Gdfsg
Determine of residual stresses around cold-worked hole based on
measured residual strains
Kh. Farhangdoost 1*, Ehsan Pooladi B 2
1 Assoc. Prof., Dept. of Mechanical Engineering, Ferdowsi
University of Mashhad, Mashhad, Iran
(Email: farhang @um.ac.ir)
2 PhD student, Dept. of Mechanical Engineering, Ferdowsi
University of Mashhad, Mashhad, Iran
(Email: ehsanpb @ yahoo.com)
*Corresponding Author
0 (0000) 0 – 0
ISBN 0-9780479
AES-ATEMA’ 2011International Conference on Advances and Trends
in Engineering Materials
and their Applications
(Montreal, Canada: August 01 – 05, 2011)
Kh. Farhangdoost, Ehsan Pooladi B.
Kh. Farhangdoost, Ehsan Pooladi B.
© AES-Advanced Engineering Solutions (Ottawa, Canada)
All rights are reserved
4
© AES-Advanced Engineering Solutions (Ottawa, Canada)
All rights are reserved
3
© AES-Advanced Engineering Solutions (Ottawa, Canada)
All rights are reserved
Abstract
Compressive residual stress due to cold working decreases
tendency of mechanical parts to initiate and growing fatigue crack
and increases fatigue life. So determining residual stress is one
of the important subjects in the last decades. There are two
categories for measuring residual stress as destructive and
nondestructive tests. In this paper distribution of residual stress
around a hole which cold worked by expansion due to interface of a
shaft into the hole and then removing it, is determined by
measuring residual strains. There is a good agreement between
residual strain measurements and residual stress in which is a good
characteristic feature of residual stress. Fourier series for
relating measured strains to residual strains in each position,
Hook's law in elastic region and power law in plastic region have
been used for derive the residual stresses. This analytical
technique is nondestructive and simple compared to other
methods.
Keywords
Residual stress, Cold working, cold-worked hole, residual
strain, total strain theory, Henkey total strain theory,
thick-walled cylinder, plasticity
1Introduction
Residual stresses can be introduced into parts during
manufacturing, e.g., cold-reducing operations, autofrettage
process,..., or can produced by the next process after
manufacturing for being them as beneficial effect. Some
complementary process after manufacturing produce compressive
residual stress which is useful for increasing fatigue life, e.g.,
shot penning, cold working,....
The fatigue life improvement of cold expanded fasten holes is
attributed to the presence of compressive residual stress induced
by cold expansion. The fatigue fracture of fasten holes account for
50-90% of structure fracture of aircrafts. Over the last 40 years,
because of its simple realization and remarkable enhancement of the
fatigue life of holes (usually 3-5 times than that of holes without
cold expansion), the cold expansion process has been widely used to
improve the fatigue life of components with fasten holes [1].
The cold-expansion, which is developed by the "Fatigue
Technology Inc. (FTI, 1994), is obtained by using increased
pressure to plasticize an annular zone around the hole. The
pressure on the surrounding material is realized by interference
generated between the drilled plate and the pressuring element,
i.e. the mandrel. When the mandrel is removed and the superficial
pressure on the hole is erased, a residual stress field is created
due to action of the elastic deformed material on that under
plastic condition [2].
In the past, analytical models, experimental techniques and
numerical simulations have been developed to predict the residual
stress field induced by a cold-expansion process of a hole.
Analytical studies have determined the closed-form solution of the
residual stress for considering the material's yield limit on
unloading step (Guo, 1993; Nadai, 1943; Hsu and Forman, 1975; Rich
and Impellizzeri, 1977) [2]. H. Jahed et al. [3] presented
elastic-plastic boundaries and residual stress fields induced by
cold expansion of fastener holes using variable material
properties. Generally there are two ways to determine residual
stress field as destructive, e.g. hole drilling method, Sach's
boring technique and non-destructive test as X-ray and neutron
diffraction, ultrasonic, Barekhausen parazit and.... These methods
have been presented by many previous investigators. A comparison
between the mentioned techniques have been considered and concluded
by P.J. Withers et al. [4]. An analytical solution using strain
gradient plasticity theory is presented for the borehole problem of
an elasto-plastic plane strain body containing a traction-free
circular hole and subjected to uniform far field stress, by X-L Gao
[5,6].
