Open Access
Issue
ESAIM: COCV
Volume 30, 2024
Article Number 50
Number of page(s) 26
DOI https://doi.org/10.1051/cocv/2024039
Published online 02 July 2024
  1. I. Ecsedi, Bounds for the effective shear modulus. Eng. Trans. 53 (2005) 415–423. [Google Scholar]
  2. Y. Kim and T. Kim, Topology optimization of beam cross sections. Int. J. Solids Struct. 37 (2000) 477–493. [CrossRef] [Google Scholar]
  3. K. Niklas, Plant Biomechanics: An Engineering Approach to Plant Form and Function. University of Chicago Press (1992). [Google Scholar]
  4. S. Vogel, Twist-to-bend ratios and cross-sectional shapes of petioles and stems. J. Exp. Bot. 43 (1992) 1527–1532. [CrossRef] [Google Scholar]
  5. S. Vogel, Living in a physical world XI. To twist or bend when stressed. J. Biosci. 32 (2007) 643–655. [CrossRef] [PubMed] [Google Scholar]
  6. S. Wolff-Vorbeck, O. Speck, M. Langer, T. Speck and P.W. Dondl, Charting the twist-to-bend ratio of plant axes. J. R. Soc. Interface 19 (2022) 20220131. [CrossRef] [Google Scholar]
  7. S. Wolff-Vorbeck, M. Langer, O. Speck, T. Speck and P.W. Dondl, Twist-to-bend ratio: an important selective factor for many rod-shaped biological structures. Sci. Rep. 9 (2019) 17182. [Google Scholar]
  8. S.S. Antman, The theory of rods, in Linear Theories of Elasticity and Thermoelasticity: Linear and Nonlinear Theories of Rods, Plates, and Shells, edited by C. Truesdell, editor. Springer, Berlin, Heidelberg (1973) 641–703. [CrossRef] [Google Scholar]
  9. S.S. Antman, Nonlinear Problems of Elasticity. Vol. 107 of Applied Mathematical Sciences, 2nd edn. Springer, New York (2005). [Google Scholar]
  10. G. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen scheibe. J. Reine Angew. Math. 1850 (1850) 51–88. [CrossRef] [Google Scholar]
  11. P. Villaggio, Mathematical Models for Elastic Structures. Cambridge University Press, Cambridge (1997). [CrossRef] [Google Scholar]
  12. M.G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by Γ-convergence. Calc. Var. Part. Diff. Equ. 18 (2003) 287–305. [CrossRef] [Google Scholar]
  13. E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 58 (1975) 842–850. [MathSciNet] [Google Scholar]
  14. E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string. J. Elasticity 25 (1991) 137–148. [CrossRef] [MathSciNet] [Google Scholar]
  15. S. Neukamm, Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity. Arch. Ration. Mech. Anal. 206 (2012) 645–706. [CrossRef] [MathSciNet] [Google Scholar]
  16. R.R. Archer, N.H. Cook, S.H. Crandall, N.C. Dahl, T.J. Lardner, F.A. McClintock, E. Rabinowicz and G.S. Reichenbach, An Introduction to the Mechanics of Solids, 2nd edn. Engineering Mechanics series. McGraw-Hill (1978). [Google Scholar]
  17. S.G. Lekhnitskii, Torsion of anisotropic and nonhomogeneous beams. Izd. Nauka, Fiz-Mat. Literaturi, Moscow (1971). [Google Scholar]
  18. M.H. Sadd, Elasticity: Theory, Applications, and Numerics, 4th edn. Academic Press (2020). [Google Scholar]
  19. G. Allaire and G. Francfort, A numerical algorithm for topology and shape optimization, in Topology Design of Structures (Sesimbra, 1992). Vol. 227 of NATO Adv. Sci. Inst. Ser. E: Appl. Sci.. Kluwer Acad. Publ., Dordrecht (1993) 239–248. [Google Scholar]
  20. M.P. Bendsøe, Optimization of Structural topology, shape, and material. Springer-Verlag, Berlin (1995). [CrossRef] [Google Scholar]
  21. R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. I, II, III. Commun. Pure Appl. Math. 39 (1986) 113–137, 139–182, 353–377. [CrossRef] [Google Scholar]
  22. A.G.M. Michell, LVIII. The limits of economy of material in frame-structures. Philos. Mag. 8 (1904) 589–597. [CrossRef] [Google Scholar]
  23. J. Thomsen, Topology optimization of structures composed of one or two materials. Struct. Optim. 5 (1992) 108–115. [Google Scholar]
  24. L. Blank, H. Garcke, C. Hecht and C. Rupprecht, Sharp interface limit for a phase field model in structural optimization. SIAM J. Control Optim. 54 (2016) 1558–1584. [CrossRef] [MathSciNet] [Google Scholar]
  25. B. Bourdin and A. Chambolle, Design-dependent loads in topology optimization. ESAIM Control Optim. Calc. Var. 9 (2003) 19–48. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  26. A. Takezawa, S. Nishiwaki and M. Kitamura, Shape and topology optimization based on the phase field method and sensitivity analysis. J. Comput. Phys. 229 (2010) 2697–2718. [CrossRef] [MathSciNet] [Google Scholar]
  27. N. Rowe, S. Isnard and T. Speck, Diversity of mechanical architectures in climbing plants: an evolutionary perspective. J. Plant Growth Regul. 23 (2004) 108–128. [CrossRef] [Google Scholar]
  28. S. Wolff-Vorbeck, O. Speck, T. Speck and P.W. Dondl, Influence of structural reinforcements on the twist-to-bend ratio of plant axes: a case study on Carex pendula. Sci. Rep. 11 (2021) 21232. [Google Scholar]
  29. L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123–142. [CrossRef] [MathSciNet] [Google Scholar]
  30. G. Dal Maso, An Introduction to Γ-convergence. Vol. 8 of Progress in Nonlinear Differential Equations and their Applications. Birkhüauser Boston, Inc., Boston, MA (1993). [Google Scholar]
  31. G. Friesecke, R.D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55 (2002) 1461–1506. [Google Scholar]
  32. S. Wolff-Vorbeck, Optimization and Uncertainty Quantification for Geometric Structures. PhD thesis, University of Freiburg, Germany (2023). [Google Scholar]
  33. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000). [CrossRef] [Google Scholar]
  34. P. Hüttl, P. Knopf and T. Laux, A phase-field version of the Faber–Krahn theorem. Interfaces Free Bound. (2024) in press. [Google Scholar]
  35. P. Sternberg, The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101 (1988) 209–260. [CrossRef] [Google Scholar]
  36. N.C. Owen, J. Rubinstein and P. Sternberg, Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition. Proc. Roy. Soc. London Ser. A 429 (1990) 505–532. [CrossRef] [MathSciNet] [Google Scholar]
  37. H. Garcke, P. Hüttl, C. Kahle, P. Knopf and T. Laux, Phase-field methods for spectral shape and topology optimization. ESAIM Control Optim. Calc. Var. 29 (2023) 57. [CrossRef] [EDP Sciences] [Google Scholar]
  38. S. Wolff-Vorbeck, C++ code for Multi-material shape optimization for bending and torsion of rods. https://zenodo.org/records/10615223 (2023). [Google Scholar]
  39. M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints. Vol. 23 of Mathematical Modelling: Theory and Applications. Springer, New York (2009). [Google Scholar]
  40. O. Speck, F. Steinhart and T. Speck, Peak values of twist-to-bend ratio in triangular flower stalks of Carex pendula: a study on biomechanics and functional morphology. Am. J. Bot. 107 (2020) 1588–1596. [CrossRef] [PubMed] [Google Scholar]

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