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We perform molecular dynamics simulations of homogeneous athermal systems of poly-disperse soft discs under shear. For purely repulsive interactions between particles, and under a confining external pressure, a monotonous flow curve (strain rate vs. stress) starting at a critical yield stress is obtained, with deformation distributing uniformly in the system, on average. Then we add a short range attractive contribution to the interaction potential that increases its intensity as particles remain in contact for a progressively longer time, mimicking an aging effect in the system. In this case the flow curve acquires a reentrant behavior, namely, a region where shear stress decreases with increasing strain. Within this region the deformation is seen to localize in a shear band with a well defined width that decreases as the global strain rate does. At very low strain rates the shear band becomes very thin and deformation acquires a prominent stick-slip behavior. This regime can be described as the system possessing a fault in which deformation occurs with an earthquake-resembling phenomenology. In this way the system we are analyzing connects a regime of uniform deformation at large strain rates, a localized deformation regime in the form of shear bands at intermediate stain rates, and seismic phenomena at very low strain rate. The unifying ingredient of this phenomenology is the existence of a reentrant flow curve, originating in the aging mechanisms present in the model.
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We investigate the down-hill creep of an inclined layer of granular material caused by quasi-static oscillatory variations of the size of the particles. The size variation is taken to be maximum at the surface and decreasing with depth, as it may be argued to occur in the case of a granular soil affected by atmospheric conditions. The material is modeled as an athermal two dimensional polydisperse system of soft disks under the action of gravity. The slope angle is below the angle of repose and therefore the system reaches an equilibrium configuration under static external conditions. However, under a protocol in which particles slowly change size in a quasistatic oscillatory way, the system is observed to creep down in a synchronized way with the oscillation. We measure the creep advance per cycle as a function of the slope angle and the degree of change in particle size. We also find that the creep rate is maximum at the surface and smoothly decreases with depth, as it is observed to occur in the field.
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We analyze a mesoscopic model of a shear stress material with a three-dimensional slab geometry, under an external quasistatic deformation of a simple shear type. Relaxation is introduced in the model as a mechanism by which an unperturbed system achieves progressively mechanically more stable configurations. Although in all cases deformation occurs via localized plastic events (avalanches), we find qualitatively different behavior depending on the degree of relaxation in the model. For no or low relaxation, yielding is homogeneous in the sample, and even the largest avalanches become negligible in size compared with the system size (measured as the thickness of the slab L_{z}) when this is increased. On the contrary, for high relaxation, the deformation localizes in an almost two-dimensional region where all avalanches occur. Scaling analysis of the numerical results indicates that in this case, the linear size of the largest avalanches is comparable with L_{z}, even when this becomes very large. We correlate the two scenarios with a qualitative difference in the flow curve of the system in the two cases, which is monotonous in the first case and velocity weakening in the second case.
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We present a two-dimensional mesoscopic model of a yield stress material that includes the possibility of local volume fluctuations coupled to shear in such a way that the shear strength of the material decreases as the local density decreases. The model reproduces a number of effects well known in the phenomenology of this kind of material. In particular, we find that the volume of the sample increases as the deformation rate increases; shear bands are no longer oriented at 45^{∘} with respect to the principal axis of the applied stress (as in the absence of volume-shear coupling); and homogeneous deformation becomes unstable at low enough deformation rates if volume-shear coupling is strong enough. We also discuss the effect of this coupling on some out-of-equilibrium configurations, which can be relevant to the study of the shear bands observed in metallic glasses.
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The thermal rounding of the depinning transition of an elastic interface sliding on a washboard potential is studied through analytic arguments and very accurate numerical simulations. We confirm the standard view that well below the depinning threshold the average velocity can be calculated considering thermally activated nucleation of defects. However, we find that the straightforward extension of this analysis to near or above the depinning threshold does not fully describe the physics of the thermally assisted motion. In particular, we find that exactly at the depinning point the average velocity does not follow a pure power law of the temperature as naively expected by the analogy with standard phase transitions but presents subtle logarithmic corrections. We explain the physical mechanisms behind these corrections and argue that they are nonpeculiar collective effects which may also apply to the case of interfaces sliding on uncorrelated disordered landscapes.
