A pseudo-dynamic paradigm for reprogramming domain-wall-lattices in architected solids

Abstract

From Renaissance drapery to tissue morphogenesis, pattern formation exemplifies how geometry and constraints generate complex structures. In soft and architected matter, motifs such as creases, kinks, and domain walls function as order-parameter textures mediating structural transitions. Yet deterministic and reprogrammable control of such patterns remains a central challenge: conventional geometry-based strategies hardwire functionality into structure, leaving deformation modes defect-sensitive and difficult to reconfigure. Here we introduce a pseudo-dynamic mapping that interprets static deformation fields as trajectories of fictitious particles evolving in engineered energy landscapes. This paradigm provides a forward design strategy in which reshaping potential symmetry and bifurcation structure prescribes diverse reprogrammable solitonic domain-wall-lattices in a single, defect-free metamaterial solely under uniform loading. We demonstrate initiation, modulation, inversion, melting, and annihilation of these patterns, governed by a tunable bifurcation landscape. Predictions are validated through simulation and experiment, culminating in a mechanical display that encodes digital information via domain-wall-bits. This approach bridges nonlinear field theory with practical pattern reprogramming, offering a versatile route for programmable design in architected and adaptive materials.

This study introduces a pseudo-dynamic framework that maps static deformations in architected solids to particle trajectories in engineered landscapes, enabling programmable phase transitions and reconfigurable domain-wall lattices under uniform loading.

Introduction

Spatial pattern formation in soft matter systems is both ubiquitous and essential, underlying diverse phenomena across physics, biology, materials science and engineering¹. From tissue morphogenesis to mechanical instabilities and phase separation, the spontaneous emergence of deformation patterns reflects the interplay between geometry, material response, and external constraints^2–4. For solids, understanding and controlling spatially organized deformations—whether they appear as creases, necks, kinks or domain walls—remains a fundamental pursuit with far-reaching implications for morphing structures, stretchable electronics, and soft robotics^5–9. From Galileo’s beam to Euler’s elastica—seminal problems where geometry and energy entwine to dictate mechanical equilibrium—canonical paradigms have long shaped the foundations of continuum mechanics¹⁰. Yet despite this classical heritage, achieving deterministic and reprogrammable control over patterned deformations in homogeneous solids remains an open challenge^11,12.

Metamaterials—architected solids with microgeometries—have emerged as a fertile platform for exploring nonlinear deformation patterns^12–14. These systems offer exceptional tunability through geometric design, but most remain constrained by a “one-structure–one-pattern” paradigm: once the architecture is fabricated, the deformation mode is essentially fixed and difficult to reconfigure^15,16. Conventional approaches often rely on embedded defects^17–19, geometric heterogeneity^20–23, textured actuation^24–28 or intricate multiphysics fields^29–31 to trigger complex responses. Moreover, along with symmetry breaking and bifurcation, phase transitions and domain-wall nucleation are widely believed to be defect-sensitive and irreproducible—posing fundamental limits on deterministic, reversible control^32–34.

Space–time analogies offer a promising perspective for reinterpreting such deformation fields: treating one spatial coordinate as a mathematical time allows static equilibrium problems to be viewed as lower-dimensional spatiotemporal dynamics. Similar reductions have been used to understand Möbius elasticity³⁵, fluctuating polymers³⁶, and static non-Hermitian mechanics^37–39. Rather than using this analogy solely to decipher mechanical behavior, we ask whether it can instead be leveraged to develop an energy-based framework that enables a homogeneous structure to generate rich, reprogrammable localized patterns on demand via landscape engineering.

Here, we propose a pseudo-dynamic paradigm that reinterprets static deformation fields as trajectories of fictitious particles moving through potential landscapes. This mapping transforms the equilibrium equations into pseudo-time dynamical systems, allowing deformation patterns to be designed by reshaping the potential and navigating the pseudo-particle motion. Within this unified paradigm, we demonstrate a forward design of Landau-type phase transitions and domain-wall-lattices in mechanical metamaterials—without the need for defects or spatial heterogeneity. By tuning global strain alone, a single structure can be deterministically reprogrammed to exhibit a rich variety of static solitonic configurations, including initiation, modulation, inversion, melting and annihilation of domain walls.

We validate this pseudo-dynamic paradigm through theory, simulation, and experiments on mechanical metamaterials. As an illustrative application, we also demonstrate a reprogrammable mechanical display that encodes digital information through strain-driven domain-wall patterns—highlighting the broader potential of pseudo-dynamics for computing, encoding, and logic in architected materials. Beyond forward design, the framework also enables inverse inference of energy landscapes and opens conceptual avenues for active matter and multiphysical morphogenesis.

Results

Pseudo-dynamic paradigm

We start by introducing the pseudo-dynamic paradigm as a unifying framework for designing and analyzing spatially inhomogeneous yet static deformation fields in solids. In this formalism, static equilibrium states are recast as trajectories of fictitious particles evolving in effective potential landscapes. This mapping endows a broad class of nonlinear equilibrium problems with a common mathematical structure, enabling their treatment through tools typically reserved for dynamical systems^7,40. The framework encompasses diverse phenomena—from topological elasticity in Möbius strips (Fig. 1a), to solitonic dislocations in layered graphene (Fig. 1b), to nonlinear excitations in biomolecular denaturation (Fig. 1c)—revealing a unifying paradigm that transcends disciplinary boundaries.

