Shaping of Elastic Sheets by Prescription of Non-Euclidean Metrics

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Science  23 Feb 2007:
Vol. 315, Issue 5815, pp. 1116-1120
DOI: 10.1126/science.1135994


The connection between a surface's metric and its Gaussian curvature (Gauss theorem) provides the base for a shaping principle of locally growing or shrinking elastic sheets. We constructed thin gel sheets that undergo laterally nonuniform shrinkage. This differential shrinkage prescribes non-Euclidean metrics on the sheets. To minimize their elastic energy, the free sheets form three-dimensional structures that follow the imposed metric. We show how both large-scale buckling and multiscale wrinkling structures appeared, depending on the nature of possible embeddings of the prescribed metrics. We further suggest guidelines for how to generate each type of feature.

Thin sheets are common in natural and man-made structures, are shaped to a huge variety of diverse three-dimensional (3D) structures, and span many length scales (1). Natural slender structures, such as flowers, lichens, and marine invertebrates, attain elaborate configurations during their unconfined (free) growth. One wonders what mechanisms lead to shaping of free sheets and whether they can be implemented with artificial materials. Thin sheets can form nontrivial 3D structures in many different ways. Confinement of flat sheets can lead to buckling (2), wrinkling (3), and crumpling (4). The construction of layered material can result in both bending (5, 6) and wrinkling (7, 8). Recent studies of wavy patterns along edges of torn plastic sheets (911) have shown that 3D wavy patterns can result from inplane deformations. Mathematically, a structure made of a thin sheet can be viewed as a two-dimensional (2D) surface in a 3D Euclidean space. Intrinsically, a surface is characterized by its metric, a tensor that specifies the local distances between points across the surface (12). The shape of a surface, its configuration in space, is a realization, an embedding, of the metric in space. In many cases, there would be many possible embeddings of a given 2D surface (metric) in space; that is, the metric alone does not determine a configuration. To select a specific shape (a specific embedding), one needs to determine, in addition to the metric, the local curvatures on the surface (13).

We present a shape selection principle based on two main ideas: The first is Gauss theorem (Theorema Egregium), which states that the metric tensor of a surface locally determines its Gaussian curvature K(x,y). The second principle, known from the study of crumpling (1416), shell collapse (17), and wrinkling (18), states that equilibria of thin elastic sheets involve only small amount of inplane strain. Combined, these two principles lead to a novel shaping mechanism: Rather than aiming at a specific embedding, one prescribes on the sheet only a 2D metric, the “target metric” gtar, one that results in a nonzero Gaussian curvature (a non-Euclidean metric). A sheet adopting a configuration (embedding) satisfying gtar would have been completely free of inplane strain, that is, stretching energy. The free sheet will settle to a 3D configuration that minimizes its elastic energy. In this mechanism, the selected configuration is set by the competition between bending and stretching energies, and its metric will be close to (but different from) gtar. We show that the construction of elastic sheets with various target metrics is possible and results in spontaneous formation of 3D structures. These structures exist in both large-scale buckling and small-scale wrinkling forms. We further suggest guidelines for how to generate each type of feature. Being free in space and not locked onto a specific embedding, these sheets undergo morphological transitions, driven by global constraints on possible embeddings of their target metrics.

We used N-isopropylacrylamide (NIPA) gels to construct sheets with inducible non-Euclidean gtar. The gels are produced by mixing NIPA monomers with bisacrylamide (BIS) (6% by weight of NIPA) cross-linker in water. The addition of catalysts initiates polymerization of a cross-linked elastic hydrogel [Supporting Online Material (SOM) text]. This gel undergoes a sharp, reversible, volume reduction transition at Tc = 33°C (19), above which its equilibrium volume decreases considerably. Calibration experiments (fig. S1) using various homogeneous (each of a different fixed NIPA concentration) gel discs provide the relation between the monomer concentration and η, the shrinkage ratio of the “activated” gel. These measurements show that dilute gels shrink a lot, whereas gels with high monomer concentrations undergo moderate shrinking.

