Section 6: Material modelling in solid mechanics - GAMM 2012

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28 Section 6: Material modelling in solid mechanics modulus. The Molecular Mechanics method originates in the investigation of the properties of big molecules which due to their size cannot be handled by quantum mechanical methods. In Molecular Mechanics the behavior of the covalent bonds in a Single Wall Carbon Nanotube is represented by a set of potentials. Here every single potential expression describes a corresponding bond deformation. As a result of the description of the bonds by potentials, the force acting between the atoms can be calculated as the derivatives of the potentials with respect to their corresponding displacement variable. Hence, the Carbon Nanotube is modeled by point masses representing the carbon atoms and a set of springs or beam elements representing the bonds. Regarding Multi Wall Carbon Nanotubes this model can be extended by adding the van-der-Waals interactions, described by another potential. During the process of modeling and also during the evaluation of the results we have to rise several challenges. Since the Molecular Mechanics method is a so called semi-empiric method we have to choose between different sets of potentials. Some of these potentials - leading to linear or nonlinear behavior of the interaction forces between the atoms - are investigated regarding their advantages and disadvantages. In order to illustrate this we compare the (theoretical) results available in literature with our numerical results, which we obtain by using the model to conduct a virtual tensile test. The dependance of the Young’s modulus on type and diameter of the Carbon Nanotubes is discussed. Dislocation Dynamics in Quasicrystals Eleni Agiasofitou, Markus Lazar (TU Darmstadt), Helmut Kirchner (Leibniz Institut für neue Materialien, Saarbrücken) In this work, we present a theoretical framework of dislocation dynamics in quasicrystals [1] according to the continuum theory of dislocations. Quasicrystals have been discovered by Shechtman et al [2] in 1982 opening a new interdisciplinary research field. A comprehensive presentation of the current state of the art of this research field, focused on the mathematical theory of elasticity, can be found in the recently published book of Fan [3]. We start presenting the fundamental theory of moving dislocations in quasicrystals giving the dislocation density tensors and introducing the dislocation current tensors for the phonon and phason fields, including the Bianchi identities. In the literature, there exist different versions of generalized linear elasticity theory of quasicrystals. The difference of these versions lies in the dynamics of phonon and phason fields. In the present work, we deal with the elastodynamic as well as the elasto-hydrodynamic model of quasicrystals. According to the first model, the equations of motion are of wave-type for both phonons and phasons while according to the second model the equations of motion for the phonons are equations of wave-type and for the phasons are equations of diffusion-type. Therefore, we give the equations of motion for the incompatible elastodynamics as well as for the incompatible elasto-hydrodynamics of quasicrystals. We continue with the derivation of the balance law of pseudomomentum thereby obtaining the generalized forms of the Eshelby stress tensor, the pseudomomentum vector, the dynamical Peach-Koehler force density and the Cherepanov force density for quasicrystals. Moreover, we deduce the balance law of energy that gives rise to the generalized forms of the field intensity vector and the elastic power density of quasicrystals. The above balance laws are produced for both models. The differences between the two models and their consequences are revealed. The influences of the phason fields as well as of the dynamical terms are also discussed. [1] E. Agiasofitou, M. Lazar and H. Kirchner, Generalized dynamics of moving dislocations in quasicrystals, J. Phys.: Condens. Matter 22 (2010), 495401 (8pp). [2] D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Metallic phase with long-range orienta-
Section 6: Material modelling in solid mechanics 29 tional order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951 – 1953. [3] T. Fan, Mathematical Theory of Elasticity of Quasicrystals and its Applications, Science Press Beijing and Springer-Verlag Berlin, 2011. On inverse form finding based on an ALE formulation Sandrine Germain, Paul Steinmann (Universität Erlangen-Nürnberg) A challenge in the design of functional parts in forming processes is the determination of the initial, undeformed shape such that under a given load a part will obtain the desired deformed shape. Two numerical methods might be used to solve this problem, which is inverse to the standard kinematic analysis in which the undeformed shape is known and the deformed shape unknown. The first method deals with the formulation of an inverse mechanical problem, where the spatial (deformed) configuration and the mechanical loads are given. Hence the objective is to find the inverse deformation map that determines the (undeformed) material configuration. The second method deals with shape optimization that predicts the initial shape in the sense of an inverse problem via successive iterations of the direct problem. In [1] a nodes-based shape optimization approach for elastoplastic materials based on logarithmic strains is presented. An update of the reference configuration is considered in order to avoid mesh distortions, which often occur in nodes-based optimization problems. The principal drawback is the high computational costs. An alternative is an Arbitrary-Lagrangian-Eulerian (ALE) formulation [2,3], which is neither purely Lagrangian (the nodes are not attached to the material) nor purely Eulerian (the nodes are not fixed in space). The nodes are free to move in space independently of the material. In this contribution we review the ALE formulation [2,3] for anisotropic hyperelastic materials and its application in shape optimization. Several examples illustrate the ALE approach in hyperelasticity. Results and computational costs are compared with the ones obtained with the approach in [1]. This work is supported by the German Research Foundation (DFG) within the Collaborative Research Centre SFB Transregio 73. [1] S. Germain and P. Steinmann. Towards form finding methods for a sheet-bulk-metal (DC04), 15th ESAFORM, Key Engineering Materials submitted (2012). [2] E. Kuhl et al., An ALE formulation based on spatial and material settings of continuum mechanics. Part 1: Generic hyperelastic formulation, Comput. Methods Appl. Mech. Engrg. 193 (2004), 4207 – 4222. [3] H. Askes et al., An ALE formulation based on spatial and material settings of continuum mechanics. Part 2: Classification and applications, Comput. Methods Appl. Mech. Engrg. 193 (2004), 4223 – 4245. On the direct connection of rheological elements in nonlinear continuum mechanics Ralf Landgraf, Jörn Ihlemann (TU Chemnitz) The direct connection of rheological elements is a widely used concept to develop material models for one-dimensional deformation processes at small strains. This concept is based on the decomposition of the total strain into several subparts and the formulation of single material laws for
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