Anisotropic buckling patterns in spheroidal film/substrate systems and their implications in some natural and biological systems

Abstract We establish a quantitative mechanics framework of elastic buckling of a spheroidal thin film/substrate system, which is highly relevant to the morphologies of quite a few natural and biological systems. The anisotropic stress-driven bifurcation is governed by the ratios between the effective size/thickness, the equatorial/polar radii, and the substrate/film moduli. The possibilities of manipulating the undulations through external constraints, anisotropic growth/material properties, and substrate geometry/structure are discussed. Analytical equations correlating the undulation characteristics with the geometry/material properties are derived. The quantitative mechanics framework established herein not only has important implications on the morphogenesis of various fruits, vegetables, nuts, eggs, tissues, and animal body parts, but also could guide the three-dimensional micro-fabrications via controlled self-assembly on curved substrate surfaces.

[1]  L. Mahadevan,et al.  Geometry and physics of wrinkling. , 2003, Physical review letters.

[2]  C. Neinhuis,et al.  Tomato (Lycopersicon esculentum Mill.) fruit growth and ripening as related to the biomechanical properties of fruit skin and isolated cuticle. , 2005, Journal of experimental botany.

[3]  P. B. Green,et al.  Expression of pattern in plants: combining molecular and calculus-based biophysical paradigms. , 1999, American journal of botany.

[4]  J Considine,et al.  Physical Aspects of Fruit Growth : THEORETICAL ANALYSIS OF DISTRIBUTION OF SURFACE GROWTH FORCES IN FRUIT IN RELATION TO CRACKING AND SPLITTING. , 1981, Plant physiology.

[5]  Charles R. Steele,et al.  Shell Stability Related to Pattern Formation in Plants , 2000 .

[6]  Rui Huang,et al.  Wrinkle patterns of anisotropic crystal films on viscoelastic substrates , 2008 .

[7]  Z. Suo,et al.  Nonlinear analyses of wrinkles in a film bonded to a compliant substrate , 2005 .

[8]  J. Considine,et al.  Physical Aspects of Fruit Growth: STRESS DISTRIBUTION AROUND LENTICELS. , 1982, Plant physiology.

[9]  M. A. Ayoub,et al.  IN VITRO ACTION OF COBRA VENOM ON GOAT SPERMATOZOA ULTRASTRUCTURE BY TRANSMISSION AND SCANNING ELECTRON MICROSCOPY , 2002 .

[10]  Alfred J. Crosby,et al.  Fabricating Microlens Arrays by Surface Wrinkling , 2006 .

[11]  S. Timoshenko,et al.  THEORY OF PLATES AND SHELLS , 1959 .

[12]  K. Nakajima,et al.  Plant-Specific Microtubule-Associated Protein SPIRAL2 Is Required for Anisotropic Growth in Arabidopsis1 , 2004, Plant Physiology.

[13]  M. Hallett,et al.  Chemotaxis and the cell surface-area problem , 2008, Nature Reviews Molecular Cell Biology.

[14]  Charles R. Steele,et al.  New Evidence for the Role of Mechanical Forces in the Shoot Apical Meristem , 2000, Journal of Plant Growth Regulation.

[15]  H. G. Allen.  Analysis and design of structural sandwich panels , 1969 .

[16]  J. Hutchinson.  Delamination of compressed films on curved substrates , 2001 .

[17]  Andreas Menzel,et al.  Modelling of anisotropic growth in biological tissues , 2005 .

[18]  Michael P Marder,et al.  Leaves, flowers and garbage bags: Making waves , 2004 .

[19]  Z. Cao,et al.  Self-assembled triangular and labyrinth buckling patterns of thin films on spherical substrates. , 2008, Physical review letters.

[20]  J. Hutchinson,et al.  A family of herringbone patterns in thin films , 2004 .

[21]  J. G. Simmonds,et al.  The Nonlinear Theory of Elastic Shells , 1998 .

[22]  Low-temperature fabrication of si thin-film transistor microstructures by soft lithographic patterning on curved and planar substrates , 2000 .

[23]  J. Rogers,et al.  Finite deformation mechanics in buckled thin films on compliant supports , 2007, Proceedings of the National Academy of Sciences.

[24]  B. Audoly,et al.  Buckling of a stiff film bound to a compliant substrate—Part I:: Formulation, linear stability of cylindrical patterns, secondary bifurcations , 2008 .

[25]  P. Liu,et al.  Yeast cells with an artificial mineral shell: protection and modification of living cells by biomimetic mineralization. , 2008, Angewandte Chemie.

[26]  E Kuhl,et al.  Computational modeling of arterial wall growth , 2007, Biomechanics and modeling in mechanobiology.

[27]  A. Newell,et al.  Phyllotactic patterns on plants. , 2004, Physical review letters.

[28]  Xingyu Jiang,et al.  Directing cell migration with asymmetric micropatterns. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[29]  George M. Whitesides,et al.  Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer , 1998, Nature.

[30]  Jie Yin,et al.  Stress-driven buckling patterns in spheroidal core/shell structures , 2008, Proceedings of the National Academy of Sciences.

[31]  R. Nuzzo,et al.  Soft lithographic fabrication of an image sensor array on a curved substrate , 2004 .

[32]  Reduced buckling in one dimension versus two dimensions of a compressively strained film on a compliant substrate , 2006 .

[33]  E. Ventsel,et al.  Thin Plates and Shells: Theory: Analysis, and Applications , 2001 .

[34]  D. Cosgrove.  Growth of the plant cell wall , 2005, Nature Reviews Molecular Cell Biology.

[35]  J. Hutchinson,et al.  Controlled formation of nanoscale wrinkling patterns on polymers using focused ion beam , 2007 .