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ARL-AERO-PROP-R- 174 AR-004-512

00

DEPARTMENT OF DEFENCE

DEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

AERONAUTICAL RESEARCH LABORATORIES

MELBOURNE, VICTORIA

Aero Propulsion Report 174

APPLICATION OF FINITE ELEMENT METHODS WITHCYCLIC ELASTO-PLASTIC STRAIN ANALYSIS TO LOW

CYCLE FATIGUE ANALYSIS OF ENGINE COMPONENTS (U)

by

S. SWANSSON!C S ELECT -LJAH 0 6 19L

Approved for Public Release

This work is copyright. Apart from any fair dealing for the purposeof study, research, criticism or review, as permitted under theCopyright Act, no part may be reproduced by any process withoutwritten permission. Copyright is the responsibility of the DirectorPublishing and Marketing, AGPS. Inquiries should be directed to theManager, AGPS Press, Australian Government Publishing Service,GPO Box 84, Canberra, ACT 2601.

DECEMBER 1986

. . . k --- , l,

AR-004-5 12

DEPARTMENT OF DEFENCEDEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

AERONAUTICAL RESEARCH LABORATORIES

Aero Propulsion Report 174

APPLICATION OF FINITE ELEMENT METHODS

WITH CYCLIC ELASTO-PLASTIC STRAIN ANALYSIS

TO LOW CYCLE FATIGUE ANALYSIS

OF ENGINE COMPONENTS

by

N.S. SWANSSON

SUMMARY

Low Cycle Fatigue (LCF) in engine components involves macroscopiccyclic plastic strains (with a stress-strain hysteresis loop) over a significantportion of the failure region. Characterising elasto-plastic behaviour inpotential failure regions is a necessary step in estimating LCF life.

The equations governing elasto plastic behaviour are summarized,and the methods of implementing them in Finite Element (FE) stress analysisprograms discussed.

An extension of the PAFEC program to include mixed isotropic-kinematic hardening is outlined, and verified by examples for whichalternative FE solutions were available.

A sample application has been made to holes in a plate with biaxialstress fields similar to those in disc webs, and the results compared with theNeuber and modified Stowell rules commonly used for design life estimation;these rules tend to overestimate the strain in biaxial stress conditions, leadingto conservative life estimates. /3

(C) COMMONWEALTH OF AUSTRALIA 1986

POSTAL ADDRESS: Director, Aeronautical Research Laboratories,P.O. Box 4331, Melbourne, Victoria, 3001, Australia.

CONTENTS

Page

NOTATION 1

1. INTRODUCTION 3

2. CYCLIC PLASTICITY ANALYSIS 3

3. PLASTICITY EQUATIONS 5

4. FINITE ELEMENT IMPLEMENTATION 8

5. COMPARISON OF SOLUTIONS 9

6. APPLICATIONS AND DISCUSSION 10

7. CONCLUSIONS 11

REFERENCES 12

APPENDIX A - STRAIGHT AND CURVED BARS

APPENDIX B MODIFICATIONS TO PAFEC CODE

FIGURES

DISTRIBUTION LIST

DOCUMENT CONTROL DATA

r * -4c

C' npyT

(;T)E

NOTATION

GeneralA areac kinematic hardening coefficientE Young's modulusf yield functionF forceg, h section dimensionsG shear modulusH plastic slopeK stress or strain concentration factorM momentV shear forceVol volumer radius or radial coordinate8 arc distancez linear coordinateu displacementa rotation of face$ rotation derivativec strain7y shear strainX shape coefficient for shear deflectionA proportionality constantA kinematic hardening constantV Poisson's ratio0 angular coordinatep isotropic fraction in mixed hardeninge' stressw total rotation

Tensors

C elastic or plastic constitutive tensorJ tensor invarianta deviatoric stress tensorA translated stress deviatora translation of stress origin6 Kronecker deltac strain tensora stress tensor

Matrices or VectorsB strain - displacement transformation matrixD stress - strain constitutive matrixF load vectoru displacement vectorf strain vector6 stress vector

• , , a III I i

Subscripts or Superscripts

A applied loadsE elasticP plasticy current yield condition

2

1. INTRODUCTIONAs engine power and speed is varied in an aircraft turbine engine, vital components

are subject to changes in thermal and mechanical loading. Each flight includes one majorstart-stop cycle, plus lesser cycles depending on the manner of operation of the engine.High stress levels are used in many components to achieve lightness and consequenthigh specific output, and cyclic variation of stresses in the components leads ultimatelyto cracking at critical stress locations, with failure by the process of low cycle fatigue(LCF).

There are no practical or economic means of determining by inspection when theLCF life of components is exhausted, so accurate and reliable means of life estimationare needed. Three categories of data are required for LCF life prediction:(i) The loading history, preferably obtained from operational records for the engine,alternatively derived from simulation or estimates.(ii) Modelling and analytical methods which use the operational loading to producethe history of stress, strain (and temperature where relevant) at critical locations incomponents where failure may originate.(iii) A damage accumulation criterion, so the cumulative effect of loading cycles canbe quantified and a measure of fatigue damage provided. This may be related eitherto the life to crack initiation (for safe life estimates), or to estimated crack propagationrate (in damage tolerant design using fracture mechanics methods).

Thermodynamic and heat flow analyses of engine operational data are used to de-ermine generalised loadings, including pressures, metal temperatures and rotor speeds.

