We　study　the　existence　and　uniqueness　of　global　strong　solutions　to　the　equations　of　an　incompressible　viscoelastic　fluid　in　a　spatially　periodic　domain,　and　show　that　a　unique　strong　solution　exists　globally　in　time　if　the　initial　deformation　and　velocity　are　small　for　the　given　physical　parameters.　In　particular,　the　initial　velocity　can　be　large　for　the　large　elasticity　coefficient.　Our　result　mathematically　verifies　that　the　elasticity　can　prevent　the　formation　of　singularities　of　strong　solutions　with　large　initial　velocity,　thus　playing　a　similar　role　to　viscosity　in　preventing　the　formation　of　singularities　in　viscous　flows.　Moreover,　for　given　initial　velocity　perturbation　and　zero　initial　deformation　around　the　rest　state,　we　find,　as　the　elasticity　coefficient　or　time　goes　toinfinity,　that　(1)　any　straight　line　segment　l(0)consisted　of　fluid　particles　in　the　rest　state,　after　being　bent　by　a　velocity　perturbation,　will　turn　into　a　straight　line　segment　that　is　parallel　to　l(0)and　has　the　same　length　as　l(0).　(2)　the　motion　of　the　viscoelastic　fluid　can　be　approximated　by　a　linear　pressureless　motion　in　Lagrangian　coordinates,　even　when　the　initial　velocity　is　large.　Moreover,　the　above　mentioned　phenomena　can　also　be　found　in　the　corresponding　compressible　fluid　case.　(C)2021　Elsevier　Inc.　All　rights　reserved.