Static and dynamic properties of inhomogeneous elastic media on disordered substrate

The pinning of an inhomogeneous elastic medium by a disordered substrate is studied analytically and numerically. The static and dynamic properties of a $D$-dimensional system are shown to be equivalent to those of the well-known problem of a $D$-dimensional random manifold embedded in $\mathrm{}(D+D)\mathrm{}$ dimensions. The analogy is found to be very robust, applicable to a wide range of elastic media, including those which are amorphous or nearly periodic, with local or nonlocal elasticity. Also demonstrated explicitly is the equivalence between the dynamic depinning transition obtained at a constant driving force, and the self-organized, near-critical behavior obtained by a (small) constant velocity drive.