  We perform a systematical investigation of the strong decay properties of the
low-lying $1P$- and $2D$-wave doubly charmed baryons with the $^3P_0$ quark
pair creation model. The main predictions include: (i) in the $\Xi_{cc}$ and
$\Omega_{cc}$ family, the $1P~\rho$ mode excitations with $J^P=1/2^-$ and
$3/2^-$ should be the fairly narrow states. (ii) For the $1P~\lambda$ mode
excitations, $|^2P_{\lambda}\frac{3}{2}^-\rangle$ and
$|^4P_{\lambda}\frac{3}{2}^-\rangle$ have a width of $\Gamma\sim150$ MeV, and
mainly decay into the $J^P=3/2^+$ ground state. Meanwhile,
$|^2P_{\lambda}\frac{1}{2}^-\rangle$ and $|^4P_{\lambda}\frac{5}{2}^-\rangle$
are the narrow states with a width of $\Gamma\sim40$ MeV, and mainly decay into
the ground state with $J^P=1/2^+$. (iii) The $2D_{\rho\rho}$ states mainly
decay via emitting a heavy-light meson if their masses are above the threshold
of $\Lambda_cD$ or $\Xi_cD$, respectively. Their strong decay widths are
sensitive to the masses and can reach several tens MeV. (iv) The
$2D_{\lambda\lambda}$ states may be broad states with a width of $\Gamma>100$
MeV. It should be emphasized that the states with $J^P=3/2^+$ and $5/2^+$
mainly decay into the ground state with $J^P=3/2^+$ plus a light-flavor meson,
while the states with $J^P=1/2^+$ and $7/2^+$ mainly decay into the ground
state with $J^P=1/2^+$ plus a light-flavor meson.