To impose nontrivial target metrics, we constructed sheets with internal lateral gradients in NIPA concentration, i.e., η = η(r). We used programmable actuated valves to inject solutions with gradients in monomer concentration into a mold (Fig. 1). Polymerization takes place within a minute, and the imposed gradients are thus frozen within the gel. The constructed sheets are flat below Tc but are programmed to shrink differentially, with ratio η(r), upon activation at T > Tc. Indeed, the sheets adopted a non-Euclidean metric and underwent large reversible shape transformations (Movie S1). To cast radially symmetric discs, we used a Hele-Shaw cell configuration. The solution is injected into the gap between two flat glass plates through a central hole in one of them (Fig. 1). Gel tubes were cast by injecting the solution into the gap between two concentric glass tubes.

Fig. 1.

The experimental system. High (∼30%) and low (∼10%) monomer concentration solutions are mixed in a programmable mixer and injected into a Hele-Shaw cell (left). Polymerization leads to the generation of a flat disc having internal lateral gradients in monomer concentration (center). Once this programmed disc is activated in a hot bath of temperature T > Tc = 33°C, it shrinks differentially, adopting a new, non-Euclidean target metric (right). As a result, it attains a 3D configuration. Illustrated is a surface of positive Gaussian curvature, generated by increasing monomer concentration during the injection.

The differential shrinking changes distances between points on the surface; that is, it defines a new target metric on the disc. Because the system is radially symmetric, we consider a closed circle of radius r on the cold disc. After the shrinking, both perimeter and radius of the circle are modified. The perimeter is now 2πrη, and the radius is ρ(r)= ∫0rη(r′)dr′. Thus, the perimeter of a circle of radius ρ on the shrunk disc is now f(ρ)2πρ, where f(ρ) is determined by η(ρ). With use of a radial coordinate system (ρ, θ), the linear element determined by gtar is dl2 = dρ2 + ρ2f(ρ)2dθ2, and the prescribed target Gaussian curvature reads Math(1) The ρ-dependent monomer concentration is thus a knob with which we can set [ρf(ρ)]ρρ and determine a target Gaussian curvature. When [ρf(ρ)]ρρ does not equal 0, Ktar also does not equal 0, implying that any embedding of gtar cannot be flat. This is demonstrated in Fig. 2A, where increasing and decreasing monomer concentrations result in Ktar < 0 and Ktar > 0, respectively. The resultant configurations of the sheets are nonflat, corresponding to gtar (Fig. 2A insets).

Fig. 2.

Shaping of non-Euclidean elastic discs. (A) A radially decreasing monomer concentration (red line) prescribes a positive Gaussian curvature on the disc, which adopts a dome shape (lower image). When the monomer concentration profile is inverted (blue line), it prescribes a negative Gaussian curvature, leading to an azimuthally oscillating shape (upper image). Discs' initial thickness t0 = 0.5 mm. (B) Disc perimeter as a function of ρ (solid lines) compared to the perimeter prescribed by gtar (dashed lines). Both positive (red) and negative (blue) Gaussian curvature discs follow, on average, gtar. The dashed black line indicates a flat disc for which the perimeter equals to 2πρ. (Images) Measured disc topography z(x,y) with a semigeodesic coordinate system. Lines of constant radius ρ = 15 mm on the curved discs are highlighted in bold.

The sheets are not ideal 2D surfaces, and their equilibrium configurations are determined by balancing stretching and bending energies: The stretching energy, which vanishes only in embeddings that fully follow gtar, scales linearly with the sheet thickness, t (20). The bending energy, which is 0 only in flat configurations (because the sheets are uniform across their thickness), leads to deviations from gtar and scales as t3. Thus, as t → 0 the stretching term dominates, and, therefore, a sheet will be willing to bend a lot in order to reduce its inplane strain. Therefore, equilibrium configurations will involve only small amounts of inplane strain, and the metric of the selected configuration g is expected to approach gtar. To check this conjecture, we compared the metrics of curved discs to their target metrics. The topography of the discs z(x,y) was measured by using an optical profilometer (Conoscan 3000, Optimet, Jerusalem, Israel) (SOM text). Radial geodesics (the equivalent of radial lines on a curved disc) were plotted for azimuthal angles θ (Fig. 2B, insets), enabling the identification of circles of radius ρ on the curved surface (in general, the projections of these curves are not circles in the x,y plane). We measured the perimeter of such circles and compared it to f(ρ)2πρ, the perimeterset by gtar. This comparison is shown for two types of discs of positive and of negative target Gaussian curvature (Fig. 2B). In both cases, the perimeter at ρ closely follows the prescribed one, and indeed the sheets' metric (averaged over θ) is very close to their target metric.

