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Fdigue Fracr . Engng M a w . Struct . Vol. 17. N o. I , pp . 77-91, 1994Printed in Great Britain. All rights reserved 8756-758X/94 $6.00+ 0.00Copyright0 994 Fatigue& Fracture ofEngineering M aterials & Structures Lt d

FATIGUE LIFE PREDICTION OF NO TCHEDCOMPOSITE COM PONENTS

G. SHEN,G. GLINKAnd A. PLUMTREEUniversity of Waterloo, Department of Mechanical Engineering, Waterloo, Ontario, Canada, N?L 3G1Abstract-The local stress/s train app roach has been used to predict the fatigue lives of notchedcomposite components. The method was based on a microstress analysis and the application of amultiaxial fatigue parameter incorporating the alternating strain components on the critical plane. Thisparameter was able to correlate the fatigue lives obtained under a variety of multiaxial loading andgeometrical configurations, enabling a generalized fatigue life curve to be determined on the basis oflimited experimental data.The ability of the multiaxial fatigue parameter to relate the fatigue behaviour of compositeswas illustrated by predicting the locations of crack initiation sites in a unidirectional silicon carbidefibre reinforced titanium plate containing a circular hole tested under constant amplitude cyclic loading.The same approach was also successfully employed to predict the fatigue lives of graphite reinforcedepoxy composite tubes with circular holes tested under several combinations of cyclic tension andtorsion.

NOMENCLATUREA , K , m =material constantsA , , Bij= coefficients relating macrostresses S, and microstresses u, , in deformationstates I, 11, I11 (using A o ) and IV , V (using 4 ) espectivelyEL,& = axial and transverse elastic moduli for a unidirectional laminate measuredwith respect to the fibre directionE: ,E: = elastic moduli for multilayer laminate in I nd y directionsG,,, GY = shear modulus for a unidirectional and multilayer laminate respectivelyk = subscript defining kth ply

n = numbers of plies in a laminateM = laminate stiffness matrixQ,= lamina stiffness coefficients(Q,), transformed stiffness coefficients in kth plyR =stress ratio ( R = Sm~o/S,,,ax)Sii, j= macro- and microstress components ( i , j = 1,2,3,R, 0, x, y )S, , = amplitudes of macrostress components

V ,= volume fraction of fibrespk= angle between loading direction and fibre axis in k th ply7,Tl3 =shear strain amplitude on critical plane and X , - X 3 plane respectivelya* = multiaxial strain parameterh , ,6, = elastic proportionality factors

1 f, thickness of a multilayer laminate and thickness of the kth ply in a laminate

E = amplitude of strain component normal to the critical planevLT , vTL= Poissons ratios in the parallel and transverse directions to fibrev : ~ Poissons ratio for a multilayer laminateR = volume

77


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78 G. SHEN t (11.INTRODUCTION

Prediction of fatigue lives o f notched com pon ents is often based on the local notch-tip stress-strainapproach [13. Recently, this method has been adopted for multiaxial stress-strain configurations [2].The aim of this paper is to apply the local notch-tip strain approach to predict the fatigue livesof notched plates and thin-walled tubes made of composite materials subjected to constantamplitude cyclic tension in addition to simultaneous torsion and tensile loading.

FATIGUE MODES OF FAILURE IN COMPOSITESIt is generally accepted that in the case of notched isotropic materials the number of cycles tocrack initiation is controlled by the maximum strain range at the notch-tip. However, this is notalways the case for composite materials where the magnitude of the local strain and its locationwith respect to the m icrostructural elements, such as fibre orientation, ar e imp orta nt. Consequently,fatigue cracks in composites need not be located at the notch-tip where the highest stresses occur,as in isotropic materials. Therefore, the most frequent mode of fatigue failure of off-axially loaded

unidirectional fibre composites occurs as a result of crack initiation and grow th in the matrix alon gin planes parallel to th e fibre direction[3-61. Initiation of these fatigue cracks is controlled by thestress-strain state associated w ith these planes, which ar e know n as critical planes. In general thestate of stress in these planes is not uniaxial and is best expressed by a multiaxial fatigueparameter[7]. Recently it has been fo und t ha t this appr oa ch used for the fatigue analysis of metallicmaterials ca n a lso be applied for fatigue life predictions of c omp osite com pon ents [8].Such a n analysis requires the macrostresses and the m icrostresses be know n in the critical region,i.e. around the notch contour. This problem will be addressed in the present work.RELATIONSHIP BETWEEN TH E MACRO- AND MICROSTRESSES

