ABSTRACT
This work aimed to implement and compare two competitive procedures for the identification of hyperelastic material parameters. Formerly, experimental tests have been conducted on fluorosilicone rubber specimens in equal-biaxial tension; the cruciform shaped-specimens underwent heterogeneous large strain distributions, which were captured by Digital Image Correlation technique; while load cells grabbed the force signals. The experimental data have been used in two different inverse techniques for material parameters estimation: the first method was based on "classic" FE model updating, which uses only the global quantities measured during the experiments (i.e. forces and boundary displacements) to define the error function to be minimized; the second method was still based on FE model updating, but experimentally determined strain fields was compared with the numerical ones in order to define a more adequate cost function; third technique was based on the Virtual Fields Method, which naturally takes into account the real strain distributions and permits to overcome the experimental difficulties represented by non-symmetry of the test/specimen, non-uniform boundary conditions, friction. The results of the three procedures are showed and compared in terms of accuracy, transferability, computational efficiency and practicability.
INTRODUCTION
Modeling the mechanical response of elastomeric materials is commonly carried out within the framework of hyperelasticity [1]. Identifying constitutive parameters that govern such a type of law is classically carried out with homogeneous tests, namely uniaxial tensile extension, pure shear and equibiaxial extension [2-3]. The aim of the present work is to develop an experimental and numerical procedure in order to characterize hyperelastic materials by means of only one single heterogeneous test in which the three different types of strain states exist. In that case, the parameters obtained are directly a weighted average of those that would be obtained from the three different tests described above. Several reliable techniques for full-field strain measurement are nowadays available (Moire [4-5], Electronic Speckle Pattern Interferometry [6], grid methods [3], Digital Image Correlation [89]), and an important research topic is to find well-suited methods to use the great amount of available data in a sharp way. The classical method for inverse problems is the so called "model updating" [10-12], mainly based on FEM numerical models. Results of experiments, e.g. in terms of load-displacement data, are compared whit results obtained with the FEM model of the experiment, using an appropriate constitutive model of the material. If
numerical and experimental data do not match properly, material parameters of the constitutive law are iteratively varied until the best set of parameters is reached. Even if this approach to inverse problems is widely known and adopted, it shows some important drawbacks. Firstly, it usually needs a great quantity of computation, that means time; inverse problems on non-linear materials using a standard PC and FEM model updating may require weeks of computation. The other disadvantage is about the assumptions made in FEM model about load distribution, constraints, boundary conditions, geometry, etc.. Usually these assumptions are rough because of the lack of information and they often make the model not faithful to the reality. Furthermore, small quantity of experimental data, usually global entities like load and displacement are exploited. With the rising of new full-field measurement techniques, much more information is available, especially about local strains at each acquired point of the specimen surface, and proper methods should be used to take advantage of this. In this context, a novel technique known as Virtual Fields Method (VFM) [13-14], developed in recent years, seems to be very effective and appropriate; its applications involve the characterization of a large variety of materials, from homogeneous to orthotropic and anisotropic (e.g. composites), from linear to non-linear (e.g. elasto-plasticity, visco-plasticity, hyperelasticity) [15-16], In this paper, a numerical procedure is proposed to find the best material parameters of Ogden and 2nd order Mooney-Rivlin hyperelastic models. The tested material is a common fluorosilicone rubber in its virgin state: it means that the stabilization procedure of cyclic loadings required to accommodate the Mullins softening effect has not been applied before testing. Hence, the data are referred to what is called ‘primary path’ in the pseudo-elastic models. Nevertheless, the same procedures can be applied on the stabilized material without significant differences. Global load and deformation data of the surface of the specimen are used to calculate the terms and coefficients of VFM equations. Two independent virtual fields are involved in the procedure to take advantage of as many data as possible; so, we can write two equations for different instants of the test in order to take into account the nonlinearity of the material and obtaining an overdetermined system of equations whose unknowns are the coefficients of the material in the constitutive models mentioned above. The parameters obtained are then compared with those obtained from the classic inverse method based on FEM, not only in terms of global load-displacement curve, but also in terms of strain and stress distributio.
EXPERIMENTAL TESTS
The experimental setup consists of a electromechanical tensile machine (Zwick ®Z050 model), a CMOS camera with resolution 1280×1024 (Pixelink® BU371F model) and a biaxial testing machine for cruciform shaped samples as illustrated in Fiq. 1a.
