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Residual Stresses in Beams (With and Without
Prior History) Numerically Assessed by the Crack
Compliance Method
G. Urriolagoitia-Sosa
†
, G. Urriolagoitia-Caldero
´n
†
, J. M. Sandoval-Pineda*,
L. H. Herna
´ndez-Go
´mez
†
, E. A. Mercha
´n-Cruz*, R. G. Rodrı
´guez-Can
˜izo*
and J. A. Beltra
´n-Ferna
´ndez
†
*Instituto Politecnico Nacional, Seccio
´n de Estudios de Posgrado e Investigacio
´n (SEPI), Escuela Superior de Ingenierı
´a Meca
´nica y
Ele
´ctrica (ESIME), Unidad profesional, Azcapotzalco, Av. de las Granjas No. 682, Col. Sta. Catarina Azcapotzalco, C.P. 02550, Me
´xico
†
Instituto Politecnico Nacional, Seccio
´n de Estudios de Posgrado e Investigacio
´n (SEPI), Escuela Superior de Ingenierı
´a Meca
´nica y
Ele
´ctrica (ESIME), Edificio 5. 2do Piso, Unidad Profesional Adolfo Lo
´pez Mateos Zacatenco Col. Lindavista, C.P. 07738, Me
´xico,
D.F. Me
´xico
ABSTRACT: This work assesses the Crack Compliance Method (CCM) by the Finite Element
Method (FEM). The CCM is a very powerful method that is based on Fracture Mechanics Theory. Its
experimental application and set up is validated by this work. The numerical assessment of the CCM
is performed on bending beams with and without prior straining history, to determine the best
position and orientation of strain gauges, as well as the optimum number of readings. The prior
straining history condition, in the analyzed components, is induced by an axial pulling before the
beam is bent. Three levels of preloading are considered: low, medium and high (which are related to
the yield strain of the simulated material); Isotropic and Kinematic hardening rules are also
considered. Additionally, an experimental evaluation is also presented by introducing a new
supporting system to cut a slot in the beams. The results obtained in this work, provide a quanti-
tative demonstration of the effect of hardening strain on the distribution of the residual stress in
beams. In the same manner, the theoretical formulation of the CCM has been evaluated validating
the application of this method for the determination of residual stress fields in mechanical
components.
KEY WORDS: crack compliance method, finite element method, prior straining history, residual
stresses
Introduction
Around the world it is well known that superficial
hardening in mechanical components induces
anisotropic behaviour on the material [1, 2]. Also, if
the hardening of the material is performed in a non-
homogenous manner, one of the consequences is the
introduction of a residual stress field. The knowledge
of the effect of residual stresses in mechanically loa-
ded materials can be helpful to select materials for
engineering applications and in determining suitable
loading in the mechanical design [3]. Several authors
have evaluated residual stresses in materials, they
have used diverse techniques [4–6]. The methods for
their evaluation are classified in three groups, which
are; non-destructive, semi-destructive and destructive
[7].
After reviewing a great deal of research information
and taking into consideration different aspects of
experimental methods for the evaluation of residual
stresses, in this paper we deal with the use and
development of the Crack Compliance Method
(CCM) [6]. The CCM is a relatively inexpensive
method for determining residual stresses in materials.
Only the elastic constants of the material and data on
strain relaxation, when a slot is introduced into the
material, are required for the application of the
method. Compared to other techniques, such as
X-ray and neutron diffraction methods, the CCM is
relatively simple and requires more commonly
available equipment such as an electric discharge
machine (EDM) and strain gauges [6]. The CCM is
however a destructive method as it requires cutting a
slot in the piece or material of interest. It is also
2010 Blackwell Publishing Ltd jStrain (2011) 47, e595–e604 e595
doi: 10.1111/j.1475-1305.2009.00663.x
An International Journal for Experimental Mechanics
sensitive to errors in measurement of strains espe-
cially when gauges are placed close to the slot. The
CCM is used worldwide and it has been applied to
various geometries. The most common specimens
examined are beams and cylinders [4, 8] and [9]
which can be either solid or hollow [10]. Other
geometries that can be tested include rectangular bars
and brackets welded to a plate [11–13].
