Keywords

Introduction

The constitutive parameters of hyperelastic models are generally identified from three homogeneous tests, basically the uniaxial tension, the pure shear and the equibiaxial tension. From about 10 years, an alternative methodology has been developed [1,2,3,4], and consists in performing only one heterogeneous test as long as the field is sufficiently heterogeneous. This is typically the case when a multiaxial loading is applied to a 3 branch [1] or a 4-branch [2] cruciform specimen, which induces a large number of strain states at the specimen surface. The Digital Image Correlation (DIC) technique is generally used to characterize the full kinematic field at the specimen surface. The induced heterogeneity is analysed through the distribution of the biaxiality ratio and the maximal eigen value of the strain. Thus, a large number of experimental data is provided for identifying the constitutive parameters of the considered model.

Several methods have been recently developed to identify parameters from experimental field measurements, typically the finite element updating method (FEMU), the constitutive equation gap (CEGM), the virtual fields method (VFM), the equilibrium gap method (EGM) and the reciprocity gap method (RGM). These methods are fully reviewed in [5].

In the present study a new methodology is proposed in order to minimize the cost function in the FEMU approach. The optimization algorithm used is based on the Particle Swarm Optimization (PSO) algorithm and the artificial smart PageRank algorithm used by the famous search engine Google. This algorithm enables us to the minimize the full kinematic field differences by modifying the constitutive parameters, while minimizing the CPU calculation time. Even though the final objective is the identification of complex constitutive models, i.e. a large number of constitutive parameters, the two-parameter Mooney’s model [6] is presented in this paper to illustrate the methodology.

Experimental Setup

The material considered here is a carbon black filled natural rubber. The specimen geometry is shown in Fig. 21.1. It is a 105 mm long and 2 mm thick cruciform specimen. Figure 21.2 presents an overview of the experimental setup composed of a home-made biaxial testing machine and an optical camera. The machine is composed of four independant RCP4-RA6C-I-56P-4-300-P3-M (IAI) electrical actuators. They were driven by a PCON-CA-56P-I-PLP-2-0 controller and four PCON-CA (IAI) position controllers. The actuators were controlled by an in-house LabVIEW program. Two cell loads, whose capacity is equal to 1094 N, store the force variation in the two perpendicular directions. In the present study, one equibiaxial loading was carried out in such a way that the specimen’s centre was motionless for the displacement measurement to be easier. The displacement and loading rate were set at 70 mm and 150 mm/min respectively for the four independent actuators.

Fig. 21.1
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Specimen geometry (dimensions in mm)

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Fig. 21.2
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Home-made biaxial testing machine

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Images of the specimen surface at increasing stretches were stored at a frequency equal to 5 Hz with a IDS camera equipped with a 55 mm telecentric objective. The charge-coupled device (CCD) of the camera has 1920 × 1200 joined pixels. The Digital Image Correlation (DIC) technique is used to determine the displacement field at the specimen surface. The software used for the correlation process was SeptD [7], and a uniform cold lighting was ensured by a home-made LED lamp. The spatial resolution, defined as the smallest distance between two independent points was equal to 4 pixels corresponding to 0.343 mm. The Region Od Interest (ROI) used for the digital correlation is represented in Fig. 21.3.

Fig. 21.3
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Region of interest for the DIC technique

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Numerical Model

A finite element calculation is performed by assuming plane stress state and material incompressibility. The four-node PLANE182 ANSYS element is used. The mesh is made of 9600 nodes, and 9353 elements. A biaxial traction load is obtained by prescribing the same displacement of 70 mm on the four branches of the specimen. The two-parameters hyperelastic Mooney model is used for the calculation. The values of the constitutive parameters are changing at each iteration of the optimization process, as described in the next section.

As the spatial resolution between the predicted and the measured kinematic field was different, the experimental kinematic field was fitted by a polynomial-based function. In this way, the predicted kinematic field was compared, for each node, with the experimental field at the same position of the specimen. With the fitting method applied, the difference between the experimental field and the polynomial-based function was less than 0.2 mm whatever the point considered.

Metaheuristic Optimization Strategy

The aim of the optimization process was here to find the constitutive parameters for the numerical kinematic field to fit the experimental one. The cost function f was the squared difference between the experimental and the numerical fields, considering the force too, as follows:

$$ f=\min {\sum}_{i=1}^N\frac{1}{N}{\left(\frac{U_{x,\mathit{\exp}}-{U}_{x, num}}{U_{x,\mathit{\exp}}}\right)}^2+\frac{F_{exp}-{F}_{num}}{F_{exp}} $$

where N is the number of nodes in the numerical ROI, Ux,exp is the polynomial-based experimental displacement, Ux,num is the numerical horizontal displacement, Fexp is the experimental horizontal force, and Fnum is the numerical horizontal force.

