Modellierung der Versprödung von elastomeren Werkstoffen
Final Report Abstract
The constitutive model developed in this research project contributes significantly to a realistic and reliable description of the fracture/rupture behaviour of aged elastomers. The research work focused on the derivation and implementation of an advanced material description accounting for elastomeric characteristics. The idealized entropic elasticity concept based on a statistical mechanics approach together with a volumetric-isochoric split of the free energy where the volumetric part is used as a penalty term to enforce incompressibility is a well established concept in rubber elasticity. The introduced model elaborates this notion of elasticity by introducing an energetic part to the free energy function responsible for bond distortion. The volumetric part of the free energy function, which acts as a penalty term in standard rubber elasticity models, is included in the energetic part of the free energy function. Direct implementation of an already existing bond potential in a serial connection enables us to account for the energetic contribution to elasticity from moderate up to large stretches below rupture. Besides, it directly includes bond rupture in terms of a certain bond energy and bond force at micro-level as material parameters. For the proposed rubber elasticity model, a novel algorithmic setting is derived and, subsequently, successfully implemented into the in-house finite element code. The algorithmic setting depends on the micro-macro transition in the kinematical context of the micro-sphere model of Miehe et al. . Moreover, the micro-scale equilibrium iterations are performed in order to realize the entropic and energetic contributions, respectively. In regard to ageing of rubber components, two observations are common: hardening in shear modulus and decrease in elongation at break. These two mechanisms of oxidative ageing commonly destroy the characteristic properties which make the rubber components popular in industrial applications. Hitherto, focus of research has been mainly on the life-time estimation of rubber components. The preceding research financed by DFG deals with the phenomenon known as Difuusion Limited Oxidation, which mainly states that oxygen depletion in the interior part of rubber components takes place as a result of surface hardening accompanied by reduced permeability. For specimens with higher than a threshold thickness, oxidative ageing stays as a surface phenomenon. Coupled mechanical-difuusion simulations revealed results in line with the experimental results on the material profile of aged rubber components. However, this work is successful in explaining one aspect of the morphological changes taking place, namely, the increase in shear modulus. The principle of mass conservation introduced as a constraint on the rubber network that enables us to explain reduced elongation at break, the second mechanism observed in aged rubber components. The proposed constitutive model leads to a non-convex potential. Bond rupture initiated in terms of material softening induces convergence problems in the global residual solution phase of the finite element method. Alternative to path-following algorithms, viscous regularization can be used in order to circumvent convergence problems encountered. Rubber-like materials exhibit highly viscoelastic material behaviour. This effect should be considered in the process zone of the crack within the micro model. Within this context, a novel algorithmic treatment for the nonlinear viscoelastic model proposed by Bergström & Boyce was described and, subsequently, successfully implemented into a finite element code. The developed scheme uses the well known predictor-corrector algorihm used traditionally in elastoplastic materials. Further information is provided in Dal & Kaliske. Moreover, a modified version of the recent large-strain viscoelasticity model of Miehe & Göktepe is adopted in order to account for the viscoelastic nature of fracture and to stabilize the global solution within the context of finite element method. The process zone of the dissipative materials shows a distinct character. In order to account for the Mullins’ type softening effects characteristic of filled rubber, the ansatz of the most recent contribution of Göktepe & Miehe has been adopted in the context of the micro-sphere model. The current research states that the aged rubber consists of shorter average chain length and higher chain density. On the other hand, network alteration due to oxidation does not alter the main characteristics of rubber-like elasticity. With proper choice of material parameters, the proposed model can be used for both aged and unaged rubber components. This makes the model applicable for a molecular description of rubber fracture. In addition to the presented model, further endeavours to model the fracture behaviour of rubber will focus on viscoelastic nature of fracture. The energy release rate of rubber-like materials is known to be very sensitive to rate effects. Classical viscoelastic formulations do not account for bond-breakage of superposed dangling free chains. However, additional to relaxation of neighbouring chains at the process zone, two possible scenarios exist for a dangling chain at the crack-tip. If the crack tip velocity is smaller than the relaxation speed of the dangling chain, the superimposed free chains will slip and the crack proceeds with less energy. Under high deformation rates, the free chains cannot disentangle and new crack surfaces form at the expense of breakage of ground state strong network and superimposed chains. This molecular description of viscoelastic fracture has to be considered in the future research. Current research focused on the elastomeric components with altered physical properties. The alteration process is considered to occur under unloaded configuration. In reality, rubber structural components are subjected to ageing under service loads. Long term relaxation of stretched rubber has been known since decades. Simultaneous chain-scission reactions are responsible for long term relaxation, whereas recombination reactions lead to modulus hardening. Long term relaxation effects accompanied by recombination reactions under deformed configuration affect the unloaded configuration leading to a permanent shape change of the components which is known as permanent set. The permanent set of the rubber-metal mounts used in automobile and high speed trains, is an example to this phenomenon. These components are subjected to long term relaxation effects with permanent set, which cannot be explained with conventional rubber viscoelasticity models. The currently developed model and its algorithmic setting can readily transferred as a material routine into commercial FE software. Simulation of rubber components at a certain stage of the alteration process can be simulated and possible surface cracks can be traced. The model, different from fracture mechanics approach, does not necessitate an existing crack in the FE model. The ultimate failure investigation of rubber components can be performed by the developed model.
