LITERATURE I,

2. 3. 4, 5.

O. T. Nazarenko and I. I. Ishchenko, ."An investigation of corrosion-fatig'~e crack development based on the fracture of samples,," Prik!. MeMh., 14, No. 9, 52-61 (1978). S. Ya. Yarema, '~An investigation of fatigue crack growth and Mnetic fatigue fracture curves," Fiz.-Khim. Mekh. Mater. No. 4, 3-22 (1977). G. V. Karpenko, The Strength of Steel in a Corrosive Medium [in Russian], Mashgiz, Moscow (1963). L. A. Glikman and L. A. Suprun, "Questions of the mechanical-corrosion strength of metals," Tr. Tsent. Nauch.-Issled. Inst. Morskogo Flora, No. 22, 16-25, Leningrad (1959). A. V. Karlashov and V. P. Tokarev, "The influence of loading frequency on the fatigue resistance of V95 aluminum alloy under conditions of the physicochemical action of a worMng medium," Fiz.-K'him. Mekh. Mater., No. 1, 67-72 (1967).

METHODOLOGY OF

CITED

THE

OF RESISTANCE

RESISTANCE) S.

OF Ya.

DETERMINING TO MATERIALS

THE

CRACK

CHARACTERISTICS

DEVELOPMENT IN

CYCLIC

Yarema

(CRACK LOADING UDC

620.178.3 : 620.191.33

The second stage of fatigue f r a c t u r e , the fatigue c r a c k development stage, has been the object of many investigations in recent decades. The m a t e r i a l s and elements of such c r i t i c a l s t r u c t u r e s as aircraft, p r e s s u r e v e s s e l s , piping, and r e a c t o r s , the providing of safety in the s e r v i c e of which is impossible without taking into consideration the p r e s e n c e of c r a c k - l i k e defects, have a t t r a c t e d special attention. Based on the data of such investigations the rate of fatigue c r a c k development p e r cycle (v) and the c o r responding p a r a m e t e r determining the mechanical condition in the p r e f r a c t u r e zone at the c r a c k tip are calculated. With known limitations, p r i m a r i l y on the size of the zone of inelastic deformations and the a p p e a r a n c e of time effects, this value is the g r e a t e s t value of Kma x or the range AK of the e l a s t i c coefficient of s t r e s s intensity [1]. These data are the points in the v vs Kmax (AK) coordinate plane (Fig. 1) on which the fatigue f r a c t u r e curve is recorded. Fatigue f r a c t u r e curves contain a large volume of information on the r e s i s t a n c e of a m a t e r i a l to fatigue f r a c t u r e (as f o r tensile curves in steady loading on the r e s i s t a n c e to deformation). The full curve (Fig. 1) to a l o g a r i t h m i c scale (log v vs log Kma x) is a steadily i n c r e a s i n g S-shaped curve bound by v e r t i c a l asymptotes c o r r e s p o n d i n g to the threshold s t r e s s intensity f a c t o r Kth below which the c r a c k does not grow and to its c r i t i c a l value Kfc (also called the cyclic f r a c t u r e toughness) upon reaching which final f r a c t u r e of the sample occurs. The e x p e r i m e n t a l points of the f u l l c u r v e of p r a c t i c a l value c o v e r a range of change in c r a c k growth rate of not less than five o r d e r s of magnitude, starting f r o m a rate of ~ 10 -~~ m / c y c l e and ending with a rate of ~10 -5 r n / c y c l e . This c o r r e s p o n d s to an i n c r e a s e in the g r e a t e s t cycle s t r e s s intensity f a c t o r of 2-50 times. The curve consists of three portions, the two e x t r e m e curved low and high c r a c k growth rates and the c e n t e r portion, which is well approximated by a straight line. Normally, the c e n t e r portion s t a r t s at a rate close to 10 -8 m / c y c l e and ends at 10 ~G m/cycle, but in specific c a s e s for brittle m a t e r i a l s there is a tendency toward a d e c r e a s e in the upper boundary, as may be seen f r o m the results of [3] and for ductile toward an inc r e a s e to 10 -5 cycles and more. The division of the curve into portions has a physical basis. The individual portions c o r r e s p o n d to f r a c ture m e c h a n i s m s c h a r a c t e r i s t i c of t h e m and reflected in the f r a c t u r e m i c r o r e l i e f ([4-6], for example). This is responsible for the r e a c t i o n specific for different portions on the action of various mechanical, metallurgical, and p h y s i e o c h e m i c a l factors, which in t u r n may lead to changes in the f r a c t u r e mechanism. The t e m p e r a t u r e change in c r a c k growth rate in cold brittle alloys is a c l e a r example of such an interaction. A reduction in G. V. Karpenko P h y s i c o m e c h a n i c a l Institute, A c a d e m y of Sciences of the Ukrainian SSR, L'vov. T r a n s lated f r o m F i z i k o - K h i m i c h e s k a y a Mekhanika Materialov, VoI. 17, No. 4, pp. 100-110, July-August, 1981. Original article submitted .April 17, 1981.