The reported advantages of the CCM over other
destructive measurement methods include; a simple
and rapid experimental implementation, since only a
single growing slot is required for a test; greater
accuracy due to the reduced degree of machining, as
repetitive processes can accumulate errors and even
more new stresses are introduced; and, a superior
performance in measuring the localised residual
stresses which result from any kind of manufacturing
process [6].
The two main disadvantages of the method arise
from the apparent difficulty in determining the
appropriate crack compliance and the manner in
which the slot is induced into a component. For the
crack compliance, the solution obtained analytically
involves quite considerable mathematics; whilst
those calculated using FEM analysis require a new
model for each depth increment [14].
When a slot is induced into the specimen, there are
several issues about the correct manner to support
the specimen before the cut is performed and the
optimum location to place the strain gauge. This
work focuses in these two issues and will suggest
some solutions. In this paper, the numerical evalua-
tion of the CCM has been presented. Until now, the
CCM has been only applied and assessed experi-
mentally. The authors’ opinion is that, there is a lack
of information about the correct manner of per-
forming the experimental procedure so as to be
convinced that the results obtained are the optimum
ones. It is thought, that numerical analysis per-
formed by FEM could give the answers to all these
issues concerning the CCM, because applying a
numerical evaluation will not introduce external
factors that can affect the recalled strain data needed
to perform it.
Methodology of the Crack Compliance
Method
In this section, it is presented a brief summary of the
theory related to the CCM. It is assumed that the
unknown residual stress distribution in a beam can
be represented by the summation of an n
th
order
polynomial series as [15]:
ryðxÞ¼X
n
i¼0
AiPiðxÞ(1)
where A
i
are constant coefficients and P
i
are a power
series, x
0
,x
1
,x
2
,….x
n
etc, Legendre polynomials are
also used. However, the crack compliance method
includes a step which assumes that a stress distribu-
tion, r
y
(x) =P
i
(x), interacting with the crack is
known. This known stress field is used to obtain the
crack compliance function Cby using Castigliano’s
approach [16]. To illustrate the determination of the
compliance functions, a strip of unit thickness and
unit width in the zdirection with an edge crack of
length a(Figure 1), is considered. In order to obtain
the horizontal displacement uat (l, s), a pair of virtual
forces Fare introduced at that location in the hori-
zontal direction. E¢is the generalised Young’s mod-
ulus (E¢=Efor plane stress condition and E¢=E/
(1 )m
2
) for plane strain). The forces may be located at
either the top or bottom surface. The change in the
strain energy due to the presence of the crack and the
virtual force is given by [4] (where Kare the stress
intensity factors due to the surface tractions K
I
and
the virtual force K
IF
):
U¼1
E0Za
0
ðKIþKIFÞ2da(2)
Applying Castigliano’s theorem, the displacement
u(a,s) can be determined by taking a derivative of the
strain energy with respect to the virtual force, as [15]:
uða;sÞ¼1
2
@U
@FjF¼0¼1
E0Za
0
KI
@KIFða;sÞ
@FdajF¼0(3)
Differentiating now with respect to the distance s
[15]:
eðaj;sÞ¼1
E0Za
0
KIðaÞ@2KIF ða;sÞ
@F@sda(4)
Figure 1: Edge-cracked strip subjected to surface loading and
virtual force [4]
e596 2010 Blackwell Publishing Ltd jStrain (2011) 47, e595–e604
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Residual Stresses Numerically Assess by the CCM : G. Urriolagoitia-Sosa et al.
