A genetic algorithm for the characterization of hyperelastic materials

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Abstract

This work deals with the characterization of a hyperelastic material and the subsequent validation in different stressed states. The well-known three-parameter Mooney–Rivlin model is chosen for the sake of simplicity. In order to obtain the mechanical properties of this material, a specimen is tested using tensile forces. Once the tests are performed, the material constants are determined using a genetic algorithm to fit the experimental curve. An accurate fitness function is defined and the procedure to obtain the generations is described. Finally, with the aim to model and validate the results in a more complex setting, physical tests are performed in combination with numerical studies and FEM simulations.

Introduction

Hyperelastic constitutive laws are used to model materials that respond elastically when subjected to very large strains. They account both for nonlinear material behaviour and large shape changes. The main applications of the theory are to model the rubbery behaviour of a polymeric material, and to model polymeric foams that can be subjected to large reversible shape changes (e.g. a sponge).

Even if this kind of materials is widely used in industry, the difficulties to obtain an accurate mathematical characterization limit the possibilities to generate feasible numerical models that predict their behaviour. As a consequence, the virtual Finite Element (FE) models using hyperelastic materials are in some cases unable to reach accurate results. Even if nowadays there exist a good number of models (see, e.g., [1], [2], [5], [7], [8], [17], [18], [19], [21]), theirs material constants usually are not simple to be obtained. The simple test performed in order to find the material data is a uniaxial tensile test [24]. Once the test is carried out, the obtained results are fitted by a three-parameter Mooney–Rivlin model [16], [23] using a fully incompressible assumption for the material [1], [15]. The first challenge in this work is to get an accurate model characterization for a complex material using only simple tests on a specimen. Once the characterization is obtained, it will be used in a FE code in order to solve a more complex problem involving different stress states.

In order to solve characterization problems, several different methods are available to find the value of parameters in hyperelastic models. Most of authors use least square fitting methods [13], very often based on iterative processes which require an initial data to initialize the method and also differentiation of the objective function. In our case, we use Genetic Algorithms (GA) to perform the searching of parameters in order to characterize the material. This technique does not require differentiation and, although there are other available and more sophisticated methods which do not require it [26], GA are methods oriented to global optimization and are less sensitive to initialization as other techniques. Moreover, using GA we do not need to provide an initial data for the parameters to run the algorithm, although a parametric domain must be defined. Finally, using this method we are able to include multi-objective optimization if needed (for example in order to use different tests for the characterization) in a very simple way.

Genetic Algorithms represent a highly efficient search procedure for the determination of global extrema of multivariable functions, imitating the patterns of genetic reproduction in living organisms. They were first introduced by Goldberg [10] and Holland [11] and differ from conventional search algorithms in the following aspects: (1) they do not work directly with the parameter set; rather, they manipulate strings of characters representing the parameters themselves, (2) they consider many points in the search space simultaneously, (3) they use random choice to guide their search, and (4) they require no derivative information. During the last twenty years, a large number of papers have been published dealing with problems solved with this type of algorithms (see, e.g., [6], [9], [12], [14], [20], [22], [25], [27], [29] and the numerous references cited therein). They work by successive modifications of a collection (often referred to as population) of parameter combinations in the search space, using a binary coded representation termed chromosome. The initial set of chromosomes evolves through a number of operators which will be detailed in the next sections, so that subsequent generations produce more and more chromosomes in the neighbourhood of the optimum, defined through a given figure of merit, or fitness, which should be previously stated.

In this work, in order to obtain the constitutive parameters of the Mooney–Rivlin model, a genetic algorithm is used to find theirs suitable values and the better fitness as possible with the material model. Once this is fitted and the hyperelastic behaviour is reproduced, new different tests are performed, based on the characterization done, and a more complex FE model is built. The new problem involves stresses in several directions (not only uniaxial tensile stresses) and the obtained characterization will be used in a FE code to solve a large deformation problem.

The paper is outlined as follows. In Section 2, the classical three-parameter Mooney–Rivlin model is presented. Then, in Section 3 the genetic algorithm implemented for the calculation of these parameters is described, providing details concerning the population initiation, the genetic codification, the selection operators and the fitness evaluation. The results obtained with this algorithm are shown in Section 4, including a basic example to show the accuracy with a known solution and a real case. Finally, the data obtained with the second example are used in Section 5 for the simulation of the deformation of a rubber plate under extensional forces.

Section snippets

The Mooney–Rivlin model

In the classical linear elasticity theory used to model metallic behaviour at small strains, the constitutive relation between stress and strain remains linear. This assumption could be right when talking about metals suffering from stresses under their yield stress but, when talking about rubbers, foams or, for example, biological tissues, the relationship between stress and strain takes a more complex form. In the case of elastomers, strong non-linearities are observed in their stress-strain

The genetic algorithm

The search of the suitable parameters for the Mooney–Rivlin model, in order to reproduce the behaviour of a given specimen, is the most important feature in this work. In order to obtain the better fitness of the curve, a genetic algorithm will be used.

This algorithm is inspired in the Darwin’s theory based on the evolution of the species. On one hand, according to this theory, the individuals better fitted to their environment are able to survive and theirs genes could be transferred to the

Results for the genetic algorithm

Once the algorithm is implemented with all the previous considerations, it can be used to fit one experimental curve.

FE analyses

Finally, in order to prove the utility of the fitting, the obtained parameters will be used in a FE model with the commercial code ANSYS. The algorithm was originally thought to obtain the Mooney–Rivlin parameters according to the definition given by ANSYS for the elastic potential. Hence, the obtained values may be introduced directly in the software. The simulated problem consists in a rubber plate 2  mm thick with a hole at the center. The plate will be a square with 160 mm. The mesh is

Conclusion

This study shows the possibility to use genetic algorithms to fit experimental data, in order to obtain the necessary parameters to model hyperelastic materials in a FE software. During the study, a functional genetic algorithm was designed to fit Mooney–Rivlin parameters according to real data. This kind of algorithms has the main advantage that no differentiation is needed in order to establish a searching direction as to Newton methods, and so the fitness function could be

Acknowledgments

The work of J.R. Fernández was partially supported by the Ministerio de Economía y Competitividad under the research project MTM2015-66640-P (with FEDER Funds).

J.A. López-Campos, A. Segade and J.A. Vilán gratefully acknowledge the funding by Xunta de Galicia, Spain, under the program Grupos de Referencia Competitiva with Ref. GRC2015/016.

J.A. López-Campos also acknowledges the funding from the University of Vigo under the program Axudas predoutorais with Ref. 00VI131H641.02.

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