Geometric frustration occurs when a locally preferred structural motif cannot be realized globally [1]. In soft matter systems, from liquid crystals [2][3][4] to colloids [5][6][7] to filamentous assemblies [8][9][10], geometric frustration gives rise to the accumulation of elastic distortions at sizes much larger than the microscopic material elements [11][12][13][14]. Geometrically-frustrated elastic membranes are a well-studied, prototypical class of such systems [15], with broad applications to engineered and natural materials ranging from nano-to macro-scales. Frustration in elastic membranes derives from the elastic incompatibility between the preferred metric (i.e. intrinsic geometry) and the preferred curvatures (i.e. extrinsic geometry) of the membrane [16]. For example, the differential growth [17] or swelling [18,19] of a sheet gives rise to a preferred non-Euclidean metric, which favors non-zero Gaussian curvature at the energetic expense of bending the sheet. These so-called non-Euclidean plates exhibit a complex spectrum of compromised equilibria, such as the undulating shapes of leaves [17] and torn plastic sheets [20]. On the other hand, for so-called hyperbolic Euclidean shells, the stress-free metric is Euclidean but the preferred curvature deforms the intrinsically flat shell to a saddle-like shape. [21].

Geometrically frustrated membranes are quite distinct from the case of externally imposed Gaussian curvature of plates and shells [22][23][24] in that the competition between metric (stretching) or shape (bending) in frustrated elastic membranes is self-generating, giving rise to remarkably non-linear, shape-adaptive, and sizedependent behavior. In hyperbolic Euclidean shells, narrow sheets accommodate the stretching cost of adopting a preferred Gaussian curvature, while wider sheets flatten to expel it to a narrow boundary layer at their edges [25,26]. This size-dependent frustration drives the shape transition between helicoids to spiral ribbons observed in diverse systems such as Bauhinia seed pods [21], inorganic nanosheet ribbons [27,28], and crystalline mem- branes of chiral amphiphiles [25,[29][30][31]. The flattening size ℓ flat [32] at which the transition occurs is set by the ratio of the elastic cost of bending to stretching in membranes, which vanishes with the microscopic thickness t [25,33]. This implies that the size range ℓ flat and ability to accumulate Gaussian curvature is especially limited [34] for self-assembled frustrated membranes, since the accumulation of internal stresses with increasing size determines the range over which assemblies can "sense" and limit their dimensions [25,29,32].

In this Letter, we study the effect of geometric frustration in a mechanical metamaterial [35][36][37] membrane, dubbed a metamembrane, characterized by a bulk floppy (i.e. zero-energy) dilational mode [38] in the absence of frustration. The existence of a floppy mode associated with the internal buckling of the metamembrane, fundamentally alters the nature and size-sensitivity of stress accumulation. We combine continuum theory and numerical simulations of 2D ordered metamembranes formed from subunits whose shape favors locally negative Gaussian curvature, with a corner-binding dilational mode permitting geometry (Fig. 1). In the absence of frustration, such structures are known as conformal metamaterials [39,40], and have been studied for their exotic 2D (in-plane) topological mechanics [41,42] as well as engineered metamaterial responses [43,44]. Here, we show that introducing geometric frustration through a preferred curvature internally stresses the otherwise floppy metamembrane. The local dilational modes screen the elastic costs of frustration [45], allowing the metamembrane to accumulate larger amounts of Gaussian curvature over a larger size-range. Unlike elastic membranes, this screening effect does not vanish with subunit size in the continuum limit, leading to a several fold increase in the thermodynamic self-limiting sizes of frustrated tubules. This opens up fundamentally distinct dimensions of structural control in programmable selfassemblies.

We consider a discrete-subunit model of hyperbolically frustrated metamembranes (Fig. 1) composed of 2D arrays of corner-binding, rigid, quasi-cuboidal subunits. In the absence of frustration, they exhibit a counter-rotation mechanism of alternating cuboids leading to 2D auxetic behavior. To introduce frustration, opposing corner edges are flared by angles +α ∥ and -α ⊥ (Fig. 1a). Ideal edge-to-edge binding along orthogonal rows gives rise to curvatures κ ∥ ≃ α ∥ /a and κ ⊥ ≃ α ⊥ /a, where a is the diagonal subunit width. Edges between neighboring units are held together by four pairs of binding sites-two vertically separated by t and two horizontally separated by w-connected by Hookean springs of stiffness k. As illustrated in Fig. 1a, distortions of these bonds determine the stiffnesses of the three modes of inter-unit mechanics: center-to-center stretching (∝ k); inter-unit bending (∝ kt 2 ); and inter-unit twisting (∝ kw 2 ).

