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 |

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 |

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 |

Slide 34 |

How compression dictates
new contacts |

Trial modes in real system |

Slide 37 |