The analysis of microstresses in unidirectional composites is usually based on the so-called unitblock ap pr oa ch [9-1 I ] taking advantage of the stress distribution symm etry aro und the compositefibres. This m ethod described in Refs [8,11] ha s been ado pted in the stress analysis presented below.The aim is to determine the relationship between the external applied load and the stress fieldin the fibre neighbourhood. Therefore two important parameters are used in the method: themicrostresses oij nd the macrostresses Sij Th e former are the actua l point t o point stresses in thematerial which are determined by considering the different properties of the fibre and matrix. Themacrostresses S,, represent the av erage values of the microstresses in the unit block an d ar e linearlyrelated to the applied loa d. The schematic structure of a m ultilayered laminate, the representativeunit block of the material and the system of coordinates used in the present work are shown inFig. 1.In the case of a notched multitayered laminate (Fig. la), the stress analysis is carried out inseveral steps. First, the laminate gross stress S, have to be calculated in the critical region usingthe anisotropic plate theory.

Secondly the macrostresses Sij cting on each lamina are determined. F inally, the macrostresses Sijat the particular lamina location (i.e. notch-tip region) and the basic block unit are then used todetermine the corresponding microstresses oii.


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80 G . SHENet al.Gross stress S;, in a notched laminate

The circumferential stress at the edge of the cutout (Fig. la) in a plate consisting of a pair ofbalanced laminates arranged symmetrically with respect to the midplane of the plate and subjectedto unidirectional loading parallel to the x axis can be calculated [12] from the following expression:

(2)In the case of shear loading, the gross stress So, along the circumference of the hole is given by:

( 1 + s , ) ( i +6,) (1+6, + ~ , - ~ , ~ 2 - 2 c o s 2 e ) s , x(1 +6: -26 ,cos28) (1+6 : -26 ,cos28)o, =

4sxy(61 , - 1)sin 28s -@ - (1 +6 : - 26 , ~0~2 8) (1 6 : -26 , ~0~28)The parameters 6, and 6, are the elastic proportionality factors given [12] as follows:

(3)

Macrostresses S , in one laminaKnowing the gross stresses Sj n a laminate, the macrostresses S , in each lamina of the laminatecan be determined by solving the constitutive equation of the lamina, provided that each laminais subjected to the same deformation and strain. At the edge of the hole, the circumferential grossstress, So,, exists only. All the other components are zero. Consequently the correspondingmacrostresses in each lamina are [13]:

< Q l l > k (Ql2)k (Q16)k(QIZ)k (Q22)k (&26)k ( 6 )( O l 6 ) k (Q26)k (066)k

Table 1. Material DroDertiesModulus of elasticity Poissons( G W ratio

Material EL E T GL T VL T v~ ~Silicon carbide Composite 207 124 55.2 0.24 0.15

0.25 0.25itanium (SCS-6/Ti 15-3) Fibre 430( V c= 0.4) Matrix I15 I15 46 0.36 0.36graphite/epoxy Composite 133 9.9 6.5 0.35 0.03(Fiberite T300/934) Fibre 220 22 22 0.35 0.35( Vr= 0.6) Matrix 3.45 3.45 1.3 0.35 0.35

-


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Fatigue life prediction of notched composite components 81

where

Subseq uently the macrostresses S , in a lamina need to be transformed to the material axis (Fig. Ic)in order to be used as the input data in the unit block model enabling the microstresses oij o becalculated.

Lamina microstresses croThe relationship between the macrostresses Sij nd the microstresses oijcan be written as[& 1 ]:

[:::]1 2 = [:::31 A 2 3 /3 3 (21)0 3 3 A,, A42 A43

and

The microstresses crlJ for given macrostresses S, or external load can be calculated by using


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82 G . SHEN et at .

