The influence of crack blunting on the thermodynamics of fatigue crack growth

Scripta METALLURGICA et M A T E R I A L I A

Vol. 25, pp. 1 9 7 5 - 1 9 7 9 , 1991 P r i n t e d in the U . S . A .

Pergamon

Press

~',]c

TIIE INFLUENCE OF CRACK BLUNTING ON THE THERMODYNAMICS OF FATIGUE CRACK GROWTH David J. Nicholls United States Air Force Academy ( R e c e i v e d J a n u a r y 29, 1991) ( R e v i s e d J u n e 11, 1991) Introduction One of the proposed mechanisms for fatigue is Laird and Smith's plastic blunting model (1). Plastic deformation at the crack tip during the tensile portion of the load cycle is thought to result both in crack extension and crack blunting. During the unloading portion of the load cycle, the crack tip is resharpened. The net result is a small increment of crack growth. Irreversibility of the process is introduced by assuming activation of different slip systems in the loading and unloading portions of the cycle. A number of models (2,3,4) for possible irreversible slip processes have been advanced. Attempts to use this mechanism to predict fatigue crack growth rates, have generally taken the increalent of crack growth to be proportional to the crack tip opening displacement (CTOB) (5). This results in a Paris law exponent of two. Unfortunately, measured Paris law increments actually range from two to four and may go as high as six (6). The purpose of this paper is to use an energy approach similar to Griffith~s (7) to describe the role of crack tip blunting on the thermodynamics of fatigue crack growth. More specifically, it will be shown that there is a release of elastic strain energy due to the blunting of the crack tip and that inclusion of this energy term in an energy balance results in a Paris law expression for fatigue crack growth with a variable Paris law exponent. Derivation

o f mQdel

Three simplifying assumptions will be made at the outset of this derivation. Firstly, it will be assumed that fatigue crack growth only occurs when the applied load is at its maximum. This assumption is made because it allows us to assume constant load conditions. It has been shown (8) that for small increments of crack growth, as is the case in fatigue, the energy terms are the same whether load or displacement is constant. Thus, similarly to Griffith, we can neglect any work done external to the specimen. Secondly, it will be assumed that the plastic zone is formed prior to crack propagation and therefore the energy associated with the formation of the plastic zone can be neglected. Bowever, the energy required for the plastic deformation directly associated with an increment of crack growth will not be neglected. Finally, it will be assumed that the minimum load is zero. Thus, AK will equal Kmax. If there is no external work done and the work required to form the plastic zone is neglected, then the remaining processes requiring energy are the creation of new surface area and the additional plasticity directly resulting from an increment of crack growth. Griffith~s expression (7) for the energy required by the new surfaces for a center crack of length 2a Us = 4 7 a This was later extended by Orowan (9) to include the the work required for materials where there was significant plasticity, flrowan's expression was U s =

Therefore,

4(Ts+7p)a

the energy required

for an increment

of crack growth

AUo = 4(7,+Tp)aa

is given

by (1)

Now, the energy released by an increment of crack growth must be calculated. With the given assumptions, the only source for energy is the release of strain energy due to the change in crack shape as the crack grows. Griffith derived Equation 2 for the strain energy

1975 0036-9748/91 $3.00

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CRACK BLUNTING

a s s o c i a t e d with an e l l i p t i c a l

IN FATIGUE

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h o l e r e m o t e l y loaded in t e n s i o n ( 7 ) .

- DTa2c 2 cosh(2%) U* = 2E

(2)

where Ue i s t h e s t r a i n energy r e l e a s e d , D i s a c o n s t a n t d e t e r m i n e d by t h e s t a t e o f s t r e s s , a i s the a p p l i e d s t r e s s , c i s t h e s e m i - f o c a l l e n g t h o f t h e e l l i p s e , ao i s an e l l i p t i c a l coordinate and E i s Young's modulus. G r i f f i t h c o n t i n u e d by e q u a t i n g c o s h ( 2 a o ) t o one s i n c e a o i s v e r y small f o r tile v e r y narrow e l l i p s e s t h a t model c r a c k s . In t h i s work, however, t h e t r i g o n o m e t r i c i d e n t i t y c o s h ( 2 % ) = 1 + 2sinh2% i s s u b s t i t u t e d i n t o E q u a t i o n 2 y i e l d i n g tle = -~D~2c2 -

D~a2c2 Esinh2a°

(3)

I f b i s t h e minor s e m i - a x i s o f t h e e l l i p s e , Dxa2c 2 u~ = - ~ 2 ~ - -

sinha o = b/c yielding

D~2b 2 --g--

(4)

S i n c e b=aq~F ( r b e i n g the c r a c k t i p r a d i u s ) and, f o r a narrow e l l i p s e , t h e s e m i - f o c a l d i s t a n c e (c) i s a p p r o x i m a t e l y equal t o t h e s e m i - m a j o r - a x i s ( a ) , E q u a t i o n 4 can be w r i t t e n a s DT~2a2 ue = - - T g The change Equation 6.

