Research

Geometry, mechanics, and dynamics of thin elastic sheets

Joint work with Shankar Venkataramani and Toby Shearman

Thin elastic sheets with lateral swelling and shrinking exhibit complex, self-similar buckling patterns, e.g., torn plastic sheets, leaves, and expanding hydrogels (Figure 1). They are particularly ubiquitous in nature. For example, growing tissue, such as in flowers, lichens, and marine invertebrates, attain elaborate structures as they undergo unconfined growth.

FIGURE 1. Examples of naturally occurring non-Euclidean elastic sheets.

One wonders about the shaping mechanisms of such free sheets and whether they can be mimicked in artificial materials. And, with recent technological advancements extending the range of mechanical structures that can be engineered, there is newfound interest in the geometry and mechanics of such elastic materials. Applications include stimuli-responsive nano-structures, shape memory materials, and furthering the understanding of biological processes, e.g., biomechanics and mechanically induced cell differentiation. 

Albeit by different mechanisms, e.g., differential growth (as in leaves) or thermal expansion (as in hydrogels), the geometry of these sheets are characterized by a growth law that permanently deforms the intrinsic distance between points on the sheet making it non-Euclidean. While a generalization of linear elasticity theory posits that such patterns arise from a non-Euclidean sheet buckling to relieve growth-induced residual strains, a full understanding of the shaping mechanism is lacking. In general, the complex morphologies we see can be understood as an embedding of a non-Euclidean surface that minimizes elastic energy, consisting of stretching, bending, and gravitational potential (or possibly other weak forces).

To explain the shaping mechanism of these sheets, we are studying the embedding of hyperbolic surfaces that minimize elastic energy. We are currently investigating (reduced formulations for) the effect of gravity on hyperbolic sheets with constant Gauss curvature. To explore this system numerically, we are developing a geometric discretization of hyperbolic surfaces (Figure 2 and 3).

Figure 2. Geometric discretization of C1,1 isometries with branch points.  Symmetric periodic surface with a single branch point at the origin with 4 wrinkles (left) and 12 wrinkles (center).  Asymmetric surface with multiple branch points away from the origin (right). All asymptotic lines are straight.

Figure 3. Geometric discretization of isometries with a single degree-4 branch point at the origin and curved asymptotic lines.

By combining geometric and variational techniques with continuum mechanics, we seek to elucidate how forces and stresses over thin membranes may play an important role in the morphogenesis of naturally growing biological tissue (e.g., leaves and flowers) as well as the biomechanics of marine invertebrates, e.g., sea slugs (Figure 4). This, in turn, would enable the prediction and control of the geometry and kinematics of carefully designed thin structures for engineering or technological applications, including soft robots.

As a whole, these research questions involve a broad and diverse spectrum of mathematics and science, including (discrete) differential geometry, variational calculus, partial differential equations, and mechanics.

Figure 4. A saddle surface whose asymptotic lines are rotating with respect to the material points satisfying a no-slip contact condition with the table below (left). The material coordinates represented by colored sectors are rotating at a slower rate than the frame in which the shape of the surface is fixed. The asymptotic frame is indicated by the white ball which rolls on the surface to remain at the minimum. A “mathematical” sea slug (right) with merging and splitting branch points is a cartoon for the motion of a true sea slug (middle).

Related Publications and Presentations

Journal articles

Unpublished reports

Seminars

Posters