Improvement of fatigue performance by cold hole expansion. Part 1: Model of fatigue limit improvement

Int J Fatigue 15 No 2 (1993) pp 93-100

Improvement of fatigue performance by cold hole expansion. Part 1: Model of fatigue limit improvement V. Kliman, M. Bil)7 and J. Proh~icka The paper is concerned with localized fatigue performance improvement by means of a hole expansion technique. In order to explain this effect a theoretical model is proposed containing potentially important factors such as a residual stress ~r (strain er) induced by an uneven plastic deformation, stress-strain curve origin transformation ~t, microstructural changes evoked by plastic deformation AS and secondary (technological) factors Atech. Based on Haigh's diagram and the material stress-strain curve it is concluded that the expanding process has an optimum that can be characterized by the optimal residual strain er, opt in the hole root material volume. A method is proposed for estimating its value, yielding the maximum fatigue life enhancement. Key words: cold expansion of holes; fatigue performance; residual stress and strain; plastic deformation

Nomenclature

e,, opt

E En, K~

~a

Ep

K~ K~ Nr AL AS Atech Eat Ccr

e~, c

Modulus of elasticity Section modulae for ¢n and wp (see Fig. 9) Strain concentration factor Fatigue strength reduction factor Stress concentration factor Theoretical stress concentration factor Number of cycles to fracture = ~cp/~rcv, relative change of the fatigue limit after prestrain Fatigue limit influenced by microstructural changes Fatigue limit influenced by expanding technology Total strain amplitude of cyclic loading Critical residual prestrain above which the fatigue performance is lowered Fatigue ductility coefficient and exponent, respectively Prestrain strain Residual strain

orC (ICE Orcp O'CR O'cv O'cy

~f,b O"m

Ohm %

O'ty, o'y The cold expansion of holes is a well-known technique for fatigue performance enhancement, which has been used for over 30 years. Extensive research into this phenomenon has resulted in various technological methods designated as roller burnishing, ballizing, coining, mandrelizing, split or solid sleeve cold expansion, bushing cold expansion, applied expansion, * and possibly some others. The common feature of all of them is that they have been developed and verified under specific circumstances and for specific design configurations and technological tools. In practice, this 0142-1123/93/020093-08 Int J Fatigue March 1993

Optimal value of residual strain yielding the maximum improvement of fatigue limit Stress amplitude of cyclic loading Fatigue limit Fatigue limit influenced by am, O~rand crt Fatigue limit influenced by ~rm, cr, ~rt and AS Resulting fatigue limit influenced by Crm, err, or, ~S and Atech Fatigue limit of virgin (unprestrained) material specimens Compressive yield stress Fatigue strength coefficient and exponent, respectively Mean stress of cyclic loading Nominal stress Fictitious nominal stress Prestrain stress Residual stress Stress-strain curve transformation value caused by prestrain Tensile yield stress

means that their generalization or transfer to other conditions is virtually impossible without additional fatigue experiments or fatigue data. In other words, the final effect, ie substantial improvement of fatigue performance, is known but the factors which condition it cannot in general be theoretically quantified and so their levels cannot be theoretically substantiated and thus intentionally selected without experiments. To improve this situation we shall try in our study to analyse and quantify the process of hole expansion and its

(~ 1993 Butterworth- Heinemann Ltd 93

consequences for fatigue performance. Thus the objectives of the present investigations are: (1) (2) (3)

to specify factors conditioning the fatigue process in a hole plastically expanded by a mandrel; to propose and quantify a model of realistic effects of these factors on the fatigue limit; to verify this model experimentally and to optimize its parameters (in Part 2).

The stress-strain curve of low-carbon steel behaves similarly after prestrain but its tensile elastic part is even enlarged after cyclic loading with ~r,l~ = 173 and cq = 136.5 MPa (a shift from point A to B in Fig. 2(b)); this process keeps going even after the material has been aged for two weeks at room temperature. One must be aware, however, that these conclusions cannot be generalized, and more materials need to be cyclically loaded after prestrain in order to make a firm statement about the stress-strain curve transformation.

