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Kh. Farhangdoost, Ehsan Pooladi B.
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
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
2
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total strain theory, thick-walled cylinder,
plasticity
1 Introduction
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
Kh. Farhangdoost, Ehsan Pooladi B.
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 elasticplastic 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.
2
Theory
2
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Kh. Farhangdoost, Ehsan Pooladi B.
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
-Strain-displacement relations:
u
du
ε = , ε rr =
θθ r
dr
-Compatibility equation:
dε
r θθ = ε rr - ε
θθ
dr
(2)
(3)
-Power law relation for plastic region:
rpo
rso
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:
dσrr σθθ - σrr
=
(1)
r
dr
Ps
σe = kεem = (σ1y mEm )εem
(4)
Where σe and εe are Von-Mises effective stress
and strain, also σy , E, m are yield stress, elastic
modulus
Figure 1. Thick-walled cylinder with internal
pressure.
3
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and
strain
respectively.
-Von-Mises effective stress:
hardening
power
Kh. Farhangdoost, Ehsan Pooladi B.
3
(σ - σ )
(5)
2 θθ rr
-(Henkey) Total strain theory for plastic
Where, D is effective strain at inner surface of
the hole. Boundary conditions of the above
problem are as follows:
region:
σ
σe =
3 εe
ε p = λσ ' = λ(σ - σ m ) where λ =
ij
ij
ij
2σ
rr r = r
so
(6)
e
(6 -1)
2
1
ε p = λ(σ - (σ rr + σ zz ))
θθ 3
θθ 2
(6 - 2)
ε
yields
(12)
=D
σ
ε
(6 - 3)
(7)
y
E
(14)
Substitution (14) into (10):
σ r2
y po
D=
E r2
so
(15)
Now put (15) & (10) into (4):
r
po 2m
σ =σ (
)
e
y r
2.1 Plastic solution (rso≤ r ≤rpo)
With substitution equation (5) into (6),
constitutive equation is derived as:
(8)
Using equation (5) in ( 8) leads to:
3
3
ε rr =
εe , ε =
ε
θθ
2
2 e
Satisfying compatibility equation causes:
=
er = r
po
Where, υ is Poision ratio.
3 εe
(σ  σ )  ε
θθ
4 σ e rr θθ
(13)
er = r
so
→
1
σ zz = (σ rr + σ )
θθ
2
Hook's law for elastic region:
E
σ rr =
[(1 υ)ε rr + υε ]
θθ
1 υ 2υ2
E
σ =
[(1 υ)ε + υε rr ]
θθ 1 υ 2υ2
θθ
σ zz = υ(σ rr + σ )
θθ
2
r
so
ε =D
e
2
r
= P
po
rr r = r
po
2
1
εp
rr = 3 λ(σ rr - 2 (σθθ + σ zz ))
ε rr 
(11)
σ
Or
2
1
εp
zz = 3 λ(σ zz - 2 (σ rr + σθθ )) = 0
= P
s
(9)
(16)
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:
σ r 2m
y po
1
1
p
σ =
[
]-P
rr
s
2
m
2
m 3 r
r m
so
(17)
Tangential and axial plastic stresses are derived
as:
(10)
σ
dσ
θθ
=
rr
+σ
rr
dr
leads to
→
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Kh. Farhangdoost, Ehsan Pooladi B.