D~a2ar

---g--

(5)

in s t r a i n energy due t o an i n c r e m e n t o f c r a c k g r o w t h i s , t h e r e f o r e , Note t h e a s s u m p t i o n t h a t t h e c r a c k t i p r a d i u s i s a f u n c t i o n o f An. AUe = -

Dxa2( (a+Aa)2-a 2 ) 2E -

D~a~[ (a+Aa) r (Aa)-ar (0) ] E

In f a t i g u e , Aa<
the

DT~2arE( A a )

crack

tip

given

by

zero

and

(6) radius

is

initially (7)

The total change in energy due to an increment of crack growth is the sum of Equations 1 and 7. AUt = 4(?'+7p )Aa -

D1r°'2a~m' --]~ -

D~a2ar(Aa) E

(8)

To find the maxima of this function we take the derivative with respect to Aa and equate it to zero. dAUt

D~.o.2a :

0 = 4(%+~p)

-

--E--

Dlro'2ar~(An) -

E

(9)

Note that if 7p<<% and the contribution of the change in crack tip radius to the change in energy is neglected then Equation 9 can be rearranged to produce Griffith~s classical expression for the maximum stress that can be applied before the brittle fracture of a cracked plate. As D o N ~

= K, Equation 9 can be rewritten as dAUt

= 0 : 4(%+7p)

-

K2 ~r -

K2r'(Aa) --g----

(i0)

Assuming t h a t t h e r a t e o f e n e r g y r e q u i r e d t o p r o p a g a t e a f a t i g u e c r a c k i s t h e same a s t h e r a t e o f e n e r g y r e q u i r e d t o p r o p a g a t e a c r a c k in s t a t i c f r a c t u r e , 4(Tm+Tp) can be e q u a t e d t o Gc. Noting t h a t Gc=Kc2/E, E q u a t i o n 10 can be s i m p l i f i e d t o y i e l d 0 = Kc2 -

K2 -

K2r'(Aa)

(11)

R e a r r a n g i n g E q u a t i o n 11 y i e l d s an e x p r e s s i o n which s t a t e s t h a t t h e l e n g t h o f a s t a b l e i n c r e m e n t o f c r a c k growth i s d e t e r m i n e d by t h e r a t e o f c r a c k b l u n t i n g v e r s u s c r a c k e x t e n s i o n r e a c h i n g a critical value. T h i s c r i t e r i o n f o r s t a b l e c r a c k growth i s g i v e n by E q u a t i o n 12. r'(Aa)

= Kc-K2

(12)

If the crack tip radius, r, is assumed to be related to A a by a polynomial function,

i.e.

Vol.

r(Aa)

25, No.

8

CRACK

= c A a p, t h e n r ' ( A a ) rc&a p-I =

Arranging Equation

= pcAa p-1.

BLUNTING

IN F A T I G U E

Substituting

this

1977

expression

into

Kc2-K2

13 t o s o l v e

Equation

12 we g e t

(13) for

h a we g e t

1

(14) Finally, noting that Aa is equal to da/dN minimum load was zero so that Kmax=AK we get

d~a = dN

l-v

pc

and

remembering

AK

the

initial

assumption

that

the

(I,5)

Discussion The first point to make is that Equation 14 is of the same form a s Paris ~ empirical expression for crack growth i.e. da/dN=AAK m. Making this comparison we can see that theoretical expressions for Paris ~ empirical terms are 1

(16) and

m = T u 2~

(17)

If AK2<
A = p[pc] l-p

(~s)

LKo'J As AK 2 approaches K c 2 , however, A increases and, therefore da/dN would increase more rapidly than predicted by the Paris law as is commonly observed at higher levels of AI(. Also, m is now predicted to be variable depending upon the relation between the increment of crack growth and the crack tip radius. For example, if p equalled 0.5, then according to Equation 17, the Paris law exponent m would equal four. Judging from Equation 16, p would have to be equal to zero to get a Paris law exponent of two. However, the previous derivation is invalid for a p equal to zero. Instead we must assume that the relationship between r and Aa is give by r = c l n ~ . This leads to a Paris law expression with m equal to two and c A = Kc2_~

(19)

Therefore, this model provides a theoretical basis for the Paris law constants. Work is currently underway to verify the proposed relationships between crack blunting behavior and the empirical Paris law constants. As this approach is so similar to Griffith~s, it is worthwhile to graphically compare the two. In Figure 1 the classical Griffith approach is illustrated showing the energy requirement given by 4a7 and the energy released given by D~a2a2/2E. When these two are summed an unstable crack size is predicted as shown. Figure 2 is analagous but is expressed in terms of the crack increment A a and includes plasticity requirements (Tp) and the energy released by crack blunting (Dfa2ar(Aa)/E. When the energy terms are summed, an energy minimum is produced and, therefore, a stable increment of crack growth is predicted. An unstable increment of crack growth is still predicted for large changes in Aa in agreement with Griffith. At large increments of crack growth the energy released by crack blunting would be relatively