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We use a continuous mesoscopic model to address the yielding properties of plastic composites, formed by a host material and inclusions with different elastic and/or plastic properties. We investigate the flow properties of the composed material under a uniform externally applied deviatoric stress. We show that due to the heterogeneities induced by the inclusions, a scalar modeling in terms of a single deviatoric strain of the same symmetry as the externally applied deformation gives inaccurate results. A realistic modeling must include all possible shear deformations. Implementing this model in a two-dimensional system, we show that the effect of harder inclusions is very weak to relatively high concentrations. For softer inclusions instead, the effect is much stronger; even a small concentration of inclusions affecting the form of the flow curve and the critical stress. We also present the details of a full three-dimensional simulation scheme and obtain the corresponding results for harder and softer inclusions.
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We consider a model of an elastic manifold driven on a disordered energy landscape, with generalized long range elasticity. Varying the form of the elastic kernel by progressively allowing for the existence of zero modes, the model interpolates smoothly between mean-field depinning and finite dimensional yielding. We find that the critical exponents of the model change smoothly in this process. Also, we show that in all cases the Herschel-Buckley exponent of the flow curve depends on the analytical form of the microscopic pinning potential. Within the present elastoplastic description, all this suggests that yielding in finite dimensions is a mean-field transition.
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We analyze the behavior of different elastoplastic models approaching the yielding transition. We propose two kinds of rules for the local yielding events: yielding occurs above the local threshold either at a constant rate or with a rate that increases as the square root of the stress excess. We establish a family of "static" universal critical exponents which do not depend on this dynamic detail of the model rules: in particular, the exponents for the avalanche size distribution P(S) â¼S-τSf(S/Ldf) and the exponents describing the density of sites at the verge of yielding, which we find to be of the form P(x) ≃P(0) + xθ with P(0) â¼L-a controlling the extremal statistics. On the other hand, we discuss "dynamical" exponents that are sensitive to the local yielding rule. We find that, apart form the dynamical exponent z controlling the duration of avalanches, also the flowcurve's (inverse) Herschel-Bulkley exponent ß ([small gamma, Greek, dot above]â¼ (σ-σc)ß) enters in this category, and is seen to differ in ½ between the two yielding rate cases. We give analytical support to this numerical observation by calculating the exponent variation in the Hébraud-Lequeux model and finding an identical shift. We further discuss an alternative mean-field approximation to yielding only based on the so-called Hurst exponent of the accumulated mechanical noise signal, which gives good predictions for the exponents extracted from simulations of fully spatial models.
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We investigate numerically the yielding transition of a two-dimensional model amorphous solid under external shear. We use a scalar model in terms of values of the total local strain, derived from the full (tensorial) description of the elastic interactions in the system, in which plastic deformations are accounted for by introducing a stochastic "plastic disorder" potential. This scalar model is seen to be equivalent to a collection of Prandtl-Tomlinson particles, which are coupled through an Eshelby quadrupolar kernel. Numerical simulations of this scalar model reveal that the strain rate versus stress curve, close to the critical stress, is of the form γ[over Ì]â¼(σ-σ_{c})^{ß}. Remarkably, we find that the value of ß depends on details of the microscopic plastic potential used, confirming and giving additional support to results previously obtained with the full tensorial model. To rationalize this result, we argue that the Eshelby interaction in the scalar model can be treated to a good approximation in a sort of "dynamical" mean field, which corresponds to a Prandtl-Tomlinson particle that is driven by the applied strain rate in the presence of a stochastic noise generated by all other particles. The dynamics of this Prandtl-Tomlinson particle displays different values of the ß exponent depending on the analytical properties of the microscopic potential, thus giving support to the results of the numerical simulations. Moreover, we find that other critical exponents that depend on details of the dynamics show also a dependence with the form of the disorder, while static exponents are independent of the details of the disorder. Finally, we show how our scalar model relates to other elastoplastic models and to the widely used mean-field version known as the Hébraud-Lequeux model.