This strain (a,s) due to the stress P
i
(x) is known as
the compliance function C
i
(a,s) so that;
Ciðaj;sÞ¼1
E0Zaj
0
KIðaÞ@2KIFða;sÞ
@F@Sda(5)
Due to the linearity of K
IF
with F, the second term
under the integral in Equation 5 is the same as Z(a) in
ZðaÞ¼B
F
@KIF
@sS¼0
j
with B=1 and a
j
denotes a growing
crack, so Equation 5 it can be written as:
Ciðaj;sÞ¼1
E0Zaj
0
KIðaÞZðaÞda(6)
By following the approach in Schindler et al. [17]
and Kang et al. [18], for the case of a beam having the
strain measurement point Mat the base, K
I
(a) and
Z(a) can be expressed as:
KIðaÞ¼Za
0
hðx;aÞryðxÞdx(7)
ZðaÞ¼4:283 Za
0
hðx;aÞð12xÞdx(8)
where r
y
(x) =P
i
(x) and h(x,a) is known as the weight
function [19]. The reference contains tabulated val-
ues of h(x,a) for given values of aand x. Therefore,
C
i
(a
j
,s) can be determined from Equation 6 by
numerically integrating this expression. Once the
C
i
(a,s) solutions have been obtained, the expected
strain due to the stress components in Equation 1 can
be determined as [15]:
eðaj;sÞ¼X
n
i¼0
AiCiðaj;sÞ(9)
The unknown terms A
i
have to be determined so
that the strains given by Equation 9 match those
strains measured in the experiment. To minimise the
average error over all data points for the n
th
order
approximation, the method of least square is used to
obtain the A
i
values. Therefore, the number of cut-
ting increments, m, is often chosen to be greater than
the order of the polynomials P
i
i.e. m>n.Typically
m=n+1 is used [15]. This work used n=8 and
m=9, the least square solution is obtained by
minimising the square of the error relative to the
unknown constant A
i
, i.e. as in Equation 10 [15]:
@
@AiP
m
j¼1
eða;sÞactual P
n
k¼0
AkCkðaj;sÞ
2
¼0i¼0;...;n
(10)
This leads to
½HfAg¼fJg(11)
where [H] =[C]
T
[C] and {J} =[C]
T
{
j
}
actual
[20]. Equa-
tion 11 gives a simple set of simultaneous linear
equations. For the problems considered in this work,
[H] is a 8 ·8 matrix. The numerical procedure was
implemented in a
FORTRAN
program using the Compac
Visual
FORTRAN
package. Therefore, Equation 11 was
solved using the LU Decompositor (LUDCMP) and
Backsubstitution (LUBKSB) routines [20]. The actual
residual stress distribution was then determined by
using Equation 1.
Numerical Modelling of the Beam
The numerical evaluation of the CCM presented in
this work is based on the residual stress induction
applying pure bending. The beams are modelled in
2D, in order to allow variations in stresses and strains
through the depth to be determined. Quadratic order
elements with eight nodes were used, in order to
obtain smooth variations of stress and strain through
the depth of the beams.
The beam was loaded (in all the cases) in a four-point
bend configuration with the force located on two sep-
arate nodes. The maximum value of the moment
applied in each case was 9 Nm. Plastic deformation was
obtained in all cases (without and with prior history).
Boundary conditions were applied at the extreme
end nodes at the bottom (Figure 2). Care was taken to
produce small elements (1 mm by 0.5 mm) in the
middle section of the beam, where the slot was sim-
ulated by deleting pair of elements and the relaxed
strain data was obtained for the numerical evaluation
of the CCM. Ten divisions were used across the depth
of the beam, to simulate nine progressive cuts, leav-
ing one pair of element so that structural integrity
would prevail.
Figure 2: Schematic representation of the four-point bending
beam by a FEA model
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G. Urriolagoitia-Sosa et al. :Residual Stresses Numerically Assess by the CCM
Plane stress analysis was carried out and the
loading was applied in a total of 100 sequential
increments. The resulting strain in the xdirection,
produced by the action of the bending loading was
recorded for each loading increment.