0038-5565/81/1704-0371507o50

9 1982 Plenum Publishing C o r p o r a t i o n

371

J

5

10

Kml~,MPa'~f~

Fig. 1. T y p i c a l fatigue f r a c t u r e curve (MA18 m a g n e s i u m alloy [2]). The n u m b e r s show the portions of the curve.

//

%

0 0 0

o

0>

9 ~ 0 -7

,0,

o o

O O [ I

I

I

Fig. 2. Fatigue f r a c t u r e curve of 3.5r a m - t h i c k sheet of V T 3 - 1 t i t a n i u m alloy.

2

3

I

5

|

Kmax, M P a "

j q~

Fig. 3. Fatigue f r a c t u r e c u r v e of 1VIA2-1 alloy.

t e m p e r a t u r e to a value below the c r i t i c a l s h a r p l y i n c r e a s e s the r a t e of f r a c t u r e , which is caused by the t r a n s i tion f r o m a dominating s t r i a t i o n ductile c r a c k d e v e l o p m e n t m e c h a n i s m to p r e d o m i n a n t l y t r a n s g r a n u l a r s h e a r [71. It is known that the c e n t e r portion, in c o n t r a s t to the e x t r e m e ones, is n o r m a l l y r e l a t i v e l y insensitive to the m a t e r i a l s t r u c t u r e [8] and also, within known limits, to cycle a s y m m e t r y if the range of the s t r e s s intensity f a c t o r is plotted on the X axis. As a r e s u l t of the l a r g e difference between the f r a c t u r e r a t e s in different p o r tions the t i m e effect, which causes, f o r e x a m p l e , a s m a l l influence of the m e d i u m in the third portion with sufficiently high loading f r e q u e n c i e s , is r e v e a l e d differently. The c o m p a r a t i v e l y l a r g e p l a s t i c zones at the c r a c k tip in the third period cause the g r e a t e s t s e n s i t i v i t y of the c r a c k growth rate to s a m p l e dimensions. F o r fatigue f r a c t u r e c u r v e s s y m m e t r y r e l a t i v e to the c e n t e r point with an a b s c i s s a of Kma x = KthKfc , a fact not a p p a r e n t l y having a p h y s i c a l basis, is c h a r a c t e r i s t i c . Small deviations f r o m s y m m e t r y a p p e a r p r i m a r i l y in different s t e e p n e s s e s of the second and fourth p o r t i o n s , F o r m a t e r i a l s with ductile c r a c k advance m e c h a n i s m s the third portion is l e s s steep. In the c e n t e r portion t h e r e is frequently convexity (Fig. 2), which is s o m e t i m e s [9] r e l a t e d to a t r a n s i t i o n f r o m plane d e f o r m a t i o n to the plane s t r a i n condition. However, t h e r e a r e many fatigue f r a c t u r e c u r v e s with m o r e s u b s t a n t i a l deviations f r o m the t y p i c a l Sshaped curve. Let us distinguish t h r e e b a s i c g r o u p s of such a n o m a l i e s . 1. C u r v e s on which the third portion does not a p p e a r at l e a s t until a r a t e of 10 -5 m / c y e l e . Such c u r v e s a r e c h a r a c t e r i s t i c of c a s e s in which at high load amplitudes l a r g e p l a s t i c zones develop and, consequently, the e l a s t i c s t r e s s intensity f a c t o r loses its d e t e r m i n i n g value. At the s a m e t i m e , e x p e r i m e n t a l points with the

372

f5 Oo

]4

7 //

//

f3 O

30

f2

o

K a,y9

it

20 i0 g 2

Fig. 4

o~o

3

r

5

6n

9

o cxO '--b o cD ~'~ ~ o o o~ o I

O i

I

2

5

4

Fig. 5

}?

