  Recent observations from weak gravitational lensing (WL) surveys indicate a
smoother Universe compared to the predictions of the Cosmic Microwave
Background (CMB). This inconsistency is commonly referred to as the $\sigma_8$
tension or $S_8$ tension, where $\sigma_8$ represents the present
root-mean-square matter fluctuation averaged over a sphere of radius $8 h^{-1}
\mathrm{Mpc}$, and $S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3}$. In this article,
we investigate a kind of general Dirac-Born-Infeld (DBI) Lagrangian referred to
as \textit{surface-type DBI} (sDBI) model. We find that up to the linear order,
the constraints on the sDBI model with high-redshift probe (CMB) and
low-redshift probes (WL and Galaxy Clustering, GC) yield $S_8=
0.7448_{-0.21}^{+0.031}$ and $0.7426_{-0.085}^{+0.054}$, respectively.
Remarkably, these values not only demonstrate self-consistency but also align
with the values obtained from the majority of low-redshift probes. Furthermore,
we present a discussion on exploring the non-linear effects of this model,
which holds the potential to address additional challenges associated with Cold
Dark Matter (CDM) on small scales.