The induction of prior history was done by pre-
pulling the beam. For this purpose, three levels with
respect to the yield strain were considered; light,
medium and heavy. These correspond to a displace-
ment in the direction xof 0.8 mm, 2.0 mm and
2.8 mm respectively. The maximum total strains
were 2, 5 and 7 times the value of the yield strain. For
the analyses, a general material model was used,
where two separately hardening rules were consid-
ered (non-linear kinematic hardening and isotropic
hardening), as given in Equations 12 and 13 respec-
tively [21].
da¼D1
r0
ðraÞd
epl cad
epl (12)
ro¼rjoþQ1ð1e
epl bÞ(13)
where
epl is the equivalent plastic strain, ais the back-
stress, Dis the initial kinematic hardening modulus, c
determines the rate at which kinematic modulus
decreases with plastic deformation, r
o
is the current
yield stress, r|
o
is the initial yield stress, Q
¥
is the
maximum change in the size of the yield surface and
bdefines the rate at which the size of the yield surface
changes as plastic straining develops. Equation 12
describes the translation of the yield surface in the
stress space due to the back-stress, a, while Equation
13 describes the change of the equivalent stress
defining the size of the yield surface, r
o
, as a function
of plastic deformation.
In this section the numerical evaluation of the
CCM is developed. For all cases [(with and without
prior history) and (non-linear kinematic and isotro-
pic hardening rules)] the same model of Figure 3 was
considered. Initially, CCM was simulated in a beam
without a prior history. On the other hand, for the
cases where prior history was considered, a homoge-
nous pre-strain of the beam was applied before
conducting the operation of bending. Moreover, in
the considered cases, different hardening rules were
applied, such as the isotropic hardening rule and the
kinematics hardening rule, with which the mechan-
ical behaviour of the component is determined. The
applied load and the boundary conditions are con-
stant in each case of study. The mechanical proper-
ties established as input data are: E = 210 GPa,
r
y
=420 MPa and m=0.29.
Determination of the Ideal Position
of the Strain Gauge in the Beam
For the experimental development of the CCM, it is
important to find the ideal position of the strain
gauge, because the accuracy of the measurement
directly depends on this position. Therefore, the
reading point in different positions with respect to
the crack was simulated. Ten points at 1 mm incre-
ments were taken into account. The first one is at the
crack plane (Figure 4).
The equivalent strain results produced by the stress
relaxation induced by the cut are shown in Figure 5.
This analysis was performed in a beam without prior
history and under the isotropic hardening rule.
Figure 6 shows the residual stress field obtained from
the readings of Figure 5. The residual stress fields
obtained by the CCM are compared against those
derived from the application of the FEM.
Figure 3: Schematic representation of the model applied in
the Finite Element Method
Figure 4: Schematic localization of the reading points
Figure 5: Strain relaxation obtained at different positions
along the beam
e598 2010 Blackwell Publishing Ltd jStrain (2011) 47, e595–e604
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Residual Stresses Numerically Assess by the CCM : G. Urriolagoitia-Sosa et al.
It is important to indicate that both numerical
results are close. It can also be concluded, and is
obvious, that the optimum location of the gauge is in
the same plane as that of the crack. Nevertheless, this
analysis proves that the application of the gauge has
to be accurate on the crack growth plane, otherwise a
placement of the gauge away, if only is 1 mm, can
produce a substantial difference. Additionally, it is
possible to obtain the residual stress profile by using
only one relaxed strain reading that it is located at
the bottom of the beam.
Numerical Evaluation of the CCM Under
Isotropic Hardening Rule
Applying the CCM method using the strain data
produced by the introduction of a cut, the original
residual stress profile in a component can be found.
Figure 7 shows a comparison of curves that represent
the residual stress fields simulated by FEM and
applying the CCM. Isotropic hardening rule is con-
sidered for all four cases and also both prior history at
different levels, Figure 7B–D and no prior history
Figure 7A was applied.
Numerical Evaluation of the CCM Under
Kinematic Hardening Rule
Figure 8 shows the residual stress results obtained
from the measurement of the relaxation of the strain
by introducing a cut and applying the CCM. They are
Figure 6: Residual stress fields at different locations on the
beam
(A)
(C)
(B)
(D)
Figure 7: Residual stress fields, isotropic hardening rule, (A) Without prior history, (B) Prior history of 2
y
, (C) Prior history of 5
y
(D) Prior history of 7
y
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G. Urriolagoitia-Sosa et al. :Residual Stresses Numerically Assess by the CCM
compared with the numerical results, which are
obtained with the kinematic hardening rule (without
and with prior history), this evaluation was
performed considering also strain hardening and
Bauschinger effect as well [1].