Finite clement (FE) models handle the complex boundary shapes commonly found inpractical components, so FE computer programs are generally used for determinationof detailed temperature, stress and strain distributions.

Since LCF by definition is characterised by macroscopic cyclic plastic strains mani-fested by a stress-strain hysteresis loop over a significant portion of the failure region, amaterial model incorporating cyclic plastic behaviour is required. Predicted stress andstrain values depend on plastic properties of the material. Further, when the componentis unloaded, plastic behaviour leaves residual stresses which also influence fatigue life.

High cycle fatigue life predictions are based on elastic stress estimates (Basquin law),but strain amplitude is regarded as the best parameter for predicting LCF life. Totalstrain values (elastic plus plastic) are used in the Coffin-Manson equation i , adoptedalmost universally for LCF prediction. Hence characterising elasto-plastic behaviour inpotential failure regions is a necessary step in estimating LCF life.

This report summarises the equations governing elasto-plastic behaviour, indicateshow they are implemented in FE stress analysis programs, evaluates and compares FEpackages available at ARL, describes the extension of the PAFEC program at ARLto include mixed isotropic kinematic hardening, and gives applications of FE elasto-plastic analysis tnder loading resembling engine components, comparing results withthe commonly used Neuber and Stowell approximations.

2. CYCLIC PLASTICITY ANALYSIS

In addition to equilibrium and compatibility (strain-displacement) relations usedin elastic stress analysis (and not changed in form if strains are assumed to be small),three concepts are required to formulate cyclic plasticity problems 2

(i) Initial Yield CriterionA number of yield criteria' ' have been formulated, but only the maximum shear

of Tresca, and the distortion energy or octahedral shear of von Mises are used for normal

3

ductile metals. For the more commonly used materials the von Mises criterion agreesbetter with test data, and is mathematically simpler as it defines a single continuousfailure surface. This surface is expressed in terms of a function of the second invariant(J2 = sa, ii) of the deviatoric stress tensor sii and can be written:

3 1f = (-i i 8i) -o0 = 02

where 0 is the initial yield stress in uniaxial tensions= Oij - 6ii ckk/3, with stress tensor oaq and Kronecker delta 64Okk - o'll + 0 22 + a33, using the tensor convention that repeated dummy

subscripts indicate summation

In 3-dimensional principal stress space, the von Mises criterion describes a cylindri-cal surface, of radius a0 = ( si si) 3 with axis through the origin and equally inclined tothe three principal stress axes (Fig. 1). For plane stress (Os = 0), this surface intersectsthe Ol, O2 principal stress plane in an ellipse.

(ii) Plastic Flow Law

Total increments in strain are assumed to be divisible into elastic and plastic frac-tions:

= + d P

The direction of the plastic strain increment tensor, according to the associated flowrule, is normal to the yield surface (requiring maximum plastic work on deformation):

Ofdtf! =dA O

where dA is a constant of proportionality. This is the normality condition and for a vonMises material, is equivalent to the Prandtl-Reuss equations dtf = dA s8q.

(iii) Strain Hardening Rule

In many materials yield strength increases progressively with increasing plasticstrain. While the von Mises yield criterion and the normality flow rule for plastic strainare generally accepted and adequately established by experiment, a similar consensusdoes not exist for rules used to describe strain hardening. Simpler rules are preferred,provided their behavioural description is reasonable, as more elaborate hardening modelslead to considerable complexity in finite element programs. Four main hardening ruleshave been used, viz. isotropic, kinematic, Mroz or multiple surface models, and sublayeror subvolume models.

(a) Isotropic Hardening

The simplest assumption is that material strengthens umiformly with increasingplastic strain, irrespective of strain direction forward and reversed loading have equaleffects (Fig. 2). This implies that the yield surface radius expands uniformly in stressspace. maintaining shape, orientation, and origin of axis (Fig. 3). Isotropic hardeningdoes not satisfactorily model reversed or cyclic loading, but its implementation is simpleand for monotonic loading results are as good as for more complex models 5.

(h) Kinematic HardeningWhen loading is reversed, most metals exhibit a reduced yield strength in the

reverse direction. known as the Bauschinger effect (Fig. 2). The kinematic hardening

4

model formulated by Prager represents this effect by assuming that the yield surfacetranslates as a rigid body in stress space, maintaining its size, shape, and orientation(Fig. 4). This translation, dai - is in the direction of the plastic strain increment, i.e.normal to the yield surface. The Prager formulation is not invariant in a subspaceof reduced dimensions - it must be modified for a two dimensional system. Ziegler' 5

formulated an alternative definition of the yield surface translation, daii namely, that itsdirection is the vector connecting the centre point (or origin) of the current yield surfaceto the existing stress point. In most problems, the difference in the results from thePrager and Ziegler formulations is negligible, with the Prager equations generally moreconvenient to use. Both satisfactorily model reversed loading with elastic-perfectlyplastic behaviour, or when the degree of strain hardening is limited, and hardeningis reasonably linear. Difficulties arise at large reversed strains when the stress-strainrelation for the material is highly nonlinear, because there is no satisfactory rule forrelating hardening coefficient to cumulative strain.

