1966-Manson-Coffin Fattigue. P.P. GILLIS
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Transcript of 1966-Manson-Coffin Fattigue. P.P. GILLIS
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MANSON-COFFIN FATIGUE*
P. P. GILLISt
The low-cycle large-amplitude fatigue law AaN!2 = constant, is derived from a dislocation model of crack growth. The crack is considered to increase its volume each half-cycle by absorption of dislo[!ations from a pseudo-torroidal region around its perimeter.
LA FATIGUE SELON LA RELATIOS DE MANSOK-COFFIN
La loi de fatigue B large amplitude et faible cycle AeN = con&ante, peut &re d&ivee B partir dun mod& de dislocations representant la croissance dune fissure.
Lauteur consid&rtt que la fissure oroit en volume B chaque demi-cycle par absorption de dislocations provenant dune region pseudo-torroidale entourant son p&im&re.
MANSON-COFFIN-~R~~DU~G
Das fiir niedrige Zyklen und grol3e Amplituden geltende Ermiidungsgesetz AEN/~ = konstant wird aus einem Versetzungsmodell fiir das Risswachstum hergeleitet. Es wird davon ausgegangen, dalj der Riss sain Volumen bei jedam Halbzyklus durch Absorption von Versetzungen eus einem pseudo-torroidalen Gebiet entlang seines Umfanges vergrliRert.
INTRODUCTION
In a recent paper (I) Grosskreutz was able to derive
the well-documented, empirical Manson-Coffin(2,3)
relation for low-cycle fatigue life. His analysis was
based on geometric assumptions regarding the cyclic
deformation of a two-dimensional elliptical crack
under plane strain.
The present analysis derives the Manson-Coffin
relation from considerations of dynamical dislocation
processes, without having to specify the crack shape.
Assuming a relatively flat crack the edge of which is
bounded by an arbitrary curve s, crack growth occurs
through absorption of dislocations into the crack from
the pseudo-torroidal region generated by a circle of radius z traversing the curve s.
such cycles since the start of the test by N. The
number of cycles to failure N, depends upon a crack nucleation and growth process in three stages: Stage
I consists of the formation of a suitable crack nucleus
by some unspecified process. Stage II consists of the
growth of this nucleus to some critical size as described
below. Stage III consists of ductile failure of the specimen during a tensile half-cycle due to the void
created by the critical crack.
The radius z is the maximum distance an edge
dislocation can move during a half-cycle, and it can be
shown to depend linearly on the plastic deformation
amplitude. This leads directly to a Manson-Coffin
type relation.
Assume that Stage I has provided a crack nucleus
t,hat is roughly disc-shaped and perpendicular to the
specimen axis as shown in Fig. 1. If the transverse
cross section of the crack is denoted by a it is oon- vcnient to define a characteristic, transverse dimension T, such that ~TP = EC.
CONSTANT STRAIN AMPLITUDE FATIGUE
Consider a cylindrical specimen of length L and
cross-sectional area A, of material characterized by an elastic modulus E and a yield stress oW. For con-
venience a transverse dimension R is defined such that .rrR2 = A although the cross section need not be circular.
In the region of the crack the st$resses and strains
will differ from the average values throllghout the bulk
of the specimen. This variation can be approximated
in the following manner. Consider that the major
effects of the crack extend an axial distance r in both
directions and treat the specimen as comprising two
segments : a solid cylinder of length I; - 2r; and a
Let the specimen be cycled at a constant deforma-
tion amplitude LAe. That is, the test begins by an
elongatiou LA.@, followed by alternating compres-
sions and elongations of LA&. Denote the number of
* Received March 18, 1966. This work supported in part by the National Science
Foundation. ? Department of Engineering Mechanics, University of
Kentucky, Lexington, Kentucky.
SCTB METALLURGICA, VOL. 14, DECEMBER 1966 1673
CRACK GROWTH DURING STAGE II
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3674 ACTA METALLURGICA, VOL. 14, 1966
cylinder of length 2r pierced by a hole of area cd. Let
subscripts 1 and 2 refer respectively to these segments and approximate the stress and strain as being constant
within each. Then the total deformation must be the
sum of the deformations in the two segments.
LF = (L - 24&l + 2YFs
Also, ~quil~briLlm must be satisfied.
(11
@,A = a,(A - (x) (21
Obviously, there will exist a transition region between
gl, c1 and g2, e2 and also there will be lateral variations
of stress and strain within the segment containing the
crack. However, for small cracks equations (1) and (2)
ought to provide a reasonable, approximate basis for
description of the macroscopic response, and it is
expected that during most of Stage II the growing crack
will indeed be small.
