  Tensor-polarized structure functions of a spin-1 hadron are additional
observables which do not exist for the spin-1/2 nucleon. They could probe novel
aspects of the internal hadron structure. Twist-2 tensor-polarized structure
functions are $b_1$ and $b_2$, and they are related by the Callan-Gross-like
relation in the Bjorken scaling limit. In this work, we theoretically calculate
$b_1$ in the standard convolution description for the deuteron. Two different
theoretical models, a basic convolution description and a virtual nucleon
approximation, are used for calculating $b_1$ and their results are compared
with the HERMES measurement. We found large differences between our theoretical
results and the data. Although there is still room to improve by considering
higher-twist effects and in the experimental extraction of $b_1$ from the spin
asymmetry $A_{zz}$, there is a possibility that the large differences require
physics beyond the standard deuteron model for their interpretation. Future
$b_1$ studies could shed light on a new field of hadron physics. In particular,
detailed experimental studies of $b_1$ will start soon at the Thomas Jefferson
National Accelerator Facility. In addition, there are possibilities to
investigate tensor-polarized parton distribution functions and $b_1$ at Fermi
National Accelerator Laboratory and a future electron-ion collider. Therefore,
further theoretical studies are needed for understanding the tensor structure
of the spin-1 deuteron, including a new mechanism to explain the large
differences between the current data and our theoretical results.
