STRAIN EVALUATION IN THE NECK OF A TENSILE TEST
SPECIMEN BY ESPI STRAIN RATE MEASUREMENT
Bruno Guelorget, Manuel François and Guillaume Montay
Université de technologie de Troyes, ICD-CNRS FRE 2848, LASMIS, 12 rue Marie Curie,
B.P. 2060, 10010 Troyes Cedex, France
ABSTRACT
In-plane Electronic Speckle Pattern Interferometry has been used during tensile test of semi-hard copper sheets for measuring
the strain rate. Strain was calculated by integrating the strain rate. The width of the strain localization zone has been evaluated
by fitting Lorentz curves on the strain rate peaks. Finally, the evolution during the tensile test of the strain localization band
width was determined.
Introduction
Many mechanical properties of materials, such as Young's modulus, yield stress, tensile stress or strain hardening coefficient
can be determined by tensile test. Many metals exhibit tensile stress-strain curves whose initial linear portion (elastic region)
passes gradually to the elasto-plastic region. In a first stage of this region, strains are homogeneous along the specimen. This
stage ends when the diffuse neck appears. Classically, it is considered [1] that the diffuse neck occurs when the force reaches
its maximum. Then strains begin to be heterogeneous up to the localized neck. In case of a sheet in uniaxial tensile test, this
localized neck is the shear band whose width is of the same order of magnitude as the specimen thickness.
As they give a strain map on relatively large areas, optical methods are often used to monitor mechanical tests. Wattrisse and
al. [2] have measured strains during a tensile test with a speckle images correlation technique. Kajberg and Lindkvist [3] have
determined the post-necking behaviour law of a material through an inverse method, using finite element modeling and strain
measurement by image correlation. Cordero et al. [4] have carried out a comprehensive whole-field analysis of a uniaxial
tensile test, using 2D-moiré interferometry, extracting strains and stress fields, ratio of anisotropy and maps of shear bands.
Thanks to its capability to perform accurate measurements of displacements, Electronic Speckle Pattern Interferometry (ESPI)
is used during the tensile test experiments performed in the present paper. Toyooka and Gong [5] have used a setup with one
sensitivity vector for recording fringes pattern and a strain gauge for correlating fringes with measured strains. In their next
paper [6], they presented a setup with two in-plane sensitivity vectors and they used four strain gauges. A completely different
approach has been followed by Panin [7] based on a synergetic methodology of physical mesomechanics, establishing
relations between the physics of dislocation-induced deformation, continuum mechanics of solids and fracture mechanics.
Several papers compare the fringes pattern obtained with ESPI and Panin's results [8-11].
Shabadi et al. [12] obtained similar results, studying Portevin-Le Châtelier bands, with an ESPI technique. They obtained a
large set of results, such as influences of strain, strain rate, thickness of the specimen or ageing on band angle, band width or
band velocity. Vial-Edwards et al. [13] have been interested in determining the true stress-true strain curve from a tensile test
and ESPI experiment, in order to have a look at the localization onset.
The aims of the present paper is to evaluate the strain in the neck of a tensile specimen by an ESPI strain rate measurement
and to determine the strain localization band width evolution during the experiment.
Experimental
The specimen (Fig. 1) was cut in a 0.8 mm thick semi-hard copper sheet, in the rolling direction. The tensile test was
performed at a constant crosshead speed equal to 0.5 mm/min. A square grid of 3 mm separation was drawn on the
specimen. The experimental setup is presented on Fig. 2. The wavelength λ of the He-Ne laser was equal to 632.8 nm. The
beam was expanded, collimated, separated into two beams by a beam-splitter and directed towards the specimen by mirrors.
The present setup belongs to an in-plane sensitive configuration, with a sensitivity vector parallel to the tensile direction (Fig.
2). In the present experiment, the incidence angle is about 53.5° and the sensitivity is 0.394 µm/fringe. Total strain can be
obtained by measuring the length change of the initial square grid:
ε t = ln
measured length
.
initial length
The relative displacement between two points A and B is:
(1)
u AB = N AB ⋅ s
(2)
where
uAB = relative between A and B;
NAB = number of interfringes between A and B;
s = sensitivity.
Figure 1: sketch of the specimen, with 0.8 mm thickness. Dimensions in mm.
Figure 2: experimental set-up.
The average strain between A and B can be expressed as:
ε AB =
u AB
N AB ⋅ s
,
=
l AB k ( x B − x A )
where
and the mean strain rate is:
εAB = average strain between A and B;
lAB = distance between A and B;
k = scale factor (which converts pixels into real size in mm);
xA, xB = coordinates of the fringes in pixels;
(3)
•
ε AB =
ε AB
∆t
=
1 N AB ⋅ s
,
∆t k ( x B − x A )
(4)
where
•
ε AB
= mean strain rate between A and B during ∆t;
∆t = time elapsed between the two subtracted pictures (see below).
Images of the speckle pattern were taken every 201 ms by a CCD camera and saved on a computer. ESPI fringes were
produced by subtracting couples of recorded pictures, with ∆t the time elapsed between these pictures. To enhance the
visibility, two filters were applied, first the Qin and al. filter [14] which was programmed with a mathematics software, and then
a Gaussian filter. More details on the experimental set-up can be found in Ref. [15].
The uncertainty on the strain rate was obtained following the ISO 07-020 procedure [16]. Taking into account the uncertainties
on the incidence angles, the sensibility, the wavelength, the scale factor which converts pixels into real size in mm and the
misorientation of the CCD, the standard uncertainty on the strain rate is [15]:
•
•
1
u  ε AB  = ε AB ⋅ 3.849 ⋅ 10 −6 +


