign. We can then in analogy with the static case find the fatigue function for the case by applying a loading in the fibre direction to extract f1(R,N). Then we proceed to test with a pure tensile or compressive fatigue load transverse to the fibre direction of the lamina to get f2(R,N) and finally for a pure shear load case to get f12(R,N). The testing necessary to implement this theory is then a set of static tests of the lamina oriented in different direction to the fibre direction. This can appear as shown in Fig.7.29.

0 10 20 30 40 50 60 70 80 90

θc θ

σ

Figure 7.29 Static off-axis strength as function of fibre angle for a unidirectional lamina.

The cross-over point, θc, is the fibre angle for which the failure mode changes from fibre dominated failure to matrix dominated failure. In the fatigue case we must then find the functions f1(R,N), f2(R,N) and f12(R,N), through fatigue tests in the fibre direction, transverse the fibre direction and for pure shear loading. If we now proceed to discuss off-axis loading, the following can be applied; Suppose that the specimen fails in the fibre dominated mode. Then, the applied fatigue stress can be written as a function of the static strength of same specimen in analogy with eq.(7.18a) and (7.19) as

) (7.21) ,,(')(ˆ θθσσ NRfN ⋅=

where the fatigue function f' may take on different form depending on the sign of the applied stress as in eq.(7.20). In the case of static loading, R = 1, then we must have that

1),,1( =θNf . Suppose now that the specimen fails in the a fibre dominated mode, then we must have that

) (7.22) ,('cosˆ),(ˆ 2111 NRfNRfN ⋅== θσσσ

where f' is the fatigue function for off-axis loading. However, following eqs.(7.21) and (7.22) the following must hold

),('),,('),(1 NRfNRfNRf == θ (7.23)

7.26

Fatigue of Composite Materials

Next, suppose that the specimen fails in a matrix dominated mode.. We can then write

) (7.24) ,,('')(ˆ θθσσ NRfN ⋅=

where f'' is the off-axis fatigue function for fibre dominated failure. Then the failure criterion is that of eq.(7.18b). Therefore,

1),(ˆ

),,(''cossinˆ),(ˆ

),,(''sinˆ2

1212

2

22

22

12

12

2

2

2 =

⋅

⋅+

⋅⋅

=

+

NRf

NRfNRf

NRfNN τ

θθθσσ

θθσττ

σσ

or

{ } 1),(ˆ

cossin),(ˆ

sin),,(''ˆ2

1212

2

22

22 =

⋅

+

⋅

⋅NRfNRf

NRfτ

θθσ

θθσ (7.25)

In the static case, when R = 1, this becomes

1cossinˆˆ

sinˆˆ 22

2

12

42

2

=

+

θθ

τσθ

σσ (7.26)

It then follows from eqs.(7.25) and (7.26) that

{ } 1),(

cossinˆ),(

sinˆ),,(''

2

1212

2

22

22 =

⋅

+

⋅ NRfNRf

NRf NN τθθσ

σθσθ

and by substituting

θθσθτ

τσσ 22

24

12

122

cosˆsinˆˆˆˆ

in+=

from eq.(7.26), the fatigue function f'' can be obtained as

θστ

θστ

22

22

1212

22

2

12

12

tanˆˆ

1

tanˆˆ

1),(''

+

+

=

ff

fNRf (7.27)

Equation (7.27) can now be used either to predict the fatigue strength of an off-axis specimen through eq.(7.24), providing the functions f2 and f12 are known for the specific load ratio R, or to find the functions f2 and f12, by means of testing off-axis specimens in at least two different off-axis angles. The transition from fibre dominated to matrix dominated failure may be defined by a critical angle θc at which both criteria are valid simultaneously at some applied stress σNC. It follows then from eqs.(7.14) and (7.16b) that

Nc

NC1

2cos σθσ =

7.27

Foundations of Fibre Composites

1cossinsin 222

12

42

2

=

+

θθ

τσθ

σσ

N

NC

N

NC

Elimination of σNC in the two above equations leads to the expression

−

+

= 1

)(1

21tan

12

212

12

22NN

N

N

N

c σστ

τσ

θ (7.28)

Since in general is much larger than both and , the later being of the same order of magnitude, the squared term under the root is very small. By series expansion of the square root, the expression becomes approximately equal to

N1σ N

2σ N12τ

),(),(

ˆˆ

tan1

12

1

12

1

12

NRfNRf

N

N

c στ

στ

θ =≈ (7.29)

The next issue is to actually extract usable data for these models. Some data from Hashin and Rotem are illustrated (without detail) in Fig.7.30. Note that the y-axis is drawn to different scales.

1000

0 1 2 3 4 5 6log N

σ

θ=5

log N

0

10

0 1 2 3 4 5 6

σθ=0 θ=10

θ=15

θ=20θ=30

θ=60

Figure 7.30 Tensile on-axis and off-axis fatigue failure stress vs. number of cycles. All values extracted at R = 0.1

7.28