Fig. 1. A unified pseudo-dynamic paradigm for engineering static deformation fields across diverse physical systems.

a Schematic illustration of topological elasticity of a Möbius ribbon, whose nontrivial states map to a pseudo-particle in a φ⁴ potential^35,41, $V_{\mathrm{L}}\left(\phi \right)=\frac{1}{4}{\left(1-{\phi }^2\right)}^2$. b Schematic illustration of strain solitons and dislocations in bilayer graphene, described by pseudo-particle trajectories in a sine-Gordon potential^57,58, $V_{\mathrm{L}}\left(\phi \right)=1-\cos \phi$. c Schematic illustration of DNA denaturation bubbles, governed by pseudo-particle dynamics in a Morse potential^42,54,59, $V_{\mathrm{L}}\left(\phi \right)={\left(e^{-\phi }-1\right)}^2$. d Conceptual schematic of a general pseudo-dynamic framework for reprogramming static deformations in solids. External stimuli and constraints ${\mathcal{G}}_i$ reshape the potential landscape V_L, thereby steering pseudo-particle motion and encompassing the cases in (a–c). Dashed lines indicate the stable paths emerging from bifurcation points. The trajectory of a pseudo-particle is equivalent to the deformation field of solids. e Application to mechanical metamaterials: reprogramming static domain walls via biaxial loading (strain ratio ε_x/ε_y) in a system of size N_x × N_y, which reshapes the Landau-type potential. f Schematic illustration of reconfigurable deformation fields φ(x) corresponding to (e), tuned by varying the strain ratio ε_x/ε_y. Here, the order parameter φ represents motif rotations in the metamaterial (e and Supplementary Fig. 2).

From the viewpoint of nonlinear field theory, the energy function of such systems takes a unified form⁴¹:

$$\mathcal{L}={\int }_{\mathcal{M}}\left[\frac{1}{2}{\left(\nabla \phi \right)}^2+V_{\mathrm{L}}\left(\phi \right)\right]dA+{\int }_{\partial \mathcal{M}}{\lambda }_i{\mathcal{G}}_i\left(\phi \right)dl,$$ 1

where φ(x) denotes the field variable (e.g., displacement, rotation, or order parameter), V_L the potential energy, and ${\mathcal{G}}_i$ the constraint with associated Lagrange multipliers λ_i.

The Euler-Lagrange equation yields the nonlinear Klein-Gordon equation⁴²:

$${\nabla }^2\phi \left(x\right)-{\partial }_{\phi }{V}_{\mathrm{L}}\left(\phi \right)=0,$$ 2

subject to

$${\mathcal{G}}_i\left(\phi \right)=0.$$ 3

Multiplying Eq. (2) by ∂_xφ and integrating yields the pseudo-dynamic equation of the static system by interpreting x as pseudo-time:

$$\frac{1}{2}{\left(\nabla \phi \right)}^2-V_{\mathrm{L}}\left(\phi \right)\equiv -{\mathcal{E}}_{\mathcal{T}}.$$ 4

It mathematically describes the motion of a unit-mass pseudo-particle with kinetic energy ${\left(\nabla \phi \right)}^2/2$ and conserved total energy $-{\mathcal{E}}_{\mathcal{T}}$, evolving in a potential landscape $-V_{\mathrm{L}}\left(\phi \right)$, as illustrated in Fig. 1d. Here, pseudo-time x is not physical time, but a mathematical parameter indexing the static equilibrium profile $\phi \left(x\right)$, which obeys the same Euler-Lagrange equation as a pseudo-particle. The dynamical terminology is thus used as a solution-classification tool, not as an assumption of temporal evolution or irreversibility.

Assuming $\phi {\mid }_{x_0}=0$, the trajectory $\phi \left(x\right)$ is given by,

$$x-x_0=\pm \frac{1}{\sqrt{2}}{\int }_0^{\phi }\frac{d\phi }{\sqrt{V_{\mathrm{L}}\left(\phi \right)-{\mathcal{E}}_{\mathcal{T}}}},$$ 5

and is mathematically equivalent to the static field of the original physical system. This mapping offers a forward design strategy: By reshaping $-V_{\mathrm{L}}\left(\phi \right)$—including its symmetry, ground states and bifurcations—and tuning $-{\mathcal{E}}_{\mathcal{T}}$ via external stimuli and/or constraints ${\mathcal{G}}_i$, one can navigate the pseudo-particle motion $\phi \left(x\right)$ (Fig. 1d), and thereby reprogram the target deformation patterns. Naturally, the pseudo-dynamic approach establishes a physically grounded pathway to engineer Landau phase transitions and domain-wall architectures⁴³. Here we focus on systems whose dominant patterns admit 1D order-parameter descriptions; potential higher-dimensional extensions are outlined in the Supplementary Note 6.

Landscape engineering and pseudo-trajectories

We now apply the pseudo-dynamic framework to design phase transitions and deformation patterns in mechanical metamaterials. Without loss of generality, we take a porous soft metamaterial as a representative system, subjected to proportional biaxial loading with a fixed strain ratio ε_x/ε_y for simplicity (Fig. 1e, Supplementary Fig. 2). In this system, the motif rotation field φ(x) serves as the order parameter governed by Eq. (4). The potential follows a quartic Landau-type free energy (see Supplementary Note 4),

$$V_{\mathrm{L}}\left(\phi \right)=f_0\left({\epsilon }_y,F_x\right)+{\sum }_{n=1}^4\frac{f_n\left({\epsilon }_y,F_x\right)}{n}{\phi }^n,$$ 6

subject to a global constraint,

$$\mathcal{G}\equiv N{L}_x{\epsilon }_x-\frac{1}{L_x}\int \left[g\left(\phi ,{\epsilon }_y\right)+\frac{F_x}{2K_x}\right]dx=0.$$ 7

As a minimal model, a quartic $V_{\mathrm{L}}\left(\phi \right)$ is sufficient for present metamaterial system. More generally, the pseudo-dynamic framework is not restricted to quartic potentials: higher-order/dimensional or non-polynomial energy landscapes^41,42 can be incorporated to describe systems with more complex microstructural and deformation mechanisms, following Eq. (4) and the derivations in Supplementary Note 6. The quartic form adopted here provides the simplest analytically tractable realization.

