A study of deformation localization in nonlinear elastic square lattices under compression

The paper investigates localized deformation patterns resulting from the onset of instabilities in lattice structures. The study is motivated by previous observations on discrete hexagonal lattices, where a variety of localized deformations were found depending on loading configuration, lattice parameters and boundary conditions. These studies are conducted on other lattice structures, with the objective of identifying and investigating minimal models that exhibit localization, hysteresis and path-dependent behaviour. To this end, we first consider a two-dimensional square lattice consisting of point masses connected by in-plane axial springs and vertical ground springs, which may be considered as a discrete description of an elastic membrane supported by an elastic substrate. Results illustrate that, depending on the relative values of the spring constants, the lattice exhibits in-plane or out-of-plane instabilities leading to localized deformations. This model is further simplified by considering the one-dimensional case of a spring–mass chain sitting on an elastic foundation. A bifurcation analysis of this lattice identifies the stable and unstable branches and sheds light on the mechanism of transition from affine deformation to global or diffuse deformation to localized deformation. Finally, the lattice is further reduced to a minimal four-mass model, which exhibits a deformation qualitatively similar to that in the central part of a longer chain. In contrast to the widespread assumption that localization is induced by defects or imperfections in a structure, this work illustrates that such phenomena can arise in perfect lattices as a consequence of the mode shapes at the bifurcation points. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.