Comparison of hyperelastic material models in the analysis of fabrics
The Authors
Manuel Julio García Ruíz, CAD/CAM/CAE Laboratory, EAFIT University, Medellín, Colombia
Leidy Yarime Suárez González, CAD/CAM/CAE Laboratory, EAFIT University, Medellín, Colombia
Acknowledgements
Project financed by COLCIENCIAS and EAFIT University. The authors thank the following people and institutions for their collaboration for in this research. EAFIT University (Prof. Dr Eng. Oscar E. Ruiz and MSc Eng Carlos Eduardo López Zapata), Colombian textile industrial – Leonisa S.A (Ing. José Fernando Duque), and the Colombian Council for Science and Technology – Colciencias.
Abstract
Purpose – This work presents a review of the application of hyperelastic models to the analysis of fabrics using finite element analysis (FEA).
Design/methodology/approach – In general, a combination of uniaxial tension (compression), biaxial tension, and simple shear is required for the characterization of a hyperelastic material. However, the use of these deformation tests to obtain the mechanical properties of a fabric may be complicated and also expensive. A methodology for characterizing the fabric employing a different experimental test is presented. The methodology consists of a comparison of the results of the fabric characterization with only a tensile test and the combination of shear, biaxial, and tension experimental tests by using FEA.
Findings – Numerical results of the fabric behavior contribute to estimate the effects of experimental limitations in the material characterization and to select the best fit material model to modeling fabrics. Finally, a comparison of hyperelastic material models is illustrated through an example of a rigid body in contact with a hyperelastic fabric in 3D.
Originality/value – Hyperelastic models are used to characterize textile materials.
Article Type:
Research paper
Keyword(s):
Textile testing; Elastic analysis; Finite element analysis.
Journal:
International Journal of Clothing Science and Technology
Volume:
18
Number:
5
Year:
2006
pp:
314-325
Copyright ©
Emerald Group Publishing Limited
ISSN:
0955-6222
1 Introduction
Linear elastic models assume a linear strain-stress relationship and small deformations. Materials with large elastic deformation (like rubber) need different constitutive models. Mooney (1940) presented a theory of large elastic deformation, Rivlin and Saunders (1951) studied large elastic deformations of rubber, in Blatz and Ko (1962) presented a new strain energy function to the deformation of rubbery materials, Yeoh (1993) proposed a strain-energy function for the characterization of carbon-black filled rubber vulcanizates in 1990 (Ogden, 1972), constructed an energy function for the characterization of rubber-like solids for nonlinear large elastic deformations based on strain energy density functions. Other constitutive relations are based on macromolecular network structure (Arruda and Boyce proposed a new constitutive model for the deformation of rubber materials, in 1992 (Arruda and Boyce, 1993). These models are characterized by a particular form of the strain energy function W. In each of these methods, a set of coefficients must be determined.
The modeling and design of hyperelastic materials consists of the selection of an appropriate strain energy function W, and accurate determination of material constants for such a function. There is no literature about the application of hyperelastic models to the deformation analysis of fabric materials. This paper deals with the study of the deformation of fabrics using different hyperelastic models.
The first section describes some of the most used hyperelastic materials models. The strain energy function for different materials is presented. The second section introduces a procedure to characterize hyperelastic fabric. Its advantages and disadvantages are evaluated and presented.
2 Hyperelastic models
This section introduces the concept of hyperelasticity and mentions some important aspects about hyperelastic models. Hyperelasticity is the capability of a material to undergo large elastic strain due to small forces, without losing its original properties. A hyperelastic material has nonlinear behavior, which means that its deformation is not directly proportional to the load applied. An elastic material is hyperelastic, if there is a scalar function, denoted by W=W(ɛ):R n×n →R called strain energy function (or stored energy function), such that: Equation 1 where, S ij are the components of the second Piola-Kirchhoff stress tensor, W is the strain energy function per unit volume undeformed, ɛ ij are the components of the strain tensor, and ɛ ij are the components of the right Cauchy-Green strain tensor. Throughly algebraical manipulation, equation (1) can determinate the components of the Cauchy (true) stress tensor (σ) Equation 2 where I 1 and I 2 are the principal invariants of the [C] tensor. The three strain invariants of the strain tensor can be expressed as: Equation 3 The strain energy functions of hyperelastic constitutive models such as Mooney-Rivlin, neo-Hookean and Arruda-Boyce are given in the next subsections. They are expressed as a function of strain invariants I 1, I 2, I 3 or in terms of the principal stretches λ 1, λ 2, λ 3 of strain tensor.