I Deformation state I

Deformation state 111f x2

Deformat ion s ta te I1

u 2 = 1 on x 2 = bu l = o on xl = O , au 2 = 0 on x 2 = 0

Deformation state IV

u l = o on x 1 20 , au 2 = 0 on x2 = 0 , bE 3 3 = 1

x2 - Deformation state V

Fig. 2. Deformation states and corresponding boundary conditions permissible for basic block.

equations (21) and (22). This is accomplished by determining the parameters A , and B , usingdeformation analysis, as outlined in Refs[& 1 11. Because of the periodic structure in the lamina,adjacent basic blocks which contain one fibre, match exactly without any overlapping. Thereforethe rectangular blocks must retain their shape under deformation. This constraint limits the basicblock unit to only five permissible deformation modes, namely I, 11, 111, IV and V, as shown in


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Fatigue life prediction of notched composite components 83Fig. 2. The microstresses for each deformation mode can be calculated by finite element analysis,using the quadrant shown in Fig. If. The corresponding macrostresses are then determined fromthe following equation

The calculation is repeated for each of the deformation modes shown in Fig. 2.Deformation modes, I, I1 and 111are associated with macrostress components S, , S, , and S,,respectively. On the other han d, the only non-zero microstress componen ts which may occur u ndersuch deformation modes are oI1 , i,,, oI2 nd o,,. Each of the deformation modes I, I1 and I11provides four equa tions (equation (21)) so tha t a total twelve equations a re available to determinetwelve unknowns A,, (i = 1 , 2 , 3 , 4 , = I , 2 , 3 ) .In the case of deformation modes IV and V, the associated non-zero macrostress componentsare S , , and S,, and the non-zero microstress com ponents are 013 and 02,. ach deformation modeIV and V provides two equations (equation (22)), given a tota l of four eq uations to be solved forfour unknowns B , ( i , j= 1 , 2).

Fig. 3. Distribution of fatigue parameter ?* around hole in unidirectional S CS-6/Ti 15-3 composite platesubjected to axial cyclic loading, R = 0.1. Crack initiation sites observed by Newaz and Majumdar [20]indicated. Calibration scale for y* = 0.01 included.


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84 G . SHENel al.

Fig. 4. Circumferential stress distribution around the edge of hole in SCS-6/Ti 15-3 unidirectional plateunder axial loading. Figure includes calibration scale for stress concentration factor of 4.

THE FATIGUE PARAMETEROne of the fundamental problems in the analysis of material failure under a cyclic multiaxialstress state is the determination of a parameter which represents the material resistance whensubjected to a variety of loading conditions. There have been several parameters proposed suchas the maximum principal stress (or strain) [14], the V on Mises equivalent stress ( or strain ) [151 andmaximum shear stress (or strain) [16]. A parameter well validated experimentally is one whichcombines the shear and normal strain amplitudes on the critical plane[7,17-191:

?* = y +KC (24)The material constant K is usually determined from multiaxial fatigue tests. A value of K = 1 hasbeen used for steel[l9] and Inconel 718[17]. The same value was adopted in the present study.In the case of an off-axially loaded unidirectional composite, the critical plane is parallel to thefibre axis, hence equation (20) is written as follows:

y* = +KCli ( 2 5 )Considering the results presented in Refs[7,17-19], the parameter y* can be related to fatigue lifeN c by the power law expression:

y* = A N ? ( 2 6 )


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Fatigue life prediction of notched composite components 8 5Expression (26) was chosen to correlate the fatigue dat a obtained under different geometrical andloading configurations. It is obviously valid for a constant mean stress which is appropriate forthe cases studied below. Naturally, the mean stress effect should be taken into account for thosecases involving variable mean stresses. Un fortun ately, the mean stress effect under multiaxial cyclicloading has not yet been clarified.

PREDICTION OF FATIGUE CRACK INITIATION SITESThe validity of y"* as a fatigue parameter can be illustrated by its ability to predict thelocation of crack initiation sites in a notched composite plane subjected to constant amplitudecyclic tensile loading. The experimental results were taken from work of Newaz andMa jum dar [20] who co nducted fatigue tests (R= 0.1, sXx 355 MPa) on unidirectional eight-ply silicon carbide fibre reinforced titanium (SCS-6/Ti 15-3) plates with central circularholes 9.53 mm in diameter. The specimens were of rectangular shape 152.4 mm long by 19 mmwide and 1.55 rnm thick. The material properties are given in Table 1. The angle between

the loading direction and the fibre axis was zero and after cycling it was observed thatfour cracks initiated a t the hole circumference, at ang les 0 = (65+72") an d 8 = f 1 55 + 162").These cracks were symmetric with respect to the x and y axes, as shown in Fig. 3.Concentration of macrostress S , near the circular hole

The circumferential stress (&) around the edge of the circular hole was calculated usingequations (2) to ( 5 ) inclusive. The resulting n ormal stress concentration (See/Sxx)roun d the edge

t p_ _254mm

I I I I

- .6mm

Fig. 5 . Composite tube with a circular hole [21].