σ
σ r 2m
y po
1
2m 1
p
=
[
]-P
θθ
s
m 3 r 2m r 2m
so
σ r 2m
y po
1
m 1
p
σ =
[
]- P
zz
s
2
m
m 3 r
r 2m
so
(18)
P r2
po po
σe =
[1 +
θθ r 2 - r 2
o po
r2
o
]
r2
2 υP r 2
po po
e
σ =
zz
2
r - r2
o po
(20 - 1)
(22 - 1)
P r2
r2
s
o
un
σ =
[1 + o ]
θθ r 2 - r 2
r2
o so
(22 - 2)
2 υP r 2
s so
un
σ =
zz
2
r -r2
o so
(22 - 3)
Radial and tangential strain during unloading
are as follows:
P rs 2 (1 - υ - 2υ2 )
r2
s
o
un
ε =
[1 - 2υ - o ]
2
2
rr
E(r - r )(1 - 2υ)
r2
o so
(23 - 1)
P rs 2 (1 - υ - 2υ2 )
r2
s
o
un
ε =
[1 - 2υ + o ]
θθ E(r 2 - r 2 )(1 - 2υ)
r2
o so
(23 - 2)
(20 - 2)
(20 - 3)
2.3
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:
σ (r 2 _ r 2 ) σ
r
y o
po
y
po 2
P =
=
[1 - (
) ]
po
2
r
3
3r
o
o
P r2
r2
s o
o
un
σ =
[1 - ]
rr
2
2
r -r
r2
o so
(19)
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:
P r2
r2
po po
o
e
σ =
[1 - ]
rr r 2 - r 2
r2
o po
relations are derived for elastic unloading;
similar to elastic loading:
(21)
2.3 Unloading (rso≤ r ≤ro)
During unloading which assumed to be elastic,
some stress created around the hole. The below
5
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Residual 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:
σ r 2m
1
1
y po
σ res =
[
]- P rr
s
m 3 r 2m r 2m
so
P r2
r2
s so (1 - o )
r2 - r2
r2
o so
(24 - 1)
Kh. Farhangdoost, Ehsan Pooladi B.
series as follows:
σ r 2m
y po
1
2m - 1
σ res =
[
]- P θθ
s
m 3 r 2m r 2m
so
P r2
r2
s so
o
(1 + )
2
2
r -r
r2
o so
(24 - 2)
σ r 2m
1 m -1
y po
res
σ =
[
]- P zz
s
m 3 r 2m r 2m
so
2 υP r 2
s so
2
r - r2
o so
(24 - 3)
And in elastic region (rpo≤ r ≤ ro), we have:
P r2
r2
P r2
r2
po po
o
s so
o
res
σ
=
[1 - ] [1 - ] (25 - 1)
rr
r2 -r2
r2 r2 -r2
r2
o po
o so
P r2
r2
P r2
r2
po po
o
s
so
res
σ =
[1 + ] [1 + o ] (25 - 2)
2
2
2
θθ r 2 - r 2
r
r -r
r2
o po
o so
2 υ P r 2 2 υP r 2
po po
s so
σ res =
θθ
2
2
2
r -r
r -r2
o po
o so
(25 - 3)
It should be noted that there are three unknown
parameters, rpo, Ppo, Ps which should be
determined.
2.4 Strain
series
expansion
with
Fourier
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
ε (r, θ) = ε 0 (r, θ) + ε1A (r, θ) cos θ +
ij
ij
ij
∞
ε1B (r, θ) sin θ + ∑ ε nA (r, θ) cos nθ +
ij
ij
n=2
∞
∑ ε nB (r, θ) sin nθ
(26)
ij
n=2
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:
2B
4B
ε 1θθA = ε 1θθB = ε θθ
= ε 3θθA = ε 3θθB = ε θθ
=0
3 New 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:
ε res = ε load - ε unload →
ε unload = ε load - ε res
(27)
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
6
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Kh. Farhangdoost, Ehsan Pooladi B.
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:
ε
0
I
II
= ε + 2ε
ε
2A
ε
4A
=
1 I 1 III
ε - ε
2
2
1 I
II 1 III
ε - 2ε + ε
2
2
=-
( 28 )
So Fourier series for relating strain at each
point to measured strains is rewritten as
follows:
1
1
( r , θ) = (ε I + 2ε II ) + ( ε I - ε II ) cos 2θ +
θθ
2
2
1
1
(- ε I - 2ε II + ε III ) cos 4θ
( 29)
2
2
ε
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:
 r
e
D
r
po
2
2
r
so

DEr

2
so
y
7
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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.
4 Conclusions
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.
Kh. Farhangdoost, Ehsan Pooladi B.
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8
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