arXiv subjects

Edan Lerner

Publications and source records attributed to Edan Lerner.

105 records · Page 6Linked to original sources

arXiv2010

Statistical Physics of the Yielding Transition in Amorphous Solids

The art of making structural, polymeric and metallic glasses is rapidly developing with many applications. A limitation to their use is their mechanical stability: under increasing external strain all amorphous solids respond elastically to small strains but have a finite yield stress which cannot be exceeded without effecting a plastic response which typically leads to mechanical failure. Understanding this is crucial for assessing the risk of failure of glassy materials under mechanical loads. Here we show that the statistics of the energy barriers ΔE that need to be surmounted changes from a probability distribution function (pdf) that goes smoothly to zero to a pdf which is finite at ΔE=0. This fundamental change implies a dramatic transition in the mechanical stability properties with respect to external strain. We derive exact results for the scaling exponents that characterize the magnitudes of average energy and stress drops in plastic events as a function of system size.

cond-mat.soft↗

arXiv2010

Statistical Physics of Elasto-Plastic Steady States in Amorphous Solids: Finite Temperatures and Strain Rates

The effect of finite temperature $T$ and finite strain rate $\dotγ$ on the statistical physics of plastic deformations in amorphous solids made of $N$ particles is investigated. We recognize three regimes of temperature where the statistics are qualitatively different. In the first regime the temperature is very low, $T<T_{\rm cross}(N)$, and the strain is quasi-static. In this regime the elasto-plastic steady state exhibits highly correlated plastic events whose statistics are characterized by anomalous exponents. In the second regime $T_{\rm cross}(N)<T<T_{\rm max}(\dotγ)$ the system-size dependence of the stress fluctuations becomes normal, but the variance depends on the strain rate. The physical mechanism of the cross-over is different for increasing temperature and increasing strain rate, since the plastic events are still dominated by the mechanical instabilities (seen as an eigenvalue of the Hessian matrix going to zero), and the effect of temperature is only to facilitate the transition. A third regime occurs above the second cross-over temperature $T_{\rm max}(\dotγ)$ where stress fluctuations become dominated by thermal noise. Throughout the paper we demonstrate that scaling concepts are highly relevant for the problem at hand, and finally we present a scaling theory that is able to collapse the data for all the values of temperatures and strain rates, providing us with a high degree of predictability.

cond-mat.stat-mech↗

arXiv2010

Predicting plastic flow events in athermal shear-strained amorphous solids

We propose a method to predict the value of the external strain where a generic amorphous solid will fail by a plastic response (i.e. an irreversible deformation), solely on the basis of measurements of the nonlinear elastic moduli. While usually considered fundamentally different, with the elastic properties describing reversible phenomena and plastic failure epitomizing irreversible behavior, we show that the knowledge of some nonlinear elastic moduli is enough to predict where plasticity sets in.

cond-mat.soft↗

arXiv2010

Athermal Nonlinear Elastic Constants of Amorphous Solids

We derive expressions for the lowest nonlinear elastic constants of amorphous solids in athermal conditions (up to third order), in terms of the interaction potential between the constituent particles. The effect of these constants cannot be disregarded when amorphous solids undergo instabilities like plastic flow or fracture in the athermal limit; in such situations the elastic response increases enormously, bringing the system much beyond the linear regime. We demonstrate that the existing theory of thermal nonlinear elastic constants converges to our expressions in the limit of zero temperature. We motivate the calculation by discussing two examples in which these nonlinear elastic constants play a crucial role in the context of elasto-plasticity of amorphous solids. The first example is the plasticity-induced memory that is typical to amorphous solids (giving rise to the Bauschinger effect). The second example is how to predict the next plastic event from knowledge of the nonlinear elastic constants. Using the results of this paper we derive a simple differential equation for the lowest eigenvalue of the Hessian matrix in the external strain near mechanical instabilities; this equation predicts how the eigenvalue vanishes at the mechanical instability and the value of the strain where the mechanical instability takes place.

