ESTIMATION OF STRESS CONCETRATION FACTOR - FESB

-ESTIMATION OF FATIGUE STRESS CONCENTRATION
FACTOR IN THE REGION OF ELASTIC-PLASTIC STRAIN
Damir Jelaska, professor
Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, R. Boškovića b.b., 21000 Split, Croatia
Tel. +385 21 305874; Fax. +385 21 563877
E-mail addres: damir.jelaska @ fesb.hr
1.Introduction
In the absence of the experimental results, designers are forced to estimate the value of stress concentration
factor in the region of elastic- plastic strain. The most popular method is based on empirical investigations of
Hardrath and Ohman [1], Neuber [2] and Wetzel [3] which propose the relation between fatigue stress
concentration factor b k * and the strain concentration factor b e * . Intersection of corresponding curve with cyclic
stress-strain curve determines the magnitude of local stress and strain and consequently the real values of stress
and strain concentration factor.
It was perceived that values of
β k
*
obtained after [1] and [2] are greater than real ones, and those
obtained after [3] less then real ones. That is why this method is unreliable. Whereas it is also time- consuming,
it became necessary to make effort to obtain simple and correct explicit formula for fast estimation of the stress
concentration factor in the region of elastic- plastic strain.
2. Derive of stress concentration factor formula in the zone of elastic- plastic strain
On the basis of entire worldwide experimental investigations, including author's, the S-N curves for the
unnotched specimens and parts (same as notched specimens) might be desplayed as straight lines (Fig. 1). This
is very good approximation for most of steels and might be taken as valid for most of metals. The limit
N gr
between finite and infinite life is taken to correspond with the limit between elastic and elastic- plastic strain
region, and it is also taken to be same for the notched and unnotched specimen. For the materials which do not
have the endurance limit, it might be taken N gr
= 10 7 . Same holds good for the steels at elevated temperatures.
The limit
N q
between region of quasistatic fracture and region of fatigue fracture is also taken to be same for
both notched and unnotched specimen.

Fig.1- Stress concentration factor computing model
In the region of finite fatigue life (Nq ≤ N ≤ Ngr)
, the equations of both S-N curves are:
m
NR
q q
= N m
a
= Ngr
R−
m
σ
1
(1)
m'
m'
m
NR
q
'
q
= Nσ an
= Ngr
( R / −1 β ) '
k
. (2)
According to definition
from (3) and (4) follows:
or
β
σ
β
* a
k
= = R σ R
an
an
1 1
* m m
k
= βk( N/ Ngr)
' −
*
β β ( R / β σ )
k
=
k −1
k an
a
, (3)
1 −m'/
m
(4)
(5)
From (3) and (4), according to Fig.1, it is easy to derive the ratio:
1/
m
m' log( Ngr
/ Nq
)
=
1/
m
m log( N / N ) + log( β / β )
gr
q
k
q
(6)

3
Whereas the ratio N / N for the most materials is close to 10 , the exponent ratio might be
approximated as
gr
q
m'
3
=
m 3 + mlog( βk / βq)
(6a)
Stress concentration factor β q
at quasistatic fracture bondary remains the only unknown in expressions
(3) to (6a). On the basis of regression analyses of great number of experimental investigations, f.e. [4,5,6,7], the
stress concentration factor at quasistatic fracture limit for steels might be estimated as
β
q
= 1+ ( 0, 00038RM
− 0, 1)( β
k
−1) . (7)
So determined fatigue stress concentration factor
β k
*
, involved in cyclic stress-strain curve
ε
a
σ
a
σ
a
= + ⎛ E ⎝ ⎜ ⎞
k
⎟
⎠
pl
1
n
(8)
enables estimation of the strain concentration factor in the region of elastic- plastic strain:
* − / n
β = β + Ek 1 ( β σ )
ε
k
pl
k
an
1/ n −1
(9)
*
For nine steels having cyclic strenghtening characteristics k pl
and n after [8] , the values of β ε
are calculated
for various magnitudes of
σ an
*
β β
. It was observed that product is not in accordance with Neuber's
k
*
ε
*
β β
* = α
2 *
β β * = β
2
hyperbola low , neither with modificated Neuber's hyperbola low .
k
ε
t
For that reason, by help of "Excel" computer program, new relation between stress and strain concentration
factor in the region of elastic- plastic strain is obtained for the steels:
k
ε
k
β β = β
* * 2+ 0, 57( σan
/ R−1 −1/ βk
)
k ε k
(10)
This equation might be generalized for all materials:
β β = β
* * 2+ a( σan
/ R−1
−1/ βk
)
k ε k
, (10a)
where a is material constant, which could be determined by experiment.
From this equation it is clear that product of stress and strain concentration factors in the region of elastic-
plastic strain increases with stress amplitude σ an
and it is equal to (after [3]) only at limit. This new
relation represents a hyperbola located between original Neuber's hyperbola and modified one, and it determines
stress and strain concentration factor if the cyclic stress- strain curve is known (Fig.2).
β k
2
N gr

