Mechanics of morphological instabilities and surface wrinkling in soft materials: a review

Morphological instabilities and surface wrinkling of soft materials such as gels and biological tissues are of growing interest to a number of academic disciplines including soft lithography, metrology, flexible electronics, and biomedical engineering. In this paper, we review some of the recent progresses in experimental and theoretical investigations of instabilities that lead to the emergence and evolution of surface wrinkling, folding and creasing under various geometrical constraints (e.g., thin films, sheets, fibers, particles, tubes, cavities, vesicles and capsules) and loading stimuli (e.g., mechanical forces, growth, atrophy, swelling, shrinkage, van der Waals interactions). Some representative theoretical and numerical approaches aimed at modelling the onset of instabilities as well as the postbuckling evolution involving multiple bifurcations and symmetry-breakings are discussed along with the main characteristics and some possible applications of this rich phenomenon.

[1]  Patrick D. Shipman,et al.  Phyllotaxis: cooperation and competition between mechanical and biochemical processes. , 2008, Journal of theoretical biology.

[2]  Luka Pocivavsek,et al.  Beyond Wrinkles: Stress and Fold Localization in Thin Elastic Membranes , 2008 .

[3]  Alain Goriely,et al.  Biomechanical models of hyphal growth in actinomycetes. , 2003, Journal of theoretical biology.

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

[5]  Pasquale Ciarletta,et al.  Swelling instability of surface-attached gels as a model of soft tissue growth under geometric constraints , 2010 .

[6]  B. Audoly,et al.  Buckling of a stiff film bound to a compliant substrate—Part III:: Herringbone solutions at large buckling parameter , 2008 .

[7]  Huajian Gao,et al.  Surface wrinkling of mucosa induced by volumetric growth: Theory, simulation and experiment , 2011 .

[8]  Ashutosh Sharma,et al.  Contact instability of thin elastic films on patterned substrates. , 2007, The Journal of chemical physics.

[9]  A. Boudaoud,et al.  Buckling of swelling gels , 2005, The European physical journal. E, Soft matter.

[10]  Jin Zhai,et al.  Super-hydrophobic surfaces: From natural to artificial , 2002 .

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

[12]  L. Mahadevan,et al.  On the growth and form of the gut , 2011, Nature.

[13]  S. Herminghaus,et al.  Elastic instability of rubber films between solid bodies , 2001 .

[14]  Rui Huang,et al.  Effect of surface tension on swell-induced surface instability of substrate-confined hydrogel layers , 2010 .

[15]  E. Sharon,et al.  Direct observation of the temporal and spatial dynamics during crumpling. , 2010, Nature materials.

[16]  David D Sampson,et al.  Airway narrowing assessed by anatomical optical coherence tomography in vitro: dynamic airway wall morphology and function. , 2010, Journal of applied physiology.

[17]  Mi Ri Kim,et al.  Facile fabrication of uniform golf-ball-shaped microparticles from various polymers , 2011 .

[18]  Michael Marder,et al.  Thin films: Wrinkling of an elastic sheet under tension , 2002, Nature.

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

[20]  Sharma,et al.  Meniscus instability in a thin elastic film , 2000, Physical review letters.

[21]  Ullrich Steiner,et al.  Electrohydrodynamic instabilities in polymer films , 2001 .

[22]  Z. Suo,et al.  Creasing instability of elastomer films , 2012 .

[23]  Arezki Boudaoud,et al.  Wrinkling of pressurized elastic shells. , 2011, Physical review letters.

[24]  Hongping Zhao,et al.  Spontaneous instability of soft thin films on curved substrates due to van der Waals interaction , 2011 .

[25]  Ashutosh Sharma,et al.  Contact Instability of Elastic Bilayers: Miniaturization of Instability Patterns , 2007 .

[26]  E. L. Starostin,et al.  The shape of a Möbius strip. , 2007, Nature materials.

[27]  Jan Genzer,et al.  Soft matter with hard skin: From skin wrinkles to templating and material characterization. , 2006, Soft matter.

[28]  C. Walsh,et al.  Molecular insights into human brain evolution , 2005, Nature.

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

[30]  Alain Goriely,et al.  On the mechanical stability of growing arteries , 2010 .

[31]  J. Hutchinson,et al.  Herringbone Buckling Patterns of Compressed Thin Films on Compliant Substrates , 2004 .

[32]  En-Jui Lee.  Elastic-Plastic Deformation at Finite Strains , 1969 .