In this paper, an analytical solution based on classic
plasticity theory using Henkey total strain theory for plastic
region is presented. The main objective of this study is to solve
analytical equations in order to derive residual stress field based
on measured strains. The material behaviour considered in this
method is linear work-hardening with power law relation in plastic
region and also Baushinger effect has been neglected. The condition
of problem is plane strain. Of course according to A.Stacy [7],
neglecting of the Baushinger effect causes the residual compression
at the bore to be overestimate. But in this paper we have not
reverse loading and there is only elastic unloading during removing
the mandrel, so Bushinger effect can be neglected. According to
D.J. Smith et al.[8,9], in this paper ,the strains are represented
by Fourier series. In fact Fourier series have been used to relate
strain in each position to measured strains. Also because of
existing symmetry, all of shear stress components are zero. The
conditions of entrance mandrel into the hole such as velocity,
friction coefficients,... affect the rate of plasticity around the
hole, but latest state which mandrel fully entered has been
considered and also for removing the mandrel out. At final state of
enter mandrel, a radial pressure, named shrunk pressure, applied to
internal surface of hole. So for simplicity, this problem has been
simulated with a thick-walled cylinder with an internal pressure.
This pressure dependent on interface value which is the difference
between mandrel and hole diameter. If this value is small enough,
the parts will be in elastic region and the shrunk pressure is
calculated by Lame's equation. But if it's larger than special
value, some part of hub and shaft will be in plastic zone and
Lame's equation won't be valid, as seen in this paper.
2Theory
As stated before, the effect of the velocity of entrance (and
removing) the mandrel, contact and Baushinger effect have been
neglected. So with omitting the contact effects, formulation
concentrated on final state of entrance, i.e., creation a radial
pressure from the mandrel to internal surface of hole and also for
final state of removing the mandrel or fully unloading. So for
simplicity, let us simulate a hole at the end of loading
(expanding) by a thick-walled cylinder with internal pressure.
Similarly the mandrel can be simulated by the mentioned cylinder
with external pressure only. We assume that the unloading process
is elastic, and plane strain condition is dominant. Also for
plastic region, power law relation and total strain theory (Henkey
strain theory) are used, as seen later. Analytical solution for
plane stress problem has been presented by other investigators
before, e.g. Nadai, of course the previous researches need some
modifications which authors will do it soon.
At first let take a thick-walled cylinder with internal pressure
only, into account for loading process as seen in Fig.1.
(ro)
(rpo)
(Ps) (rso)
Figure 1. Thick-walled cylinder with internal pressure.
Where, rso is inner radius of the hole, Ps internal (shrunk)
pressure, Ppo the radial pressure created at plastic radius
(elasto-plastic limit). When the interface is more than a special
value, some material around the hole, up to plastic radius (rpo),
pass the elastic limit and will be in the plastic region meanwhile
the outer ring, with radius bigger than the plastic radius, is in
elastic region yet. Removing the mandrel leads to create tendency
for spring back the elastic zone to previous (initial) position.
This tendency is precluded by the inner plastic zone. So a
compressive force is applied from material in outer ring to that of
inner part. This compression enhances the fatigue life and fatigue
cracks can't initiate or grow in compressed region which is
immediately around the hole. Now let solve thick-walled cylinder
with an internal pressure in both plastic and elastic states.
-Equilibrium equation in cylinder coordinates:
-Strain-displacement relations:
-Compatibility equation:
-Power law relation for plastic region:
Where σe and εe are Von-Mises effective stress and strain, also
σy , E, m are yield stress, elastic modulus and strain hardening
power respectively.
-Von-Mises effective stress:
-(Henkey) Total strain theory for plastic region:
Or
Hook's law for elastic region:
Where, υ is Poision ratio.
2.1Plastic solution (rso≤ r ≤rpo)
With substitution equation (5) into (6), constitutive equation
is derived as:
Using equation (5) in ( 8) leads to:
Satisfying compatibility equation causes:
Where, D is effective strain at inner surface of the hole.
Boundary conditions of the above problem are as follows:
Substitution (14) into (10):
Now put (15) & (10) into (4):
Using latest relation and equation (5), solve equation (1); with
considering (16) , (5) and integrating over the radius, a statement
for radial stress in plastic zone is derived which satisfying
boundary conditions (11) , (12) leads to:
Tangential and axial plastic stresses are derived as:
2.2 Elastic solution (rpo≤ r ≤ro)
There are many papers which introduced elastic solution of
cold-worked holes. So we use only the results as follows:
For calculating radial pressure at elasto-plastic limit,
according to validation of both elastic and plastic relations at
exactly elastic boundary (rpo), equations (20-1) should be
substituted into (5) and finally satisfying σe(rpo)=σy leads
to:
2.3 Unloading (rso≤ r ≤ro)
During unloading which assumed to be elastic, some stress
created around the hole. The below relations are derived for
elastic unloading; similar to elastic loading:
Radial and tangential strain during unloading are as
follows:
2.3Residual stress field
Residual stress field is obtained by subtracting
the unloading stresses from loading parts. So
in plastic region (rso≤ r ≤ rpo), they are as:
And in elastic region (rpo≤ r ≤ ro), we have:
It should be noted that there are three unknown parameters, rpo,
Ppo, Ps which should be determined.
2.4Strain expansion with Fourier series
If some strain gages are erected on some special points around
the hole, the measured strains recorded by the mentioned gages, can
be related to strain profile using Fourier series. In fact
according to [8, 9], strains (similarly to each function) can be
expanded by the Fourier series as follows:
Which (r, θ) is the position of point in cylindrical coordinates
and n is the amount of gauges; each at angle θ. With neglecting of
high order coefficients and due to symmetry we have:
3New analytical procedure
When entire the mandrel moves into the hole, the strain gauges
record loading strains and when entire the mandrel removes out the
measured value introduces residual strain. Strain due to unloading
can derived as subtracting loading strain and residual strain,
as:
For determining residual stress field here, 3 pieces strain
gages used at angles 0º, 120º, 270º. For simplicity, it's
recommended that the gages erected in plastic zone. A criterion to
find out a point is in plastic zone or not, according to equations
(8), (9), is equality and opposite sign of radial and tangential
strains. Because of symmetry, the strains recorded by gages at
angle 90º and 270º should be the same. So at first a gage used at
angle 90º and a known radius in radial direction and another at
angle 270º and at same radius. After loading, if the values showed
by the used gages are equal and in opposite sign, then that radius
would be in plastic zone, otherwise is in elastic zone.
Now three gages are erected at the mentioned angles and in
plastic zone tangentially. If the measured values by each gage are
named εθθI, εθθII, εθθIII respectively,then Fourier coefficients of
strain series will be derived as:
So Fourier series for relating strain at each point to measured
strains is rewritten as follows:
After fully loading, the tangential strain of a desired point in
the same radius of strain gages position is calculated by using
equation (29).
Because the point is in plastic zone, the effective strain is
derived through second equation of (9). One of unknown parameters
is obtained by equations (10) & (15), i.e. plastic radius:
Where, r is the point radius. And the radial pressure at
elasto-plastic limit (Ppo) is calculated by equation (21). Also by
measuring residual strain and using equation (27), unloading strain
will be derived. By second equation of (23) the shrunk pressure can
be calculated.
Finally with knowing plastic radius (rpo), shrunk pressure (Ps)
and elasto-plastic pressure (Ppo) and using equations (24) , (25)
residual stress field is determined.
4Conclusions
As stated before there are two ways for determining residual
stress field as destructive and non-destructive tests which have
some advantages or disadvantages. Approximately there is not an
analytical background in form of existed paper in previous
researches.
In this paper an analytical formulation which can verifies
finite element models, have been presented. Simplicity, take
reality material behaviour into consideration and accuracy compared
with numerical solution are some of benefits of these
relations.
Also the suggested technique for measuring strains is
non-destructive, commodious, and fast.
Determining residual stresses through measuring residual strains
is a new method which can performed quickly and online with doing
test, i.e., there is no need to waste time for measuring strains
and no complex relations for estimating strain by e.g., micro
structural changes such as X-ray diffraction and ....Of course in
this method just surface strains are measured.
Take stochastic properties of residual stresses and affected
parameters into consideration is the next authors' research.
References
[1]Liu Yongshou,Shao Xiaojun,Liu Jun,Yue Zhufeng (2009), Finite
element method and experimental investigation on the residual
stress fields and fatigue performance of cold expansion hole,
Materials and Design, Article In Press.
[2]V.Nigrelli,S.Pasta (2008),Finite-element simulation of
residual stress induced by split-sleeve cold-expansion process of
holes, Journal of Material Processing Technology, 205, 290-296
[3]H.Jahed, S.B.Lambert, RN.Dubey (2000), Variable material
property method in the analysis of cold-worked fastener holes,
Journal of Strain Analysis,35(2),137-142.
