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200 Ilare Bordeasu et al.
In conformity with Fig. 4, after the crack initiation, the maximum danger is
represented by the zone III, characterized by great propagation velocities
leading to instable increases of the failure. In this zone, the velocity of crack
propagation is correlated with the variation of the crack intensity factor, through
the relation proposed by Forman [2]:
(1)
da
dN
n
C(
ΔK
)
( − R)
K − ΔK
= 1
C
= f (ΔK, R),
where: C and n are constant values depending on the used material;
ΔK = Kmax − Kmin
- represents the variation of the stress intensity factor; KC
- is the critical value of the stress intensity factor (breaking tenacity); R -
represents the asymmetry ratio of the stress cycle.
Expressed by Nr, the necessary number of cycles for the extension of the
crack, the lifetime can be obtained by solving the equation for dN.
Integrating both members it results:
(2) ∫ dN = N − N = N = ∫
N
N
f
d
f
d
r
a
a
cr
d
da
.
f ( ΔK,
R)
With the equation (2) it is possible to calculate the number of cycles Nr
necessary for the extension of the crack from the detectable length ad (for which
corresponds the number of cycles Nd) till the critical length acr (which
corresponds to the Nf number of cycles).
For the lifetime computation (the final number of cycles Nr) it was used the
specialized program AFGROW, developed by H a r t n e r at WRIGHT-
PATTERSON AIR FORCE BASE [3] in order to estimate the lifetime of some
components for fighter planes. It was taken into consideration a ring section,
having an elliptical crack as can be seen in Fig. 5. For such geometry, the
intensity factor was proposed by Raju and Newman [4] as being:
a
= t b , i e ,
Q
(3) K ( σ + H σ ) π F(
a c,
D , D , φ)
I
where σt - represents the axial stress applied to the shaft, σb - represents the
bending stress, H and F depends on the crack geometry (crack depths and
length), the shaft thickness and the frontal position of the crack, a is the depth
and c the half-length of the crack,