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Chakraborty S, Ramola K. Long-range correlations in elastic moduli and local stresses at the unjamming transition. SOFT MATTER 2024; 20:4895-4904. [PMID: 38860707 DOI: 10.1039/d4sm00328d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2024]
Abstract
We explore the behaviour of spatially heterogeneous elastic moduli as well as the correlations between local moduli in model solids with short-range repulsive potentials. We show through numerical simulations that local elastic moduli exhibit long-range correlations, similar to correlations in the local stresses. Specifically, the correlations in local shear moduli exhibit anisotropic behavior at large lengthscales characterized by pinch-point singularities in Fourier space, displaying a structural pattern akin to shear stress correlations. Focussing on two-dimensional jammed solids approaching the unjamming transition, we show that stress correlations exhibit universal properties, characterized by a quadratic p2 dependence of the correlations as the pressure p approaches zero, independent of the details of the model. In contrast, the modulus correlations exhibit a power-law dependence with different exponents depending on the specific interaction potential. Furthermore, we illustrate that while affine responses lack long-range correlations, the total modulus, which encompasses non-affine behavior, exhibits long-range correlations.
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Affiliation(s)
| | - Kabir Ramola
- Tata Institute of Fundamental Research, Hyderabad 500046, India.
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Maharana R, Nampoothiri JN, Ramola K. First-contact-breaking distributions in strained disordered crystals. Phys Rev E 2022; 106:064901. [PMID: 36671178 DOI: 10.1103/physreve.106.064901] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2022] [Accepted: 10/12/2022] [Indexed: 06/17/2023]
Abstract
We derive exact probability distributions for the strain (ε) at which the first stress drop event occurs in uniformly strained disordered crystals, with quenched disorder introduced through polydispersity in particle sizes. We characterize these first stress drop events numerically as well as theoretically and identify them with the first-contact-breaking event in the system. Our theoretical results are corroborated with numerical simulations of quasistatic volumetric strain applied to disordered near-crystalline configurations of athermal soft particles. We develop a general technique to determine the distribution of strains at which the first stress drop events occur, through an exact mapping between the cumulative distribution of first-contact-breaking events and the volume of a convex polytope whose dimension is determined by the number of defects N_{d} in the system. An exact numerical computation of this polytope volume for systems with small numbers of defects displays a remarkable match with the distribution of strains generated through direct numerical simulations. Finally, we derive the distribution of strains at which the first stress drop occurs, assuming that individual contact-breaking events are uncorrelated, which accurately reproduces distributions obtained from direct numerical simulations.
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Affiliation(s)
- Roshan Maharana
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
| | - Jishnu N Nampoothiri
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
- Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02454, USA
| | - Kabir Ramola
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
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Acharya P, Das D, Sengupta S, Ramola K. Emergent power-law interactions in near-crystalline membranes. Phys Rev E 2022; 106:L052902. [PMID: 36559404 DOI: 10.1103/physreve.106.l052902] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2021] [Accepted: 10/29/2022] [Indexed: 06/17/2023]
Abstract
We derive exact results for the fluctuations in energy produced by microscopic disorder in near-crystalline athermal systems. Our formalism captures the heterogeneity in the elastic energy of polydisperse soft disks in energy-minimized configurations. We use this to predict the distribution of interaction energy between two defects in a disordered background. We show that this interaction energy displays a disorder-averaged power-law behavior 〈δE〉∼Δ^{-4} at large distances Δ between the defects. These interactions upon disorder average also display the sixfold symmetry of the underlying reference crystal. Additionally, we show that the fluctuations in the interaction energy encode the athermal correlations introduced by the disordered background. We verify our predictions with energy-minimized configurations of polydisperse soft disks in two dimensions.
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Affiliation(s)
- Pappu Acharya
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
| | - Debankur Das
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
| | - Surajit Sengupta
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
| | - Kabir Ramola
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
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Acharya P, Das D, Ramola K. Disorder perturbation expansion for athermal crystals. Phys Rev E 2021; 104:034608. [PMID: 34654106 DOI: 10.1103/physreve.104.034608] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2021] [Accepted: 09/03/2021] [Indexed: 11/07/2022]
Abstract
We introduce a perturbation expansion for athermal systems that allows an exact determination of displacement fields away from the crystalline state as a response to disorder. We show that the displacement fields in energy-minimized configurations of particles interacting through central potentials with microscopic disorder can be obtained as a series expansion in the strength of the disorder. We introduce a hierarchy of force-balance equations that allows an order-by-order determination of the displacement fields, with the solutions at lower orders providing sources for the higher-order solutions. This allows the simultaneous force-balance equations to be solved, within a hierarchical perturbation expansion to arbitrary accuracy. We present exact results for an isotropic defect introduced into the crystalline ground state at linear order and second order in our expansion. We show that the displacement fields produced by the defect display interesting self-similar properties at every order. We derive a |δr|∼1/r and |δf|∼1/r^{2} decay for the displacement fields and excess interparticle forces at large distances r away from the defect. Finally, we derive nonlinear corrections introduced by the interactions between defects at second order in our expansion. We verify our exact results with displacement fields obtained from energy-minimized configurations of soft disks.