Residual stress effect Potentially important factors influencing fatigue performance after hole expansion Hole expansion sets up permanent plastic deformation in the annular material volume around the hole. Its consequent effect on the fatigue performance can be formulated, therefore, as a problem of the influence of localized plastic deformation on fatigue performance, taking into account a residual stress pattern brought about by an uneven deformation. In order to explain the difference between the virgin component behaviour under cyclic loading and that with an expanded hole a few groups of potentially important factors will be considered:

Figure l(a) also offers the explanation of the residual stress effect which is especially important for the material volumes around points A of the net section in Fig. l(b), critical from the point of view of fatigue crack initiation. If the component is cyclically loaded with tensile mean stress (Tm and amplitude o'a, for example, then the compressive residual stress cr', is added to o'm and as a result zero or even compressive mean stress o,~ may result (Fig. l(a)); for most materials such a change of the stress cycle asymmetry has a beneficial effect 3 as is also obvious from the Manson-Coffin equation with the mean stress correction Crm, ie c.,t =

(a) (b) (c) (d) (e) (f)

a material stress-strain curve transformation; a residual stress pattern; the fatigue (cyclic) properties of the material used; microstructural material changes evoked by plastic deformation; surface properties; secondary factors (such as static and dynamic strain ageing).

Material stress-strain curve transformation This can be explained following deformation of the material volume inside the hole during expansion (Fig. 1). When the stress o- is gradually increased above the yield point O'ty up to the stress value % and the corresponding strain value ep (point B in Fig. l(a)) then, after unloading, the plastically deformed material is compressed by elastically stressed surrounding material volumes and owing to the unavoidable stress equilibrium it is exposed to residual stress Or and strain e, (point A in Fig. l(b)). Providing that the Massing relation Oty + ~cy = constant = 2 O~y holds, the stress-strain curve origin O becomes O ' , shifted in the direction of the prestraining force to the tensile stress crt. Thus compared with the original stress-strain curve OB the new one AO'B (dashed line) has a higher tensile yield stress O',y. Under certain circumstances (see below) the improved tensile elastic properties may bring about better fatigue properties. The natural question which can be immediately raised in this connection concerns the stability of the transformed stress-strain curve during subsequent cyclic loading or at least during the fatigue crack initiation stage. The published evidence does not offer exhaustive information but from our previous experiments we can conclude 2 that the stress-strain curve shift is not altered by cyclic loading and, above all, can even be enhanced when the loaded material is aged. This is obvious from Fig. 2, which shows the stress-strain curve of A I - C u 4 - M g alloy in the virgin state (curve 1) and after 15000 cycles with Or. = 203 and o~ = 155 MPa (curve 2); almost the same curve was also obtained after 40000 cycles with o~ = 95.4 and ~ra = 161 MPa.

94

~f--

O'm

~E

b

(2Nf) + c~ (2N~) c

(1)

As for the stress-strain curve transformation, the stability of residual stresses during subsequent cyclic loading may also be questionable. Here the situation is far better, however, because various experiments prove that the induced residual negative stresses act during the whole fatigue crack initiation stage (at least) and obviously improve the fatigue performance. However, this may not be entirely true for tests in which the specimens with residual stresses are additionally prestrained (by a high peak of operating loading exceeding the yield stress, for example) and as a result the residual stress pattern is changed. Also, in cases when the cyclic loading generates vibrations at resonant frequencies evoking internal inertia forces and high local cyclic stresses, certain thermal ageing processes may take place and residual stresses are then diminished or even washed out. 4 This phenomenon is sometimes used as a kind of technological operation for diminishing residual stresses by vibration at resonant frequencies. s Considering, however, that the practical range of operating vibrations does not usually exceed 50 Hz, the resonant effects at higher frequencies do not usually occur in cases with expanded holes (as an exception one can mention a wing covering, or car body).