The odd-order terms in Eq. (6) arise naturally from the horizontal constraint, which introduces an effective mechanical bias that breaks the φ ↔ −φ symmetry (see Supplementary Note 4). The strain ratio ε_x/ε_y and the effective lateral force F_x (Supplementary Eq. (7)) together shape V_L(φ), while $\mathcal{G}$ implicitly sets the total energy $-{\mathcal{E}}_T$ (see Supplementary Notes 4, 5). Within the pseudo-dynamic framework, these factors jointly steer the pseudo-particle trajectories and thereby prescribe domain-wall patterns in a deterministic and reprogrammable manner, as illustrated in Figs. 1f, 2a.

Fig. 2. Pseudo-dynamic framework for reprogrammable structural phase transition and domain-wall evolution in metamaterials.

a Schematic illustration of mapping between pseudo-particle trajectories in a tunable potential landscape (red curve) and deformation fields in a metamaterial (green curve). The total energy $-{\mathcal{E}}_{\mathcal{T}}$, initial position φ₂ and pseudo-dynamics of the fictitious particle are controlled by the global constraint $\mathcal{G}$. The curve labeled Δ = 0 marks the phase transition, and the blue point denotes the bifurcation. b Calculated Landau free-energy landscapes under biaxial loading, showing modulation of symmetry, bifurcations, and ground states. By tuning ε_x/ε_y, first-order, second-order, and suppressed phase transitions can be reprogrammed, as illustrated by ε_x/ε_y = 0.6, 0, and  − 2.1 from (i) to (iii). c Schematic illustration of the evolution space of domain walls, spanned by pulse-soliton, kink/antikink, and inverted pulse-soliton-lattices. Their types, wavelengths, and amplitudes are dictated by the asymmetry of  − V_L(φ) and the energy level $-{\mathcal{E}}_{\mathcal{T}}$. This asymmetry, characterized by the peak-height difference s, is essentially determined by the auxiliary coefficients f₁, f₋, f₊, and f_c of V_L (Eqs. (18, 19) in “Methods” and Supplementary Note 5), and is tuned through the effective lateral force F_x. Source data are provided as a Source Data file.

We begin with visualizing the motion signatures of a pseudo-particle in a typical asymmetric landscape −V_L(φ) (Fig. 2a). Before bifurcation (the discriminant Δ > 0, see “Methods”), the landscape is monostable and the pseudo-particle remains stationary on the only stable path φ₀, resulting in uniform deformation. Otherwise, the pseudo-particle motion would diverge (∇φ → ∞). As the loading drives the pseudo-particle smoothly forward to the transition point (Δ = 0, Fig. 2a), −V_L(φ) switches from concave to convex with two stable branches φ_± separated by a valley. This bifurcation allows richer pseudo-particle dynamics such as reciprocation, heteroclinic and homoclinic orbits⁴⁴, and consequently a broad design space of deformation.

The pseudo-particle trajectory φ(x) is inherently governed by the symmetry and bifurcation structure of −V_L(φ). We characterize its asymmetry near bifurcation by the peak-height difference: $s\triangleq \mathrm{sgn}\left(\mid V_{-}\mid -\mid V_{+}\mid \right)$ (Fig. 2a). According to Eq. (7), loading and constraint regulate this asymmetry through F_x, such that F_x >, =, <0 ⇒ s = −1, 0, +1, which correlates with the coefficients in Eq. (6) (see Supplementary Note 5). Accordingly, Figure 2b displays three representative potentials regulated by ε_x/ε_y: Asymmetric potentials yield discontinuous bifurcations and first-order transitions (panel i), symmetric ones yield continuous bifurcations and second-order transitions (panel ii), and single-well (trivial) ones suppress bifurcations and phase transitions entirely (panel iii).

To enable domain-wall formation, the pseudo-particle must undergo bounded motions between two phases. We thus layout −V_L(φ) with two local maxima V_± and a global minimum V₀ after bifurcation (Δ≤0), and set $V_0\le -{\mathcal{E}}_{\mathcal{T}}\le \min \left\{V_{-},V_{+}\right\}$. Let ϕ₁≤ϕ₂≤ϕ₃≤ϕ₄ denote the roots of $-{\mathcal{E}}_{\mathcal{T}}=-V_{\mathrm{L}}\left(\phi \right)$ (Fig. 2a). Assigning different global loadings and boundary conditions, then we can design four classes of deformation patterns according to Eq. (5):

The case s = −1 corresponds to a pulse-soliton-lattice. With ${\mathcal{E}}_{\mathcal{T}}={lim}_{x\to \pm \infty }{V}_{\mathrm{L}}\left({\varphi }_2\right)$,

$$\phi \left(x\right)={\varphi }_4+\frac{{\varphi }_3-{\varphi }_4}{1-{\alpha }^2{\mathrm{sn}}^2\left(\frac{x-x_0}{\zeta },k\right)}.$$ 8

It produces asymmetric pseudo-particle oscillations between ϕ₂ and ϕ₃ with period T = 2ζK(k) and $\mathrm{sgn}\left({\varphi }_2\right)=-\mathrm{sgn}\left({\varphi }_3\right)=\mathrm{sgn}\left({\varphi }_0\right)$, representing periodic transitions between parent and product phases in space—the pulse-soliton-lattice (Fig. 2c).

