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

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

Improving the initial guess for the Newton-Raphson protocol in time-dependent simulations

A general linearisation procedure for the consistent tangent of a small-strain visco-plastic material model is presented in this note. The procedure is based on multi-variable linearisation around a so-called 'reference state'. In particular, the linerarisation of the time integration scheme is found to yield an extra term compared to classical expressions, which only appears because the material response is time-dependent. It has the effect of yielding a very accurate initial guess for the Newton-Raphson protocol based on the ongoing viscous flow. It is shown, using a modern variational FFT-based solver, that the extra term reduces both the CPU time and the number of Newton-Raphson iterations by around a factor two.Comment: Journal of Computational Physics, 202

Stick-slip synchronization in stack of elastically coupled frictional interfaces

We perform physical and numerical experiments to study the stick-slip response of a stack of slabs in contact through dry frictional interfaces driven in quasistatic shear. The ratio between the drive's stiffness and the slab's shear stiffness controls the presence or absence of slip synchronization. A sufficiently high stiffness ratio leads to synchronization, comprising periodic slip events in which all interfaces slip simultaneously. A lower stiffness ratio leads to asynchronous slips and, experimentally, to the stick-slip amplitude being broadly distributed as the number of layers in the stack increases. We interpret this broadening in light of the combined effect of surface disorder, complex loading paths of the asynchronous slips, and creep. Consequently, the ageing rate can be readily extracted from the stick-slip cycle. The extracted aging rate is found to be of the same order of magnitude as existing experimental results on a similar material. Finally, we discuss the emergence of slow slips and an increase in creep-rate variations when more slabs are added to the stack

Thermal origin of quasi-localised excitations in glasses

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 quasi-localised modes of the Hessian that grows upon cooling, while others predict a pseudo-gap ${D_L(\omega)} \sim \omega^\alpha$. To unify these views and observations, we generate glassy configurations of controlled gap magnitude $\omega_c$ at temperature ${T=0}$, using so-called `breathing' particles, and study how such gapped states respond to thermal fluctuations. We find that \textit{(i)}~the gap always fills up at finite $T$ with ${D_L(\omega) \approx A_4(T) \, \omega^4}$ and ${A_4 \sim \exp(-E_a / T)}$ at low $T$, \textit{(ii)}~$E_a$ rapidly grows with $\omega_c$, in reasonable agreement with a simple scaling prediction ${E_a\sim \omega_c^4}$ and \textit{(iii)}~at larger $\omega_c$ excitations involve fewer particles, as we rationalise, and eventually become string-like. 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 pseudo-gap 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 string-like relaxation occurring near the glass transition