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Fatigue life prediction of notched composite components 87PREDICTION OF FATIGUE LIFE UNDER CYCLIC TENSION AND TORSION

Un$cation of fatig ue life data fo r various loading conditionsThe aim of the present stud y is to pred ict the fatigue lives of notched tubes subjected to a varietyof loading modes such as tension, torsion, and a com bination of bo th by using material d ata obtained

from simple experim ents associated with o ne loading m ode, i.e. cyclic tension. T herefore, the first stepwas to determine w hether the experimental fatigue dat a ob tained und er cyclic tension and torsionmay be unified by using the fatigue parameter y'*. The tension and torsion fatigue data for agraphite/epoxy ( [*45'], lay up) tube con taining a hole (Fig. 5) were take n from Ref. [21]. Th eappropriate material properties a re included in Table 1. Th e specimens were in the form of a lam inatetube 254 mm long, 25.4 mm d iameter a nd 0 .6 mm w all thickness. All tubes contained a single throu ghthickness circular hole of 4.8 mm diame ter. The distribution of the param eter y'* aroun d the edge ofthe hole in the -45" laminae obta ined und er cyclic tension with the maxim um tension stress of108 M Pa is shown in F ig. 6. Th at obtained under cyclic torsion with the m aximum torsion stress of1 10 M Pa is shown in Fig. 7. Both series of tests were carried ou t under the sam e stress ratio of R = 0.1.Th e distribution of parameter y* in the +45" am inae can be obtained by a 180" rotation of thatdetermined for the -45" ply. The maximum values of param eter y'* were found a t angles 0 = f 0"in the -45" laminae subjected to cyclic tensile loading. Ho wev er, in the case of torsion lo ading f ou rmaxima located a t angles 0 = + 123", + 1 4 7 , - 3" and - 7" were found in the - 5" taminae. Th emaximum values of param eter y'* at these locations plotted against the expe rimental fatigue lives ar eshown in Fig. 8. It is app are nt that the y'* para me ter is quite cap able of norma lizing the fatigue livesobtain ed unde r entirely different loading mo des allowing the relationship t o be given in a logarithmicform:

Y -* - 1.168N;O.OS8 (27)

~ -2''\\. _-/--i_.---.../--__-Fig. 7. Distribution of fatigue parameter y* around edge of hole in [ +45 ' ] s grdphite/epoxy tube undercyclic torsion, R = 0.1. A calibration scale for q* of 0.05 is included.


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88 G . SHENet al.

Fig. 8 . Fatigue life ?* -N,curve of notched [+45"]s graphite/epoxy tube. Cyclic tension and to rsion dataonly. R = 0.1.The numerical coefficients in equation (27) were determined by the least squares method using theexperimental tensile and torsion data[21] shown in Fig. 8. They agree very closely with those resultsobtained under combined cyclic axial tension and internal pressure given in Ref. [21] which can beexpressed by:

y'* = 1.19fjN;0.053 ( 2 8 )Equation (28) was subsequently used to predict fatigue lives under other modes of loading, asdescribed below.Prediction of fatigue liues under combined cvclic tension and torsion

The notched composite tube (Fig. 5) as reported in Ref.[21] was also tested under combinedcyclic tension and torsion loading. For these loading conditions, the ?* parameter around theTable 2. Location of crack initiation sites ( 8 ) in graphite/epoxy (Fiberite T3001934) tubeRatio(tensionitorsion) l : o 1 :0.5 1 : l 1 : 2 0 :

+45" ply + 90" + 6 8 ' +62 " +60 " + 33+ 57- 0 - 12' - 18" - 20" - 23- 47-45 ply +9 0 + 112' +I18 + 120 + 147+ I23- 0" - 8- - 2 - 0' - 7- 3


