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Equilibria of a clamped Euler beam (Elastica) with distributed load: large deformations. (English) Zbl 1368.74031

Summary: We present some novel equilibrium shapes of a clamped Euler beam (Elastica from now on) under uniformly distributed dead load orthogonal to the straight reference configuration. We characterize the properties of the minimizers of total energy, determine the corresponding Euler-Lagrange conditions and prove, by means of direct methods of calculus of variations, the existence of curled local minimizers. Moreover, we prove some sufficient conditions for stability and instability of solutions of the Euler-Lagrange, that can be applied to numerically found curled shapes.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74B20 Nonlinear elasticity
74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010)
74G35 Multiplicity of solutions of equilibrium problems in solid mechanics
74G55 Qualitative behavior of solutions of equilibrium problems in solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
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[1] Abali, B. E., Müller, W. H. and Eremeyev, V. A., Strain gradient elasticity with geometric nonlinearities and its computational evaluation, Mech. Adv. Mater. Mod. Process.1 (2015).
[2] Alibert, J. J., Seppecher, P. and dell’Isola, F., Truss modular beams with deformation energy depending on higher displacement gradients, Math. Mech. Solids8 (2003) 51-73. · Zbl 1039.74028
[3] Antman, S. S., Nonlinear Problems of Elasticity (Springer, 1995). · Zbl 0820.73002
[4] Bernoulli, D., The 26th letter to Euler, in Correspondence Mathmatique et Physique, Vol. 2 (P. H. Fuss, 1742).
[5] Bernoulli, J., Quadratura curvae, e cujus evolutione describitur inflexae laminae curvatura, in Die Werke von Jakob Bernoulli (Birkhäuser, 1692), pp. 223-227.
[6] Bisshopp, K. E. and Drucker, D. C., Large deflection of cantilever beams, Quart. Appl. Math.3 (1945) 272-275. · Zbl 0063.00418
[7] M. Born, Untersuchungen uber die Stabilitat der elastischen Linie in Ebene und Raum, under verschiedenen Grenzbedingungen, Ph.D. thesis, University of Göttingen (1906). · JFM 38.0984.03
[8] Dacorogna, B., Direct Methods in the Calculus of Variations, Vol. 78 (Springer Science & Business Media, 2007). · Zbl 0703.49001
[9] dell’Isola, F., Giorgio, I., Pawlikowski, M. and Rizzi, N. L., Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenization, experimental and numerical examples of equilibrium, Proc. Roy. Soc. A472 (2016) 20150790.
[10] dell’Isola, F., Lekszycki, T., Pawlikowski, M., Grygoruk, R. and Greco, L., Designing a light fabric metamaterial being highly macroscopically tough under directional extension: First experimental evidence, Z. Math. Phys.66 (2015) 3473-3498. · Zbl 1395.74002
[11] dell’Isola, F., Steigmann, D. and Della Corte, A., Synthesis of fibrous complex structures: Designing microstructure to deliver targeted macroscale response, Appl. Mech. Rev.67 (2016) 21 pp.
[12] L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti (Marc-Michel Bousquet et Soc., 1744). Additamentum 1. · Zbl 0788.01072
[13] Evans, L. C., Partial Differential Equations (Amer. Math. Soc., 2010). · Zbl 1194.35001
[14] Faulkner, M. G., Lipsett, A. W. and Tam, V., On the use of a segmental shooting technique for multiple solutions of planar elastica problems, Comput. Meth. Appl. Mech. Eng.110 (1993) 221-236. · Zbl 0847.73073
[15] Fertis, D. G., Nonlinear Structural Engineering (Springer-Verlag, 2006). · Zbl 1102.74002
[16] Fonseca, I. and Leoni, G., Modern Methods in the Calculus of Variations: \(L^p\) Spaces (Springer Science & Business Media, 2007). · Zbl 1153.49001
[17] Forest, S. and Sievert, R., Nonlinear microstrain theories, Int. J. Solids Struct.43 (2006) 7224-7245. · Zbl 1102.74003
[18] Fried, I., Stability and equilibrium of the straight and curved elastica-finite element computation, Comput. Meth. Appl. Mech. Eng.28 (1981) 49-61. · Zbl 0466.73089
[19] Germain, P., The method of virtual power in continuum mechanics. Part 2: Microstructure, SIAM J. Appl. Math.25 (1973) 556-575. · Zbl 0273.73061
[20] Kudrila, A. J. and Zabarankin, M., Convex Functional Analysis (Springer Science & Business Media, 2006).
[21] Ladevèze, P., Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation (Springer Science & Business Media, 2012).
[22] Lagrange, J. L., Mécanique Analytique, Vols. 1-2 (Mallet-Bachelier, 1853).
[23] Milton, G. W., Adaptable nonlinear bimode metamaterials using rigid bars, pivots, and actuators, J. Mech. Phys. Solids61 (2013) 1561-1568.
[24] Mindlin, R. D. and Eshel, N. N., On first strain-gradient theories in linear elasticity, Int. J. Solids Struct.4 (1968) 109-124. · Zbl 0166.20601
[25] Mora, M. G. and Müller, S., A nonlinear model for inextensible rods as a low energy \(\Gamma \)-limit of three-dimensional nonlinear elasticity, Ann. l’IHP Anal. Nonlinaire21 (2004) 271-293. · Zbl 1109.74028
[26] Pideri, C. and Seppecher, P., A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium, Contin. Mech. Thermodynam.9 (1997) 241-257. · Zbl 0893.73006
[27] Pideri, C. and Seppecher, P., Asymptotics of a non-planar rod in nonlinear elasticity, Asympt. Anal.48 (2006) 33-54. · Zbl 1157.74314
[28] Pipkin, A. C., Plane traction problems for inextensible networks, Quart. J. Mech. Appl. Math.34 (1981) 415-429. · Zbl 0466.73037
[29] Raboud, D. W., Faulkner, M. G. and Lipsett, A. W., Multiple three-dimensional equilibrium solutions for cantilever beams loaded by dead tip and uniform distributed loads, Int. J. Nonlinear Mech.31 (1996) 297-311. · Zbl 0862.73029
[30] Rivlin, R. S., Networks of inextensible cords, in Collected Papers of R. S. Rivlin (Springer, 1997), pp. 566-579.
[31] Scerrato, D., Giorgio, I. and Rizzi, N. L., Three-dimensional instabilities of pantographic sheets with parabolic lattices: Numerical investigations, Z. Math. Phys.67 (2016) 1-19. · Zbl 1464.74028
[32] Sullivan, D., Combinatorial invariants of analytic spaces, in Proc. Liverpool Singularities Symp. I (Springer, 1971), p. 165. · Zbl 0227.32005
[33] Tonelli, L., Opere scelte: Calcolo delle variazioni, Vol. 2 (Edizioni Cremonese, 1961). · Zbl 0131.24103
[34] Turco, E., dell’Isola, F., Cazzani, A. and Rizzi, N. L., Hencky-type discrete model for pantographic structures: Numerical comparison with second gradient continuum models, Z. Math. Phys.67 (2016) 1-28. · Zbl 1432.74158
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