Fatigue (cyclic) properties of the material These represent a set of various factors such as the stability of the stress-strain curve (static, cyclic, of plastically deformed material), material sensitivity to mean stresses of cyclic loading (slope of Haigh's diagram), slope of the S I N (W6hler) curve, etc. They can be estimated only for a given material.

Microstructural material changes The changes evoked by plastic deformation and their consequences for fatigue damage accumulation present a problem which has not been clarified yet; moreover, the results are often contradictory. Some authors have deduced from their experiments that a small plastic deformation (say, up to 3-5%) increases the fatigue limit; others have found just the opposite. This is documented in Fig. 3, illustrating relations between the relative change of the fatigue limit AL = Ocp/Crcv vs the

Int J Fatigue March 1993

0

/

! °t

i

/(1)

% /

/!

Op

¢

P

Off

%

•20ty

O" I*

0

/

m

~-~ A

a

b

Fig. 1 Loading and unloading of the material volume inside the hole: (a)stress-strain curve and cyclic loading parameters; (b) stresses during loading (expansion) at the peak stress level Crp and unloading reaching ~r in the hole root

total material prestrain e. It is obvious that some materials increase their fatigue limit after prestrain but others may behave differently. Nevertheless, one should be careful when comparing these data because they are not homogeneous. For example, curves 26 and 27 belonging to A1-Cu-Mg alloy were obtained for a relatively high mean stress O"m = 191.0 MPa and cyclic amplitudes ~a = 183 and 140 MPa (high peak tensile stresses may bring about additional plastic deformation or wash out the preceding effect), whereas other specimens were tested with ~m = 0. Curve 20 corresponds to plastically deformed rods (from which fatigue specimens were subsequently machined), whereas other curves were obtained for prestrained specimens. Some specimens were dynamically and others statically loaded; certain specimens were strain aged after prestrain and before testing; others were prestrained in the first load cycle of cyclic loading, etc. Furthermore, most prestrains in Fig. 3 are too high for hole expansion applications because normally we shall be concerned with prestrain up to 3-4% only, as mentioned above. Nevertheless, despite these contradictions, the results from Fig. 3 seem to lead to the conclusion that most metallic materials improve their fatigue limit after a small prestrain.

Surface properties Fatigue is understood to be a surface phenomenon and so the surface properties play an important role in fatigue crack initiation. In the hole expansion process it is crucial, therefore, to control the state of the inside hole surface during and after prestrain, avoid formation of sharp burrs which can initiate fatigue cracks, etc.

Int J Fatigue March 1993

Secondary factors As far as the secondary factors are concerned, the adjective 'secondary' means that they may be secondary by their rank but not by their final consequences for the fatigue performance of components with prestrained holes. Above we mentioned, for example, strain ageing processes (static or dynamic). Another factor may be represented by the transition temperature, as it is known that, in general, plastic deformation changes (mostly increases) its level2 and so the prestrained material may become brittle at room temperature.

Model of hole expansion effect on fatigue performance It is undoubtedly difficult to look for a unique explanation of the influence of potentially important factors on fatigue performance; the more so, since they may be correlated and their experimental effects cannot be separated from each other. It is indispensable, therefore, to formulate a working hypothesis and verify its validity. For this reason we shall try to explain the role of the aforementioned factors and their influence on the fatigue performance of a component with an expanded hole using Haigh's (master) diagram (Fig. 4) constructed for the annular material volume in the hole root (and especially for the locations A in Fig. l(b)). Suppose that cyclic loading of a virgin component has the mean level O"m and its fatigue limit is ~cv. The effect of expansion and the corresponding compressive residual stress crr and stress-strain curve transformation wt. (Fig. l(a)) on the fatigue limit is now treated in the same way as the effect of mean stresses. This means that by subtracting their

95

B

,,0

j.

