  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 the onset of
non-uniform, quasi-static deformation patterns was associated with the loss of
convexity of the interaction potential, and where a variety of localized
deformations were found depending on loading configuration, lattice parameters
and boundary conditions. These observations are here conducted on other lattice
structures, with the goal of identifying models of reduced complexity that are
able to provide insight into the key parameters that govern the onset of
instability-induced localization. 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. Results illustrate that depending on
the choice of spring constants and their relative values, the lattice exhibits
in-plane or out-of plane instabilities leading to folding and unfolding. 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 illustrates its
hysteretic and loading path-dependent behaviors. Finally, the lattice is
further reduced to a minimal four mass model which undergoes a
folding/unfolding process qualitatively similar to the same process in the
central part of a longer chain, helping our understanding of localization in
more complex systems. 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.