When averaged over θ, the two types of discs follow gtar; however, they present two qualitatively different physical behaviors. The surfaces of Ktar > 0 preserve the radial symmetry of gtar, generating surfaces of revolution (Fig. 2, lower insets). The surfaces of Ktar < 0 break this symmetry, forming wavy structures (Fig. 2 upper insets). To understand the nature of this qualitative difference, we compared the magnitude and the distribution of bending and stretching energy densities across the sheets. The stretching energy density results from inplane strain, that is, differences between g and gtar. Thus (according to Gauss' theorem), local differences between K(ρ,θ) and Ktar(ρ) indicate a nonzero stretching energy density. The bending energy density is Eb(ρ,θ) = DB(ρ,θ), where D is the bending stiffness of the sheet (SOM text) and B(ρ,θ)= 4H2(ρ,θ)–K(ρ,θ), with H(ρ,θ) the local mean curvature. Thus, the bending energy density can be studied by analyzing B(ρ,θ).

The distributions of H2(ρ,θ) and K(ρ,θ) are presented in Fig. 3. For discs of Ktar > 0 (Fig. 3, A and B, left), both K and H2 are distributed in a radially symmetric manner and are of the same magnitude. The symmetric distribution of K indicates that gtar is obeyed locally and not just on average (Fig. 2). This indicates that the surfaces' configuration is very close to an embedding of gtar, and thus its stretching energy is close to 0. H2(ρ,θ) ≈ K(ρ,θ) implies that B is close to its minimal locally possible value, B = 3K (SOM text). Thus, the bending energy density is minimal as well. The selected configuration locally minimizes both bending and stretching energy densities, thus forming a very low energy solution. In contrast, for surfaces with Ktar <0 (Fig. 3, A and B, right), H2 attains large values (larger than |Ktar|), and the condition for minimal bending energy density [H(ρ,θ)=0] is far from being fulfilled, resulting in high bending energy. Indeed, the average of B(ρ,θ) over this surface is twice as large as that of the surface of Ktar > 0 (whereas its minimally possible value, B = K, is three times smaller than that of the surface of Ktar >0).

Fig. 3.

Distribution of Gaussian and mean curvatures on the discs. (A) The squared local mean curvature on discs of Ktar > 0 (left) and Ktar < 0 (right). The first is radially symmetric (except for few defects), whereas the other oscillates. (B) The measured Gaussian curvature on the positive curvature disc (left) is radially symmetric and of magnitude similar to that of H2. In the case of Ktar < 0, the Gaussian curvature oscillates, breaking the radial symmetry. (C) The local Gaussian curvature along a line of ρ = 15 mm is plotted versus the angle for the discs with Ktar > 0 (red) and Ktar < 0 (blue). The dashed lines indicate Ktar at ρ = 15 mm. For the disc of Ktar > 0, the generated Gaussian curvature is positive and closely distributed around its mean. In contrast, the Gaussian curvature of the disc of Ktar < 0 oscillates with amplitudes larger than its mean, in correlation with surface undulations.

A more surprising observation is the asymmetric distribution of the Gaussian curvature. Instead of the negative, rotationally symmetric Ktar (Eq. 1), K(ρ,θ) varies periodically in θ, attaining positive and negative values (Fig. 3B, right). The comparison with Ktar at a fixed ρ (Fig. 3C) shows that K oscillates, with amplitudes larger than its mean (Ktar). These fluctuations imply a periodic deviation from gtar, i.e., significant modulations in stretching energy density. Compared with the case of Ktar >0, the selected configuration is not successful in reducing both bending and stretching energies.