[4]P.J.Withers, M.Turski, L.Edwards, P.J.Bouchard, D.J.Buttle
(2008), Recent advances in residual stress measurement,
International Journal of Pressure Vessels and Piping,
85,118-127.
[5] X.-L.Gao (2002), Analytical solution of a borehole problem
using strain gradient plasticity, Journal of Engineering Materials
and Technology, 124,365-370.
[6] X.-L.Gao (2003), Elasto-plastic analysis of an internally
pressurized thick-walled cylinder using a strain gradient
plasticity theory, International Journal of Solids and Structures,
40, 6445-6455.
[7] A.Stacy, G.A.Webster (1988), Determination of residual
stress distribution in autofrettaged tubing, International Journal
of Pressure Vessels and Piping, 31, 205-220.
[8] A.A.Garcia-Granada, D.J.Smith, M.J.Pavier (2000), A new
procedure based on Sach's boring for measuring non-axisymmetric
residual stresses, International Journal of Mechanical Science, 42,
1027-1047.
[9] A.A.Garcia-Granada,V.D.Lacarac, D.J.Smith, M.J.Pavier
(2001), A new procedure based on Sach's boring for measuring
non-axisymmetric residual stresses: experimental application,
International Journal of Mechanical Science, 43, 2753-2768.
[10] M.Su, A.Amrouche, G.Mesmacque, N.Beneseddiq (2008),
Numerical study of double cold expansion of the hole at the crack
tip and the influence of the residual stresses field, Computational
Materials Science, 41, 350-355.
)
28
(
III
ε
2
1
+
II
ε
2
-
I
ε
2
1
-
=
A
4
ε
)
29
(
θ
4
cos
)
III
ε
2
1
+
II
ε
2
-
I
ε
2
1
-
(
+
θ
2
cos
)
II
ε
2
1
-
I
ε
2
1
(
+
)
II
ε
2
+
I
ε
(
=
)
θ
,
r
(
θθ
ε
2
so
r
2
e
D
r
e
=
y
so
DEr
po
s
2
r
=
(1)
r
rr
σ
-
θθ
σ
=
dr
rr
d
σ
(2)
dr
du
=
rr
ε
,
r
u
=
θθ
ε
(3)
θθ
ε
-
rr
ε
=
dr
θθ
ε
d
r
(4)
m
e
ε
)
m
E
m
1
y
σ
(
=
m
e
ε
k
=
e
σ
(5)
)
rr
σ
-
θθ
σ
(
2
3
=
e
σ
(6)
e
σ
e
ε
2
3
=
λ
where
)
m
σ
-
ij
σ
(
λ
=
'
ij
λσ
=
p
ij
ε
1)
-
(6
))
zz
σ
+
θθ
σ
(
2
1
-
rr
σ
(
λ
3
2
=
p
rr
ε
2)
-
(6
))
zz
σ
+
rr
σ
(
2
1
-
θθ
σ
(
λ
3
2
=
p
θθ
ε
→
yields
0
=
))
θθ
σ
+
rr
σ
(
2
1
-
zz
σ
(
λ
3
2
=
p
zz
ε
3)
-
(6
)
θθ
σ
+
rr
σ
(
2
1
=
zz
σ
)
7
(
)
θθ
σ
+
rr
σ
(
υ
=
zz
σ
]
rr
υε
+
θθ
ε
)
υ
1
[(
2
υ
2
υ
1
E
=
θθ
σ
]
θθ
υε
+
rr
ε
)
υ
1
[(
2
υ
2
υ
1
E
=
rr
σ
(8)