Fig. 6

Fig. 4. ]Plan of the fatigue fracture curve of an alloy tested in a corrosive and in air (broken line). Fig. 5. The relationship between the parameters and literature data for 90 types of steels).

5

medium

(solid line)

n and C for steels (from the data of our tests

Fig. -6. The distribution of the value of K* in relation to the parameter data as in Fig. 5).

n (according to the same

f o r m a l l y calculated a b s c i s s a Kma x may lie on the extension of the straight center portion (Fig. 2) and also deviate f r o m it upward and even downward [10]. 2. Curves with local anomalies. This type includes the curve shown in Fig. 3, at the end of the first p o r tion of which c r a c k growth a l m o s t stops in a s m a l l range of Kmax and then i n c r e a s e s sharply. 3. A large class is made up of curves obtained in tests made in c o r r o s i v e media, the action of which is v e r y varied. In the overwhelming m a j o r i t y of cases the influence of the medium is revealed in the formation of a typical hump, as shown in Fig. 4, which g e n e r a l i z e s the results of testing steels in hydrogen, w a t e r [12], and a 3% aqueous NaC1 solution [13]. F o r fatigue f r a c t u r e curves a r e l a t i v e l y large spread in e x p e r i m e n t a l points, which may be explained by the following factors, is c h a r a c t e r i s t i c . 1. The d i s c r e t e n e s s of c r a c k growth and microinhomogeneity of material, leading to local fluctuations in the r e s i s t a n c e to c r a c k development. Nonuniformities in c r a c k tip advance both with r e g a r d to time and to the contour caused by this are inherent in the p r o c e s s itself and b e a r useful information on the o c c u r r e n c e of f r a c ture in the m i c r o v o l u m e s . 2. A p p r o x i m a t e n e s s and limitations in the concepts of f r a c t u r e mechanics based on some mechanical model. Small v a r i a t i o n s in c r a c k growth rate with specified p a r a m e t e r s of the cycle of the s t r e s s intensity f a c t o r s caused by this are inseparable f r o m v a r i a t i o n s related to inhomogeneity in the material. However, we must p r o t e c t the conditions of applicability of f r a c t u r e mechanics, p a r t i c u l a r l y of its linear variation, f r o m large changes of this type. 3. Attendant phenomena such as a delay in a c r a c k a f t e r a change (especially a reduction) in load, loss of stability of the f o r m of the sample in g e n e r a l and locally in the vicinity of the crack, a macrodeviation of the c r a c k t r a j e c t o r y f r o m its plane, m a c r o b r a n c h i n g of it, etc. 4. Changes during testing of its conditions, including a s y m m e t r y , frequency and f o r m of the cycle, v a r i a tions in t e m p e r a t u r e and m e d i u m composition, etc. 5. I n a c c u r a c y in m e a s u r i n g c r a c k length and number of cycles. 6. Choice of a method of calculating c r a c k growth rate and s t r e s s intensity factors inadequate for the expe ri ment.

373

5'O

1970

Erdogan F., Ratwani M. [20]

Author

)

I/Ko=inf(Kta, R K ~

]~

time

2

2

/c

1(2

K . . . . --Ko

v=voln.~[lg(K1dKt~)lg-l(KtdKmax)]

Kmax=dKmax/dt, t -

(9)

[28]

A. E. Andreikiv

1976 Dover W. D. [29]

(lO) 1976

Branco C. M-, 1975 Radon J. C., Culver L. E. [27]

[26]

S. I. Mikitishin

S. Ya. Yarema,

1975

(8)

(K1r

max

TM

(Kmax--Kth)q (Ks~--Kmax) r

q--( Kmax--Kth) q

(Kmax--Ktn) q

fc

KS _K2

2

2

Kmax--Kth ]q

(Kfc--Kmax)

v~A(l--R)~+r-qKmax

U~Uo

n

Kmax--Kth

n

(Kyc--Kmax) m

(Kmax--Kth) '~

'Ktc Kmax

v=vo(1--R2) q

V=Vo [

v = A ( I - - R ) .....

