The "analogue gravity formalism", an interdisciplinary theoretical scheme developed in the past for studying several non relativistic classical and quantum systems through effective relativistic curved space-times, is here applied to largely deformable elastic bodies described by the nonlinear theory of solid mechanics. Assuming the simplest nonlinear constitutive relation for the elastic material given by a Kirchhoff-St Venant strain-energy density function, it is possible to write for the perturbations an effective space-time metric if the deformation is purely longitudinal and depends on one spatial coordinate only. Theoretical and numerical studies of the corresponding dynamics are performed in selected cases and physical implications of the results obtained are finally discussed.