  In this work, we systematically study the OZI-allowed two-body strong decay
properties of $1P$- and $1D$-wave $\Sigma_c$ baryons within the $j $-$j$
coupling scheme in the framework of the quark pair creation model. For a
comparison, we also give the predictions of the chiral quark model. Some model
dependencies can be found in the predictions of two models. The calculations
indicate that: (i) The $1P$-wave $\lambda$-mode $\Sigma_c$ states most likely
to be relatively narrow states with a width of $\Gamma<80$ MeV. Their main
decay channels are $\Lambda_c\pi$, or $\Sigma_c\pi$, or $\Sigma_c^*\pi$. The
$1P$-wave $\rho$-mode states most might be broad states with a width of
$\Gamma\sim 100-200$ MeV. They dominantly decay into $\Sigma_c\pi$ and
$\Sigma_c^*\pi$ channels. Some evidences of these $1P$-wave states are most
likely to be observed in the $\Lambda_c\pi$ and $\Lambda_c\pi\pi$ invariant
mass spectra around the energy range of $2.75-2.95$ GeV. (ii) The $1D$-wave
$\lambda$-mode $\Sigma_c$ excitations may be moderate states with a width of
about dozens of MeV. The $1D$-wave $\lambda$-mode states mainly decay into the
$1P$-wave charmed baryon via the pionic decay processes. Meanwhile, several
$1D$-wave $\lambda$-mode states have significant decay rates into $DN$ or
$D^*N$. Hence, the $DN$ and $D^*N$ are likely to be interesting channels for
experimental exploration. (iii) Furthermore, the two $1D$-wave $\rho$-mode
excitations $\Sigma_c|J^P=5/2^+,3\rangle_{\rho\rho}$ and
$|J^P=7/2^+,3\rangle_{\rho\rho}$ are most likely to be fairly narrow state with
a width of dozens of MeV, and they mainly decay into $\Lambda_c\pi$. Some
evidences of them might be observed in the $\Lambda_c\pi$ invariant mass
spectra around the energy range of $3.1-3.2$ GeV.