Publications
- [2006]: An Approach to the Modeling of Physical Ageing in Rubbery Polymers. Proceedings in Applied Mathematics and Mechanics, 6: 363–364
Dal, H.; Kaliske, M.
- [2007] Finite Nonlinear Viscoelastic Modelling and Fracture Mechanical Investigations. ANSYS Conference & 25. CADFEM Users Meeting, Dresden
Kaliske, M.; Dal, H.; Näser, B.; Schmidt, J.
- [2007] Formulierung und Simulation bruchmechanischer Eigenschaften viskoelastischer Werksto?e. 11. Problemseminar ”Deformation und Bruchverhalten von Kunststo?en”, Halle, CD-ROM
Näser, B.; Dal, H.; Kaliske, M.
- [2007] Fracture mechanical investigations of rate-dependent inelastic materials at large strains. Proceedings in Applied Mathematics and Mechanics, 7: 1090103-1090104
Näser, B.; Dal, H.; Kaliske, M.
- [2007] Numerical determination of the energy release rate for endochronic plastic material. 2nd GACM Colloquium on Computational Mechanics, TU Munich Dal, H.; Kaliske, M. [2008]: A micro-macro approach to the failure of rubber-like materials. Proceedings in Applied Mathematics and Mechanics, 8: 10413-10414
Morgner, S.; Kaliske, M.; Näser, B.; Dal, H.
- [2007]: Computational aspects of Bergström- Boyce ?nite viscoelasticity model. in Boukamel, A., L. Laiarinandrasana, S. Méo, E. Verron (Editors): Constitutive Models for Rubber V, pages 241–248. Taylor & Francis Group, London
Dal, H.; Kaliske, M.; L. Nasdala
- [2007]: Fracture of Viscoelastic Materials. in Boukamel, A., L. Laiarinandrasana, S. Méo, E. Verron (Editors): Constitutive Models for Rubber V, pages 185–190. Taylor & Francis Group, London
Kaliske, M.; Näser, B.; Dal, H.
- [2008]: A Micromechanical model for failure analysis of rubber-like materials. World Congress on Computational Mechanics, Venice, CD- ROM
Dal, H.; Kaliske, M.
- [2009]: Time-dependent cohesive zone modelling for discrete fracture simulation. Engineering Fracture Mechanics
Geissler, G., M. Kaliske
(See online at https://doi.org/10.1016/j.engfracmech.2009.09.013) - [2009]: A micro-continuum-mechanical material model for failure of rubber-like materials. in Heinrich, G.; Kaliske, M.; Lion, A.; Reese, S. (Editors): Constitutive Models for Rubber VI, Taylor & Francis Group, London
Dal, H.; Kaliske, M.; Nasdala, L.
- [2009]: A micro-continuum-mechanical material model for failure of rubber-like materials: Application to ageing induced fracturing. Journal of the Mechanics and Physics of Solids 57, 1340-1356
Dal, H., Kaliske, M.
- [2009]: Bergström-Boyce model for nonlinear ?nite rubber viscoelasticity: Theoretical aspects and algorithmic treatment for FE method. Computational Mechanics 44, 809-823
Dal, H., Kaliske, M.
- [2009]: Failure Analysis of Elastomers within the Framework of Continuum Mechanics. 12. Problemseminar ”Deformation und Bruchverhalten von Kunststo?en”, Merseburg, CD-ROM
Dal, H.; Kaliske, M.
- [2009]: Fracture mechanical behaviour of visco-elastic materials: Application to the so-called dwell-e?ect. Zeitschrift für Angewandte Mathematik und Mechanik 89, 677-686
Näser, B., Kaliske, M., Dal, H., Netzker, C.
- [2009]: The cohesive crack tip model within the ?nite element method: Implementation, enhancements, applications. Institut für Statik und Dynamik der Tragwerke, Technische Universität Dresden, Dissertation
Geißler, G.
- [2009]: Thermo-visco-elasticity of ?bre-reinforced elastomers at ?nite strains. Int. Rubber Conference, 29 June - 2 July 2009, Nürnberg
Dal, H.; Kaliske, M.
