DETERMINATION OF THE ELONGATIONAL PROPERTIES OF POLYMERS
USING A MIXED NUMERICAL-EXPERIMENTAL METHOD
L. Robert, S. Hmida-Maamar, V. Velay and F. Schmidt
Research Centre on Tools, Materials and Processes (CROMeP)
Ecole des Mines d Albi-Carmaux, 81013 ALBI Cedex 9, France
ABSTRACT
The biaxial properties of materials such as rubbers or polymers, used in mechanical or thermoplastic applications, are often
difficult to identify. However, these biaxial properties are useful for the simulation of plastic-processing operations such as blow
moulding, stretch blow moulding, or thermoforming. The rheological behaviour and the mechanical properties of rubbers and
polymers can be obtained by using a bubble inflation rheometer. In this paper, the approach is to identify the hyperelastic
behaviour of this type of materials from contour measurements and full field measurements by inverse analysis using Finite
Element numerical simulations. A detailed description of the experimental part is made. Two optical measurement methods
are used: (i) the bubble shape contour measurement can be obtained using one CCD camera; (ii) the displacement / strain
field can be obtained by Digital Image Stereo-Correlation (DISC) both on planar or 3-D shape. Numerical Modelling are
realized by using Finite Element software Abaqus®. Preliminary results of the shape contour based identification (taken into
account of the boundary conditions, function cost, parameters of the behaviour model) and validation of these results by
carrying out tensile tests are presented.
Introduction
The biaxial properties of materials such as rubbers or polymers are often difficult to identify, and generally a simple isotropic
behaviour derived from classical 1D tensile test is considered. However, biaxial properties are useful for the simulation of
plastic-processing operations such as blow moulding [1], stretch blow moulding, or thermoforming. As previously reported, the
rheological behaviour and the mechanical properties of rubbers and polymers can be obtained by using a bubble inflation
rheometer [2-4]. Reuge et al. [2] have used a bubble inflation rheometer for natural rubber and found that for this material the
biaxial behaviour was very similar to the uniaxial behaviour. Li et al. [3] have proposed a study based on a polymer membrane
inflation highlighted that attaining a homogeneous temperature by infrared heating was problematic in their case. However
they have used a technique of grid projection to obtain the shape of the bubble during the inflation. More recently, Jones et al.
[4] have developed an inflated rheometer with thermal facilities. In their work, the shape is measured by photogrammetry and
the strain field over the bubble can be obtained by differentiating the displacement field using of finite element software.
Another way to access to the bi-axial properties is to perform multi-axial tensile test. For instance, Chevalier and Marco [5]
have proposed the use of a multiaxial testing machine coupled with Digital Image Correlation (DIC). Due to high strain
heterogeneities, they have used the measured displacement field as realistic boundary conditions for their simulation, and thus
have validated their behaviour model.
In this work, we present a mixed numerical-experimental study based on the use of several experimental devices coupled with
numerical simulations. Experimental data are provided by optical measurements (bubble shape contour measurement, and
displacement / strain field by Digital Image Stereo-Correlation (DISC)) on tensile tests and bubble inflation tests. These
experimental data are included into an inverse analysis procedure using Finite Element Method simulations and optimisation
loop for identifying the rheological behaviour. This work is actually based on natural rubber and will be extend to thermoplastic
(PP and PET) materials.
Experimental rheometer
A bubble inflation rheometer has been developed in the laboratory (see Figure 1, left). It allows to blow under controlled
pressure rubber or thermoplastic membranes [2]: a circular membrane, clamped at the rim, is inflated by applying air pressure
to its bottom face (see Figure 1, right). In the case of thermoplastic, a heating step is necessary before applying the pressure.
The heating can be performed by air convection because the inflated device is positioned in a convection oven, by conduction
with a heating cartridge near the rim controlled by a thermocouple with proportional integral derivative (PID) controllers, and by
infrared radiation with IR lamps.
In this work, natural rubber circular membranes of 1.5 mm in thickness and 90 mm in diameter (50 mm for the active initial
bubble diameter) are used.
Figure 1. Bubble inflation rheometer (left) and detail of a inflated rubber bubble (right).
Optical techniques
Two experimental optical techniques based on non-contact measurement by CCD cameras have been developed for in situ
measurements: (i) images acquisition of the 2-D projection of the bubble is done during the inflation process, giving shape
contours versus inflated pressure (see Figure 2); (ii) Digital Image Stereo-Correlation (DISC) [6] is applied using a calibrated
stereo rig in order to obtain the three-dimensional description of the strain fields on the surface of the bubble or in the surface
of the tensile test specimen.
