×

zbMATH — the first resource for mathematics

Post-buckling solutions of hyper-elastic beam by canonical dual finite element method. (English) Zbl 1298.74085
Summary: The post-buckling problem of a large deformed beam is analyzed using the canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional space. Based on the canonical duality theory and the associated triality theorem, a primal-dual algorithm is proposed, which can be used to find all possible solutions of this non-convex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam, while the local extrema of the pure-complementary energy present unstable deformation states. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified through numerical examples and some interesting phenomena in post-bifurcation of this large deformed beam are observed.

MSC:
74G60 Bifurcation and buckling
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Washizu K, Variational methods in elasticity and plasticity (1968)
[2] Gao DY, Mech Res Commun 23 pp 11– (1996) · Zbl 0843.73042 · doi:10.1016/0093-6413(95)00071-2
[3] Gao DY, Appl Mech Rev 50 pp S64– (1997) · doi:10.1115/1.3101852
[4] Gao DY, Int J Non Linear Mech 35 pp 103– (2000) · Zbl 1068.74569 · doi:10.1016/S0020-7462(98)00091-2
[5] Bengue FM, Electron J Differ Equ 27 pp 1– (2008)
[6] Machalova J, Appl Comput Mech 5 pp 45– (2011)
[7] Andrews KT, Z Angew Math Phys 63 pp 1005– (2012) · Zbl 1261.35093 · doi:10.1007/s00033-012-0233-9
[8] Kuttler KL, Q J Mech Appl Math 65 pp 1– (2012) · Zbl 1248.74032 · doi:10.1093/qjmam/hbr018
[9] Ahn J, Electron J Differ Equ 194 pp 1– (2012)
[10] Gao DY, Z Angew Math Phys 59 pp 498– (2008) · Zbl 1143.74018 · doi:10.1007/s00033-007-7047-1
[11] Gao DY, Advances in applied mathematics and global optimization pp 249– (2009)
[12] Levinson M, ASME Trans J Appl Mech 32 pp 826– (1965) · doi:10.1115/1.3627322
[13] Fraeijs de Veubeke B, Int J Eng Sci 10 pp 745– (1972) · Zbl 0245.73031 · doi:10.1016/0020-7225(72)90079-1
[14] Koiter WT, Trends in applications of pure mathematics to mechanics pp 207– (1975)
[15] Ogden RW, Math Proc Cambridge Philos Soc 77 pp 609– (1975) · Zbl 0312.73052 · doi:10.1017/S0305004100051422
[16] Guo ZH, Arch Mech 32 pp 577– (1980)
[17] Gao DY, Q J Mech Appl Math pp 487– (1989)
[18] Gao DY, Mech Res Commun 26 pp 31– (1999) · Zbl 0992.74013 · doi:10.1016/S0093-6413(98)00096-2
[19] Gao DY, Meccanica 34 pp 169– (1999)
[20] Li SF, J Elast 84 pp 13– (2006) · Zbl 1103.74013 · doi:10.1007/s10659-005-9047-8
[21] Gao DY, Duality principles in nonconvex systems: theory, methods and applications (2000) · doi:10.1007/978-1-4757-3176-7
[22] DOI: 10.2514/3.2546 · doi:10.2514/3.2546
[23] Pian THH, Int J Numer. Methods Eng 12 pp 891– (1978) · doi:10.1002/nme.1620120514
[24] Pian THH, Hybrid and incompatible finite element methods (2006) · Zbl 1110.65003
[25] Hodge PG, J Appl Mech 35 pp 796– (1968) · Zbl 0172.52003 · doi:10.1115/1.3601308
[26] Belytschko T, J Eng Mech Div ASCE 96 pp 931– (1970)
[27] Glaum L, Int J Solids Struct 11 pp 1023– (1975) · Zbl 0315.73064 · doi:10.1016/0020-7683(75)90045-1
[28] Gao DY, Comput Struct 28 pp 749– (1988) · Zbl 0632.73031 · doi:10.1016/0045-7949(88)90415-4
[29] Qin QH, Mech Res Commun 31 pp 321– (2004) · Zbl 1079.74648 · doi:10.1016/j.mechrescom.2003.11.003
[30] Qin QH, Wang H. Matlab and C programming for Trefftz finite element methods. Boca Raton, FL: Taylor & Francis, 2009.
[31] Gao DY, Int J Solids Struct 45 pp 3660– (2008) · Zbl 1169.74518 · doi:10.1016/j.ijsolstr.2007.08.027
[32] Santos HAFA, Int J Non Linear Mech 47 pp 240– (2012) · doi:10.1016/j.ijnonlinmec.2011.05.012
[33] Gao DY, J Ind Manage Optim 8 pp 229– (2012) · Zbl 1364.90271 · doi:10.3934/jimo.2012.8.229
[34] Gao DY, Q J Mech Appl Math 61 pp 497– (2008) · Zbl 1153.74032 · doi:10.1093/qjmam/hbn014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.