STRAIN LOCALIZATION ANALYSIS BY A COMBINATION OF THE
ESPI WITH A BULGE TEST
Guillaume Montay1, Bruno Guelorget1, Ignacio Lira2, Marie Tourneix1,
Manuel François1 and Cristián Vial-Edwards2
1
Université de Technologie de Troyes,
Laboratoire des Systèmes Mécaniques et d’Ingénierie Simultanée
(ICD/LASMIS FRE CNRS 2848)
12 rue Marie Curie, BP 2060, 10010 Troyes, France.
2
Pontificia Universidad Católica de Chile, Department of Mechanical and Metallurgical
Engineering, Vicuña Mackenna 4860, Santiago, Chile.
[email protected]
ABSTRACT
The problem of strain localization is important in sheet metal forming, as it determines the forming limit
diagram of the material. To analyze this process, engineers use a variety of tests. One of them is the hydraulic
bulge test. In it, an initially flat specimen is placed between a matrix and a blank holder, and hydraulic pressure is
applied on one of its surfaces. An approximately equi-biaxial strain loading path is obtained.
In this paper we report on the application of electronic speckle-pattern interferometry (ESPI) to determine strain
localization during bulge test. A video sequence of images was captured and stored. The video allowed a
posteriori analysis of the test. By subtracting pairs of images, fringes were obtained at load steps close to fracture.
In this way, the progress of local strain rate at various positions on the apex of the dome was followed. The
stages of diffuse and localized necking were clearly seen.
Introduction
The problem of strain localization is important in sheet metal forming, as it determines the forming limit diagram of
the material. To analyze this process, engineers use a variety of tests. One of them is the bulge test. An initially
flat specimen is placed between a matrix and a blank holder, and hydraulic pressure is applied on one of its
surfaces. An approximately equi-biaxial strain loading path is obtained [1-2].
The aim of this study is to detect the localization of plastic strain (diffuse and localized necking) during the bulge
test combined with a common speckle interferometer. This paper describes an original technique to detect the
localization using the strain rate measured on the sample at different positions and different times.
A classical speckle interferometer [3] with in-plane sensitivity is placed over the specimen to measure the strain
rate during the bulging. The LASER beam is divided into two beams by the beam splitter. The two beams produce
speckle patterns in the reference state and in a subsequent state.
The bulge test: presentation of the experimental setup
The test specimen was placed between a matrix and a blank holder. A hydraulic pressure was applied on one
side of the specimen. Figure 1 describes the experimental device used to load the structure. The metal sheet was
initially flat. The deformation of the structure is performed through an equi-biaxial strain.
Screw
Blank holder
Specimen
Matrix
Oil to generate the pressure
Figure 1. Scheme of the bulge test.
The hydraulic pressure has been applied with a tensile testing machine through a hydraulic Jack. Figure 2
represents a scheme of the experimental device.
Beam
Application of a
constant velocity
From the LASER
source
Mirror
Mirror
Jack
Oil
Pressure
sensor
Test
specimen
Figure 2. Scheme of the optical device mounted on the bulge test.
The oil is under compression during the totality of the experiment. The oil flow was varied from 6333 mm3/s at the
3
beginning of the test to 3166 mm /s near the localization stage. With this device it is not necessary to stop the
load to perform measurements. A pressure sensor pmrovides the pressure and thus the stress applied on the
plate.
A classical speckle interferometer with in plane sensitivity is placed over the specimen to measure the strain rate
during the bulging. The LASER beam is divided into two beams by the beam splitter. The two beams produce
speckle patterns in the reference state and in a subsequent state of the surface. The fringe pattern is obtained by
a pixel to pixel subtraction of the two speckle patterns images which are separated by a time interval of about 2
seconds.
Total strain measurement
To measure the total strain a grid of perpendicular lines, initially (5 mm × 5 mm) was drawn on the sample. The
displacement of the lines was measured vertically and horizontally. This information (in pixels) is transformed in
mm with the magnification factor of the camera. Because of height variation, the magnification factor changes
during the test. This was monitored with the help of a piece of graph paper glued on the specimen near its apex
(Figure 3).
The strains in the
v
x
Figure 3. Grid drawn on the plate to measure the strain.
and
v
y directions are directly obtained with the equations (1):
l
ε xx = ln 
 l0 
L
ε yy = ln 
 L0 
(1)
L0 and l0 represent the initial length of one or several squares of the grid. The strain along the thickness of the
sheet εzz is obtained assuming incompressibility of the material.
Strain rate measurement
The speckle interferometer presented in Figure 2 is a full-field experimental device to determine in plane
displacements. Speckle patterns are acquired by a CCD camera and then subtracted with an images processing
software to create fringe patterns.
Traditionally speckle interferometry is used to measure only small displacement because of decorrelation
phenomena. To determine the strain rate, small strain increment are measured. In this case, the two speckle
patterns remain correlated. This plastic strain increment is controlled with the time between two states of
deformation. However strain increments are not quantity that are easily interpreted, so we propose the use of
strain rates, which are simply strain increment divided by the time between two pictures.
With the fringe patterns an increment of displacement ∆uy is measured by
∆u y = N y .s ,where Ny is the number
of fringes in the direction y and s the sensitivity of the interferometer.
