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The paper investigates localized deformation patterns resulting from theonset of instabilities in lattice structures. The study is motivated byprevious observations on discrete hexagonal lattices, where the onset ofnon-uniform, quasi-static deformation patterns was associated with the loss ofconvexity of the interaction potential, and where a variety of localizeddeformations were found depending on loading configuration, lattice parametersand boundary conditions. These observations are here conducted on other latticestructures, with the goal of identifying models of reduced complexity that areable to provide insight into the key parameters that govern the onset ofinstability-induced localization. To this end, we first consider atwo-dimensional square lattice consisting of point masses connected by in-planeaxial springs and vertical ground springs. Results illustrate that depending onthe choice of spring constants and their relative values, the lattice exhibitsin-plane or out-of plane instabilities leading to folding and unfolding. Thismodel is further simplified by considering the one-dimensional case of aspring-mass chain sitting on an elastic foundation. A bifurcation analysis ofthis lattice identifies the stable and unstable branches and illustrates itshysteretic and loading path-dependent behaviors. Finally, the lattice isfurther reduced to a minimal four mass model which undergoes afolding/unfolding process qualitatively similar to the same process in thecentral part of a longer chain, helping our understanding of localization inmore complex systems. In contrast to the widespread assumption thatlocalization is induced by defects or imperfections in a structure, this workillustrates that such phenomena can arise in perfect lattices as a consequenceof the mode-shapes at the bifurcation points.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

A study of deformation localization in nonlinear elastic lattices