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Fatigue life prediction of notched composite components 89

2oo

401

R = 0.1

0- redicted, eq.(28)0 I 1 I 1 1 1 1 1 1 ~ 1 l 1 1 1 1 1 1 , I , 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 I , I 1 1 1 1 , I , l l , J1 10 10 10 10 10 ag *Fig. 9. Fatigue life predictions for notched composite tube under combined tension and torsion. R = 0.1circumference of the hole was first calculated to determine the locations of the fatigue crackinitiation sites due to the different tension t o torsion load ratios. The initiation sites were assumedto coincide with the sites of maximum y* and are given in Table 2 for five different tension totorsion ratios. Unfortun ately, since the experimentally determined locations w ere not given in Ref.[21], a direct comparison with prediction cannot be made.The generalized y* -N,curve (equation (28)) was used for the de termination of fatigue lives forthree series of tubes tested under combined cyclic tension and torsion with tension : orsion loadratios of 1 :0.5, 1 :1 and 1 :2. The fatigue lives were predicted using equation (28) and the maximumvalues of y* found on the notch circumference. Comparison of the experimental and calculatedfatigue lives is given in Fig. 9. Good agreement between the predicted and experimental fatiguelives is apparent for the whole range of load combinations.

DISCUSS 0NThe analysis described above was based on linear elastic behaviour of the fibre and matrix.However, the maximum microstresses and microstrains shown in Figs 3-7 may exceed the yieldlimit of the matrix in the composite materials considered. The data shown in Figs 3 and 4 dealswith the prediction of the location of the maximum local strain parameter and the location of t h emaximum normal stresses. It is anticipated that due to localized plastic yielding the microstrainsmay be higher than those shown in Fig. 3. However, du e to the local nature of plastic yielding thelocation of the maximum hypothetical elastic strains should coincide with the location of themaximum true elastic-plastic microstrains. Therefore, the location of the crack initiation sites


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90 G . SHENel al .should be predicted adequately in spite of the fact that the microstrains (Fig. 3) calculated for thesilicon carbide/titanium (SCS-B/Ti 15-3) composite may be lower tha n the actual.A similar situation could be encountered in the case of the glass/epoxy tube with the circularhole (Fig. 5 ) . The values of the strain parameters shown in Figs 6 and 7 appear to be relativelyhigh and above the yield limit. How ever, the da ta shown in Figs 8 and 9 have been w ell correlatedusing the linear elastic parameter 7* for both the short and long fatigue life regime because thecritical plane parameter concentrates on a very localized and small region surrounded by arelatively large elastic volume of material. Therefore, the material in the plastic zone is controlledby the surro und ing elastic field which can be characterized by the elastic param eter y"*. The secondreason is that the cyclic yield limit is approximately twice the monotonic, thus plastic yielding ismore restricted than monotonic yielding. It is also known that the epoxy materials behave linearlyfor the major part of the stress/strain curve. Additionally, the basic reference data shown in Fig. 8and the predicted curves given in Fig. 9 are consistently related to the hypothetical elasticmicrostrains and stresses. Because of the localized nat ur e of the plastic zo ne the hypothetical elasticstrains can be used a s the controlling param eter. W hen these are consistently applied good fatiguelife predictions result. It should also be me ntioned tha t this analysis was conduc ted using a cons tantmean stress. In the case of variable mean stresses an extra strain term accounting for the meanstress (omean/E)ould be added to the critical plane parameter.

CONCLUSIONSThe fatigue life of notched composite components is controlled mainly by the local microstrainswhich are a function of the applied load, microstructure an d mechanical properties of the fibre andmatrix materials. The location of fatigue crack initiation sites can be predicted by using the localfatigue parameter incorporating the multiaxial alternating strains on the critical plane parallel to

the fibre axis. The application of this parameter enables the generalized strain-life relationship tobe determined from a limited am ou nt of experimental da ta, which can then be used to predict thelife of notched composite components subjected to complex multiaxial loading conditions.

