Elastic rods with incompatible strain: Macroscopic versus microscopic buckling

We consider the buckling of a long prismatic elastic solid under the combined effect of a pre-stress that is inhomogeneous in the cross-section, and of a prescribed displacement of its endpoints. A linear bifurcation analysis is carried out using different structural models (namely a double beam, a rectangular thin plate, and a hyper-elastic prismatic solid in 3-d): it yields the buckling mode and the wavenumber qc that are first encountered when the end-to-end displacement is progressively decreased with fixed pre-stress. For all three structural models, we find a transition from a long-wavelength (qc=0qc=0) to a short-wavelength first buckling mode (qc ≠ 0) when the inhomogeneous pre-stress is increased past a critical value. A method for calculating the critical inhomogeneous pre-stress is proposed based on a small-wavenumber expansion of the buckling mode. Overall, our findings explain the formation of multiple perversions in elastomer strips, as well as the large variations in the number of perversions as a function of pre-stress and cross-sectional geometry, as reported by Liu et al. (2014).

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

[2]  Hamid Zahrouni,et al.  Post-buckling modeling for strips under tension and residual stresses using asymptotic numerical method , 2015 .

[3]  A. M. A. Heijden.  W. T. Koiter-s Elastic Stability of Solids and Structures , 2012 .

[4]  M. Cicalese,et al.  On global and local minimizers of prestrained thin elastic rods , 2016, 1606.04524.

[5]  Martin Golubitsky,et al.  Boundary conditions and mode jumping in the buckling of a rectangular plate , 1979 .

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

[7]  H. Aben,et al.  Photoelasticity of Glass , 1993 .

[8]  David R. Clarke,et al.  Spontaneous and deterministic three-dimensional curling of pre-strained elastomeric bi-strips , 2012 .

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

[10]  G. M.,et al.  A Treatise on the Mathematical Theory of Elasticity , 1906, Nature.

[11]  M D Thouless,et al.  Surface instability of an elastic half space with material properties varying with depth. , 2008, Journal of the mechanics and physics of solids.

[12]  Giles W Hunt,et al.  Mode jumping in the buckling of struts and plates: a comparative study , 2000 .

[13]  N. Triantafyllidis,et al.  ASYMPTOTIC ANALYSIS OF STABILITY FOR PRISMATIC SOLIDS UNDER AXIAL LOADS , 1998 .

[14]  Kaushik Bhattacharya,et al.  Plates with Incompatible Prestrain , 2014, 1401.1609.

[15]  S. Timoshenko.  Theory of Elastic Stability , 1936 .

[16]  David R. Clarke,et al.  Structural Transition from Helices to Hemihelices , 2014, PloS one.

[17]  Stanislav Y Shvartsman,et al.  Three-dimensional epithelial morphogenesis in the developing Drosophila egg. , 2013, Developmental cell.

[18]  A. Raoult,et al.  The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity , 1995 .

[19]  A. Goriely,et al.  Morphoelastic rods Part II: Growing birods , 2017 .

[20]  Kazutake Komori,et al.  Analysis of cross and vertical buckling in sheet metal rolling , 1998 .

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

[22]  Vicente Hernández,et al.  SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems , 2005, TOMS.

[23]  G. Friesecke,et al.  Mathematik in den Naturwissenschaften Leipzig A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence , 2005 .

[24]  M. Ahmer Wadee,et al.  Mode interaction in thin-walled I-section struts with semi-rigid flange-web joints , 2015 .

[25]  Kékéli Kpogan,et al.  Simulation numérique de la planéité des tôles métalliques formées par laminage , 2014 .

[26]  Anders Logg,et al.  Automated Code Generation for Discontinuous Galerkin Methods , 2008, SIAM J. Sci. Comput..

[27]  Timothy J. Healey,et al.  Multiple Helical Perversions of Finite, Intristically Curved Rods , 2005, Int. J. Bifurc. Chaos.

[28]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[29]  W. Wieser,et al.  Buckling phenomena related to rolling and levelling of sheet metal , 2000 .

[30]  L. Mahadevan,et al.  The Föppl-von Kármán equations for plates with incompatible strains , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[31]  C.-S. Chien,et al.  Mode Jumping In The Von Kármán Equations , 2000, SIAM J. Sci. Comput..

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

[33]  Franz G. Rammerstorfer,et al.  Buckling of Free Infinite Strips Under Residual Stresses and Global Tension , 2001 .

[34]  Alain Goriely,et al.  Tendril Perversion in Intrinsically Curved Rods , 2002, J. Nonlinear Sci..

[35]  M. Marder,et al.  Theory of edges of leaves , 2003 .

[36]  H. Stone,et al.  Directed assembly of fluidic networks by buckle delamination of films on patterned substrates , 2007 .

[37]  Kevin Chiou,et al.  Emergent perversions in the buckling of heterogeneous elastic strips , 2016, Proceedings of the National Academy of Sciences.

[38]  Y. Tomita,et al.  BUCKLING BEHAVIOR IN THIN SHEET METAL SUBJECTED TO NONUNIFORM MEMBRANE-TYPE DEFORMATION , 1993 .

[39]  M. Tabor,et al.  Spontaneous Helix Hand Reversal and Tendril Perversion in Climbing Plants , 1998 .

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

[41]  Karl Meerbergen,et al.  The Quadratic Eigenvalue Problem , 2001, SIAM Rev..

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