In order to deduce the strain energy functions, it is assumed, unless indicated, that the material is isotropic and with constant volume (isometric deformation λ 1 λ 2 λ 3=1). Also, unless indicated, hyperelastic materials are assumed to be nearly or purely incompressible. The most common functions of deformation energy are as follows:
- Mooney-Rivlin model. Mooney and Rivlin proposed a strain energy function W as an infinite series in powers of (I 1−3) and (I 2 −3) of the form: Equation 4 where c ij are constants. For example, the Mooney-Rivlin form with two parameters is: Equation 5
- Neo-Hookean model. Equation (3) shows a series of powers which are usually truncated in the first terms. Then, taking only the first term of equation (3), the neo-Hookean model is obtained: Equation 6
- Ogden model. Ogden (1972) deduced a hyperelastic constitutive model for large deformations of incompressible rubber-like solids. The strain energy is expressed as a function of principal stretches as: Equation 7 with μ r and α r as material constants, which can be determined by experimental tests.
- Yeoh model. The Yeoh (1993) model depends only on the first strain invariant I 1. The strain energy function W is obtained by Equation 8 It applies to the characterization of elastic properties of carbon-black filled rubber vulcanizates.
- Arruda-Boyce model. The constitutive model for the large stretch behavior of rubber elastic materials is presented by Arruda and Boyce (1993). Also sometimes called the eight-chain model because it was derived by idealizing a polymer as eight elastic chains inside a volume element. The strain stress function is based on an eight chain representation of the macromolecular structure of the rubber: Equation 9 where the module G=nkΘ, n is the chain density, k is Boltz-mann's constant, N is the number of rigid links composing a single chain, and Θ the temperature.
- Gent model. The strain energy density in the Gent model (Gent, 1996) is a simple logarithmic function of the first invariant I 1, and involves two material constants, the shear modulus μ and I m which measures a limiting value for I 1−3. Gent proposed strain energy density as: Equation 10
- Blatz-Ko model. An application of finite elastic theory to the deformation of rub-bery materials is given in (Blatz and Ko, 1962). Incompressibility is not assumed. The strain energy function is cast in terms of the constants ν, μ and f, which can be determined experimentally: Equation 11 where ν is similar to the Poisson's ratio, μ is the shear modulus, and f is a material constant. Also, a new set of invariants is defined by J 1=∑λ i 2, J 2=∑λ i −2 and J 3=Πλ i .
Successful modeling and characterization of hyperelastic materials depend on the selection of an appropriate strain energy function, and the accurate determination of coefficients in the function. The next section describes the procedure used to identify the minimum number of standard tests needed to obtain a good characterization of a hyperelastic fabric.
3 Problem description
The types of experimental tests to determine the constants of the hyperelastic model are: uniaxial tension, uniaxial compression, planar shear, biaxial tension, and volumetric test (Figure 1). However, the textile testing equipment utilized to measure these properties has a lot of features, unusual for general-purpose mechanical testing machines. Test method variations reported in the literature and proper interpretation of test results add to the uncertainty. In this section, we discuss standardization of three types of tests and the application of the results of these tests to the numerical modeling by finite element models (FEM).
The simplest deformation mode from the experimental point-of-view, is the uniaxial tension. For this purpose, the norms ASTM 412 (ASTM D412 98a, 2002) for elastomer and rubbers and the norm ISO 527 (ISO 527-5, 1997) for plastics were used. Figure 2 shows the stress-strain curve of the hyperelastic fabric at 20, 80 and 105°C.
