  We describe a general constitutive framework for a teleparallel extension of
the general theory of relativity. This approach goes beyond the teleparallel
equivalent of general relativity (TEGR) by broadening the analogy with the
electrodynamics of media. In particular, the main purpose of this paper is to
investigate in detail a local constitutive extension of TEGR that is the local
limit of nonlocal gravity (NLG). Within this framework, we study the modified
FLRW cosmological models. Of these, the most cogent turns out to be the
modified Cartesian flat model which is shown to be inconsistent with the
existence of a positive cosmological constant. Moreover, dynamic dark energy
and other components of the modified Cartesian flat model evolve differently
with the expansion of the universe as compared to the standard flat
cosmological model. The observational consequences of the modified Cartesian
flat model are briefly explored and it is shown that the model is capable of
resolving the H_0 tension.
