Newly observed Λ_c(2860)^+ at LHCb and its D-wave partners Λ_c(2880)^+, Xi_c(3055)^+ and Xi_c(3080)^+

New resonance $\Lambda_c(2860)^+$, reported by LHCb, is a key to establish \emph{D}-wave charmed and charmed-strange baryon families. In this work, we first carry out an analysis of mass spectra of the $\lambda$-mode excited charmed and charmed-strange baryon states, which may reveal the relation of $\Lambda_c(2860)^+$, $\Lambda_c(2880)^+$, $\Xi_c(3055)^+$ and $\Xi_c(3080)^+$, $i.e.$, $\Lambda_c(2860)^+$ and $\Lambda_c(2880)^+$ can form a \emph{D}-wave doublet $[3/2^+,5/2^+]$ while $\Xi_c(3055)^+$ and $\Xi_c(3080)^+$ are the strange partners of $\Lambda_c(2860)^+$ and $\Lambda_c(2880)^+$, respectively. Further study of their two-body Okubo-Zweig-Iizuka-allowed decays supports $\Lambda_c(2860)^+$ and $\Xi_c(3055)^+$ as the \emph{D}-wave charmed and charmed-strange baryons, respectively, with $J^P=3/2^+$. For $\Lambda_c(2880)^+$ and $\Xi_c(3080)^+$, however, there exist some difficulties under the \emph{D}-wave assignment with $J^P=5/2^+$ since the experimental widths and some ratios of partial width of $\Lambda_c(2880)^+$ and $\Xi_c(3080)^+$ cannot be reproduced in our calculation.