Computational micromechanics and homogenization require the solution of the mechanical equilibrium of a periodic cell that comprises a (generally complex) microstructure. Techniques that apply the Fast Fourier Transform have attracted much attention as they outperform other methods in terms of speed and memory footprint. Moreover, the Fast Fourier Transform is a natural companion of pixel-based digital images which often serve as input. In its original form, one of the biggest challenges for the method is the treatment of (geometrically) non-linear problems, partially due to the need for a uniform linear reference problem. In a geometrically linear setting, the problem has recently been treated in a variational form resulting in an unconditionally stable scheme that combines Newton iterations with an iterative linear solver, and therefore exhibits robust and quadratic convergence behavior. Through this approach, well-known key ingredients were recovered in terms of discretization, numerical quadrature, consistent linearization of the material model, and the iterative solution of the resulting linear system. As a result, the extension to finite strains, using arbitrary constitutive models, is at hand. Because of the application of the Fast Fourier Transform, the implementation is substantially easier than that of other (Finite Element) methods. Both claims are demonstrated in this paper and substantiated with a simple code in Python of just 59 lines (without comments). The aim is to render the method transparent and accessible, whereby researchers that are new to this method should be able to implement it efficiently. The potential of this method is demonstrated using two examples, each with a different material model

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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 Ac governed by a Griffith criterion. We find that after slip, the density of asperities at a local distance to yielding xσ presents a pseudo-gap P (xσ) ∼ (xσ) θ , where θ 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 Ac diverges as a θ-dependent power law of the system size. We discuss how these predictions can be tested experimentally. Significance statementUnderstanding how slip at a frictional interface initiates is important for a range of problems including earthquake prediction and precision engineering. The force needed to start sliding a solid object over a flat surface is classically described by a 'static friction coefficient': a constant established by measurements. It was recently questioned if such constant exists, as it was shown to be poorly reproducible. We provide a model supporting that it is stochastic even for very large system sizes: sliding is nucleated when, by chance, an avalanche of microscopic detachments reaches a critical radius, beyond which slip becomes unstable and propagates along the interface. It leads to testable predictions on key observables characterising the stability of the interface.

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, DL(ω) ∼ ω α , 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) ∼ x θ (with x the additional stress needed to trigger a shear transformation). If θ > 1/4, α = 4 and a finite fraction of quasi-localised modes form shear transformations, whose amplitudes vanish at low frequencies. If θ < 1/4, α = 3 + 4θ and all quasi-localised modes form shear transformations with a finite amplitude at vanishing frequencies. We confirm our predictions numerically. arXiv:1806.01561v2 [cond-mat.soft]

Key aspects of glasses are controlled by the presence of excitations in which a group of particles can rearrange. Surprisingly, recent observations indicate that their density is dramatically reduced and their size decreases as the temperature of the supercooled liquid is lowered. Some theories predict these excitations to cause a gap in the spectrum of quasilocalized modes of the Hessian that grows upon cooling, while others predict a pseudogap D L (ω) ∼ ω α . To unify these views and observations, we generate glassy configurations of controlled gap magnitude ω c at temperature T = 0, using so-called breathing particles, and study how such gapped states respond to thermal fluctuations. We find that (i) the gap always fills up at finite T with D L (ω) ≈ A 4 (T ) ω 4 and A 4 ∼ exp(−E a /T ) at low T , (ii) E a rapidly grows with ω c , in reasonable agreement with a simple scaling prediction E a ∼ ω 4 c and (iii) at larger ω c excitations involve fewer particles, as we rationalize, and eventually become stringlike. We propose an interpretation of mean-field theories of the glass transition, in which the modes beyond the gap act as an excitation reservoir, from which a pseudogap distribution is populated with its magnitude rapidly decreasing at lower T . We discuss how this picture unifies the rarefaction as well as the decreasing size of excitations upon cooling, together with a stringlike relaxation occurring near the glass transition.

We analyse the equilibrium pile-up configurations of infinite periodic walls of edge dislocations which are forced against an impenetrable obstacle by a constant applied shear stress. Numerically generated density distributions exhibit two distinct regions, for each of which we provide an interpretation and an analytical prediction. Near the obstacle, the influence of neighbouring slip planes may be neglected and the classical solution for a single slip plane applies. At a larger distance a linear decay is obtained. The characteristic length scales of the two parts of the pile-up are shown to depend differently on the parameters of the problem.

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 compare the two methods using the energetic norm of local fields, which has the clear physical interpretation as being the error in the homogenized properties. This enables the comparison of memory and computational requirements at the same level of approximation accuracy. We show that the methods' effectiveness rely 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 linear system, independent of the number of degrees of freedom, but generally needs more degrees of freedom. More studies are needed for other FFT-based schemes, non-linear problems, and dual problems (which require special treatment in FEM). keywords:energy-based comparison, finite element method, Fourier-Galerkin method, fast Fourier transform, numerical homogenisation, computational effectiveness arXiv:1709.08477v1 [math.NA]

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