1
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Duan YJ, Nabahat M, Tong Y, Ortiz-Membrado L, Jiménez-Piqué E, Zhao K, Wang YJ, Yang Y, Wada T, Kato H, Pelletier JM, Qiao JC, Pineda E. Connection between Mechanical Relaxation and Equilibration Kinetics in a High-Entropy Metallic Glass. PHYSICAL REVIEW LETTERS 2024; 132:056101. [PMID: 38364152 DOI: 10.1103/physrevlett.132.056101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2023] [Revised: 11/08/2023] [Accepted: 11/17/2023] [Indexed: 02/18/2024]
Abstract
The slow transition from an out-of-equilibrium glass towards a supercooled liquid is a complex relaxation phenomenon. In this Letter, we study the correlation between mechanical relaxation and equilibration kinetics in a Pd_{20}Pt_{20}Cu_{20}Ni_{20}P_{20} high-entropy metallic glass. The evolution of stress relaxation with aging time was obtained with an unprecedented detail, allowing us to pinpoint new interesting features. The long structural relaxation towards equilibrium contains a wide distribution of activation energies, instead of being just associated to the β relaxation as commonly accepted. The stress relaxation time can be correlated with the equilibration rate and we observe a decrease of microstructural heterogeneity which contrasts with an increase of dynamic heterogeneity. These results significantly enhance our insight of the interplay between relaxation dynamics and thermodynamics in metallic glasses.
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Affiliation(s)
- Y J Duan
- School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an 710072, China
- Department of Physics, Institute of Energy Technologies, Universitat Politècnica de Catalunya, Barcelona 08019, Spain
| | - M Nabahat
- Department of Physics, Institute of Energy Technologies, Universitat Politècnica de Catalunya, Barcelona 08019, Spain
| | - Yu Tong
- CAS Key Laboratory of Magnetic Materials and Devices, and Zhejiang Province Key Laboratory of Magnetic Materials and Application Technology, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China
| | - L Ortiz-Membrado
- Department of Materials Science, Universitat Politècnica de Catalunya, Barcelona 08019, Spain
| | - E Jiménez-Piqué
- Department of Materials Science, Universitat Politècnica de Catalunya, Barcelona 08019, Spain
| | - Kun Zhao
- State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China
- School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Yun-Jiang Wang
- State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China
- School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Y Yang
- Department of Mechanical Engineering, College of Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong SAR, China
- Department of Materials Science and Engineering, College of Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong SAR, China
| | - T Wada
- Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
| | - H Kato
- Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
| | - J M Pelletier
- Université de Lyon, MATEIS, UMR CNRS5510, Bâtiment Blaise Pascal, INSA-Lyon, F-69621 Villeurbanne Cedex, France
| | - J C Qiao
- School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an 710072, China
| | - E Pineda
- Department of Physics, Institute of Energy Technologies, Universitat Politècnica de Catalunya, Barcelona 08019, Spain
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2
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Abstract
Materials with random microstructure are characterized by additional thermodynamic parameters, entropy and temperature of microstructure. It has been argued that there is one more law of thermodynamics: entropy of microstructure decays in isolated systems. In this paper, we check this assertion experimentally for the process of grain growth. We show that entropy of grain structure decays indeed as expected. We study also the equation of state for microstructure entropy. In general, microstructure entropy should be a function of microstructure energy and the average grain size. We observed that the equation of state degenerates, and there is a universal dependence of microstructure entropy on microstructure energy, at least at the stage of self similar grain growth.
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3
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Boffi NM, Rycroft CH. Coordinate transformation methodology for simulating quasistatic elastoplastic solids. Phys Rev E 2020; 101:053304. [PMID: 32575210 DOI: 10.1103/physreve.101.053304] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/11/2019] [Accepted: 02/03/2020] [Indexed: 11/07/2022]
Abstract
Molecular dynamics simulations frequently employ periodic boundary conditions where the positions of the periodic images are manipulated in order to apply deformation to the material sample. For example, Lees-Edwards conditions use moving periodic images to apply simple shear. Here, we examine the problem of precisely comparing this type of simulation to continuum solid mechanics. We employ a hypoelastoplastic mechanical model, and develop a projection method to enforce quasistatic equilibrium. We introduce a simulation framework that uses a fixed Cartesian computational grid on a reference domain, and which imposes deformation via a time-dependent coordinate transformation to the physical domain. As a test case for our method, we consider the evolution of shear bands in a bulk metallic glass using the shear transformation zone theory of amorphous plasticity. We examine the growth of shear bands in simple shear and pure shear conditions as a function of the initial preparation of the bulk metallic glass.
