Disarrangements and instabilities in 1D hyperelasticity

In the present work, the overall nonlinear elastic behavior of a 1D multi-modular structure incorporating possible imperfections at the discrete (micro-scale) level, is derived with respect to both tensile and compressive applied loads. The model is built up through the repetition of n units, each one comprising two rigid rods having equal lengths, linked by means of pointwise constraints capable to elastically limit motions in terms of relative translations (sliders) and rotations (hinges). The mechanical response of the structure is analyzed by varying the number n of the elemental moduli, as well as in the limit case of infinite number of infinitesimal constituents, in light of the theory of (first order) Structured Deformations (SDs), that interprets the deformation of any continuum body as the projection, at the macroscopic scale, of geometrical changes occurring at the level of its sub-macroscopic elements. In this way, a wide family of nonlinear elastic behaviors is generated by tuning internal microstructural parameters, the tensile buckling and the classical Euler Elastica under compressive loads resulting as special cases in the so-called continuum limit, say when n tends to 1. Finally, by plotting the results in terms of first Piola-Kirchhoff stress versus macroscopic stretch, it is for the first time demonstrated that such SDs-based 1D models can be helpfully used to generalize some standard hyperelastic behaviors by additionally taking into account instability phenomena and concealed defects.