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Constitutive Modeling of Elastomers – Accuracy of Predictions and Numerical Efficiency

J. S. Bergström

jorgen@polymerFEM.com

ABSTRACT: The mechanical behavior of elastomers is characterized by rate- and temperature-dependence, and the stress-strain response is known to be strongly non-linear. These experimental features are well recognized and important, and have been extensively studied for more than 50 years. The understanding of the micromechanisms controlling the macroscopic mechanical behavior is much more recent, and advanced modeling tools allowing for accurate predictions of arbitrary deformation histories have only started to become available during the last few years. This paper outlines the current state of the art in finite element modeling of elastomers, and exemplifies the predictive capabilities of modern constitutive theories for filled elastomers.

Introduction In today’s competitive business environment it has become increasingly important to improve,

streamline, and shorten the design process of technical and non-technical products. This trend is relevant for many industries, including the tire industry, where the need to produce the best possible product using the least amount of resources is paramount for prosperity. As a result there is an increased need to perform and rely upon computational tools early in the design cycle, and rely as little as possible on trial-and-error and real mechanical experiments. One of the most important computational tools for this purpose is the Finite Element (FE) method. The FE method is an established procedure that enables predictions of deformations and stresses of products in normal or accelerated loading environments. Although the FE procedures continue to evolve, there are numerous FE packages that are commercially available and capable of performing advanced simulations.

There are three types of input that are needed to perform a FE simulation: (1) the geometry of the part(s) of interest; (2) the applied loading and boundary conditions; and (3) the material behavior of each of the different materials. The first two of these required inputs are often easy to accurately specify by CAD software and knowledge of the loading environment. The third input, the specification of the material models, is typically the most difficult and challenging part of performing FE simulations. The goal of this paper is to survey a select set of material models (constitutive equations) that are suitable for simulating tire materials. Commercial FE packages typically come with a selection of material models that are suitable for elastomers, under certain loading conditions. Most of the built-in models are hyperelastic models, which are effective at predicting the average behavior of an elastomer, but do not capture rate-effects, dynamic loading, hysteresis, or Mullins effect. Recently, Ogden and Roxburgh [1] and Qi and Boyce [2] have proposed models for augmenting the classical hyperelastic models to include predictive capabilities of the Mullins effect [3]. The rate-effect, dynamic loading, and hysteresis behavior of elastomers can be captured using the Bergstrom-Boyce (BB) model [4-5].

This study compares predictions from these different material model combinations with experimental data for two filled elastomers. Two different chloroprene rubbers are studied, but the applied theory and results are applicable also for other elastomers. The theory of the different models is briefly reviewed, and the predictive capabilities of each of the different models are

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quantified by the coefficient of determination, R2. When selecting a material model for a FE simulation, it is not only the predictive capability that is of interest but also the numerical efficiency and computational time. For this reason, the simulation times resulting from using the different material models are also quantified. This information enables intelligent decisions related to the tradeoff between simulation accuracy and required simulation time.

Experimental Data The material models examined in this work were compared to experimental data obtained

from chloroprene rubber filled with 7 vol% and 25 vol% carbon black N600. These materials were used and quantified in earlier studies [4-5]. In the experiments the samples were sized according to ASTM D575, and the experiments were performed at room temperature using strain control loading. A summary of the experimental results for these two types of chloroprene rubber is given in the next two sections.

Chloroprene Rubber with 7 vol% Carbon Black

The material was loaded in displacement control following the strain history shown in the inset in Fig. 1. The specimen was loaded using four load-unload cycles with gradually increasing strain amplitude, using an applied strain rate of 0.01/s. The figure shows that the material exhibits significant amounts of hysteresis during the cyclic loading, and that the amount of Mullins softening is only about 10%1.

Chloroprene Rubber with 25 vol% Carbon Black

A second chloroprene rubber with higher carbon black concentration was loaded using the same applied strain history (see the inset in Fig. 2). This figure shows that the material response is characterized by large hysteresis and significant amount of strain hardening (stiffening) at large compressive strains. For this material, the amount of Mullins softening is approximately 25%1.

Material Models There are three major classes of material models that can be used to simulate the response of

elastomeric materials: hyperelastic models, hyperelastic models incorporating the Mullins effect, and advanced time-dependent models. In this study, the following material models have been studied: the Yeoh model [6], the eight-chain model [7], the Yeoh model with Ogden-Roxburgh Mullins [1] effect, the eight-chain model with Ogden-Roxburgh Mullins effect, the Yeoh model with Qi-Boyce Mullins effect [2], the eight-chain model with Qi-Boyce Mullins effect, the Bergstrom-Boyce model [4-5], the Bergstrom-Boyce model [4-5] with Ogden-Roxburgh Mullins effect, and the Bergstrom-Boyce (BB) model with Qi-Boyce Mullins effect. All of these models have been discussed in detail in the references cited above and only brief summaries of the individual model components are included. Note that the models for Mullins effect can be activated in combination with any of the hyperelastic models or the BB-model.