If oJE < AE ihe elastic deformations in the speci-
men earn be neglectsed. Then during a half-cycle
equation (1 f asserts that :
AE = (1 - Z+C)&,P -j- (3Y/L)&, (3)
Here the superscript p denotes plastic deformation.
According to Gillis and Gilman4) the plastic
deformation rate due to dislocation glide processes can
be written as:
dep/dt = ~?bpv (4
Here p is an orientation factor, b is the di5Iocation strength, p is the length of dislocation line per unit
volume, and 21 is an average dislocation velocity
specified by : 21 = w* exp {-223/o) (8
Here u* and D are material parameters.
After the first few strain cycles the dislocation
population will nor vary greatly during any given half-cycle. Treating p as a constant, during one
half-cycle, ~rp and .QP can be written as:
El p = ybp,v* f exp {-ZD/qj dt (6)
% P = phpiv j exp {-Bz)/cQ dt (3
Bath integrals are evaluated over a half-cycle.
To relate the integral involving b2 to As consider the
two limiting cases a = 0 and a = A. In the first case
d1 = oz so that 1/x = ZJ~ and equation (3) gives:
p5 j UJ & = A+$& (31
In the second case cur = 0 so that vu1 = 0 and:
~2.f v2dt = (~+$JW+~) (9)
Thus p2jz+ dt depends linearly on AC and can be
written in terms of some unspecified? fur&ion of (r/R)
which will be denoted a5 p.
In the immediate vicinity of the crack there will be
dislocations moving towards the crack during each
half-cycle. In fact, for an isotropic dislooation distri-
bution, just half of the mobile dislocations will move
towards the crack during one half of the strain cycle and the other half of the dislocations will move
towards the crack during the other half-cycle. Some
of these dislocations will pass out of the specimen by
reaching the boundary surface af the crack. Con-
sidering that the edge dislocations are equivalent to a
string of vacancies, ~1 those that are absorbed by the
crack increase its volume by an amount a,pproximately
equal to b21 where 1 is the dislacation line length. Very low stresses exist near the surface of the crack
except in the vicinity of its edge so that dislocations
are expected to feed inta the crack primarily from the vicinity of the crack front. If the crack front is
described by a curve s, at any point, along this curve
the element of volume from which dislocatlions may
empty into the crack is approximately 7~2~ d.s where z = f~s dt. (See Fig. 2.) Call S the total length of crack
front, or the crack perimeter. Then the volume
emptying dislocations into the crack is &S and the
total length of edge dislocation line reaching the crack
per half-cycle is pm%/2 where p is the edge disloca- tion density in the region of the crack front. The
earresponding volume change is ~z~~~2S~2. For a full
cycle the volume change is twice this.
dVldN = bzpm2S (111
Now both the crack volume and perimeter can be expressed as functions of the parameter r for any particular crack shape, For the convenience of
FIG. 2. An element of the volume from which disbca- tions can reach the crack during a half-cycle.
t From physical arguments a monotonic transition is expected between these two values. That is, the deformation per cycle in the crack region is expected to increase continu- ously as the crack grows. Another way of saying this is that more and more of the total deformation will occur in the region of hhe crack.
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GILLIS: MANSON-COFFIN FATIGUE 1675
non-dimensionalization they may be alternatively
expressed as functions of c = rlR. Equation (I 1) can
then be rewritten as :
(d V/d~)(d~ld~) = b2pm2X(
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1676 ACTA METALLURGICA, VOL. 14, 1966
3. For the case of constant plastic-strain amplitude 2. S. S. MANSON, SACA TN 2933 (1953). See also Fatigue-
dynamical dislocation theory predicts a frequency- An Inte~disci$inary Approach Proc. 10th Sagamore A. M. R. Conf., edited by J. J. Burke, N. L. Reed and
independent specimen response. V. Weiss, p. 133. Syracuse University Press, Syracuse, N.Y. (1964) and Exp. Mech. 5, 193 (1965).
ACKNOWLEDGMENT 3. L. F. COFFIN, JR., Tram Am. Sot. Mech. Engrs 76, 923
The author wishes to thank Dr. J. C. Grosskreutz (1954). See also AppZ. Mater. Res. 1, 129 (1962), and
for stimulating an interest in this problem. Financial Fatigzse-An Interdisciplinary Approach, Proceedings of the 10th Sagamore Army Materials Research Conference,
support was provided in part by the National Science Edited by J. J. Burke, N. L. Reed and V. Weiss, p. 173. Syracuse University Press, Syracuse, N.Y. (1964).
Foundation. 4. P. P. GILL~S and J. J. GILRZAN, J. appl. Phys. 36, 3370 REFERENCES (1965).
I. J. C. GROSSKREITTZ, USAFML Report 10883 (1964). 5. W. BOAS, Dislocations and Mechanical Properties of
Crystals, p. 333. Wiley, New York (1957).