cond-mat.stat-mech↗

arXiv2010

Plasticity-Induced Anisotropy in Amorphous Solids: the Bauschinger Effect

Amorphous solids that underwent a strain in one direction such that they responded in a plastic manner `remember' that direction also when relaxed back to a state with zero mean stress. We address the question `what is the order parameter that is responsible for this memory?' and is therefore the reason for the different subsequent responses of the material to strains in different directions. We identify such an order parameter which is readily measurable, we discuss its trajectory along the stress-strain curve, and propose that it and its probability distribution function must form a necessary component of a theory of elasto-plasticity.

cond-mat.stat-mech↗

arXiv2009

Time Scales in the Theory of Elasto-Plasticity of Amorphous Solids

Developing a macroscopic theory of elasto-plasticity in amorphous solids calls for (i) identifying the relevant macro state-variables and (ii) discriminating the different time-scales which characterize these variables. In current theories it is assumed that the stress reaches its elasto-plastic steady state value on the same time-scale as the configurational variables (be they the configurational energy, configurational entropy or the effective temperature). By examining numerical simulations in two and three dimensions we show that this is generally not the case, the configurational degrees of freedom may reach the elasto-plastic steady state on the time scales which can be very different from the time scale of the stress. We provide a physical discussion to rationalize these findings.

cond-mat.stat-mech↗

arXiv2009

Size of Plastic Events in Strained Amorphous Solids at Finite Temperatures

We address the system-size dependence of typical plastic flow events when an amorphous solid is put under a fixed external strain rate at a finite temperature. For system sizes that are accessible to numerical simulations at reasonable strain rates and at low temperatures the magnitude of plastic events grows with the system size. We explain however that this must be a finite size effect; for larger systems there exist two cross-over length-scales $ξ_1$ and $ξ_2$, the first determined by the elastic time-scale and the second by the thermal energy-scale. For system of linear size $L$ larger than any of these scales the magnitude of plastic events must be bounded. For systems of size $L\gg ξ$ there must exist $(L/ξ)^d$ uncorrelated plastic events which occur simultaneously. We present a scaling theory that culminates with the dependence of the cross-over scales on temperature and strain rate. Finally we relate these findings to the temperature and size dependence of the stress fluctuations. We comment on the importance of these considerations for theories of elasto-plasticity.

cond-mat.stat-mech↗

arXiv2009

Predictive Statistical Mechanics for Glass Forming Systems

Using two extremely different models of glass formers in two and three dimensions we demonstrate how to encode the subtle changes in the geometric rearrangement of particles during the scenario of the glass transition. We construct a statistical mechanical description that is able to explain and predict the geometric rearrangement, the temperature dependent thermodynamic functions and the $α$-relaxation time within the measured temperature range and beyond. The theory is based on an up-scaling to proper variables (quasi-species) which is validated using a simple criterion. Once constructed, the theory provides an accurate predictive tool for quantities like the specific heat or the entropy at temperatures that cannot be reached by measurements. In addition, the theory identifies a rapidly increasing typical length scale $ξ$ as the temperature decreases. This growing spatial length scale determines the $α$-relaxation time as $τ_α\sim \exp(μξ/T)$ where $μ$ is a typical chemical potential per unit length.

cond-mat.stat-mech↗

arXiv2009

Scaling Theory for Steady State Plastic Flows in Amorphous Solids

Strongly correlated amorphous solids are a class of glass-formers whose inter-particle potential admits an approximate inverse power-law form in a relevant range of inter-particle distances. We study the steady-state plastic flow of such systems, firstly in the athermal, quasi-static limit, and secondly at finite temperatures and strain rates. In all cases we demonstrate the usefulness of scaling concepts to reduce the data to universal scaling functions where the scaling exponents are determined a-priori from the inter-particle potential. In particular we show that the steady plastic flow at finite temperatures with efficient heat extraction is uniquely characterized by two scaled variables; equivalently, the steady state displays an equation of state that relates one scaled variable to the other two. We discuss the range of applicability of the scaling theory, and the connection to density scaling in supercooled liquid dynamics. We explain that the description of transient states calls for additional state variables whose identity is still far from obvious.

cond-mat.mtrl-sci↗

arXiv2009

Relations between the Material Mechanical Parameters and the Inter-particle Potential in Amorphous Solids