(c) Mixed Isotropic-Kinematic Hardening

Mixed hardening formulations 7 ," have been developed combining isotropic andkinematic hardening, which means that the yield surface both expands and translates.By extending the scope of the kinematic model, better agreement with test data is ob-tained for some materials. A particularly simple and readily implemented formulationof mixed hardening has been developed and is presented below.

(d) Multi-Surface Models

Non-linear hardening, reduced in the uniaxial case to a piecewise linear stress-straincurve, is represented by the Mroz and subvorume models. The Mroz model 9 comprisesa series of (initially concentric) yield surfaces, each related to a particular yield stresscorresponding to one segment of the curve. When plastic strain occurs, the surfacetranslates until it touches the next bounding surface, which is translated in turn. Contactbetween the surfaces is maintained until unloading occurs (Fig. 5).

A similar piecewise linear modelling is achieved by the sublayer or subvolumemodels 10-11 These postulate that the material comprises a number of subvolumes(physically analogous to material grain structure). Each subvolume has elastic-perfectlyplastic properties with a different yield stress. All are subject to the same strain, withtheir individual stresses combining to give total stress.

The Mroz and subvolume models result in the Masing description 12 of the Bausch-inger effect, viz. that on reversed loading the shape of initial loading curve is maintained,magnified by a factor of two. This is a satisfactory representation for many engineeringmaterials. More elaborate models 13,14, 23 have been formulated to describe fine de-tails of behaviour, but generally these are too complex to implement in a finite elementcomputer program, or too restricted in the materials represented.

3. PLASTICITY EQUATIONS

Failure Criterion

A general form of the von Mises yield criterion, incorporating the various hardeningrules, can be expressed as:

f 3 (") - e((P) = (1)

where ii= 8ii - aii (2)

5

and sii =aii - 6i o,/3 as previously defined. a.(cp ) is the yield stress as a functionof the plastic strain. The translation aij of the stress origin is also a function of plasticstrain, depending on the hardness model adopted, and determines hi, the translateddeviatoric stress.

Plastic Flow Equations

Strain Partitioningdfi= dcf + dcf (3)

Normality Condition

dPd f =3d"dA (4)caii 2 7

The proportionality constant dA is found by taking the inner product of equation (4):

2 d M!= 3 dA2

3 1) 'J 2O

so: dA dcd d) =dcP (5)

For a von Mises material dA = dcp is equal to the strain in tuiaxial tension.

Strain HardeningIsotropic a, = 0do', = Hd P

where H is the plastic slope in simple tension, so a, = ay, the yield stress in tension.

Kinematic da, = 0 so o , = o0, the initial yield stress.dai = c dtf (Prager)daii= d1 (o - ai) (Ziegler)

Equivalence with uniaxial tension requires that the constants c 2H, or

d = H dtp so with equations (4) and (5):t'w

da= 84 H dtp (Prager) (6)

o'.

or: daii = dc (Ziegler) (7)

Mixed Hardening

The hardening effect H dP in equations for isotropic and kinematic hardening canbe divided into fractions p defining yield surface expansion (isotropic), and (I - p)yield surface translation (kinematic). Notionally either the strain hardening rate Hor the plastic strain increment d1P can be so partitioned, and this makes subsequentimplementation in computer code particularly straightforward.

da, = pHd(i (8)

so: oa, = ea0 + p(ay - ea0 )

dai= (I - p)HdYp (Prager) (9)Ges

. mmm m m • -- am| m l i l•[]•6

dctii -a I*(I - p)HdEp (Ziegler)as

Constitutive Equations

Constitutive equations relating increments of stress and elastic strain, using equa-

tion (3), can be written:do, i = C Ck(dCk, - dn) (10)

where the elastic stiffness tensor is:

Ci7 t = 2G(6 ik 6 i + l b-ij 6 kt )ijkl 2v

with shear modulus G. Poisson's ratio v.Constitutive equations for plastic deformation are derived by noting that during

plastic flow the stress point stays on the yield surface, so that df = 0 in equation (1):

f=Of a Of dcl + Ofd =

df= daa ij + o da,+0do,.= (1)OIi Oaqi

Substituting in equation (1) and using (2), (8) and the Prager result of (9):

3 -8i (daj - daii) - pHd E = 02 o',

P { _ P1- Hdf } - pHdc=02 a-, or,

with equations (3) and (10):

3 jjiF dkl- 3 ko JdA - HdA = 0 (12)

Since hij = 0, for an elastically isotropic material only the "diagonal" components

i = k, j = I of the tensor CEikl produce non-zero terms, and equation (12) can bereduced

to:

dA = Ski dtkI (13)o, (1 + H/3G)

Using equations (4) and (13) with (10), with dummy subscripts k,l changed to m,n:

= CEId 3 8klsmn dtmndo i 1 2 ,2 (I + H/3G)

= (CF 1 - Ct'kt)dkl (14)

By a reduction similar to equation (13), the plastic constitutive tensor becomes:

3G Aiskt (15)ikl I + H3G 7

,

4. FINITE ELEMENT IMPLEMENTATION

Finite Element Analysis

For a continuum, the constitutive equation (14) can be used in standard finiteelement equations of incremental form. Using matrix notation:

f BT(D - DP)B dVol bu = bF4 (16)

(K - KP)6u = bFA (17)

where K,K P are elastic and plastic structural stiffness matricesbFA is vector of applied loadsB is displacement strain transformation matrix so bc = B 6uD, D P are material elastic and plastic constitutive matrices,

corresponding to tensors CE P in equatio

Equation (17) provides an incremental solution for bu using the tangential stiffnessmatrix (K - KP). Because K t is a function of stress, the solution is obtained by suc-cessive Newton-Raphson approximations, and the stiffness matrix has to be re-evaluatedand reduced at each iteration.