Experimental Implementation of the CCM
Due to Numerical Results
In this section a new cutting procedure is presented
by taking in consideration the numerical simulation
of the CCM. It has been shown by the numerical
simulation performed in this research, that the
specimen to be cut must almost be free to relax when
the slot is introduced. So, the numerical information
presented in this article shows, that the cutting pro-
cedure must consist of a mechanical component
which permits the specimens to be cut and allow to
move or relax freely. The experimental supporting
system designed consists of a cylindrical piece sec-
tioned in half [5]. The new cutting procedure devel-
oped focused on producing a straight cut along a
desired plane and simulates a crack. Firstly, it was
important to design a new specimen supporting
device that could support different specimen shapes
without clamping and ensure that the cut remains in
plane. It is also necessary that the weight of the
specimen should be prevented from influencing the
strain readings and the specimen should displace
freely to give accurate strain relaxation readings. This
research work considered the analysis of uniform
rectangular cross-section beams. The weight of the
beam can be considered to be uniformly distributed
as illustrated in Figure 9. The beam is assumed to be
of length 2L and the uniformly distributed weight is
w(N/m).In order to maintain symmetry, that will
assist balance, the supports are assumed to be placed
(A) (B)
(D)(C)
Figure 8: Residual stress fields, kinematic hardening rule, (A) Without prior history, (B) Prior history of 2
y
, (C) Prior history of 5
y
,
(D) Prior history of 7
y
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Residual Stresses Numerically Assess by the CCM : G. Urriolagoitia-Sosa et al.
at distance afrom the two ends of the beams. The
bending moment at the centre of the beam where
strain gauges are located is given by:
Mðx¼LÞ¼wLðL
2aÞ(14)
It is therefore obvious that if a is taken to be equal
to L/2 in Equation 14 then the moment at the centre
of the beam will be zero. Therefore no strains will be
induced at the centre of the beam due to bending
from self-weight. Although zeroing or initializing
strain gauge reading could be seen as acceptable if
there is a moment at the centre of the beam; addi-
tional strain readings will occur as the cutting pro-
gresses and the second moment of area, I, at the
section changes. The supports were therefore located
at distance a = L/2 in the approach used. Figure 10
shows the new supporting rig which consists of two
semi-cylindrical pieces that allow the specimen to
rotate freely as the stress is relaxed and prevent the
weight of the parts to influence the strain reading.
The rotation of the supports also has the effect of
keeping the slot in the same plane at all times. It is
assumed that the rotations due to stress relief will not
change the moment distribution significantly. EDM
methods use a dielectric liquid that helps to produce
the spark to erode the component. This adds an
additional problem to the cutting procedure. The
dielectric fluid (kerosene) erodes the coating on
gauges and causes short circuits to take place. The
solution is to seal the strain gauges with a high tear
strength silicon rubber (DOW CORNING 3145 RTV
MIL-A-46146) and additionally with a nitrile rubber
coating (M-COAT B), which are special products
resistant to kerosene. At the time of performing the
cuts on the first set of annealed specimens that had
been bent, two important issues were noticed that
affect the strain results as highlighted below. It was
observed that there was a pushing action of the
electrode on the specimen when the feeding speed
was relatively high. The instruction manual of the
EDM specifies that the electrode would not touch the
specimen when the cut is performed, which in theory
is true. But a weight sensor in the machine controls
the closeness and location of the electrode to the
specimen to produce the cut automatically. Thus,
actually there is a small interrupted touch between
plate and specimen at all times as the cut is per-
formed, but no cut is produced when there is an
actual contact. The contact effect can be minimised if
the control is set to light feed position. By taking this
action the electrode plate pushes the specimen with
such a small force that the strain reading is not sig-
nificantly affected. The pushing action of the elec-
trode can be so severe that at times it can alter the
original residual stress of the specimen as the cut is
introduced. In fact, when the feeding setting is not
adjusted correctly, plastic deformation can be seen in
the electrode plate. Heavy feeding can introduce
additional residual stresses into the specimen at some
stage of the cutting by re-bending the specimen into
the plastic region. It was therefore important to set
the control to the light feeding position before any
cut was done.
