Laplacian transfer across a rough interface: Numerical resolution in the conformal plane

We use a conformal mapping technique to study the Laplacian transfer across a rough interface. Natural Dirichlet or Von Neumann boundary condition are simply read by the conformal map. Mixed boundary condition, albeit being more complex can be efficiently treated in the conformal plane. We show in particular that an expansion of the potential on a basis of evanescent waves in the conformal plane allows to write a well-conditioned 1D linear system. These general principle are illustrated by numerical results on rough interfaces

Mechanical noise dependent Aging and Shear Banding behavior of a mesoscopic model of amorphous plasticity

We discuss aging and localization in a simple "Eshelby" mesoscopic model of amorphous plasticity. Plastic deformation is assumed to occur through a series of local reorganizations. Using a discretization of the mechanical fields on a discrete lattice, local reorganizations are modeled as local slip events. Local yield stresses are randomly distributed in space and invariant in time. Each plastic slip event induces a long-ranged elastic stress redistribution. Mimicking the effect of aging, we focus on the behavior of the model when the initial state is characterized by a distribution of high local yield stress values. A dramatic effect on the localization behavior is obtained: the system first spontaneously self-traps to form a shear band which then only slowly widens. The higher the "age" parameter the more localized the plastic strain field. Two-time correlation computed on the stress field show a divergent correlation time with the age parameter. The amplitude of a local slip event (the prefactor of the Eshelby singularity) as compared to the yield stress distribution width acts here as an effective temperature-like parameter: the lower the slip increment, the higher the localization and the decorrelation time

Modeling the mechanics of amorphous solids at different length and time scales

We review the recent literature on the simulation of the structure and deformation of amorphous glasses, including oxide and metallic glasses. We consider simulations at different length and time scales. At the nanometer scale, we review studies based on atomistic simulations, with a particular emphasis on the role of the potential energy landscape and of the temperature. At the micrometer scale, we present the different mesoscopic models of amorphous plasticity and show the relation between shear banding and the type of disorder and correlations (e.g. elastic) included in the models. At the macroscopic range, we review the different constitutive laws used in finite element simulations. We end the review by a critical discussion on the opportunities and challenges offered by multiscale modeling and transfer of information between scales to study amorphous plasticity.Comment: 58 pages, 14 figure

Front propagation in random media: From extremal to activated dynamics

Front propagation in a random environment is studied close to the depinning threshold. At zero temperature we show that the depinning force distribution exhibits a universal behavior. This property is used to estimate the velocity of the front at very low temperature. We obtain a Arrhenius behavior with a prefactor depending on the temperature as a power law. These results are supported by numerical simulations.Comment: 6 pages, 2 figures, accepted in Int. J. Mod. Phys.

Quantitative prediction of effective toughness at random heterogeneous interfaces

The propagation of an adhesive crack through an anisotropic heterogeneous interface is considered. Tuning the local toughness distribution function and spatial correlation is numerically shown to induce a transition between weak to strong pinning conditions. While the macroscopic effective toughness is given by the mean local toughness in case of weak pinning, a systematic toughness enhancement is observed for strong pinning (the critical point of the depinning transition). A self-consistent approximation is shown to account very accurately for this evolution, without any free parameter

Avalanches, precursors and finite size fluctuations in a mesoscopic model of amorphous plasticity

We discuss avalanche and finite size fluctuations in a mesoscopic model to describe the shear plasticity of amorphous materials. Plastic deformation is assumed to occur through series of local reorganizations. Yield stress criteria are random while each plastic slip event induces a quadrupolar long range elastic stress redistribution. The model is discretized on a regular square lattice. Shear plasticity can be studied in this context as a depinning dynamic phase transition. We show evidence for a scale free distribution of avalanches $P(s)\propto S^{-\kappa}$ with a non trivial exponent $\kappa \approx 1.25$ significantly different from the mean field result $\kappa = 1.5$. Finite size effects allow for a characterization of the scaling invariance of the yield stress fluctuations observed in small samples. We finally identify a population of precursors of plastic activity and characterize its spatial distribution

Avalanches, thresholds, and diffusion in meso-scale amorphous plasticity

We present results on a meso-scale model for amorphous matter in athermal, quasi-static (a-AQS), steady state shear flow. In particular, we perform a careful analysis of the scaling with the lateral system size, $L$, of: i) statistics of individual relaxation events in terms of stress relaxation, $S$, and individual event mean-squared displacement, $M$, and the subsequent load increments, $\Delta \gamma$, required to initiate the next event; ii) static properties of the system encoded by $x=\sigma_y-\sigma$, the distance of local stress values from threshold; and iii) long-time correlations and the emergence of diffusive behavior. For the event statistics, we find that the distribution of $S$ is similar to, but distinct from, the distribution of $M$. We find a strong correlation between $S$ and $M$ for any particular event, with $S\sim M^{q}$ with $q\approx 0.65$. $q$ completely determines the scaling exponents for $P(M)$ given those for $P(S)$. For the distribution of local thresholds, we find $P(x)$ is analytic at $x=0$, and has a value $\left. P(x)\right|_{x=0}=p_0$ which scales with lateral system length as $p_0\sim L^{-0.6}$. Extreme value statistics arguments lead to a scaling relation between the exponents governing $P(x)$ and those governing $P(S)$. Finally, we study the long-time correlations via single-particle tracer statistics. The value of the diffusion coefficient is completely determined by $\langle \Delta \gamma \rangle$ and the scaling properties of $P(M)$ (in particular from $\langle M \rangle$) rather than directly from $P(S)$ as one might have naively guessed. Our results: i) further define the a-AQS universality class, ii) clarify the relation between avalanches of stress relaxation and diffusive behavior, iii) clarify the relation between local threshold distributions and event statistics

Cracks in random brittle solids: From fiber bundles to continuum mechanics

Statistical models are essential to get a better understanding of the role of disorder in brittle disordered solids. Fiber bundle models play a special role as a paradigm, with a very good balance of simplicity and non-trivial effects. We introduce here a variant of the fiber bundle model where the load is transferred among the fibers through a very compliant membrane. This Soft Membrane fiber bundle mode reduces to the classical Local Load Sharing fiber bundle model in 1D. Highlighting the continuum limit of the model allows to compute an equivalent toughness for the fiber bundle and hence discuss nucleation of a critical defect. The computation of the toughness allows for drawing a simple connection with crack front propagation (depinning) models.Comment: The European Physical Journal Special Topics Special Topics, 201