The difference pixel to pixel of these two images creates an image of fringe patterns as shown in Figure 4.
Figure 4: Images of fringes patterns obtained by subtracting two images taken at time t and t+∆t.
The lines numbered 1 to 3 are used to measure the strain increment
∆ε y
The sensitivity vector is along axis y.
The increment of strain in the direction y is given by
∆ε y =
∆u y N y . s
=
L
L
(2)
Where L is the length separating the Ny fringes.
The length of the line in pixels (Lp) was measured with an image processing software. With the magnification
factor (G) of the viewing system, the length in mm, L=Lp×G was determined. Knowing the time ∆t between the two
speckle patterns, we determine the strain rate with
•
εy =
∆ε y
∆t
=
∆u y
L.∆t
=
N y .s
(3)
L.∆t
∆t is given by the video record of the speckle patterns.
The strain rate is determined at three positions on the image. The line numbered 2 represents the centre of the
dome. This position represents the future fracture, known with the last recorded image. The lines 1 and 3 are
positioned at equal distance from the centre line. The lines 1, 2 and 3 do not have the same length, because the
strain rate is different at the centre of the dome and on the right and left side. In this case, the strain rate for the
three positions is given with:
•i
εy =
Practically
N 1y = N y2 = N y3
and
∆ε iy
∆u i
N i .s
= i y = iy
∆t
L .∆t L .∆t
(i = 1 to 3)
(4)
L1 ≠ L2 ≠ L3
This procedure is repeated for subsequent increment of deformation.
Obviously, as the length L is large compared to the size of the picture, the measured quantities (strain increment
and strain rate) are averages. They are space average over L and time average over ∆t.
Result
Total strain result
Figure 5 represents the evolution of the strain in the 3 directions
function of the displacement of the crosshead of the tensile machine.
r
v v
x , y and z . This evolution is plotted as a
80
Strain (%)
30
-20
0
20
40
60
80
100
120
-70
εxx
εyy
εzz
-120
-170
Displacement of the crossbeam (mm)
Figure 5. Total strain in the
r
v v
x , y and z directions.
v
The maximum strain obtained is about 65% and is equal in directions x and
isotropic behaviour of the material. In the third direction, the strain is about 130%.
v
y . This indicates an in plane
Strain rate as a function of strain
Figure 6 represents the evolution of the strain rate for the three positions on the specimen depicted in Figure 4,
this positions are x=2,5 mm (on the left of the image fringe patterns), x= 16,75 mm (at the centre of the plate
which is the position of the fracture) and x=31 mm (on the right of the image).
2,0E-04
1,8E-04
x=2,5 mm
1,6E-04
x=16,75 mm
x=31 mm
Strain rate s
-1
1,4E-04
1,2E-04
1,0E-04
8,0E-05
6,0E-05
4,0E-05
2,0E-05
0,0E+00
0
5
10
15
20
25
30
35
40
45
Total strain (% )
Figure 6: Strain rate as a function of the total strain
for a as received copper plate
On this figure, it can be observed that the strain rate is almost the same on the left and on the right of the future
fracture. The strain rate is higher at the centre of the sample than on the left and right side.
At the beginning of the bulge, the three curves are almost superposed, but when the deformation reaches 20%30%, the strain rate at centre accelerate suddenly. This change in the strain rate is shown in Figure 7 where the
strain rate heterogeneity is presented as a function of the total strain. The strain rate heterogeneity A(%) is
computed with
 •1 • 3
ε y+ε y
ε y−
 2

A=
•2
•2




 * 100
(5)
εy
It can be seen that the strain rate heterogeneity increases gradually and rapidly after a strain of 20% until the final
fracture. This acceleration of the strain can be interpreted as the onset of the localization of strain. This strain
indicates an instability point in the deformation process [4-6].
20
Stra in r a te he te ro g e ne ity (% )
18
16
14
As received
12
10
8
6
4
2
0
0
10
20
30
Total strain %
40
50
Figure 7: Strain rate heterogeneity as a function of the total strain
Strain rate as a function of the position
On the images of fringe patterns, the fringe width was measured for 10 positions (Figure 8) in the x
direction instead of three positions (in the precedent section). The strain rate is presented as a function of the
position x for several states of strain.
Figure 8: Image of fringes patterns with the measurement lines
The strain rate is determined on each of the lines plotted on the image of fringe pattern. This procedure is
repeated for subsequent state of strain. The result is presented on Figure.9.
0,00025
38,26%
Strain rate (s-1)
0,0002
37,8%
0,00015
37,37%
35,74%
0,0001
0,00005
22,8%
0
0
5
10
15
20
25
30
35
x position (mm)
Figure 9: Strain rate as a function of the x position
The strain rate is maximum at the centre of the sample and increase with the strain.
Conclusions
In this paper, an original application of ESPI in materials engineering has been described. The technique was
used to analyze a bulge test in order to study strain localization by following the strain rate progress. Results
show that ESPI allows detecting clearly the two stages of localization [7-8], namely, the diffuse and localized
necks. Using this technique, forming limit diagrams can thus be established accurately.
Acknowledgments
The support of Conicyt (Chile) through Fondecyt 1030399 and ECOS/CONICYT C01E04 research grants is
gratefully acknowledged.
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