Repeated matrix reductions can be avoided by re-arranging equation (16) so thatonly the constant elastic matrix K need be decomposed, which is required once onlyduring the solution:

f BTD B dVol 6u = 6 FA + f BTDPB dVol bu (18)

K bu =6FA + 6Fp (19)

The plastic pseudo-forces 6Fp are found by noting that when equations (10) and (14)are written in matrix form:

6u = (D - D")6t = D(6t - 6&p) (20)

Hence DPbe = D bep and since bf = B 6u the RHS integral in equation (18) becomes:

bFp f BTD dVol bEp (21)

Partitioning 6u = 6 1uE + 6UP in equation (19) leads to:

K(OUE + 6uP) = 6F.. + bFp (22)

The equation K 6uE = 6FA, is simply a scaled elastic solution which can be subtractedfrom equation (22) giving:

K6up 6Fp (23)

In this equation 6 up is found by iterative solution, with successive approximations tobFp given by equation (21).

Computer Programs

The commercial FE program package PAFEC was available at ARL as a generalpurpose stress analysis, heat flow and dynamics program when this work commenced.

.. . . . -,.- - - - . - ,, m . md =,. m I mm m m8

Subsequently access to NASTRAN was acquired for special applications, and the plas-ticity capabilities of both program packages are now briefly compared.

PAFEC uses the development just presented; a flow chart of the 1AFEC plasticityroutine is shown in Figure 6. This approach is simple, but at times it results in conver-gence difficulties. NASTRAN by comparison uses the tangential stiffness approach ofequation (17), though a flexible strategy is employed by updating the matrix only afteran optimum number of iterations. Convergence is assured, but computational effort isgreater.

Regarding types of elements available for elasto-plastic problems, PAFEC is muchmore versatile, and NASTRAN is highly restricted in its ability to model continua ofcomplex shape, typical of aircraft engine components. For 2--dimensional models, NAS-TRAN has only straight-sided linear elements, with plastic conditions represented onlyat the centroid. In three dimensions more capability is offered by linear brick elementswith 2 x 2 x 2 Gauss point integration, and equivalent triangular prism elements. Bycomparison PAFEC has, as well as linear elements, quadratic and cubic isoparametricelements in two and three dimensions, with 2 x 2 or 2 x 2 x 2 Gauss integration. Bothtriangular and rectangular (or brick and prism) shapes are available, and the curvedshapes obtainable with isoparametric elements allow better modelling of practical com-ponents.

Hardening rules available are similar. Both programs have isotropic and kinematicmodels. NASTRAN also offers "mixed" hardening, but this is restricted to the specialcase of p = 0.5 in equations (8) and (9), which implies that the yield stress alwaysremains equal to the initial value despite reversal of loading. The writer has extendedPAFEC code to incorporate the more general case of mixed hardening described bythese equations. Nonlinear hardening is available with the isotropic model, but withkinematic hardening PAFEC expresses the yield surface translation in an equation ofthe form ai = 2 Ht, which requires constant H (i.e. linear hardening) when integratingequation (6).

5. COMPARISON OF SOLUTIONS

As a preliminary to using PAFEC in practical analysis of cyclic plasticity, suit-able verification problems were sought. The only reasonably accessible example whichincluded linear strain hardening, was the proving ring problem ing ; an alternative so-lution to this problem showing slightly different results is also found in . The sameproblem with mixed hardening is treated in 7 . The proving ring and its material prop-erties are shown in Figure 7, and preliminary results with PAFEC level 5.2 are shownin Figuire 8.

The initial results reveal significant discrepancies, and when after discussion nosatisfactory reason could be found 1 a detailed examination of PAFEC code was un-dertaken. Various differences between implemented code and plasticity equations werefound (listed in Appendix B), most of them having only minor effects on results. Howeverchanges in the way of satisfying convergence, either by imposing stricter requirements,or by adopting a self-correcting scheme 17, 1, produced significant changes in resultsfor this problem. It is expected that updated levels of PAFEC code will incorporateappropriate modifications L6

Alternative FE programs such as NASTRAN were not accessible at ARL at thistime. anti no satisfactory Iul)lished results for suitable test cases were available. Soin order to test the modifications, original theoretical solutions were developed for testproblems, given in Appendix A. Starting with the simple case of a rectangular beam in

--9 -,-w mmmmm m mm m m[ ]m

pure bending, solutions are extended to a curved rectangular bar in pure bending, andfinally the proving ring which is a curved bar under both bending and tension.

Results for the straight beam and curved bar in pure bending are shown in Figure9, and for the proving ring in Figure 8. In all cases excellent agreement is obtainedbetween the modified PAFEC and the theoretical solutions. NASTRAN had by thistime become available at ARL, so it was used for the proving ring, giving a solutionvirtually coincident with modified PAFEC.

Cyclic loading for the proving ring is shown in Figure 10, comparing isotropic,kinematic and mixed hardening (for p = 0.25), with results qualitatively similar tothose given in 6 .