1 The amount of Mullins effect was calculated from the stress level at a strain of –0.4 during the first and forth loading cycle.

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Yeoh Model

The Yeoh model [6] is a hyperelastic material model that is based on a representation of the strain energy density in a 3-term expansion of the first strain invariant, I1. For incompressible uniaxial loading 2

1 2 /I λ λ= + , where λ is the applied stretch. The true stress is given by:

( ) ( )2210 20 1 30 1

12 2 3 3 3C C I C Iσ λ

λ = − + − + −

, (1)

where C10, C20, and C30, are material parameters.

The Yeoh model has been proven to accurately predict the equilibrium response of elastomers in different loading modes. Additionally, the Yeoh model is easy to apply and provides robust (Drucker stable) predictions. The Yeoh model is available in all major Finite Element (FE) codes.

Eight-Chain Model

The eight-chain model [7] is a physically inspired hyperelastic model that is based on a non-linear representation of the strain energy density in terms of the first invariant I1. For uniaxial incompressible loading the true stress is given by:

1 2 1chain

chain lockLµ λσ λ

λ λ λ− = −

, (2)

where and lockλ are material parameters, ( )2 2 / / 3chainλ λ λ= + , and ( )1L x− is the inverse

Langevin function [7], where ( ) coth( ) 1/L x x x= − . The eight-chain model is always

unconditionally stable and has been shown to give accurate predictions of large-strain multiaxial deformation states. The eight-chain model is available in all major FE codes.

Ogden-Roxburgh Model for Mullins Effect

To account for the Mullins effect, Ogden and Roxburgh [1] proposed an extension of hyperelastic models in which the strain energy density, U, is taken to be a function not only of the applied deformation state, but also an internal state variable, , tracking the damage state in the material. For incompressible loading, the following form is used:

( ) ( )1 2 3 1 2 3, , , , ,U Uλ λ λ η η λ λ λ= , (3)

where iλ are the principal stretches. In the Ogden-Roxburgh (OR) model the damage variable is taken to evolve with applied deviatoric strain energy as follows:

max

max

11 erf dev dev

dev

U Ur m U

ηβ

−= − + , (4)

where r, , and m are material parameters, erf(x) is the error function, devU is the current

deviatoric strain energy density, and maxdevU is the evolving maximum deviatoric strain energy

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density for the material point in its deformation history. The OR-model can be applied to any hyperelasticity model.

Qi-Boyce Model for Mullins Effect

A different approach to account for the Mullins effect has been developed by Qi and Boyce [2]. In their model the virgin elastomer is considered to consist of two phases: soft domains and hard domains. The Qi-Boyce (QB) model introduces a state variable to track the transformation of hard domains to soft domains with the applied deformation history. For incompressible loading, the following form is used:

( ) ( )11 fU v U I= − ⋅ (5)

The volume fraction of hard domains, vf, evolves with the applied chain stretch as follows:

( ) ( )max

max

1exp

chainf ff ff fi

lock chain

Av v v v

λ

− Λ − = − −

− Λ , (6)

where vff is the final volume fraction of hard domains, vfi is the initial volume fraction of hard domains, and max

chainΛ is the maximum chain stretch the during the deformation history of the material point. The Qi-Boyce model for Mullins effect is not yet available as a built-in feature of commercial FE codes, but has been implemented by the author as a user material subroutine (UMAT). Most FE codes have the capability to augment the built-in material models with user-defined material models, typically coded as external Fortran subroutines that are linked into the FE software during simulations. The use of external UMATs can provide significant advantages in certain circumstances. The QB-model can be applied to any hyperelastic model.

Bergstrom-Boyce Model

The hyperelastic models presented above provide useful information regarding the equilibrium behavior of elastomers, but they are all inherently incapable of predicting rate-dependence, hysteresis, and the response during cyclic loading. To overcome these shortcomings, Bergstrom and Boyce [4-5] developed a physically inspired model that captures most of the experimentally observed characteristics of the rubbery response. The Bergstrom-Boyce (BB) model is based on a conceptual representation of the molecular microstructure into two networks acting in parallel: one network gives the equilibrium behavior, and one network gives the time-dependent deviation from the equilibrium state. The key of the BB-model is a reptation-inspired equation capturing the rate of viscoelastic flow as a function of the deformation state and the applied stress state:

( )1m

Cchain

base

τγ λτ

= −

� , (7)

where γ� is the viscoelastic flow rate in shear, C, m, and baseτ are material parameters, and τ is the effective shear stress. Additional details of the BB-model are given in the original publications [4-5].

