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

TL;DR

This work addresses how discreteness in a plate–bistable foundation system shapes domain-wall stability and morphing capabilities. It develops a collective-coordinate ROM with a tanh-domain-wall ansatz to derive a cohesive energy landscape that captures both continuum nucleation and PN-type pinning, then validates predictions against high-fidelity FE analyses. The study identifies a critical nucleation size $r_{cr}$ and a discreteness threshold $\gamma_{cr}(\alpha)$ that separate expanding, shrinking, and metastable pinned walls for axisymmetric geometries, and extends the framework to irregular polygons using local stability criteria and complementary hexagon arguments. Collectively, the results provide design rules for multistable reconfigurable surfaces and offer general insights into domain-wall stability in elastically coupled bistable metamaterials.

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.

Figures (15)

• Figure 1: (a) Plate-bistable foundation system in original configuration. The bistable units are arranged in a hexagonal pattern. Their stable states are indicated by O (with displacement in z-direction $w=0$) and B ($w=w_b$). (b) Displacement boundary condition (perpendicular to the paper) is applied to a circular region, causing the interior region to deform from the original state O to the other stable B, forming a domain wall (in green) with radius $r_{d0}$ and width $W$. The insert shows the edge size of the hexagonal pattern is $d$. The relative density of the foundation is defined as $\gamma=d/W$. (c) FEA results: (i) and (iii) the domain wall moves when the foundation density $\gamma$ is relatively small ($\gamma=1/8$). It shrinks when the initial domain wall size $r_{d0}$ is smaller than a critical size $r_{cr}$ and expands when $r_{d0}>r_{cr}$; (ii) and (iv) the domain wall stabilizes and stays near its initial radius when the foundation density is relatively large with $\gamma=1/4$. All the FEA results in this plot are with parameters: $h=1, D=1, \rho=0.1, k=1, w_b=1, \alpha=0.48$ and the corresponding characteristic length is $L=1$. All the dimensions shown here are dimensionless, normalized by $L$ if in $x-y$ plane and by $w_b$ in $z$ direction. Moderate damping is used to obtain the metastable configurations.
• Figure 2: The energy distribution in the bistable foundation for (a) small $\gamma$ (continuous foundation) and (b) large $\gamma$ (discrete foundation), respectively. The red color denotes state O, blue denotes state B, and green denotes the domain wall (intermediate region between O and B).
• Figure 3: Two-stage shrinking process of a domain wall. Dots mark the domain wall center positions. FEA results shown for $\alpha=0.48$. w-displacement and r-coordinate are normalized by $w_b$ and characteristic length $L$, respectively.
• Figure 4: The potential energy $V(r_d,H)$ of the reduced order model (ROM) with $\alpha=0.48$. The two stars represent the global maximum in stage 1 (domain wall expanding or shrinking) and the local minimum in stage 2 (domain wall disappearing), respectively. The trajectory FEDCBA is a possible evolution of the domain wall. The insert shows the profiles of the plate with different domain wall positions.
• Figure 5: (a) the potential energy landscape $U_{unit}(w)$ of a single bistable unit with different $\alpha$ values; (b) the potential energy landscape $V(r_d)$ of the whole system by ROM with different $\alpha$ values; (c) the critical nucleation size $r_{cr}$ as a function of $\alpha$ by the ROM Eq. (\ref{['r_cr']}); (d) comparison between the ROM and the FEA simulation results with different $\alpha$ and initial domain wall size $r_{\text{d0}}$. In FEA, $\gamma=\frac{1}{16}$.