  It is known that the Cardassian universe is successful in describing the
accelerated expansion of the universe, but its dynamical equations are hard to
get from the action principle. In this paper, we establish the connection
between the Cardassian universe and $f(T, \mathcal{T})$ gravity, where $T$ is
the torsion scalar and $\mathcal{T}$ is the trace of the matter energy-momentum
tensor. For dust matter, we find that the modified Friedmann equations from
$f(T, \mathcal{T})$ gravity can correspond to those of Cardassian models, and
thus, a possible origin of Cardassian universe is given. We obtain the original
Cardassian model, the modified polytropic Cardassian model, and the exponential
Cardassian model from the Lagrangians of $f(T,\mathcal{T})$ theory.
Furthermore, by adding an additional term to the corresponding Lagrangians, we
give three generalized Cardassian models from $f(T,\mathcal{T})$ theory. Using
the observation data of type Ia supernovae, cosmic microwave background
radiation, and baryon acoustic oscillations, we get the fitting results of the
cosmological parameters and give constraints of model parameters for all of
these models.
