Abstract
A general mathematical framework of bifurcation analysis that is to be employed throughout the book is presented. In particular, the Liapunov–Schmidt reduction is introduced as a tool to derive bifurcation equation. Perfect and imperfect bifurcation behaviors at simple critical points are investigated asymptotically in view of the leading terms of the power series expansion of this equation. This chapter lays a theoretical foundation of Chaps. 3–6 and is extended to a system with group symmetry in Chaps. 8 and 9.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This term, “Liapunov–Schmidt reduction,” is widely accepted in nonlinear mathematics (e.g., Sattinger, 1979 [167]; Chow and Hale, 1982 [26]; and Golubitsky and Schaeffer, 1985 [55]). In structural mechanics, it is called the “Liapunov–Schmidt–Koiter reduction ” (e.g., Peek and Kheyrkhahan, 1993 [155]) or the “elimination of passive coordinates ” (e.g., Thompson and Hunt, 1973 [181]; Thompson, 1982 [180]; El Naschie, 1990 [43]; and Godoy, 2000 [54]).
- 2.
- 3.
ε can be negative until Chap. 3. In later chapters, ε is assumed to be nonnegative in considering the worst imperfection and random imperfection.
- 4.
In structural mechanics, the Jacobian matrix is called the tangent stiffness matrix and the imperfection is called the initial imperfection .
- 5.
See, for example, Oden and Ripperger, 1981 [145, Theorem VIII, page 305].
- 6.
It is assumed in (2.11) that F = (F i∣i = 1, …, N) is a column vector and ∂U∕∂ u = (∂U∕∂u i∣i = 1, …, N) is a row vector.
- 7.
The influence of the weight of structural members is ignored here and in the remainder of this book.
- 8.
It is intended that {η j} be a basis of the space of the vectors u, and that {ξ j} be a basis of the (dual) space of the values of the function F.
- 9.
Such is almost always the case with practical examples. Mathematically, however, (2.60) represents a restrictive assumption. For example, the bifurcation equation \(w^{2} - {\tilde {f\,}}^{4} = 0\) gives \( \tilde {f}= \pm \sqrt {|w|}\), which is not smooth at w = 0.
- 10.
Stated precisely, it is necessary to distinguish column and row vectors in the expression (2.66). Nevertheless it seems more comprehensive as it is.
- 11.
By the definition (2.71), we may replace \(\tilde {F}\) with \(\hat {F}\).
- 12.
- 13.
The point is considered degenerate if A 200 = 0; in particular, it is a stationary point if A 200 = 0 and A 300 ≠ 0.
- 14.
- 15.
The relation between bifurcation and symmetry is treated systematically in Part II.
- 16.
The imperfection parameter vector v is suppressed in (2.124), as it is kept fixed in the following argument.
References
Budiansky, B. (1974) Theory of buckling and post-buckling behavior of elastic structures. In: Advances in Applied Mechanics, Vol. 14, edited by C.-S. Yih, pp. 1–65. Academic Press, New York.
Chow, S., Hale, J.K. (1982) Methods of Bifurcation Theory. Grundlehren der mathematischen Wissenschaften, Vol. 251. Springer-Verlag, New York.
Chow, S., Hale, J.K., Mallet-Panet, J. (1975) Applications of generic bifurcations. I. Arch. Rational Mech. Anal. 59, 159–188.
Chow, S., Hale, J.K., Mallet-Panet, J. (1976) Applications of generic bifurcations. II. Arch. Rational Mech. Anal. 62, 209–235.
El Naschie, M.S. (1990) Stress, Stability and Chaos in Structural Engineering: An Energy Approach. McGraw-Hill, London.
Godoy, L.A. (2000) Theory of Elastic Stability—Analysis and Sensitivity. Taylor & Francis, Philadelphia.
Golubitsky, M., Schaeffer, D.G. (1985) Singularities and Groups in Bifurcation Theory, Vol. 1. Applied Mathematical Sciences, Vol. 51. Springer-Verlag, New York.
Iooss, G., Joseph, D.D. (1990) Elementary Stability and Bifurcation Theory, 2nd ed. Undergraduate Texts in Mathematics. Springer-Verlag, New York.
Keener, J.P. (1974) Perturbed bifurcation theory at multiple eigenvalues. Arch. Rational Mech. Anal. 56, 348–366.
Keener, J.P., Keller, H.B. (1973) Perturbed bifurcation theory. Arch. Rational Mech. Anal. 50, 159–175.
Koiter, W.T. (1945) On the Stability of Elastic Equilibrium. Dissertation. Delft, Holland (English translation: NASA Technical Translation F10: 833, 1967).
Marsden, J.E., Hughes, T.J.R. (1983) Mathematical Foundations of Elasticity. Prentice-Hall Civil Engineering and Engineering Mechanics Series. Prentice-Hall, Englewood Cliffs, NJ.
Matkowsky, B.J., Reiss, E.L. (1977) Singular perturbations of bifurcations. SIAM J. Appl. Math. 33 (2), 230–255.
Oden, J.T. and Ripperger, E.A. (1981) Mechanics of Elastic Structures, 2nd ed. Hemisphere, Washington, DC.
Peek, R., Kheyrkhahan, M. (1993) Postbuckling behavior and imperfection sensitivity of elastic structures by the Lyapunov–Schmidt–Koiter approach. Comput. Methods Appl. Mech. Engrg. 108, 261–279.
Reiss, E.L. (1977) Imperfect bifurcation. In: Applications of Bifurcation Theory, ed. P.H. Rabinowitz, pp. 37–71. Academic Press, New York.
Sattinger, D.H. (1979) Group Theoretic Methods in Bifurcation Theory. Lecture Notes in Mathematics, Vol. 762. Springer-Verlag, Berlin.
Seydel, R. (1994) Practical Bifurcation and Stability Analysis, 2nd ed. Interdisciplinary Applied Mathematics, Vol. 5. Springer-Verlag, New York; 3rd ed., 2009.
Thompson, J.M.T. (1982) Instabilities and Catastrophes in Science and Engineering. Wiley Interscience, Chichester.
Thompson, J.M.T., Hunt, G.W. (1973) A General Theory of Elastic Stability. Wiley, New York.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Ikeda, K., Murota, K. (2019). Local Behavior Around Simple Critical Points. In: Imperfect Bifurcation in Structures and Materials. Applied Mathematical Sciences, vol 149. Springer, Cham. https://doi.org/10.1007/978-3-030-21473-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-21473-9_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-21472-2
Online ISBN: 978-3-030-21473-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)