The case s = 0 corresponds to a kink/antikink-lattice. With ${\mathcal{E}}_{\mathcal{T}}={lim}_{x\to -\infty }{V}_{\mathrm{L}}\left({\varphi }_2\right)={lim}_{x\to +\infty }{V}_{\mathrm{L}}\left({\varphi }_3\right)$,

$$\phi \left(x\right)={\varphi }_0\pm \left({\varphi }_3-{\varphi }_0\right)\mathrm{sn}\left(\frac{x-x_0}{\zeta },k\right).$$ 9

This layouts symmetric pseudo-particle oscillations across ϕ₂ and ϕ₃ with period T = 4ζK(k) and ϕ₂ = −ϕ₃, ϕ₀ = 0, representing the kink/antikink-lattice in configuration space (Fig. 2c).

The case s = +1 corresponds to an inverted pulse-soliton-lattice. With ${\mathcal{E}}_{\mathcal{T}}={lim}_{x\to \pm \infty }{V}_{\mathrm{L}}\left({\varphi }_3\right)$,

$$\phi \left(x\right)={\varphi }_1+\frac{{\varphi }_2-{\varphi }_1}{1-{\beta }^2{\mathrm{sn}}^2\left(\frac{x-x_0}{\zeta },k\right)}$$ 10

It designs an asymmetric pseudo-particle oscillating between ϕ₃ and ϕ₂, which is mirror-symmetric to the situation of Eq. (8): the inverted pulse-soliton-lattice (Fig. 2c).

The case Δ > 0 corresponds to the uniform phase. With ${\mathcal{E}}_{\mathcal{T}}=\min V_{\mathrm{L}}$,

$$\phi \left(x\right)={\phi }_0$$ 11

This describes a pseudo-particle resting at the bottom of a single-well landscape, where no phase transition occurs. The particle traces a single stable path φ₀(ε_y) as ε_y increases, designing uniform patterns (Fig. 2b, iii).

Each reciprocating trajectory of the pseudo-particle prescribes a periodic spatial transition between two phases, forming a static domain-wall-lattice. Here, $\mathrm{sn}\left(x,k\right)$ denotes the Jacobi elliptic function with modulus k, and K(k) is the complete elliptic integral of the first kind. Increasing ${\mathcal{E}}_{\mathcal{T}}$ induces both the growth of k and period T. In the limit ${\mathcal{E}}_{\mathcal{T}}\to \min \left\{V_{-},V_{+}\right\}$, k → 1 and T → ∞, the reciprocation degenerates into heteroclinic or homoclinic motions. We therefore reduce soliton-lattices into isolated static solitons (Eqs. (14, 16) in “Methods”). Although we frequently use dynamical intuition, none of the conclusions in this work rely on dynamical properties that do not hold in spatial fields. Beyond a purely illustrative analogy, the pseudo-dynamic mapping provides a structured route for reprogramming spatial patterns, including solitonic domain walls (Eqs. (8, 10)) and uniform phases (Eq. (11)), through the language of pseudo-particle trajectories in an engineered landscape (Supplementary Movie 1).

Reprogrammable domain-wall-lattices

To demonstrate the regulatory power of the pseudo-dynamic framework, we apply biaxial loading ε_x/ε_y to a family of defect-free metamaterials of varying sizes (Fig. 1d, Supplementary Fig. 9). Within this framework, uniaxial loading—either ε_y or ε_x—yields a fixed potential landscape and a uniquely determined pseudo-particle trajectory. In particular, ε_y alone produces a symmetric landscape (s = 0) that induces a spontaneous phase transition¹⁵, while ε_x alone yields a single-well landscape that precludes any phase transition (Supplementary Figs. 5d, 7). In both cases, the deformation configuration is not reprogrammable: once the loading is set, the resulting pseudo-particle motion and the deformation field are fully prescribed. By contrast, proportional biaxial loading ε_x/ε_y provides a continuous tuning knob that reshapes the potential landscape and modulates the total energy—thereby enabling dynamic control of pseudo-particle trajectories (Fig. 2) and reprogrammable domain-wall morphologies in a single, defect-free system (Fig. 3, Supplementary Fig. 5c).

Fig. 3. Reprogramming domain-wall-lattices in metamaterials via proportional biaxial loading with strain ratio ε_x/ε_y.

a Phase diagram of Landau phase transitions and domain-wall evolution for a metamaterial with N_x × N_y = 10 × 3. The vertical critical strain ${\epsilon }_y^{\mathrm{c}}$ for phase transition (see “Methods”) and 6 representative pattern regimes are predicted by theory, simulations, and experiments. Error bars represent average measurement uncertainty from three independent samples. By varying ε_x/ε_y (see −III – + III), domain-wall processes, including creation, modulation, inversion, melting, and annihilation, are programmably controlled, corresponding to different pseudo-particle trajectories. For sufficiently large lateral strains, e.g., ε_x/ε_y < −1 or ε_x/ε_y > 1.7, the phase transitions are suppressed (${\epsilon }_y^{\mathrm{c}}\to -\infty$), yielding trivial uniform phases (∓III). b Simulated and experimental deformed configurations corresponding to (a). Phase structures in ∓I, ∓II and ∓III reveal a hidden antisymmetry in metamaterials: opposite loadings (ε_x/ε_y < 0 vs. ε_x/ε_y > 0) generate opposite polarization phases (↑ vs. ↓) and inverted domain walls. The point ε_x = 0 marks the inversion of domain walls. All configurations are taken at the same vertical strain level ε_y = − 0.10, with −III – −I extracted from simulations without colormaps for clarity. Source data are provided as a Source Data file.

