"Random-Graph Models and Characterization of Granular Networks"

A paper of mine from 2020 now has its final coordinates listed on the published file itself. Here are some details.

Title: Random-Graph Models and Characterization of Granular Networks

Authors: Silvia Nauer, Lucas Böttcher, and Mason A. Porter

Abstract: Various approaches and measures from network analysis have been applied to granular and particulate networks to gain insights into their structural, transport, failure-propagation and other systems-level properties. In this article, we examine a variety of common network measures and study their ability to characterize various two-dimensional and three-dimensional spatial random-graph models and empirical two-dimensional granular networks. We identify network measures that are able to distinguish between physically plausible and unphysical spatial network models. Our results also suggest that there are significant differences in the distributions of certain network measures in two and three dimensions, hinting at important differences that we also expect to arise in experimental granular networks.

"Forecasting Failure Locations in 2-Dimensional Disordered Lattices"

One of my papers (which we published in PNAS) now has its final coordinates (volume, page numbers, etc.), after previously being posted in 'advanced access'. Here are some details about the article.

Title: Forecasting Failure Locations in 2-Dimensional Disordered Lattices

Authors: Estelle Berthier, Mason A. Porter, and Karen E. Daniels

Abstract: Forecasting fracture locations in a progressively failing disordered structure is of paramount importance when considering structural materials. We explore this issue for gradual deterioration via beam breakage of 2-dimensional (2D) disordered lattices, which we represent as networks, for various values of mean degree. We study experimental samples with geometric structures that we construct based on observed contact networks in 2D granular media. We calculate geodesic edge betweenness centrality, which helps quantify which edges are on many shortest paths in a network, to forecast the failure locations. We demonstrate for the tested samples that, for a variety of failure behaviors, failures occur predominantly at locations that have larger geodesic edge betweenness values than the mean one in the structure. Because only a small fraction of edges have values above the mean, this is a relevant diagnostic to assess failure locations. Our results demonstrate that one can consider only specific parts of a system as likely failure locations and that, with reasonable success, one can assess possible failure locations of a structure without needing to study its detailed energetic states.

Significance Statement: Disordered lattices are used widely for mechanical applications because they are lightweight and robust. Due to their heterogeneous structure, it is a complicated task to understand and forecast their progressive degradation. To safely use these materials and design structures with optimized mechanical properties, it is crucial to understand where failures occur. We show that a simple test that consists of comparing the importance of a beam with respect to the other beams in a lattice permits a successful forecast of the locations of failures. It allows one to consider only a small fraction of the beams as likely failure locations. Our approach also provides a roadmap for studies of failures in other spatial networks.

What Happens in Dresden Stays in Dresden (2019 Edition)

I am here in Dresden for our workshop on granular and particulate networks.

This is the first stop on my (now significantly shortened) 2019 European summer tour.

"Network Analysis of Particles and Grains"

Our review article on granular and particulate networks, which appeared in advanced access a few months ago, is now out in final form (with its page numbers and other coordinates). Here are some details.

Title: Network Analysis of Particles and Grains

Authors: Lia Papadopoulos, Mason A. Porter, Karen E. Daniels, and Danielle S. Bassett

Abstract: The arrangements of particles and forces in granular materials have a complex organization on multiple spatial scales that range from local structures to mesoscale and system-wide ones. This multiscale organization can affect how a material responds or reconfigures when exposed to external perturbations or loading. The theoretical study of particle-level, force-chain, domain and bulk properties requires the development and application of appropriate physical, mathematical, statistical and computational frameworks. Traditionally, granular materials have been investigated using particulate or continuum models, each of which tends to be implicitly agnostic to multiscale organization. Recently, tools from network science have emerged as powerful approaches for probing and characterizing heterogeneous architectures across different scales in complex systems, and a diverse set of methods have yielded fascinating insights into granular materials. In this article, we review work on network-based approaches to studying granular matter and explore the potential of such frameworks to provide a useful description of these systems and to enhance understanding of their underlying physics. We also outline a few open questions and highlight particularly promising future directions in the analysis and design of granular matter and other kinds of material networks.

Ordering Dice by Stirring Them (Not Shaking Them)

As stated in the Physics Focus blurb about this article, "A jumble of thousands of cubic dice, agitated by an oscillating rotation, can rapidly become completely ordered, a result that is hard to produce with more conventional shaking."

Also, take a look at the very short video.

Very cool!

(Tip of the cap to Andrea Welsh.)

"Nonlinear Coherent Structures in Granular Crystals"

My review article on granular crystals (and especially work on it during the last decade) just came out in final form. Here are the details.

Title: Nonlinear Coherent Structures in Granular Crystals

Authors: Chris Chong, Mason A. Porter, Panayotis G. Kevrekidis, and Chiara Daraio

Abstract: The study of granular crystals, which are nonlinear metamaterials that consist of closely packed arrays of particles that interact elastically, is a vibrant area of research that combines ideas from disciplines such as materials science, nonlinear dynamics, and condensed-matter physics. Granular crystals exploit geometrical nonlinearities in their constitutive microstructure to produce properties (such as tunability and energy localization) that are not conventional to engineering materials and linear devices. In this topical review, we focus on recent experimental, computational, and theoretical results on nonlinear coherent structures in granular crystals. Such structures—which include traveling solitary waves, dispersive shock waves, and discrete breathers—have fascinating dynamics, including a diversity of both transient features and robust, long-lived patterns that emerge from broad classes of initial data. In our review, we primarily discuss phenomena in one-dimensional crystals, as most research to date has focused on such scenarios, but we also present some extensions to two-dimensional settings. Throughout the review, we highlight open problems and discuss a variety of potential
engineering applications that arise from the rich dynamic response of granular crystals.

"Scattering of Waves by Impurities in Precompressed Granular Chains"

You may have heard of the Ramsaur–Townsend resonance from scattering problems. The simplest version of it (by examing a particle in a square well) is one of the canonical textbook problems in quantum mechanics.

This is one of the important effects illustrating the need for a notion of wave mechanics.

It turns out that one can also get an RT resonance in a macroscopic, classical system.

The discovery of this classical RT effect (in granular crystals) is the subject of a paper by my collaborators and me, out in final form in Physical Review E today. Here are the details of the article.


Title: "Scattering of Waves by Impurities in Precompressed Granular Chains"

Authors: Alejandro J. Martínez, Hiromi Yasuda, Eunho Kim, P. G. Kevrekidis, Mason A. Porter, and Jinkyu Yang

Abstract: We study scattering of waves by impurities in strongly precompressed granular chains. We explore the linear scattering of plane waves and identify a closed-form expression for the reflection and transmission coefficients for the scattering of the waves from both a single impurity and a double impurity. For single-impurity chains, we show that, within the transmission band of the host granular chain, high-frequency waves are strongly attenuated (such that the transmission coefficient vanishes as the wavenumber k → ±π), whereas low-frequency waves are well-transmitted through the impurity. For double-impurity chains, we identify a resonance—enabling full transmission at a particular frequency—in a manner that is analogous to the Ramsauer–Townsend (RT) resonance from quantum physics. We also demonstrate that one can tune the frequency of the RT resonance to any value in the pass band of the host chain. We corroborate our theoretical predictions both numerically and experimentally, and we directly observe almost complete transmission for frequencies close to the RT resonance frequency. Finally, we show how this RT resonance can lead to the existence of reflectionless modes in granular chains (including disordered ones) with multiple double impurities.

March Meeting Chatchkes

I was pretty selective about chatchkes, but here is the stuff I have gotten so far –– most of them are appropriately GSNP-themed. GSNP is the Group on Statistical and Nonlinear Physics, which is my main home in this community. (I did get roped into signing up for the Soft Matter group, and then I was given appropriately granular mints.)

The rattleback was my reward for the Plinko game. It was what I wanted; there were multiple possible chatchkes that one could get, but the game was less random than a good Plinko construction would have been.

The coolest chatchke by far is the hot-pack/cold-pack, which is also appropriately granular (and which I think can be appropriated for many other uses in addition to the intended ones).

By the way: I plan to use everything in this picture, with the possible exception of the brain.

At registration, everybody received a keychain with a small bottle of antibacterial hand sanitizer (because of the scientific reputation for hygiene).

"Superdiffusive Transport and Energy Localization in Disordered Granular Crystals"

A new paper of mine just came out in final form today. Here are the details. (Also see a paper by others published as a consecutive article with ours. Scientifically, it's really good that these articles have appeared as back-to-back papers.)

Title: Superdiffusive Transport and Energy Localization in Disordered Granular Crystals

Authors: Alejandro J. Martínez, P. G. Kevrekidis, and Mason A. Porter

Abstract: We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to differ fundamentally from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder—an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements)—and for two types of initial conditions (displacement excitations and velocity excitations). We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics depend strongly on the type of initial condition. In particular, for displacement excitations, the long-time asymptotic behavior of the second moment ˜m2 of the energy has oscillations that depend on the type of disorder, with a complex trend that differs markedly from a power law and
which is particularly evident for an Anderson-like disorder. By contrast, for velocity excitations, we find that a standard scaling m_2 ∼ t^γ (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, m_2 and the inverse participation ratio P^{−1} satisfy scaling relations m_2 ∼ t^γ and P^{−1} ∼ t^{−η}, and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with η < 0.1) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of a propagating wave packet. For an Anderson-like disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly γ ≈ 1.7 and η ≈ 1) for all three types of disorder and for both types of initial conditions.

"Granular Crystals: Nonlinear Dynamics Meets Materials Engineering"

Chiara Dario, Panos Kevrekidis, and I have written an expository article about granular crystals for the magazine Physics Today.

"Extraction of Force-Chain Network Architecture in Granular Materials Using Community Detection"

A new paper by my collaborators and me came out last week in the journal Soft Matter, and the inside front cover picture of the associated issue of the journal goes with our article. Here are the details about the article.


Title: Extraction of Force-Chain Network Architecture in Granular Materials Using Community Detection

Authors: Danielle S. Bassett, Eli T. Owens, Mason A. Porter, M. Lisa Manning, and Karen E. Daniels

Abstract: Force chains form heterogeneous physical structures that can constrain the mechanical stability and acoustic transmission of granular media. However, despite their relevance for predicting bulk properties of materials, there is no agreement on a quantitative description of force chains. Consequently, it is difficult to compare the force-chain structures in different materials or experimental conditions. To address this challenge, we treat granular materials as spatially-embedded networks in which the nodes (particles) are connected by weighted edges that represent contact forces. We use techniques from community detection, which is a type of clustering, to find sets of closely connected particles. By using a geographical null model that is constrained by the particles' contact network, we extract chain-like structures that are reminiscent of force chains. We propose three diagnostics to measure these chain- like structures, and we demonstrate the utility of these diagnostics for identifying and characterizing classes of force-chain network architectures in various materials. To illustrate our methods, we describe how force-chain architecture depends on pressure for two very different types of packings: (1) ones derived from laboratory experiments and (2) ones derived from idealized, numerically-generated frictionless packings. By resolving individual force chains, we quantify statistical properties of force-chain shape and strength, which are potentially crucial diagnostics of bulk properties (including material stability). These methods facilitate quantitative comparisons between different particulate systems, regardless of whether they are measured experimentally or numerically.

New Article in Physical Review E: "Flow and Clogging of a Sheep Herd Passing Through a Bottleneck"

Yes, really.

The first person who told me about this granular sheep project, which has finally appeared in published form, is Karen Daniels, who points out that there are all sorts of wonderful (baaaaaaaad?) puns that one can make about shearing forces, sheared sheep, and so on.

Sweet Dreams are Made of Granular Packings (and Who am I to Disagree?)

On Friday morning, I had a strange dream dream. It included an ice-cream parlor, which was rumored to be wild and crazy. I backtracked to this parlor after leaving my friends in our hotel. The parlor had two flavors named after one of my Oxford colleagues (though I decided in my dream that it must be a coincidence), and their method of choosing mix-ins --- which were required to come with the ice cream (one couldn't opt out) were based on which part of their large vat with a mixture toppings had the largest local packing density (and one had no choice in the algorithm either). That's right: granular packings are now officially showing up in my dreams.

There were some other parts as well, but that was the most vivid. Clearly, I need to get some ice cream --- though I still don't feel well enough to do that (I've been really sick with the flu for a week). I also need to work on reviewing a paper on granular force chains (which is on my desk) and finishing up a new paper of my own on granular force chains.

As Matt Sullivan reminded me on Facebook, some candies have been used to study granular packings. Given that things like M&Ms and the other things were all mixed together in one container, I think the dream store's idea was that the property of particular items having a higher local packing fraction than others was going to systematically make specific items show up more often among the mix-ins. I also suspect that having seen that project before had an influence on some of the particulars of my dream.

"Geometric Cohesion in Granular Materials"

I just read a really fascinating expository article called "Geometric Cohesion in Granular Materials". It concerns how the shape of granular materials can make them stick together. A good example is clumping in staples. Way cool!

"Influence of Network Topology on Sound Propagation in Granular Materials"

One of my papers just came out today. Here are the details.


Title: Influence of Network Topology on Sound Propagation in Granular Materials

Authors: Danielle S. Bassett, Eli T. Owens, Karen E. Daniels, and Mason A. Porter

Abstract: Granular media, whose features range from the particle scale to the force-chain scale and the bulk scale, are usually modeled as either particulate or continuum materials. In contrast with each of these approaches, network representations are natural for the simultaneous examination of microscopic, mesoscopic, and macroscopic features. In this paper, we treat granular materials as spatially embedded networks in which the nodes (particles) are connected by weighted edges obtained from contact forces.We test a variety of network measures to determine their utility in helping to describe sound propagation in granular networks and find that network diagnostics can be used to probe particle-, curve-, domain-, and system-scale structures in granular media. In particular, diagnostics of mesoscale network structure are reproducible across experiments, are correlated with sound propagation in this medium, and can be used to identify potentially interesting size scales. We also demonstrate that the sensitivity of network diagnostics depends on the phase of sound propagation. In the injection phase, the signal propagates systemically, as indicated by correlations with the network diagnostic of global efficiency. In the scattering phase, however, the signal is better predicted by mesoscale community structure, suggesting that the acoustic signal scatters over local geographic neighborhoods. Collectively, our results demonstrate how the force network of a granular system is imprinted on transmitted waves.



As a side note, my friend and coauthor Karen Daniels and I have talked about various scientific things on and off for over a decade since we were grad students together at Cornell. This is the first paper we've published together.

Home Experiments with Granular Materials

I wouldn't even dream of playing with my food in order to look at the packing properties of the grains in my chicken soup.

"Nonlinear Waves in Disordered Diatomic Granular Chains"

Every scientist has written papers that have interesting stories. Like everybody else, I have several of these, and my paper that was published in final form today is one such paper. You can see the title of this blog entry, and I will give the abstract and some other comments below, but let me first tell you a story.

I have been working on nonlinear waves in granular crystals since November 2006, and this has become the main component of my nonlinear waves research. It's been a fun and challenging adventure and I am looking forward to continuing in it. The idea for this paper arose from three different occurrences: a question from an audience member in the seminar that I gave to OCIAM (the research group I'm in) when I first joined the faculty in Fall 2007, a seminar I gave the same term at University of Cambridge, and a similar idea by e-mail when I was showing a collaborator (from a different set of projects) a draft of a much earlier paper of mine on granular crystals. In each of these cases, I was asked about the possibility of Anderson localization in disordered granular crystals.

I hadn't thought about it before nor did I know too much about Anderson localization (though I had heard about it), but it sounded interesting, and eventually my collaborators and I designed an undergraduate student project to study disordered granular crystals. The student project started in summer 2008 and eventually became this paper, which was first submitted to a journal in April 2009 and had a bit of a rough path. The first referees asked us to remove language related to Anderson localization, and the fifth and final referee (five---count 'em---five referees, though I do actually have one paper that needed 6 referees) asked us to put that stuff back in. [In fact, that referee all but called one of the previous ones an idiot.] It got a bit frustrating at times, but the published version of the paper is so much better than the original version, so in many senses the pain was worth it. And now the paper is finally out!

OK, so what did we find? Well, I was asked several times about the possibility of Anderson localization in granular crystals. What actually occurs is a different and seemingly novel form of localization, and in my view our paper opens up the problem of what exactly this phenomenon is. I have no idea what kind of impact this paper will ultimately have and I have several papers in more prestigious journals, but I do feel like my collaborators and I have opened up a pretty damned interesting problem with this paper. Alex, I'll take "nonlinear localization" for the win!

Oh, and here is the formal paper information:

Title: Nonlinear Waves in Disordered Diatomic Granular Chains

Authors: Laurent Ponson, Nicholas Boechler, Yi Ming Lai, Mason A. Porter, P. G. Kevrekidis, and Chiara Daraio

Abstract: We investigate the propagation and scattering of highly nonlinear waves in disordered granular chains composed of diatomic (two-mass) units of spheres that interact via Hertzian contact. Using ideas from statistical mechanics, we consider each diatomic unit to be a "spin," so that a granular chain can be viewed as a spin chain composed of units that are each oriented in one of two possible ways. Experiments and numerical simulations both reveal the existence of two different mechanisms of wave propagation: in low-disorder chains, we observe the propagation of a solitary pulse with exponentially decaying amplitude. Beyond a critical level of disorder, the wave amplitude instead decays as a power law, and the wave transmission becomes insensitive to the level of disorder. We characterize the spatiotemporal structure of the wave in both propagation regimes and propose a simple theoretical interpretation for a transition between the two regimes. Our investigation suggests that an elastic spin chain can be used as a model system to investigate the role of heterogeneities in the propagation of highly nonlinear waves.

"Discrete Breathers in One-Dimensional Diatomic Granular Crystals"

A new paper by my collaborators and me has just appeared in Physical Review Letters. This is my 4th paper in PRL. The concerns discrete breathers in granular crystal, and it includes theory, numerical simulations, and (especially!) experiments. In fact, this paper gives the first experimental demonstration of intrinsic localized modes (these are the discrete breathers) in granular crystals, which is a rather exciting result. I believe that Caltech is going to be issuing a press release, so I'll pass along that and any ensuing press coverage later.

Title: Discrete Breathers in One-Dimensional Diatomic Granular Crystals

Authors: N. Boechler, G. Theocharis, S. Job, P. G. Kevrekidis, Mason A. Porter, and C. Daraio

Abstract: We report the experimental observation of modulational instability and discrete breathers in a one-dimensional diatomic granular crystal composed of compressed elastic beads that interact via Hertzian contact. We first characterize their effective linear spectrum both theoretically and experimentally. We then illustrate theoretically and numerically the modulational instability of the lower edge of the optical band. This leads to the dynamical formation of long-lived breather structures, whose families of solutions we compute throughout the linear spectral gap. Finally, we experimentally observe the manifestation of the modulational instability and the resulting generation of localized breathing modes with quantitative characteristics that agree with our numerical results.

"Optimal Design of Composite Granular Protectors"

This paper, which was accepted for publication in October 2008, has finally appeared in print. About bloody time.

Title: Optimal Design of Composite Granular Protectors

Authors: Fernando Fraternali, Mason A. Porter, and Chiara Daraio

Abstract: We employ an evolutionary algorithm to investigate the optimal design of composite protectors using one-dimensional granular chains composed of beads of various sizes, masses, and stiffnesses. We define a fitness function using the maximum force transmitted from the protector to a "wall" that represents the body to be protected and accordingly optimize the topology (arrangement), size, and material of the chain. We obtain optimally randomized granular protectors characterized by high-energy equipartition and the transformation of incident waves into interacting solitary pulses. We consistently observe that the pulses traveling to the wall combine to form an extended (long-wavelength), small-amplitude pulse.


Note that this paper is in many senses an engineering paper. Some of the results are definitely things that would be nice to understand in a more "fundamental" fashion.

"Localized Breathing Modes in Granular Crystals with Defects"

My latest paper has just come out in Physical Review E. Here are the specs:

Title: Localized Breathing Modes in Granular Crystals with Defects

Authors:: G. Theocharis, M. Kavousanakis, P. G. Kevrekidis, Chiara Daraio, Mason A. Porter, and I. G. Kevrekidis

Abstract: We study localized modes in uniform one-dimensional chains of tightly packed and uniaxially compressed elastic beads in the presence of one or two light-mass impurities. For chains composed of beads of the same type, the intrinsic nonlinearity, which is caused by the Hertzian interaction of the beads, appears not to support localized, breathing modes. Consequently, the inclusion of light-mass impurities is crucial for their appearance. By analyzing the problem’s linear limit, we identify the system’s eigenfrequencies and the linear defect modes. Using continuation techniques, we find the solutions that bifurcate from their linear counterparts and study their linear stability in detail. We observe that the nonlinearity leads to a frequency dependence in the amplitude of the oscillations, a static mutual displacement of the parts of the chain separated by a defect, and for chains with two defects that are not in contact, it induces symmetry-breaking bifurcations.


By the way, this is the paper that gave me a bi-Kevrekidis number of 1.