6. APPLICATION AND DISCUSSION

Simple semi-empirical rules for dealing with plastic deformation at stress concen-trations when the plastic properties of the material are known, have been developed byextending solutions obtained for particular geometries to more general cases. At a notchroot with shear stresses, Neuber 23 derived the relation:

t a(K? (24)Kt2 = KK( or K, = K02(4

where K, = theoretical elastic stress concentration factor

K, = stress factor a/&K,= strain factor t/t

Stowell 24 obtained a series solution for elasto plastic deformation around a holein an infinite plate umder iuiaxial stress, which when modified to cover general stressconcentrations 25 is expressed:

K0K, = (Kt - 1) K(25)

Compressor and turbine discs subject to rotational, other mechanical and possiblyalso thermal loading are generally the most critical aircraft engine components subjectto low cycle fatigue. A major failure source is at holes in the disc web, where stressconditions surrounding the hole are substantially biaxial.

For a circular hole in an infinite thin plate, where stress conditions resemble thosein a turbine disc. finite element, Neuber and modified Stowell solutions are compared.Principal stress ratios (72 /ej = 0 (uniaxial), 0.5 and 1.0 (pure biaxial) are shown inFigures 11. 12 and 13 respectively. Little difference between the three stress solutionscan be seen, largely because of the low plastic hardening slope (typical Ti-8-1-1 materialproperties were used). Strain solutions differ significantly, depending on the principalstress ratio. For uniaxial stress (0 2 = 0, Kt = 3.0), the modified Stowell and finiteelement solutions coincide, at least for lower values of plastic strain (as would be ex-pected, since the Stowell equation derives from a series solution for this case). As stressincreases, the Stowell equation tends to overestimate strain, and the Neuber solution issimilar, excepting that it overestimates strain throughout the stress range.

As stresses become more nearly biaxial, both Neuber and n.dified Stowell rulescontinue to overestimate strain. With pure biaxial stresses ( r2 = (I, Kt = 2.0),Neuber and Stowell give similar moderate o(verestimates at low plastic strains, withStowell giving much higher excess estimates of strain at higher stresses. Fortunatelysuch estimates lead to c(nservative LCF life pre(lictions when using the Coffin Mansonlaw.

10

The limitations of the Neuber and modified Stowell rules have been consideredelsewhere 21 . Generally speaking they overpredict plastic strain, but this depends onthe particular geometry. Most investigations have been concerned with their accuracyfor LCF fatigue life prediction 19 . 20 rather than for strain estimation, and in this respectthey are widely accepted as useful design rules.

A thin disc with negligible stresses through its thickness has been assumed in thecurrent analysis, and the effect of biaxial stresses in the thickness direction at the holesurface has not been considered. With surface biaxial stresses a correction should beapplied , which tends to reduce predicted strains. The preliminary applications de-scribed herein are intended to be followed by a fuller explorations of thick discs, plasticcrack growth and other practical LCF applications in the next phase of this investigation.

7. CONCLUSIONS

Application of PAFEC to cyclic elasto-plastic calculation has been established atARL. Careful investigation after finding discrepancies between results and publishedsolutions led to the development and verification of modified code. This code extendedthe available strain hardening models by adding mixed hardening, for which a simpleformulation was developed, to the existing isotropic and kinematic hardening rules.Verification entailed the determination of original alternative solutions to problems ofbending and stretching of straight and curved bars, the latter exemplified by a provingring for which several alternative FE solutions were available.

Sample applications were made to holes in plates with biaxial strets similar to discwebs, and results compared with the Neuber and modified Stowell rules, commonly usedfor design life estimation. Compared with finite element solutions, these rules tend tooverestimate strain in biaxial stress conditions, which would lead to a conservative lifeestimate.

A further complexity ensues when the disc or plate is of sufficient thickness thatstress levels are significant in the thickness direction. Stress conditions at the holesurface are then biaxial, and corrections need to be applied to give equivalent uniaxialstrains. Multiaxial fatigue is a complex topic and is not considered here; it has beenaddressed in numerous papers elsewhere 2 . Practical discs generally exhibit biaxial andoften triaxial effects. and analysis of these discs is proposed in the next phase of thiswork.

11

REFERENCES

1. Coffin,L.F.Jr. (1974) "Fatigue at High Temperature - Prediction and Interpreta-tion" Proc. I.Mech.E., Vol.188, p.10.9

2. Kalev,I. (1981) "Cyclic Plasticity Models and Application in Fatigue Analysis"Computers 8( Structure. Vol.18 p. 709

3. Nayak,G.C. & Zienklewicz,O.C. (1972) "Elasto-Plastic Stress Analysis. A Gen-eralisation for Various Constitutive Relations Including Strain Softening"Int.J.Num.Meth.Eng. Vol.5 p.1 1 8

4. Marques,J.M.M. & Owen,D.R.J. (1984) 'Some Reflections on ElastoplasticStress Calculation in Finite Element Analysis" Computers 8( Structures Vol.18p.1185

5. Hunsaker,B.Jr., Vaughan,D.K., Strlcklln,J.A. & HailierWA. (1973) "AComparison of Current Work Hardening Models Used in the Analysis of PlasticDeformations" Texas Engineering Experiment Station AD-776667