Pairwise interactions determine the collective mechanics of 2D sheets of these units (Fig. 1b) when targeting minimal surface shapes (e.g. κ ∥ = -κ ⊥ ). Notably, the inter-unit twist stiffness dramatically impacts the ability of frustrated metamembranes to accommodate non-zero Gaussian curvature. This is shown for the tubular assemblies in Fig. 1c with a self-closed hoop radius R = κ -1 ⊥ and an axial trumpet-like flare [46]. When the twist mode is stiff (e.g. w ≈ t), the mechanism is effectively quenched and the equilibrium shape is only weakly perturbed from a uniform cylindrical shape near the edges. In contrast, for a floppy (e.g. w = 0) twist mode, equilibrium shapes exhibit a hyperbolic curvature over the bulk of the assembly, as well as larger scale gradients in strain energy.

We can understand the shape and stress accumulation in frustrated metamembranes by way of a continuum Föppl-von Kármán like description of dilational metamaterial sheets [44] combined with the theory of hyperbolic Euclidean shells [25] (see Supplemental Material [47]). This model is described by an elastic energy W composed of elastic strains ε within the 2D array, the curvature tensor κ of its out-of-plane shape, and a scalar field Ω characterizing the amount of local dilation. The energy takes the form

The bulk modulus (λ + µ) couples the dilational strain ε dil = Tr ε to the twist mechanism, while deviatoric strains ε dev = ε -1ε dil are uncoupled from the mechanism and penalized by a shear modulus µ. The third term describes the bending cost with modulus B of a membrane deviating from its target curvatures

The last two terms describe uniform and non-uniform actuation costs of the twist mechanism, where ∇Ω is its in-plane gradient. It is simple to show (see Supplemental Material) that the moduli are related to the subunit model (Fig. 1), with λ ∝ µ ∝ k, B ∝ kt 2 , C mech ∝ k(w/a) 2 , and C ∇ ∝ ka 2 , consistent with planar dilational metamaterials [37,40]. While it is straightforward to analytically solve for the shape equilibria of axisymmetric trumpets (see Supplemental Material), simple energy arguments suffice to explain the differences in how the elastic energy density, ω ≡ W/A, accumulates with length L for stiff (C mech ≈ k) versus floppy (C mech = 0) metamembrane trumpets. For the stiff case (C mech ≈ k), the mechanism is quenched (Ω = 0) at all relevant scales, and the bending energy dominates to maintain the target hyperbolic shape with Gaussian curvature K G ≈ -κ ⊥ κ ∥ . Gauss' theorem prescribes the dilational strain gradients to accommodate the non-Euclidean geometry ∇ 2 ε dil ≃ -K G , which is solved by the quadratic growth of strain with axial distance z from the mid-point of the trumpet, ε dil (z) = ε dil (0) -K G z 2 /2, and a stretching energy density that grows as

, where Y ∝ 4µ(λ + µ)/(λ + 2µ) is the (bare) 2D Young's modulus. This quartic growth with length (Fig. 2a) proceeds until trumpets reach a size where stretching exceeds the finite energy cost ω(L → ∞) ∼ Bκ 2 ∥ /2 to unbend, or flatten the shape along the axial direction. This size is called the flattening length

tR, which is the well-known narrow size of the curved boundary layer in frustrated membranes [33]. For the floppy case (C mech = 0), the twist mechanism is free to screen dilational strains via Ω(z) ≃ ε dil (z), leading to a residual cost for twist gradients ∂ z Ω ≈ -K G z that grows only quadratically with size as ω ∼ C ∇ K 2 G L 2 . Dilational screening by twist (Fig. 2a) occurs for L ≳ ℓ mech = C ∇ /Y ∝ a, beyond which the soft accumulation of elastic energy leads to a larger flattening size ℓ flat (floppy) = (B/C ∇ ) 1/2 κ -1 ⊥ ∝ (t/a)R. Interestingly, the screening of dilational strains in metamembranes leads to a quadratic power-law accumulation of frustration cost, akin to orientational strains in frustrated liquid crystalline membranes [13,49].

