Material instability and subsequent restabilization from homogenization of periodic elastic lattices

TL;DR

This paper introduces two classes of homogenized nonlinear elastic materials derived from periodic lattices, revealing how stability can be lost and regained under compression, enabling the design of reconfigurable architected materials.

Contribution

It presents a novel homogenization approach for elastic lattices showing stability loss and restabilization, and explores their potential for creating tunable and reconfigurable materials.

Findings

Elasticity tensor loses positive definiteness under compression.

Materials can re-enter stable regimes after initial instability.

Structural elements' axial compliance influences stability behavior.

Abstract

Two classes of non-linear elastic materials are derived via two-dimensional homogenization. These materials are equivalent to a periodic grid of axially-deformable and axially-preloaded structural elements, subject to incremental deformations that involve bending, shear, and normal forces. The unit cell of one class is characterized by elements where deformations are lumped within a finite-degrees-of-freedom framework. In contrast, the other class involves smeared deformation, modelled as flexurally deformable rods with sufficiently high axial compliance. Under increasing compressive load, the elasticity tensor of the equivalent material loses positive definiteness and subsequently undergoes an ellipticity loss. Remarkably, in certain conditions, this loss of stability is followed by a subsequent restabilization; that is, the material re-enters the elliptic regime and even the positive definiteness domain and simultaneously, the underlying elastic lattice returns to a stable state. This effect is closely related to the axial compliance of the elements. Our results: (i.) demonstrate new possibilities for exploiting structural elements within the elastic range, characterized by a finite number of degrees of freedom, to create architected materials with tuneable instabilities, (ii.) introduce reconfigurable materials characterized by 'islands' of stability or instability.