Fig. 1 cross-shaped specimen
Fig. 2 dispersion of experimental points on invariant plane
A cruciform sample is glued onto ‘L’-shaped brackets, which are put in proper clamps in the biaxial testing machine; two load cells allow to read load values in the vertical and horizontal directions. Though this system is designed with different possibilities of adjustment, in this work only a 45° diagonal configuration is used, because the shape of the sample allows to generate a strain field sufficiently heterogeneous. Fig.2 shows the dispersion of experimental points obtained through experimental test; x-axis represents the first strain invariant, while y-axis represents the second strain invariant. Since Mooney-Rivlin hyperelastic model expresses the strain energy function as a polynomial combination of these two invariants, it is useful to cover as much area as possible on invariant plane; it can be seen that just one test made it possible to cover most of the space between the two curves corresponding to ideal uniaxial and equibiaxial tests. In figure 3 is shown a set of images of the specimen in the last step of the test, superimposed with color maps of the distribution of strain ex, ey, exy, e-j, e2, £3. Figure 4 shows the load-displacement curve read by the two load cells; the x axis indicates the vertical displacement (almost coincident with the horizontal one). It can be seen that the horizontal force becomes slightly smaller than the vertical force from a certain point; this aspect, probably due to failure in the gluing of the horizontal grips and/or to an asymmetry in the deformation of the specimen.
Fig. 3 Strain maps of the specimen at the last step of the test (112%)
Fig.4 Experimental load-displacement curves
APPLICATION OF VFM
In a generic inverse problem, where the geometry, the strain field and the applied load are given, the parameters governing the constitutive equations have to be identified. However, these unknown parameters that depend on the material and on the form of the adopted constitutive equations are not directly related to measured strain field; this means that a closed-form solution is not available. Besides constitutive laws, also other equations of continuum mechanics of solids must be verified at each point of the specimen; they are the equilibrium and congruence equations. Among recent methods that exploit full-field deformation data to solve inverse problems, one of the most prominent techniques is represented by the VFM [13-14], It is based on the principle of virtual work; for a given solid of volume V, this principle can be written as follows:
where a is the first Piola-Kirchhoff stress tensor, s is the virtual displacement gradient tensor, T is the distribution vector of loadings acting on the boundary, dV is the part of solid boundary where the loading forces are applied, u is the virtual displacement vector,/is the distribution of volume forces acting on V , p is the density and a is the acceleration. An important feature is that the above equation is verified for any kinematically admissible virtual field («/*, £*). In a general case the constitutive equation can be written as a=g{s), where g is a function of the actual strain components and is defined involving the material dependent parameters. Considering also a static case, whit no acceleration, and no volume forces, equation (1) reduces to
It is important to note that, if the actual strain field is heterogeneous, any new virtual field in equation (2) leads to a new equation involving the constitutive parameters. Hence, the VFM is based on equation (2) written with a given set of virtual fields, and the resulting system of equations is used to extract the unknown constitutive parameters. The Ogden model adopted in this paper requires knowledge of six parameters; we chose 2 different virtual fields used for 25 test steps, leading to at an over-determined system of 50 equations in six unknowns. In Figure 5 are represented the two displacement maps corresponding to the two virtual fields used, the blue color indicates no displacement, while red indicates the maximum displacement.
Fig. 5 displacements associated with two virtual fields
Considering a point on the surface of the specimen of x and y coordinates in a reference system with origin at the midpoint of the specimen, the first virtual field applied is expressed by the following equation:
and consists of an expansion / contraction of the specimen with the edges stuck; this virtual field is equivalent to the imposition of that the global balance of the work of internal actions, including shear, in the absence of external work; the virtual deformation field is expressed by:
and the equation (2) becomes:
For the second virtual field, represented in the right map of fig. 5, the area occupied by the specimen was divided into five different areas, and has been assigned a different displacement law for each area, respecting the constraints of congruence between neighboring areas; in this case the displacements of the edges are not null but the displacements are such that it generates a virtual work by external forces vertically and horizontally. However it’ important to note, that the upper and lower edges of the virtual specimen have only vertical displacements, and the left and right edges have only horizontal displacements; in this way, the VFM method is able to filter out any effects that are unknown or unmeasured in the clamps areas.