The shear-modulus and yield-stress of amorphous solids are important material parameters, with the former determining the rate of increase of stress under external strain and the latter being the stress value at which the material flows in a plastic manner. It is therefore important to understand how these parameters can be related to the inter-particle potential. Here a scaling theory is presented such that given the inter-particle potential, the dependence of the yield stress and the shear modulus on the density of the solid can be predicted in the athermal limit. It is explained when such prediction is possible at all densities and when it is only applicable at high densities. These results open up exciting possibilities for designing in principle new materials with desirable mechanical properties.

cond-mat.mtrl-sci↗

arXiv2009

Statistical Mechanics and Dynamics of a 3-Dimensional Glass-Forming System

In the context of a classical example of glass-formation in 3-dimensions we exemplify how to construct a statistical mechanical theory of the glass transition. At the heart of the approach is a simple criterion for verifying a proper choice of up-scaled quasi-species that allow the construction of a theory with a finite number of 'states'. Once constructed, the theory identifies a typical scale $ξ$ that increases rapidly with lowering the temperature and which determines the $α$-relaxation time $τ_α$ as $τ_α\sim \exp(μξ/T)$ with $μ$ a typical chemical potential. The theory can predict relaxation times at temperatures that are inaccessible to numerical simulations.

cond-mat.stat-mech↗

arXiv2009

Locality and Non-locality in Elasto-plastic Responses of Amorphous Solids

A number of current theories of plasticity in amorphous solids assume at their basis that plastic deformations are spatially localized. We present in this paper a series of numerical experiments to test the degree of locality of plastic deformation. These experiments increase in terms of the stringency of the removal of elastic contributions to the observed elasto-plastic deformations. It is concluded that for all our simulational protocols the plastic deformations are not localized, and their scaling is sub-extensive. We offer a number of measures of the magnitude of the plastic deformation, all of which display sub-extensive scaling characterized by non-trivial exponents. We provide some evidence that the scaling exponents governing the sub-extensive scaling laws are non-universal, depending on the degree of disorder and on the parameters of the systems. Nevertheless understanding what determines these exponents should shed considerable light on the physics of amorphous solids.

cond-mat.stat-mech↗

arXiv2008

Quantitative Theory of a Time-Correlation Function in a One-Component Glass-Forming Liquid with Anisotropic Potential

The Shintani-Tanaka model is a glass-forming system whose constituents interact via anisotropic potential depending on the angle of a unit vector carried by each particle. The decay of time-correlation functions of the unit vectors exhibits the characteristics of generic relaxation functions during glass transitions. In particular it exhibits a 'stretched exponential' form, with the stretching index beta depending strongly on the temperature. We construct a quantitative theory of this correlation function by analyzing all the physical processes that contribute to it, separating a rotational from a translational decay channel. Interestingly, the separate decay function of each of these processes is temperature independent. Taken together with temperature-dependent weights determined a-priori by statistical mechanics one generates the observed correlation function in quantitative agreement with simulations at different temperatures. This underlines the danger of concluding anything about glassy relaxation functions without detailed physical scrutiny.

cond-mat.stat-mech↗

arXiv2008

Quantitative Theory of a Relaxation Function in a Glass-Forming System

We present a quantitative theory for a relaxation function in a simple glass-forming model (binary mixture of particles with different interaction parameters). It is shown that the slowing down is caused by the competition between locally favored regions (clusters) which are long lived but each of which relaxes as a simple function of time. Without the clusters the relaxation of the background is simply determined by one typical length which we deduce from an elementary statistical mechanical argument. The total relaxation function (which depends on time in a nontrivial manner) is quantitatively determined as a weighted sum over the clusters and the background. The `fragility' in this system can be understood quantitatively since it is determined by the temperature dependence of the number fractions of the locally favored regions.

cond-mat.mtrl-sci↗

arXiv2007

Statistical Mechanics of the Glass Transition in One-Component Liquids with Anisotropic Potential

We study a recently introduced model of one-component glass-forming liquids whose constituents interact with anisotropic potential. This system is interesting per-se and as a model of liquids like glycerol (interacting via hydrogen bonds) which are excellent glass formers. We work out the statistical mechanics of this system, encoding the liquid and glass disorder using appropriate quasi-particles (36 of them). The theory provides a full explanation of the glass transition phenomenology, including the identification of a diverging length scale and a relation between the structural changes and the diverging relaxation times.

cond-mat.mtrl-sci↗