Compared to the symmetric case ε_x/ε_y = 0, introducing a small lateral compression breaks the potential symmetry and increases the domain-wall density: two pairs of walls emerge once the system reaches the critical vertical strain ${\epsilon }_y^{\mathrm{c}}$ that triggers the structural phase transition (Fig. 3a, Regime +I). Further increasing the loading ratio gradually modifies the amplitude, wavelength, and position of the domain walls. When ε_x/ε_y ∈ (0.7, 1.7), only one pair of domain walls remains (Regime +II), which gradually annihilates as the system approaches the critical loading ratio ε_x/ε_y = 1.7. Beyond this threshold (Regime +III), the system is locked in a uniform negative phase, and the phase transition is fully suppressed (${\epsilon }_y^{\mathrm{c}}\to +\infty$).

Labeling the polarization of motifs by ↑ (φ > 0) and ↓ (φ < 0), we find that lateral tension (ε_x/ε_y < 0) leads to a dual evolution with reversed domain-wall polarity. That is, tensive Regimes −I to −III mirror the sequence of compressive Regimes +I to +III, but with inverted polarization states and domain-wall orientations (Fig. 3). The point ε_x/ε_y = 0 marks the sharp inversion in polarization, underscoring a self-dual structure in the metamaterial response (Fig. 3b). Notably, near the regime boundaries, competing deformation modes may have comparable energies and exhibit richer bifurcation behaviors; away from these boundaries, the dominant configuration becomes increasingly robust.

Guided by the pseudo-dynamic framework, these results show that a single, pristine metamaterial can be deterministically reprogrammed to realize a broad spectrum of domain-wall-lattices—encompassing their initiation, modulation, inversion, melting and annihilation—simply by tuning a global loading ε_x/ε_y (Fig. 3b, Supplementary Movie 2). We validate this reconfigurability through consistent theoretical predictions, numerical simulations, and experimental measurements, all in strong agreement in overall trends (Fig. 3a, Supplementary Fig. 10). Small quantitative deviations between experiment and theory are expected due to material nonlinearity, finite-size effects, and fabrication imperfections. The revealed antisymmetry and hidden duality in rotation-polarized states—opposite loading (±ε_x/ε_y) lead to inverted polarization phases (↓, ↑) and domain-wall patterns—underscore the geometric origin of programmable deformations in mechanical metamaterials.

Validation of the paradigm

To quantitatively validate our theoretical model, we compute polarization fields and domain-wall evolution using the discrete pseudo-dynamic framework, given the geometry and material parameters (see “Methods”, Supplementary Note 3). The predicted domain-wall configurations exhibit strong size dependence: as shown in Fig. 4a–c, the accessible deformation patterns expand as the system length N_x increases. This sensitivity stems from the interplay between the intrinsic reciprocation period of the pseudo-particle and the finite spatial extent of the system.

Fig. 4. Typical polarization fields and validation of reprogrammable domain walls.

a–c Theoretically predicted polarization fields of metamaterials with sizes N_x × N_y = 3, 6, 10 × 3, exhibiting 2, 4, and 6 distinct pattern regimes, respectively. Representative deformed unit cells with uniform polarization are shown in (a). Arrows indicate the polarization direction of each motif. The colormap applied to the curves and arrows quantitatively represents the loading ratio (ε_x/ε_y): orange denotes tensile loading in the x-direction with compression in the y-direction (ε_x > 0, ε_y < 0), while blue denotes compression in both directions (ε_x < 0, ε_y < 0). d–f Experimental, simulated, and theoretical rotation fields for the systems in (a–c) at strain ratios ε_x/ε_y = 2.1, 0.6, and 0.3, respectively. Excellent agreement is observed across all approaches. No stable domain-wall patterns form in metamaterials with $N_x<N_x^{\mathrm{p}}=4$. All results are obtained at the same vertical strain level, ε_y = −0.10. Corresponding experimental and simulated configurations are shown in Supplementary Fig. 9. Source data are provided as a Source Data file.

According to the pseudo-dynamic framework, a pseudo-particle undergoing bounded oscillations in a bifurcated landscape induces a periodic domain-wall-lattice in an infinitely extended system. In finite systems, however, this oscillation must be accommodated within the available spatial domain. If the system is too short, the pseudo-particle cannot complete a full cycle of motion and instead becomes confined to a stable branch (φ_±), leading to uniform polarization. This behavior parallels that of a single-well landscape, where oscillation is entirely suppressed.

The necessary condition for exciting domain walls is that the system length exceeds a critical threshold: $N_x>N_x^{\mathrm{p}}=\lceil \min N_x^{\mathrm{c}}\rceil$, as derived from perturbation analysis (see Supplementary Note 6). For systems with $N_x<N_x^{\mathrm{p}}$, the deformation stabilizes into uniform polarized states, and nontrivial domain-wall structures cannot be sustained (Fig. 4a, N_x = 3). As the system length grows (N_x = 6, 10), multiple domain-wall configurations emerge (Figs. 4b, c), enriching the phase space with two, four, or more distinct polarization patterns in addition to two trivial uniform states. These configurations align closely with those predicted by the phase diagrams (Fig. 3, Supplementary Figs. 5, 8–9), where the ratio ε_x/ε_y governs a spectrum of domain-wall behaviors—initiation, modulation, inversion, fusion, and annihilation—across different regimes.

