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 $A_c$ governed by a Griffith criterion. We find that after slip, the density of asperities at a local distance to yielding $x_\sigma$ presents a pseudo-gap $P(x_\sigma) \sim (x_\sigma)^\theta$, where $\theta$ 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 $A_c$ diverges as a $\theta$-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, $D_{L}(\omega)\sim\omega^{\alpha}$, that establishes this link for systems at zero temperature under quasi-static loading. It predicts two regimes depending on the density of shear transformations $P(x)\sim x^{\theta}$ (with $x$ the additional stress needed to trigger a shear transformation). If $\theta>1/4$, $\alpha=4$ and a finite fraction of quasi-localised modes form shear transformations, whose amplitudes vanish at low frequencies. If $\theta<1/4$, $\alpha=3+ 4 \theta$ 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