[33]  P D Paré,et al.  Airway narrowing and internal structural constraints. , 2000, Journal of applied physiology.

[34]  Rui Huang,et al.  Kinetic wrinkling of an elastic film on a viscoelastic substrate , 2005 .

[35]  T. Witten.  Stress focusing in elastic sheets , 2007 .

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

[37]  N. Talley,et al.  Barium studies in diffuse eosinophilic gastroenteritis , 2005, Gastrointestinal Radiology.

[38]  Xi-Qiao Feng,et al.  Spinodal surface instability of soft elastic thin films , 2008 .

[39]  H. Stone,et al.  Hierarchical folding of elastic membranes under biaxial compressive stress. , 2011, Nature materials.

[40]  Jun Young Chung,et al.  Roles of discontinuities in bio-inspired adhesive pads , 2005, Journal of The Royal Society Interface.

[41]  A. Goriely,et al.  Possible role of differential growth in airway wall remodeling in asthma. , 2011, Journal of applied physiology.

[42]  Vittorio Cristini,et al.  Predictions of tumour morphological stability and evaluation against experimental observations , 2011, Journal of The Royal Society Interface.

[43]  Hai-Chao Han,et al.  A biomechanical model of artery buckling. , 2007, Journal of biomechanics.

[44]  Hai-Chao Han,et al.  Blood vessel buckling within soft surrounding tissue generates tortuosity. , 2009, Journal of biomechanics.

[45]  J. Burdick,et al.  Solvent induced transition from wrinkles to creases in thin film gels with depth-wise crosslinking gradients , 2010 .

[46]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[47]  John A Rogers,et al.  Semiconductor wires and ribbons for high-performance flexible electronics. , 2008, Angewandte Chemie.

[48]  T. Witten,et al.  Compression induced folding of a sheet: an integrable system. , 2011, Physical review letters.

[49]  Gang Wang,et al.  Surface effects in various bending-based test methods for measuring the elastic property of nanowires , 2010, Nanotechnology.

[50]  Qunyang Li,et al.  Pattern instability of a soft elastic thin film under van der Waals forces , 2006 .

[51]  V. Tsukruk,et al.  Spontaneous Self‐Folding in Confined Ultrathin Polymer Gels , 2010, Advanced materials.

[52]  Jay D. Humphrey,et al.  Review Paper: Continuum biomechanics of soft biological tissues , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[53]  Ashutosh Sharma,et al.  Electric-field induced instabilities and morphological phase transitions in soft elastic films. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  Z. Suo,et al.  Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load , 2009 .

[55]  T. Russell,et al.  Electrically induced structure formation and pattern transfer , 2000, Nature.

[56]  Pasquale Ciarletta,et al.  Pattern formation in fiber-reinforced tubular tissues: Folding and segmentation during epithelial growth , 2012 .

[57]  Rudolf Podgornik,et al.  Stability of elastic icosadeltahedral shells under uniform external pressure: application to viruses under osmotic pressure. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  M. Ben Amar,et al.  Shape transition in artificial tumors: from smooth buckles to singular creases. , 2011, Physical review letters.

[59]  Roberto Lazzaroni,et al.  Confined wrinkling: impact on pattern morphology and periodicity , 2011 .

[60]  Xi-Qiao Feng,et al.  Buckling and postbuckling of a compressed thin film bonded on a soft elastic layer: a three-dimensional analysis , 2010 .

[61]  Benny Davidovitch,et al.  Smooth cascade of wrinkles at the edge of a floating elastic film. , 2009, Physical review letters.

[62]  Zhigang Suo,et al.  Periodic patterns and energy states of buckled films on compliant substrates , 2011 .

[63]  Zhigang Suo,et al.  Creases in soft tissues generated by growth , 2011 .

[64]  Charles R Steele,et al.  An anisotropic-viscoplastic model of plant cell morphogenesis by tip growth. , 2006, The International journal of developmental biology.

[65]  Peter Sternberg,et al.  Energy estimates for the von Kármán model of thin-film blistering , 2001 .

[66]  Zhigang Suo,et al.  Formation of creases on the surfaces of elastomers and gels , 2009 .

[67]  I. Heath.  Tip growth in plant and fungal cells , 1992 .

[68]  Ashutosh Sharma,et al.  Instabilities of soft elastic microtubes filled with viscous fluids: pearls, wrinkles, and sausage strings. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[69]  Xi-Qiao Feng,et al.  Mucosal wrinkling in animal antra induced by volumetric growth , 2011 .

