Contribution to the modelling of the viscoelastic behavior of elastomers with Payne effect

Résumé

It is well known that rubber-like materials exhibit nonlinear viscoelastic behavior over a wide range of strain and strain rates confronted in several engineering applications such as civil engineering, automotive and aerospace industries. This is due to their capacity to undergo high strain and strain rates without exceeding the elastic range of behavior. Further, the time dependent properties of these materials, such as shear relaxation modulus and creep compliance, are, in general, functions of the history of the strain or the stress. Therefore, in a wide range of strain, a linear viscoelasticity theory is no longer applicable for such material and new models are required to fully depict the behavior of rubber-like materials for quasi-static and dynamic configurations of huge interest in engineering applications. Despite the multitude of nonlinear viscoelastic models developed over the years, there is a lack of models capable of depicting the nonlinear behavior of rubber-like materials with ease of identification and implementation into commercial software. In this work, a nonlinear viscoelastic model at finite strain is developed to describe nonfactorizable behavior of isotropic incompressible rubber-like materials. The model is developed within the framework of rational thermodynamics and internal state variable approach such that the second law of thermodynamics in the form of Clausius-Duhem inequality is satisfied. From experimental results on Bromobutyl (BIIR) a dependence of the shear relaxation modulus upon strain has been observed and introduced in the model via a strain dependent relaxation times which led to a reduced time similar to the thermorheologically simple material’s formulation. Then, a systematic identification procedure have been developed to identify the model’s parameters. A separation of the instantaneous elastic and viscoelastic contributions to the stress was employed which led to a separate identification of the characteristic functions of the model. This procedure was applied to experimental data and generated data from the Pipkin-Rogers model and a good capacity of the model to predict both static and dynamic behaviors of the material was observed. Thereafter, the nonlinear viscoelastic model was implemented into Abaqus software using a Umat subroutine. To this end, the discrete form of the model was written and the tangent stiffness was calculated (required for the Umat) using the objective rate derivatives of Jaumann. The implementation was validated using homogeneous transformations of simple shear and simple extension for monotonic, sinusoidal and relaxation strain histories. The non vanishing components of the Cauchy stress tensor were calculated for the strain history considered and compared to the numerical results of the model. Finally, a non homogeneous transformation was considered. Namely, the problem of simple v torsion of a hollow viscoelastic cylinder for several strain histories. From the equilibrium equations, the indeterminate pressure arising from the incompressibility was computed and then the components of the Cauchy stress were calculated along the radius of the cylinder. The analytic results showed a total agreement with the simulations performed with the implemented model.

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