The　unified　displacement　function　(UDF)　is　presented　to　describe　the　deformation　behaviours　of　the　tunnel　profile　along　with　time　under　the　surface　slope　condition.　Based　on　the　discrete　Fourier　method,　the　third-order　UDF　in　the　physical　plane　is　expanded　to　the　Laurent　series　in　the　complex　variable　plane.　The　complex　variable　method　is　employed　to　derive　the　elastic　analytical　solution　of　stratum　displacement,　when　the　third-order　UDF　is　taken　as　the　displacement　boundary　condition　of　tunnel　cross-section　(DBCTC).　The　proposed　elastic　solution　agrees　well　with　the　results　of　the　finite　element　method　for　the　consistent　model,　which　verifies　the　correctness　of　the　proposed　analytical　solution.　Combining　the　corresponding　principle　and　fractional　Generalized　Kelvin　viscoelastic　constitutive　model,　the　fractional　viscoelastic　solution　under　the　surface　slope　condition　is　determined.　The　time　effect　of　stratum　displacement　is　presented　in　two　aspects:　time-dependent　DBCTC　and　time-dependent　material　parameters.　The　parameter　analysis　is　performed　to　investigate　influences　of　deformation　modes　of　the　third-order　UDF,　slope　angle,　tunnel　radius　and　fractional　order　on　the　time　effect　of　stratum　vertical　and　horizontal　displacement.　©　2023　Tongji　University