The procedure for the shape contour measurement is as follow: a CCD camera (Micam VHR 2000 with a CCD resolution of
752x582 pixels) is located in front of and parallel to the bubble. The pinhole projection of the contour is obtained by lighting the
background behind the bubble (Figure 2 on the left). A contour extraction procedure implemented into Matlab® allows to
access to the contour in pixel (Figure 2 on the centre). The true contour curve is finally obtained knowing the magnification
factor, as presented in Figure 2 on the right.
Figure 2. Rubber bubble and shape contour extraction for a pressure inflation of 0.0595 MPa.
The Digital Image Stereo-Correlation (DISC) is done with the Vic-3D® commercial software [7]. The basic principle is as
follows: the technique uses a DIC algorithm to determine point correspondences between two stereoscopic images of a
specimen acquired from two cameras rigidly bounded [8]. In this work, two 8-bit Qimaging Qicam digital cameras with CCD
resolution of 1360x1036 pixels and Nikkor Nikon 60 mm f2.8 macro lenses are used. The correlation scores are computed by
measuring the similarity of a fixed subset window in the first image to a shifting subset window in the second one. A first-order
two-dimensional shape function and a zero normalized sum of square difference correlation criterion are used. Sub-pixel
correlation is performed using a quintic B-spline grey level interpolation. After determining the calibration parameters for each
camera as well as the 3-D relative position/orientation of the two cameras (pinhole model and radial distortion of 3 orders), the
3-D specimen shape can be reconstructed from the point correspondence using triangulation. To determine the 3-D
displacement field, DIC is also used to determine point correspondences between the stereo pairs acquired before and after
deformation, and the 3-D displacement field is then computed by triangulation [6]. The strain field (Green Lagrange) is
obtained from the displacement field by derivation.
It should be noted that DISC has been used in this work also for 2-D planar deformation, instead of classical 2-D DIC, for two
main reasons: (a) in 2-D DIC, the specimen must be positioned parallel to the camera sensor and must undergo a planar
deformation without any out-of-plane displacement. In practice, this is impossible to guaranty, and an apparent strain could be
added, and (b) thanks to the need of calibration of the stereo rig, spatially varying lens and sensor distortions are also
corrected for better accuracy measurements.
To perform DISC directly on the rubber samples, several difficulties are to be solved (speckle pattern of black pencil pen, large
level of deformation, semi-transparent aspect of the materials, lighting, reflection, etc.).
Uniaxial tension test
Tensile tests have been performed on natural rubber tension specimens (4 mm in thickness and 25 mm in width). For this kind
of material, the DISC is used due to the high level of strain that gives the use of mechanical extensometer impossible. Tensile
test performed on standard specimen allow obtaining the stress/strain curve. The Mooney-Rivling hyperelastic behaviour has
been selected for its simplicity (only 2 parameters C10 and C01) and because it is representative of natural rubbers behaviour,
at least for reasonable stretch ratio (no strain hardening). Energy function is formulated as follow:
w
C10 (I1 3) C01(I 2 3)
(1)
where I1 and I2 are the first and the second invariant of the left Cauchy-Green tensor, respectively.
The 1-D stress/strain expression derived from Eq. (1) is giving by:
F
S0
2C10
1
2
2C 01 1
1
(2)
3
where
F = tensile force
S0 = cross section
= stretch ratio
Figure 3. Nominal stress/strain curve.
The fitting of the stress/strain curve is presented in Figure 3. A very good agreement is obtained in this stretch ration range,
and the Mooney-Rivling behaviour is validated. The fitting gives a set of initial parameters P0 using eq. (2), and the numerical
values obtained are C10 = 0.136 MPa and C01 = 0.097 MPa.
Identification on shape contours
The identification of the Mooney-Rivling coefficients is now performed on experimental shape contours during the bubble
inflation, using a Finite Element Model Updated (FEMU) procedure as presented in Figure 4. The cost function is define as
the norm (in a least squares sense) between the experimental and simulated bubble shape contour as:
N
i
y sim
2
i
f x sim
i 1
N
i
f x sim
2
(3)
i 1
where f(x) is the 4th order polynomial interpolation of the experimental contour.
Figure 4. Algorithm of the identification problem.