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Affiliation(s)
- Nicholas M Boffi
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Chris H Rycroft
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA.,Computational Research Division, Lawrence Berkeley Laboratory, Berkeley, California 94720, USA
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4
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Ma X, Elbanna A. Strain localization in dry sheared granular materials: A compactivity-based approach. Phys Rev E 2018; 98:022906. [PMID: 30253526 DOI: 10.1103/physreve.98.022906] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2018] [Indexed: 11/07/2022]
Abstract
Shear banding is widely observed in natural fault zones as well as in laboratory experiments on granular materials. Understanding the dynamics of strain localization under different loading conditions is essential for quantifying strength evolution of fault gouge and energy partitioning during earthquakes and characterizing rheological transitions and fault zone structure changes. To that end, we develop a physics-based continuum model for strain localization in sheared granular materials. The grain-scale dynamics is described by the shear transformation zone (STZ) theory, a nonequilibrium statistical thermodynamic framework for viscoplastic deformation in amorphous materials. Using a finite strain computational framework, we investigate the initiation and growth of complex shear bands under a variety of loading conditions and identify implications for strength evolution and the ductile to brittle transition. Our numerical results show similar localization patterns to field and laboratory observations and suggest that shear zones show more ductile response at higher confining pressures, lower dilatancy, and loose initial conditions. Lower pressures, higher dilatancy, and dense initial conditions favor a brittle response and larger strength drops. These findings shed light on a range of mechanisms for strength evolution in dry sheared granular materials and provide a critical input to physics-based multiscale models of fault zone instabilities.
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Affiliation(s)
- Xiao Ma
- Department of Civil and Environmental Engineering, University of Illinois, Urbana-Champaign, Illinois, USA
| | - Ahmed Elbanna
- Department of Civil and Environmental Engineering, University of Illinois, Urbana-Champaign, Illinois, USA
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5
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Langer JS. Thermodynamic analysis of the Livermore molecular-dynamics simulations of dislocation-mediated plasticity. Phys Rev E 2018; 98:023006. [PMID: 30253488 DOI: 10.1103/physreve.98.023006] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2018] [Indexed: 11/07/2022]
Abstract
Results of recent large-scale molecular dynamics simulations of dislocation-mediated solid plasticity are compared with predictions of the statistical thermodynamic theory of these phenomena. These computational and theoretical analyses are in substantial agreement with each other in both their descriptions of strain-rate-dependent steady plastic flows and of transient stress peaks associated with initially small densities of dislocations. The comparisons between the numerical simulations and basic theory reveal inconsistencies in some conventional phenomenological descriptions of solid plasticity.
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Affiliation(s)
- J S Langer
- Department of Physics, University of California, Santa Barbara, California 93106-9530, USA
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6
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Zeng F, Jiang MQ, Dai LH. Dilatancy induced ductile-brittle transition of shear band in metallic glasses. Proc Math Phys Eng Sci 2018; 474:20170836. [PMID: 29740259 DOI: 10.1098/rspa.2017.0836] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2017] [Accepted: 03/12/2018] [Indexed: 11/12/2022] Open
Abstract
Dilatancy-generated structural disordering, an inherent feature of metallic glasses (MGs), has been widely accepted as the physical mechanism for the primary origin and structural evolution of shear banding, as well as the resultant shear failure. However, it remains a great challenge to determine, to what degree of dilatation, a shear banding will evolve into a runaway shear failure. In this work, using in situ acoustic emission monitoring, we probe the dilatancy evolution at the different stages of individual shear band in MGs that underwent severely plastic deformation by the controlled cutting technology. A scaling law is revealed that the dilatancy in a shear band is linearly related to its evolution degree. A transition from ductile-to-brittle shear bands is observed, where the formers dominate stable serrated flow, and the latter lead to a runaway instability (catastrophe failure) of serrated flow. To uncover the underlying mechanics, we develop a theoretical model of shear-band evolution dynamics taking into account an atomic-scale deformation process. Our theoretical results agree with the experimental observations, and demonstrate that the atomic-scale volume expansion arises from an intrinsic shear-band evolution dynamics. Importantly, the onset of the ductile-brittle transition of shear banding is controlled by a critical dilatation.