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Affiliation(s)
- Pappu Acharya
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
| | - Debankur Das
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
| | - Kabir Ramola
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
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Acharya P, Sengupta S, Chakraborty B, Ramola K. Athermal Fluctuations in Disordered Crystals. PHYSICAL REVIEW LETTERS 2020; 124:168004. [PMID: 32383939 DOI: 10.1103/physrevlett.124.168004] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2019] [Revised: 01/30/2020] [Accepted: 04/09/2020] [Indexed: 06/11/2023]
Abstract
We analyze the fluctuations in particle positions and interparticle forces in disordered crystals composed of jammed soft particles in the limit of weak disorder. We demonstrate that such athermal systems are fundamentally different from their thermal counterparts, characterized by constrained fluctuations of forces perpendicular to the lattice directions. We develop a disorder perturbation expansion in polydispersity about the crystalline state, which we use to derive exact results to linear order. We show that constrained fluctuations result as a consequence of local force balance conditions, and are characterized by non-Gaussian distributions, which we derive exactly. We analytically predict several properties of such systems, including the scaling of the average coordination with polydispersity and packing fraction, which we verify with numerical simulations using soft disks with one-sided harmonic interactions.
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Affiliation(s)
- Pappu Acharya
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
| | - Surajit Sengupta
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
| | - Bulbul Chakraborty
- Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02454, USA
| | - Kabir Ramola
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
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Tong H, Hu H, Tan P, Xu N, Tanaka H. Revealing Inherent Structural Characteristics of Jammed Particulate Packings. PHYSICAL REVIEW LETTERS 2019; 122:215502. [PMID: 31283321 DOI: 10.1103/physrevlett.122.215502] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/26/2018] [Indexed: 06/09/2023]
Abstract
We look for inherent structural characteristics hidden behind amorphous solid formation by using zero-temperature jammed packings of frictionless particles as models. Differently from previous geometrical approaches, we introduce a microscopic mechanical or vibrational order parameter Ψ, which characterizes the susceptibility of particle motion to infinitesimal thermal excitation. We show that (i) the distribution of Ψ has a power-law tail toward high Ψ and (ii) the spatial organization of Ψ is characterized by a nontrivial scale-free correlation. Both findings (i) and (ii) are regarded as a real-space manifestation of marginal stability due to critical self-organization of jammed packings toward mechanical equilibrium. Furthermore, we find that the power-law exponent of the Ψ distribution tail shows a critical-like scaling behavior toward the unjamming transition, which unveils an intriguing interplay between jamming criticality and marginal stability. Our microscopic order parameter provides new structural insights into the marginal stability and instability of jammed packings and may shed light on the important common structural feature of amorphous solids.
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Affiliation(s)
- Hua Tong
- Department of Fundamental Engineering, Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
- CAS Key Laboratory of Soft Matter Chemistry, Hefei National Laboratory for Physical Sciences at the Microscale, and Department of Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Hao Hu
- School of Physics and Materials Science, Anhui University, Hefei 230601, People's Republic of China
| | - Peng Tan
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, People's Republic of China
| | - Ning Xu
- CAS Key Laboratory of Soft Matter Chemistry, Hefei National Laboratory for Physical Sciences at the Microscale, and Department of Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Hajime Tanaka
- Department of Fundamental Engineering, Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
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Behringer RP, Chakraborty B. The physics of jamming for granular materials: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2019; 82:012601. [PMID: 30132446 DOI: 10.1088/1361-6633/aadc3c] [Citation(s) in RCA: 90] [Impact Index Per Article: 18.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Granular materials consist of macroscopic grains, interacting via contact forces, and unaffected by thermal fluctuations. They are one of a class systems that undergo jamming, i.e. a transition between fluid-like and disordered solid-like states. Roughly twenty years ago, proposals by Cates et al for the shear response of colloidal systems and by Liu and Nagel, for a universal jamming diagram in a parameter space of packing fraction, ϕ, shear stress, τ, and temperature, T raised key questions. Contemporaneously, experiments by Howell et al and numerical simulations by Radjai et al and by Luding et al helped provide a starting point to explore key insights into jamming for dry, cohesionless, granular materials. A recent experimental observation by Bi et al is that frictional granular materials have a a re-entrant region in their jamming diagram. In a range of ϕ, applying shear strain, γ, from an initially force/stress free state leads to fragile (in the sense of Cates et al), then anisotropic shear jammed states. Shear jamming at fixed ϕ is presumably conjugate to Reynolds dilatancy, involving dilation under shear against deformable boundaries. Numerical studies by Radjai and Roux showed that Reynolds dilatancy does not occur for frictionless systems. Recent numerical studies by several groups show that shear jamming occurs for finite, but not infinite, systems of frictionless grains. Shear jamming does not lead to known ordering in position space, but Sarkar et al showed that ordering occurs in a space of force tiles. Experimental studies seeking to understand random loose and random close packings (rlp and rcp) and dating back to Bernal have probed granular packings and their response to shear and intruder motion. These studies suggest that rlp's are anisotropic and shear-jammed-like, whereas rcp's are likely isotropically jammed states. Jammed states are inherently static, but the jamming diagram may provide a context for understanding rheology, i.e. dynamic shear in a variety of systems that include granular materials and suspensions.
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Affiliation(s)
- Robert P Behringer
- Department of Physics & Center for Non-linear and Complex Systems, Duke University, Durham, NC, United States of America. Dr Robert Behringer passed away in July 2018
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