For brevity we give here only the equations that describe the field of virtual displacement of the left region (left):
which, by differentiation, lead to a virtual strain field described by:
In this case the equation (2) becomes:
Writing equations (5) and (8) for N different test steps, leads to an overdetermined system of 2xN equations where the unknowns are the constants of the material present in the analytical expression of terms Oj. These stress components are expressed as a function of the constitutive model chosen; for hyperelastic models, the principal Kirchhoff stresses are expressed by:
where W is the strain energy function expressed by:
or by
VFM RESULTS AND COMPARISON WITH FEM
The coefficients of Ogden and Mooney-Rivlin identified with the two inverse methods implemented are shown in Table 1. It can see that it is actually quite difficult, especially for the Mooney-Rivlin model, compare the parameters obtained, while a certain regularity, at least for the algebraic signs, is observed for the model of Ogden. This is probably related to the polynomial nature of both the constitutive laws, and could suggest a further difficulty due to the presence of multiple solutions, i.e. set of parameters different from each other but which can provide a similar matching of experimental results. Interesting considerations can be deduced from graphs of the load-displacement curves and from comparison of the displacement maps and deformation.
Table 1: Ogden coefficients
|
coeff. |
MU |
VFM |
 |
1.934 |
1.687 |
 |
-2.755 |
-0.511 |
 |
2.278 |
0.146 |
 |
1.029 |
0.647 |
 |
2.987 |
-0.545 |
 |
3.241 |
4.176 |
 |
2.07 |
2.80 |
Table 2: Mooney-Rivlin coefficients
|
coeff. |
MU |
VFM |
|
C10 [MPa] |
0.3953 |
1.0114 |
|
C01 [Mpa] |
0.3354 |
-0.1940 |
|
C20 [Mpa] |
0.1278 |
-0.2447 |
|
C11 [Mpa] |
-0.4098 |
0.2370 |
|
C02 [Mpa] |
0.3097 |
-0.0552 |
|
err [%] |
1.98 |
2.51 |
Fig. 6 a global load-displacement curves exp-FEM
Fig. 6 b global load-displacement curves exp-VFM
It can be seen from figures 6a and 6b that both VFM and MU coefficients can correctly reproduce the experimental global variables, with standard error shown in Table 1. Nevertheless, for a proper comparison of the two methods, next figure 7 shows the results of a FEM simulation, carried out using the two sets of different coefficients (VFM and MU) for the Ogden model, in terms of strain maps.
Fig. 7 strain maps compured by FE simulation using coeff. of table 1: MU at left, VFM at right
It can be seen in the images to the left of Figure 7 that the deformation and the form assumed by the specimen simulated with the coefficients of the MU method diverges from with experimental observations; in particular, it seen the first straight edge of the specimen undergoes a lateral contraction much greater than the experimental evidence (Fig. 3). The VFM technique obviously does not have this kind of problem because the shape and strain distributions experimentally measured are input values; this positive aspect is also present in the simulation performed with the coefficients obtained by the VFM method (right images in Fig. 7), where the deformed shape is similar to the experimental one.
Fig. 8 comparison of load-displacement curves
Returning to the issue of multiple solutions, figure 8 shows the load-displacement curves obtained with a FEM model in which the material coefficients obtained by the VFM were used; the diversity of these curves compared to those of figures 6, although not totally excludes the possibility that we can achieve similar results with different sets of coefficients, shows that the coefficients previously identified by the two methods actually describe different mechanical behaviors. The difference between the mechanical behavior estimated with the VFM and the FEM can be attributed to the phenomena that finite element model does not consider, mainly non perfect boundary conditions and asymmetry in the test. These defects in the FE modeling leads to erroneous estimates of the stiffness of the material in various steps of test, also because only the global load was used, and the FE model was not imposed to deform locally as the real specimen.
CONCLUSIONS
Through a simple device specially designed, it was possible to perform biaxial tensile test on cross-shaped rubber specimens; the deformation of the specimen, observed experimentally by the technique of digital image correlation, was highly heterogeneous, covering almost the entire the region in the plane of invariants between the uniaxial and equi-biaxial limits. The strain distribution obtained experimentally allowed to apply the VFM technique to identify the material parameters using two of the most common hyperelastic constitutive models. In particular, it was shown that these parameters differ significantly from those determined by a standard FEM based inverse procedure. The sets of parameters obtained by the two methods, although providing a good representation of load-displacement curves, describe the mechanical behavior of materials in very different ways. This shows that the traditional technique of model updating based on FEM introduces simplifications that can lead to erroneous assessments of the real behavior of the material.
