\n Besides the well-known landmark models for the hyperelastic response of rubberlike materials, many new hyperelastic constitutive models have emerged over the last decade. Despite many reviews on constitutive modeling or elastomers, it is still a challenging endeavor for engineers to decide for a constitutive model for the specific rubber compound and application. In this work, we have reviewed 44 hyperelastic constitutive models for elastomers and assessed their strength and weaknesses under uniaxial, pure shear, and (equi)biaxial deformations. To this end, we first present a novel parameter identification methodology based on various multi-objective optimization strategies for the selection of the best constitutive models from a given set of uniaxial tension, pure shear, and (equi)biaxial tension experiments. We utilize a hybrid multi-objective optimization procedure using a genetic algorithm to generate multiple initial points for gradient-based search algorithm, Fmincon utility in matlab. The novelty of the approach is (i) simultaneous fitting with variable weight factors for uniaxial, equibiaxial, and pure shear data, and (ii) the sorting of the models based on objective normalized quality of fit metric. For the models incapable of simultaneously fitting the three distinct deformation data, the validity range is assessed through a threshold value for the quality of fit measure. Accordingly, 44 hyperelastic models are sorted with respect to their simultaneous fitting performance to the experimental dataset of Treloar and Kawabata. Based on the number of material parameters, and their fitting performance to experimental data, a detailed discussion is carried out.

On the Performance of Isotropic Hyperelastic Constitutive Models for Rubber-Like Materials: A State of the Art Review

Efficient modeling of filled rubber assuming stress-induced microscopic restructurization

Rubberlike materials exhibit strong rate-dependent mechanical response which manifests itself in creep and relaxation tests as well as in the hysteresis curves under cyclic loading. Unlike linear viscoelasticity, creep and relaxation response of rubber is nonlinear and amplitude-dependent. Within this context, the contribution of this work is three fold. (i) On the experimental side, the characterization of equilibrium and non-equilibrium responses are carried out by means of uniaxial and equibiaxial extension tests. Also performed are the creep and relaxation experiments at various stress and strain levels. (ii) On the theoretical side, we extend the well-known eight-chain model via incorporating a simple yet instrumental tube-constraint term composed of the second invariant into the non-affine network contribution reflecting the ground state equilibrium response. For the non-equilibrium response, we propose a new evolution equation for the creep/relaxation behavior of rubberlike materials based on a relaxation kinetics of a single polymer chain. The geometric non-linearity is incorporated via the finite deformation kinematics based on the multiplicative split of the deformation gradient into elastic and viscous parts, whereas the volumetric and isochoric split of the deformation gradient is entirely discarded. The rheology uses a nonlinear spring responsible for equilibrium elastic response in parallel to n number of Maxwell elements, leading to a generalized Maxwell-Wiechert viscoelastic solid. (iii) The algorithmic implementation of the model features the spectral decomposition of the respective terms and is demonstrated within the context of the finite element method. The developed model is validated by fitting both the elastic and viscoelastic model responses with respect to the experimental data in the sense of uniaxial, (equi)biaxial extensions, and pure shear tests. Relaxation and creep behavior of the model are thoroughly assessed. Also presented in the manuscript is the capability and the performance of the model in the face of a relevant non-homogeneous boundary value problem. The quality of the findings earns the model vast utilization areas from an engineering perspective.

An extended eight-chain model for hyperelastic and finite viscoelastic response of rubberlike materials: Theory, experiments and numerical aspects

Nonlinear finite element analysis is widely used for structural optimization of the design and the reliability analysis of complex elastomeric components. However, high-precision numerical results cannot be achieved without reliable strain energy functions (SEFs) of the rubber or rubber nanocomposite material. Although hyperelastic constitutive models have been studied for nearly 80 years, selecting one that accurately describes rubber's mechanical response is still a challenge. This work reviews 85 isotropic SEFs based on both the phenomenological theory and the micromechanical network theory proposed from the 1940s to 2019. A fitting algorithm which can realize the automatic fitting optimization and determination of the parameters of all SEFs reviewed is developed. The ability of each SEF to reproduce the experimental data of both the unfilled and highly filled rubber nanocomposite is quantitatively assessed based on a new proposed evaluation index. The top 30 SEFs for the unfilled rubber and the top 14 SEFs for the highly filled rubber nanocomposite are presented in the ranking lists. Finally, some suggestions on how to select an appropriate hyperelastic constitutive model are given, and the perspective on the future progress of constitutive models is summarized.

A comparative study of 85 hyperelastic constitutive models for both unfilled rubber and highly filled rubber nanocomposite material

Strain energy functions (SEFs) are used to model the hyperelastic behavior of rubberlike materials. In tension, the stress–strain response of these materials often exhibits three character...

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A constitutive model for both low and high strain nonlinearities in highly filled elastomers and implementation with user-defined material subroutines in abaqus

The micromechanically based network models are known to be superior over the purely phenomenological models in the analysis of unfilled rubber. However, various factors play a crucial role in the selection of the appropriate constitutive model for the analysis of technical rubbers. These are; (i) the number of available experiments under various loading conditions, (ii) maximum stretch level expected in the critical loading scenarios of the rubber component, and (iii) percentage of fillers and additives that might distort the mechanical response from that of an ideal rubber. Moreover, the number of material parameters to be identified should be as low as possible and the parameters should be physically interpretable. 12 years have passed since the celebrated work of Marckmann and Verron [Rubb. Chem. Tech. 79.5, 835-858, 2006] where they have compared 20 hyperelastic constitutive models for rubber. Since then, the rubber community witnessed emergence of many more hyperelastic models. Although the recent reviews have been very useful for the assessment of the strengths and weaknesses of various constitutive models, especially on the modeling of uncrosslinked ideal rubber, it is still a challenging endeavor for engineers to decide the best constitutive model for the specific rubber compound at hand. This paper presents a novel parameter identification toolbox based on various multi-objective optimization strategies for the selection of the best constitutive models from a given set of uniaxial tension, pure shear and equibiaxial tension experiments. The toolbox aims at providing an objective model selection procedure along with the material parameters for the rubber compound at hand. To this end, we utilize the multi-objective optimization using a successive genetic algorithm and gradient based search through the FMINCON utility in MATLAB. The novelty of our approach is (i) simultaneous fitting - the use of variable weight factors for uniaxial, equibiaxial, and pure shear data which accommodates for the (in)capability of the specific constitutive model to the specific deformation modes (ii) the sorting of the models based on an objective normalized quality of fit metric and (iii) simultaneous fittings are conducted for the models that are inherently incapable of catching large deformations, starting from zero stretch, and with 5% stretch increments until the model breaks where its quality of fit value goes above a predetermined set value. This allows identification of their validity region, i.e., the region they can work within an acceptable quality of fit margin. Accordingly, over 40 hyperelastic models are sorted with respect to the experimental data of Treloar and Kawabata.

Yashar Badienia

Papers

On the Performance of Isotropic Hyperelastic Constitutive Models for Rubber-Like Materials: A State of the Art Review

A comparative study on hyperelastic constitutive models on rubber: State of the art after 2006