Beyond the discrete model, we further validate the paradigm through experiments, simulations, and continuum theory under representative loading ratios ε_x/ε_y = 2.1, 0.6, 0.3 at fixed ε_y = −0.10 (Fig. 4d–f, Supplementary Fig. 9). The observed domain-wall profiles, symmetries, wavelengths, and amplitudes show excellent agreement across all methods. The small quantitative deviations between experiment and theory in wavelength and amplitude are expected due to material nonlinearity and finite-size effects. Notably, the continuum theory predicts the configurations in Figs. 4e, f as inverted pulse-soliton-lattices arising from first-order phase transitions. Full-field comparisons between theory and simulation (Supplementary Fig. 10) further confirm this consistency, underscoring the predictive power of the pseudo-dynamic paradigm.

Taken together, these results validate the framework not only as a coherent theoretical construct but also as a robust and versatile methodology for controlling structural phase transitions and domain-wall architectures in metamaterials.

Mechanical display with domain-wall-bits

The unified physical paradigm inspires the design of mechanical displays that encode digital information using reprogrammable domain-wall-bits (DW-bits). Each display consists of M × N pixels, where each pixel is a metamaterial supercell comprising N_x × N_y unit cells. We define the DW-bit as the number of domain walls formed within a single pixel.

To facilitate fabrication, square pixels satisfying 2N_xL_x = N_yL_y are selected. Furthermore, pixel orientations can be programmed during assembly as either vertical (↕) or horizontal (↔), enriching the combinatorial design space (Fig. 5a). This dual encoding—DW-bit count and orientation—allows massive amounts of digital information to be screened through a mechanical display under different global loading ratios ε_x/ε_y.

Fig. 5. Reprogrammable mechanical displays enabled by domain-wall-bits (DW-bits).

a Schematic illustration of a mechanical display consists of M × N metamaterial pixels. A square pixel is a metamaterial super-cell with the size N_y/N_x = 2L_x/L_y, whose orientation (↕, ↔) is defined as the major direction of the large elliptical hole. The display stores, encrypts and screens digital information---domain-wall excitations in each pixel, by the macroscopic loading ε_x/ε_y. b Calculated critical pixel size $N_x^{\mathrm{c}}$ required to support non-uniform deformation as a function of ε_x/ε_y. The threshold $N_x^{\mathrm{p}}=\lceil \min N_x^{\mathrm{c}}\rceil =4$ ensures the reliability of nontrivial display. Pixels smaller than this size remain homogeneous—only trivial information can be screened. c Experimental mechanical barcodes (M × N = 3 × 1) consisting of pixels oriented horizontally (↔) or vertically (↕), respectively (see Eq. (12)). DW-bit values—"0” for uniform deformation and “1” for domain-wall excitation—are encoded under uniaxial compression ε_y, producing outputs $\left[010\right]$ and $\left[101\right]$. d Simulated mechanical QR-codes (M × N = 2 × 2) encoded by pixel orientations and proportional biaxial loadings. DW-bit matrices $\left[01;10\right]$ and $\left[11;11\right]$ correspond to loading ratios ε_x/ε_y = 5/9 and 9/9, respectively (see Eq. (13)). Pixels are assembled within flexible (c) rigid (d) frames, enabling modular design and reprogrammable mechanical displays. Additional encoding results and examples are provided in Supplementary Figs. 11, 12. Source data are provided as a Source Data file.

To guarantee the display reliability for domain-wall imaging, the pixel size should exceed the minimum critical threshold across all loading ratios: $N_x>N_x^{\mathrm{p}}$. For our current metamaterial system, we determine $N_x^{\mathrm{p}}=4$ (Fig. 5b).

We fabricate prototype displays using flexible or rigid frames to assemble the pixels, allowing their orientations to be easily assigned (Fig. 5a). Based on this modular design, two mechanical encoders are demonstrated: a barcode (M × N = 3 × 1) and a QR-code (M × N = 2 × 2), as shown in Fig. 5c, d, respectively.

We denote the pixel orientation array by O. In our barcode examples, two orientation arrays are given by

$${\mathbf{O}}_1=\left[\leftrightarrow \updownarrow \leftrightarrow \right],{\mathbf{O}}_2=\left[\updownarrow \leftrightarrow \updownarrow \right].$$ 12

Each pixel has dimensions N_x × N_y = 6 × 5, which are insufficient to activate stable domain walls under uniaxial loading (ε_x or ε_y) due to boundary effects (Supplementary Fig. 5d). Nevertheless, within the barcode, domain walls can still be excited in the vertically oriented pixels (↕) when subject to sufficient vertical compression ε_y. Assigning “0” to uniform deformation and “1” to domain-wall patterns, the barcodes encode sequences such as $\left[010\right]$ and $\left[101\right]$ (Fig. 5c).

In our QR-code, the orientation array is defined as

$${\mathbf{O}}_{\mathrm{QR}}=\left[\begin{array}{cc}\hfill \leftrightarrow \hfill & \hfill \updownarrow \hfill \\ \hfill \updownarrow \hfill & \hfill \leftrightarrow \hfill \end{array}\right]$$ 13

and subjected to biaxial loading ε_x/ε_y. By varying the loading ratio, different binary patterns are selectively activated. For example, ε_x/ε_y = 5/9 and 9/9 correspond to digital outputs of [01; 10] and [11; 11], respectively (Fig. 5d). Thus, by tailoring both (microscopic) pixel orientation and (macroscopic) global loadings, mechanical displays offer a powerful platform for encoding, storing, and screening vast digital information through reprogrammable deformation states. Additional examples are provided in Supplementary Figs. 11, 12.

