Abstract
The strength of structures undergoing bifurcation is affected by imperfections. The mechanism of such a dependence of the strength has been conventionally described by “imperfection sensitivity laws.” We follow the Liapunov–Schmidt–Koiter approach and introduce imperfection sensitivity laws for simple critical points of various kinds that are described in Chap. 2, which is a prerequisite of this chapter. This chapter lays a foundation of Chaps. 4–6 and is extended to a system with group symmetry in Chap. 9.
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Notes
- 1.
A direct method to compute the relation (3.2) is to solve simultaneously the extended system consisting of the nonlinear governing equation (3.1) and the criticality condition \(\det \left (\partial \boldsymbol {F}/\partial \boldsymbol {u}\right )=0 \). For the analysis of the extended system, see, for example, Seydel, 1979, [173, 174]; Werner and Spence, 1984 [197]; and Wriggers and Simo, 1990 [201].
- 2.
The imperfection sensitivity of simple structures was observed experimentally by Roorda, 1965 [163]. Thompson and Hunt, 1973 [181] formulated the imperfection sensitivity law of a system with a single imperfection parameter through the perturbation to the total potential energy function of the system. Hunt, 1977 [70] combined this approach with catastrophe theory to determine imperfection sensitivity. See also textbooks by Godoy, 2000 [54] and Ohsaki and Ikeda, 2007 [148].
- 3.
A nearly coincidental pair of points was observed for a stressed atomic crystal lattice (Thompson and Schorrock, 1975 [183]; and Ikeda, Providéncia, and Hunt, 1993 [96]) and for steel specimens (Needleman, 1972 [141]; and Hutchinson and Miles, 1974 [72]). A more account of hilltop bifurcation points, which are parametric critical points, can be found in Ikeda, Oide, and Terada, 2002 [94]; Ikeda, Ohsaki, and Kanno, 2005 [93]; and Ohsaki and Ikeda, 2006 [147], 2007 [148].
- 4.
This analysis is based on Okazawa et al., 2002 [150].
References
Godoy, L.A. (2000) Theory of Elastic Stability—Analysis and Sensitivity. Taylor & Francis, Philadelphia.
Hunt, G.W. (1977) Imperfection-sensitivity of semi-symmetric branching. Proc. Roy. Soc. London Ser. A 357, 193–211.
Hutchinson, J.W., Miles, J.P. (1974) Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension. J. Mech. Phys. Solids 22, 61–71.
Ikeda, K., Kitada, T., Matsumura, M., Yamakawa Y. (2007) Imperfection sensitivity of ultimate buckling strength of elastic–plastic square plates under compression. Internat. J. Nonlinear Mechanics 42, 529–541.
Ikeda, K., Murota, K. (1990) Critical initial imperfection of structures. Internat. J. Solids Structures 26 (8), 865–886.
Ikeda, K., Ohsaki, M., Kanno, Y. (2005) Imperfection sensitivity of hilltop branching points of systems with dihedral group symmetry. Internat. J. Nonlinear Mechanics 40 (5), 755–774.
Ikeda, K., Oide, K., Terada, K. (2002) Imperfection sensitive strength variation at hilltop bifurcation point. Internat. J. Engrg. Sci. 40 (7), 743–772.
Ikeda, K., Providéncia, P., Hunt, G.W. (1993) Multiple equilibria for unlinked and weakly-linked cellular forms. Internat. J. Solids Structures 30 (3), 371–384.
Koiter, W.T. (1945) On the Stability of Elastic Equilibrium. Dissertation. Delft, Holland (English translation: NASA Technical Translation F10: 833, 1967).
Needleman, A. (1972) A numerical study of necking in circular cylindrical bars. J. Mech. Phys. Solids 20, 111–127.
Ohsaki, M., Ikeda, K. (2006) Imperfection sensitivity analysis of hill-top branching with many symmetric bifurcation points. Internat. J. Solids Structures 43 (16), 4704–4719.
Ohsaki, M., Ikeda, K. (2007) Stability and Optimization of Structures—Generalized Sensitivity Analysis. Mechanical Engineering Series, Springer-Verlag, New York.
Okazawa, S., Oide, K., Ikeda, K., Terada, K. (2002) Imperfection sensitivity and probabilistic variation of tensile strength of steel members. Internat. J. Solids Structures, 39 (6), 1651–1671.
Roorda, J. (1965) The Instability of Imperfect Elastic Structures. Ph.D. Dissertation. University of London.
Seydel, R. (1979) Numerical computation of branch points in ordinary differential equations. Numer. Math. 32, 51–68.
Seydel, R. (1979) Numerical computation of branch points in nonlinear equations. Numer. Math. 33, 339–352.
Thompson, J.M.T., Hunt, G.W. (1973) A General Theory of Elastic Stability. Wiley, New York.
Thompson, J.M.T., Schorrock, P.A. (1975) Bifurcation instability of an atomic lattice. J. Mech. Phys. Solids 23, 21–37.
van der Waerden, B.L. (1955) Algebra. Springer-Verlag, New York.
von Kármán, T., Dunn, L.G., Tsien, H.S. (1940) The influence of curvature on the buckling characteristics of structures. J. Aero. Sci. 7 (7), 276–289.
Werner, B., Spence, A. (1984) The computation of symmetry-breaking bifurcation points. SIAM J. Numer. Anal. 21 (2), 388–399.
Wriggers, P., Simo, J.C. (1990) A general procedure for the direct computation of turning and bifurcation points. Internat. J. Numer. Methods Engrg. 30, 155–176.
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Ikeda, K., Murota, K. (2019). Imperfection Sensitivity Laws. In: Imperfect Bifurcation in Structures and Materials. Applied Mathematical Sciences, vol 149. Springer, Cham. https://doi.org/10.1007/978-3-030-21473-9_3
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