The BB-model is a built-in material model in ABAQUS/Standard. It has been implemented by the author as a user material subroutine (UMAT) for ABAQUS and LS-DYNA, and also extended to include the two Mullins effect models discussed above.

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Results The quality of the predictions of the different material models presented in the previous

section was assessed by direct comparison with the experimental data for the chloroprene rubber. For this comparison, the appropriate material parameters for each material were first obtained by a trial-and-error procedure followed by a non-linear minimization approach [8]. The results from this evaluation are presented in the next two subsections.

Chloroprene Rubber with 7 vol% Carbon Black

A direct comparison between the experimental data for the chloroprene rubber with 7 vol% CB and the predictions from the eight-chain model is presented in Fig. 3. The corresponding results for the combined eight-chain and OR-model is shown in Fig. 4, and the results from the combined eight-chain and QB-model is shown in Fig. 5. The predictions from the combined BB and QB-models are shown in Fig. 6.

Chloroprene Rubber with 25vol% Carbon Black

The experimental data for the chloroprene rubber with 25 vol% CB are compared in Figs. 7 through 9 with the eight-chain model, the combined eight-chain and OR-model, and the combined BB and QB-model, respectively.

Conclusions The different material models studied in this work all provide reasonable first-order

predictions of the experimentally observed behavior of the two types of chloroprene rubber. It is clear, however, that some of the models provide a much more detailed and accurate description of the material response, and therefore are more suitable for FE simulations. Direct comparisons between the predictive qualities are presented in Table 1 and in Fig. 10. The (1-R2) term shown in Fig. 10 can be interpreted as the average error of the model predictions. For each material model, the given data is the average response for the two types of chloroprene rubber. It is interesting to note that the Yeoh model and the eight-chain model give predictions of similar accuracy. The relative error of the predictions of these pure hyperelastic material models is about 25%. Augmenting either of these hyperelastic models with one of the models for Mullins effect reduces the relative error in the predictions to about 7%. Note that the hyperelastic models with Mullins effect (Figs. 4, 5, and 8) do not capture time-dependence and hysteresis. The history-dependent response that is depicted in these figures is caused by damage accumulation and not viscoelastic flow. Also, even though the OR-model and the QB-model for Mullins effect are significantly different in background and implementation, both give similar quality of predictions.

The most accurate model in this study is the Bergstrom-Boyce (BB) model combined with one of the models for Mullins effect. Both the OR-model and QB-model work well and give predictions with a relative error of about 2%. The BB-model accurately predicts [4-6] the response of elastomers after the Mullins effect has been removed. It is shown here that the BB-model combined with a Mullins effect model can accurately predict the complete set of experimental behavior of the elastomers.

When performing FE analysis of elastomeric components it is not only the accuracy that is important but also the computational costs and total simulation time. To examine the numerical efficiency of the different material models a simple uniaxial benchmarking simulation was performed. The results from this benchmarking study are presented in Table 2 and in Fig. 11. For completeness, some of the material models were benchmarked both with the built-in model in

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ABAQUS/Standard and a user material subroutine (UMAT) implementation. Some of the more advanced models are not available as a standard feature of commercial FE codes, for these models only the results from the UMAT simulation are given. Figure 11 shows that combining the OR-model or QB-model for Mullins model with a hyperelastic model does not cause a reduction in the required simulation time. The figure also shows that for the hyperelastic models where both a UMAT and a built-in model for ABAQUS exist, the UMAT-based simulation is about 35% slower than the built-in ABAQUS implementation. The main reason for this is that the UMAT is using a less efficient algorithm to calculate the Jacobian stiffness matrix needed for the implicit simulation. Figure 11 also shows that the built-in model in ABAQUS for the BB-model is very inefficient, it requires at least twice the run time compared to the UMAT implementation of the same model. It is important to note that the increase in simulation time between the hyperelastic models and the BB-model with Mullins effect is only 14%. The largest difference in simulation time between the most advanced UMAT simulation and the fastest built-in hyperelastic model is about 55%. This is a rather small difference considering that the BB-model combined with a Mullins effect model gives predictions that have 10 times smaller error than the traditional hyperelastic models, and 3 times smaller error than the hyperelastic models combined with a Mullins effect model. Hence, using an advanced material model can enable simulations with outstanding accuracy, at a relatively small computational cost. A future study will focus on other elastomers of interest in tire applications.

References [1] Ogden, R. W., Roxburgh D. G., “A Pseudo-Elastic Model for the Mullins Effect in Filled

Rubber,” Proceedings of the Royal Society of London, Series A, Vol. 455, 1999, pp. 2861-2877.

