  Grain-grain collisions shape the 3-dimensional size and velocity distribution
of the inner Zodiacal Cloud and the impact rates of dust on inner planets, yet
they remain the least understood sink of zodiacal dust grains. For the first
time, we combine the collisional grooming method combined with a dynamical
meteoroid model of Jupiter-family comets (JFCs) that covers four orders of
magnitude in particle diameter to investigate the consequences of grain-grain
collisions in the inner Zodiacal Cloud. We compare this model to a suite of
observational constraints from meteor radars, the Infrared Astronomical
Satellite (IRAS), mass fluxes at Earth, and inner solar probes, and use it to
derive the population and collisional strength parameters for the JFC dust
cloud. We derive a critical specific energy of $Q^*_D=5\times10^5 \pm
4\times10^5 R_\mathrm{met}^{-0.24}$ J kg$^{-1}$ for particles from
Jupiter-family comet particles, making them 2-3 orders of magnitude more
resistant to collisions than previously assumed. We find that the differential
power law size index $-4.2\pm0.1$ for particles generated by JFCs provides a
good match to observed data. Our model provides a good match to the mass
production rates derived from the Parker Solar Probe observations and their
scaling with the heliocentric distance. The higher resistance to collisions of
dust particles might have strong implications to models of collisions in solar
and exo-solar dust clouds. The migration via Poynting-Roberson drag might be
more important for denser clouds, the mass production rates of astrophysical
debris disks might be overestimated, and the mass of the source populations
might be underestimated. Our models and code are freely available online.