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Affiliation(s)
- F Zeng
- State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China.,Software Center for High Performance Numerical Simulation, Chinese Academy of Engineering Physics, Beijing 100088, People's Republic of China
| | - M Q Jiang
- State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China.,School of Engineering Science, University of Chinese Academy of Sciences, Beijing 101408, People's Republic of China
| | - L H Dai
- State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China.,School of Engineering Science, University of Chinese Academy of Sciences, Beijing 101408, People's Republic of China
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7
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Hinkle AR, Rycroft CH, Shields MD, Falk ML. Coarse graining atomistic simulations of plastically deforming amorphous solids. Phys Rev E 2017; 95:053001. [PMID: 28618619 DOI: 10.1103/physreve.95.053001] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2017] [Indexed: 06/07/2023]
Abstract
The primary mode of failure in disordered solids results from the formation and persistence of highly localized regions of large plastic strains known as shear bands. Continuum-level field theories capable of predicting this mechanical response rely upon an accurate representation of the initial and evolving states of the amorphous structure. We perform molecular dynamics simulations of a metallic glass and propose a methodology for coarse graining discrete, atomistic quantities, such as the potential energies of the elemental constituents. A strain criterion is established and used to distinguish the coarse-grained degrees-of-freedom inside the emerging shear band from those of the surrounding material. A signal-to-noise ratio provides a means of evaluating the strength of the signal of the shear band as a function of the coarse graining. Finally, we investigate the effect of different coarse graining length scales by comparing a two-dimensional, numerical implementation of the effective-temperature description in the shear transformation zone (STZ) theory with direct molecular dynamics simulations. These comparisons indicate the coarse graining length scale has a lower bound, above which there is a high level of agreement between the atomistics and the STZ theory, and below which the concept of effective temperature breaks down.
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Affiliation(s)
- Adam R Hinkle
- Department of Materials Science & Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Chris H Rycroft
- Paulson School of Engineering & Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Michael D Shields
- Department of Materials Science & Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA
- Department of Civil Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Michael L Falk
- Department of Materials Science & Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA
- Department of Mechanical Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA
- Department of Physics & Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
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8
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Langer JS. Statistical thermodynamics of strain hardening in polycrystalline solids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:032125. [PMID: 26465444 DOI: 10.1103/physreve.92.032125] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/26/2015] [Indexed: 06/05/2023]
Abstract
This paper starts with a systematic rederivation of the statistical thermodynamic equations of motion for dislocation-mediated plasticity proposed in 2010 by Langer, Bouchbinder, and Lookman [Acta Mat. 58, 3718 (2010)ACMAFD1359-645410.1016/j.actamat.2010.03.009]. It then uses that theory to explain the anomalous rate-hardening behavior reported in 1988 by Follansbee and Kocks and to explore the relation between hardening rate and grain size reported in 1995 by Meyers et al. A central theme is the need for physics-based, nonequilibrium analyses in developing predictive theories of the strength of polycrystalline materials.
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Affiliation(s)
- J S Langer
- Department of Physics, University of California, Santa Barbara, California 93106-9530, USA
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9
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Lieou CKC, Elbanna AE, Langer JS, Carlson JM. Stick-slip instabilities in sheared granular flow: The role of friction and acoustic vibrations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022209. [PMID: 26382396 DOI: 10.1103/physreve.92.022209] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2015] [Indexed: 06/05/2023]
Abstract
We propose a theory of shear flow in dense granular materials. A key ingredient of the theory is an effective temperature that determines how the material responds to external driving forces such as shear stresses and vibrations. We show that, within our model, friction between grains produces stick-slip behavior at intermediate shear rates, even if the material is rate strengthening at larger rates. In addition, externally generated acoustic vibrations alter the stick-slip amplitude, or suppress stick-slip altogether, depending on the pressure and shear rate. We construct a phase diagram that indicates the parameter regimes for which stick-slip occurs in the presence and absence of acoustic vibrations of a fixed amplitude and frequency. These results connect the microscopic physics to macroscopic dynamics and thus produce useful information about a variety of granular phenomena, including rupture and slip along earthquake faults, the remote triggering of instabilities, and the control of friction in material processing.
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Affiliation(s)
- Charles K C Lieou
- Department of Physics, University of California, Santa Barbara, California 93106, USA
| | - Ahmed E Elbanna
- Department of Civil and Environmental Engineering, University of Illinois, Urbana-Champaign, Illinois 61801, USA
| | - J S Langer
- Department of Physics, University of California, Santa Barbara, California 93106, USA
| | - J M Carlson
- Department of Physics, University of California, Santa Barbara, California 93106, USA
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10
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Langer JS. Shear-transformation-zone theory of yielding in athermal amorphous materials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012318. [PMID: 26274172 DOI: 10.1103/physreve.92.012318] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2015] [Indexed: 06/04/2023]
Abstract
Yielding transitions in athermal amorphous materials undergoing steady-state shear flow resemble critical phenomena. Historically, they have been described by the Herschel-Bulkley rheological formula, which implies singular behaviors at yield points. In this paper, I examine this class of phenomena using an elementary version of the thermodynamic shear-transformation-zone (STZ) theory, focusing on the role of the effective disorder temperature, and paying special attention to scaling and dimensional arguments. I find a wide variety of Herschel-Bulkley-like rheologies but, for fundamental reasons not specific to the STZ theory, conclude that the yielding transition is not truly critical. In particular, for realistic many-body models with short-range interactions, there is a correlation length that grows rapidly but ultimately saturates near the yield point.
