3,079 research outputs found
Germain Curvature: The Case for Naming the Mean Curvature of a Surface after Sophie Germain
How do we characterize the shape of a surface? It is now well understood that
the shape of a surface is determined by measuring how curved it is at each
point. From these measurements, one can identify the directions of largest and
smallest curvature, i.e. the principal curvatures, and construct two surface
invariants by taking the average and the product of the principal curvatures.
The product of the principal curvatures describes the intrinsic curvature of a
surface, and has profound importance in differential geometry - evidenced by
the Gauss's Theorema Egregium and the Gauss-Bonnet theorem. This curvature is
commonly referred to as the Gaussian Curvature after Carl Friedrich Gauss,
following his significant contributions to the emerging field of differential
geometry in his 1828 work. The average, or mean curvature, is an extrinsic
measure of the shape of a surface - that is, the shape must be embedded in a
higher dimensional space to be measured. Beginning in 1811, and culminating in
efforts in 1821 and 1826, the mathematician Sophie Germain identified the mean
curvature as the appropriate measure for describing the shape of vibrating
plates. Her hypothesis leads directly to the equations describing the behavior
of thin, elastic plates. In letters to Gauss, she described her notion of a
"sphere of mean curvature" that can be identified at each point on the surface.
This contribution stimulated a period of rapid development in elasticity and
geometry, and yet Germain has not yet received due credit for deducing the
mathematical and physical significance of mean curvature. It is clear from the
primary source evidence that this measure of mean curvature should bear the
name of Sophie Germain.Comment: 6 pages, 3 figure
Morphing of Geometric Composites via Residual Swelling
Understanding and controlling the shape of thin, soft objects has been the
focus of significant research efforts among physicists, biologists, and
engineers in the last decade. These studies aim to utilize advanced materials
in novel, adaptive ways such as fabricating smart actuators or mimicking living
tissues. Here, we present the controlled growth--like morphing of 2D sheets
into 3D shapes by preparing geometric composite structures that deform by
residual swelling. The morphing of these geometric composites is dictated by
both swelling and geometry, with diffusion controlling the swelling-induced
actuation, and geometric confinement dictating the structure's deformed shape.
Building on a simple mechanical analog, we present an analytical model that
quantitatively describes how the Gaussian and mean curvatures of a thin disk
are affected by the interplay among geometry, mechanics, and swelling. This
model is in excellent agreement with our experiments and numerics. We show that
the dynamics of residual swelling is dictated by a competition between two
characteristic diffusive length scales governed by geometry. Our results
provide the first 2D analog of Timoshenko's classical formula for the thermal
bending of bimetallic beams - our generalization explains how the Gaussian
curvature of a 2D geometric composite is affected by geometry and elasticity.
The understanding conferred by these results suggests that the controlled
shaping of geometric composites may provide a simple complement to traditional
manufacturing techniques
Geometry and Mechanics of Thin Growing Bilayers
We investigate how thin sheets of arbitrary shapes morph under the isotropic
in-plane expansion of their top surface, which may represent several stimuli
such as nonuniform heating, local swelling and differential growth. Inspired by
geometry, an analytical model is presented that rationalizes how the shape of
the disk influences morphing, from the initial spherical bending to the final
isometric limit. We introduce a new measure of slenderness that
describes a sheet in terms of both thickness and plate shape. We find that the
mean curvature of the isometric state is three fourth's the natural curvature,
which we verify by numerics and experiments. We finally investigate the
emergence of a preferred direction of bending in the isometric state, guided by
numerical analyses. The scalability of our model suggests that it is suitable
to describe the morphing of sheets spanning several orders of magnitude.Comment: 5 pages, 4 figure
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