From the pseudo-dynamic perspective, each pixel functions as a loaded nonlinear system, where a pseudo-particle follows a bounded trajectory within a reprogrammable landscape. The resulting pattern—whether uniform or domain-wall-rich—encodes binary states, allowing the entire display to act as a physical compiler of digital DW-bits. This conception highlights how the mechanical encoder operates from orchestrated pseudo-dynamic motions across scales.

Discussion

We have introduced a pseudo-dynamic framework as a unifying paradigm for designing and controlling static, spatially inhomogeneous deformation of solids. By reinterpreting the static field as the trajectory of a fictitious particle in a reshaped potential landscape, this approach bridges nonlinear field theory with dynamical systems concepts. It offers a forward design strategy: target deformation patterns emerge from pseudo-particle trajectories, which in turn are programmable through reshaping the underlying energy landscape and tuning global constraints.

Using this framework, we constructed programmable soliton-lattice solutions, revealed their dependence on bifurcation asymmetry, and demonstrated how global loading regulates the pseudo-particle trajectory and thereby the deformation morphology. Importantly, we showed that the same defect-free metamaterial system can be deterministically reprogrammed to exhibit a wide range of phase transitions and spatial configurations—including domain-wall initiation, modulation, inversion, melting and annihilation—simply by tuning the ratio of applied strain—a feat long considered challenging in solid mechanics.

To illustrate its design potential, we further demonstrate a mechanical display composed of pixelated metamaterials. Each pixel functions as a constrained nonlinear system, whose deformation state—uniform or domain-wall-rich—reflects the underlying pseudo-particle trajectory. By assigning orientations and tuning global loadings, digital patterns such as barcodes and QR codes can be encoded and mechanically displayed, highlighting how microscopic deformation mechanisms can be harnessed for macroscopic information representation^45–47.

Beyond static systems, the pseudo-dynamic framework opens avenues for interpreting and programming dynamical phenomena. In active or driven systems⁴⁸, time-dependent or high-frequency forcing can effectively reshape the energy landscape, alter pseudo-particle dynamics, and induce emergent behaviors inaccessible in passive settings⁴⁹. A classic example is the inverted pendulum, where rapid vertical excitation stabilizes an otherwise unstable inverted state by modifying the effective potential⁵⁰. Similar ideas may apply to rate-dependent viscoelasticity⁵¹, multiphysical morphing in thin sheets^52,53, biomolecular denaturation⁵⁴, ferroic phase transitions³³, and more—pointing to a broader utility of this framework in spatiotemporal pattern control¹.

Finally, while this work focused on forward design—reshaping the landscape to guide deformation—the framework also admits inverse strategies. Given a target deformation field φ(x), one can interpret it as a pseudo-particle trajectory and infer the corresponding potential V_L(φ) via Eq. (4). Constraining V_L to a physically admissible class, such as polynomial Landau-type free energies, yields candidate constitutive laws that realize the desired configuration. This pathway connects to ongoing efforts in materials-by-design and physics-informed learning^34,55,56, offering a physically grounded alternative to black-box approaches.

Collectively, these results establish the pseudo-dynamic framework as a versatile and physically transparent tool for analyzing, designing, and reprogramming static field configurations—providing both a theoretical lens and a practical platform for reprogrammable shape-morphing metamaterials^8,12.

Methods

Sample preparation

All metamaterial samples were fabricated using a two-step molding process (Supplementary Fig. 1). First, rigid master molds were 3D-printed using epoxy resin to define the metamaterial geometry and an injection reservoir. After surface polishing and application of a release agent, silicone prepolymer was poured into the master mold and cured at room temperature for 12 h to yield flexible secondary molds. The resulting silicone molds were cleaned and coated with a thin layer of release agent. Subsequently, a commercial polyurethane elastomer (Hei-Cast 800 A/B) was mixed at the desired ratio and cast into the secondary mold. The entire structure was cured at room temperature for another 12 h before demolding. This process yields a soft metamaterial of size N_x × N_y (Fig. 1e), whose geometry is defined in Supplementary Fig. 2. The stiffness of the samples could be tuned by adjusting the A:B mixing ratio of the casting resin.

Experimental setup

All samples were tested using a custom-built in-situ biaxial testing platform, as shown in Supplementary Fig. 2 (IPBF-5000, CARE Testing Systems, Tianjin). During the experiments, samples were subjected to uniaxial or biaxial compressive displacements at a constant strain rate of 10⁻⁵ – 10⁻⁴s⁻¹ to ensure quasi-static conditions. High-resolution digital imaging (FDR-AX40, Sony Corporation) was used to record the deformation process, enabling tracking of discrete marker points across the metamaterial structure. To investigate domain-wall evolution under programmable loading, we applied either uniaxial compression or proportional biaxial displacements by controlling the horizontal and vertical actuators. During biaxial loading, only normal displacements are prescribed on each boundary, while tangential sliding is left free to maintain compatibility and avoid boundary over-constraint. The same boundary conditions are used in the simulations. The lateral-to-vertical displacement ratio U_x/U_y, or equivalently the strain ratio ε_x/ε_y, was prescribed prior to each experiment to realize different loading regimes and programmable phase transitions. Each sample size was tested in six parallel trials to confirm the robustness of the observed behaviors. To minimize friction effects, the inner surfaces of the steel gripping plates were finely sanded to a mirror-like finish. Before each test, a thin layer of Vaseline was applied to the contact surfaces to ensure low sliding resistance.