Finite element simulation is done with Abaqus® with 2-D quadratic CAX8H elements in an axisymmetric configuration, as
presented in Figure 5: there is one element in the thickness of the membrane and 20 elements for the radius. In this model,
imposed boundary conditions are Ui = 0
M
M
4,
2 where P is the applied pressure, and an
ij ni = P
i
i
axisymmetric condition on
3. Node coordinates on
1 are noted ( x sim , y sim ) and their displacements are calculated by the
FE computation and used in Eq. 3.
1
4
3
2
Figure 5. Finite Elements modelisation of the rubber membrane
A script allows extracting node displacements so the main program automatically runs the finite element calculation, extracts
calculated and experimental data for the cost function evaluation and updates behaviour model parameters to be identified.
Cost function minimization is performed with a Sequential Quadratic Programming (SQP) optimisation algorithm implemented
in Matlab®. This robust method attempts to minimize a scalar-valued nonlinear function of several variables using only
function values. In this investigation, this method is suitable as the cost function cannot be explicitly formulated, and because
under-constraint optimisation is needed for convergence convenience of the Finite Element calculations.
This identification procedure was applied for several pressure inflation steps, and the mean value for the parameters are 0.135
MPa and 0.098 MPa respectively for C10 and C01 with very small discrepancies. Numerical values are very similar to those
provided by the 1-D fitting. A new global identification procedure based on multi-steps pressure inflations was preformed,
giving only one set of parameters considering all the pressure steps, giving C10=0.133 MPa and C01=0.1 MPa. Results of the
experimental and numerical shape contours after global optimisation are presented in Figure 6. We show that the MooneyRivling behaviour can represented the biaxial behaviour of natural rubber at least for small stretch ratio (< 200%).
Identification with open-hole tensile specimen
The second identification approach is based on a Finite Element Method Updated (FEMU) procedure using full field
measurements by DISC during the open-hole tensile test of a drilled rubber specimen. This test has been considered because
it exhibits heterogeneous strain fields. However, the heterogeneous strain fields are very localized near the hole thus
experimental data are not easy to evaluate quantitatively with a good accuracy. Recently, the remarkable improvement in
metrological qualities of optical full field methods renders the open-hole tensile test a configuration of interest for the
identification of material properties. Firstly, FE direct simulations of an open-hole tensile test on rubber with Mooney-Rivling
behaviour are done. As an illustration is showing in Figure 7 the result of the vertical displacement field provided by a
multistage numerical simulation.
The identification strategy can be divided into several stages. First, some preliminary works have to be performed on the
experimental files in order to keep only the good matched points and to avoid mismatched informations. Then, locations where
experimental data are measured have to be projected onto the finite element mesh. Figure 8 on the left presents a comparison
between nodes considered in the experimental and the numerical meshes. At this stage of the work, no interpolation
procedure has been developed, and the experimental data points are simply selected with a criterion of minimum gap between
experimental points and finite element nodes. This point will be studied in future works.
Figure 6. Experimental (strait line) and numerical (cross) shape contours after global optimisation.
Figure 7. Vertical displacement fields provided by a multistage direct numerical simulation of an open-hole tensile test on a
Mooney-Rivlin material.
Only the investigation area is modelised in the numerical simulation. Experimental and calculated displacements are
considered to evaluate the cost function. Moreover, experimental displacements measured at the upper and lower parts of the
investigation area are introduced into the simulation to define the boundary conditions (Figure 8 on the right). Thus, rigid body
motions are taken into account in the simulation. Finite element simulation is performed with Abaqus®, a script allows to
extract node displacements and global strength necessary to the analysis. The main program automatically runs the finite
element calculation, extracts calculated and experimental data for the cost function evaluation and updates behaviour model
parameters to be identified. Cost function minimization is performed with a Sequential Quadratic Programming (SQP)
optimisation algorithm implemented in Matlab®, as previously discussed.
The cost function
depends on experimental and calculated displacements and strength. Its formulation includes spatial
(nodes) and time (images) informations, as:
(4)
w
1 w
d
f
with w a weight parameter, and
d
and
f
the displacement and strength part, respectively, as:
1
d
N image
1
f
N image
Nimage
i 1
Nimage
i 1
1
Nnode
N node
j 1
Fsimu,i
i
U simu
,j
Fexp,i
i
U exp,
j
(5)
i
U exp,
j
(6)
Fexp,i
Nnode is the number of nodes considered for the analysis; Nimage the number of images obtained at various time increments; U
the displacement vector; and F the global strength in the loading direction.
