Thermo-mechanical behaviour of rubber materials during vulcanization
Introduction
The numerical simulation of production processes plays an important role in the improvement of product quality, in reducing the product development time and in generating a deeper understanding of industrial processes. Although the performance of hardware and software used in industrial simulation tasks has increased continuously during the last years, there is still a huge need for suitable material models and for reliable procedures to determine material parameters for refined constitutive models. In process simulation, like the production of rubber sealings that is considered in this work, usually coupled field equations have to be solved and often the material behaviour cannot be considered to be constant during the process. This requires the introduction of complex material models that include inelastic effects with a strongly rate-dependent behaviour and the dependency on additional variables like temperature, state of chemical reactions, etc.
In this work we will develop a constitutive framework for the computational prediction of the production process of rubber sealings including the associated parameter identification procedure from a minimum number of experimental tests. Please refer to Fig. 1 for an overview of the production process. The process starts with the still uncured material that has just left the die of the extruder. Then due to the applied heating, the vulcanization process is initiated. This also yields a deformation of the rubber structure in the heating lines due to gravity forces. The constitutive model should be applicable to predict the final shape of the rubber sealing. Since the state of vulcanization (curing) can be predicted during the heating process, adjustment of the heating parameters is possible in order to optimize the curing process and the tolerances of the final shape of the rubber sealings. During the vulcanization process chemical and electrochemical reactions can occur. However, this paper is focused on chemical reactions only.
For this purpose a chemical reaction model is developed in Section 2 that describes the vulcanization process on the basis of internal variables Coleman (1964) and Coleman and Gurtin (1967). These variables can be used within a continuum mechanical frame and, thus, allow to model the vulcanization within a computational analysis. Afterwards, in Section 3 a suitable material model is introduced that shows visco-elastic effects, plastic effects and that includes the dependence on temperature and the state of the vulcanization process. Experiments that are needed to determine the set of material parameters of this model are described in Section 4. Experimental results are shown and used for an exemplary identification of material parameters.
Section snippets
Chemical reactions
Rubber material mainly consists of a large amount of long polymer chains, stochastically distributed and orientated in space. In the uncured state, shown in Fig. 2a, these chains are not connected by cross-links between each other and movement between them is possible. On a macroscopic basis this can be observed as inelastic (plastic) behaviour that is intentionally used during the molding process of rubber products. Nevertheless, physical links and entanglements between the molecules and links
Material model
The well known nonlinear stress-strain relationship of rubber materials in the large strain range is usually adequately described by a hyperelastic material model expressed in terms of a strain energy potential. This approach is sufficient when restricting to static and quasi-static problems and when looking at unfilled rubber compounds only. Some popular examples for such strain energy functions are the Neo-Hooke, the Mooney–Rivlin, the Arruda–Boyce, the Yeoh or the Ogden material model. All
Experimental results and parameter identification
In this section we describe the experiments that are needed to determine all constitutive parameters introduced so far for the chemo-thermo-mechanical material equations. Due to the complexity of the experiments involving the curing process in dependency on temperature and mechanical state we targeted to use only a minimal number of different experimental setups. This is the vulkameter test equipment, see e.g. Haupt and Olt (1989), which is commonly applied in rubber industry and a special
Conclusion
In this paper we have developed a constitutive model for the coupled chemo-thermo-mechanical process of vulcanization. This material model can be applied to predict the vulcanization time and the associated state as well as the mechanical response (deformations and residual stresses) within a rubber part. For these the governing field equations and the evolution equations which are highly nonlinear have to be solved using a numerical method. In this connection the authors have developed a
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