Nonuniform growth and topological defects in the shaping of elastic sheets.

@article{Bende2014NonuniformGA,
  title={Nonuniform growth and topological defects in the shaping of elastic sheets.},
  author={Nakul Prabhakar Bende and Ryan C. Hayward and Christian Santangelo},
  journal={Soft matter},
  year={2014},
  volume={10 34},
  pages={
          6382-6
        }
}
We demonstrate that shapes with zero Gaussian curvature, except at singularities, produced by the growth-induced buckling of a thin elastic sheet are the same as those produced by the Volterra construction of topological defects in which edges of an intrinsically flat surface are identified. With this connection, we study the problem of choosing an optimal pattern of growth for a prescribed developable surface, finding a fundamental trade-off between optimal design and the accuracy of the… 
Disclinations, e-cones, and their interactions in extensible sheets.
TLDR
The system is found to show a transition from a regime where the wavelength was given by the ribbon geometry, to where it is given by its elasticity as a function of the ratio of the applied tension to the elastic modulus and cross-sectional area of the ribbon.
Geometry and mechanics of thin growing bilayers.
TLDR
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, and introduces a new measure of slenderness that describes a sheet in terms of both thickness and plate shape.
Distortion-controlled isotropic swelling: numerical study of free boundary swelling patterns.
TLDR
This work proposes a computational algorithm to produce optimal growth patterns by introducing cuts into the target surfaces, and proposes that the patterns requiring the fewest or shortest cuts produce the best approximations to the target shape at finite thickness.
Shape transformations of soft matter governed by bi-axial stresses.
TLDR
This work explored experimentally and using simulations how simultaneous or consecutive application of two orthogonal perturbations to thin patterned stimuli-responsive hydrogel sheets affects their three-dimensional shape transformations.
Non-Euclidean Shells: A Study of Growth-Induced Fabrication and Mechanical Multi-Stability
NON-EUCLIDEAN SHELLS: A STUDY OF GROWTH-INDUCED FABRICATION AND MECHANICAL MULTI-STABILITY SEPTEMBER 2017 NAKUL PRABHAKAR BENDE B.TECH., INDIAN INSTITUTE OF TECHNOLOGY ROORKEE M.TECH., INDIAN
Biasing Buckling Direction in Shape‐Programmable Hydrogel Sheets with Through‐Thickness Gradients
A photocrosslinkable poly(N, N′‐diethylacrylamide) copolymer allows for the photolithographic fabrication of hydrogel sheets with nonuniform crosslinking density and swelling ratio. Using this
Stimuli‐responsive buckling mechanics of polymer films
Thin polymer films may undergo a wide variety of elastic instabilities that include global buckling modes, wrin- kling and creasing of surfaces, and snapping transitions. Tradi- tionally, these
Grayscale gel lithography for programmed buckling of non-Euclidean hydrogel plates.
TLDR
A simple method of 'grayscale gel lithography' that relies on a digital micromirror array device (DMD) to control the dose of ultraviolet (UV) light, and therefore the extent of swelling of a photocrosslinkable poly(N-isopropyl acrylamide) (PNIPAm) copolymer film, with micrometer-scale spatial resolution is described.
The MoSeS dynamic omnigami paradigm for smart shape and composition programmable 2D materials
TLDR
A new approach, combining phase/strain engineering with shape programming, to form 3D objects by patterned alloying of 2D transition metal dichalcogenide (TMD) monolayers, which provide control of both bending and stretching deformations, are reversibly actuatable with electric fields, and possess the extraordinary and diverse properties of TMDs.
...
...

References

SHOWING 1-10 OF 52 REFERENCES
Conical defects in growing sheets.
A growing or shrinking disc will adopt a conical shape, its intrinsic geometry characterized by a surplus angle phi(e) at the apex. If growth is slow, the cone will find its equilibrium. Whereas this
Programmed buckling by controlled lateral swelling in a thin elastic sheet.
TLDR
This work develops a solution to the design problem suggested by such systems, namely, if and how one can generate particular three-dimensional shapes from thin elastic sheets by mere imposition of a two-dimensional pattern of locally isotropic growth.
Mechanics: Buckling cascades in free sheets
TLDR
The edge of a torn plastic sheet forms a complex three-dimensional fractal shape that results from a simple elongation of the sheet in the direction along its edge.
Elastic platonic shells.
TLDR
This work shows that by deflating a crystalline shell with defects, it is possible to create elastic shell analogs of the classical platonic solids and suggests methods to engineer shape into soft spherical shells using a frozen defect topology.
Asymptotic Shape of a Fullerene Ball
We infer scaling of the shape and energy of a space-enclosing elastic sheet such as a large fullerene ball of linear dimension R. Stretching deformation is crucial in determining the optimal shape,
Minimal resonances in annular non-euclidean strips.
  • B. ChenC. Santangelo
  • Engineering
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2010
TLDR
This work uses the class of "conical" closed strips with a prescribed metric tensor on their center line as a variational ansatz to obtain the minimal energy shapes of closed strips and finds excellent agreement with the results of a numerical bead-spring model.
Geometry and Elasticity of Strips and Flowers
We solve several problems that involve imposing metrics on surfaces. The problem of a strip with a linear metric gradient is formulated in terms of a Lagrangian similar to those used for spin
How paper folds : bending with local constraints
A variational framework is introduced to describe how a surface bends when it is subject to local constraints on its geometry. This framework is applied to describe the patterns of a folded sheet of
Dipoles in thin sheets
TLDR
This work considers a “charge-neutral” dipole composed of two conical singularities of opposite sign, and determines the shapes of the minima and evaluates their energies in the thin-sheet regime where bending dominates over stretching.
Morphogenesis of growing soft tissues.
TLDR
Using the theory of finite elasticity, Rodriguez has postulated a multiplicative decomposition of the geometric deformation gradient into a growth-induced part and an elastic one needed to ensure compatibility of the body.
...
...