Open Access
Volume 30, 2024
Article Number 50
Number of page(s) 26
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. (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]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.