Sum Rules for the Quasi-Static and Visco-Elastic Response of Disordered Solids at Zero Temperature

Lemaı̂tre, Anaël; Maloney, Craig

doi:10.1007/s10955-005-9015-5

Cited by 184 publications

(325 citation statements)

References 65 publications

(155 reference statements)

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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: 90%

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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confidence: 90%

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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“…Moreover, the situation is further complicated by the presence of disorder that makes the response of a fragile material inherently nonaffine [5,6].…”

Section: Introductionmentioning

confidence: 99%

Soliton Attenuation and Emergent Hydrodynamics in Fragile Matter

2014

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Disordered packings of soft grains are fragile mechanical systems that lose rigidity upon lowering the external pressure toward zero. At zero pressure, we find that any infinitesimal strain impulse propagates initially as a nonlinear solitary wave progressively attenuated by disorder. We demonstrate that the particle fluctuations generated by the solitary-wave decay can be viewed as a granular analogue of temperature. Their presence is manifested by two emergent macroscopic properties absent in the unperturbed granular packing: a finite pressure that scales with the injected energy (akin to a granular temperature) and an anomalous viscosity that arises even when the microscopic mechanisms of energy dissipation are negligible. Consistent with the interpretation of this state as a fluidlike thermalized state, the shear modulus remains zero. Further, we follow in detail the attenuation of the initial solitary wave, identifying two distinct regimes-an initial exponential decay, followed by a longer power-law decay-and suggest simple models to explain these two regimes.

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“…In order to study the differences and similarities of the vibrational and elastic properties of the bonddepleted fcc and of the fcc with vacancies, we shortly introduce the framework of non-affine lattice dynamics 8 .…”

Section: Non-affine Lattice Dynamicsmentioning

confidence: 99%

“…Then bonds or particles are removed uniformly at random from the system to reach the desired bondoccupation p or vacancy concentration c, respectively. The resulting configuration is then used to solve the equations of motion of non-affine lattice dynamics 8 from which the shear modulus can be extracted.…”

Section: Comparison With Simulationsmentioning

confidence: 99%

Non-affine lattice dynamics of defective fcc crystals

2017

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The mechanical, thermal and vibrational properties of defective crystals are important in many different contexts, from metallurgy and solid-state physics to, more recently, soft matter and colloidal physics. Here we study two different models of disordered fcc crystal lattices, with randomly-removed bonds and with vacancies, respectively, within the framework of non-affine lattice dynamics. We find that both systems feature the same scaling of the shear modulus with the newly defined inversion-symmetry breaking (ISB) parameter, which shows that local inversion-symmetry breaking around defects is the universal root source of the non-affine softening of the shear modulus. This finding allows us to derive analytical relations for the non-affine (zero-frequency) shear modulus as a function of vacancy concentration in excellent agreement with numerical simulations. Nevertheless, due to the different microstructural disorder, the spatial fluctuations of the local ISB parameter are different in the vacancy and bonddepleted case. The vacancy fcc exhibits comparatively a more heterogenous microstructural disorder (due to the broader distribution of coordination number Z), which is reflected in a different scaling relation between boson peak frequency in the DOS and the averageZ. These differences are less important at low vacancy concentrations, where the numerical DOS of the vacancy fcc can be well described theoretically by coherent-potential approximation, presented here for the bond-depleted fcc lattice in 3d.

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Sum Rules for the Quasi-Static and Visco-Elastic Response of Disordered Solids at Zero Temperature

Abstract: We study exact results concerning the non-affine displacement fields observed by Tanguy et al

Cited by 184 publications

References 65 publications

Shear bands as manifestation of a criticality in yielding amorphous solids

Shear bands as manifestation of a criticality in yielding amorphous solids

Soliton Attenuation and Emergent Hydrodynamics in Fragile Matter

Non-affine lattice dynamics of defective fcc crystals

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