Section 6: Material modelling in solid mechanics - GAMM 2012

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-