θθ
ε
)
θθ
σ
rr
(
σ
e
σ
e
ε
4
3
rr
ε
-
=
-
=
(9)
e
ε
2
3
=
θθ
ε
,
e
ε
2
3
=
rr
ε
(10)
2
r
2
so
r
D
=
e
ε
)
11
(
s
P
=
so
r
=
r
rr
σ
)
12
(
po
P
=
po
r
=
r
rr
σ
)
13
(
D
=
so
r
=
r
e
ε
)
14
(
E
y
σ
=
po
r
=
r
e
ε
)
15
(
2
so
r
E
2
po
r
y
σ
=
D
)
16
(
m
2
)
r
po
r
(
y
σ
=
e
σ
)
17
(
s
P
-
]
m
2
r
1
-
m
2
so
r
1
[
3
m
m
2
po
r
y
σ
=
p
rr
σ
→
to
leads
rr
σ
+
dr
rr
σ
d
=
θθ
σ
)
18
(
s
P
-
]
m
2
r
1
m
2
-
m
2
so
r
1
[
3
m
m
2
po
r
y
σ
=
p
θθ
σ
)
19
(
s
P
-
]
m
2
r
1
m
-
m
2
so
r
1
[
3
m
m
2
po
r
y
σ
=
p
zz
σ
1)
-
(20
]
2
r
2
o
r
-
1
[
2
po
r
-
2
o
r
2
po
r
po
P
=
e
rr
σ
2)
-
(20
]
2
r
2
o
r
+
1
[
2
po
r
-
2
o
r
2
po
r
po
P
=
e
θθ
σ
3)
-
(20
2
po
r
-
2
o
r
2
po
r
po
P
υ
2
=
e
zz
σ
(21)
]
2
)
o
r
po
r
(
-
1
[
3
y
σ
=
2
o
r
3
)
2
po
r
_
2
o
r
(
y
σ
=
po
P
1)
-
(22
]
2
r
2
o
r
-
1
[
2
so
r
-
2
o
r
2
o
r
s
P
=
un
rr
σ
2)
-
(22
]
2
r
2
o
r
+
1
[
2
so
r
-
2
o
r
2
o
r
s
P
=
un
θθ
σ
3)
-
(22
2
so
r
-
2
o
r
2
so
r
s
P
υ
2
=
un
zz
σ
1)
-
(23
]
2
r
2
o
r
-
υ
2
-
1
[
)
υ
2
-
1
)(
2
so
r
-
2
o
r
(
E
)
2
υ
2
-
υ
-
1
(
2
o
rs
s
P
=
un
rr
ε
2)
-
(23
]
2
r
2
o
r
+
υ
2
-
1
[
)
υ
2
-
1
)(
2
so
r
-
2
o
r
(
E
)
2
υ
2
-
υ
-
1
(
2
o
rs
s
P
=
un
θθ
ε
)
1
-
24
(
)
2
r
2
o
r
-
1
(
2
so
r
-
2
o
r
2
so
r
s
P
-
s
P
-
]
m
2
r
1
-
m
2
so
r
1
[
3
m
m
2
po
r
y
σ
=
res
rr
σ
)
2
-
24
(
)
2
r
2
o
r
+
1
(
2
so
r
-
2
o
r
2
so
r
s
P
-
s
P
-
]
m
2
r
1
-
m
2
-
m
2
so
r
1
[
3
m
m
2
po
r
y
σ
=
res
θθ
σ
)
3
-
24
(
2
so
r
-
2
o
r
2
so
r
s
P
υ
2
-
s
P
-
]
m
2
r
1
-
m
-
m
2
so
r
1
[
3
m
m
2
po
r
y
σ
=
res
zz
σ
1)
-
(25
]
2
r
2
o
r
-
1
[
2
so
r
-
2
o
r
2
so
r
s
P
-
]
2
r
2
o
r
-
1
[
2
po
r
-
2
o
r
2
po
r
po
P
=
res
rr
σ
2)
-
(25
]
2
r
2
o
r
+
1
[
2
so
r
-
2
o
r
2
so
r
s
P
-
]
2
r
2
o
r
+
1
[
2
po
r
-
2
o
r
2
po
r
po
P
=
res
θθ
σ
3)
-
(25
2
so
r
-
2
o
r
2
so
r
s
P
υ
2
-
2
po
r
-
2
o
r
2
po
r
po
P
υ
2
=
res
θθ
σ
(26)
θ
n
sin
��
∞
2
=
n
)
θ
,
r
(
nB
ij
ε
+
θ
n
cos
��
∞
2
=
n
)
θ
,
r
(
nA
ij
ε
+
θ
sin
)
θ
,
r
(
B
1
ij
ε
+
θ
cos
)
θ
,
r
(
A
1
ij
ε
+
)
θ
,
r
(
0
ij
ε
=
)
θ
,
r
(
ij
ε
0
=
ε
=
ε
=
ε
=
ε
=
ε
=
ε
B
4
θθ
B
3
θθ
A
3
θθ
B
2
θθ
B
1
θθ
A
1
θθ
)
27
(
res
ε
-
load
ε
=
unload
ε
→
unload
ε
-
load
ε
=
res
ε
II
ε
2
+
I
ε
=
0
ε
III
ε
2
1
-
I
ε
2
1
=
A
2
ε