/

I I

300

¢

~

cycles

II

l 120

°

0 Co

200

31"'D

..-~"

30

j____

':

281

..0

"

~"~':'" ':

/

P ~lO000

:ycles ....j <3

10(] II E ~

d

"1"• g

6I

4

a

e

0

12

7

(g) 6

I

.-0""

80

>, "'

~'8 :o_..-o"" 13 14 ,26

D C

60 J27

400

0

7

II o tO -

200

100

16

24

(%)

300 O

8

A

Fig. 3 Relative change of the fatigue limit &L = CcP/Ccv vs the material prestrain (collected from various sources). 1, carbon steel; 2, 3, 4, 5, 7, 17, 18, l o w carbon steels; 10, 12, 19, 20, Cr steels; 6, 8, 11, 21, 22, 24, 30, C r - N i steels; 13, 14, refractory steels; 9, 15, 16, 23, 25, Mn steels; 26, 27, 28, a l u m i n i u m alloys; 29, copper

-

7,

E o

2

q

10

I

~ _ _ . L ~ 14 16 18 °C

e (%)

b

Fig. 2 Influence of cyclic loading on the stress-strain curve:

(a) A I - C u 4 - M g 2, after cyclic with (rr~ = 95.4 with (~r~ = 173

alloy; (b) low-carbon steel. 1, virgin material; loading with (rm = 203 and (r. = 155 MPa, or and or. = 161 MPa; 3, after 3.33 x 10 e cycles and (~. = 136.5 MPa and two-week ageing at

I OE c °Cp

°CR

room t e m p e r a t u r e ; 4, after additional t w o - w e e k ageing at room temperature

values from

Orm

we obtain the new fatigue limit ~CE- The

microstructural changes due to plastic deformation may now have either positive or negative influence on the fatigue limit ~cE- If they are quantified by AS they should be added to or subtracted from ~rcr. Suppose, for example, that their effect is negative (material is damaged during plastic deformation), so we get wcv. Finally, we consider the effect of surface properties and secondary factors A~echas being related to the expanding technology. It is our natural intention to make this part positive and thus to choose a procedure which would guarantee that the resulting fatigue limit crcR is higher than Crcv.

Optimization of hole expansion parameters It follows from the foregoing analysis that there are two groups of factors which could be varied for a given material, design and loading parameters in order to obtain the optimal increase of the fatigue performance: (1)

the peak (prestrain) stress %, residual stress cr~ and transformation stress crt values, respectively and, to a lesser extent

96

o=

m

Ot

Or

-I

Fig. 4 Interpretation of the influence of plastic deformation

(prestrain) on the fatigue limit (fatigue life) in Haigh's diagram

(2)

the method (type of technology) of hole expansion (plastic deformation) which at least partially affects the ~lS and ~ltech values (Fig. 4).

Consider first the o'r + o'~ value according to Fig. l(a) corresponding to the annular material volume behaviour inside the hole. When the static prestrain ep increases from %1 to eps (Fig. 5) the corresponding value of the transformation stress crt also increases in the same sense. When a certain value of %, say ~p4, is exceeded the unloading part of the stress-strain curve becomes non-linear, the o't value decreases and may even become negative (errs in Fig. 5). It is, therefore, obvious that after exceeding a certain limit value ep the sum crr + wt remains constant and approximately equals o'ty (the consequence of the Massing ratio). This means

Int J Fatigue March 1993

Oc~ Oy

°ty

°CP4 O

cy

0

%

Om

c

Fig. 5 Stress-strain diagram of the material volume inside the hole for loading and unloading

that the shift err and crt in Haigh's diagram to the negative mean stress shown in Fig. 4 is limited by the value ~ cry as schematically shown in Fig. 6. Suppose, further, that the microstructural material changes caused by plastic deformation have a negative influence and so reduce the fatigue limit (if they have a positive effect the final fatigue limit improvement is more pronounced). Thus by increasing the level of prestrain e v the fatigue limit changes from thc fatigue limit of the virgin material crcv along the curve P (dashed line in Fig. 7) up to the value, say, CroP4. It is obvious that this curve has its maximum related to the yield stress Cry. A higher prestrain does not result in a further shift to the left, but may have a detrimental effect on the fatigue limit which may decrease. In our model this is obviously due to the unfavourable microstructural damage induced by plastic deformation but for other materials, whose fatigue properties are not degraded by plastic deformation, the fatigue limit may stay constant or become even higher. Because this fact is not known a priori it is, therefore, reasonable to expand the hole to only such an optimal value for the residual strain in its inside annular material volume to become er, opt (Fig. 8). If the expansion is larger and the residual strain exceeds ~cr (point C), the component fatigue

+--]---

°t4

io I -

2 J~

+

°r4

l°.f

b,l°Erl Er2

I

~oy I

~3 ..

