Analytic characterization and control of nonlinear nonuniform boundary modes in isostatic systems

Lattices with balanced numbers of mechanical degrees of freedom and constraints (called Maxwell, mechanically critical or isostatic) acquire rigid, zero-energy modes from absent bonds at their boundaries. While some uniform nonlinear and spatially varying linear modes of such lattices have been characterized analytically, the nonuniform, nonlinear reference states which they are hypothesized to contain remain largely uncharted. Here, we show analytically that the nonlinear boundary modes for the classic example of the Kagome lattice may be understood to correspond with the conformal, or angle-preserving, maps and we further present an algorithm to generate these configurations. This observation enables the construction and numeric confirmation of a unique elastic energy functional which displays the signature of a recently identified structural duality, along with bulk-boundary correspondence which enables the zero energy deformations to be activated using boundary control. We further introduce a diagram-based approach to reveal an infinite, but subextensive, number of uniform collapse pathways. We find that all of these pathways are pure-dilational, suggesting the continued role of the conformal maps, and overall bringing the complete characterization and control of the zero energy deformation pathways of this lattice into view for the first time.