arXiv subjects

Mark D. Shattuck

Publications and source records attributed to Mark D. Shattuck.

At least 55 records · Page 3Linked to original sources

arXiv2018

Stress anisotropy in shear-jammed packings of frictionless disks

We perform computational studies of repulsive, frictionless disks to investigate the development of stress anisotropy in mechanically stable (MS) packings. We focus on two protocols for generating MS packings: 1) isotropic compression and 2) applied simple or pure shear strain $γ$ at fixed packing fraction $ϕ$. MS packings of frictionless disks occur as geometric families (i.e. parabolic segments with positive curvature) in the $ϕ$-$γ$ plane. MS packings from protocol 1 populate parabolic segments with both signs of the slope, $dϕ/dγ>0$ and $dϕ/dγ<0$. In contrast, MS packings from protocol 2 populate segments with $dϕ/dγ<0$ only. For both simple and pure shear, we derive a relationship between the stress anisotropy and dilatancy $dϕ/dγ$ obeyed by MS packings along geometrical families. We show that for MS packings prepared using isotropic compression, the stress anisotropy distribution is Gaussian centered at zero with a standard deviation that decreases with increasing system size. For shear jammed MS packings, the stress anisotropy distribution is a convolution of Weibull distributions that depend on strain, which has a nonzero average and standard deviation in the large-system limit. We also develop a framework to calculate the stress anisotropy distribution for packings generated via protocol 2 in terms of the stress anisotropy distribution for packings generated via protocol 1. These results emphasize that for repulsive frictionless disks, different packing-generation protocols give rise to different MS packing probabilities, which lead to differences in macroscopic properties of MS packings.

cond-mat.soft↗

arXiv2018

Critical scaling near the yielding transition in granular media

We show that the yielding transition in granular media displays second-order critical-point scaling behavior. We carry out discrete element simulations in the low inertial number limit for frictionless, purely repulsive spherical grains undergoing simple shear at fixed nondimensional shear stress $Σ$ in two and three spatial dimensions. To find a mechanically stable (MS) packing that can support the applied $Σ$, isotropically prepared states with size $L$ must undergo a total strain $γ_{\rm ms}(Σ,L)$. The number density of MS packings ($\propto γ_{\rm ms}^{-1}$) vanishes for $Σ> Σ_c \approx 0.11$ according to a critical scaling form with a length scale $ξ\propto |Σ- Σ_c|^{-ν}$, where $ν\approx 1.7-1.8$. Above the yield stress ($Σ>Σ_c$), no MS packings that can support $Σ$ exist in the large system limit, $L/ξ\gg 1$. MS packings generated via shear possess anisotropic force and contact networks, suggesting that $Σ_c$ is associated with an upper limit in the degree to which these networks can be deformed away from those for isotropic packings.

cond-mat.soft↗

arXiv2018

Jamming of Deformable Polygons

There are two main classes of physics-based models for two-dimensional cellular materials: packings of repulsive disks and the vertex model. These models have several disadvantages. For example, disk interactions are typically a function of particle overlap, yet the model assumes that the disks remain circular during overlap. The shapes of the cells can vary in the vertex model, however, the packing fraction is fixed at $ϕ=1$. Here, we describe the deformable particle model (DPM), where each particle is a polygon composed of a large number of vertices. The total energy includes three terms: two quadratic terms to penalize deviations from the preferred particle area $a_0$ and perimeter $p_0$ and a repulsive interaction between DPM polygons that penalizes overlaps. We performed simulations to study the onset of jamming in packings of DPM polygons as a function of asphericity, ${\cal A} = p_0^2/4πa_0$. We show that the packing fraction at jamming onset $ϕ_J({\cal A})$ grows with increasing ${\cal A}$, reaching confluence at ${\cal A} \approx 1.16$. ${\cal A}^*$ corresponds to the value at which DPM polygons completely fill the cells obtained from a surface-Voronoi tessellation. Further, we show that DPM polygons develop invaginations for ${\cal A} > {\cal A}^*$ with excess perimeter that grows linearly with ${\cal A}-{\cal A}^*$. We confirm that packings of DPM polygons are solid-like over the full range of ${\cal A}$ by showing that the shear modulus is nonzero.

cond-mat.soft↗

arXiv2017

Hypostatic jammed packings of frictionless nonspherical particles

We perform computational studies of static packings of a variety of nonspherical particles including circulo-lines, circulo-polygons, ellipses, asymmetric dimers, and dumbbells to determine which shapes form hypostatic versus isostatic packings and to understand why hypostatic packings of nonspherical particles can be mechanically stable despite having fewer contacts than that predicted from naïve constraint counting. To generate highly accurate force- and torque-balanced packings of circulo-lines and -polygons, we developed an interparticle potential that gives continuous forces and torques as a function of the particle coordinates. We show that the packing fraction and coordination number at jamming onset obey a master-like form for all of the nonspherical particle packings we studied when plotted versus the particle asphericity ${\cal A}$, which is proportional to the ratio of the squared perimeter to the area of the particle. Further, the eigenvalue spectra of the dynamical matrix for packings of different particle shapes collapse when plotted at the same ${\cal A}$. For hypostatic packings of nonspherical particles, we verify that the number of "quartic" modes along which the potential energy increases as the fourth power of the perturbation amplitude matches the number of missing contacts relative to the isostatic value. In addition, we calculate the principal curvatures of the inequality constraints for each contact in circulo-line packings and identify specific types of contacts with inequality constraints that possess convex curvature. These contacts can constrain multiple degrees of freedom and allow hypostatic packings of nonspherical particles to be mechanically stable.

cond-mat.soft↗

arXiv2017

Determining the Onset of Hydrodynamic Erosion in Turbulent Flow

We revisit the longstanding question of the onset of sediment transport driven by a turbulent fluid flow via laboratory measurements. We use particle tracking velocimetry to quantify the fluid flow as well as the motion of individual grains. As we increase the flow speed above the transition to sediment transport, we observe that an increasing fraction of grains are transported downstream, although the average downstream velocity of the transported grains remains roughly constant. However, we find that the fraction of mobilized grains does not vanish sharply at a critical flow rate. Additionally, the distribution of the fluctuating velocities of non-transported grains becomes broader with heavier tails, meaning that unambiguously separating mobile and static grains is not possible. As an alternative approach, we quantify the statistics of grain velocities by using a mixture model consisting of two forms for the grain velocities: a decaying-exponential tail, which represents grains transported downstream, and a peaked distribution centered at zero velocity, which represents grains that fluctuate due to the turbulent flow but remain in place. Our results suggest that more sophisticated statistical measures may be required to quantify grain motion near the onset of sediment transport, particularly in the presence of turbulence.

physics.flu-dyn↗

arXiv2017

Particle rearrangement and softening contributions to the nonlinear mechanical response of glasses

Amorphous materials such as metallic, polymeric, and colloidal glasses, exhibit complex preparation-dependent mechanical response to applied shear. We perform numerical simulations to investigate the mechanical response of binary Lennard-Jones glasses undergoing athermal, quasistatic pure shear as a function of the cooling rate $R$ used to prepare them. The ensemble-averaged stress versus strain curve $\langleσ(γ)\rangle$ resembles the spatial average in the large size limit, which appears smooth and displays a putative elastic regime at small strains, a yielding-related peak at intermediate strain, and a plastic flow regime at large strains. In contrast, for each glass configuration in the ensemble, the stress-strain curve consists of many short nearly linear segments that are punctuated by particle-rearrangement-induced rapid stress drops. We quantify the shape of the small stress-strain segments, as well as the frequency and size of the stress drops in each glass configuration. We decompose the stress loss into the loss from particle rearrangements and from softening i.e., the reduction of the slopes of the linear segments in $σ(γ)$), and then compare the two contributions as a function of $R$ and $γ$. For the current studies, the rearrangement-induced stress loss is larger than the softening-induced stress loss, however, softening stress losses increase with decreasing cooling rate. We also characterize the structure of the potential energy landscape along the strain direction for glasses prepared with different $R$, and observe a dramatic change of the properties of the landscape near the yielding transition. We then show that the rearrangement-induced energy loss per strain can serve as an order parameter for the yielding transition, which sharpens for slow cooling rates and in the large system-size limit.

cond-mat.stat-mech↗

arXiv2017

The role of grain dynamics in determining the onset of sediment transport

Sediment transport occurs when the nondimensional fluid shear stress $Θ$ at the bed surface exceeds a minimum value $Θ_c$. A large collection of data, known as the Shields curve, shows that $Θ_c$ is primarily a function of the shear Reynolds number ${\rm{Re}}_*$. It is commonly assumed that $Θ>Θ_c({\rm{Re}}_*)$ occurs when the ${\rm Re}_*$-dependent fluid forces are too large to maintain static equilibrium for a typical surface grain. A complimentary approach, which remains relatively unexplored, is to identify $Θ_c({\rm{Re}}_*)$ as the applied shear stress at which grains cannot stop moving. With respect to grain dynamics, ${\rm{Re}}_*$ can be viewed as the viscous time scale for a grain to equilibrate to the fluid flow divided by the typical time for the fluid force to accelerate a grain over the characteristic bed roughness. We performed simulations of granular beds sheared by a model fluid, varying only these two time scales. We find that the critical Shields number $Θ_c({\rm Re}_*)$ obtained from the model mimics the Shields curve and is insensitive to the grain properties, the model fluid flow, and the form of the drag law. Quantitative discrepancies between the model results and the Shields curve are consistent with previous calculations of lift forces at varying ${\rm Re}_*$. Grains at low ${\rm Re}_*$ find more stable configurations than those at high ${\rm{Re}}_*$ due to differences in the grain reorganization dynamics. Thus, instead of focusing on mechanical equilibrium of a typical grain at the bed surface, $Θ_c({\rm{Re}}_*)$ may be better described by the stress at which mobile grains cannot find a stable configuration and stop moving.

cond-mat.soft↗

arXiv2016

The effects of cooling rate on particle rearrangement statistics: Rapidly cooled glasses are more ductile and less reversible

Amorphous solids, such as metallic, polymeric, and colloidal glasses, display complex spatiotemporal response to applied deformations. In contrast to crystalline solids, during loading, amorphous solids exhibit a smooth crossover from elastic response to plastic flow. In this study, we investigate the mechanical response of binary Lennard-Jones glasses to athermal, quasistatic pure shear as a function of the cooling rate used to prepare them. We find several key results concerning the connection between strain-induced particle rearrangements and mechanical response. We show that more rapidly cooled glasses undergo more frequent and larger particle rearrangements than slowly cooled glasses. We find that the ratio of the shear to bulk moduli decreases with increasing cooling rate, which suggests that more rapidly cooled glasses are more ductile than slowly cooled samples. In addition, we characterized the degree of reversibility of particle motion during cyclic shear. We find that irreversible particle motion occurs even in the putative linear regime of stress versus strain. However, slowly cooled glasses, which undergo smaller rearrangements, are more reversible under cyclic shear than rapidly cooled glasses. Thus, we show that more ductile glasses are also less reversible.

cond-mat.soft↗

arXiv2016

Reconstruction of Ordinary Differential Equations From Time Series Data

We develop a numerical method to reconstruct systems of ordinary differential equations (ODEs) from time series data without {\it a priori} knowledge of the underlying ODEs using sparse basis learning and sparse function reconstruction. We show that employing sparse representations provides more accurate ODE reconstruction compared to least-squares reconstruction techniques for a given amount of time series data. We test and validate the ODE reconstruction method on known 1D, 2D, and 3D systems of ODEs. The 1D system possesses two stable fixed points; the 2D system possesses an oscillatory fixed point with closed orbits; and the 3D system displays chaotic dynamics on a strange attractor. We determine the amount of data required to achieve an error in the reconstructed functions to less than $0.1\%$. For the reconstructed 1D and 2D systems, we are able to match the trajectories from the original ODEs even at long times. For the 3D system with chaotic dynamics, as expected, the trajectories from the original and reconstructed systems do not match at long times, but the reconstructed and original models possess similar Lyapunov exponents. Now that we have validated this ODE reconstruction method on known models, it can be employed in future studies to identify new systems of ODEs using time series data from deterministic systems for which there is no currently known ODE model.

physics.data-an↗

arXiv2015

Onset and cessation of motion in hydrodynamically sheared granular beds

We performed molecular dynamics simulations of granular beds driven by a model hydrodynamic shear flow to elucidate general grain-scale mechanisms that determine the onset and cessation of sediment transport. By varying the Shields number (the nondimensional shear stress at the top of the bed) and particle Reynolds number (the ratio of particle inertia to viscous damping), we explore how variations of the fluid flow rate, particle inertia, and fluid viscosity affect the onset and cessation of bed motion. For low to moderate particle Reynolds numbers, a critical boundary separates mobile and static states. Transition times between these states diverge as this boundary is approached both from above and below. At high particle Reynolds number, inertial effects become dominant, and particle motion can be sustained well below flow rates at which mobilization of a static bed occurs. We also find that the onset of bed motion (for both low and high particle Reynolds numbers) is described by Weibullian weakest-link statistics, and thus is crucially dependent on the packing structure of the granular bed, even deep beneath the surface.

cond-mat.soft↗

arXiv2015

Beyond packing of hard spheres: The effects of core softness, non-additivity, intermediate-range repulsion, and many-body interactions on the glass-forming ability of bulk metallic glasses

When a liquid is cooled well below its melting temperature at a rate that exceeds the critical cooling rate $R_c$, the crystalline state is bypassed and an amorphous glassy state forms instead. $R_c$ (or the corresponding critical casting thickness $d_c$) characterizes the glass-forming ability (GFA) of each material. While silica is an excellent glass-former with small $R_c<10^{-2}$ K/s, pure metals and most alloys are poor glass-formers with large $R_c>10^{10}$ K/s. Only in the past thirty years have bulk metallic glasses (BMGs) been identified with $R_c$ approaching that for silica. Recent simulations have shown that hard-sphere models are able to identify the atomic size ratio and number fraction regime where BMGs exist with critical cooling rates more than 13 orders of magnitude smaller than those for pure metals. However, there are many other features of interatomic potentials beyond hard-core interactions. How do these other features affect the glass-forming ability of BMGs? We perform molecular dynamics simulations to determine how variations in the softness and non-additivity of the repulsive core and form of the interatomic pair potential at intermediate distances affect the GFA of binary alloys. These variations in the interatomic pair potential allow us to introduce geometric frustration and change the crystal phases that compete with glass formation. We also investigate the effect of tuning the strength of the many-body interactions from zero to the full embedded atom model on the GFA for pure metals. We then employ the full embedded atom model for binary BMGs and show that hard-core interactions play the dominant role in setting the GFA of alloys, while other features of the interatomic potential only change the GFA by one to two orders of magnitude.

cond-mat.mtrl-sci↗

arXiv2015

Outcome prediction in mathematical models of immune response to infection

Clinicians need to predict patient outcomes with high accuracy as early as possible after disease inception. In this manuscript, we show that patient-to-patient variability sets a fundamental limit on outcome prediction accuracy for a general class of mathematical models for the immune response to infection. However, accuracy can be increased at the expense of delayed prognosis. We investigate several systems of ordinary differential equations (ODEs) that model the host immune response to a pathogen load. Advantages of systems of ODEs for investigating the immune response to infection include the ability to collect data on large numbers of `virtual patients', each with a given set of model parameters, and obtain many time points during the course of the infection. We implement patient-to-patient variability $v$ in the ODE models by randomly selecting the model parameters from Gaussian distributions with variance $v$ that are centered on physiological values. We use logistic regression with one-versus-all classification to predict the discrete steady-state outcomes of the system. We find that the prediction algorithm achieves near $100\%$ accuracy for $v=0$, and the accuracy decreases with increasing $v$ for all ODE models studied. The fact that multiple steady-state outcomes can be obtained for a given initial condition, i.e. the basins of attraction overlap in the space of initial conditions, limits the prediction accuracy for $v>0$. Increasing the elapsed time of the variables used to train and test the classifier, increases the prediction accuracy, while adding explicit external noise to the ODE models decreases the prediction accuracy. Our results quantify the competition between early prognosis and high prediction accuracy that is frequently encountered by clinicians.

q-bio.QM↗

arXiv2015

Asymmetric crystallization during cooling and heating in model glass-forming systems

We perform molecular dynamics (MD) simulations of the crystallization process in binary Lennard-Jones systems during heating and cooling to investigate atomic-scale crystallization kinetics in glass-forming materials. For the cooling protocol, we prepared equilibrated liquids above the liquidus temperature $T_l$ and cooled each sample to zero temperature at rate $R_c$. For the heating protocol, we first cooled equilibrated liquids to zero temperature at rate $R_p$ and then heated the samples to temperature $T > T_l$ at rate $R_h$. We measured the critical heating and cooling rates $R_h^*$ and $R_c^*$, below which the systems begin to form a substantial fraction of crystalline clusters during the heating and cooling protocols. We show that $R_h^* > R_c^*$, and that the asymmetry ratio $R_h^*/R_c^*$ includes an intrinsic contribution that increases with the glass-forming ability (GFA) of the system and a preparation-rate dependent contribution that increases strongly as $R_p \rightarrow R_c^*$ from above. We also show that the predictions from classical nucleation theory (CNT) can qualitatively describe the dependence of the asymmetry ratio on the GFA and preparation rate $R_p$ from the MD simulations and results for the asymmetry ratio measured in Zr- and Au-based bulk metallic glasses (BMG). This work emphasizes the need for and benefits of an improved understanding of crystallization processes in BMGs and other glass-forming systems.

cond-mat.soft↗

arXiv2014

The glass-forming ability of model metal-metalloid alloys

Bulk metallic glasses (BMGs) are amorphous alloys with desirable mechanical properties and processing capabilities. To date, the design of new BMGs has largely employed empirical rules and trial-and-error experimental approaches. Ab initio computational methods are currently prohibitively slow to be practically used in searching the vast space of possible atomic combinations for bulk glass formers. Here, we perform molecular dynamics simulations of a coarse-grained, anisotropic potential, which mimics interatomic covalent bonding, to measure the critical cooling rates for metal-metalloid alloys as a function of the atomic size ratio $σ_S/σ_L$ and number fraction $x_S$ of the metalloid species. We show that the regime in the space of $σ_S/σ_L$ and $x_S$ where well-mixed, optimal glass formers occur for patchy and LJ particle mixtures coincides with that for experimentally observed metal-metalloid glass formers. Our simple computational model provides the capability to perform combinatorial searches to identify novel glass-forming alloys.

cond-mat.mtrl-sci↗

arXiv2014

Can the packing efficiency of binary hard spheres explain the glass-forming ability of bulk metallic glasses?

We perform molecular dynamics simulations to compress binary hard spheres into jammed packings as a function of the compression rate $R$, size ratio $α$, and number fraction $x_S$ of small particles to determine the connection between the glass-forming ability (GFA) and packing efficiency in bulk metallic glasses (BMGs). We define the GFA by measuring the critical compression rate $R_c$, below which jammed hard-sphere packings begin to form "random crystal" structures with defects. We find that for systems with $α\gtrsim 0.8$ that do not de-mix, $R_c$ decreases strongly with $Δϕ_J$, as $R_c \sim \exp(-1/Δϕ_J^2)$, where $Δϕ_J$ is the difference between the average packing fraction of the amorphous packings and random crystal structures at $R_c$. Systems with $α\lesssim 0.8$ partially de-mix, which promotes crystallization, but we still find a strong correlation between $R_c$ and $Δϕ_J$. We show that known metal-metal BMGs occur in the regions of the $α$ and $x_S$ parameter space with the lowest values of $R_c$ for binary hard spheres. Our results emphasize that maximizing GFA in binary systems involves two competing effects: minimizing $α$ to increase packing efficiency, while maximizing $α$ to prevent de-mixing.

cond-mat.mtrl-sci↗

arXiv2014

The statistics of frictional families

We develop a theoretical description for mechanically stable frictional packings in terms of the difference between the total number of contacts required for isostatic packings of frictionless disks and the number of contacts in frictional packings, $m=N_c^0-N_c$. The saddle order $m$ represents the number of unconstrained degrees of freedom that a static packing would possess if friction were removed. Using a novel numerical method that allows us to enumerate disk packings for each $m$, we show that the probability to obtain a packing with saddle order $m$ at a given static friction coefficient $μ$, $P_m(μ)$, can be expressed as a power-series in $μ$. Using this form for $P_m(μ)$, we quantitatively describe the dependence of the average contact number on friction coefficient for static disk packings obtained from direct simulations of the Cundall-Strack model for all $μ$ and $N$.

cond-mat.soft↗

arXiv2013

Vibrations of Jammed Disk Packings with Hertzian Interactions

Contact breaking and Hertzian interactions between grains can both give rise to nonlinear vibrational response of static granular packings. We perform molecular dynamics simulations at constant energy in 2D of frictionless bidisperse disks that interact via Hertzian spring potentials as a function of energy and measure directly the vibrational response from the Fourier transform of the velocity autocorrelation function. We compare the measured vibrational response of static packings near jamming onset to that obtained from the eigenvalues of the dynamical matrix to determine the temperature above which the linear response breaks down. We compare packings that interact via single-sided (purely repulsive) and double-sided Hertzian spring interactions to disentangle the effects of the shape of the potential from contact breaking. Our studies show that while Hertzian interactions lead to weak nonlinearities in the vibrational behavior (e.g. the generation of harmonics of the eigenfrequencies of the dynamical matrix), the vibrational response of static packings with Hertzian contact interactions is dominated by contact breaking as found for systems with repulsive linear spring interactions.

cond-mat.soft↗

arXiv2013

Vibrations in jammed solids: Beyond linear response

We propose a `phase diagram' for particulate systems that interact via purely repulsive contact forces, such as granular media and colloidal suspensions. We identify and characterize two distinct classes of behavior as a function of the input kinetic energy per degree of freedom $T_0$ and packing fraction deviation above and below jamming onset $Δϕ=ϕ- ϕ_J$ using numerical simulations of purely repulsive frictionless disks. Iso-coordinated solids (ICS) only occur above jamming for $Δϕ> Δϕ_c(T_0)$; they possess average coordination number equal to the isostatic value ($< z> = z_{\rm iso}$) required for mechanically stable packings. ICS display harmonic vibrational response, where the density of vibrational modes from the Fourier transform of the velocity autocorrelation function is a set of sharp peaks at eigenfrequencies $ω_k^d$ of the dynamical matrix evaluated at $T_0=0$. Hypo-coordinated solids (HCS) occur both above and below jamming onset within the region defined by $Δϕ> Δϕ^*_-(T_0)$, $Δϕ< Δϕ^*_+(T_0)$, and $Δϕ> Δϕ_{cb}(T_0)$. In this region, the network of interparticle contacts fluctuates with $< z> \approx z_{\rm iso}/2$, but cage-breaking particle rearrangements do not occur. The HCS vibrational response is nonharmonic, {\it i.e} the density of vibrational modes $D(ω)$ is not a collection of sharp peaks at $ω_k^d$, and its precise form depends on the measurement method. For $Δϕ> Δϕ_{cb}(T_0)$ and $Δϕ< Δϕ^*_{-}(T_0)$, the system behaves as a hard-particle liquid.

cond-mat.soft↗