Applying the experimental displacement fields as boundary conditions (upper and lower parts of the investigation area), rigid
body displacement fields are naturally taken into account into the calculation. The reaction forces induced by the previous
applied boundary conditions are considered for the calculation of the global strength F used for the cost function evaluation.
Figure 8. Experimental data projection (crosses) onto the finite element mesh (left), and view of the mesh and of the boundary
conditions for the FE simulation (imposed displacement is represented by arrows) (right).
The identification procedure is applied using experimental data based on an open-hole tensile test on a specimen of width 25
mm, length 50 mm and thickness 4 mm with a 10 mm in diameter hole. Both total tensile force and pairs of images of the
specimen are recorded during the experiment. A randomly speckle pattern is manually applied with a specific black pencil in
order to avoid light reflection and to guaranty the integrity of the pattern even for large level of deformation. DISC is then
performed on the images.
The multi-stage identification procedure is performed only on the middle left part of the specimen with a weight value w = 0.1
which seems to be a good compromise and allows to maximize the effect of the induced force. However, preliminary results of
the optimised parameters after convergence (25 iterations) differ significantly from those obtained in the case of contour
optimisation and 1-D fitting. Numerical values founded are 0.175 and 0.035 for C10 and C01 respectively. A comparison
between experimental and calculated shear strain field is presented in Figure 9 (left). Whereas the experimental strain fields
allow assessing the global heterogeneous characteristics of the field, the heterogeneous strains are very localized near the
hole thus experimental values are not equal to the calculated ones. However, as it can be seen in Figure 9 (right), a
comparison between the plot of the nominal stress/strain curve using Eq. (2) within the values actually founded and those
previously obtained from the conventional tensile test (1-D fitting) show a relatively small divergence and only for stretch ratio
upper than 2. Work is in progress concerning that point and will be addressed during the presentation.
Figure 9. Comparison between experimental shear strain 12 and calculated angular shear strain 12 = 2 12 fields (left), and
nominal stress/strain curve using Eq. (2) within the values founded with the FEMU on the open-hole tensile test and those
previously obtained from the conventional tensile test (right).
Strain field on the bubble: first experimental results
An original item of our proposal woks is to use directly the 3-D displacement fields during the inflation in order to identify the
behaviour parameters.
Figure 10. DISC on the surface of an inflated rubber membrane. On the left is showing the 3-D vertical displacement field. For
several areas located from the edge to the pole of the bubble are plotted the max principal strain versus the min principal strain
(right).
Figure 10 presents preliminary results of DISC measurements on an inflated bubble: in the left is presented the out-of-plane
displacement field. It appears that in some areas, the DISC measurement did not succeed because of lightning and speckle
pattern degradation. For several areas located from the edge to the pole of the bubble (see Fig. 10 on the left), are plotted in
Figure 10, right, the evolution of the maximum principal strain ( I) versus the minimum principal strain ( II) during the pressure
inflation.
As for the surface part close to the pole the behaviour is purely equi-biaxial ( I = II), we show here that near the edge (where
the membrane is clamped at the rim), the behaviour becomes of plane strain type ( I > II), depending of the inflated pressure.
This homogeneous strain field will be considered in our next work as an experimental surface strain field on a 3-D shape
included in a global FEMU procedure.
Conclusions and perspectives
In this investigation, an efficient methodology was carried out in order to identify the multiaxial Mooney-Rivlin behaviour model
based on the use of several experimental devices coupled with numerical simulations. Experimental data are provided by
optical measurements (bubble shape contour measurement and displacement / strain field by Digital Image StereoCorrelation) on both tensile tests and bubble inflation tests. These experimental data are integrated in an inverse analysis
procedure using Finite Element Method simulations and optimisation loop for identifying the hyperelastic rheological behaviour.
A cost function was formulated in term of (i) points of the shape contour or (ii) displacement fields and reaction forces, and
minimized by a SQP algorithm. Identification based on shape contour succeeds and will be used for study polymer-like
behaviour at blowing temperature. In the case of the identification using the open-hole tensile test, some experimental
problems occur and work have to be done for a better robustness of the FEMU procedure. At this stage, works are in progress
concerning the choice of the investigation area on the specimen, as well as the interpolation procedure of the measured
values at the FE nodes that give no bias or smoothing. This work will be extended on 3-D surface specimens in order to
identify the hyperelastic behaviour of bubbles during pressure inflation.
Acknowledgments
We are grateful to Dr. D. Mercier, J.-P. Arcens, J.-M. Mouys, D. Ade and S. Le Roux for their cooperation in this work.
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