[2] Qi H. J., Boyce M. C., “Constitutive Model for Stretch-Induced Softening of the Stress-Strain Behavior of Elastomeric Materials,” Journal Mech. Physics Solids, Vol. 52, 2004, pp. 2187-2205.

[3] Mullins, L., “Softening of Rubber by Deformation,” Rubber Chem. Technol., Vol., 42, pp. 339-362.

[4] Bergström, J. S., Boyce M. C., “Mechanical Behavior of Particle Filled Elastomers,” Rubber Chem. Technol., Vol 72, 1999, pp. 633-656.

[5] Bergström, J. S. Boyce M. C., “Large Strain Time-dependent Behavior of Filled Elastomers,” Mech. Mater., Vol. 32, 2000, pp. 627-644.

[6] Yeoh, O. H., “Some Forms of the Strain Energy Function for Rubber,” Rubber Chem,. Technol., Vol. 66, 1993, pp. 754-771.

[7] Arruda, E. M., Boyce M. C., “A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials,” Journal Mech. Physics Solids, Vol. 41, 1993, pp. 389-412.

[8] Bergström, J. S., Rimnac, C. M., Kurtz, S. M., “An Augmented Hybrid Constitutive Model for Simulation of Unloading and Cyclic Loading Behavior of Conventional and Highly Crosslinked UHMWPE,” Biomaterials, Vol. 25, 2004, pp. 2171-2178.

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TABLE 1 - Summary of R2 values for predictions of different constitutive models for two chloroprene rubbers with different amounts of carbon black.

Material Model R2-value for CR

7vol% CB R2-value for CR

25vol% CB Yeoh model 0.825 0.659

8-chain model 0.822 0.677

Yeoh model, Ogden-Roxburgh Mullins effect 0.946 0.924

8-chain model, Ogden-Roxburgh Mullins effect 0.935 0.933

Yeoh model, Qi-Boyce Mullins effect 0.943 0.897

8-chain model, Qi-Boyce Mullins effect 0.939 0.905

BB-model 0.975 0.928

BB-model, Ogden-Roxburgh Mullins effect 0.986 0.968

BB-model, Qi-Boyce Mullins effect 0.990 0.962

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TABLE 2 - Run time for the different constitutive models for an ABAQUS/Standard benchmarking test case.

Material Model Normalized Run

Time

Yeoh model (ABAQUS built-in) 1.01

Yeoh model (UMAT implementation) 1.35

8-chain model (ABAQUS built-in) 1.00

8-chain model (UMAT implementation) 1.37

Yeoh model, Ogden-Roxburgh Mullins effect (ABAQUS built-in) 1.03

Yeoh model, Ogden-Roxburgh Mullins effect (UMAT implementation) 1.36

8-chain model, Ogden-Roxburgh Mullins effect (ABAQUS built-in) 1.03

8-chain model, Ogden-Roxburgh Mullins effect (UMAT implementation) 1.39

Yeoh model, Qi-Boyce Mullins effect 1.39

8-chain model, Qi-Boyce Mullins effect 1.39

BB-model (ABAQUS built-in) 3.04

BB-model (UMAT implementation) 1.52

BB-model, Ogden-Roxburgh Mullins effect 1.56

BB-model, Qi-Boyce Mullins effect 1.55

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FIG. 1 – Uniaxial compression data for chloroprene rubber with 7 vol% of CB. The figure shows the material response during the first four load cycles.

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FIG. 2 – Uniaxial compression data for chloroprene rubber with 25 vol% CB. The figure shows the material response during the first four load cycles.

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FIG. 3 – Comparison between experimental data for chloroprene rubber with 7 vol% CB and predictions from the eight-chain model.

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FIG. 4 – Comparison between experimental data for chloroprene rubber with 7 vol% CB and predictions from the eight-chain model with Ogden-Roxburgh Mullins effect.

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FIG. 5 – Comparison between experimental data for chloroprene rubber with 7 vol% CB and predictions from the eight-chain model with Qi-Boyce Mullins effect.

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FIG. 6 – Comparison between experimental data for chloroprene rubber with 7 vol% CB and predictions from the BB-model with Qi-Boyce Mullins effect.

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FIG. 7 – Comparison between experimental data for chloroprene rubber with 25 vol% CB and predictions from the eight-chain model.

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FIG. 8 – Comparison between experimental data for chloroprene rubber with 7 vol% CB and predictions from the eight-chain model with Ogden-Roxburgh Mullins effect.

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FIG. 9 – Comparison between experimental data for chloroprene rubber with 7 vol% CB and predictions from the BB-model with Qi-Boyce Mullins effect.

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FIG. 10 – Summary of relative error of different constitutive models to predict the behavior of two types of chloroprene rubber.

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FIG. 11 – Summary of normalized simulation time of a three-dimensional benchmarking case using different material models.