RECORD
Geometrically incompatible confinement of solids
Soft Condensed Matter · Submitted 18 Sep 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This emergent rule states that - despite the unavoidable strain in such an incompatible confinement - the ratio between the energies stored in straining and bending the solid may be arbitrarily small.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
The complex morphologies exhibited by spatially confined thin objects have long challenged human efforts to understand and manipulate them, from the representation of patterns in draped fabric in Renaissance art to current day efforts to engineer flexible sensors that conform to the human body. We introduce a theoretical principle, broadly generalizing Euler's {\emph{elastica}} -- a core concept of continuum mechanics that invokes the energetic preference of bending over straining a thin solid object and has been widely applied to classical and modern studies of beams and rods. We define a class of {\emph{geometrically incompatible confinement}} problems, whereby the topography imposed on a thin solid body is incompatible with its intrinsic ("target") metric and, as a consequence of Gauss' {\emph{Theorema Egregium}}, induces strain. Focusing on a prototypical example of a sheet attached to a spherical substrate, numerical simulations and analytical study demonstrate that the mechanics is governed by a principle, which we call the "Gauss-Euler {\emph{elastica}}". This emergent rule states that - despite the unavoidable strain in such an incompatible confinement - the ratio between the energies stored in straining and bending the solid may be arbitrarily small. The Gauss-Euler {\emph{elastica}} underlies a theoretical framework that greatly simplifies the daunting task of solving the highly nonlinear equations that describe thin solids at mechanical equilibrium. This development thus opens new possibilities for attacking a broad class of phenomena governed by the coupling of geometry and mechanics.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The complex morphologies exhibited by spatially confined thin objects have long challenged human efforts to understand and manipulate them, from the representation of patterns in draped fabric in Renaissance art to current day efforts to engineer flexible sensors that conform to the human body. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | We introduce a theoretical principle, broadly generalizing Euler's {\emph{elastica}} -- a core concept of continuum mechanics that invokes the energetic preference of bending over straining a thin solid object and has been widely applied to classical and modern studies of beams and rods. | contribution 0.92 | 0.87 | 0.05 | asserted 0.98 |
| s2 | We define a class of {\emph{geometrically incompatible confinement}} problems, whereby the topography imposed on a thin solid body is incompatible with its intrinsic ("target") metric and, as a consequence of Gauss' {\emph{Theorema Egregium}}, induces strain. | contribution 0.95 | 0.69 | 0.06 | asserted 0.99 |
| s3 | Focusing on a prototypical example of a sheet attached to a spherical substrate, numerical simulations and analytical study demonstrate that the mechanics is governed by a principle, which we call the "Gauss-Euler {\emph{elastica}}". | contribution 0.58 | 0.95 | 0.13 | asserted 1.00 |
| s4 | This emergent rule states that - despite the unavoidable strain in such an incompatible confinement - the ratio between the energies stored in straining and bending the solid may be arbitrarily small. | result 0.97 | 0.89 | 0.75 | asserted 0.60 |
| s5 | The Gauss-Euler {\emph{elastica}} underlies a theoretical framework that greatly simplifies the daunting task of solving the highly nonlinear equations that describe thin solids at mechanical equilibrium. | interpretation 0.68 | 0.84 | 0.13 | asserted 1.00 |
| s6 | This development thus opens new possibilities for attacking a broad class of phenomena governed by the coupling of geometry and mechanics. | interpretation 0.86 | 0.59 | 0.12 | hedged possibility 0.50 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
The complex morphologies exhibited by spatially confined thin objects have long challenged human efforts to understand and manipulate them, from the representation of patterns in draped fabric in Renaissance art to current day efforts to engineer flexible sensors that conform to the human body. We introduce a theoretical principle, broadly generalizing Euler's {\emph{elastica}} -- a core concept of continuum mechanics that invokes the energetic preference of bending over straining a thin solid object and has been widely applied to classical and modern studies of beams and rods. We define a class of {\emph{geometrically incompatible confinement}} problems, whereby the topography imposed on a thin solid body is incompatible with its intrinsic ("target") metric and, as a consequence of Gauss' {\emph{Theorema Egregium}}, induces strain. Focusing on a prototypical example of a sheet attached to a spherical substrate, numerical simulations and analytical study demonstrate that the mechanics is governed by a principle, which we call the "Gauss-Euler {\emph{elastica}}". This emergent rule states that - despite the unavoidable strain in such an incompatible confinement - the ratio between the energies stored in straining and bending the solid may be arbitrarily small. The Gauss-Euler {\emph{elastica}} underlies a theoretical framework that greatly simplifies the daunting task of solving the highly nonlinear equations that describe thin solids at mechanical equilibrium. This development thus opens new possibilities for attacking a broad class of phenomena governed by the coupling of geometry and mechanics.
Human review
How the literature uses it
1 arXiv paper cite this in 1 sentence
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.