237 research outputs found
Approximation of Periodic PDE Solutions with Anisotropic Translation Invariant Spaces
We approximate the quasi-static equation of linear elasticity in translation
invariant spaces on the torus. This unifies different FFT-based discretisation
methods into a common framework and extends them to anisotropic lattices. We
analyse the connection between the discrete solution spaces and demonstrate the
numerical benefits. Finite element methods arise as a special case of
periodised Box spline translates
Geometrically nonlinear Cosserat elasticity in the plane: applications to chirality
Modelling two-dimensional chiral materials is a challenging problem in
continuum mechanics because three-dimensional theories reduced to isotropic
two-dimensional problems become non-chiral. Various approaches have been
suggested to overcome this problem. We propose a new approach to this problem
by formulating an intrinsically two-dimensional model which does not require
references to a higher dimensional one. We are able to model planar chiral
materials starting from a geometrically non-linear Cosserat type elasticity
theory. Our results are in agreement with previously derived equations of
motion but can contain additional terms due to our non-linear approach. Plane
wave solutions are briefly discussed within this model.Comment: 22 pages, 1 figure; v2 updated versio
A review of nonlinear FFT-based computational homogenization methods
Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution methods, leading to improvements of the original approach and extending the applications. This article provides a condensed overview of results scattered throughout the literature and guides the reader to the current state of the art in nonlinear computational homogenization methods using the fast Fourier transform
On the critical nature of plastic flow: one and two dimensional models
Steady state plastic flows have been compared to developed turbulence because
the two phenomena share the inherent complexity of particle trajectories, the
scale free spatial patterns and the power law statistics of fluctuations. The
origin of the apparently chaotic and at the same time highly correlated
microscopic response in plasticity remains hidden behind conventional
engineering models which are based on smooth fitting functions. To regain
access to fluctuations, we study in this paper a minimal mesoscopic model whose
goal is to elucidate the origin of scale free behavior in plasticity. We limit
our description to fcc type crystals and leave out both temperature and rate
effects. We provide simple illustrations of the fact that complexity in rate
independent athermal plastic flows is due to marginal stability of the
underlying elastic system. Our conclusions are based on a reduction of an
over-damped visco-elasticity problem for a system with a rugged elastic energy
landscape to an integer valued automaton. We start with an overdamped one
dimensional model and show that it reproduces the main macroscopic
phenomenology of rate independent plastic behavior but falls short of
generating self similar structure of fluctuations. We then provide evidence
that a two dimensional model is already adequate for describing power law
statistics of avalanches and fractal character of dislocation patterning. In
addition to capturing experimentally measured critical exponents, the proposed
minimal model shows finite size scaling collapse and generates realistic shape
functions in the scaling laws.Comment: 72 pages, 40 Figures, International Journal of Engineering Science
for the special issue in honor of Victor Berdichevsky, 201
Stochastic 3D microstructure modeling of twinned polycrystals for investigating the mechanical behavior of -TiAl intermetallics
A stochastic 3D microstructure model for polycrystals is introduced which
incorporates two types of twin grains, namely neighboring and inclusion twins.
They mimic the presence of crystal twins in -TiAl polycrystalline
microstructures as observed by 3D imaging techniques. The polycrystal grain
morphology is modeled by means of Voronoi and -- more generally -- Laguerre
tessellations. The crystallographic orientation of each grain is either sampled
uniformly on the space of orientations or chosen to be in a twinning relation
with another grain. The model is used to quantitatively study relationships
between morphology and mechanical properties of polycrystalline materials. For
this purpose, full-field Fourier-based computations are performed to
investigate the combined effect of grain morphology and twinning on the overall
elastic response. For -TiAl polycrystallines, the presence of twins is
associated with a softer response compared to polycrystalline aggregates
without twins. However, when comparing the influence on the elastic response, a
statistically different polycrystalline morphology has a much smaller effect
than the presence of twin grains. Notably, the bulk modulus is almost
insensitive to the grain morphology and exhibits much less sensitivity to the
presence of twins compared to the shear modulus. The numerical results are
consistent with a two-scale homogenization estimate that utilizes laminate
materials to model the interactions of twins
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