We test this accumulation of elastic energy density with length N ∥ , the number of subunits along the axial direction, for stiff versus floppy frustrated trumpets with a fixed geometry κ ∥ = (3/4)κ ⊥ in Fig. 2b (top). The comparison between continuum theory and discrete metamembranes shows a non-linear softening of elastic energy growth for floppy trumpets relative to stiff ones, with both smoothly saturating as L → ∞. The inset shows an L 4 growth of stretching energy for stiff trumpets, while for floppy trumpets the discrete subunit model suggests a power-law slightly weaker than the predicted L 2 . We characterize the range of frustration accumulation in terms of integrated Gaussian curvature (Fig. 2b, bottom), which saturates just beyond a weak local maximum, marked by ℓ flat . Not only is the ℓ flat much larger for floppy trumpets compared to stiff ones, but so too is the total accumulated Gaussian curvature as L → ∞ (Fig. 2c). We highlight that, while ℓ flat (stiff) ∼ t 1/2 implies that the size scale of the K G ̸ = 0 zone vanishes as a → 0 for fixed aspect ratio a/t, the (larger) ℓ flat (floppy) ∼ t/a, is independent of microscopic subunit size in this same limit (Fig. 2d). Notably, for the distinct limit of a ≫ t, arguably relevant to macroscopic "rigid panel" kirigami realizations of metasheets [37,39], this theory suggests a much smaller range for ℓ flat (floppy), implying that unbending largely preempts the anomalous softening and curvature accumulation in such structures. In contrast, molecular or self-assembled membranes [27][28][29][30]34], where the in-plane size and thickness are comparable (a ≈ t), constitute a unique class of frustrated 2D materials possessing the ability to "absorb" Gaussian curvature. This behavior is summarized in the 3D parameter space shown in Fig. 2e, where continuum theory is used to delineate conditions where non-zero Gaussian curvature is extensive in frustrated metatrumpets (i.e. at least 50% of its target value, see Supplemental Material). For standard 2D elastic solids lacking floppy modes (e.g. C mech ≈ Y ), Gaussian curvature is expelled to a boundary-layer of size ℓ flat (stiff) ∼ t 1/2 , akin to the Meissner effect of expelling magnetic fields from the interior of a superconductor. Hence, in the 2D limit of vanishing thickness t/R, the fraction of the elastic solid that absorbs Gaussian curvature vanishes for any macroscopic size. In the floppy limit (C mech → 0), however, the metamembrane lacks one of the two characteristic moduli of a 2D elastic solid, exhibiting extensivity of Gaussian curvature even in the strictly 2D (t → 0) limit.

The much softer accumulation of elastic energy with size in frustrated floppy metamembranes has important consequences on their thermodynamic self-limiting states. In general, equilibrium self-limitation derives from a local minimum in the per-subunit free energy. This behavior has been reported in simulations of frustrated, trumpet-shaped crystalline membranes [46], notably inspired by a recently developed class of shape- programmable DNA origami building blocks [50,51]. However, the narrow range of elastic energy accumulation limits their self-limiting sizes to ≲ 4 -8 subunits in width [34,46]. Here, we consider the (ground state) persubunit free energy of defect-free, axisymmetric metatrumpets, relevant in finite-T simulations [46], which is parameterized by a binding free energy -u 0 per bound corner,

where the elastic energy density ω(N ∥ , N ⊥ ) is minimized over hoop size (N ⊥ ) for each trumpet length (N ∥ ). As shown in the Supplemental Material, stiff trumpets favor hoop sizes close to those of the open structures (i.e. N ⊥ = 2πR/a = 2π/(κ ⊥ a)), while floppy trumpets prefer larger hoop sizes that favor the maximal closure of the mechanism, increasing the selected hoop size to N ⊥ = 2πR sec ψ max /a, where ψ max is the maximal twist angle (π/4 for square-twist), which requires excluded volume interactions to prevent subunit overlap. The per subunit cohesive cost u 0 /N ∥ of the open boundary in the second term drives the assembly to larger lengths, competing with the monotonically increasing per subunit elastic cost of frustration ω with length. Figs. 3a-b show normalized per subunit free energy for stiff and floppy axisymmetric trumpets predicted by Eq. ( 3), for increasing binding strengths. While we are not considering finite-T simulations of the length-selection process illustrated schematically in Fig. 3c, it is well-known that equilibrium finite-size assembly is controlled by the local minimum in the per-subunit assembly free energy [32]. For sufficiently small u 0 , f (N ∥ ) has local minima at N * ∥ , indicating a thermodynamically favored self-limiting length.