[

£ r3 c roOpt E:r4

Fig. 6 Relation between o"r -I- ort and residual strain cr

Int J Fatigue March 1993

t_

ar3

ar4

Fig. 7 Determination of the optimal value of prestrain and interpretation of the fatigue limit change with mean stress cm

.a 1 <~

'

I

'

~.,,.....

I

i

I

I

i .oo, I Fig. 8 Relative change of the fatigue limit AL vs residual strain cr with its optimal value cr. opt

life may even drop below the original value of virgin specimens. Thus in order to propose the optimal technology of hole expansion it is essential to determine the whole stress-strain pattern during expanding and subsequently, or as a minimum only the peak stress cry (strain ev) and residual stress crr (strain er) in the most critical hole root material volume.

Stress and strain inside a hole

i ~3

I

°t3

°t4

--

Er

Various approaches have been proposed for the determination of stresses (strains) in the annular material volume inside a hole during expansion and after unloading. However, not all of them give reliable results which are not in contradiction with experiments. In the first instance we have used the elastic-plastic analysis of a radially stressed annular plate by Chen 6 and of an infinite Sheet having a circular hole under pressure by Hsu and Forman, 7 which were reworked and programmed by Ko~fit, 8 but the computed stresses and strains were too high for our case of a finite aluminium alloy strip used for

97

the experimental verification of the proposed model (see Part 2) and exceeded even the tensile strength. Thus these methods were considered unacceptable. The next approach, theoretically rather suspect but commonly accepted in fatigue analysis, is based on the assumption that the material volumes around points A in Fig. 1, critical from the fatigue point of view, behave in similar fashion to the material of a plain specimen loaded in tension-compression. Two possible variants can be proposed.

oi

0 - 9 l - - -

Hardrath-Ohman

p

m

m

-+

P-= O'

method

Hardrath and Ohman 9 proposed an expression for the elastic-plastic stress K,~ and strain K~ concentration factors, respectively, in the form K,~ = 1 + (Kt - 1)(Ep/En)

/

/

/

/

(2)

II

/#. /

E

and

P

t

/

1 + (K~ - l)(Ep/En) Kc

-

(3)

!

Ep/En

The peak stress ¢p in the hole root of a specimen loaded by the nominal stress trn then equals o'p = OrnK,~ = cry[1 + ( K - 1 ) ( E p / E . ) ]

,//

//]

(4)

Thus, knowing the relation cr = fie) for a given material, one can easily determine %. If the peak stress % and strain ep are bound by the relation % = Ep ep

/

cn E

~3

-~nn (Kt-1)

(5)

then after substituting for Ep into Equation (4) we get O"n

%-Enep-(Kt-

E. Cp

(6) 1)Crn

expressing (for a given nominal stress en) a hyperbola in the coordinate system ~-e. Consequently, this hyperbola is a geometrical location of points representing possible stresses and strains in the hole root loaded with a nominal stress ~n. Its asymptotes are (Fig. 9): for ep going to infinity, % = ~n; and for ~p going to infinity Cp = ( T n (K~ - 1)/E,. Provided that the loaded material volume in the hole root behaves analogously to the material of a plain specimen, ie the functional relationship cr = fie) for a plain specimen coincides with the relationship % = flep), one can determine % and ep for a given nominal stress t~, as the intersection P of the material stress-strain curve with the curve described by Equation (6). In a similar way one can proceed during unloading or compression from the previous tension. Here point P becomes the origin O ' of the new coordinate system ~p-% in which a new intersection Pl of the unloading curve 4 with the hyperbola (6) is again determined. When the specimen with a hole is only unloaded, ie when the nominal stress Wn becomes zero, then point P1 determines the residual stress err and strain er in the hole root (Fig. 9). As is known, a material may change its mechanical properties during cycling owing to its softening or hardening. It may be, therefore, more appropriate to use the cyclic stress-strain curve instead of the static one. But this problem is still open to discussion because for random loading the stabilized cyclic stress-strain curve has an unclear meaning (if any). Neuber method