3( xB − x A ) 2
,
(5)
where
•
u  ε AB 


= standard uncertainty on the strain rate;
xA, xB = coordinates of the fringes in pixels.
According to Eq. 5, the main contribution to uncertainty is due to those on xB - xA. Consequently, it is extremely important to
have a determination of the fringes coordinates as accurate as possible.
Results
Applying Eq. 4, strain rates were measured in three different zones (Fig. 3) during the test. Results are plotted on Fig. 4.
Figure 3: fringe pattern just before fracture.
Stain rate along line 314 (Fig. 3) was extracted at different moments and plotted on Fig. 5. In order to be able to determine
accurately the width of the band in which the strain rate is significant, the curves were fitted with Lorentz functions (Fig. 6),
which can be written:
  π ( x − x0 )  2 
f ( x) = A 1 + 
  ,
B

 

where A is the maximum of
f
(6)
and B is the integral width. In the case of Fig. 6, the integral width is equal to 4.28 mm (the
integral width is the width of a rectangle whose height is A and which surface is equal to those under the Lorentz function).
Figure 4: strain rates and tensile force versus time.
Figure 5: time evolution of the strain rate along line 314 (Fig. 3).
To express the evolution of width versus total strain requires measuring the strain at each step of the experiment, but Eq. 1
only holds till the force maximum. Another way to evaluate the total strain is to integrate the strain rate curve of Fig. 4 (i.e. to
calculate the area under the strain rate curve). Results are plotted on Fig. 7, where values called “Grid” were calculated by Eq.
1. The advantage of the integration method is its capability to determine the strain when the grid method cannot any more.
Maximum strain can be evaluated at 46% just before fracture (Fig. 7).
The widths of the last 4 strain rate distributions of Fig. 5 were evaluated and corresponding strains were deduced from Fig. 7.
Results are plotted on Fig. 8. Several features can be outlined. First, the width is decreasing with increasing strain. Second,
just before fracture, the width of the localization band is 4 mm. The initial thickness of the sample is 0.8 mm and the width of
the shear band is about 1 mm, which is much smaller than 4 mm. So, even just before fracture, strains are concentrating in a
zone whose width is 4 or 5 times larger than the shear band. For sake of comparison, a 4 mm wide band was drawn on the
fringe pattern of Fig. 9. Moreover, the curve on Fig. 8 seems to be linear. To our knowledge, this was never reported before.
This experimental result is to be confirmed with other measurements, on other samples or with copper in other metallurgical
states.
Figure 6: fitting by a Lorentz function of one peak of Fig. 5,
with a 4.28 mm wide band.
Figure 7: comparison of strains determined by grid or integration method.
Figure 8: strain localization band width.
Figure 9: strain localization band.
Conclusion
Strain rate during a tensile test was determined using ESPI. This strain rate was used to calculate strain by integration. The
maximum strain in the localized neck, just before fracture, is 46%. Strain localization band width was shown to be decreasing
with increasing strain, but the final width is evaluated as 4 mm, which is 4 or 5 times larger than the thickness of the specimen
or the width of the shear band.
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