Chu H. P. [25]

1974

(7)

v = A ( 1 - - R ) ......

Chu H. P. [25]

19i4

(11)

--K z o

q

(KIr

(Kfe-J-Kmax) (Kf e--K0)

/(max--/(0

In

fz

Kz

JJ

2

-t-~-ax

2

K/e--Kin a x

2

Kyc--Kmax

2 2 /(m a x - - K / h

K)' e--Kin a x

(/(max--K*h) m

/7

I

i[

v=A(1--R)q

Equation

-+

J. E., Da1972 Collipriest 1974 vies K B., F e d d e r - v = v o l g q [ ( 1 - - R ) / ( . . . . /Kth]lg-q[Kso/((1--R)Km~x)] sen C. E. [23, 241

(6)

(5)

1973 Freudenthal A. [14]

[22]

(4)

Wnuk M. P.

1973

(3)

(2) t 1973 MeEvily A, J. [21]

1

Eq. Year No.

TA BLE 1

r

Annis C, O., jr., 1977 Wallace R, M., Si,s D, L. [31)

Kt~--Kma.~

b

Kmax--/Gh

max

if,

n

)~'~ }~

may

K,~

In a l a t e r w o r k [35] t h e a u t h o r of Eq. (2) w r o t e it i n t h e (1(.... --K~I~)~ ~=~(~-~}~ (K,~-~ .... ), 2, Tn Eq, (13) the c o n s t a n t the

1.

n

l o g a r i t h m o f w h i c h i s t h e a b s c i s s a of t h e c e n t e r of s y m m e t r y of the f a t i g u e f r a c t u r e c u r v e t o a l o g a r i t h m i c s c a l e i s d e s i g n a t e d as K m. 3) In [34] E q . (16} w a s w r i t t e n w i t h an e r r o r w h i c h h a s b e e n c o r r e c t e d he r e .

form

Notes:

[1--

(16) 1981 / P~176

z3==vo

n

/

K m~

~ A [K .... KK/c__Kmax--Kth] q

Saxena A., Iv=~'~ (]5)} 1979 I Hudak S. J,, jr. [33]

(14) t 1977 Duggan T. V. [32]

03)

r K v~voexp [ (1 R) .....

Bowie O. E., Hoep-I ~176 ...... )]-|--lgq[Kj~/(Kl~--Kth)]}t

(12) 1976 pner D. W. [30]

TABLE 2 Eq. No. of paramNo. eters (11) (6) (21) (22) (9) (23) (8) (5) (t)

