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The formation of singularity and breakdown of classical solutions to the threedimensional compressible viscoelasticity and inviscid elasticity are considered. For the compressible inviscid elastic fluids,the finite-time formation of singularity in classical solutions is proved for certain initial data.For the compressible viscoelastic fluids,a criterion in term of the temporal integral of the velocity gradient is obtained for the breakdown of smooth solutions.