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Geometry, mechanics, and dynamics of thin elastic sheets
Geometry, mechanics, and dynamics of thin elastic sheets
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.ย
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.ย Symmetric periodic surface with a single branch point at the origin with 4 wrinkles (left) and 12 wrinkles (center).ย 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 โmathematicalโ 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
Related Publications and Presentations
Journal articles
Journal articles
K. K. Yamamoto, T. S. Shearman, E. J. Struckmeyer, J. A. Gemmer, and S. C. Venkataramani. Natureโs forms are frilly, flexible, and functional. Eur. Phys. J. E 44, 95 (2021). https://doi.org/10.1140/epje/s10189-021-00099-6
Unpublished reports
Unpublished reports
The role of weak forces in the self-similar buckling of non-Euclidean elastic sheets, PhD Comprehensive Exam Paper, October 2017.
The role of gravity in energy-minimizing isometric immersions of hyperbolic elastic sheets, RTG paper, May 2017.
Seminars
Seminars
Sea slugs to soft robots: Computation, discrete geometry, and soft mechanics in non-Euclidean elasticity, Computational Science Seminar, University of TexasโDallas, Nov 2022.
From sea slugs to robots: Computation, discrete geometry, and soft mechanics in non-Euclidean elasticity, USNC TAM (U.S. National Congress on Theoretical and Applied Mechanics), Austin, TX, Jun 2022.
Sea slugs to soft robots: Computation, discrete geometry, and soft mechanics in non-Euclidean elasticity, Mathematics Colloquium, Southern Methodist University, Sep 2020.
Discrete geometry and PDE-constrained optimization for mechanics of hyperbolic elastic sheets, SIAM Conference on Analysis of Partial Differential Equations (PD19), La Quinta, CA, Dec 2019.
From sea slugs to robots: Soft mechanics, discrete geometry, and computation of hyperbolic elastic sheets, Modeling and Computation Seminar, University of Arizona, Oct 2019.
Discrete geometry and mechanics of leaves, flowers, and sea slugs, Analysis and its Applications Seminar, University of Arizona, Apr 2019.
Optimization and data analysis for non-Euclidean elastic sheets, Arizona โ Los Alamos Days, University of Arizona, Apr 2019.
Geometric defects, weak forces, and self-similar buckling in non-Euclidean elastic sheets, APS March Meeting (American Physical Society), Boston, MA, Mar 2019.
The role of weak forces in the self-similar buckling of non-Euclidean elastic sheets, SIAM Conference on Mathematical Aspects of Materials Science, Portland, OR, Jul 2018.
The role of weak forces in the self-similar buckling of non-Euclidean elastic sheets, APS March Meeting (American Physical Society), Los Angeles, CA, Mar 2018.
The role of weak forces in the self-similar buckling of non-Euclidean elastic sheets, Analysis and its Applications Seminar, University of Arizona, Feb 2018.
The role of weak forces in the self-similar buckling of non-Euclidean elastic sheets, Program in Applied Mathematics PhD Comprehensive Exam Presentation, University of Arizona, Nov 2017.
The role of gravity in the self-similar buckling of non-Euclidean elastic sheets, Program in Applied Mathematics Student Brown Bag Seminar, University of Arizona, Oct 2017.
The role of gravity in energy-minimizing isometric immersions of hyperbolic elastic sheets, Second-Year RTG Conference, University of Arizona, Dec 2016.
Topology and computation for real-world problems, First-Year Term Paper Workshop, University of Arizona, May 2016.
Posters
Posters
From sea slugs to robots: Computation, discrete geometry, and soft mechanics in non-Euclidean elasticity, APS March Meeting (American Physical Society), Chicago, IL, Mar 2022.
From sea slugs to robots: Computation, discrete geometry, and soft mechanics in non-Euclidean elasticity, 4th Annual Meeting of the SIAM Texas-Louisiana Section, University of Texas Rio Grande Valley, South Padre Island, TX, Nov 2021.
Geometry, mechanics, and dynamics of leaves, flowers, and sea slugs, 2019 Workshop on Mathematical Models for Pattern Formation, Center for Nonlinear Analysis, Carnegie Mellon University, Mar 2019.
Geometry, mechanics, and dynamics of leaves, flowers, and sea slugs, Arizona Center for Mathematical Sciences (ACMS) 30th Anniversary Workshop, University of Arizona, Nov 2018.
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