Effect of discreteness on domain wall stability in a plate coupled to a foundation of bistable elements

Abstract: Surfaces and structures capable of multiple stable configurations have attracted growing interest for on-demand shape morphing. In this work, we consider an elastic compliant plate coupled to a two-dimensional foundation comprising an array of bistable elements, a system that can form and retain complex continuous morphologies without sustained actuation via creation of stable domain walls separating regions in different stable states. These domain walls exhibit three distinct behaviors: expansion, shrinking, and metastable pinning. These arise from two limits of foundation discreteness. In the continuum limit, where bistable units are strongly coupled, domain walls undergo global phase transitions analogous to first-order phase transitions. In the anti-continuum limit, discreteness introduces Peierls-Nabarro-type energy modulations that lead to metastable pinning. To quantify these behaviors and the transition between the two limits, we develop a reduced-order model that captures the total potential energy of configurations with domain walls and validate it using finite element analysis (FEA). For axisymmetric domain walls, the model yields phase diagrams identifying the regimes of expansion, shrinking, and pinning as functions of bistable-potential asymmetry, relative foundation discreteness, and domain-wall size. We then extend the analysis to non-axisymmetric geometries and establish local geometric criteria that predict the stability of convex and concave polygonal domain walls, confirmed by simulations. Together, these results clarify how discreteness enables stability through energy-landscape modulation, provide predictive design rules for multistable reconfigurable surfaces, and offer insights into domain-wall stability more generally in elastically coupled multistable metamaterials.