[70]  Hai-Chao Han,et al.  Mechanical Buckling of Veins Under Internal Pressure , 2010, Annals of Biomedical Engineering.

[71]  J. Joanny,et al.  Instabilities of monolayered epithelia: shape and structure of villi and crypts. , 2011, Physical review letters.

[72]  Z. Suo,et al.  Wrinkling of a compressed elastic film on a viscous layer , 2002 .

[73]  E. Cerda,et al.  Capillary wrinkling of elastic membranes , 2010 .

[74]  Hillel Aharoni,et al.  Spontaneous buckling of elastic sheets with a prescribed non-Euclidean metric , 2007 .

[75]  A. McCulloch,et al.  Stress-dependent finite growth in soft elastic tissues. , 1994, Journal of biomechanics.

[76]  David Reguera,et al.  Mechanical properties of viral capsids. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[77]  Ashutosh Sharma,et al.  Stability of a thin elastic film interacting with a contactor , 2002 .

[78]  Lallit Anand,et al.  A coupled theory of fluid permeation and large deformations for elastomeric materials , 2010 .

[79]  B. Audoly,et al.  Self-similar structures near boundaries in strained systems. , 2003, Physical review letters.

[80]  Xuanhe Zhao,et al.  Creasing to cratering instability in polymers under ultrahigh electric fields. , 2011, Physical review letters.

[81]  Christopher M. Stafford,et al.  Surface Wrinkling: A Versatile Platform for Measuring Thin‐Film Properties , 2011, Advanced materials.

[82]  L. Taber.  Biomechanics of Growth, Remodeling, and Morphogenesis , 1995 .

[83]  M. Fayed,et al.  Localization of Sugar Residues in the Stomach of Three Species of Monkeys (Tupaiidae glis, Nycticebus cocang and Callithrix jacchus) by Lectin Histochemistry , 2010 .

[84]  Stephen C Cowin,et al.  Tissue growth and remodeling. , 2004, Annual review of biomedical engineering.

[85]  Xi Chen,et al.  Buckling of anisotropic films on cylindrical substrates: insights for self-assembly fabrication of 3D helical gears , 2010 .

[86]  Alain Goriely,et al.  Tissue tension and axial growth of cylindrical structures in plants and elastic tissues , 2008 .

[87]  Arezki Boudaoud,et al.  Multiple-length-scale elastic instability mimics parametric resonance of nonlinear oscillators , 2010, 1006.2404.

[88]  R. Vannix,et al.  Kinking of the internal carotid artery. Clinical significance and surgical management. , 1977, American journal of surgery.

[89]  Yanping Cao,et al.  From wrinkles to creases in elastomers: the instability and imperfection-sensitivity of wrinkling , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[90]  M. Ben Amar,et al.  Morphogenesis of growing soft tissues. , 2007, Physical review letters.

[91]  Xi Chen,et al.  Modeling and simulation of curled dry leaves , 2011 .

[92]  William M Gelbart,et al.  Elasticity theory and shape transitions of viral shells. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[93]  D A Weitz,et al.  Onset of buckling in drying droplets of colloidal suspensions. , 2005, Physical review letters.

[94]  K. Suh,et al.  Physical Self-Assembly of Microstructures by Anisotropic Buckling , 2002 .

[95]  R. Hayward,et al.  Creasing instability of surface-attached hydrogels. , 2008, Soft matter.

[96]  Zhigang Suo,et al.  Modeling and simulation of buckling of polymeric membrane thin film gel , 2010 .

[97]  B. Lin,et al.  Geometric tools for complex interfaces: from lung surfactant to the mussel byssus , 2009 .

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

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

[100]  S. Minko,et al.  Stimuli-responsive hydrogel thin films , 2009 .

[101]  Xi-Qiao Feng,et al.  Surface patterning of soft polymer film-coated cylinders via an electric field , 2009, Journal of physics. Condensed matter : an Institute of Physics journal.

[102]  L. Mahadevan,et al.  Unfolding the sulcus. , 2010, Physical review letters.

[103]  D. Ambrosi,et al.  On the mechanics of a growing tumor , 2002 .

[104]  Rui Huang,et al.  Swell-induced surface instability of confined hydrogel layers on substrates , 2010 .

[105]  K. Garikipati.  The Kinematics of Biological Growth , 2009 .

[106]  H. Herrmann,et al.  Self-contact and instabilities in the anisotropic growth of elastic membranes. , 2010, Physical review letters.

[107]  C. Ru.  Surface wrinkling of two mutually attracting elastic thin films due to van der Waals forces , 2001 .