The optimization algorithm used is an adaptation of the classical Particle Swarm Optimization algorithm. In this version, all the particles are influenced by all the others, by considering this influence to be adapted as a function of the respective performance of the particles. The population of PSO particles is then considered as a Markov chain, in which the particles are the nodes, and the transition probabilities between them are the links between them. For each particle, the PageRank value—that is the steady state of the considered Markov chain—is given by the following equation (2). In this way, the PageRank value of each particle is deduced from its performance compared to the best one.

$$ {\pi}_{target}^T=\left|\frac{fitness\left({\boldsymbol{G}}_{\boldsymbol{best}}\right)\times 100}{fitness\left({\boldsymbol{G}}_{\boldsymbol{best}}\right)- fitness\left(\boldsymbol{P}\right)+\varepsilon}\right| $$

It is then possible to deduce the transition connectivity matrix C giving the influence of all the particles on all the others by using a pseudo-random process. The classical equations of PSO are then modified, weighing the influence of all the particles by using the components of C, as follow:

$$ {\boldsymbol{V}}_i^{t+1}=\omega \times {\boldsymbol{V}}_i^t+{c}_1\times {\mathit{\operatorname{rand}}}_1\times \left({\boldsymbol{P}}_{i, best}^{t+1}-{\boldsymbol{X}}_i^t\right)+{c}_2\times {\mathit{\operatorname{rand}}}_2\times {\sum}_{j=1}^n{\boldsymbol{C}}_{ij}\times \left({\boldsymbol{P}}_{j, best}^{t+1}-{\boldsymbol{X}}_i^t\right) $$
$$ {\boldsymbol{X}}_i^{t+1}={\boldsymbol{X}}_i^t+{\boldsymbol{V}}_i^{t+1} $$

where \( {\boldsymbol{V}}_i^{t+1} \) is the speed of the ith particle at iteration t + 1, c1 and c2 are confident parameters, ω is the inertia weight, \( {\boldsymbol{X}}_i^{t+1} \) is the position of particle i at iteration t + 1, rand1 and rand2 are random numbers given in [0, 1], \( {\boldsymbol{P}}_{j, best}^{t+1} \) is the personal best position of particle i at iteration t + 1, and C is the transition connectivity matrix of the considered Markov chain. This Inverse-PageRank-PSO algorithm is fully described in [8].

Results and Discussion

As the particles are initially randomly defined, the optimization was launched 10 times, to compare the obtained solutions, and be sure that the global minimum of the cost function was reached. The convergence curves of the 10 launched optimization calculation are represented in Fig. 21.4. The obtained values of the cost function and constitutive parameters are given in Table 21.1.

Fig. 21.4
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Convergence curves of the 10 optimization calculation launched

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Table 21.1 Obtained results
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The validation of the optimized results is checked by comparing the experimental displacements and efforts in the sample with the numerical optimized one. In the final numerical model, the values of the constitutive parameters have been set to the mean of the obtained optimized values found in the 10 different calculations launched. Figure 21.5 shows the difference between the experimental polynomial-based kinematic field, and the optimized numerical one, for every point in the ROI.

Fig. 21.5
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Comparison between the kinematic fields after the optimization process

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This difference is presented in a quantitative way in Fig. 21.6 showing the difference between the two fields for every point of the ROI. One can note that the difference is always less than 6.5% of the experimental kinematic field. Considering the force, the experimental value was 176.02 N, while the numerical value obtained with the optimized values of the constitutive parameters is 176.37 N, which leads to a difference up to 0.4%.

Fig. 21.6
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Difference between the kinematic fields for every point of the ROI

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Conclusions

This work is proposing a new inverse identification method based on the coupling of experimental kinematic fields retrieved by DIC, and the using of a PSO-based parametric optimization algorithm. Experimental and numerical kinematic fields are compared to finally be fitted through the optimization process, while the constitutive parameters are smartly modified. Applied on a Mooney model, this process is able to find the constitutive parameters reproducing the mechanical response of the specimen, while minimizing the number of optimization iterations. The constitutive parameters found by the optimization process are actually giving a numerical model that retrieves precisely the entire kinematic experimental field.