Material instability and subsequent restabilization from homogenization of periodic elastic lattices

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.

The lumped structural model is homogenized using a purely mechanical approach (whose results are also confirmed via formal homogenization based on variational calculus), resulting in an analytical closed-form solution that serves as a reference model. Despite its simplicity, the model exhibits a surprisingly rich mechanical behaviour. Specifically, for certain radial paths in stress space: (i.) stability is always preserved; (ii.) compaction, shear, and mixed-mode localization bands emerge; (iii.) shear bands initially form, but later ellipticity is recovered, and finally, mixed-mode localization terminates the path. This lumped structural model is (at least in principle) realizable in practice and offers an unprecedented and vivid representation of strain localization modes, where the corresponding equations remain fully ‘manageable by hand’. The structural model with smeared deformability behaves similarly to the discrete model but introduces a key distinction: ‘islands’ of instability emerge within a broad zone of stability. This unique feature leads to unexpected behaviour, where shear bands appear, vanish and reappear along radial stress paths originating from the unloaded state.

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.