Fig.2- Determining of local stress and strain
For the supposed exponential relation between stress and number of cycles of the notched and unnotched
specimens in the region of quasistatic fracture, it was easy to derive the expression for the approximate
estimation of stress concentration factor:
*
β = β ( β / β )
q s q s
log4
N
log4
For materials that fail in brittle manner, β s
factor depends on α t
and notch sensitivity
N q
(11)
at static fracture
related to the level of the plastic deformation at static fracture, and there is inside limits 1≤β
≤β
. For fully
sensitive materials (such as titanium, beryllium and most of its alloys) is β
s
= β
q
= α
t
.
s
q
For ductile materials, including mild steels with
σ M
< 700N/
mm
2
, the plastic strain is so high at
static fracture (N=1/4) that stress concentration effect is relieved by yielding, i.e., it might be taken that β s
=1,
and consequently
*
β
q
= β
log4
N
log N
q q
4 . (11a)

N q
3 5
is an exsclusive characteristic of a material and vary from 10 to 10 for the metals, just like
particularly for the steels.
3. Concluding remarks
Stress concentration factor calculated in the way described above, is as close to its true value, as suggested
computing model is close to the real one. It also depends on basic data reliability, particularly on stress
concentration factor boundary values β
q, β
k and β s . The special attention has to be paid to β q , the
suggested calculation of which is good approximation for only the steels. For other materials the additional
investigations have to be done to obtain the applicable formula for determination of β q
. For that reason the
author proposes to apply the known procedure after Fig.2 using obtained formula (10), because this method
becomes now much more precise. In such a way the usage of β q
is unnecessary, and results are not worse then
using it.
NOMENCLATURE
a- material constant
E- modulus of elasticity, MPa
k pl
- coefficient of cyclic strengthening, MPa
m - Wöhler-curve exponent of the unnotched specimen
m'- Wöhler-curve exponent of the notched specimen
N - number of endurable cycles
N gr
- boundary number of cycles corresponding to endurance limit R −1
N q
- boundary number of cycles corresponding to quasistatical fracture limit
n - cyclic strengthening exponent
R M
- ultimate strength of a material, Mpa
R q
- unnotched specimen strength at quasistatic fracture boundary, MPa
R q
'
- notched specimen strength at quasistatic fracture boundary, MPa
R −1
- endurance limit of material at -1 stress cycle asymmetry number, MPa
α t
- theoretical stress concentration factor
β k
- fatigue stress concentration factor in the zone of elastic strains
β k * - fatigue stress concentration factor in the zone of elastic-plastic strains
β q
- stress concentration factor at quasistatic fracture limit
β q * - stress concentration factor at quasistatic fracture zone

β s
- stress concentration factor at static fracture
σ a
- local stress amplitude, MPa
σ an
- nominal stress amplitude , Mpa
REFERENCES
[1] Hardrath F.H.,Ohman L.:"A Study of Elastic and Plastic Stress Concentration Factors Due
to Notches and Fillets in Flat Plates" , NACA Rept. 1117 , 1951.
[2] Neuber H.:"Theory of Stress Concentration for Shear-Strained Prismatical Bodies with
Arbitrary Nonlinear Stress-Strain Low" ASME Trans.,N.York , 28(1961) , p.544-550.
[3] Wetzel R. M. :"Smooth specimen simulation of fatigue behavior of notches", J. Materials
(1968), p.646- 657.
[4] "Fatigue Investigations of Higher Strength Structural Steels in Notched and in Welded
Conditions", Komission der Europäischen Gemeinschaften, Luxemburg, EUR- Bericht Nr.
5337 (1975).
[5] Heywood, R. B. "Designing Against Fatigue", Chapman and Hall Ltd. , London 1962.
[6] Olivier R., Ritter W.,:"Wöhlerlinien katalog für Schweiβverbindungen aus Baustählen",
Fraunhofer Institut für Betriebfestigkeit, Berichr Nr. FB-151 (1979-1985)
[7] Osterman H., Grubišić, V., Einflus des Werkstoffes auf die ertragbare
Schwingbeanspruchung, Chapter in "Verhalten von Stahl bei Schwingender
Beanspruchung", Hrsq. Verein D. E., Düsseldorf 1979.
[8] Troshchenko V.T. et al. :"Cyclic Strains and Fatigue of Metals" (in Russian) Vol 1., p.77 ,
Naukova Dumka , Kiev , 1985.