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In this paper we analyze the free longitudinal vibrations of a kind of nonlinear one dimensional structured solid, modeling it as a discrete chain of masses interacting through nonlinear springs. The motion equations of this discrete system are solved numerically and the size effect associated to the structure of solid arises. In order to derive continuum models which capture this scale effect, we perform a non-standard continualization of the nonlinear lattice, as well as standard Taylor-based continualization up to different approximation orders. After we propose an axiomatic generalized continuum model, based on a version of the Mindlin general model but extended to finite deformations. Both formulations lead to the same continuous equation, which depends on a scale parameter. Meanwhile in the axiomatic model the scale constant need to be fixed independently, with the continualized one it is possible to obtain its value from the microstructural characteristic of the solid. The nonlinear equations corresponding to the continuum models are solved and, in contrast to other works, we compare the results obtained from the discrete system with those obtained with the continuum ones. This comparison pointed out the capability of the proposed axiomatic model to capture the size effects present in the structured 1D solid, both in the linear and nonlinear regimes. Moreover, the inability of the classical model to capture the scale features when they play a role has been clearly stated.