Subextensive Scaling in the Athermal, Quasistatic Limit of Amorphous Matter in Plastic Shear Flow

Maloney, Craig; Lemaı̂tre, Anaël

doi:10.1103/physrevlett.93.016001

Cited by 248 publications

(142 citation statements)

References 19 publications

(32 reference statements)

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“…ΔΣ is limited by the next plastic event and thus is inversely proportional to the rate at which avalanches are triggered. Although one might think that if the system were twice as large a plastic event would occur twice as soon (implying ΔΣ ∼ 1/L d ), this is not the case, and ΔΣ depends on system size with a nontrivial exponent (9,10,35,36). As argued in refs.…”

Section: Scaling Relationsmentioning

confidence: 99%

“…The comparison between these two phenomena has led to the proposition that the yielding transition is in the universality class of mean-field depinning (33, 34). However, experiments find a reological exponent β > 1 against the β ≤ 1 predicted for elastic depinning, and numerical simulations display intriguing finite size effects that differ from depinning (9,10,(35)(36)(37).Formally, elasto-plastic models are very similar to automaton models known to capture the depinning transition well (17); the key difference lies in the interaction kernel G, long-ranged and of variable sign for elasto-plastic models while essentially a Laplacian for depinning with short-range elasticity. We recently showed (20) that in the presence of long-ranged interaction with variable sign, the distribution of shear transformations at a distance x from instability, P(x), is singular with P(x) ∼ x θ , unlike in depinning for which θ = 0.…”

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Scaling description of the yielding transition in soft amorphous solids at zero temperature

Lin

Lerner

Rosso

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

252

370

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Yield stress materials flow if a sufficiently large shear stress is applied. Although such materials are ubiquitous and relevant for industry, there is no accepted microscopic description of how they yield, even in the simplest situations in which temperature is negligible and in which flow inhomogeneities such as shear bands or fractures are absent. Here we propose a scaling description of the yielding transition in amorphous solids made of soft particles at zero temperature. Our description makes a connection between the Herschel-Bulkley exponent characterizing the singularity of the flow curve near the yield stress Σ c , the extension and duration of the avalanches of plasticity observed at threshold, and the density P(x) of soft spots, or shear transformation zones, as a function of the stress increment x beyond which they yield. We argue that the critical exponents of the yielding transition may be expressed in terms of three independent exponents, θ, d f , and z, characterizing, respectively, the density of soft spots, the fractal dimension of the avalanches, and their duration. Our description shares some similarity with the depinning transition that occurs when an elastic manifold is driven through a random potential, but also presents some striking differences. We test our arguments in an elastoplastic model, an automaton model similar to those used in depinning, but with a different interaction kernel, and find satisfying agreement with our predictions in both two and three dimensions. M any solids will flow and behave as fluids if a sufficiently large shear stress is applied. In crystals, plasticity is governed by the motion of dislocations (1, 2). In amorphous solids, there is no order, and conserved defects cannot be defined. However, as noticed by Argon (3), plasticity consists of elementary events localized in space, called shear transformations, in which a few particles rearrange. This observation supports that there are special locations in the sample, called shear transformation zones (STZs) (4), in which the system lies close to an elastic instability. Several theoretical approaches of plasticity, such as STZ theory (4) or soft glassy rheology (5), assume that such zones relax independently or are coupled to each other via an effective temperature. However, at zero temperature and small applied strain rate _ γ, computer experiments (6-11) and very recent experiments (12, 13) indicate that local rearrangements are not independent: plasticity occurs via avalanches in which many shear transformations are involved, forming elongated structures in which plasticity localizes. If conditions are such that flow is homogeneous (as may occur, for example, in foams or emulsions), one finds that the flow curves are singular at small strain rate and follow a Herschel-Bulkley law Σ − Σ c ∼ _ γ 1=β (14, 15). These features are reproduced qualitatively by elasto-plastic models (16)(17)(18)(19)(20) in which space is discretized. In these models, a site that yields plastically affects the stress in its surroundi...

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Section: Scaling Relationsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Scaling description of the yielding transition in soft amorphous solids at zero temperature

Lin

Lerner

Rosso

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

252

370

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“…Here, we focus on the mechanical yield of amorphous materials, such as molecular and colloidal glasses, foams, and granular matter. The phenomenology exhibited by the yielding point within this vast class of materials, as reported in countless strain-controlled simulations (2)(3)(4)(5)(6)(7)(8) and experiments (9)(10)(11), shows a remarkable degree of universality, despite the highly varied nature of the model systems involved. Among these universal features is the presence, at the onset of flow at yielding, of system-spanning excitations referred to as shear bands (12,13), wherein the shear strongly localizes, leaving the rest of the material unperturbed.…”

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confidence: 99%

Shear bands as manifestation of a criticality in yielding amorphous solids

Parisi

Procaccia

Rainone

et al. 2017

Proc. Natl. Acad. Sci. U.S.A.