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We study the slow stochastic dynamics near the depinning threshold of an elastic interface in a random medium by solving a particularly suited model of hopping interacting particles that belongs to the quenched-Edwards-Wilkinson depinning universality class. The model allows us to compare the cases of uniformly activated and Arrhenius activated hops. In the former case, the velocity accurately follows a standard scaling law of the force and noise intensity with the analog of the thermal rounding exponent satisfying a modified "hyperscaling" relation. For the Arrhenius activation, we find, both numerically and analytically, that the standard scaling form fails for any value of the thermal rounding exponent. We propose an alternative scaling incorporating logarithmic corrections that appropriately fits the numerical results. We argue that this anomalous scaling is related to the strong correlation between activated hops that, alternated with deterministic depinning-like avalanches, occur below the depinning threshold. We rationalize the spatiotemporal patterns by making an analogy of the present model in the near-threshold creep regime with some well-known models with extremal dynamics, particularly the Bak-Sneppen model.
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We study the yielding transition of a two-dimensional amorphous system under shear by using a mesoscopic elasto-plastic model. The model combines a full (tensorial) description of the elastic interactions in the system and the possibility of structural reaccommodations that are responsible for the plastic behavior. The possible structural reaccommodations are encoded in the form of a "plastic disorder" potential, which is chosen independently at each position of the sample to account for local heterogeneities. We observe that the stress must exceed a critical value σ_{c} in order for the system to yield. In addition, when the system yields a flow curve (relating stress σ and strain rate γ[over Ì]) of the form γ[over Ì]â¼(σ-σ_{c})^{ß} is obtained. Remarkably, we observe the value of ß to depend on some details of the plastic disorder potential. For smooth potentials a value of ß≃2.0 is obtained, whereas for potentials obtained as a concatenation of smooth pieces a value ß≃1.5 is observed in the simulations. This indicates a dependence of critical behavior on details of the plastic behavior. In addition, by integrating out nonessential, harmonic degrees of freedom, we derive a simplified scalar version of the model that represents a collection of interacting Prandtl-Tomlinson particles. A mean-field treatment of this interaction reproduces the difference of ß exponents for the two classes of plastic disorder potentials and provides values of ß that compare favorably with those found in the full simulations.
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We discuss the size distribution N(S) of avalanches occurring at the yielding transition of mean-field (i.e., Hebraud-Lequeux) models of amorphous solids. The size distribution follows a power law dependence of the form N(S)â¼S(-τ). However (contrary to what is found in its depinning counterpart), the value of τ depends on details of the dynamic protocol used. For random triggering of avalanches we recover the τ=3/2 exponent typical of mean-field models, which, in particular, is valid for the depinning case. However, for the physically relevant case of external loading through a quasistatic increase of applied strain, a smaller exponent (close to 1) is obtained. This result is rationalized by mapping the problem to an effective random walk in the presence of a moving absorbing boundary.
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We study analytically and by numerical simulations the statistics of the aftershocks generated after large avalanches in models of interface depinning that include viscoelastic relaxation effects. We find in all the analyzed cases that the decay law of aftershocks with time can be understood by considering the typical roughness of the interface and its evolution due to relaxation. In models where there is a single viscoelastic relaxation time there is an exponential decay of the number of aftershocks with time. In models in which viscoelastic relaxation is wave-vector dependent we typically find a power-law dependence of the decay rate that is compatible with the Omori law. The factors that determine the value of the decay exponent are analyzed.
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In many complex systems a continuous input of energy over time can be suddenly relaxed in the form of avalanches. Conventional avalanche models disregard the possibility of internal dynamical effects in the interavalanche periods, and thus miss basic features observed in some real systems. We address this issue by studying a model with viscoelastic relaxation, showing how coherent oscillations of the stress field can emerge spontaneously. Remarkably, these oscillations generate avalanche patterns that are similar to those observed in seismic phenomena.
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The Drössel-Schwabl model of forest fires can be interpreted in a coarse-grained sense as a model for the stress distribution in a single planar fault. Fires in the model are then translated to earthquakes. I show that when a second class of trees that propagate fire only after some finite time is introduced in the model, secondary fires (analogous to aftershocks) are generated, and the statistics of events becomes quantitatively compatible with the Gutenberg-Richter law for earthquakes, with a realistic value of the b exponent. The change in exponent is analytically demonstrated in a simplified percolation scenario. Experimental consequences of the proposed mechanism are indicated.
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We address several questions on the behavior of a numerical model recently introduced to study seismic phenomena, which includes relaxation in the plates as a key ingredient. First, we make an analysis of the scaling of the largest events with system size and show that, when parameters are appropriately interpreted, the typical size of the largest events scale as the system size, without the necessity to tune any parameter. Second, we show that the temporal activity in the model is inherently nonstationary and obtain from here justification and support for the concept of a "seismic cycle" in the temporal evolution of seismic activity. Finally, we ask for the reasons that make the model display a realistic value of the decaying exponent b in the Gutenberg-Richter law for the avalanche size distribution. We explain why relaxation induces a systematic increase in b from its value b≃0.4 observed in the absence of relaxation. However, we have not been able to justify the actual robustness of the model in displaying a consistent b value around the experimentally observed value b≃1.
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I adapted a model recently introduced in the context of seismic phenomena to study creep rupture of materials. It consists of linear elastic fibers that interact in an equal load sharing scheme, complemented with a local viscoelastic relaxation mechanism. The model correctly describes the three stages of the creep process; namely, an initial Andrade regime of creep relaxation, an intermediate regime of rather constant creep rate, and a tertiary regime of accelerated creep toward final failure of the sample. In the tertiary regime, creep rate follows the experimentally observed creep rate over time-to-failure dependence. The time of minimum strain rate is systematically observed to be about 60%-65 % of the time to failure, in accordance with experimental observations. In addition, burst size statistics of breaking events display a -3/2 power law for events close to the time of failure and a steeper decay for the all-time distribution. Statistics of interevent times shows a tendency of the events to cluster temporarily. This behavior should be observable in acoustic emission experiments.
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The Olami-Feder-Christensen model describes a limiting case of an elastic surface that slides on top of a substrate and is one of the simplest models that display some features observed in actual seismicity patterns. However, temporal and spatial correlations of real earthquakes are not correctly described by this model in its original form. I propose and study a modified version of the model, which includes a mechanism of structural relaxation. With this modification, realistic features of seismicity emerge, which are not obtained with the original version, mainly: aftershocks that cluster spatially around the slip surface of the main shock and follow the Omori law, and averaged frictional properties similar to those observed in rock friction, in particular the velocity-weakening effect. In addition, a Gutenberg-Richter law for the decaying of number of earthquakes with magnitude is obtained, with a decaying exponent within the range of experimentally observed values. Contrary to the original version of the model, a realistic value of the exponent appears without the necessity to fine tune any parameter.
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I study the average deformation rate of an amorphous material submitted to an external uniform shear strain rate, in the geometry known as the split-bottom configuration. The material is described using a stochastic model of plasticity at a mesoscopic scale. A shear band is observed to start at the split point at the bottom, and widen progressively towards the surface. In a two-dimensional geometry the average statistical properties of the shear band look similar to those of the directed polymer model. In particular, the surface width of the shear band is found to scale with the system height H as H;{alpha} with alpha=0.68+/-0.02 . In more realistic three-dimensional simulations the exponent changes to alpha=0.60+/-0.02 and the bulk profile of the width of the shear band is closer to a quarter of a circle, as it was observed to be the case in recent simulations of granular materials.
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A two dimensional amorphous material is modeled as an assembly of mesoscopic elemental pieces coupled together to form an elastically coherent structure. Plasticity is introduced as the existence of different minima in the energy landscape of the elemental constituents. Upon the application of an external strain rate, the material shears through the appearance of elemental slip events with quadrupolar symmetry. When the energy landscape of the elemental constituents is kept fixed, the slip events distribute uniformly throughout the sample, producing on average a uniform deformation. However, when the energy landscape at different spatial positions can be rearranged dynamically to account for structural relaxation, the system develops inhomogeneous deformation in the form of shear bands at low shear rates, and stick-slip-like motion at the shear bands for the lowest shear rates. The origin of strain localization is traced back to a region of negative correlation between strain rate and stress, which appears only if structural relaxation is present. The model also reproduces other well known effects in the rheology of amorphous materials, as a stress peak in a strain rate controlled experiment staring from rest, and the increase of the maximum of this peak with sample age.