This method is based on the relation between the stress K~ and strain K~ concentration factors, respectively, derived by Neuber, l° which lead to the formula

98

Fig. 9 Determination of %, Cp, ~rr and er by means of Equation (6): 1, material stress-strain curve; 2, curve expressed by Equation (6), origin in point 0; 3, the same but origin in point O'; 4, unloading part of the material stress-strain curve

Crp ep =/~F o~oen

(7)

also representing a hyperbola in the coordinate system ¢-e. Further steps are similar to those above. Estimation expansion

of fatigue

limit after hole

The model of hole expansion effects proposed above can now be used for estimation of the resulting fatigue limit or, alternatively, for proposal of parameters of a hole expanding technological process. In order to simplify this procedure two additional assumptions are made, as follows. (1)

(2)

Haigh's diagram is linearized and supposed to grow continuously for negative mean stresses (this seems to be quite well experimentally proved and supported by Equation (1)). The effect of surface properties and secondary factors after prestrain At~ch is neglected (Fig. 4) as it is practically impossible to quantify it without specific experiments; this leads to the estimation of the minimum fatigue limit crcR provided that the hole expanding technology is reasonably optimized.

In accord with Fig. 4 we can write for the material volume in the hole root.

Int J Fatigue M a r c h 1993

where ¢cepl is the plain specimen fatigue limit after prestrain at el. In practice this means that a set of plain material specimens is prestrained at el and then the averaged value of the stress-strain curve transformation wp~ and corresponding fatigue limit ~C~p~ are determined. After inserting into Equation (1) the relation AS1 = f(el) is calculated. Repeating this procedure for various ei the whole curve ~S = fie) (Fig. 11) can be quantified.

oC

As[

°ct 0

ot

0

m

m

Discussion

or

Fig. 10 Experimentally determined (non-linear) Haigh's diagram

L/3 I

¢rq

Er

Fig. 11 Relation between AS and residual strain er

Crcv + h (~r~ + o-J - AS = ~rci,

(8)

O'CR = OrCp -[- Atech

(9)

and

where h is the linearized Haigh's diagram slope. If this is not available then we need the experimentally determined master diagram similar to Fig. 10. In order to make use of Equation (8) it is indispensable, however, to know the relation between AS and strain e. Provided the microstructural changes evoked by plastic deformation have a negative effect this may look like the curve in Fig. 11 and is to be determined experimentally at plain specimens, analogously to Haigh's diagram. In a certain sense, it also represents a material characteristic. In accordance with the proposed model modified for plain specimens one can write (Fig. 12) ~rcv + h

(10)

(J'tl -- A S I -~" O ' c p p l

oC

o

~CPpl

OCv

0

o~

0

E, c

0 m

Fig. 12 Haigh's diagram for plain material specimens and the material stress-strain curve