3.1 Methodology to determine the constants of the hyperelastic textile
Different levels of experimental test can be used to characterize a hyperelastic fabric. The following procedure was developed to investigate the approximation error when only few of these tests are available. Also a computational experiment was designed to test the accuracy of the models. It consists of a rigid sphere in contact with a hyperelastic fabric as shown in Figure 3(a). The fabric is initially horizontal and it is fixed all around its edges. The solid sphere is then moved a distance towards the fabric.
The results of these analysis undertaken using different hyperelastic material models (Mooney-Rivlin, Ogden, Neo-Hookean, Yeoh, Blazt-Ko and Arruda-Boyce) were determined using input data determined from two different combinations of test data:Case A only uniaxial tension data is employed. Case B shear, biaxial and tension test data are employed. An finite element analysis (FEA) is used to obtain the hyperelastic coefficients of fabric and apply them to the both cases A and B. The procedure that followed to analyze cases A and B is shown in Figure 4. First, the experimental data are introduced and a hyperelastic model is selected. A nonlinear regression routine is used to determine the coefficients of the selected model so as to obtain the model that best fits to the experimental data. The least-squares error to be minimized during data fitting can be based on absolute or relative errors and are defined as follows: Equation 12 The next step consists of comparing the residual of least squared error with an e value, if e is lower than least-squared error, then the constants are taken for modeling the hyperelastic material.
Finally, the FEA is undertaken. The problem was solved by applying nonlinear analysis with FEM (Figure 3(b)). Although the hyperelastic model selected fit the strain-stress data, some models do not converge to the solution. Then, stability of the hyperelastic model was checked, i.e. the stretch ratio must be in a permissible range. If the material is deemed stable, then the strain-stress results for both cases A and B are stored for takes comparison.
3.2 Comparison of the results for the cases A and B
- Case A. Determination of the properties of a hyperelastic material only using experimental data of uniaxial tension.The hyperelastic models analyzed in this case are shown in Table II. Models with a least squared error value greater than ɛ=30 percent are not acceptable for this study. Acceptable error maximum between experimental and fitted data is assumed because this correlation represents few variations in the results of stress state of the fabric. The results of the Mooney-Rivlin and Yeoh models of higher order give the best fit with uniaxial tension. The Arruda-Boyce model also gives an acceptable result. However, the solution of the problem converges only for Yeoh and Arruda-Boyce. Figure 5 shows the results of fitting the hyperelastic constants using only the uniaxial tension test.
- Case B. Determination of the properties of a hyperelastic material using experimental data of uniaxial tension, biaxial and shear tension. Hyperelastic models are compared in Table I. The Mooney-Rivlin model (with 9 parameters) best fits the experimental data. The strain-stress results are shown in Figure 6. Results obtained by using all three types of test data adjust exactly to the real problem.
3.3 Normalized error value
Let the value of stress for case A be defined byσ i A , i=1, … ,n and n is the total number of nodes in the domain. Let σ i A correspondingly be the stress value for case B. The normalized error value e can be calculated using the expression: Equation 13 The error found when employing only the uniaxial tension test data is approximately 15 percent (Table II).
4 Hyperelastic constants to characterize the fabric
Mooney-Rivlin (nine parameters), Yeoh (order 2 and 3), Arruda-Boyce and Ogden constitutive models of hyperelastic fabric have been established on the basis of uniaxial test results (Figure 7). The high-order Mooney-Rivlin and Yeoh models best fit the test data. The Arruda-Boyce and order 1 Ogden models also achieve an acceptable fit. In contrast, Ogden (order 2 and 3), neo-Hookean, Gent, Blazt-Ko, and the lower-order Mooney-Rivlin and Yeoh models show large differences between the numerical and test data.
5 Conclusions
The derivation of models such as the Mooney-Rivlin and Gent is difficult because of the amount of experimental data required to obtain the model coefficients. To improve the accuracy of predictions, it is best to use experimental data from a range of experimental tests (uniaxial, biaxial, and planar tension). The Arruda-Boyce, neo-Hookean and Yeoh models offer a physical interpretation and provide a better description of general deformation modes when the parameters are based only on one test. In all cases, it is best to obtain experimental data over the range of strain of interest (this is especially true of the Ogden and polynomial models), and to select the model coefficients carefully to ensure stability.
