Localized and extended deformations of elastic shells

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Entry by Haifei Zhang, AP 225, Fall 2009

Soft matter keywords

Deformation, Elastic


The dried raisin, the crushed soda can, and the collapsed bicycle inner tube exemplify the nonlinear mechanical response of naturally curved elastic surfaces with different intrinsic curvatures to a variety of different external loads. To understand the formation and evolution of these features in a minimal setting, the authors consider a simple assay: the response of curved surfaces to point indentation. The authors find that for surfaces with zero or positive Gauss curvature, a common feature of the response is the appearance of faceted structures that are organized in intricate localized patterns, with hysteretic transitions between multiple metastable states. In contrast, for surfaces with negative Gauss curvature the surface deforms nonlocally along characteristic lines that extend through the entire system. These different responses may be understood quantitatively by using numerical simulations and classified qualitatively by using simple geometric ideas. The authors' ideas have implications for the behavior of small-scale structures.

What's the problem

Thin naturally curved shells arise on a range of length scales: from nanometer-sized viruses to carbon nanotubes, from the micrometer-sized cell wall to bubbles with colloidal armor, and from architectural domes to the megameterscale earth’s crust. In each of these examples, the underlying curved geometry of the object leads to enhanced mechanical stability relative to that of naturally flat sheets. In particular, whereas a naturally flat sheet can almost always be bent weakly without stretching, almost any deformation of a curved shell causes its mid-surface to bend and stretch simultaneously. This fact is a simple consequence of a far-reaching concept from differential geometry, Gauss’s Theorema Egregium and its application to determine the conditions for the isometric deformation of a surface. Indeed, our everyday experience playing with thin flat and curved sheets of similar materials such as sheets of plastic suggests that the natural geometry of the surface dominates its mechanical response: a surface with positive Gauss curvature (e.g., an empty plastic bottle) has a qualitatively different response from that of a surface that is either flat (e.g., a plastic sheet) or has negative Gauss curvature. To understand this we must combine the geometry of idealized surfaces and the effects of a small but finite thickness on the mechanical response of these slender structures.

Indentation of a spherical shell with positive Gauss curvature

To simplify the problem, the authors used a simple indentation assay, the method of choice to probe the properties of solid interfaces at various length scales and connect geometry to mechanics. The nonlinear character of the governing equations that arises from the effects of large deformations precludes the use of purely analytical techniques to solve them. It is nevertheless possible to get a qualitative view of the mechanical response of a doubly curved thin shell (thickness t, radii of curvature R1, R2; R= min[R1, R2]; <math>\epsilon</math>=t/R<<1) subjected to a point indentation load by a consideration of the linearized equations of equilibrium. For such a shallow shell, where the authors may use the Cartesian coordinates (x, y) to describe any material point on the shell rather than any more elaborate intrinsic coordinate system, these are given by


A complete formulation of the problem additionally requires the specification of boundary conditions on the displacements and stresses associated with support or lack thereof along a boundary curve. The above equations are globally elliptic, i.e., following the usual classification of linear partial differential equations, they have imaginary characteristics and thus require the prescription of conditions along all boundaries. However they can exhibit behaviors associated with hyperbolic or parabolic systems because of the fact that, when scaled appropriately, the termwith highest derivative in the first equation is in general very small.

