We present sub-kpc-scale mapping of the 870 $mu$m ALMA continuum emission in six luminous ($L_{rm IR}~sim~5~times10^{12}$ L$_{odot}$) submillimeter galaxies (SMGs) from the ALESS survey of the Extended Chandra Deep Field South. Our high-fidelity 0.07$$-resolution imaging ($sim$500 pc) reveals robust evidence for structures with deconvolved sizes of $lesssim$0.5-1 kpc embedded within (dominant) exponential dust disks. The large-scale morphologies of the structures within some of the galaxies show clear curvature and/or clump-like structures bracketing elongated nuclear emission, suggestive of bars, star-forming rings, and spiral arms. In this interpretation, the ratio of the `ring and `bar radii (1.9$pm$0.3) agrees with that measured for such features in local galaxies. These potential spiral/ring/bar structures would be consistent with the idea of tidal disturbances, with their detailed properties implying flat inner rotation curves and Toomre-unstable disks (Q<1). The inferred one-dimensional velocity dispersions ($sigma_{rm r}lesssim$ 70-160 km s$^{-1}$) are marginally consistent with the limits implied if the sizes of the largest structures are comparable to the Jeans length. We create maps of the star formation rate density ($Sigma_{rm SFR}$) on $sim$500 pc scales and show that the SMGs are able to sustain a given (galaxy-averaged) $Sigma_{rm SFR}$ over much larger physical scales than local (ultra-)luminous infrared galaxies. However, on 500 pc scales, they do not exceed the Eddington limit set by radiation pressure on dust. If confirmed by kinematics, the potential presence of non-axisymmetric structures would provide a means for net angular momentum loss and efficient star formation, helping to explain the very high star formation rates measured in SMGs.
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