Disarrangements and instabilities in augmented one-dimensional hyperelasticity

Abstract

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'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 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.

Mechanobiology predicts raft formations triggered by ligand-receptor activity across the cell membrane

“…11 (see e.g. Deseri and Owen, 2003;Deseri and Owen, 2010;Deseri and Owen, 2012;Deseri and Owen, 2015;Deseri and Owen, 2019;Palumbo et al, 2018 ), as discussed in details in Appendix C .…”

“…Structured Deformations (see e.g. Deseri and Owen, 2003;Deseri and Owen, 2010;Deseri and Owen, 2012;Deseri and Owen, 2015;Deseri and Owen, 2019;Palumbo et al, 2018 ) is a multiscale geometric framework that permits to account for both conformational changes and remodelling arising at the submacroscopic level. Such a framework does allow for envisioning three full configurations for the membrane.…”

Bulky auxeticity, tensile buckling and deck-of-cards kinematics emerging from structured continua

“…Now, it is natural to explore what happens if the number of discrete elements becomes infinite while their size tends to zero. This can be effectively achieved by adopting the so-called first-order SD theory, which gives a unique way to pass from discrete to continuum without loss of information about displacements and their spatial gradients, allowing one to interpret the classical (smooth) macroscopic motion of a body as the projection of non-classical (piecewise-smooth) deformations occurring at the sub-macroscopic scale [15,19,26–30]. We will call these media—resulting from an initially discrete structure through a limit procedure and preserving internal discontinuities (disarrangements)— structured continua .…”

“…Motivated by the vast variety of inputs and challenges posed by the current research in materials and by the opportunity to unveil some possible logics underlying the response of metamaterials, in this work, we exploit a simple— ad hoc designed—mechanism of mutual sliding between rigid tesserae, interconnected via pre-tensed hyperelastic cables, to obtain uniaxial and biaxial structures capable of offering simultaneously various uncommon mechanical responses. Multi-scale kinematics with local disarrangements [15], tensile buckling [16], instability-induced auxetic properties [17,18] related to chiral deformation modes arising from initially achiral (bulky and geometrically symmetrical) configurations, non-symmetrical responses to compressive and tensile loads as well as deck-of-cards deformation modes producing macroscopic shear [19] are in fact all replicated in the paradigmatic system studied here. In particular, we start from the case of a uniaxial multi-element macro-system comprising an arbitrary number of sub-units, so obtaining tunable critical buckling loads and stiffness if macroscopically stretched under tensile forces, reproducing instead overall shear as a sort of complementary version of the kinematics exhibited by Timoshenko beams if forces are applied orthogonally to the axis of the uniaxial system.…”

“…The theory of structured deformations (SD) is then recalled to trace the kinematical response of the systems under analysis in the limit case in which they comprise an infinite number of infinitesimal constituents. Following this way we de facto propose a first paradigm of biaxially structured medium, demonstrating that a SD-based homogenization strategy—which at the first order does not require the introduction of any characteristic length—is the most effective way for transiting from discrete to continuum models by taking into account sub-macroscopic disarrangements that lead to non-standard macroscopic behaviours [15]. Finally, to validate the theoretical outcomes, numerical finite-element simulations and some qualitative experimental tests have been performed and shown synoptically.…”