REFERENCES1. N . E. Dowling, W. R. Brose and W. K . Wilson (1977) Notched member fatigue life predictions by thelocal strain app roa ch, fatigue under com plex loading: analyses an d experiments, R. M . Wetzel ed., TheSociety of Automotive Engineers, Inc, Warrendale, pp. 55-84.2. D. F. Socie (1992) Discriminating experiments for multiaxial fatigue damage model, VT T Sym posium 131on Fatigue Design, Vol. 2 , eds. J. Solin. G . Marquis, A . Siljander and S. Sipila, Technical Research Centre

of Finland, pp. 9-20.3. Z. Hashin and A. Rotem (1973) A fatigue failure criterion for fibre reinforced materials. J . CompositeMaterials 7, 4 4 8 4 6 4 .4. J . Awerbuch and H. T. Hahn (1981) Off-axis fatigue of graphite/epoxy composite, Fatigue of FibrousComposite Materials, ST P 723, J. B. Wheeler ed., ASTM, Philadelphia, pp. 243-273.5. P . T. Curtis (1986) Fatigue of comp osite materials, fatigue of engineering m aterials and structures, Vol. 11.Proceedings of the Institution o Mechanical Engineers, Lo ndo n, pp. 297-306.6. R. Talreja ( 1987) Fatigue of Composite Materials. Technomic Publishing Company, Inc., Lancaster,Pennsylvania.7. M . W . Brown and K . J . Miller (1973) A theo ry for fatig ue unde r multiaxial stress-strain conditions. Proc.Inst . Mech. Engrs 187, 745-155.8 . G . Shen, G. Gl inka and A. Plum tree (1993) Com posi te life prediction using a multiaxial fatigue param eter.


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Fatigue life prediction of notched composite com ponents 91Proceedings o ft h e Ninth International Conference o n Composite M aterials ( ICCM/9) , Vol. V , CompositesBehaviour, ed. A. Miravete, Woodhead Pub. Ltd, Cambridge, pp. 707-714.9. D. F. Adams (1970) Inelastic analysis of a unidirectional composite subjected to transverse normalloading. J . Composite Materials 4, 3 10-328.

10. R. L. Foye (1973) Theoretical post yielding behaviour of composites, Pa rt I-Inelastic micromechanics.J . Composite Materials 7, 178-193.1 1 . T. H. Lin, D. Salinas and Y . M. Ito (1972) Initial yield surface of a unidirectionally reinforced composite.J . Applied Mechanics, A S M E 39, 321-326.12 . L . B. Greszczuk (1972) Stress concentrations and failure criteria for orthotropic and anisotropic plateswith circular opening, composite materials: testing and design (Second Conference). STP 497, ASTM,13. R. M. Jones (1975) Mechanics o Composite Materials. Scripta Book Co mpa ny, Washington, D.C .,14. G. F. Libertiny (1967) Short life fatigue under combined stress. Journal o Strain Analysis 2, 91-95.15. S. Taira , T. Inoue and M. Takahashi (1967) Low cycle fatigue under multiaxial stresses (in the case ofcombined cyclic tension-compression and cyclic torsion in the same phase a t elevated te mp eratu re). TenthCongress o Japan SOC.Testing and Materials.16 . H. 0. Fuc hs (1979) Fatig ue research w ith discriminating specimens. Fatigue o Eng. Mat. and Struct. 2,207-21 5.17. D. F. Socie, L. A. Wail and D. F. Dittmer (1985) Biaxial fatigue of Inconel 718 including mean stresseffects, K . J. Miller and M. W. Brown eds. ST P 853 ASTM, Philadephia, pp. 463481 .18. R. D. Lohr and E. G. Ellison (1980) A simple theory for lo w cycle multiaxial fatigue. Fatigue ofEngineering Materials and Structures 3, 1-1 7.19. F. A. Kandil, M. W. Brown and K. J. Miller (1982) Biaxial low cycle fatigue fracture of 316 stainlesssteel at elevated temperature. Book 280, The Metals Society, London, pp. 203-210.20. G. M. Newaz and B. S. M ajum dar (1992) Technical N ote, Crack initiation ar oun d holes in aunidirectional MMC under fatigue loading. Engineering Fracture Mechanics 42, 697-71 1.21. D. H . Francis, D. E. Walrath, D. F. Sims and D. N. Weed (1977) Biaxial fatigue loading of notchedcomposites. NASA Report No. Cr-145198.

pp. 363-381.pp. 31-84.

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