<p>Time-dependent deformation of an elastic open cylindrical shell resting on a viscoelastic foundation under constant loading is considered. The viscoelastic foundation is represented by a fractional standard linear solid model formulated using the Riemann-Liouville fractional derivative. Based on classical shell theory, the governing equations of the shell-foundation system are derived. Application of the Laplace transform and its inversion leads to an analytical expression for the shell displacement in terms of the Mittag–Leffler function. Numerical examples for a simply supported shell illustrate the spatial distribution and temporal evolution of the deformation. Results are compared with those corresponding to the integer-order viscoelastic model and the elastic foundation model. The effects of the fractional order and viscosity parameters on the deformation response is examined. The fractional-order foundation model provides a greater flexibility in representing the time-dependent interaction between the shell and the supporting medium and captures the evolution of deformation more effectively than classical integer-order models.</p>