6. Prager,W. (1955) "The Theory of Plasticity: A Survey of Recent Achievements"Proc.L.Mech.E. Vol.169 P-41

7. Axelsson,K. & SamuelssonA. (1979) "Finite Element Analysis of Elastic-Plastic Materials Displaying Mixed Hardening" Int.J.Nurn.Meth.Eng. Vol.14 P.211

8. SowerbyR. & Tomita,Y. (1977) "On the Bauschinger Effect and its Influenceon the U.O.E. Pipe Making Process" Int.J.Mech.Sci. Vol1.19 p.3 5 1

9. Mroz,Z. (1969) "An Attempt to Describe the Behaviour of Metals under CyclicLoads Using a More General Work-Hardening Model" Acta Mechanica Vol. 7 p.199

10. Owen.D.R.J., PrakashA. & Zienkiewicz,O.C. (1974) "Finite Element Anal-ysis of Non-Linear Composite Materials by Use of Overlay Systems" Computer. 8Structures Vol.4 P.1 2 5 1

11. McKnigbt,R.L. & Sobel,L.H. (1977) "Finite Element Cyclic Thermoplasticityby the Method of Subvolumes" Computers 8f Structure, Vol. 7 p. 1 8 9

12. Masing,G. (1926) "Eigenspannungen und Verfestigung beim Messing'Proc.2nd.Int. Cong.App.Mech. p. 8 22

13. Asaro,R.J. (1975) "Elastic- Plastic Memory and Kinematic Type Hardening"Acta Metallurgica V61.23 p.1255

14. Popov,E.P. & Petersson,H. (1978) -Cyclic Metal Plasticity: Experiments andTheory" Proc.ASCE J.Eng.Mech.Div. No.EM4 p.1371

15. Ziegler,H. (1958) 'A Modification of Prager's Hardening Rule" Quart.App.Math.Vol. 17 p. 55

16. Owen,D.R.J. & Salonen,E.M. (1975) "Three Dimensional Elasto -Plastic FiniteElement Analysis' Int. J.Num.Mcth. Eng. Vol.9 p.209

17. Platt,J. (1985) Private Comimunicat ions re PAFEC18. Stricklin,J.A., Haisler,W.E. & von Riesemnann,W.A. (1971) "Geometrically

Nonlinear Structural Analysis by Direct Stiffness Method" Proc.ASGE J.Struct.DivNo. ST9 p. 22 9 9

19. Hotmeister,L.D., Greenbaum,G.A. & Evensen,D.A. (1971) 'Large StrainElasto Plastic Finite Element Analysis- AJAA Journ. Vol.9 p.l2 48

20. Wundt,B.M. (1972) -Effects of Notches on Low-Cycle Fatigue" ASTM STP 49021. Leis,B.N., Gowda,C.V.B & Topper,T.H. (1973) -Cyclic Inelastic Deformation

and the Fatiguie Notch Factor- Cyclic Stress Strain Behaviour Analysis. Experi-menttton and Faiilure Predliction ASTM STP 510 p.188

12

;WW

22. Mowbray,D.F. & McConnelee,J.E. (1973) "Applications of Finite ElementStress Analysis Techniques and Stress- Strain Properties in Determining Notch Fa-tigue Specimen Deformation and Life" ibid. p.151

23. Jhansale,H.R. & Topper,T.H. (1973) 'Engineering Analysis of the InelasticStress Response of a Structural Metal under Variable Cyclic Strains" ibid. p.2 46

24. Neuber,H. (1961) "Theory of Stress Concentrations for Shear Strained Prismati-cal Bodies with Arbitrary Nonlinear Stress Strain Law" Tran.ASME J.App.Mech.Vol. 28 p. 544

25. Stowell,E.Z. (1950) "Stress and Strain Concentration at a Circular Hole in aInfinite Plate" NACA Tech. Note 2078

26. Hardrath,H.F. & Ohman,L. (1953) "A Study of Elastic and Plastic Stress Con.centration Factors Due to Notches and Fillets in Flat Plates" NACA Rept. 1117

27. Beaver,P.W. (1984) "Multiaxial Fatigue and Fracture A Literature Review"Dept. of Defence, ARL Struc -R 410

13

APPENDIX A - STRAIGHT AND CURVED BARS

Pure Bending of Straight Rectangular Bar

Adopting the standard beam assumption that plane sections remain plane, andusing notation as in Figure 14, strains are given by:

2z _=c,.ox - and h - ,,

if the elastic region is of depth g, yield strain t.. Stress-strain relations for a bilinearmaterial with elastic modulus E, total plastic modulus H are:

- 5 < a = E( (26)

~> y } d'=(E- H)t1 + H( (27)

Moment M per unit width of bar is:

M= 2 az dz

= 2 Et hna 2xdZ+2 {(E - H)ty + H,,,.z2x dZ

At initial yield My = Etyh2 /6 so the integral becomes:

M 1 H Y taMv 2(1

- E C2 3 az.' ..E'

Bending and Stretching of Curved Bar

Geometry of DeformationA small segment do of a rectangular bar with circular centreline is shown in Fig-

ure 15, and the following geometrical relations apply:

Total rotation of face w di + cdb (29)

where t is uniform strain, dci is rotation, so:

dw da=-- +i = +E (30)

Rotation of the bar centreline is greater by the increase of shear deflection from -Y to- + d-y, so the equation for change of curvature is:

ro{do(1 + t) + da + d-y} = rdO(1 + i)

so since i is small:

r ro r( + t) d

I + ) (31)

= (3d)

Stresses and Strains

Fibre strain I rd + (r - r)d.rdo

=c+fl( - (32)r

Yield in tension and compression occurrs where t = + ty, with corresponding radii:

;; } (33)re )9+± t

Fibre stresses are found with equations (26) and (27):

Elastic region r, S r < r, a = Et

=Ep+#(Ir), (34)

Plastic region, assuming r, > r,.

