| Slide 1 |
| Last time: focused, nonlocal response from embedding singularities |
| Force propagation in a simple solid: two pictures |
| Why study solids like this? |
| packed hard particles: solidity without elasticity |
| Bead-by-bead packing makes minimal connectivity |
| Stress-balanced medium has ray-like force propagation |
| Simulation verifies stress balance condition |
| Simulation reveals wild heterogeneity |
| Simulation confirms ray-like propagation |
| Summary: forces in jammed solids |
| Solid is uniform, but forces are heterogeneous | |
| Forces grow exponentially with distance from source [Moukarzel Phys. Rev. Lett. 81, 1634 (1998)] | |
| They can propagate asymmetrically, unlike an elastic solid | |
| These properties arise from their minimal connectivity, which requires a delicate balance of stresses. |
| Vibrations: another anomalous feature of jammed materials |
| Squeeze-jammed grains Þexcess slow vibrations |
| OÕHern simulation: isotropic hard particle pack |
| How many lowest modes in a solid of size L? |
| Marginally jammed particles are isostatic |
| Threshold: some particles feel forces | ||
| All N forced* particles must have balanced forces. | ||
| d N constraints on contact forces in d dimensions | ||
| Requires** at least d N contacts. | ||
| Adiabatic jamming adds contacts one by one: stops when forces balance | ||
| É expect marginally jammed state to have just d N contacts: isostatic | ||
| i.e. minimal number of contacts to fix particle positions. | ||
| Observed in simulation | ||
| Nearly isostatic packings have free modes |
| Energy ¨ dynamical matrix ¨ normal modes |
| Constructing slow modes of |
| Trial modes account for excess slow modes |
| Deformed free mode picture agrees with marginally jammed simulation |
| Further implications of deformed free modes |
| Properties of marginal modes |
| Frequency spacing probes vibrational coupling |
| Marginal modes without packing: randomized square network |
| randomized square lattice
reproduces properties of packed spheres vibrations |
| How do marginal modes transmit energy, momentum |
| energy current around a particle |
| Spatial distribution of energy current in randomized square lattice |
| Profile of energy current lacks correlation |
| Ray-like force propagation and isotropic packs |
| Randomized square lattice shows ray propagation |
| Conclusions |
| Packed, frictionless particles are solid, but qualitatively different from elastic solids. | ||
| Distinctive features can be explained by a single property: isostaticity | ||
| ray-like force propagation | ||
| marginal vibrational modes | ||
| randomized square lattice reproduces properties of particle packs | ||
| details of the packing process seem unimportant | ||
| How are particle packs distinct from general isostatic networks? | ||
| eg. reconnection of network from stress | ||
| Slide 34 |
| How compression dictates new contacts |
| Trial modes in real system |
| Slide 37 |