Theoretical model for FCGR near the threshold
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semicircular notch with a finite radius p. The development of the model requires the evaluation of the stress and strain distributions ahead of the crack tip. By the use of the linear elastic fracture mechanics (LEFM), the total strain range is found to be a quadratic function of the stress intensity range. The incremental crack growth occurs within the distance 4p from the crack tip, after a number of cycles given by the Manson-Coffin rule have elapsed. The onset of the continuum regime is triggered by a reversed plastic strain reaching the fatigue ductility e).
ANALYSIS 1. Monotonic Stress and Strain Analysis Let us consider the crack tip. As a result of the first loading cycle, the tip blunts to a semicircular notch of radius p. Figure 1 shows the normal stress distribution along the x-axis. It can be seen that three different regions describe the behavior of this stress component: Region I: The fully plastic region treated by the theory of plasticity." In plane strain, the normal stress is given by: o = Oy[1 + l n ( l + x / p ) ]
At the fullest extent of this zone, x = 3.8p (Ref. 1 l), and therefore o = 2.57Oy. This approach assumes a rigidplastic solid, for which 2.57 is the maximum triaxiality factor. Region II: The elastic-plastic region treated by LEFM. The equivalent elastic crack length is given by a + R m(Ref. 12) where R m -- (1/2~z) ( K / o ) 2. In plane strain, Irwin 12assumes at RM~=~68e~, that is Rm = (1/6~r) ( K / o y ) 2 Region III: The elastic region treated by LEFM. Westergaard 13has established: K ~ = 2X/f~ It has been theoretically established TMand experimentally confirmed 15that the normal strain distribution
ISSN 0360-2133/81/0311-0459500.75/0 9 1981 AMERICAN SOCIETY FOR METALS AND THE METALLURGICAL SOCIETY OF AIME
VOLUME 12A, MARCH 1981--459
F r o m Eq. [4], the elastic strain is given by Oo/Oy:
~a~v n o
I
K 2~[c~
with0 =0(r
Z f
]2p
I'
I
9
=
from
the
= x),
(1 + v)(1 - 2v) K E ~
(x > R~)
[81
At x = R,,, Eqs. [6] and [8] respectively give
crack hp
Fig. 1--Normal stress distribution ahead of the crack.
e =
4 V ~ ( 1 - v2)d
ey
pc
[91
c = (1 + v) (1 - 2v) "v~%
in the yielded zone was of the form: k = -
[1]
x
In order to avoid the singularity at x = 0, we assume that the d e f o r m a t i o n is held c o n s t a n t within a distance d from the tip of the notch. If in this region, the material initially of length 20 is elongated by 6, we should have according to T e t e l m a n ~6 8 ce= ~(x
[71
~'X
Rm
Dmtance
- 2v) + (1 + v)
• (sin 0 / 2 sin 3 0 / 2 ) ) ] (x > Rm)
Tr
E
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