Int J F a t i g u e

March

1993

The proposed model generates one natural question: what is the actual reason for the fatigue limit improvement after hole expansion? The foregoing derivation makes it clear that the effect of the stress-strain curve transformation and residual stress is to reduce the level of mean stress acting in the critical material volume in the hole root around points A (Fig. 1). Because, according to Conlon and Reid's findings, i I the time spent to initiate a short crack from an expanded hole up to its effective 'technical' length of about 0.3 mm represents a substantial portion of the total fatigue life, it must be agreed that the benefit of the cold hole expansion consists in closing of existing short cracks and reducing their stress concentration effects and/or retardation of the short-crack propagation rate. This fact leads to the conclusion, however, that the hole expansion technique for fatigue performance enhancement is the most effective in cases of cyclic loading with high positive mean levels, otherwise the negative cyclic amplitude added to the negative residual stress may exceed the yield stress and the effect of cold expansion will be washed out (the fatigue limit may even drop below that of virgin specimens owing to the negative influence of plastically deformed microstructure: point C in Fig. 8). The starting information for estimation of residual stresses and strains is the static material stress-strain curve. It can be argued that this curve is related to the first cycle only and subsequent cyclic loading may change the material properties because of hardening or softening. This is an undisputable fact but the residual stress (strain) redistribution may only be expected when the material substantially softens and decreases its elastic range of straining. Such a case can occur especially for thermomechanically treated and hardened materials but this is not usually the kind of technological operation planned for material sheets containing drilled holes. If, however, the material stays stable or hardens cyclically, no substantial differences are to be anticipated between the use of static or cyclic stress-strain curves; naturally, we are not concerned with low-cycle fatigue with rather large cyclic plastic deformations as the hole expansion technique is not intended for such applications. A questionable assumption used in the proposed model is the equivalence of behaviour of material under tension-compression and the material volume in the hole root (around points A in Fig. l(b)). Whereas the material of a plain specimen is stressed uniaxially, the hole root material volume is stressed biaxially (hoop and radial stresses). Considering, however, that the hoop stress is substantially higher than the radial one which, besides, has a zero value at the inside hole surface, it is wise to neglect the effect of biaxiality and suppose analogous behaviour in both cases. The same assumption was, moreover, used in the derivation of the Neuber hyperbola (Equation (7)) whose validity has been experimentally verified many times.

99

References 1.

2.

Chempoux, R.L. 'An overview of hole cold expansion methods' Proc /nt Conf Fatigue, Prevention and Design, Amsterdam, The Nether~ands, 1986 (Chameleon Press Ltd, London, UK, 1986) pp 35-61 Bily, M., Kliman, V. and Terentev, V.F./nfluence of P/astic Deformation and Strain Ageing on Fatigue Strength (Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977) (in Czech)

3.

Heywood, R.B. Designing Against Fatigue (Chapman & Hall, London, 1962)

4.

Rottvel, F. On Residua/ Stress During Random Load Fatigue (Danish Center for Applied Mathematics and Mechanics, Lyngby, 1971)

5.

Rappen, A. 'Vibration zur Eigenspannungsreduzierung' Mashinenbautechnik 21 (1972) pp 40- 42

6.

Chen, P.C.T. 'Elastic-plastic analysis of a radially stressed annular plate' J Appl Mech 44 (1977) pp 167-169

7.

Hsu, Y.C. and Forman, R.G. 'Elastic-plastic analysis of

100

8. 9.

10.

11.

an infinite sheet having a circular hole under pressure' J Appl Mech 42 (1976) pp 347-351 Ko,~dt,J. 'Computational program OKO.Fort' (developed in the authors' Institute; private communication) Hardrath, F.H. and Ohman, L. 'A study of elastic and plastic stress concentration factor due to notches and fillets in flat plates' NACA TN 1117 (1971) Neuber, H. 'Uber die Ber0cksichtigung der Spannungskonzentration bei Festigkeitsberechnung' Konstruktion 2D (1968) pp 245-249 Conlon, J. and Reid, C.N. 'The effect of cold-expanded holes on the fatigue properties of aluminium alloy 6082' Proc Fatigue 84, Birmingham, UK, 1984 (EMAS, Chameleon Press, London, UK, 1984) pp 1683-1693

Authors The authors arc with the Department of Fatigue, Institute of Materials and Machine Mechanics of the Slovak Academy of Science, Ra~ianska 75, 836 06 Bratislava, Czechoslovakia. Received 15 June 1992; revised 2 September 1992.

Int J F a t i g u e M a r c h 1993