title: Curvature-induced symmetry breaking determines elastic surface patterns.
abstract: Symmetry-breaking transitions associated with the buckling and folding of curved multilayered surfaces-which are common to a wide range of systems and processes such as embryogenesis, tissue differentiation and structure formation in heterogeneous thin films or on planetary surfaces-have been characterized experimentally. Yet owing to the nonlinearity of the underlying stretching and bending forces, the transitions cannot be reliably predicted by current theoretical models. Here, we report a generalized Swift-Hohenberg theory that describes wrinkling morphology and pattern selection in curved elastic bilayer materials. By testing the theory against experiments on spherically shaped surfaces, we find quantitative agreement with analytical predictions for the critical curves separating labyrinth, hybrid and hexagonal phases. Furthermore, a comparison to earlier experiments suggests that the theory is universally applicable to macroscopic and microscopic systems. Our approach builds on general differential-geometry principles and can thus be extended to arbitrarily shaped surfaces.
authors: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, Massachusetts 02139-4307, USA.
authors: Norbert
authors: N
authors: Stoop
authors: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, Massachusetts 02139-4307, USA.
authors: Romain
authors: R
authors: Lagrange
authors: Department of Civil &Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, Massachusetts 02139-4307, USA.
authors: Denis
authors: D
authors: Terwagne
authors: 1] Department of Civil &Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, Massachusetts 02139-4307, USA [2] Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, Massachusetts 02139-1713, USA.
authors: Pedro M
authors: PM
authors: Reis
authors: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, Massachusetts 02139-4307, USA.
authors: Jörn
authors: J
authors: Dunkel
references: Phys Rev Lett. 2001 Oct 8;87(15):158102
references: 11580727
references: Phys Rev Lett. 2010 Aug 6;105(6):068101
references: 20868017
references: Science. 2004 Oct 29;306(5697):828-33
references: 15514148
references: Chaos. 2007 Sep;17(3):037103
references: 17903010
references: Proc Natl Acad Sci U S A. 2010 Jun 29;107(26):11653-4
references: 20562344
references: Phys Rev E Stat Nonlin Soft Matter Phys. 2006 May;73(5 Pt 2):056211
references: 16803030
references: Phys Rev Lett. 2011 Jun 10;106(23):234301
references: 21770507
references: Adv Mater. 2014 Oct;26(38):6608-11
references: 24956072
references: Sci Rep. 2014 Jul 16;4:5710
references: 25028198
references: Nat Mater. 2005 Apr;4(4):293-7
references: 15750598
references: Proc Natl Acad Sci U S A. 2007 Oct 2;104(40):15607-12
references: 17898178
references: Proc Natl Acad Sci U S A. 2009 Dec 29;106(52):22085-9
references: 20018703
references: Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Jan;71(1 Pt 2):015205
references: 15697647
references: Science. 1975 Jul 4;189(4196):18-21
references: 1135626
references: Phys Rev Lett. 2003 Feb 21;90(7):074302
references: 12633231
references: Phys Rev Lett. 2013 Jul 5;111(1):014301
references: 23863002
references: Proc Natl Acad Sci U S A. 2013 Apr 9;110(15):6015-20
references: 23530223
references: Phys Rev Lett. 1988 Jun 20;60(25):2638-2640
references: 10038410
references: Science. 2008 May 16;320(5878):912-6
references: 18487188
references: Phys Rev Lett. 2008 Jan 25;100(3):036102
references: 18233005
references: Proc Natl Acad Sci U S A. 2012 Jun 19;109(25):9716-20
references: 22679292