Tension r > r a = ±(E - H)(V + H(( +fl) -Hfle (35)('onpression r < r, r

Force aid moment per unit width are found by integrating stress over the cross section.using the appropriate equation for elastic or plastic stress regions.

F = jadr (36)

M = j a rdr (37)

Explicit integrals for radius limits a, , can be found from expressions (34) or (35):

Elastic Region Forces F = E[(fl + )r - fir log, rj]' (38)

Elastic Region - Moments M = E[(#l + t)r 2/2 - flrrlbo (39)

Plastic Region Forces F = +(E- H) + H( + ()}r - Hfrlogr] (40)

Plastic Region Moments M = [{+(E - H), + H(fl + [)}r 2 /2 - Hflrr] (41)L Ja

Using radius limits found from equation (33), values of F and M for given #? and t canbe found from equations (38) - (41). To find / and c as functions of F and M requiressolution of nonlinear simultaneous equations. Since the diagonal terms predominate (Fis principally a function of c and M of fl), simple linear inverse interpolation can beused. (A Newton-Raphson solution was first programmed but was not necessary.)

Pure Bending of Curved BarShear is zero, and for each value of fl, a value of i is found giving no net force

(F = 0). The resultant value of M is expressed in the form:

MM = f ( ai/ia e t (42)where M., fly are values of moment and rotation at initial yield (at the intrados).

Circular Proving RingFor symmetrical loading, the ring can be represented by one quadrant (Fig. 16).

Forces and moments at angle 0 for load 2P are:

Normal Force F = Pcoso (43)

Shear Force V = -Psin# (44)

Moment M=MO+Pfcoso (45)

Deflection Equations(i) Rotation

Symmetry requires a boundary condition such that there is no net rotation over thequadrant:

fi do 3=(9 +E)do = 0 (4)

The value of Mo for given P can be found when this integral is satisfied.

(ii) Centreline Deflection of RingGeometry of deformed ring is shown in Figure 17, and neglecting second order

terms: 1 _d$

Initial curvature -

ds

I do+Aod_ S2dDeflected curvature +

rt is + Ads ds (1 - ujr)

I d2u

Change in curvature 1_ 1 = r +s 2 1r, r I - U/f" f

I d 2 + U) (47)

Using equation (31), and noting that opposite sense of M in Figures 15 and 16 reversessigns, the differential equation for deflection becomes:

1 d 2u 1 d'r2 ( do2 + U) = r(I - )(W + )

d 2 u= r( - i)(# + d-) (48)

Shear deflection -y is small and is assumed equal to the elastic deflection:

"= Awhere xt is the shape coefficient for shear and A cross-se-tional area. Substituting Vfrom equation (44):

d-y _ -P cos- (49)do GA

so the deflection equation becomes:d2 u cod 2 + U = r(1 - )(fl -PcOS ) (50)

d02 GAThe differential equation (50) is integrated using any standard numerical integrationtechnique, with terms i and fl for each angle 0 calculated as previously indicated.

APPENDIX B - MODIFICATIONS TO PAFEC CODE

The modifications listed were applied to PAFEC plasticity code at level 5.2. Itis expected 17 that later releases of PAFEC will include these or similar modificationshaving the same effect.

(i) Damping. When oscillatory behaviour occurs and convergence is uncer-tain, a damping routine may be invoked. This may lead to incorrect implementationof the Prandtl-Reuss equations and has been discarded. Section (iii) describes othermeans used for securing convergence.

(ii) Kinematic Hardening. Implementation of the Prandtl-Reuss equationshas been modified to properly incorporate yield surface translation.

(iii) Convergence. Convergence is accelerated and stability improved byadaptive adjustment of a convergence parameter.

(a) Convergence is accelerated when too slow.(b) When maximum error increases (instability), previous converged valuesare restored and convergence slowed until error reduces.(c) After convergence is re-established, it is again accelerated to hasten at-tainment of solution within tolerance.

(iv) Iteration Loop. Exit conditions have been changed so that a self-correcting load adjustment 18 is applied at the start of the next load increment. Thisgreatly improves solution accuracy for given error tolerance, and may allow tolerancelevel to be relaxed while maintaining satisfactory accuracy.

(v) Mixed Hardening. Mixed kinematic -isotropic hardening with variableproportioning has been included.

~- sii S.