6 5 5 5 4 4 4 4 4

0,096 0,097 0,097 0.098 0,102 0,103 0,104 0,106 0,108

The last four groups of f a c t o r s d i s t o r t the information, but in principle m a y be d e c r e a s e d to a specified allowable value. Yet, recently, d i f f e r e n c e s of 10 t i m e s between c r a c k growth r a t e s in n o r m a l l y s i m i l a r m a t e r r i a l s with a specified range of s t r e s s intensity f a c t o r have been a s s u m e d n o r m a l [14]. Such a l a r g e s p r e a d is undoubtedly the r e s u l t of i n c o r r e c t n e s s in coriducting t e s t s caused, f o r e x a m p l e , by the use of l i n e a r f r a c t u r e m e c h a n i c s beyond the limits of its applicability and by o t h e r r e a s o n s given above. In connection with f a c t o r s which cannot be t a k e n into c o n s i d e r a t i o n o c c u r r i n g in t e s t i n g of t h e m (such as the complex configuration of the c r a c k contour) many s a m p l e s do not p o s s e s s sufficient m e t r o l o g i c a l capacity, and, t h e r e f o r e , the i n f o r m a t i o n obtained with the use of t h e m is to a significant d e g r e e only relative. In recent y e a r s , as a r e s u l f of much work on e s t a b l i s h i n g the conditions of c o r r e c t n e s s and s t a n d a r d s f o r conducting t e s t s [15, 16], the evaluation of the allowable s p r e a d has changed significantly. F r o m an a n a l y s i s [17] of the data of a base e x p e r i m e n t made on the initiative of the A m e r i c a n Society f o r T e s t i n g and M a t e r i a l s by 15 leading l a b o r a t o r i e s of the USA and England it was e s t a b l i s h e d that today a t h r e e fold v a r i a t i o n in the values obtained in different l a b o r a t o r i e s and in a single l a b o r a t o r y of the c r a c k growth r a t e in a m a c r o h o m o g e n e o u s m a t e r i a l of a single heat with a specified cycle a s y m m e t r y R and Kma x within the l i m i t s of the c e n t e r portion of the curve is typical. If, in addition, data lying outside the g e n e r a l rule a r e e x cluded, then the amount of the v a r i a t i o n d r o p s to two. T h e s e conclusions c o m p l e t e l y a g r e e with o u r r e s u l t s , which show that the r a t i o of the m a x i m u m and m i n i m u m c r a c k growth r a t e s obtained during t e s t s using the known method of [16] with a constant Kmax and cycle a s y m m e t r y n o r m a l l y do not exceed 1.6 and onlyin e x t r e m e c a s e s r e a c h 2. The i n f o r m a t i o n included on the curve on the r e s i s t a n c e of a m a t e r i a l to fatigue c r a c k d e v e l o p m e n t is e x p r e s s e d quantitatively through the c h a r a c t e r i s t i c s of cyclic c r a c k r e s i s t a n c e , the role of which is s i m i l a r to the role of the e l a s t i c constants, the e l a s t i c limit, the yield strength, etc., in d e s c r i b i n g the d e f o r m a t i o n p r o p e r t i e s of a m a t e r i a l using the t e n s i l e curve. Such c h a r a c t e r i s t i c s must be as few as possible, but sufficient to provide c l e a r r e p r o d u c t i o n of the curve with an a c c u r a c y inherent in its s p r e a d in e x p e r i m e n t a l points. F r o m this r e s u l t two b a s i c p r o b l e m s . The f i r s t is to d e t e r m i n e f r o m the e x p e r i m e n t a l data the c h a r a c t e r i s t i c s of cyclic c r a c k r e s i s t a n c e of the m a t e r i a l , and the r e v e r s e is to r e p r o d u c e the curve of the m a t e r i a l f r o m the known c h a r a c t e r i s t i c s . We p r o p o s e d [15, 18] a s s u m i n g as the b a s i c c h a r a c t e r i s t i c s the t h r e s h o l d Kth and c r i t i c a l Kfc s t r e s s int e n s i t y f a c t o r s of the m a t e r i a l and the p a r a m e t e r s n and K* of the exponential equation d e s c r i b i n g the c e n t e r portion of the curve in the range of values of Kmax f r o m KI_ 2 to K2_~. The l a t t e r values a r e considered as s e c o n d a r y c h a r a c t e r i s t i c s of the m a t e r i a l . The c h a r a c t e r i s t i c Kth is d e t e r m i n e d d i r e c t l y f r o m the e x p e r i m e n t as the g r e a t e s t value of Kmax at which the c r a c k does not grow or, in p r a c t i c e , at which its rate is l e s s than 10 -l~ m / c y c l e . Since d i r e c t d e t e r m i n a t i o n of it is v e r y laborious and is always a c c o m p a n i e d by a high p r o b a b i l i t y of slowing down of c r a c k growth i n t r o duced by the reduction in load unavoidable in such c a s e s , it is obviously n e c e s s a r y to give p r e f e r e n c e to d e t e r mination of Kth by analytical e x t r a p o l a t i o n using the points of the f i r s t portion. The cyclic f r a c t u r e toughness Kfc is found e x p e r i m e n t a l l y . Strictly speaking, the b a s i c c h a r a c t e r i s t i c of the m a t e r i a l is the value of KIfe c o r r e s p o n d i n g to b r i t t l e final f r a c t u r e a c c o r d i n g to the m e c h a n i s m of n o r m a l rupture. However, in this case, as has b e c o m e c l e a r now [19], it to a c e r t a i n d e g r e e depends upon the condition of the m a t e r i a l in the p r e f r a c t u r e zone reached during the loading p r e h i s t o r y and the g e o m e t r i c d i m e n s i o n s of the s a m p l e , i.e., upon f a c t o r s only weakly capable of s t r i c t accounting for. The outcome of such a situation m a y be dropping of s e a r c h e s f o r the ~true" Kfc and d e t e r m i n a t i o n of its conditional value by a p p r o p r i a t e e x t r a p o l a tion of the e x p e r i m e n t a l r e s u l t s in the third portion of the curve. F o r a reliable and c l e a r d e t e r m i n a t i o n of the c h a r a c t e r i s t i c s Kth and Kfc by the e x t r a p o l a t i o n method it is n e c e s s a r y to have e x p e r i m e n t a l c r a c k growth r a t e s 376

in the f i r s t and third portions 50 to 100 t i m e s less or g r e a t e r than the rates at the left and right bands of the curve, respectively.

