Advertisement

Minimal energy configurations of strained multi-layers

  • Bernd SchmidtEmail author
Original Article

Abstract

We derive an effective plate theory for internally stressed thin elastic layers as are used, e.g., in the fabrication of nano- and microscrolls. The shape of the energy minimizers of the effective energy functional is investigated without a priori assumptions on the geometry. For configurations in two dimensions (corresponding to Euler-Bernoulli theory) we also take into account a non-interpenetration condition for films of small but non-vanishing thickness.

Keywords

Plate theory Geometry of Energy minimizers Nanoscrolls 

Mathematics Subject Classification (2000)

49J45 74K20 74G65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Friesecke G., James R.D. and Müller S. (2002). Rigorous derivation of nonlinear plate theory and geometric rigidity. C. R. Acad. Sci. Paris 334: 173–178 zbMATHGoogle Scholar
  2. 2.
    Friesecke G., James R.D. and Müller S. (2002). A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55: 1461–1506 zbMATHCrossRefGoogle Scholar
  3. 3.
    Friesecke G., James R.D. and Müller S. (2006). A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Rational Mech. Anal. 180: 183–236 zbMATHCrossRefGoogle Scholar
  4. 4.
    Grundmann M. (2003). Nanoscroll formation from strained layer heterostructures. Appl. Phys. Lett. 83: 2444–2446 CrossRefGoogle Scholar
  5. 5.
    Pakzad M.R. (2004). On the Sobolev space of isometric immersions. J. Differ. Geom. 66: 47–69 zbMATHGoogle Scholar
  6. 6.
    Paetzelt H., Gottschalch V., Bauer J., Herrnberger H. and Wagner G. (2006). Fabrication of A(III)-B(II) nano- and microtubes using MOVPE grown materials. Phys. Stat. Sol. (A) 203: 817–824 CrossRefGoogle Scholar
  7. 7.
    Schmidt B.(2006). Effective theories for thin elastic films. PhD thesis, University of Leipzig, LeipzigGoogle Scholar
  8. 8.
    Schmidt, B.: Plate theory for stressed heterogeneous multilayers of finite bending energy. Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, preprint 86/2006Google Scholar
  9. 9.
    Schmidt B. (2006). Effective theories for thin elastic films. University of Leipzig, Leipzig CrossRefGoogle Scholar
  10. 10.
    Schmidt O.G. and Eberl K. (2001). Thin solid films roll up into nanotubes. Nature 410: 168 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations