  We experimentally study nonlinear force propagation into granular material
during impact from an intruder, and we explain our observations in terms of the
nonlinear grain-scale force relation. Using high-speed video and photoelastic
particles, we determine the speed and spatial structure of the force response
just after impact. We show that these quantities depend on a dimensionless
parameter, $M'=t_c v_0/d$, where $v_0$ is the intruder speed at impact, $d$ is
the particle diameter, and $t_c$ is the collision time for a pair of grains
impacting at relative speed $v_0$. The experiments access a large range of $M'$
by using particles of three different materials. When $M' \ll 1$, force
propagation is chain-like with a speed, $v_f$, satisfying $v_f \propto d/t_c$.
For larger $M'$, the force response becomes spatially dense and the force
propagation speed departs from $v_f\propto d/t_c$, corresponding to collective
stiffening of a strongly compressed packing of grains.

Soft Condensed Matter