In the absence of a clearly expressed third portion and also in the presence of developed plastic zones when the conditions of applicability of the stress intensity factor as characteristics of the stressed state are not fulfilled, Kfc is generally not determined. The characteristic n is the exponent in the Paris exponential equation V=CAK n, and K* is related clearly to the coefficient C of the equation K* = (I07C)-I/n and corresponds to writing this equation in the form v = 10-TAKn/[(I_R)K, in, in m/cycle. From this it may be seen that the value of K* is equal ~co the greatest cycle stress intensity factor with a crack growth rate of 10 -7 m/cycle. The choice of the parameter K* and not C as the characteristic of crack resistance is the resuR of the fact that C is not an independent value, which is expressed in its dimension and is dependent upon n, and also in the known regression equation log C =an+b, which is characterized by a high (more than 0.96) correlation factor (Fig. 5). This equation~ despite the general interest caused by it, does not in essence carry any useful information and expresses only the known fact that a change in crack growth rate caused by a change in chemical composition and structure of the material, the medium, the loading frequency, etc., i.e., by factors the influence of which is the subject of our investigations, is insignificantly small in comparison with a change in the coefficient C. The coefficient C, numerically equal to the crack growth rate formally calculated using the Paris equation with AI4ma ~ =I, may, as follows from literature data, vary for steels by !6 orders of magnitude, 2 _rl f r o m 10- 5 to 10 -9 (MPa. 4-~) , m / c y c l e . A l l o f t h i s is true with the eondition that the point A K = I is sufficiently distant f r o m ~ h e c e n t e r portion of the curve. This o c c u r s if K is m e a s u r e d in the generally used units, M'Pak g / m m y2, ksi - ~ o r finer. If K is measured, f o r example, in t o n s / c m 3/2, then the c o r r e l a t i o n between n and C d i s a p p e a r s since the point A K = I then falls in the c e n t e r portion of the curve or v e r y close to it. In connection with this the p r o b l e m a r i s e s of finding a second c h a r a c t e r i s t i c which is independent in a s t a t i s t i c a l sense. The solution of this p r o b l e m has led us to the above mentioned value K*. The absence of a c o r r e l a t i o n be'tween it and n may be seen well in Fig. 6. The c h a r a c t e r i s t i c s n and K* are determined f r o m the P a r i s equation by the method of least squares using the points of the c e n t e r portion, the s t a r t of whichis f i r s t found visually on the curve [16] and then, after calculation of n and K* is d e t e r m i n e d more accurately. Taking into consideration the above differences between the portions of the curve each of the basic c h a r a c t e r i s t i c s of cyclic c r a c k r e s i s t a n c e was selected so as to r e f e r only to a certain portion, and only the a~txiliary ones are c o m m o n to the two adjoining ones. F r o m this we a r r i v e at the conclusion that in spite of the e s t a b lished p r a c t i c e each of the c h a r a c t e r i s t i c s must be determined not f r o m all of the points but only f r o m the points of that portion to which it r e f e r s . The fulfillment of this r e q u i r e m e n t is also important because of the following considerations. If the results of t e s t s only within the limits of a single portion exist (the center, for example, ~,hich is most frequently true), then the values of the c h a r a c t e r i s t i c s found (n and K* in this ease) will be final and do not change a f t e r obtaining information on the other portions. The possibility of the influence of points of one portion on the c h a r a c t e r i s t i c s of another is eliminated, which appears e s p e c i a l l y in those cases when the e x p e r i m e n t a l points are distributed nonuniformly by portions and the spread and a c c u r a c y in determining the c r a c k growth rate in t h e m is different. F o r example, in determining by extrapolation the c h a r a c t e r i s t i c s Kth and Kfc using all of the points the points of the c e n t e r portion, which do not have a relationship to t h e m and a r e n o r m a l l y m a r k e d l y m o r e than all the others, may have a substantial influence on t h e i r values. Finally, using an analytical e x p r e s s i o n s y m m e t r i c relative to the c e n t e r portion of the curve we artificially relate the values of these two completely different c h a r a c t e r i s t i c s . T h e r e f o r e , we have outlined in g e n e r a l t e r m s the method of solving the problem. However, to accomplish it and also to solve the r e v e r s e p r o b l e m it is n e c e s s a r y to select an adequate m a t h e m a t i c a l model of fatigue c r a c k growth, i.e., an analytical e x p r e s s i o n d e s c r i b i n g the fatigue f r a c t u r e curve, also called the c r a c k growth rule. Such an e x p r e s s i o n is n o r m a l l y constructed f o r m a l l y on the basis of the above- considered p r o p e I i i e s of the curves. At present, to the best of our knowledge, 16 m a t h e m a t i c a l models (Table 1) have been proposed for d e s c r i b i n g all t h r e e portions of the curve and containing f r o m four to eight p a r a m e t e r s f o r e x p e r i m e n t a l d e t e r r~Anation. Five of them, (3), (4), (10), (11), and (14), apparently to some degree because of mathematical difficulties o c c u r r i n g in calculating t h e i r p a r a m e t e r s have n e v e r been used in practice. All of these equations, other than (13), require the p r e s e n c e of the v e r t i c a l asymptote Kma x =Kfc and, with the exception of (13) and (15), of the threshold s t r e s s intensity f a c t o r different f r o m zero. One q u a r t e r of the equations, (5), (8), (9), and