When using only the strain gauge opposite the
front of the crack, it was found that the calculation of
the residual stress field could benefit from performing
the calculation on two different bent specimens. One
cut is done on the concave side of a beam and the
other cut from the convex side of another beam with
the same residual stress field (Figure 11).
By obtaining the strain data in this manner the
most sensitive results from the cuttings can be
used. It is presented in this paper an experimental
Figure 10: Cutting procedure of a beam using a plate EDM
Figure 9: Consideration for the supporting of uniformly
weighted beams
Figure 11: Different cutting procedures (concave and convex)
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evaluation of the new cutting support system. This
experimental assessment is performed on plastically
bent beams, where the residual stress introduced will
be calculated by the CCM. The cut has been done by
an electro discharge machine using a copper plate [3].
The specimens used for this purpose were beams of
250 mm long, and a cross section of 38.1 mm by
12.7 mm, manufactured from EN 8 steel. All the
specimens were annealed and strain gauges attached
to each piece.
The bars were plastically bent to introduce a pre-
dictable residual stress field based on well proven
superposition of loading and unloading stresses [22].
The bending operation was controlled by the strain
reading obtained from the strain gauges attached to
the surfaces of the bar. Using the knowledge of the
stress strain behaviour of the material, the beams
were bent to reach the plastic range making sure that
they did not make contact with the lower surface of
the four point bending rig. Figure 12 shows a bend-
ing moment versus strain results obtained from one
of several tests.
This result is representative of tests carried out on
this type of steel that had been stress released by
annealing. From the results presented in Figure 12
the mechanical properties of the material can be
obtained [5] and the residual stress profile can be
determined by using the method of superposition of
loading and unloading stresses [22]. Results presented
in Figure 12 show an almost perfect bending test,
because the top and bottom strain gauges gave vir-
tually identical values during loading and unloading.
When the bending procedure was completed, the
specimen was cut using the EDM and the cutting
supporting system described. Several cutting tests
were performed, alternating between cutting from
concave and convex sides (top or bottom surfaces).
Although the supports are placed to minimise
moment at the gauges at the middle, the actual effect
of the curvature of the plastically bent beam is not
accounted for. This is due to the practical difficulty of
measuring and making allowance for the effect of it
during the placement of the supports. This means
that some effect of self-weight on the strain results
still occurs towards the end of the cut. There is still a
final rocking to allow each part to balance in its
support. This effect was slightly more severe in the
convex arrangement. In Figure 13 the experimental
strain results from both kinds of cutting procedures
(concave and convex) are presented. The strain
results of a FE model for the process of bending and
cutting are also indicated in the figure.
The results of the residual stress field determined
using the crack compliance method and strain data
for different cases are given in Figure 14. The results
obtained by FEA are also shown in the figure. It can
be seen that the results from data based on cutting
from the concave side (top to bottom) agree reason-
ably well with the expected results (as predicted by
FEA) over the range 0 < x/t < 0.8. The agreement is
particularly good in the early parts of the range. It
can also be seen that the results from the convex
cutting (bottom to top) data agrees well in the range
0.2 < x/t < 1.0 and are particularly good towards the
later part of the range. It therefore appears that a
form of averaging in the central part would give a
Figure 12: Bend test of stress relief annealed EN 8 steel
Figure 13: Strain results by cutting for stress released annealed
EN 8 steel
Figure 14: Residual stress calculation for stress released EN 8
steel by different cutting direction
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Residual Stresses Numerically Assess by the CCM : G. Urriolagoitia-Sosa et al.
better overall result across the full range. The aver-
aging was carried out at the middle of the range as
this position maximised the input from the two
gauges.
