  The elastic backbone is the set of all shortest paths. We found a new phase
transition at $p_{eb}$ above the classical percolation threshold at which the
elastic backbone becomes dense. At this transition in $2d$ its fractal
dimension is $1.750\pm 0.003$, and one obtains a novel set of critical
exponents $\beta_{eb} = 0.50\pm 0.02$, $\gamma_{eb} = 1.97\pm 0.05$, and
$\nu_{eb} = 2.00\pm 0.02$ fulfilling consistent critical scaling laws.
Interestingly, however, the hyperscaling relation is violated. Using Binder's
cumulant, we determine, with high precision, the critical probabilities
$p_{eb}$ for the triangular and tilted square lattice for site and bond
percolation. This transition describes a sudden rigidification as a function of
density when stretching a damaged tissue.