For a spherical cap shell that is clamped along its edge, the underlying positive Gauss curvature implies that the deformation is localized in the neighborhood of the indentation but decays isotropically away from it, and earlier results show that once the sphere is weakly deformed near the localized load, it flattens before eventually becoming partly inverted. Even larger indentations lead to a faceting behavior of the spherical shell as seen when a plastic bottle is poked with a pen (Fig. 1A). For a cylinder that has zero Gauss curvature, the behavior under indentation loads is more complex and subtle owing to the anisotropy in initial curvature along and perpendicular to the principal axis. Previous work has shown that both the location of the indentation and the nature of the boundary conditions are crucial in determining the cylindrical shell’s response. For a long free cylinder that is pinched at an edge, the deformations persist over many diameters owing to the dominant role of nearly inextensible deformations. In contrast, for a cylinder that is clamped along its later edges, the deformation is strongly localized near the point of indentation, but decays anisotropically away from it, slowly along the axis but much more rapidly in the direction perpendicular to it, eventually leading to the formation of localized structures that themselves bifurcate. Finally, for a shell with negative Gauss curvature such as the inner part of a toroidal shell, the leading order solution is wave-like with characteristics making an angle arctan(sqrt(R1/R2)) with the principal axes of the shell, so that the deformation is nonlocal and extends all of the way to the shell boundary or up to the intersection of the nodal lines of zero Gauss curvature with these characteristics. Indeed, for this last case, the indentation problem is the spatial analogue of the Cauchy initial-value problem for wave propagation, although there are important differences caused by the presence of boundary layers near the point of indentation and along the nodal lines.

Fig. 1. Indentation of a spherical cap.

The above approximate analysis is valid only for small deformation because of the limitations posed by the asymptotic analysis of the linearized equations and cannot be easily extended to explain the rich behavior afforded by poking a plastic bottle with the point of a pen, as shown in Fig. 1A. This simple experiment shows that as the indentation displacement is increased, the bottle first deforms to form a circular dimple, which then loses symmetry to a polygonal shape with three vertices attached by ridges to each other as well as to the indentation point. Further indentation leads to the formation of additional vertices and ridges. To understand the formation of these faceted structures, the authors use detailed numerical simulations based on the finite element method. The authors restrict their material choice to that of an isotropic linear elastic material for two reasons: simplicity and generality. The computations were carried out by using ABAQUS (Dassault Syste`mes), a commercial finite element package, with the following material parameter values: Young’s modulus E=1 GPa and Poisson ratio v=0.3. Four-node shell elements with reduced integration were used in all calculations. A single element spanned the thickness and no initial geometric or material imperfection was included in the computational model. To follow the postbuckling response of the structure, the authors used a stabilizing mechanism based on automatic addition of volume-proportional damping, which was decreased systematically to ensure that the response is insensitive to this change.

The first numerical experiments explored the point indentation of a segment of a spherical shell with thickness t and natural curvature R (here the authors considered the range 0.0005<t/R<0.01) that is clamped at its boundary, shown in Fig. 1B. This simulation qualitatively mimics the simple experiment of indenting a plastic bottle shown in Fig. 1A. Because the spherical cap has positive Gauss curvature it responds initially by deforming axisymmetrically with an approximately linear force–indentation response (Fig. 1E), but once the deformation is of the order of the thickness of the shell, the response becomes nonlinear. Further indentation leads to the appearance of an axisymmetric dimple with a strongly localized region of deformation along a circular ridge, about which the cap is approximately mirror-symmetric relative to its original shape, so that this mode of deformation is sometimes termed mirror-buckling. When the indentation is increased even farther, this dimpled axisymmetric mode loses stability to an asymmetric mode, which starts out with threefold symmetry and then through a series of transitions moves through polyhedral shapes with a varying number of vertices, as shown in Fig. 1C[see also supporting information (SI) Movie S1]. Each of these transitions is marked by the bifurcation of a single vertex defect into two, which then move apart just as when a cylindrical shell is indented along an edge.

Indentation of a cylindrical shell with zero Gauss curvature

To contrast the behavior of spherical shells, which have positive Gauss curvature, with that of other curved shells, we now turn to the indentation of a segment of a cylindrical shell, which has zero Gauss curvature and is clamped along its lateral edges. Indenting the shell (length L, thickness t, radius R) shown in Fig. 2A at its center leads to the formation of two vertices; further indentation eventually leads to a final configuration that is a simple deformed developable surface. However, the intermediate configurations leading to the final developable state depend on the geometrical parameters L/R and t/R.We find two generic scenarios: (I) the shell never breaks symmetry in the span-wise or longitudinal directions of the cylinder; and (II) the shell breaks symmetry in both the span-wise and the longitudinal directions. A phase diagram characterizing the parameter regime for these scenarios is shown in Fig. 2C; shells with large t/R and L/R always follow scenario I, whereas those with small t/R and intermediate L/R follow scenario II. For shells with very small L/R the behavior is like that of a planar elastica, which also deforms symmetrically. In Fig. 2D, we show the detailed indentation response in each of these scenarios: in scenario I we see a single jump in the force when the cylinder snaps through to the final configuration, whereas in scenario II, we see multiple jumps corresponding to the different metastable states (which are dependent on t/R) that lie between the initial and final configurations. As one might expect, these transitions are also strongly hysteretic (Fig. 2C).

Fig. 2. Indentation of a cylindrical shell.

Indentation of a toroidal shell with both positive and negative Gauss curvature

The authors finally turn to the case of the indentation of a segment of a toroidal shell clamped along its edges, as shown in Fig. 3A. This scenario is geometrically (and thus physically) interesting because the outer (inner) halves of toroidal shells have positive (negative) Gauss curvature, with two nodal lines of vanishing Gauss curvature that separate these regions. In Fig. 3B, we show the response of the toroidal shell as it is indented at a point along the line with positive Gauss curvature. The response qualitatively has the same features as the response of the spherical shell studied in Fig. 1. In Fig. 3C, we show the response of the shell when indented at a point on its nodal line of vanishing Gauss curvature. For small deformations, localized vertices form on either side of the nodal line. Increasing the indentation causes the deformation in the outer half of the shell, which has positive Gauss curvature, to remain localized, whereas in the inner part of the toroidal shell the deformation extends along a narrow zone all of the way to the two nodal lines of zero Gauss curvature along characteristics as suggested by the linearized analysis.

Fig. 3. Indentation of a toroidal shell.

Soft matter details

The response of a curved thin shell depends on its Gaussian curvature

A plastic bottle, a can, and a stiff torus pose rich problems in geometry and mechanics to anyone who pokes them with a pencil. The applied pressure forms indentations in these rounded objects, producing a variety of faceted structures, either localized or extensive. The study shows that the mechanical behavior of naturally curved thin shell structures, which are soft by virtue of their geometry, is very rich. The indentation response of these objects generally leads to multifaceted multistable polyhedral structures, but the extent of these deformations depends on the underlying geometry of the surface. The authors show that the response of the system depends on the Gaussian curvature of the original surface. Although a simplified linear model predicts the qualitative response in individual cases, a detailed model requires a numerical solution of the governing equations. For example, a sphere forms an inverted pyramidal structure of three-, then four-, then fivefold rotational symmetry, with strongly localized deformations at the vertices of the pyramid. New facets appear in the indentation as the surface alters to minimize stretching. A simple geometric argument gives approximate expressions for the size of the pyramid and the force required to form it. A torus responds very differently depending on whether force is applied from the inside (where the Gaussian curvature is negative) or outside (where the Gaussian curvature is positive). Different multifaceted states can coexist under certain conditions according to the authors.


The analysis has been restricted to purely elastic shells, i.e., systems where there are no irreversible effects. This is not as restrictive as might seem, because on small length scales, in polymersomes, nanotubes, virus shells, graphene sheets, and other thin shell structures inelastic effects are often relatively unimportant. Therefore, these symmetrybreaking elastic bifurcations that lead to polyhedral localized structures should be easily realizable in them.

Potential impact to soft matter studies

The studies provide a stimulus to the design of mesoscopic structural materials with geometry-dominated responses that can serve as mechanical memories (using the geometrically determined multistability of the shape that the shells can take), exhibit long-range force transmission (using toroidal shells), and form the basis for surfaces with controllable frictional, wetting, and adhesion properties.


[1] Localized and extended deformations of elastic shells A. Vaziri and L. Mahadevan, Proceedings of the National Academy of Sciences (USA), 105, 7913, 2008.