2,884 research outputs found
Energy-based comparison between the Fourier--Galerkin method and the finite element method
The Fourier-Galerkin method (in short FFTH) has gained popularity in
numerical homogenisation because it can treat problems with a huge number of
degrees of freedom. Because the method incorporates the fast Fourier transform
(FFT) in the linear solver, it is believed to provide an improvement in
computational and memory requirements compared to the conventional finite
element method (FEM). Here, we systematically compare these two methods using
the energetic norm of local fields, which has the clear physical interpretation
as being the error in the homogenised properties. This enables the comparison
of memory and computational requirements at the same level of approximation
accuracy. We show that the methods' effectiveness relies on the smoothness
(regularity) of the solution and thus on the material coefficients. Thanks to
its approximation properties, FEM outperforms FFTH for problems with jumps in
material coefficients, while ambivalent results are observed for the case that
the material coefficients vary continuously in space. FFTH profits from a good
conditioning of the linear system, independent of the number of degrees of
freedom, but generally needs more degrees of freedom to reach the same
approximation accuracy. More studies are needed for other FFT-based schemes,
non-linear problems, and dual problems (which require special treatment in FEM
but not in FFTH).Comment: 24 pages, 10 figures, 2 table
Enantiomer fractions instead of enantiomer ratios
The use of enantiomer ratios (ERs) to indicate the relative amounts of a pair of enantiomers in a sample has some disadvantages. Enantiomer fractions (EFs) are proposed as all alternative expression to eliminate the difficulties. (C) 2000 Elsevier Science Ltd
Microstructural topology effects on the onset of ductile failure in multi-phase materials - a systematic computational approach
Multi-phase materials are key for modern engineering applications. They are
generally characterized by a high strength and ductility. Many of these
materials fail by ductile fracture of the, generally softer, matrix phase. In
this work we systematically study the influence of the arrangement of the
phases by correlating the microstructure of a two-phase material to the onset
of ductile failure. A single topological feature is identified in which
critical levels of damage are consistently indicated. It consists of a small
region of the matrix phase with particles of the hard phase on both sides in a
direction that depends on the applied deformation. Due to this configuration, a
large tensile hydrostatic stress and plastic strain is observed inside the
matrix, indicating high damage. This topological feature has, to some extent,
been recognized before for certain multi-phase materials. This study however
provides insight in the mechanics involved, including the influence of the
loading conditions and the arrangement of the phases in the material
surrounding the feature. Furthermore, a parameter study is performed to explore
the influence of volume fraction and hardness of the inclusion phase. For the
same macroscopic hardening response, the ductility is predicted to increase if
the volume fraction of the hard phase increases while at the same time its
hardness decreases
How collective asperity detachments nucleate slip at frictional interfaces
Sliding at a quasi-statically loaded frictional interface can occur via
macroscopic slip events, which nucleate locally before propagating as rupture
fronts very similar to fracture. We introduce a novel microscopic model of a
frictional interface that includes asperity-level disorder, elastic interaction
between local slip events, and inertia. For a perfectly flat and homogeneously
loaded interface, we find that slip is nucleated by avalanches of asperity
detachments of extension larger than a critical radius governed by a
Griffith criterion. We find that after slip, the density of asperities at a
local distance to yielding presents a pseudo-gap , where is a non-universal exponent that depends on
the statistics of the disorder. This result makes a link between friction and
the plasticity of amorphous materials where a pseudo-gap is also present. For
friction, we find that a consequence is that stick-slip is an extremely slowly
decaying finite size effect, while the slip nucleation radius diverges as
a -dependent power law of the system size. We discuss how these
predictions can be tested experimentally
Theory for the density of interacting quasi-localised modes in amorphous solids
Quasi-localised modes appear in the vibrational spectrum of amorphous solids
at low-frequency. Though never formalised, these modes are believed to have a
close relationship with other important local excitations, including shear
transformations and two-level systems. We provide a theory for their frequency
density, , that establishes this link for
systems at zero temperature under quasi-static loading. It predicts two regimes
depending on the density of shear transformations (with
the additional stress needed to trigger a shear transformation). If
, and a finite fraction of quasi-localised modes form
shear transformations, whose amplitudes vanish at low frequencies. If
, and all quasi-localised modes form shear
transformations with a finite amplitude at vanishing frequencies. We confirm
our predictions numerically
Fracture initiation in multi-phase materials: a systematic three-dimensional approach using a FFT-based solver
This paper studies a two-phase material with a microstructure composed of a
hard brittle reinforcement phase embedded in a soft ductile matrix. It
addresses the full three-dimensional nature of the microstructure and
macroscopic deformation. A large ensemble of periodic microstructures is used,
whereby the individual grains of the two phases are modeled using equi-sized
cubes. A particular solution strategy relying on the Fast Fourier Transform is
adopted, which has a high computational efficiency both in terms of speed and
memory footprint, thus enabling a statistically meaningful analysis. This
solution method naturally accompanies the regular microstructural model, as the
Fast Fourier Transform relies on a regular grid.
Using the many considered microstructures as an ensemble, the average
arrangement of phases around fracture initiation sites is objectively
identified by the correlation between microstructure and fracture initiation --
in three dimensions. The results show that fracture initiates where regions of
the hard phase are interrupted by bands of the soft phase that are aligned with
the direction of maximum shear. In such regions, the hard phase is arranged
such that the area of the phase boundary perpendicular to the principal strain
direction is maximum, leading to high hydrostatic tensile stresses, while not
interrupting the shear bands that form in the soft phase. The local
incompatibility that is present around the shear bands is responsible for a
high plastic strain. By comparing the response to a two-dimensional
microstructure it is observed that the response is qualitatively similar (both
macroscopically and microscopically). One important difference is that the
local strain partitioning between the two phases is over-predicted by the
two-dimensional microstructure, leading to an overestimation of damage
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