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Affiliation(s)
- J S Langer
- Department of Physics, University of California, Santa Barbara, California 93106-9530, USA
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11
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Benzi R, Sbragaglia M, Scagliarini A, Perlekar P, Bernaschi M, Succi S, Toschi F. Internal dynamics and activated processes in soft-glassy materials. SOFT MATTER 2015; 11:1271-1280. [PMID: 25560202 DOI: 10.1039/c4sm02341b] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Plastic rearrangements play a crucial role in the characterization of soft-glassy materials, such as emulsions and foams. Based on numerical simulations of soft-glassy systems, we study the dynamics of plastic rearrangements at the hydrodynamic scales where thermal fluctuations can be neglected. Plastic rearrangements require an energy input, which can be either provided by external sources, or made available through time evolution in the coarsening dynamics, in which the total interfacial area decreases as a consequence of the slow evolution of the dispersed phase from smaller to large droplets/bubbles. We first demonstrate that our hydrodynamic model can quantitatively reproduce such coarsening dynamics. Then, considering periodically oscillating strains, we characterize the number of plastic rearrangements as a function of the external energy-supply, and show that they can be regarded as activated processes induced by a suitable "noise" effect. Here we use the word noise in a broad sense, referring to the internal non-equilibrium dynamics triggered by spatial random heterogeneities and coarsening. Finally, by exploring the interplay between the internal characteristic time-scale of the coarsening dynamics and the external time-scale associated with the imposed oscillating strain, we show that the system exhibits the phenomenon of stochastic resonance, thereby providing further credit to the mechanical activation scenario.
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Affiliation(s)
- R Benzi
- Department of Physics and INFN, University of "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Rome, Italy.
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12
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Daub EG, Klaumünzer D, Löffler JF. Effective temperature dynamics of shear bands in metallic glasses. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062405. [PMID: 25615110 DOI: 10.1103/physreve.90.062405] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2014] [Indexed: 06/04/2023]
Abstract
We study the plastic deformation of bulk metallic glasses with shear transformation zone (STZ) theory, a physical model for plasticity in amorphous systems, and compare it with experimental data. In STZ theory, plastic deformation occurs when localized regions rearrange due to applied stress and the density of these regions is determined by a dynamically evolving effective disorder temperature. We compare the predictions of STZ theory to experiments that explore the low-temperature deformation of Zr-based bulk metallic glasses via shear bands at various thermal temperatures and strain rates. By following the evolution of effective temperature with time, strain rate, and temperature through a series of approximate and numerical solutions to the STZ equations, we successfully model a suite of experimentally observed phenomena, including shear-band aging as apparent from slide-hold-slide tests, a temperature-dependent steady-state flow stress, and a strain-rate- and temperature-dependent transition from stick-slip (serrated flow) to steady-sliding (nonserrated flow). We find that STZ theory quantitatively matches the observed experimental data and provides a framework for relating the experimentally measured energy scales to different types of atomic rearrangements.
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Affiliation(s)
- Eric G Daub
- Geophysics Group and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA and Laboratory of Metal Physics and Technology, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland and Center for Earthquake Research and Information, University of Memphis, Memphis, Tennessee 38152, USA
| | - David Klaumünzer
- Laboratory of Metal Physics and Technology, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland
| | - Jörg F Löffler
- Laboratory of Metal Physics and Technology, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland
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13
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Lieou CKC, Elbanna AE, Langer JS, Carlson JM. Shear flow of angular grains: acoustic effects and nonmonotonic rate dependence of volume. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032204. [PMID: 25314434 DOI: 10.1103/physreve.90.032204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2014] [Indexed: 06/04/2023]
Abstract
Naturally occurring granular materials often consist of angular particles whose shape and frictional characteristics may have important implications on macroscopic flow rheology. In this paper, we provide a theoretical account for the peculiar phenomenon of autoacoustic compaction-nonmonotonic variation of shear band volume with shear rate in angular particles-recently observed in experiments. Our approach is based on the notion that the volume of a granular material is determined by an effective-disorder temperature known as the compactivity. Noise sources in a driven granular material couple its various degrees of freedom and the environment, causing the flow of entropy between them. The grain-scale dynamics is described by the shear-transformation-zone theory of granular flow, which accounts for irreversible plastic deformation in terms of localized flow defects whose density is governed by the state of configurational disorder. To model the effects of grain shape and frictional characteristics, we propose an Ising-like internal variable to account for nearest-neighbor grain interlocking and geometric frustration and interpret the effect of friction as an acoustic noise strength. We show quantitative agreement between experimental measurements and theoretical predictions and propose additional experiments that provide stringent tests on the new theoretical elements.
