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A WKB analysis of the buckling of an everted neo-Hookean cylindrical tube. (English) Zbl 1072.74027
Summary: It is well-known that a circular cylindrical tube may not stay circular cylindrical after eversion and it may prefer to adopt a wrinkled configuration. A linear analysis using the method of adjacent equilibria followed by a numerical computation shows that for a neo-Hookean tube, a wrinkled configuration is possible when the ratio of the inner radius to the outer radius is approximately equal to 0.4232. The wrinkles have a circumferential mode number approximately equal to 14. In this paper the WKB method is used to derive an asymptotic expression for the critical ratio of the inner radius to the outer radius, with the mode number used as a large parameter. A turning point is found to exist in the eigenmodes, and is used to explain the difficulties experienced in previous numerical computations.

74G60 Bifurcation and buckling
74B20 Nonlinear elasticity
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
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[1] Rivlin, R. S., Phil. Trans. Roy. Soc. Lond. 242 pp 173– (1949) · Zbl 0035.41503
[2] Varga, O. H., Stress-strain behaviour of elastic materials (1966) · Zbl 0166.43201
[3] Chadwick, P., J. Elasticity 2 pp 123– (1972)
[4] Chadwick, P., J. Inst. Math. Applic. 10 pp 258– (1972) · Zbl 0247.73091
[5] Adeleke, S. A., J. Elasticity 13 pp 63– (1983) · Zbl 0516.73059
[6] Szeri, A. J., Appl. Math. 47 pp 49– (1990)
[7] Antman, S. S., J. Elasticity 50 pp 129– (1998) · Zbl 0921.73204
[8] Antman, S. S., J. Elasticity 63 pp 171– (2001) · Zbl 1052.74539
[9] Haughton, D. M., Int. J. Non-linear Mech. 30 pp 81– (1995) · Zbl 0820.73033
[10] Fu, Y. B., SIAM J. Appl. Maths 62 pp 1856– (2002) · Zbl 1053.74016
[11] Wolfram, S., The Mathematica Book, 4. ed. (1999) · Zbl 0924.65002
[12] Steele, C. R., Mechanics Today 3 pp 243– (1976)
[13] Abramowitz, M., Handbook of mathematical functions (1970)
[14] Dowaikh, M. A., IMA J. Appl. Maths 44 pp 261– (1990) · Zbl 0706.73018
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