Two different combinations of input data were evaluated: uniaxial tension only and combined uniaxial, biaxial and planar tension, as described in Section 3.1. The normalized error calculated shows that there is little difference between the accuracy of pre-dictions made using uniaxial data only and those made using the combined uniaxial, biaxial, and planar tension data. Therefore, it appears that the biaxial and planar tests can be omitted with little effect on accuracy, thereby simplifying testing requirements. The use of only uniaxial test data will induce an error of 15 percent. In general, it is better to obtain data from several experiments involving different kinds of deformation. The range of deformation should be restricted to the interest of application, in order to determine the model coefficients. For example, the Ogden and Mooney-Rivlin models of higher order present some instabilities when only limited test data are available.
Arruda-Boyce and Yeoh (order 3) models provide the best fit when the constants are calculated using only the uniaxial tension test.
Numerical simulations of forming processes can be powerful tools for materials selection, tool design, and process optimization. A critical component of these simulations is the correct representation of the material response, as well as robust, standardized methods to characterize the materials and validate the simulations. It should be noted that the experimental and numerical approaches discussed here are applicable to the forming processes. For example, the behavior of a hyperelastic fabric in a thermoforming process could be simulated by a exact characterization, as is shown in Figure 8(b).
Equation 1
Equation 2
Equation 3
Equation 4
Equation 5
Equation 6
Equation 7
Equation 8
Equation 9
Equation 10
Equation 11
Equation 12
Equation 13
Figure 1Stress-strain experimental curves for an elastomer
Figure 2Stress-strain curves for hyperelastic fabric at different temperatures
Figure 3Model of the fabric in contact with rigid body
Figure 4Methodology of the analysis hyperelastic materials
Figure 5Comparison of material models case A
Figure 6Stress-strain fitting with Mooney-Rivlin (9 parameters, T = 20°C)
Figure 7Comparison of hyperelastic material models
Figure 8Modeling of a thermoformed mold of hyperelastic fabrics
Table IResults of fitting using three tests data
Table IIResults of fitting using uniaxial tension data
References
Arruda, E.M., Boyce, M.C. (1993), "A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials", Journal of the Mechanics and Physics of Solids, Vol. 41 No.2, pp.389-412.
ASTM D412 98a (2002), "Standard test methods for vulcanized rubber and thermoplastic elastomers-tension", Annual Book of ASTM Standards, December, pp.1-14.
Blatz, P.J., Ko, W.L. (1962), "Application of finite elastic theory to the deformation of rubbery materials", Transactions of the Society of Rheology, Vol. VI pp.223-52.
Gent, A.N. (1996), "A new constitutive relation for rubber", Rubber Chem. Technol., Vol. 69 pp.59-61.
ISO 527-5 (1997), "Plastics determination of tensile properties-part 5: test conditions for unidirectional fibre-reinforced plastic composites", Annual Book of ASTM Standards, ISO, Geneva, pp.1-9.
Mooney, M. (1940), "A theory of large elastic deformation", Journal of Applied Physics, January, Vol. 11 pp.582-92.
Ogden, R.W. (1972), "Large deformation isotropic elasticity on the correlation of theory and experiment for incompressible rubberlike solids", Proc. R. Soc. Lond. A., .
Rivlin, R.S., Saunders, D.W. (1951), "Large elastic deformations of isotropic materials, VII, experiments on the deformation of rubber", Transactions of the Royal Society of London. Series A (Mathematical and Physical Sciences), Vol. Vol. 243 pp.251-88.
Yeoh, O.H. (1993), "Some forms of the strain energy function for rubber", Rubber Chem. Technol., Vol. 66 pp.754-71.
Corresponding author
Manuel Julio Garcıa Ruız can be contacted at: mgarcia@eafit.edu.co