Section 6: Material modelling in solid mechanics - GAMM 2012

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32 Section 6: Material modelling in solid mechanics [1] Homberg, W., Dau, J., Damerow, U. Combined Forming of Steel Blanks with Local CFRP Reinforcement, in: G. Hirt, A. E. Tekkaya (Eds.), steel research int., Special Edition: Proceedings of the 10th International Conference on Technology of Plasticity, Wiley-VCH, Weinheim, 2011, pp. 441-446 [2] Menzel, A., Modelling and Computation of Geometrically Nonlinear Anisotropic Inelasticity, Dissertation, University of Kaiserlautern, 2002 [3] Tschoegl, N.W. The Phenomenologial Theory of Linear Viscoelastic Behaviour, Springer Verlag, 1989 [4] Lion, A. and Höfer, P., On the phenomenological representation of curing phenomena in continuum mechanics, Arch. Mech. 59, 2007 On the axially compressed multiple walled carbon nanotubes Ligia Munteanu (Institute of Solid Mechanics of Romanian Academy) A new approach is proposed in this paper, which combines elements of Toupin-Mindlin strain gradient theory and the Molecular Mechanics to include the inlayer van der Waals atomistic interactions for axially compressed multiple walled carbon nanotubes. The neighboring walls of a multiwalled nanotube are coupled through van der Waals interactions, and the shell buckling would initiate in the outermost shell, when nanotubes are short. The load-unloaded-displacement curve, the critical buckling and the appropriate values for elastic moduli are obtained. The theoretical results obtained show a good agreement with the experimental data reported by Waters, Gudury, Jouzi and Xu (2005). The size dependence of the hardness with respect to the depth and the radius of the indenter is also investigated. Results show a peculiar size influence on the hardness, which is explained via the shear resistance between the neighboring walls during the buckling of the multiwalled nanotubes. The present method can be further extended to investigate the stress-strain relations and the fracture behaviors of S/MWCNT From vortices to dislocations: How fluid mechanics can inspire solid mechanics Markus Scholle (Hochschule Heilbronn) Although it is known for nearly a century that production and movement of dislocations are the elementary processes responsible for the plastic deformation of a solid body, all attempts to derive a continuum theory of plasticity from the discrete micro–theory of dislocations did not succeed in universally valid field equations like e.g. Navier–Stokes equations in fluid dynamics. In this paper an approach is presented based on an analogy between dislocations and vortex lines in fluid flow. Since the dynamics of the latter ones is well understood in terms of the Navier– Stokes equations, it is near at hand to make use of this analogy in order to establish a kind of ’Navier–Stokes equations for the distorsion tensor’. Starting from Clebsch’s variational formulation for inviscid flow, a relation beween the symmetries of the Lagrangian and the balance of vorticity resulting from it is shown giving rise for the construction of another Lagrangian from which the respective dislocation balance results. The present state of the theory is discussed. On the connection of dissipation and deviatoric stress in the continuum theory of defects Johannes Schnepp (RWTH Aachen)
Section 6: Material modelling in solid mechanics 33 Continuous distributions of defects (e. g. dislocations) can be described as differential geometric properties of the material manifold. Material forces and the Eshelby tensor are also quantities in this three-dimensional material space. The relation to defects has been treated often in the literature and various theories for the dynamics of defects have been proposed. Moving defects lead to differential geometric quantities changing with time. This has motivated some authors to augment the three space-like coordinates in the material space by a time-like coordinate. In this way one gets a four-dimensional material space-time manifold, analogous to the four-dimensional physical space-time in the theory of relativity. A connection of a time-like material coordinate to temperature can be found in the literature on relativistic elasticity. These ideas will be brought together in this contribution and some implications will be developed. A time-like vector field in the material space-time can be associated with temperature and heat flux. Together with the three lattice vectors a tetrad field is defined and this field determines the differential geometric properties of the material manifold. If a variational formulation for a hyper-elastic solid is adopted, the derivatives of the Lagrangian density with respect to the components of the tetrad can be arranged in a four-dimensional second-order tensor. This tensor unifies the information about the three-dimensional Eshelby tensor (material momentum current), the entropy current, and the entropy density. The time-like component of the divergence of this tensor is the entropy production. Besides the entropy production due to heat transfer the theory yields entropy production due to defect movement which is proportional to deviatoric stress. Stress Reduction Factor of Ceramic Plate to Thermal Shock Weiguo Li, Tianbao Cheng (College of Resources and Environmental Science, Chongqing), Daining Fang (Peking University) In this work, through introducing the analytical solution to transient conduction problem for the thin rectangular plate with convection into the thermal stress field model of the elastic plate, the stress reduction factor which is useful in determining the thermal stresses in the ceramic plate subjected to thermal shock was presented in the dimensionless form. The properties and appropriate conditions of the stress reduction factor were analyzed. Then the definitions, origins and limitations of the first and second thermal stress fracture resistance parameters which characterize the thermal shock resistance of brittle ceramics were discussed. Because of the demands in the calculation of thermal stress, a new stress reduction factor was defined and compared with the previous one. For the given Biot number, the previous stress reduction factor first increases and then reaches a maximum value, and afterwards decreases as the Fourier number increases. However, the new stress reduction factor decreases as the Fourier number increases. The new factor is more convenient for calculating the thermal stress in the plate subjected to thermal shock. The presented results are useful for the calculating of the thermal stress and choosing of the appropriate thermal stress fracture resistance parameter when the ceramic plate is subjected to thermal shock. Modeling Deformation Twinning in FeMn Alloys Shyamal Roy, Rainer Glüge, Albrecht Bertram (Universität Magdeburg) Multiple twinning [1] in single crystals is modeled by studying the response at each material point in order to understand micro-structural evolution and mechanical properties of FeMn (TWIP steel). Here the minimum elastic strain energy approach is used. It is well known that the elastic strain energy of an individual material is convex. Since twinning leads to isomorphic structures, the elastic strain energy of a twin material is convex too and hence, forms a non-convex energy landscape. A smooth strain energy landscape is constructed to avoid the discontinuity at the transition point from parent to twin material. A controlled viscous relaxation scheme is
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Section 6: Material modelling in solid mechanics - GAMM 2012

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