Nonaffine Displacements Research Articles

Entanglements in Glassy Polymer Crazing: Cross-Links or Tubes?

Models of the mechanical response of glassy polymers commonly assume that entanglements inherited from the melt act like chemical cross-links. The predictions from these network models and the physical picture they are based on are tested by following the evolution of topological constraints in simulations of model polymer glasses. The same behavior is observed for polymers with entanglement lengths Ne that vary by a factor of 3. A prediction for the craze extension ratio Λ based on the network model describes trends with Ne, but polymers do not have the taut configurations it assumes. There is also no evidence of the predicted geometrically necessary entanglement loss. While the number of entanglements remains constant, the identity of the chains forming constraints changes. The same relation between the amount of entanglement exchange and nonaffine displacement of monomers is found for crazing and thermal diffusion in end-constrained melts. In both cases, about 1/3 of the constraints change when monomer...

Understanding Atomic-Scale Features of Low Temperature-Relaxation Dynamics in Metallic Glasses

Being a key feature of a glassy state, low temperature relaxation has important implications on the mechanical behavior of glasses; however, the mechanism of low temperature relaxation is still an open issue, which has been debated for decades. By systematically investigating the influences of cooling rate and pressure on low temperature relaxation in the Zr50Cu50 metallic glasses, it is found that even though pressure does induce pronounced local structural change, the low temperature-relaxation behavior of the metallic glass is affected mainly by cooling rate, not by pressure. According to the atomic displacement and connection mode analysis, we further demonstrate that the low temperature relaxation is dominated by the dispersion degree of fast dynamic atoms rather than the most probable atomic nonaffine displacement. Our finding provides the direct atomic-level evidence that the intrinsic heterogeneity is the key factor that determines the low temperature-relaxation behavior of the metallic glasses.

Correlation between Viscoelastic Moduli and Atomic Rearrangements in Metallic Glasses.

Dynamical moduli, such as storage and loss moduli, characterize the viscoelasticity of materials (i.e., time-dependent elasticity) and convey important information about the relaxation processes of glasses and supercooled liquids. A fundamental question is what ultimately determines them in glassy materials. Here, for several model metallic glasses, we demonstrate that both the storage and loss moduli are uniquely determined by the most probable atomic nonaffine displacements, regardless of temperature or frequency. Moreover, the fast-moving atoms (which contribute to dynamical heterogeneity) do not contribute explicitly to the moduli. Our findings provide a physical basis for the origin of viscoelasticity in metallic glasses.

Nonaffine displacements and the nonlinear response of a strained amorphous solid

We demonstrate that irreversible structural reorganization is not necessary for the observation of yield behavior in an amorphous solid. While the majority of solids strained to their yield point do indeed undergo an irreversible reorganization, we find that a significant fraction of solids exhibits yield via a reversible strain. We also demonstrate that large instantaneous strains in excess of the yield stress can result in complete stress relaxation, a result of the large nonaffine motions driven by the applied strain. The empirical similarity of the dependence of the ratio of stress over strain on the nonaffine mean-square displacement to that for the shear modulus obtained from quiescent liquid at nonzero temperature supports the proposition that rigidity depends on the size of the sampled configurational space only and is insensitive to how this space is sampled.

Nonaffine rearrangements of atoms in deformed and quiescent binary glasses.

The influence of periodic shear deformation on nonaffine atomic displacements in an amorphous solid is examined via molecular dynamics simulations. We study the three-dimensional Kob-Andersen binary mixture model at a finite temperature. It is found that when the material is periodically strained, most of the atoms undergo repetitive nonaffine displacements with amplitudes that are broadly distributed. We show that particles with large amplitudes of nonaffine displacements are organized into compact clusters. With increasing strain amplitude, spatial correlations of nonaffine displacements become increasingly long-ranged, although they remain present even in a quiescent system due to thermal fluctuations.

Correlation between strain rate sensitivity and α relaxation of metallic glasses

An inherent correlation between the strain rate sensitivity and α relaxation of metallic glasses (MGs) is observed. This correlation can be attributed to the secondary term which incorporates the nonaffine displacements of atoms in the analytical expression of the elastic modulus of amorphous solids. The observed correlation supports the proposition that stress and temperature play equivalent role in the glass transition of MGs. Besides, an ideal liquid state of MGs is observed in the supercooled liquid region when they are deformed below a critical loading rate. This observation would benefit the application of MGs in the fabrication of micro parts for MEMS (Micro Electro-Mechanical Systems).

Microstructure evolution of granular soils in cyclic mobility and post-liquefaction process

Understanding the evolution of microstructure in granular soils can provide significant insights into constitutive modeling of soil liquefaction. In this study, micromechanical perspectives of the liquefaction process are investigated using the Discrete Element simulation. It is observed that during various stages of undrained cyclic loading, the soil exhibits definitive change in the load-bearing structure, indicated by evolution of the coordination number and non-affine displacements. A new particle-void fabric, termed as “centroid distance”, is also proposed to quantify the evolution of particles and voids distribution in the granular packing. The fabric index is found to have strong correlation with cyclic mobility and post-liquefaction deformation of granular soils. Evolution of the fabric index indicates that particles and voids redistribute irreversibly before and after liquefaction. A highly anisotropic particle-void structure and loading-bearing capacity can be formed in the post liquefaction stage.

Shuffling-controlled versus strain-controlled deformation twinning: The case for HCP Mg twin nucleation

The atomistic pathways of deformation twinning can be computed ab initio, and quantified by two variables: strain which describes shape change of a periodic supercell, and shuffling which describes non-affine displacements of the internal degrees of freedom. The minimum energy path involves juxta-position of both. But if one can obtain the same saddle point by continuously increasing the strain and relaxing the internal degrees of freedom by steepest descent, we call the path strain-controlled, and vice versa. Surprisingly, we find the {101¯2}〈101¯1¯〉 twinning of Mg is shuffling-controlled at the smallest lengthscale of the irreducible lattice correspondence pattern, that is, the reaction coordinate at the level of 4 atoms is dominated by non-affine displacements, instead of strain. Shuffling-controlled deformation twinning is expected to have different temperature and strain-rate sensitivities from strain-controlled deformation twinning due to relatively weaker strength of long-range elastic interactions, in particular at the twin nucleation stage. As the twin grows large enough, however, elastic interactions and displacive character of the transformation should always turn dominant.

Deformation of metallic glasses: Recent developments in theory, simulations, and experiments

We review recent research into elastic and plastic deformation of metallic glasses, with an emphasis on making connections between developments in theory and simulation (largely from the physics community) and experimental results (largely from the metallurgy community). Topics covered include strain measurement via scattering techniques, non-affine atomic displacements during elastic deformation, shear transformations, constitutive equations, shear bands, and strain hardening. Where possible we connect the observed behavior and properties to the structure of the glass on the atomic- and nano-scales.

Nonlinear plastic modes in disordered solids.

We propose a theoretical framework within which a robust micromechanical definition of precursors to plastic instabilities, often termed soft spots, naturally emerges. They are shown to be collective displacements (modes) z[over ̂] that correspond to local minima of a barrier function b(z[over ̂]), which depends solely on inherent structure information. We demonstrate how some heuristic searches for local minima of b(z[over ̂]) can a priori detect the locus and geometry of imminent plastic instabilities with remarkable accuracy, at strains as large as γ_{c}-γ∼10^{-2} away from the instability strain γ_{c}. Our findings suggest that the a priori detection of the entire field of soft spots can be effectively carried out by a systematic investigation of the landscape of b(z[over ̂]).

Reversible plastic events during oscillatory deformation of amorphous solids.

The effect of oscillatory shear strain on nonaffine rearrangements of individual particles in a three-dimensional binary glass is investigated using molecular dynamics simulations. The amorphous material is represented by the Kob-Andersen mixture at the temperature well below the glass transition. We find that during periodic shear deformation of the material, some particles undergo reversible nonaffine displacements with amplitudes that are approximately power-law distributed. Our simulations show that particles with large amplitudes of nonaffine displacement exhibit a collective behavior; namely, they tend to aggregate into relatively compact clusters that become comparable with the system size near the yield strain. Along with reversible displacements there exist a number of irreversible ones. With increasing strain amplitude, the probability of irreversible displacements during one cycle increases, which leads to permanent structural relaxation of the material.

Analysis of Low-Frequency Vibrational Modes and Particle Rearrangements in Marginally Jammed Amorphous Solid under Quasi-Static Shear**Supported by the National Natural Science Foundation of China under Grant Nos 11272048 and 51239006, and the European Commission Marie Curie Actions under Grant No IRSES-294976.

We present the numerical simulation results of a model granular assembly formed by spherical particles with Hertzian interaction subjected to a simple shear in the athermal quasi-static limit. The stress-strain curve is shown to separate into smooth, elastic branches followed by a subsequent plastic event. Mode analysis shows that the lowest-frequency vibrational mode is more localized, and eigenvalues and participation ratios of low-frequency modes exhibit similar power-law behavior as the system approaches plastic instability, indicating that the nature of plastic events in the granular system is also a saddle node bifurcation. The analysis of projection and spatial structure shows that over 75% contributions to the non-affine displacement field at a plastic instability come from the lowest-frequency mode, and the lowest-frequency mode is strongly spatially correlated with local plastic rearrangements, inferring that the lowest-frequency mode could be used as a predictor for future plastic rearrangements in the disordered system jammed marginally.

Deformation-driven diffusion and plastic flow in amorphous granular pillars.

We report a combined experimental and simulation study of deformation-induced diffusion in compacted quasi-two-dimensional amorphous granular pillars, in which thermal fluctuations play a negligible role. The pillars, consisting of bidisperse cylindrical acetal plastic particles standing upright on a substrate, are deformed uniaxially and quasistatically by a rigid bar moving at a constant speed. The plastic flow and particle rearrangements in the pillars are characterized by computing the best-fit affine transformation strain and nonaffine displacement associated with each particle between two stages of deformation. The nonaffine displacement exhibits exponential crossover from ballistic to diffusive behavior with respect to the cumulative deviatoric strain, indicating that in athermal granular packings, the cumulative deviatoric strain plays the role of time in thermal systems and drives effective particle diffusion. We further study the size-dependent deformation of the granular pillars by simulation, and find that different-sized pillars follow self-similar shape evolution during deformation. In addition, the yield stress of the pillars increases linearly with pillar size. Formation of transient shear lines in the pillars during deformation becomes more evident as pillar size increases. The width of these elementary shear bands is about twice the diameter of a particle, and does not vary with pillar size.

Stress-driven crystallization via shear-diffusion transformations in a metallic glass at very low temperatures

At elevated temperatures, glasses crystallize via thermally activated diffusion. However, metallic glasses can also undergo deformation-induced crystallization at very low temperatures. Here we demonstrate the crystallization of $\mathrm{A}{\mathrm{l}}_{50}\mathrm{F}{\mathrm{e}}_{50}$ metallic glasses under cyclic deformation at 50 K using molecular dynamics simulations and reveal the underlying atomic-scale processes. We demonstrate that stress-driven nonaffine atomic rearrangements, or shear diffusion transformation (SDT) events, lead to successive metabasin-to-metabasin transitions and long-range ordering. We also illustrate that the nucleation and growth of the crystal proceed via collective attachment of ordered clusters, advancing the amorphous/crystal interface in an intermittent manner. The cooperative nature of the steplike crystallization is attributed to the large activation volume of Eshelby transformations which generate as a by-product nonaffine diffusive atomic displacements that accumulate over loading cycles. The dual nature of shear (affine) and diffusion (nonaffine) in low-temperature stress-driven SDT events thus unifies inelasticity with crystallization.

Fluctuations and diffusion in sheared athermal suspensions of deformable particles

We analyze fluctuations of particle displacements and stresses in a sheared athermal suspension of elastic capsules (red blood cells). Upon variation of the volume fraction from the dilute up to the highly concentrated regime, our numerical simulations reveal different characteristic power-law regimes of the fluctuation variances and relaxation times. In the jammed phase and at high shear rates, anomalous scaling exponents are found that deviate from pure dimensional predictions. The observed behavior is rationalized via kinetic arguments and a dissipation balance model that takes into account the local fluid flows between the particles. Our findings support the view that the rheology of dense suspensions is essentially governed by the non-affine displacements.

Evolution of atomic rearrangements in deformation in metallic glasses.

Atomic rearrangements induced by shear stress are fundamental for understanding deformation mechanisms in metallic glasses (MGs). Using molecular dynamic simulation, the atomic rearrangements characterized by nonaffine displacements (NADs) and their spatial distribution and evolution with tensile stress in Cu50Zr50 MG were investigated. It was found that in the elastic regime the atomic rearrangements with the largest NADs are relatively homogeneous in space, but exhibit strong spatial correlation, become localized and inhomogeneous, and form large clusters as strain increases, which may facilitate the so-called shear transformation zones. Furthermore, initially they prefer to take place around Cu atoms which have more nonicosahedral configurations. As strain increases, the preference decays and disappears in the plastic regime. The atomic rearrangements with the smallest NADs are preferentially located around Cu atoms, too, but with more icosahedral or icosahedral-like atomic configurations. The preference is maintained in the whole deformation process. In contrast, the atomic rearrangements with moderate NADs distribute homogeneously, and do not show explicit preference or spatial correlation, acting as matrix during deformation. Among the atomic rearrangements with different NADs, those with largest and smallest NADs are nearest neighbors initially, but separating with increasing strain, while those with largest and moderate NADs always avoid to each other. The correlations in the fluctuations of the NADs confirm the long-range strain correlation and the scale-free characteristic of NADs in both elastic and plastic deformation, which suggests a universality of the scaling in the plastic flow in MGs.

Soft spots and their structural signature in a metallic glass.

In a 3D model mimicking realistic Cu64Zr36 metallic glass, we uncovered a direct link between the quasi-localized low-frequency vibrational modes and the local atomic packing structure. We also demonstrate that quasi-localized soft modes correlate strongly with fertile sites for shear transformations: geometrically unfavored motifs constitute the most flexible local environments that encourage soft modes and high propensity for shear transformations, whereas local configurations preferred in this alloy, i.e., the full icosahedra (around Cu) and Z16 Kasper polyhedra (around Zr), contribute the least.

Surface shear-transformation zones in amorphous solids.

We perform a systematic study of the characteristics of shear transformation zones (STZs) that nucleate at free surfaces of two-dimensional amorphous solids subject to tensile loading using two different atomistic simulation methods, the standard athermal, quasistatic (AQ) approach and our recently developed self-learning metabasin escape (SLME) method, to account for the finite temperature and strain-rate effects. In the AQ, or strain-driven limit, the nonaffine displacement fields of surface STZs decay exponentially away from their centers at similar decay rates as their bulk counterparts, though the direction of maximum nonaffine displacement is tilted away from the tensile axis due to surface effects. Using the SLME method at room temperature and at the high strain rates that are seen in classical molecular dynamics simulations, the characteristics for both bulk and surface STZs are found to be identical to those seen in the AQ simulations. However, using the SLME method at room temperature and experimentally relevant strain rates, we find a transition in the surface STZ characteristics where a loss in the characteristic angular tensile-compression symmetry is observed. Finally, the thermally activated surface STZs exhibit a slower decay rate in the nonaffine displacement field than do strain-driven surface STZs, which is characterized by a larger drop in potential energy resulting from STZ nucleation that is enabled by the relative compliance of the surface as compared to the bulk.

Elastic and plastic characteristics of a model Cu–Zr amorphous alloy

Athermal quasistatic simulation of shear deformation has been conducted for a realistic model Cu–Zr amorphous alloy to investigate characteristic features of elasticity and plasticity of the material. Significant reduction of the shear modulus by nonaffine atomic displacements and appreciable nonlinearity of elasticity have been observed. The fourth-order elastic constant in shear deformation and the ideal shear strength have been evaluated. Plastic deformation has been observed to start with isolated local shear transformations (LSTs) followed by collective LSTs leading to the formation of a shear band. Participation-ratio analysis (PRA) has demonstrated how the nonaffine displacement field converges as the system approaches the critical point of losing structural stability. PRA has also evaluated quantitatively the numbers of atoms participating in LSTs – the average number is about 30. Spatially anisotropic development of nascent shear band on a plane has been shown, attributable to anisotropic internal stress field induced by an LST. The evaluated stresses for the shear-band nucleation and for its propagation have indicated that the yielding in real materials is controlled by the shear-band propagation, as previously pointed out.

Computer simulation of rearrangements in chains of glassy polymethylene subjected at low temperature inelastic deformation

Molecular dynamics simulation of glassy polymethylene (PM) plastic deformation is performed up to ɛ = 30% in uniaxial compression regime at a temperature of 50 K, which is ∼140 K below T g of the polymer. All atoms of PM chains are represented explisitly (all-atom model). Calculations were performed for two series of samples with different molecular mass distribution of chains: Samples have average degree of polymerization DP ≈ 212 with Mn ≈ 3000 and Mw ≈ 9500 (the first series) and DP ≈ 350, Mn ≈ 5000 and Mw ≈ 9500 (the second series). Each sample contains 12288 -CH2- monomeric units per computational sell. Nonaffine displacements of carbon atoms and conformational rearrangements in chains during deformation are visualized and analyzed. The transformation of relatively fragments of chains up to 16–20 monomer units length are basic structural units, non-conformational displacements of which controls plastic process. Relatively large nonaffine displacements are observed even in the range of low strains, which are usually interpreted as Hookean strains. In the range of yield tooth and steady plastic flow, the number of these displacements increases along with their amplitude. Conformational set of PM chains does not show a serious change during deformation. Analysis had shown that the number of conformational rearrangements of trans-gauche type in PM chains during deformation is small and such rearrangements do not play decisive role in the considered range of PM plasticity, even at ɛ > 15%, at the stage of the developed plastic flow.