377

(13), a r e s y m m e t r i c relative to t h e i r c e n t e r and half of them, (1), (2), (6)-(9), (11), and (14), with 1~ =0 m a y be considered as a p a r t i c u l a r case of the g e n e r a l e x p r e s s i o n v = AK~,~

(K~. x --

K;i)e/(K}c -- K ~ ) ' .

(17)

Giving to the p a r a m e t e r s of this g e n e r a l e i g h t - p a r a m e t e r equation the values d e t e r m i n e d o r r e l a t i n g t h e m to each other by ce rtain re lationships, we obtain many new equations still not p r o p o s e d by anyone. F r o m it also r e s u l t the e x p r e s s i o n s 7)

A 1K

(g ....

-

K~,) q~ with Kmax << Kfc,

r = A 2 K ~,'noqo with t(~ (( ](max<< Kfc, v =AaI--K'~,) ", with K~,~: >.'2Kth,

(18) (19) (20)

d e s c r i b i n g a portion of the full fatigue f r a c t u r e curve, Eq, (18) the f i r s t and second portions, Eq. (19), which a g r e e s with the P a r i s rule, the second, and Eq. (20), f r o m which with na = r 3 =1 follows the known F o r m a n e q u a tion [36], the second and third. Possibly, these equations, a f t e r s o m e simplifications of the f i r s t and last, could be used f o r d e t e r m i n i n g the c h a r a c t e r i s t i c s Kth (18), n and K* (18), and Kfc (20) using the e x p e r i m e n t a l points ef the c o r r e s p o n d i n g portions, as was suggested above. In o r d e r to d e t e r m i n e which of the p r o p o s e d equations is b e s t suited f o r actual fatigue f r a c t u r e c u r v e s , 26 groups of p a i r s of values of v and K m a x of v a r i o u s s t e e l s and also of aluminum, titanium, and m a g n e s i u m alloys w e r e d e s c r i b e d by nine equations. In all c a s e s the quantity of e x p e r i m e n t a l points in the group exceeded 70, they covered a range of change in c r a c k growth rate of about five o r d e r s of magnitude, and the c u r v e s had c l e a r l y e x p r e s s e d e x t r e m e portions. All of the groups w e r e obtained as a r e s u l t of t e s t s with a s t r e s s cycle s t a r t i n g a l m o s t f r o m z e r o in l a b o r a t o r y a i r or in vacuum. The r a t i o of the m a x i m u m and m i n i m u m c r a c k growth r a t e s with a given value of Kmax, which c h a r a c t e r i z e s the width of the s p r e a d band of the e x p e r i m e n t a l points, was no g r e a t e r than 3 in the p o o r e s t case. Equations (1), (5), (6), (8), (9), and (11) (Table 1) w e r e inv e s t i g a t e d and a l s o the p a r t i c u l a r c a s e s of Eq. 0 7 ) not yet d e s c r i b e d by anyone: (21) 'V: ~'o (Km~ax-- ](~)l'(Kfc --/(max) q, m

v = Vo ( i < ~

__

m

m

tn

I<,,~)/(Kj~ - K~.~).

