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

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.

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(

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).