Numerical simulation

To simulate the static deformation of mechanical metamaterials under various displacement loading conditions, we conducted finite element (FE) analysis using ABAQUS/2017(Dassault Systèmes). Based on the experimentally characterized geometry and material properties, all simulations were performed in the plane strain regime using hybrid 4-node elements with reduced integration (CPE4RH). A mesh sensitivity study confirmed that a resolution of ~1.5 × 10⁴ elements per unit cell ensures numerical convergence. In the FE model, the metamaterial structure was clamped between two parallel rigid planar surfaces, implemented via analytical rigid-body interactions with frictionless contact. Self-contact between internal pore walls was neglected, as all simulations were conducted under moderate compressive loading. Quasi-static displacement boundary conditions were imposed along single or dual axes, and a small artificial damping factor (with an order of 10⁻⁵ – 10⁻⁴) was applied to stabilize the numerical solution. This damping coefficient was calibrated to suppress instabilities without introducing artifacts in the deformation patterns.

Discrete pseudo-dynamic solver

To solve the discrete pseudo-dynamic model, we numerically solve the nonlinear control equations (Supplementary Eqs. (6, 10)), which comprise N + 2 equations for N + 2 unknowns and thus admit a unique solution. Alternatively, the problem can be formulated as a constrained optimization: minimizing the discrete Lagrangian functional (Supplementary Eq. (5)) subject to the constraint equations. Given appropriate boundary and initial conditions, we solve this constrained minimization problem using the InteriorPoint algorithm implemented in Mathematica 14.0 (Wolfram Research, Inc.), which enforces constraints through a barrier-function formulation and iteratively minimizes the objective within the feasible region. This approach robustly yields pseudo-particle trajectories φ_i that satisfy both mechanical equilibrium and global compatibility constraints. Full-field rotation profiles corresponding to Fig. 4d, e are presented in the second row of Supplementary Fig. 10, where the sharp transition in φ identifies the critical strain ${\epsilon }_y^{\mathrm{c}}$ in the vertical direction (Fig. 3a).

Static single-soliton trajectories

In the limit ${\mathcal{E}}_{\mathcal{T}}\to \min \left\{V_{-},V_{+}\right\}$, the elliptic modulus approaches unity (k → 1) and the oscillation period diverges (T → ∞). Under this condition, the reciprocating pseudo-particle motion described in Eqs. (8, 10) degenerates into heteroclinic or homoclinic trajectories.

For the pulse-soliton-lattice in Eq. (8), as k → 1, ϕ₂ → ϕ₁, and T → + ∞, the solution reduces to an isolated pulse-soliton trajectory:

$$\phi \left(x\right)={\varphi }_4+\frac{{\varphi }_3-{\varphi }_4}{1-{\alpha }^2{\tanh }^2\left(\frac{x-x_0}{\zeta }\right)}.$$ 14

For the kink/antikink-lattice in Eq. (9), as k → 1, ϕ₂ → ϕ₁, ϕ₃ → ϕ₄, and T → ∞, the solution degenerates into an isolated kink- or antikink-soliton trajectory:

$$\phi \left(x\right)={\varphi }_0\pm \left({\varphi }_3-{\varphi }_0\right)\tanh \left(\frac{x-x_0}{\zeta }\right).$$ 15

For the inverted pulse-soliton-lattice in Eq. (10), as k → 1, ϕ₃ → ϕ₄, and T → ∞, the solution reduces to an isolated inverted pulse-soliton trajectory:

$$\phi \left(x\right)={\varphi }_1+\frac{{\varphi }_2-{\varphi }_1}{1-{\beta }^2{\tanh }^2\left(\frac{x-x_0}{\zeta }\right)}.$$ 16

For compactness, we denote the discriminant of the Landau free energy V_L(φ) as

$$\mathrm{\Delta }\triangleq {\left(\frac{f_3^3}{27f_4^3}-\frac{f_2{f}_3}{6f_4^2}+\frac{f_1}{2f_4}\right)}^2+{\left(\frac{f_2}{3f_4}-\frac{f_3^2}{9f_4^2}\right)}^3,$$ 17

and the auxiliary coefficients,

$$f_{\pm }\triangleq \frac{f_2{f}_3}{3f_4}-\frac{2f_3^3}{27f_4^2}\pm 2f_4{\left(\frac{f_3^2}{9f_4^2}-\frac{f_2}{3f_4}\right)}^{3/2},$$ 18

$$f_{\mathrm{c}}\triangleq \frac{-2f_3^3+9f_2{f}_3{f}_4}{27f_4^2}.$$ 19

Explicit expressions for parameters such as α, β, k and ζ, as well as the coefficients f_i, are provided in the Supplementary Note 5.

Author contributions

C.Q.C and Y.F.Z. designed research; Y.F.Z., Y.Z., and Z.X.W. performed research; and contributed new reagents/analytic tools; Y.F.Z., Y.Z., Z.X.W., M.C.L., and C.Q.C. analyzed data; Y.F.Z., M.C.L., and C.Q.C. wrote the paper; C.Q.C. supported this research.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Funding

C.Q.C. discloses support for the research of this work from the National Natural Science Foundation of China [grant numbers 12132007 and T2488101]. M.C.L. discloses support for the publication of this work from the start-up funding from the University of Birmingham.

Data availability

All the data supporting this study are available in a public Zenodo repository at 10.5281/zenodo.19428407. Source data are provided with this paper.

Code availability

The codes supporting this study are available in a public Zenodo repository at 10.5281/zenodo.19428407.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-73096-y.