This approach assumes that more than one sample
with the representative residual stress field is avail-
able. The results obtained in this manner are pre-
sented in Figure 15. In Figure 15A symmetrical
residual stress field can be seen, which is positioned
almost exactly at the central depth position of the
beam. The trend of this result is more symmetrical
than those obtained by other researchers such as
Nowell et al. [11] and Schindler and Bertschinger [23].
Figure 15 shows good agreement with expected
results across the depth of the beam. The extrapola-
tion of the curves to the surface also show that good
results can be obtained.
Discussions and Conclusions
In this paper, the numerical evaluation of the CCM
has been presented. Until now, the CCM has been
only applied and assessed experimentally. In that
sense, it is the authors’ opinion that there is a lack of
information about the correct manner of performing
the experimental procedure and to be convinced that
the results obtained are the optimum ones. It is
considered that, FEM could be the best option for the
evaluation of the CCM, because there are not exter-
nal factors involved that can affect the recalled strain
data needed to perform it. Also, since the FEM sim-
ulation of the induction of the crack will not incor-
porate further stress into the component, due to the
cutting process.
On the other hand, Prime [24] and Nowell et al.
[11] have experimentally practiced the introduction
of the slot into the component by clamping one side
of the specimen. But FEM results presented within
this work, suggest that the experimental procedure
on the beam or strip should be a better one if the
component is positioned in a two point rolling con-
figuration. Urriolagoitia-Sosa [5] has developed a new
and simple supporting system. This consists of two
semi-cylindrical pieces upon which the beams anal-
ysed were placed. The pieces were simply placed on a
smooth flat surface in the EDM cutting machine. The
centres of the pieces were located at quarter distances
from the two ends of a beam which should be fully
parted in two halves. Each one is capable of being
supported by a semi cylindrical piece. The cylindrical
shape of the pieces also allow them to rock as
the beam is being cut and the internal residual stress
released causes movement.
The numerical evaluation of the CCM will con-
tribute in the selection of the optimal location for the
strain gauges so it is possible to use only one strain
reading to calculate the entire residual stress field.
This will reduce the cost and time to develop and
perform experimental work.
This numerical evaluation was made by applying
different hardening rules and different prior history
stress levels, to a rectangular beam under four points
bending, onto which later residual stress fields were
induced. Residual stress fields induced under the
isotropic rule (with and without prior hardening) still
have their axisymmetric shape and it can also be
observed a magnitude reduction of the stress peaks as
the prior history tends to be higher. On the other
hand, kinematic hardening rule, shows a decline of
the axisymmetric shape of the residual stress fields (a
modification of the neutral axis) and an increase of
the tensile stress magnitude (also a decrease on the
compressive stress peak) as the effect of the prior
history is elevated. This effect can be extremely bad
for a component, because it can promote the nucle-
ation of cracks.
The paper has presented the numerical evaluation
and experimental application of the CCM developed
for the study of residual stresses in beams that were
pre-strained before residual stresses were mechani-
cally induced through bending. The results obtained
from the CCM agree closely with those obtained
using the FEM of superposition of loading and
unloading stresses. As expected, residual stresses
developed in beams with prior straining, show
asymmetry in magnitude of the stresses on the tensile
and compressive sides.
ACKNOWLEDGEMENT
The authors gratefully acknowledge the financial sup-
port from the Mexican government by de Consejo
Nacional de Ciencia y Tecnologı
´a and the Instituto
Polite
´cnico Nacional.
Figure 15: Residual stress calculation in stress relief annealed
EN 8 steel
2010 Blackwell Publishing Ltd jStrain (2011) 47, e595–e604 e603
doi: 10.1111/j.1475-1305.2009.00663.x
G. Urriolagoitia-Sosa et al. :Residual Stresses Numerically Assess by the CCM
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doi: 10.1111/j.1475-1305.2009.00663.x
Residual Stresses Numerically Assess by the CCM : G. Urriolagoitia-Sosa et al.