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CRACK BLUNTING IN FATIGUE

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i n s i g n i f i c a n t and so this model i s c o n s i s t e n t with G r i f f i t h ~ s e x p r e s s i o n . F i n a l l y , i t would be a p p r o p r i a t e to comment as to the i m p l i c a t i o n s of t h i s approach to the p r e d i c t i o n of environmental e f f e c t s . As has been pointed out elsewhere (10), chemical environment should change the s u r f a c e energy r e q u i r e d f o r crack growth but not the energy required f o r p l a s t i c i t y . This poses a d i f f i c u l t y f o r energy based f a t i g u e models in general because the p l a s t i c i t y component of the energy r e q u i r e d f o r crack growth (7p) is so much g r e a t e r than the s u r f a c e energy component (7s). T h e r e f o r e , most energy based models would p r e d i c t l i t t l e e f f e c t of a chemical environment on f a t i g u e crack growth. This p r e d i c t i o n is c o n t r a r y to experimental evidence where, f o r example, f a t i g u e cracks have been found to grow an o r d er of magnitude slower in vacuum than in moist a i r . In the model proposed in t h i s paper, however, the environment could s i g n i f i c a n t l y impact the d r i v i n g f o r c e i . e the s t r a i n energy r e l e a s e d . The key c r i t e r i o n of t h i s model (Equation 12) was t h a t f a t i g u e crack growth would continue until r'(Aa) = Kc-K2 K2 In words this equation says that when ratio of the rate at which the crack blunts and the rate at which it extends drops below a critical value, it will stop growing. The relative rates of crack blunting versus crack extension would clearly be a function of the material properties which would be changed locally by the action of the chemical environment. Therefore, this model allows significant environmental effects.

Conclusions 1. Inclusion of an energy term due to crack prediction of a stable increment of crack growth. 2.

blunting

in an

energy

balance

leads

to the

A Paris law type expression is predicted for fatigue crack growth.

3. The model results in expressions for the Paris law empirical exponents based on an experimentally measurable relation between the crack tip radius and the increment of crack growth. 4. The model has the advantage that it can accommodate the observed variations in the Paris law exponent. The crack blunting behavior is shown to control the value of the exponent. 5.

The model is consistent with significant environmental effects on fatigue crack growth. AcknowledEeme~t

I would like to acknowledge Seller Laboratory.

the financial

support

of the United States

Air Force Frank J

References ( i ) Laird, C. and Smith, G.C., P h i l o s o p h i c a l Magazine, 8, 1945 (1963) (2) McEvily, A.J. and Boettner, R.C., Acta N e t a l l u r g i c a , 11, 725 (1963) (3) Pelloux, R.H.N., Trans. of ASM, 62, 281 (1969) (4) Neumann, p., Acta M e t a l l u r g i c a , 17, 1219 (1969) (5) McEvily, A.J. ASTM STP 811, J. Lankford, D.L. Davidson, W.L. Morris and R.P. ASTM, 283 (1983)

Wei, Eds,

(6) Lindley, T.C., Richards, C.E. and R i t c h i e , R.O., M e t a l l u r g i a and Metal Forming, 268 (1976) (7) G r i f f i t h , A.A., Trans. Royal S o c i e t y of London, Vol. 221, 1920, p. 163 (8) P a r i s , P.C. and Sih, G.C.M. ASTM STP 381, 30, (1965) (9) Orowan, E . , Reports on Progress in Physics, 12, 185 (1948) (10) Weertman, J . , Acta N e t a l l u r g i c a , 26, 1731 (1978)

Vol.

25, No,

8

CRACK BLUNTING

IN FATIGUE

1979

f

.J

i

/. j 1 j. J / ./

/

/

f

J

."

I

UNSTABLECRACK GROWTH

~,

Au

\ \ \ \

\ x

]~ '--

FIG. i .

\

SURFACEENERGY ENERGYDUETOCRACKEXTENSION SUMMATION

I

Graphical illustration o f the Griffith criterion

f.l j" j"

STABLECRACK . , " / ~ GROWTH ,NCR~ENT f' /,

I

. / •/

w, UNSTABLE CRACK GROWTH

Au

\ \

\ \ \ \ \

l~

FIG. 2

SURFACEENERGY ENERGYDUETOCRACKEXTENSION ENERGYDUETOCRACKBLUNTING SUMMATION

Graphical illustration of energy balance including energy term due to crack blunting