Disarrangements and instabilities in augmented one-dimensional hyperelasticity

Abstract

In the present work, the overall nonlinear elastic behaviour of a one-dimensional multi-modular structure incorporating possible imperfections at the discrete (microscale) 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 of elastically limiting motions in terms of relative translations (sliders) and rotations (hinges). The mechanical response of the structure is analysed by varying the number n of the elemental moduli, as well as in the limit case of an infinite number of infinitesimal constituents, in light of the theory of (first-order) structured deformations (SDs), which 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 behaviours is generated by tuning internal microstructural parameters, the tensile buckling and the classical Euler's elastica under compressive loads resulting as special cases in the so-called continuum limit —say when n → ∞ . Finally, by plotting the results in terms of the first Piola–Kirchhoff stress versus macroscopic stretch, it is for the first time demonstrated that such SD-based one-dimensional models can be used to generalize some standard hyperelastic behaviours by additionally taking into account instability phenomena and concealed defects.