Section 6: Material modelling in solid mechanics - GAMM 2012

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16 Section 6: Material modelling in solid mechanics asphalt cover layer, i.e. the top layer of the entire asphalt-construction. In order to predict the macro-scale effective material properties of asphalt we apply a numerical homogenisation scheme based on volume averaging techniques. The main advandtage of this numerical approach is, that the effective material properties can be determined in knowledge of the micro-scale properties. Hence, the first step towards an overall model for bituminous asphalt is the quantification of the micro-scale mechanical properties. Against the background of solid-borne acoustical properties we restrict ourselves to a geometrically linear description involving only small deformations on both, macro- and micro-scale. Doing so, the linear-elastic properties of the stiff, granular filler can be adopted from relevant literature. The viscoelastic behaviour of the bituminous phase is characterized by rheological experiments (dynamic shear rheometer with plate-plate geometry). The numerical implementation on the micro-scale is realized using a generalized Zener model (3d). Making use of the numerical homogenization approach, the effective viscoelastic properties on the macro-scale are investigated within transient experiments (relaxation/creep test). In particular we are interested in the particular relaxation mechanisms and the related characteristic frequencies to be observed on the macro-scale. The influence of micro-scale boundary conditions will be taken into account. In order to study the interaction between micro-constituents as well as their geometrical morphology on the one hand and the effective viscoelastic properties on the other, we introduce artificially produced periodic unit cells based on simplified geometries with varying volume and surface fractions of the mineral filler. Jumps of the critical tracking loadings for viscoelastic beams with vanisihing internal viscosity S.A.Agafonov (Moscow State Technical University), D.V.Georgievskii (Moscow State University) Transverse vibrations of a viscoelastic beam under action of a tracking loading are considered. The constitutive relation represents the following non-linear connection of stress σ(t), strain ε(t) and strain rate ˙ε(t): σ = Eε + where E is the Young modulus, k (n) i N n n=1 i=1 k (n) i ε2(n−i) ˙ε 2i−1 , N ≥ 1 > 0 are the coefficients of internal viscosity. Analytical and numerical investigation of dynamic stability in case N = 3, k (1) 1 = 0, k (2) 1 = 0, k (1) 2 = 0 shows that if k (3) α → 0 (k (3) β = 0, k(3) γ = 0; (α, β, γ) may be some permutation of (1, 2, 3)) then three critical values of tracking loading corresponding to each nonzero coefficient of viscosity k (3) α less than the value for elastic system by a finite quantity. Numerical modeling of a non-linear viscous flow in order to determine how parameters in constitutive relations influence the entropy production Wolfgang H. Müller, B. Emek Abali (TU Berlin) Some rheological materials like melting polymers, cosmetic creams, ketchup, toothpaste can be modeled as non-Newtonian fluids by using a non-linear constitutive relation. Flow of this kind of amorphous matter can be considered as a thermodynamic process, and a solution of pressure, velocity and temperature fields describe it fully. Since flow processes are generally irreversible, entropy is produced leading to dissipation in the system. This energy loss can be measured indirectly in a cone/plate viscometer which is used to determine viscosity of a Bingham fluid. While dissipation is a measurable quantity we want to be able to calculate it. Thus the goal of this work is to explain how to calculate entropy production using balance equations in a spatial
Section 6: Material modelling in solid mechanics 17 frame. Starting from balance of mass, linear momentum, internal energy and employing method of weighted residuals, we get a non-linear coupled set of partial differential equations in space and time. A space discretization in finite elements method and a time discretization in finite difference method leads to an approximation after a successful linearization. We achieve to solve the problem without any stabilization or usage of specific type of elements and show here the entropy production and its variation subject to material parameters for the sake of a better intuitive understanding of dissipation. This may lead to an inverse problem where the calculated dissipation is measured and material parameters are determined out of it, which is left to future research. S6.7: Phase Transformations II Wed, 16:00–18:00 Chair: Wolfgang Dreyer S1|01–A03 Nonsingular Dislocation Loops in Gradient Elasticity Markus Lazar (TU Darmstadt) This work studies the fundamental problem of nonsingular dislocations in the framework of the theory of gradient elasticity. A general theory of nonsingular dislocations is developed for linearly elastic, infinitely extended, homogeneous, isotropic media. Using gradient elasticity, we give the nonsingular fields produced by arbitrary dislocation loops. We present the ‘modified’ Mura, Peach- Koehler and Burgers formulae in the framework of gradient elasticity theory. These formulae are given in terms of an elementary function, which regularizes the classical expressions, obtained from the Green tensor of generalized Navier equations. Using the mathematical method of Green’s functions and the Fourier transform, we found exact, analytical and nonsingular solutions. The obtained dislocation fields are nonsingular due to the regularization of the classical singular fields. On a paradox within the phase field modeling of storage systems and its resolution Clemens Guhlke, Wolfgang Dreyer (WIAS Berlin) We study two import storage problems: The storage of lithium in an electrode of a lithium-ion battery and the storage of hydrogen in hydrides. When foreign atoms are reversibly stored in a crystal, there may be a regime where two coexisting phases with low and high concentration of the stored atoms occur. Furthermore hysteretic behavior can be observed, i.e. the processesof loading and unloading follow different paths. We apply a viscous Cahn-Hilliard model with mechanical coupling to calculate the voltagecharge diagram of the battery, respectively the pressure-charge diagram of a hydrogen system. The diagrams exhibit phase transition and hysteresis. However, we show that the model can only describe the observed phenomena for fast but nor for slow loading. We relate the reason for failure to the microstructure of modern storage systems. In fact these consist of an ensemble of nano-sized interconnected storage particle. Each particle is described by a non-monotone chemical potential function but on the time scale of slow loading of the ensemble, the coexisting phases are unstable within an individual particle and cannot be observed on the time scale of the loading. In the slow loading regime, the occurence of two coexisting phases phases is a many-particle effect of the ensemble. The nucleation and evolution of the phases is embodied by a nonlocal conservation law of Fokker-Planck type. Numerical Simulation of Coarsening in Metallic Alloys Uli Sack, Carsten Gräser, Ralf Kornhuber (FU Berlin)
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Section 6: Material modelling in solid mechanics - GAMM 2012

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