Increasing binding strength increases that self-limiting size as well as the value of f (N * ∥ ). For sufficiently large u 0 , the flattening of ω(N ∥ ) at large sizes leads either to f (N * ∥ ) > f (N ∥ → ∞) (i.e. metastable finite length) or the vanishing of a local minimum, resulting in the thermodynamically unlimited growth of flattened trumpet assemblies. The dependence of the equilibrium selflimiting lengths on cohesive binding for stiff and floppy assemblies is shown in Fig. 3d. Notably, the softer and longer range of elastic energy accumulation in floppy assemblies results in a significantly enhanced self-limitation range. For the same target curvatures, the maximal selflimiting lengths for floppy trumpets extend up to 16 subunits in comparison to the modest upper limit of 5 subunits for stiff trumpets.

We compare the maximal self-limiting lengths for stiff and floppy metatrumpet assemblies of cuboidal subunits as a function of target curvature (for fixed ratio κ ∥ /κ ⊥ ) in Fig. 3e. Despite the absence of excluded volume in the continuum model, the upper limit of the self-limiting assembly size follows the same scaling as the flattening length ℓ flat , i.e., the maximal length grows more rapidly as the target curvature tends to zero for floppy trumpets (∼ κ -1 ∥ ) than for stiff trumpets (∼ κ -1/2 ∥

). At the lowest curvature considered (κ ∥ a ≈ 0.094), the maximalsize structures are composed of 1890 subunits for floppy trumpets, compared to only 432 subunits in the stiff ones.

While excluded volume effects do not apparently alter the scaling of elastic energy accumulation, we do find that micro-structural effects of saturating the dilational mechanism alter the quantitative enhancement of self-limiting size because the ratio κ ∥ /κ ⊥ controls the twist gradients that screen dilational strains (see Supplemental Mate-rial). For sufficiently large κ ∥ /κ ⊥ , dilational screening exceeds the maximal twist ψ max = π/4 for square-twist assemblies, leading to self-contacting zones. With this in mind, we consider a Kagome structure of rigid triangular units, which also exhibits a bulk floppy mode [52], but with an enhanced dilational range (4:1) relative to that of the square-twist (2:1). Hence, a simple redesign to flared triangular prismoidal sub-unit shapes (see Supplemental Material) exhibits the same frustrated metamembrane behavior (Fig. 3f) with enhanced self-limiting lengths for the same κ ∥ /κ ⊥ , up to a ∼ 6-fold increase in the maximal length relative to the stiff case, with the largest structures considered possessing 5040 subunits.

In summary, the introduction of bulk "floppy" metamaterial modes into positionally ordered 2D membranes qualitatively changes the accumulation of internal stresses, and significantly extends the size range over which frustrated membranes accumulate Gaussian curvature. Hence, while frustrated "trumpets" [46] with standard solid elasticity exhibit hyperbolic curvature over a fraction ∼ √ tR 0 /L of their length, which vanishes in the strictly 2D (t → 0) limit, frustrated metatrumpets exhibit curvature over a far more extensive, thicknessindependent fraction ∼ R 0 /L. Although our specific model considers a dilational (i.e. auxetic) behavior, the elastic response of metasheets with a floppy shear mechanism [44] to Gaussian curvature implies that a more general class of metamembranes will similarly exhibit an anomalous absorption of Gaussian curvature, provided at least one of either their effective in-plane dilation or shear moduli vanish. The extensivity of curvature absorption has significant implications for molecular, self-assembled membranes, which are especially thin, and where elastic costs of in-plane and out-of-plane orientational gradients can be expected to be comparable (i.e. C ∇ ∼ B). Hence these predictions apply to self-assembled metastructures of geometrically engineered nanoscale building blocks, such as auxetic square-twist assemblies of de novo proteins [53]. These results also demonstrate the significant potential to engineer assemblies that self-limit at large dimensions through not only the design of the "misfit" geometry of the subunits [54,55], but also the introduction of floppy intra-assembly modes via their binding geometries. Notably, the triangular DNA origami subunit designs used in recent works demonstrating the curvatureprogrammable assembly of spheres [50] and cylinders [51] are ideally suited for realizing Kagome metamembrane assemblies of the type shown in (Fig. 3) implemented via vertex-to-vertex binding.