Section 6: Material modelling in solid mechanics - GAMM 2012

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2 Section 6: Material modelling in solid mechanics and incremental algorithmic variational principles. We demonstrate the modeling capabilities and algorithmic performance by means of representative numerical examples for multislip scenarios in fcc single crystals. Numerical Implementation of a Three-Dimensional Continuum Dislocation Microplasticity Theory Stephan Wulfinghoff, Thomas Böhlke (KIT) The modelling of the mechanical behaviour of micro-devices can not be established by classical continuum mechanical plasticity models without internal length scale. Depending on the system size, ab-initio simulations or Discrete Dislocation Dynamics serve as mature tools to predict the mechanical response of very small systems. Anyhow, the gap between these discrete and classical continuum mechanical methods is not yet filled. Besides phenomenological gradient plasticity models, dislocation density based approaches have emerged which account explicitly for dislocation transport and line length increase. The presentation is based on the kinematical continuum mechanical dislocation framework of Hochrainer et al. [1] which can be considered as a generalization of Nye’s theory [3]. The theory transfers the kinematics of three-dimensional systems of discrete dislocations to a continuum mechanical representation in terms of continuously distributed dislocations. An averaged version of the theory by Hochrainer et al. [2] (see also Sandfeld et al. [4]) leads to a significant reduction of the computational simulation effords and will mainly be addressed. The kinematical dislocation theory is coupled to a crystal plasticity framework. The presentation will focus on the numerical implementation of the theory, especially on promising schemes for the numerical coupling of the set of PDEs. Finite Element simulation results demonstrate the performance of the implementation in three-dimensional applications. [1] T. Hochrainer, M. Zaiser and P. Gumbsch, A three-dimensional continuum theory of dislocations: kinematics and mean field formulation. Philosophical Magazine 87 (2007), 1261–1282. [2] T. Hochrainer, M. Zaiser and P. Gumbsch, Dislocation transport and line length increase in averaged descriptions of dislocations. arXiv:1010.2884v1 [3] J. F. Nye, Some geometrical relations in dislocated crystals. Acta Metallurgica 1 (1953), 153–162. [4] S. Sandfeld, T. Hochrainer, M. Zaiser and P. Gumbsch, Continuum modeling of dislocation plasticity: Theory, numerical implementation, and validation by discrete dislocation simulations. J. Mater. Res. 26 (2011), 623–632. Rigorous derivation of a dissipation for laminate microstructures Sebastian Heinz (WIAS Berlin) We study models for finite plasticity in the framework of rate-independent evolutionary systems. The corresponding incremental minimization problems, in general, admit no solutions due to the creation of microstructure, see [1]. We focus on the case where the microstructure is made of simple laminates only. We show in a mathematical rigorous way how the incremental minimization problem can be relaxed and identify the relaxed dissipation as the one introduced in [2]. [1] C. Carstensen, K. Hackl, A. Mielke. Non-convex potentials and microstructures in finite-strain plasticity, Proc. Royal Soc. London, Ser. A 458 (2002), 299 – 317.
Section 6: Material modelling in solid mechanics 3 [2] K. Hackl, D. M. Kochmann. Relaxed potentials and evolution equations for inelastic microstructures. In B. Daya Reddy, editor, IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media, pages 27 – 39. Springer-Verlag, 2008. Thermodynamically and variationally consistent modeling of distortional hardening: application to magnesium Baodong Shi (Helmholtz-Zentrum Geesthacht), Jörn Mosler (Helmholtz-Zentrum Geesthacht, TU Dortmund) To capture the complex elastoplastic response of many materials, classical isotropic and kinematic hardening alone are often not sufficient. Typical phenomena which cannot be predicted by the aforementioned hardening models include, among others, cross hardening or more generally, the distortion of the yield function. However, such phenomena do play an important role in several applications in particular, for non-radial loading paths. Thus, they usually cannot be ignored. In the present contribution, a novel macroscopic model capturing all such effects is proposed. In contrast to most of the existing models in the literature, it is strictly derived from thermodynamical arguments. Furthermore, it is the first macroscopic model including distortional hardening which is also variationally consistent. More explicitly, all state variables follow naturally from energy minimization within advocated framework. A viscosity-limit approach to the evolution of microstructures in finite plasticity Christina Günther, Klaus Hackl (Universität Bochum) Material microstructures in finite single-slip crystal plasticity occur and evolve due to deformation. Their formation is not arbitrary, they tend to form structured spatial patterns. This hints at a universal underlying mechanism, in the same manner as the minimization of the global energy governs the behavior of purely elastic materials. For non-quasiconvex potentials, the minimizers are small scale fluctuations, related to probability distributions of the deformation gradients, which can be found by the relaxation of the potential. As in the approach of D.Kochmann and K.Hackl, we use a variational framework, focusing on the Lagrange functional. For the treatment of the laminates, the potentials have to be adapted. The new approach rests on the introduction of a small smooth transition zone between the laminates in order to avoid a global minimization. This makes the evolution equations more handable for numerical calculations. We present explicit time-evolution equations for the volume fractions and the internal variables. We outline a numerical scheme and show some examples. S6.2: Polymers and Elastomers I Tue, 13:30–15:30 Chair: Mikhail Itskov, Vladimir Kolupaev S1|01–A1 Constitutive modelling of chemical ageing Alexander Lion, Michael Johlitz (Universität der Bundeswehr München) In order to model chemical ageing of rubber a three-dimensional theory is proposed. The fundamentals of this approach are different decompositions of the deformation gradient in combination with an additive split of the Helmholtz free energy into three parts. Its first part belongs to the volumetric material behaviour. The second part is a temperature-dependent hyperelasticity model which depends on an additional internal variable to consider the long-term degradation of the primary rubber network. The third contribution is a functional of the deformation history and a further internal variable; it describes the creation of a new network which remains free of stress
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