02

/0

Yil ou03 \i Pla e (aj= 0

Yield surface

FIG. 1 REPRESENTATION OF VON MISES YIELD SURFACE

OITT lOy 2oy

Kinematic hardening(showing Bauschinger effect) pad

Mixed hardening

Isotropic hardening(no Bauschinger effect)

FIG. 2 STRAIN HARDENING BEHAVIOUR

Hardening curve

Initial yield locus

FIG. 3 ISOTROPIC HARDENING

Hardening curve

. 2

Initial yield fous

FIG. 4 KINEMATIC HARDENING

FIG. 5 MROZ HARDENING

Increment Load(Elastic Displacement Increment)

U1 =Uo+SUE

Zero 8uP, &P

U~U1 +bUpfP=CP+8fP

0

=Element & Gauss Point Loop

f=BuCE=(-( P

or=DcE

Calculate Oeq , eq

JCalculate error from difference

between aeq, feq and truestress-strain curveFind maximum errorI

Recalculate &e using errorCalculate &W from Prandtl-Reuss equations

Calculate 6FP=B& P

FIG. 6 PAFECyPLASTICITY ALGORITHM

k--~~~~~~~ Tolerancemm im =-- m

(L0CL 2 2

L co c-2L

ui0

C)C

0zLU

a-I

.(.

d

CL. LL

35 -Rf1 AE .

30 -Ter

25 -PAFEC (modified)

20 200

15

10

5

0 _jII0 2 4 6 8 10

Displacement mm

FIG. 8 COMPARISON OF SOLUTIONS FOR PROVING RING (SEE FIG. 7)

LUD

U-U04 CA z

Co zE LU

CL

E <0 0

LO CV) 0

LU

U., 0z

7-j

0) CD

LU <

LL-L

rC)- LO NL 1

(An)q 0li/la

35

30

25

z 20

o 15

10

5

0-10 -8 -6 -4 -2 0 2 4 6 8 10

Displacement mm -

FIG.~~~~1 10CCI ODN FPOIGRN

PLASTIC STRESS/STRAIN CONCENTRATIONS

6- Kt =3.0

2 Strain%.. Stowell

5-.

c,

4-

3-.- Finite element

'Ln, Neuber -,.

0 Stress01

1 .2 3 .4 .5 .6 .7 .8 .9 1.0

mean stress/yield stress

FIG. 11 HOLE IN PLATE - UNIAXIAL STRESSES

6- Kt =2.5(1 2 c(1/2

c Stowell/

/ Strain4-

N Neuber - .

>- 3 - "

Finitec 2- element

o Stress

.1 .2 .3 .4 .5 ,6 .7 .8 .9 1.0

mean stress/yield stress

FIG. 12 HOLE IN PLATE - PARTIAL BIAXIAL STRESS

U). .. ...1n -

,, , Finit,

PLASTIC STRESS/STRAIN CONCENTRATIONS

6 Kt =2.02

-. 5Cfl Stowell/

_ Strain4/

3 Neuber looU, --

2Finite element --.

0 Stress

.1 .2 .3 .4 .5 .6 7 .8 .9 1.0

mean stress/yield stress

FIG. 13 HOLE IN PLATE BIAXIAL STRESSES

r c----- I -, -I1+ s ! |

x

1 e max Yield stress

Strains Stresses

FIG. 14 RECTANGULAR BAR - STRESSES AND STRAINS

Original t of bar

r t with shear deflection

Rotation of facer dadc

Shear Shear dy

slope y

Shear slope% y +- dy

- i(shear deflectionnot illustrated)

FIG. 15 CURVED BAR - DEFLECTION AND STRAIN

~Mo

P/0

FIG 16POIGRN mODN

wLzLU

+ z

U-0z0

U-LUI

-LJ

c-

w)

d.- ~.-

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DOCUMENT CONTROL DATA

I .-AR I Jb. L-stubishnent No)2 ouetDoe3 oct&

AR-004- 512 - ARL-AERO-PROP-R-1 74 DECE MBER 1986 DST/860404.Fmle --- S Secnty6. lb FirAPPLICATION OF FINITE ELEMENT METHODS a. documriient 34

WITH CYCLIC ELASTO-PLASTIC ANALYSIS UNCLASSTO LOW CYCLE FATIGUE ANALYSIS OF b. title c. obstruct 7. W Refs

ENGINE COMPONENT'S U U _ 27

N.S. SWANSSON

o.Sporssor b.Security c.Downgroding d.Approvol

Aeronautical Research LaboratoriesP.O. Box 4331,MELBOURNE, VIC. 3001

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4. DecripionsIS. COSATI Grrxq

Finite element analysis 11130Elastic-plastic analysis 21100Biaxial stressesStress concentration

Low Cycle Fatigue (LCF) in engine components involves macroscopic cyclicplastic strains (with a stress-strain hysteresis loop) over a significant portion of thefailure region. Characterising elasto-plastic behaviour in potential failure regions is anecessary step in estimating LCF life.

The equations governing elasto-plastic behaviour are summarized, and themethods of implementing them in Finite Element (FE) stress analysis programsdiscussed.

An extension of the PAFEC program to include mixed isotropic-kinematichardening is outlined, and verified by examples for which alternative FE solutions wereavailable.

This paper is to be used to record information which is required by the Establishmrent for its own use but which will not be added tothe DISTIS data base unless specifically requested.

16. Abstract (cootd-A sample application has been made to holes in a plate with biaxial stress fieldssimilar to those in disc webs, and the results compared with the Neuber and modifiedStowell rules commonly used for design life estimation; these rules tend tooverestimate the strain in biaxial stress conditions, leading to conservative lifeestimates.

I1. 6print

Aeronautical Research Laboratories, Melbourne

18. Document Series and Nmrber -1. Cost C0. Type of Iort and Period Covere

AERO PROPULSION 41 3151REPORT 174

2 I. Computer Proqrnms Used

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