Intrinsically, the discs in Fig. 2 differ only by the sign of Ktar. What is the mechanism that causes the disc with Ktar < 0 to break the symmetry, bend a lot, and localize inplane strain? Because the discs are free in space and energy is minimized globally over the entire disc, limitations on possible global embeddings of gtar play a central role in setting the shape of the disc. For metrics with Ktar(ρ) > 0, radially symmetric global embeddings with small bending do exist (21). Such theoretical configurations are good minimizers of the sheet's energy; they fully follow gtar (are free of stretching) and would possess low bending energy. The physical sheets select such embeddings as a basis for their equilibrium configurations. The finite thickness of the sheets will lead to configuration that are close to the mathematical (2D) ones, with both bending and stretching energies small, as we have shown. In contrast, embeddings of radially symmetric metrics, with Ktar < 0 (hyperbolic metrics) are nontrivial, do not preserve the radial symmetry of the metric, and must include small-scale structure (22, 23). The larger the sheet is, the smaller this scale gets. Such embeddings of a physical sheet would have large (bending) energy and thus are not candidates for sheets' equilibrium shapes. Indeed, the substantial localized stretching energy, together with the large bending energy (Fig. 3, A and B, right), indicates that the sheets do not select an embedding of gtar as a basis for their equilibrium configuration but follow a wrinkling-type behavior. In wrinkling, stressed small-scale (18) and multiscale (24) wavy structures are formed because of the inability to facilitate stretch-free configurations with low bending energy. Our experiments show that such conditions can occur with free sheets, depending on their target metric.

The prescription of smooth symmetric metrics can thus lead to the formation of both symmetric large-scale and oscillating small-scale structures. This tool can be used as a basis for a shaping principle. Different types of shapes are constructed (Fig. 4) by combining regions of different curvatures and controlling sheet thickness and sheet topology. In contrast to the disc topology, in cylindrical topology, symmetric, low bending embeddings of Ktar < 0 do exist (22, 25) in a trumpet form. A physical sheet will thus be able to select such an embedding as a basis for its equilibrium configuration, resulting (Fig. 4 E) in a configuration that is symmetric and feature-free. However, such a symmetric surface can accommodate only up to –2π negative Gaussian curvature (22, 25). Beyond this limit, the symmetric solution no longer exists. This is seen in Fig. 4, F to H, where cylindrical sheets with Ktar < 0 adopt wavy configurations, as with the radial discs.

Fig. 4.

Different structures of sheets with radially symmetric target metrics. (A) A thick sheet (t = 0.75 mm) with relatively flat hyperbolic metric adopts a configuration with only three waves. Thinner (t = 0.3 mm) sheets with larger gradients in monomer concentration form two generations of waves (B). Symmetric surfaces of positive curvature, such as in (C), can be combined with negative curvature margins to obtain a wavy sombrero-like structure (D). Axially symmetric metrics can be applied to cylindrical sheets. (E) A tube with K < 0 preserves the radial symmetry because the amount of Gaussian curvature integrated over the tube is less than –2π. When too much negative curvature is accumulated over a tube, it develops a wavy edge. Tubes with two (F), four (G), and six (H) waves are obtained, depending on the sheet thickness and the metric profile.

We suggest that large-scale buckling of unconstrained elastic sheets occurs when a non-Euclidean target metric gtar can be symmetrically embedded, with low associated bending energy. When no such embedding exists, energy minimization of the sheet is achieved via a wrinkling-type behavior. This shaping principle might play a role during developmental processes in naturally growing tissues, where the local nature of the growth provides a mechanism for the formation of non-Euclidean metrics. In our experimental system, gtar can be turned “on” and “off” by environmental conditions, having an applicative potential. This approach can be implemented by using other artificial materials that undergo large volume reduction. Such new materials are being developed to respond to different external stimuli, such as light (26), pH (27), glucose level (28), and other chemical signals (29). Further study of the principles of shaping by metric prescription can extend the types and variety of structures that can be formed by using thin sheets, as well as improve our understanding of developmental processes.

Supporting Online Material

Materials and Methods

Fig. S1

Movie S1

References and Notes

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