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

[109]  Alain Goriely,et al.  Differential growth and residual stress in cylindrical elastic structures , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[110]  John A Rogers,et al.  Molecular scale buckling mechanics in individual aligned single-wall carbon nanotubes on elastomeric substrates. , 2008, Nano letters.

[111]  Antonio Siber.  Buckling transition in icosahedral shells subjected to volume conservation constraint and pressure: relations to virus maturation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[112]  J. Drazen,et al.  Mucosal folding in biologic vessels. , 2002, Journal of biomechanical engineering.

[113]  Willi Volksen,et al.  A buckling-based metrology for measuring the elastic moduli of polymeric thin films , 2004, Nature materials.

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

[115]  Alain Goriely,et al.  Instabilities in elastomers and in soft tissues , 2006, 0711.4664.

[116]  Shull,et al.  Fingering instabilities of confined elastic layers in tension , 2000, Physical review letters.

[117]  Carlos E. Castro,et al.  Growth strain-induced wrinkled membrane morphology of white blood cells , 2011 .

[118]  Arezki Boudaoud,et al.  An introduction to the mechanics of morphogenesis for plant biologists. , 2010, Trends in plant science.

[119]  J. Dunlop,et al.  A theoretical model for tissue growth in confined geometries , 2010 .

[120]  Xi-Qiao Feng,et al.  Growth and surface folding of esophageal mucosa: a biomechanical model. , 2011, Journal of biomechanics.

[121]  Cyprien Gay,et al.  Wrinkling hierarchy in constrained thin sheets from suspended graphene to curtains. , 2010, Physical review letters.

[122]  W. Klug,et al.  Mechanical modeling of viral capsids , 2007 .

[123]  Alain Goriely,et al.  Differential growth and instability in elastic shells. , 2005, Physical review letters.

[124]  L. Qiao,et al.  Pre-tension regulates buckling patterns of soft films with interactions , 2007 .

[125]  Pedro M Reis,et al.  Thin films: Folded in hierarchy. , 2011, Nature materials.

[126]  B. Davidovitch,et al.  Elastic building blocks for confined sheets. , 2010, Physical review letters.

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

[128]  Jie Yin,et al.  Anisotropic buckling patterns in spheroidal film/substrate systems and their implications in some natural and biological systems , 2009 .

[129]  P. Tracqui,et al.  Biophysical models of tumour growth , 2009 .

[130]  R Skalak,et al.  Kinematics of surface growth , 1997, Journal of mathematical biology.

[131]  Alfred J Crosby,et al.  Draping films: a wrinkle to fold transition. , 2010, Physical review letters.

[132]  Lei Jiang,et al.  Smart responsive surfaces switching reversibly between super-hydrophobicity and super-hydrophilicity , 2009 .

[133]  Jie Yin,et al.  Mechanical self-assembly fabrication of gears , 2009 .

[134]  M. Boyce,et al.  Swelling and Mechanical Stretching of Elastomeric Materials , 2001 .

[135]  M. Chaudhury,et al.  Soft and Hard Adhesion , 2005 .

[136]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[137]  A modeling study of the effect of environmental ionic valence on the mechanical characteristics of pH-electrosensitive hydrogel. , 2009, Acta biomaterialia.

[138]  Sean X. Sun,et al.  Morphology, growth, and size limit of bacterial cells. , 2010, Physical review letters.

[139]  S. Torquato,et al.  Pattern of self‐organization in tumour systems: complex growth dynamics in a novel brain tumour spheroid model , 2001, Cell proliferation.

[140]  H. Greenspan.  Models for the Growth of a Solid Tumor by Diffusion , 1972 .

[141]  R. Sutherland.  Cell and environment interactions in tumor microregions: the multicell spheroid model. , 1988, Science.

[142]  M. Chaudhury,et al.  Adhesion-Induced Instability Patterns in Thin Confined Elastic Film , 2003 .

[143]  Yan Lu,et al.  Thermosensitive core–shell microgels: From colloidal model systems to nanoreactors , 2011 .

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

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

[146]  Yonggang Huang,et al.  Biaxially stretchable "wavy" silicon nanomembranes. , 2007, Nano letters.

[147]  Xi-Qiao Feng,et al.  Three-dimensional analysis of spontaneous surface instability and pattern formation of thin soft films , 2008 .

[148]  L. Lyon,et al.  Soft nanotechnology with soft nanoparticles. , 2005, Angewandte Chemie.

[149]  Hai-Chao Han,et al.  Nonlinear buckling of blood vessels: a theoretical study. , 2008, Journal of biomechanics.

[150]  Rui Huang,et al.  Buckling modes of elastic thin films on elastic substrates , 2007 .

[151]  A. Boudaoud,et al.  The Buckling of a Swollen Thin Gel Layer Bound to a Compliant Substrate , 2007, cond-mat/0701640.

[152]  M. Aleksić,et al.  Surgical approach to kinking and coiling of the internal carotid artery. , 2004, The Journal of cardiovascular surgery.

[153]  Xi-Qiao Feng,et al.  Interfacial slippage effect on the surface instability of a thin elastic film under van der Waals force , 2009 .

[154]  Toyoichi Tanaka,et al.  Mechanical instability of gels at the phase transition , 1987, Nature.

[155]  S. Alben,et al.  Foldable structures and the natural design of pollen grains , 2010, Proceedings of the National Academy of Sciences.

[156]  Xi-Qiao Feng,et al.  Superior water repellency of water strider legs with hierarchical structures: experiments and analysis. , 2007, Langmuir : the ACS journal of surfaces and colloids.

[157]  Xi-Qiao Feng,et al.  Surface wrinkling and folding of core–shell soft cylinders , 2012 .

[158]  A. Imhof,et al.  Anisotropic colloids through non-trivial buckling , 2008, The European physical journal. E, Soft matter.

[159]  Shu Yang,et al.  Strain responsive concave and convex microlens arrays , 2007 .

[160]  D. Clemens,et al.  Lipid multilayered particles: the role of chitosan on structure and morphology , 2010 .

[161]  R. Hayward,et al.  Nucleation, growth, and hysteresis of surface creases on swelled polymer gels , 2010 .

[162]  O. Schmidt,et al.  Principles and applications of micro and nanoscale wrinkles , 2010 .

[163]  M. Chaudhury,et al.  Confinement-induced instability and adhesive failure between dissimilar thin elastic films , 2006, The European physical journal. E, Soft matter.

[164]  R. Kupferman,et al.  Geometry and Mechanics in the Opening of Chiral Seed Pods , 2011, Science.

[165]  B. Audoly.  Localized buckling of a floating elastica. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[166]  Alain Goriely,et al.  Circumferential buckling instability of a growing cylindrical tube , 2011 .

[167]  Brendan A O McCormack,et al.  A three-layer model of skin and its application in simulating wrinkling , 2009, Computer methods in biomechanics and biomedical engineering.

[168]  Huimin Xie,et al.  Buckling modes of polymer membranes restricted by metal wires , 2011 .

[169]  Stephan Herminghaus,et al.  Dynamical instability of thin liquid films between conducting media. , 1999 .

[170]  Pasquale Ciarletta,et al.  Morphogenesis of thin hyperelastic plates: A constitutive theory of biological growth in the Föppl-von Kármán limit , 2009 .

[171]  M. Biot.  Surface instability of rubber in compression , 1963 .

[172]  Eran Sharon,et al.  Geometrically driven wrinkling observed in free plastic sheets and leaves. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[173]  Haiyi Liang,et al.  Growth, geometry, and mechanics of a blooming lily , 2011, Proceedings of the National Academy of Sciences.

[174]  Yong-Wei Zhang,et al.  Surface instability maps for soft materials , 2010 .

[175]  Huajian Gao,et al.  Surface wrinkling patterns on a core-shell soft sphere. , 2011, Physical review letters.

[176]  Todd Emrick,et al.  Capillary Wrinkling of Floating Thin Polymer Films , 2007, Science.

[177]  Y. Pomeau,et al.  Surface instability of soft solids under strain , 2011 .

[178]  E. L. Starostin,et al.  Triangular buckling patterns of twisted inextensible strips , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[179]  J. Andersons,et al.  Buckling of a coating strip of finite width bonded to elastic half-space , 2008 .

[180]  Rui Huang,et al.  Viscoelastic properties of confined polymer films measured via thermal wrinkling , 2009 .

[181]  K. Lam,et al.  A chemo-electro-mechanical model for simulation of responsive deformation of glucose-sensitive hydrogels with the effect of enzyme catalysis , 2009 .

[182]  Gang Wang,et al.  Morphological instability of spherical soft particles induced by surface charges , 2009 .

[183]  C K Chong,et al.  Instability of the two-layered thick-walled esophageal model under the external pressure and circular outer boundary condition. , 2007, Journal of biomechanics.