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Affiliation(s)
- Charles K C Lieou
- Department of Physics, University of California, Santa Barbara, California 93106, USA
| | - Ahmed E Elbanna
- Department of Civil and Environmental Engineering, University of Illinois, Urbana-Champaign, Illinois 61801, USA
| | - J S Langer
- Department of Physics, University of California, Santa Barbara, California 93106, USA
| | - J M Carlson
- Department of Physics, University of California, Santa Barbara, California 93106, USA
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14
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Luo AM, Öttinger HC. Nonequilibrium thermodynamics of the shear-transformation-zone model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:022137. [PMID: 25353452 DOI: 10.1103/physreve.89.022137] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/2013] [Indexed: 06/04/2023]
Abstract
The shear-transformation-zone (STZ) model has been applied numerous times to describe the plastic deformation of different types of amorphous systems. We formulate this model within the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) framework, thereby clarifying the thermodynamic structure of the constitutive equations and guaranteeing thermodynamic consistency. We propose natural, physically motivated forms for the building blocks of the GENERIC, which combine to produce a closed set of time evolution equations for the state variables, valid for any choice of free energy. We demonstrate an application of the new GENERIC-based model by choosing a simple form of the free energy. In addition, we present some numerical results and contrast those with the original STZ equations.
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Affiliation(s)
- Alan M Luo
- ETH Zürich, Department of Materials, Polymer Physics, CH-8093, Zürich, Switzerland
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15
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Lieou CKC, Elbanna AE, Carlson JM. Grain fragmentation in sheared granular flow: weakening effects, energy dissipation, and strain localization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:022203. [PMID: 25353463 DOI: 10.1103/physreve.89.022203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2013] [Indexed: 06/04/2023]
Abstract
We describe the shear flow of a disordered granular material in the presence of grain fracture using the shear-transformation-zone theory of amorphous plasticity adapted to systems with a hard-core interparticle interaction. To this end, we develop the equations of motion for this system within a statistical-thermodynamic framework analogous to that used in the analysis of molecular glasses. For hard-core systems, the amount of internal, configurational disorder is characterized by the compactivity X = ∂V / ∂S(C), where V and S(C) are, respectively, the volume and configurational entropy. Grain breakage is described by a constitutive equation for the temporal evolution of a characteristic grain size a, based on fracture mechanics. We show that grain breakage is a weakening mechanism, significantly lowering the flow stress at large strain rates, if the material is rate strengthening in character. We show in addition that if the granular material is sufficiently aged, spatial inhomogeneity in configurational disorder results in strain localization. We also show that grain splitting contributes significantly to comminution at small shear strains, while grain abrasion becomes dominant at large shear displacements.
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Affiliation(s)
- Charles K C Lieou
- Department of Physics, University of California, Santa Barbara, California 93106, USA
| | - Ahmed E Elbanna
- Department of Civil and Environmental Engineering, University of Illinois, Urbana-Champaign, Illinois 61801, USA
| | - Jean M Carlson
- Department of Physics, University of California, Santa Barbara, California 93106, USA
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16
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Langer JS, Egami T. Glass dynamics at high strain rates. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:011502. [PMID: 23005420 DOI: 10.1103/physreve.86.011502] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2012] [Indexed: 06/01/2023]
Abstract
We present a shear-transformation-zone (STZ) theoretical analysis of molecular-dynamics simulations of a rapidly sheared metallic glass. These simulations are especially revealing because, although they are limited to high strain rates, they span temperatures ranging from well below to well above the glass transition. With one important discrepancy, the simplified STZ theory used here reproduces the simulation data, including the way in which those data can be made to collapse approximately onto simple curves by a scaling transformation. The STZ analysis implies that the system's behavior at high strain rates is controlled primarily by effective-temperature thermodynamics, as opposed to system-specific details of the molecular interactions. The discrepancy between theory and simulations occurs at the lower strain rates for temperatures near the glass transition. We argue that this discrepancy can be resolved by the same multispecies generalization of STZ theory that has been proposed recently for understanding frequency-dependent viscoelastic responses, Stokes-Einstein violations, and stretched-exponential relaxation in equilibrated glassy materials.
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Affiliation(s)
- J S Langer
- Department of Physics, University of California, Santa Barbara, California 93106-9530, USA
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17
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Lieou CKC, Langer JS. Nonequilibrium thermodynamics in sheared hard-sphere materials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:061308. [PMID: 23005087 DOI: 10.1103/physreve.85.061308] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2012] [Indexed: 06/01/2023]
Abstract
We combine the shear-transformation-zone (STZ) theory of amorphous plasticity with Edwards' statistical theory of granular materials to describe shear flow in a disordered system of thermalized hard spheres. The equations of motion for this system are developed within a statistical thermodynamic framework analogous to that which has been used in the analysis of molecular glasses. For hard spheres, the system volume V replaces the internal energy U as a function of entropy S in conventional statistical mechanics. In place of the effective temperature, the compactivity X=∂V/∂S characterizes the internal state of disorder. We derive the STZ equations of motion for a granular material accordingly, and predict the strain rate as a function of the ratio of the shear stress to the pressure for different values of a dimensionless, temperature-like variable near a jamming transition. We use a simplified version of our theory to interpret numerical simulations by Haxton, Schmiedeberg, and Liu, and in this way are able to obtain useful insights about internal rate factors and relations between jamming and glass transitions.
