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CN 34-1054/N

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Morphology of non-Euclidean disks

  • Department of Modern Mechanics, CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei 230027, China

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https://doi.org/10.3969/j.issn.0253-2778.2015.09.009
  • Received Date: 31 March 2015
  • Accepted Date: 29 May 2015
  • Rev Recd Date: 29 May 2015
  • Publish Date: 30 September 2015
    Abstract

  • Abstract

    The out-of-plane displacements of plates are influenced by factors such as nonuniform thermal expansion which corresponds the research of 3D deformation of elastomer controlled by the target metric in non-Euclidean plates. The well-known mass-spring model was used to simulate the deformation of disks driven by given swelling functions corresponding positive and negative target Gaussian curvature K. The evolution of waves under negative target Gaussian curvature was analyzed with different thickness t. The impact of swelling factor and strain gradient on the ratio of bending energy to strain energy was studied with the same thicknesses. The ratio transforms rapidly when the number of waves changes for K<0. Scaling of stretching and bending energy about thickness was obtained while dicks were driven by the same swelling function. Stretching and bending energy both vary like t2.5 with K>0, compared with t4 of bending energy in disks with K<0.

    Abstract

    The out-of-plane displacements of plates are influenced by factors such as nonuniform thermal expansion which corresponds the research of 3D deformation of elastomer controlled by the target metric in non-Euclidean plates. The well-known mass-spring model was used to simulate the deformation of disks driven by given swelling functions corresponding positive and negative target Gaussian curvature K. The evolution of waves under negative target Gaussian curvature was analyzed with different thickness t. The impact of swelling factor and strain gradient on the ratio of bending energy to strain energy was studied with the same thicknesses. The ratio transforms rapidly when the number of waves changes for K<0. Scaling of stretching and bending energy about thickness was obtained while dicks were driven by the same swelling function. Stretching and bending energy both vary like t2.5 with K>0, compared with t4 of bending energy in disks with K<0.
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    • References(12)
  • [1]
    Liang H Y, Mahadevan L. Growth, geometry, and mechanics of a blooming lily[J]. Proceedings of the National Academy of Sciences, 2011, 108(14): 5 516-5 521.
    [2]
    Sharon E, Efrati E. The mechanics of non-Euclidean plates[J]. Soft Matter, 2010, 6(22): 5 693-5 704.
    [3]
    Klein Y, Venkataramani S, Sharon E. Experimental study of shape transitions and energy scaling in thin non-Euclidean plates[J]. Physical Review Letters, 2011, 106(11): 118303.
    [4]
    Efrati E, Sharon E, Kupferman R. Non-Euclidean plates and shells[EB/OL].[2015-01-01] http://math.huji.ac.il/~razk/Publications/PDF/ESK09b.pdf.
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    Kim J, Hanna J A, Byun M, et al. Designing responsive buckled surfaces by halftone gel lithography[J]. Science, 2012, 335(6 073): 1 201-1 205.
    [6]
    Byun M, Santangelo C D, Hayward R C. Swelling-driven rolling and anisotropic expansion of striped gel sheets[J]. Soft Matter, 2013,9:8 264-8 273.
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    Kirchhoff G. ber das Gleichgewicht und die Bewegung einer elastischen Scheibe[J]. Journal für die reine und angewandte Mathematik, 1850, 40: 51-88.
    [8]
    Truesdell C. The Mechanical Foundations of Elasticity and Fluid Dynamics[M]. New York: Gordon & Breach Science Pub, 1966.
    [9]
    Oneill B. Elementary Differential Geometry[M]. New York: Academic Press, 1966.
    [10]
    Spivak M. A Comprehensive Introduction to Differential Geometry, Volume Ⅰ[M]. Berkeley: Publish or Perish, 1979.
    [11]
    Efrati E, Sharon E, Kupferman R. Buckling transition and boundary layer in non-Euclidean plates[J]. Physical Review E, 2009, 80(1): 016602.
    [12]
    Li J, Liu M, Xu W, et al. Boundary-dominant flower blooming simulation[J]. Computer Animation and Virtual Worlds, 2015, 26: 433-443.
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    [1]
    Liang H Y, Mahadevan L. Growth, geometry, and mechanics of a blooming lily[J]. Proceedings of the National Academy of Sciences, 2011, 108(14): 5 516-5 521.
    [2]
    Sharon E, Efrati E. The mechanics of non-Euclidean plates[J]. Soft Matter, 2010, 6(22): 5 693-5 704.
    [3]
    Klein Y, Venkataramani S, Sharon E. Experimental study of shape transitions and energy scaling in thin non-Euclidean plates[J]. Physical Review Letters, 2011, 106(11): 118303.
    [4]
    Efrati E, Sharon E, Kupferman R. Non-Euclidean plates and shells[EB/OL].[2015-01-01] http://math.huji.ac.il/~razk/Publications/PDF/ESK09b.pdf.
    [5]
    Kim J, Hanna J A, Byun M, et al. Designing responsive buckled surfaces by halftone gel lithography[J]. Science, 2012, 335(6 073): 1 201-1 205.
    [6]
    Byun M, Santangelo C D, Hayward R C. Swelling-driven rolling and anisotropic expansion of striped gel sheets[J]. Soft Matter, 2013,9:8 264-8 273.
    [7]
    Kirchhoff G. ber das Gleichgewicht und die Bewegung einer elastischen Scheibe[J]. Journal für die reine und angewandte Mathematik, 1850, 40: 51-88.
    [8]
    Truesdell C. The Mechanical Foundations of Elasticity and Fluid Dynamics[M]. New York: Gordon & Breach Science Pub, 1966.
    [9]
    Oneill B. Elementary Differential Geometry[M]. New York: Academic Press, 1966.
    [10]
    Spivak M. A Comprehensive Introduction to Differential Geometry, Volume Ⅰ[M]. Berkeley: Publish or Perish, 1979.
    [11]
    Efrati E, Sharon E, Kupferman R. Buckling transition and boundary layer in non-Euclidean plates[J]. Physical Review E, 2009, 80(1): 016602.
    [12]
    Li J, Liu M, Xu W, et al. Boundary-dominant flower blooming simulation[J]. Computer Animation and Virtual Worlds, 2015, 26: 433-443.
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