Programmed buckling by controlled lateral swelling in a thin elastic sheet.

Recent experiments have imposed controlled swelling patterns on thin polymer films, which subsequently buckle into three-dimensional shapes. We develop 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. Not every shape is possible. Several types of obstruction can arise, some of which depend on the sheet thickness. We provide some examples using the axisymmetric form of the problem, which is analytically tractable.

[1]  Sergei Nechaev,et al.  On the plant leaf's boundary, 'jupe ` a godets' and conformal embeddings , 2001 .

[2]  Raz Kupferman,et al.  Buckling transition and boundary layer in non-Euclidean plates. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Udo Seifert,et al.  Configurations of fluid membranes and vesicles , 1997 .

[4]  W. Flügge,et al.  Tensor Analysis and Continuum Mechanics , 1972 .

[5]  T. Banchoff,et al.  Differential Geometry of Curves and Surfaces , 2010 .

[6]  Arezki Boudaoud,et al.  ‘Ruban à godets': an elastic model for ripples in plant leaves , 2002 .

[7]  Laplace pressure as a surface stress in fluid vesicles , 2006, cond-mat/0602289.

[8]  N. Papanicolaou,et al.  Geometry and Elasticity of Strips and Flowers , 2004, cond-mat/0409135.

[9]  E. Sharon,et al.  The mechanics of non-Euclidean plates , 2010 .

[10]  N. Hicks.  Notes on Differential Geometry , 1967 .

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  L. Carroll.  The Complete Works of Lewis Carroll , 1936 .

[13]  Alain Goriely,et al.  Growth and instability in elastic tissues , 2005 .

[14]  M. Garland,et al.  NOTES ON DIFFERENTIAL GEOMETRY , 2004 .

[15]  R. Kupferman,et al.  Elastic theory of unconstrained non-Euclidean plates , 2008, 0810.2411.

[16]  Christian D. Santangelo,et al.  Buckling thin disks and ribbons with non-Euclidean metrics , 2008, 0808.0909.

[17]  E. M. Lifshitz,et al.  Theory of Elasticity: Vol. 7 of Course of Theoretical Physics , 1960 .

[18]  The Shape of the Edge of a Leaf , 2002, cond-mat/0208232.

[19]  J. Guven,et al.  How paper folds: bending with local constraints , 2007, 0712.0978.

[20]  Shiing-Shen Chern,et al.  An elementary proof of the existence of isothermal parameters on a surface , 1955 .