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Affiliation(s)
- Charles K C Lieou
- Department of Physics, University of California, Santa Barbara, California 93106, USA
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Bouchbinder E, Langer JS. Shear-transformation-zone theory of linear glassy dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061503. [PMID: 21797368 DOI: 10.1103/physreve.83.061503] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2011] [Indexed: 05/31/2023]
Abstract
We present a linearized shear-transformation-zone (STZ) theory of glassy dynamics in which the internal STZ transition rates are characterized by a broad distribution of activation barriers. For slowly aging or fully aged systems, the main features of the barrier-height distribution are determined by the effective temperature and other near-equilibrium properties of the configurational degrees of freedom. Our theory accounts for the wide range of relaxation rates observed in both metallic glasses and soft glassy materials such as colloidal suspensions. We find that the frequency-dependent loss modulus is not just a superposition of Maxwell modes. Rather, it exhibits an α peak that rises near the viscous relaxation rate and, for nearly jammed, glassy systems, extends to much higher frequencies in accord with experimental observations. We also use this theory to compute strain recovery following a period of large, persistent deformation and then abrupt unloading. We find that strain recovery is determined in part by the initial barrier-height distribution, but that true structural aging also occurs during this process and determines the system's response to subsequent perturbations. In particular, we find by comparison with experimental data that the initial deformation produces a highly disordered state with a large population of low activation barriers, and that this state relaxes quickly toward one in which the distribution is dominated by the high barriers predicted by the near-equilibrium analysis. The nonequilibrium dynamics of the barrier-height distribution is the most important of the issues raised and left unresolved in this paper.
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Affiliation(s)
- Eran Bouchbinder
- Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel
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Warren M, Rottler J. Deformation-induced accelerated dynamics in polymer glasses. J Chem Phys 2010; 133:164513. [DOI: 10.1063/1.3505149] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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21
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Hermundstad AM, Daub EG, Carlson JM. Energetics of strain localization in a model of seismic slip. ACTA ACUST UNITED AC 2010. [DOI: 10.1029/2009jb006960] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Daub EG, Manning ML, Carlson JM. Pulse-like, crack-like, and supershear earthquake ruptures with shear strain localization. ACTA ACUST UNITED AC 2010. [DOI: 10.1029/2009jb006388] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Krüger M, Fuchs M. Nonequilibrium fluctuation-dissipation relations of interacting Brownian particles driven by shear. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:011408. [PMID: 20365374 DOI: 10.1103/physreve.81.011408] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2009] [Indexed: 05/29/2023]
Abstract
We present a detailed analysis of the fluctuation-dissipation theorem (FDT) close to the glass transition in colloidal suspensions under steady shear using mode coupling approximations. Starting point is the many-particle Smoluchowski equation. Under shear, detailed balance is broken and the response functions in the stationary state are smaller at long times than estimated from the equilibrium FDT. An asymptotically constant relation connects response and fluctuations during the shear driven decay, restoring the form of the FDT with, however, a ratio different from the equilibrium one. At short times, the equilibrium FDT holds. We follow two independent approaches whose results are in qualitative agreement. To discuss the derived fluctuation-dissipation ratios, we show an exact reformulation of the susceptibility which contains not the full Smoluchowski operator as in equilibrium, but only its well defined Hermitian part. This Hermitian part can be interpreted as governing the dynamics in the frame comoving with the probability current. We present a simple toy model which illustrates the FDT violation in the sheared colloidal system.
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Affiliation(s)
- Matthias Krüger
- Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany
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Daub EG, Carlson JM. Stick-slip instabilities and shear strain localization in amorphous materials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:066113. [PMID: 20365237 DOI: 10.1103/physreve.80.066113] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2009] [Indexed: 05/29/2023]
Abstract
We study the impact of strain localization on the stability of frictional slipping in dense amorphous materials. We model the material using shear transformation zone (STZ) theory, a continuum approximation for plastic deformation in amorphous solids. In the STZ model, the internal state is quantified by an effective disorder temperature, and the effective temperature dynamics capture the spontaneous localization of strain. We study the effect of strain localization on stick-slip instabilities by coupling the STZ model to a noninertial spring slider system. We perform a linear stability analysis to generate a phase diagram that connects the small scale physics of strain localization to the macroscopic stability of sliding. Our calculations determine the values of spring stiffness and driving velocity where steady sliding becomes unstable and we confirm our results through numerical integration. We investigate both homogeneous deformation, where no shear band forms, and localized deformation, where a narrow shear band spontaneously forms and accommodates all of the deformation. Our results show that at a given velocity, strain localization leads to unstable frictional sliding at a much larger spring stiffness compared to homogeneous deformation, and that localized deformation cannot be approximated by a homogeneous model with a narrower material. We also find that strain localization provides a physical mechanism for irregular stick-slip cycles in certain parameter ranges. Our results quantitatively connect the internal physics of deformation in amorphous materials to the larger scale frictional dynamics of stick-slip.
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Affiliation(s)
- Eric G Daub
- Geophysics Group and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
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Rottler J. Fracture in glassy polymers: a molecular modeling perspective. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2009; 21:463101. [PMID: 21715863 DOI: 10.1088/0953-8984/21/46/463101] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
Over the past 25 years, molecular modeling and simulations have provided important insights into the physics of deformation and fracture of glassy polymers. This review presents an overview of key results discussed in the context of experimentally observed polymer behavior. Both atomistic and coarse-grained polymer models have been used in different deformation protocols to study elastic properties, shear yielding, creep, physical aging, strain hardening and crazing. Simulations reproduce most of the macroscopic features of plasticity in polymer glasses such as stress-strain relations and creep response, and reveal information about the underlying atomistic processes. Trends of the shear yield stress with loading conditions, temperature and strain rate, and the atomistic dynamics under load have been systematically explored. Most polymers undergo physical aging, which leads to a history-dependent mechanical response. Simulations of strain hardening and crazing demonstrate the nature of polymer entanglements in the glassy state and the role of local plasticity and provide insight into the origin of fracture toughness of amorphous polymers.
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Affiliation(s)
- Jörg Rottler
- Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC, V6T 1Z1, Canada
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Bouchbinder E, Langer JS. Nonequilibrium thermodynamics of driven amorphous materials. III. Shear-transformation-zone plasticity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:031133. [PMID: 19905088 DOI: 10.1103/physreve.80.031133] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2009] [Indexed: 05/28/2023]
Abstract
We use the internal-variable, effective-temperature thermodynamics developed in two preceding papers to reformulate the shear-transformation-zone (STZ) theory of amorphous plasticity. As required by the preceding analysis, we make explicit approximations for the energy and entropy of the STZ internal degrees of freedom. We then show that the second law of thermodynamics constrains the STZ transition rates to have an Eyring form as a function of the effective temperature. Finally, we derive an equation of motion for the effective temperature for the case of STZ dynamics.
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Affiliation(s)
- Eran Bouchbinder
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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Bouchbinder E, Langer JS. Nonequilibrium thermodynamics of driven amorphous materials. II. Effective-temperature theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:031132. [PMID: 19905087 DOI: 10.1103/physreve.80.031132] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2009] [Indexed: 05/28/2023]
Abstract
We develop a theory of the effective disorder temperature in glass-forming materials driven away from thermodynamic equilibrium by external forces. Our basic premise is that the slow configurational degrees of freedom of such materials are weakly coupled to the fast kinetic-vibrational degrees of freedom and therefore that these two subsystems can be described by different temperatures during deformation. We use results from the preceding paper on the nonequilibrium thermodynamics of systems with internal degrees of freedom to derive an equation of motion for the effective temperature and to learn how this temperature couples to the dynamics of the system as a whole.
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Affiliation(s)
- Eran Bouchbinder
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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Manning ML, Daub EG, Langer JS, Carlson JM. Rate-dependent shear bands in a shear-transformation-zone model of amorphous solids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:016110. [PMID: 19257110 DOI: 10.1103/physreve.79.016110] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2008] [Indexed: 05/27/2023]
Abstract
We use shear transformation zone (STZ) theory to develop a deformation map for amorphous solids as a function of the imposed shear rate and initial material preparation. The STZ formulation incorporates recent simulation results [T. K. Haxton and A. J. Liu, Phys. Rev. Lett. 99, 195701 (2007)] showing that the steady state effective temperature is rate dependent. The resulting model predicts a wide range of deformation behavior as a function of the initial conditions, including homogeneous deformation, broad shear bands, extremely thin shear bands, and the onset of material failure. In particular, the STZ model predicts homogeneous deformation for shorter quench times and lower strain rates, and inhomogeneous deformation for longer quench times and higher strain rates. The location of the transition between homogeneous and inhomogeneous flow on the deformation map is determined in part by the steady state effective temperature, which is likely material dependent. This model also suggests that material failure occurs due to a runaway feedback between shear heating and the local disorder, and provides an explanation for the thickness of shear bands near the onset of material failure. We find that this model, which resolves dynamics within a sheared material interface, predicts that the stress weakens with strain much more rapidly than a similar model which uses a single state variable to specify internal dynamics on the interface.
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Affiliation(s)
- M L Manning
- Center for Theoretical Science, Princeton University, New Jersey 08544, USA.
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Abate AR, Durian DJ. Effective temperatures and activated dynamics for a two-dimensional air-driven granular system on two approaches to jamming. PHYSICAL REVIEW LETTERS 2008; 101:245701. [PMID: 19113632 DOI: 10.1103/physrevlett.101.245701] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2008] [Revised: 09/22/2008] [Indexed: 05/27/2023]
Abstract
We present experiments on several distinct effective temperatures in a granular system at a sequence of increasing packing densities and at a sequence of decreasing driving rates. This includes single-grain measurements based on the mechanical energies of both the grains and an embedded oscillator, as well as a collective measurement based on the Einstein relation between diffusivity and mobility, which all probe different time scales. Remarkably, all effective temperatures agree. Furthermore, mobility data along the two trajectories collapse when plotted vs effective temperature and exhibit an Arrhenius form with the same energy barrier as the microscopic relaxation time.
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Affiliation(s)
- A R Abate
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396, USA
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Rottler J, Maass P. Aging in the shear-transformation-zone theory of plastic deformation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:056109. [PMID: 19113192 DOI: 10.1103/physreve.78.056109] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2008] [Indexed: 05/27/2023]
Abstract
Aging phenomena in the plastic response of amorphous solids are studied within the theory of shear transformation zones (STZs), which describes the kinetic rearrangement of localized defects in response to external stress. To account for the slow, nonequilibrium dynamics after a quench below the glass transition temperature, two possible models are considered. In the first model, transition rates between the internal states of STZs decrease with time, while in the second model aging occurs due to the relaxation of an effective temperature that determines the number density of STZs and other out-of-equilibium degrees of freedom. It is shown that for reasonable choices of parameters, both models capture qualitatively typical aging features seen in computer simulations and experiments: (i) compliance curves measured for different waiting times t_{w} after the quench can be superimposed, when the observation times are rescaled with relaxation times proportional, variantt_{w};{mu} , 0<micro< or =1 , and (ii) stress-strain curves show a stationary plateau stress independent of t_{w} and a peak stress that increases logarithmically with both t_{w} and the strain rate. Trends of the aging behavior with the quench temperature are also discussed.
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Affiliation(s)
- Jörg Rottler
- Department of Physics and Astronomy, The University of British Columbia, 6224 Agricultural Road, Vancouver, BC, Canada V6T 1Z1.
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Bouchbinder E, Lo TS. Front propagation at the onset of plastic yielding. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:026119. [PMID: 18850909 DOI: 10.1103/physreve.78.026119] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2007] [Indexed: 05/26/2023]
Abstract
The existence of a finite threshold, the yield stress, for the onset of plastic yielding is a universal feature of plasticity. This jamming-unjamming transition is naturally accounted for by the dynamics of a bistable internal state field. We show, within the athermal shear transformation zone theory of amorphous plasticity, that the transition is accompanied by the propagation of plastic fronts. We further show that the mean-field theory cannot select the velocity of these fronts, and go beyond the mean-field description to include fluctuations and correlations effects, resulting in additional nonlocal terms in the equations. Finally, we demonstrate that these terms, with an associated intrinsic length scale, provide a velocity selection mechanism for the plastic fronts.
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Affiliation(s)
- Eran Bouchbinder
- Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
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Bouchbinder E. Effective temperature dynamics in an athermal amorphous plasticity theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:051505. [PMID: 18643072 DOI: 10.1103/physreve.77.051505] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2008] [Revised: 04/13/2008] [Indexed: 05/26/2023]
Abstract
Recent developments in the theory of amorphous plasticity point to the central role played by the concept of an effective disorder temperature T_{eff} . An athermal dynamics for T_{eff} is proposed in the framework of a deformation theory and discussed in light of recent steady-state simulations by Haxton and Liu [T. K. Haxton and A. J. Liu, Phys. Rev. Lett. 99, 195701 (2007)]. The structure of the resulting theory, its parameters and transient dynamics are discussed and compared to available data.
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Affiliation(s)
- Eran Bouchbinder
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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Langer JS. Shear-transformation-zone theory of plastic deformation near the glass transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:021502. [PMID: 18352029 DOI: 10.1103/physreve.77.021502] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2007] [Indexed: 05/26/2023]
Abstract
The shear-transformation-zone (STZ) theory of plastic deformation in glass-forming materials is reformulated in light of recent progress in understanding the roles played by the effective disorder temperature and entropy flow in nonequilibrium situations. A distinction between fast and slow internal-state variables reduces the theory to just two coupled equations of motion, one describing the plastic response to applied stresses and the other the dynamics of the effective temperature. The analysis leading to these equations contains, as a by-product, a fundamental reinterpretation of the dynamic yield stress in amorphous materials. In order to put all these concepts together in a realistic context, I conclude with a reexamination of the experimentally observed rheological behavior of a bulk metallic glass. That reexamination serves as a test of the STZ dynamics, confirming that system parameters obtained from steady-state properties such as the viscosity can be used to predict transient behaviors.
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Affiliation(s)
- J S Langer
- Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA
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