(22) (23)

All of t h e s e equations contain f r o m four to eight p a r a m e t e r s subject to determination, whichis in a c cordance with the n u m b e r of c h a r a c t e r i s t i c s of the m a t e r i a l . The p a r a m e t e r s w e r e calculated based on the condition of the m i n i m u m of the s u m of the s q u a r e s of the l o g a r i t h m s of the r a t i o s of the e x p e r i m e n t a l Ve and calculated v c values of the c r a c k growth rate and t h e i r suitability was judged f r o m the standard deviation M

where M is the n u m b e r of p a i r s in the group, and m is the n u m b e r of p a r a m e t e r s in the equation. F o r each equation the a v e r a g e a r i t h m e t i c value S of the value of S c o r r e s p o n d i n g to individual groups w e r e calculated. On the b a s i s of an a n a l y s i s of the r e s u l t s of calculations* the following m a y be stated. The band between the lines c o r r e s p o n d i n g to the calculated line of the curve and p a s s i n g below and above it at a distance of 2S in all c a s e s a p p r o x i m a t e l y c o r r e s p o n d e d with the band of s p r e a d of the e x p e r i m e n t a l points. This indicated good a g r e e m e n t between the a c c u r a c y of the anal3~ical d e s c r i p t i o n of the fatigue f r a c t u r e curve and the s p r e a d c h a r a c t e r i s t i c of c r a c k growth rate. The a v e r a g e value S (Table 2) v a r i e d f r o m 0.09 to 0.108, depending upon the choice of equation, and it d e c r e a s e d with an i n c r e a s e in the n u m b e r of p a r a m e t e r s . In c o n c r e t e c a s e s the o r d e r of a r r a n g e m e n t of the equations p r e s e n t e d in Table 2 is substantially violated. Equations containing f e w e r p a r a m e t e r s m a y be b e t t e r (have a l o w e r value of S) than those with a l a r g e r n u m b e r of them. The m o s t stable position in the s e r i e s of f o u r - p a r a m e t e r equations is occupied by EcL (8). Models (9) and (23) a r e b e t t e r than it if the curve has a long c e n t e r portion (higher K2_3/Kl_2 ratios, and in our selection such c u r v e s w e r e in the m a j o r i t y ) and p o o r e r in the r e m a i n i n g c a s e s . * The a p p r o p r i a t e p r o g r a m s in F o r t r a n IV language f o r the E8 c o m p u t e r w e r e set up by E n g i n e e r L. S. M e l ' nichok, who aiso made all of the calculations and actively p a r t i c i p a t e d in t h e i r analysis.

378

The only four-parameter Eq. (I) taking into consideration asymmetl-y of the curve does not provide an adequate description of it. it also does not stand up to the competition of the so-called equation of the inverse hyperbolic tangent function widely used abroad. Consequently, if we are limited to a single equation for describing fatigue fracture curves as it we may recommend, as was done in the method instructions [16], Eq. (8), which has a comparatively simple structure and is convenient in use. However, if a more accur~ate approach is required, the mathematical model mast be selected differently- from several of the most suitable expressions depending upon the actual form of the fatigue fracture curve~ To develop the method for such a choice, further investigations are necessary covering a wider range of materials and equations and also taking into consideration the convenience of use of the equations, their predicting capacity outside the range in which their experimental points are located, etc. For a full solution of the above formulated reverse problem it is necessary to establish the relationship between the characteristics of cyclic crack resistance, On the one hand, arid the parameters of the mathematical model, on the other, and also to develop such an algorithm for determining the latter so that these characteristics remain constant regardless of choice of the model. LITERATURE

I. 2. 3. 4. 5. 6.

CITED

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