[184]  D'arcy W. Thompson.  On Growth and Form , 1945 .

[185]  Jie Yin,et al.  Buckling patterns of thin films on curved compliant substrates with applications to morphogenesis and three-dimensional micro-fabrication , 2010 .

[186]  M. Ortiz,et al.  The morphology and folding patterns of buckling-driven thin-film blisters , 1994 .

[187]  Huajian Gao,et al.  Some general properties of stress-driven surface evolution in a heteroepitaxial thin film structure , 1994 .

[188]  D. C. Drucker,et al.  Mechanics of Incremental Deformation , 1965 .

[189]  J. Rogers,et al.  Stretchable Electronics: Materials Strategies and Devices , 2008 .

[190]  E. Sharon,et al.  Shaping of Elastic Sheets by Prescription of Non-Euclidean Metrics , 2007, Science.

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

[192]  Eliot Fried,et al.  A theory for species migration in a finitely strained solid with application to polymer network swelling , 2010 .

[193]  N. Johnson.  The Longitudinal Muscle in Esophageal Disease , 1997 .

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

[195]  Xi-Qiao Feng,et al.  Determination of the elastic modulus of micro- and nanowires/tubes using a buckling-based metrology , 2009 .

[196]  J. Burdick,et al.  Swelling‐Induced Surface Patterns in Hydrogels with Gradient Crosslinking Density , 2009 .

[197]  G. Gompper,et al.  Compression, crumpling and collapse of spherical shells and capsules , 2011 .

[198]  J. Dervaux,et al.  Buckling condensation in constrained growth , 2011 .

[199]  Mauro Ferrari,et al.  Morphologic Instability and Cancer Invasion , 2005, Clinical Cancer Research.

[200]  R D Kamm,et al.  On the mechanism of mucosal folding in normal and asthmatic airways. , 1997, Journal of applied physiology.

[201]  Hwa,et al.  Evolution of surface patterns on swelling gels. , 1988, Physical review letters.

[202]  Sharma,et al.  Pattern Formation in a Thin Solid Film with Interactions. , 2001, Physical review letters.

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

[204]  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 .

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

[206]  M. Boyce,et al.  Wrinkled surface topographies of electrospun polymer fibers , 2009 .

[207]  J. Kopeček.  Hydrogel biomaterials: a smart future? , 2007, Biomaterials.

[208]  C. Ru,et al.  Surface instability of a bilayer elastic film due to surface van der Waals forces , 2005 .

[209]  Michael P Marder,et al.  Crumpling, buckling, and cracking: Elasticity of thin sheets , 2007 .

[210]  L. Mahadevan,et al.  The shape of a long leaf , 2009, Proceedings of the National Academy of Sciences.

[211]  Z. Suo,et al.  A theory of coupled diffusion and large deformation in polymeric gels , 2008 .

[212]  L. Mahadevan,et al.  Localized and extended deformations of elastic shells , 2008, Proceedings of the National Academy of Sciences.

[213]  Alfred J. Crosby,et al.  Effect of stress state on wrinkle morphology , 2011 .

[214]  Fu-Shu Jeng,et al.  Buckling folds of a single layer embedded in matrix - Theoretical solutions and characteristics , 2008 .

[215]  P. Lenz,et al.  Mechanical limits of viral capsids , 2007, Proceedings of the National Academy of Sciences.

[216]  A. Rachev.  A theoretical study of mechanical stability of arteries. , 2009, Journal of biomechanical engineering.

[217]  J. Hogg,et al.  The pathology of asthma , 1997, APMIS : acta pathologica, microbiologica, et immunologica Scandinavica.

[218]  D'arcy W. Thompson.  On growth and form i , 1943 .

[219]  J. Rogers,et al.  Finite width effect of thin-films buckling on compliant substrate : Experimental and theoretical studies , 2008 .

[220]  Zhigang Suo,et al.  Osmotic collapse of a void in an elastomer: breathing, buckling and creasing , 2010 .

[221]  Xi-Qiao Feng,et al.  Determining the elastic modulus of thin films using a buckling-based method: computational study , 2009 .

[222]  B. Audoly,et al.  Buckling of a stiff film bound to a compliant substrate—Part II:: A global scenario for the formation of herringbone pattern , 2008 .

[223]  David R Nelson,et al.  Virus shapes and buckling transitions in spherical shells. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[224]  K. Pinder,et al.  Mechanical properties of the tracheal mucosal membrane in the rabbit. I. steady-state stiffness as a function of age. , 2000, Journal of applied physiology.