109

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Amorphous solids increase their stress as a function of an applied strain until a mechanical yield point whereupon the stress cannot increase anymore, afterward exhibiting a steady state with a constant mean stress. In stress-controlled experiments, the system simply breaks when pushed beyond this mean stress. The ubiquity of this phenomenon over a huge variety of amorphous solids calls for a generic theory that is free of microscopic details. Here, we offer such a theory: The mechanical yield is a thermodynamic phase transition, where yield occurs as a spinodal phenomenon. At the spinodal point, there exists a divergent correlation length that is associated with the system-spanning instabilities (also known as shear bands), which are typical to the mechanical yield. The theory, the order parameter used, and the correlation functions that exhibit the divergent correlation length are universal in nature and can be applied to any amorphous solids that undergo mechanical yield.shear bands | yielding | glass | spinodal | criticality A solid, be it crystalline or amorphous, is operatively defined as any material capable to respond elastically to an externally applied shear deformation (1). However, any solid material, when subject to a large-enough shear strain, finally undergoes a mechanical yield. Here, we focus on the mechanical yield of amorphous materials, such as molecular and colloidal glasses, foams, and granular matter. The phenomenology exhibited by the yielding point within this vast class of materials, as reported in countless strain-controlled simulations (2-8) and experiments (9-11), shows a remarkable degree of universality, despite the highly varied nature of the model systems involved. Among these universal features is the presence, at the onset of flow at yielding, of system-spanning excitations referred to as shear bands (12, 13), wherein the shear strongly localizes, leaving the rest of the material unperturbed. This phenomenon is of capital importance for engineering applications, because it is responsible for the brittleness typical of glassy materials, in particular metallic glasses (14), whose potential for practical use is stymied by their tendency to shear band and fracture (13,15,16).In athermal amorphous solids, the phenomenon has universal features. For strains γ smaller than some critical value denoted as γY , the stress in the material grows on the average when the strain is increased. After yield, the stress cannot grow on the average, no matter how much the strain is increased. The universality of the basic phenomenology of yielding begs a picture of its characteristics in terms of a universal theory, in the sense that such a theory should rely on a statistical-mechanical framework and be independent of details, such as chemical composition and production process of the material. This need was addressed in a recent work (17), wherein building up from ideas first advanced in ref. 18, there emerged a picture of mechanical yielding as a first-order phenomenon [i.e., as a discontinuous ph...

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“…The -process becomes active within the glass transition range with increasing temperature, and is assumed as a spatio-temporal correlation between plastic events, which leads to irreversibility of the rearrangements [88]. Argon specied a plastic rearrangements of atomic regions of tens of atoms as a shear transformation zone (STZ) [5,6,129].…”

Section: Microscopic Description Of Rearrangementsmentioning

confidence: 99%

Nonlinear response and avalanche behavior in metallic glasses

Riechers

Samwer

2017

Eur. Phys. J. Spec. Top.

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The response to dierent stress amplitudes at temperatures below the glass transition temperature is analyzed by mechanical oscillatory excitation of Pd 40 Ni 40 P 20 metallic glass samples in single cantilever bending geometry. The strain response of the material is well below the critical yield stress even for highest stress amplitudes, implying the expectation of a linear relation between stress and strain according to Hook's Law. However, a deviation from the linear behavior is evident, which is evaluated in terms of temperature dependence and inuence of the applied stress amplitude by two dierent approaches of evaluation. The nonlinear approach is based on a nonlinear expansion of the stress-strain-relation, assuming an intrinsic nonlinear character of the shear or elastic modulus. The degree of nonlinearity is extracted by a period-by-period Fourier-analysis and connected to nonlinear coecients, describing the intensity of nonlinearity at the fundamental and higher harmonic frequencies. While rather small stress amplitudes are connected to a linear response behavior, higher stress amplitudes result in nonlinear behavior, which is enhanced with increasing temperature. The characteristic timescale to adapt to a signicant change in stress amplitude in terms of a recovery timescale to a steady state value is connected to the structural relaxation time of the material, suggesting a connection between the observed nonlinearity and primary relaxation processes. The second approach of evaluation is termed the incremental analysis and relates the observed response behavior to avalanches, which occur due to the activation and correlation of local microstructural rearrangements consisting of a few tens of atoms. These rearrangements are termed as shear transformation zones and correspond to localized plastic events, which are superimposed on the linear response behavior of the material. Temperature and stress enhance the occurrence of intervals of monotonously increasing or decreasing strain. These are connected to avalanche behavior according to the power-law observed in the distributions of strain response. Despite the onset of nonlinearity observed in connection with the nonlinear analysis, the power-law behavior is extracted for all applied stress amplitudes, high and low, suggesting activated plastic events throughout the so called Hookean response regime. The intensity of strain response itself shows a direct relation to the strain rate of the experiment. Both approaches of evaluation are compared regarding common aspects and limits. Response phenomena connected to plastic events in the linear response regime are barely reported in literature even though their existence is implied. The incremental analysis yields experimental evidence supporting their occurrence. Moreover it reects the limits of the nonlinear approach, which neglects strain response on the small scale by the period-wise analysis of the data, resulting in an apparent linear strain response. Still, the nonlinear approach illustrates clearl...

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Subextensive Scaling in the Athermal, Quasistatic Limit of Amorphous Matter in Plastic Shear Flow

Cited by 248 publications

References 19 publications

Scaling description of the yielding transition in soft amorphous solids at zero temperature

Scaling description of the yielding transition in soft amorphous solids at zero temperature

Shear bands as manifestation of a criticality in yielding amorphous solids

Nonlinear response and avalanche behavior in metallic glasses

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