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Geometrically incompatible confinement of solids.

Benny Davidovitch1, Yiwei Sun2, Gregory M Grason3

  • 1Department of Physics, University of Massachusetts, Amherst, MA 01003; bdavidov@physics.umass.edu grason@mail.pse.umass.edu.

Proceedings of the National Academy of Sciences of the United States of America
|December 29, 2018
PubMed
Summary
This summary is machine-generated.

Scientists developed the Gauss-Euler elastica, a new principle for understanding thin solids. This principle simplifies analyzing complex shapes and mechanics in confined objects, like flexible sensors.

Keywords:
elasticitypattern formationvariational calculuswrinkles

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Area of Science:

  • Continuum mechanics
  • Geometric mechanics
  • Materials science

Background:

  • Spatially confined thin objects exhibit complex morphologies, challenging understanding and manipulation.
  • Euler's elastica principle describes the energetic preference of bending over straining in thin solids.
  • Engineering flexible sensors requires understanding how thin materials conform to surfaces.

Purpose of the Study:

  • Introduce a theoretical principle generalizing Euler's elastica for geometrically incompatible confinement problems.
  • Define and analyze a class of problems where imposed topography conflicts with intrinsic material metrics.
  • Develop a framework to simplify the analysis of thin solids under mechanical equilibrium.

Main Methods:

  • Definition of geometrically incompatible confinement problems.
  • Numerical simulations of a sheet attached to a spherical substrate.
  • Analytical study applying Gauss' Theorema Egregium.

Main Results:

  • Demonstrated that incompatible confinement induces strain due to the Theorema Egregium.
  • Introduced the Gauss-Euler elastica principle governing the mechanics of such systems.
  • Showed that the ratio of strain to bending energy can be arbitrarily small.

Conclusions:

  • The Gauss-Euler elastica provides a simplified theoretical framework for analyzing thin solids.
  • This principle facilitates solving highly nonlinear equations describing thin solids at mechanical equilibrium.
  • Opens possibilities for studying phenomena coupling geometry and mechanics in confined thin objects.