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Analytic solutions of Von Kármán plate under arbitrary uniform pressure – equations in differential form. (English) Zbl 1367.74008

Summary: The large deflection of a circular thin plate under uniform external pressure is a classic problem in solid mechanics, dated back to T. Von Kármán [“Festigkeitsprobleme im Maschinenbau”, in: Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen 4, Teil 4. 348–351. Leipzig: Teubner (1910)]. This problem is reconsidered in this paper using an analytic approximation method, namely, the homotopy analysis method (HAM). Convergent series solutions are obtained for four types of boundary conditions with rather high nonlinearity, even in the case of \(w(0)/h>20\), where \(w(0)/h\) denotes the ratio of central deflection to plate thickness. Especially, we prove that the previous perturbation methods for an arbitrary perturbation quantity (including the J. J. Vincent’s [Philos. Mag., VII. Ser. 12, 185–196 (1931; Zbl 0002.21601; JFM 57.1060.03)] and W. Z. Chien’s [“Large deflection of a circular clamped plate under uniform pressure”, Acta Phys. Sin. 7, No. 2, 102–113 (1947)] methods) and the modified iteration method [K. Y. Yeh et al., “Nonlinear stabilities of thin circular shallow shells under actions of axisymmetrical uniformly distributed line loads”, J. Lanzhou Univ., Nat. Sci. 18, 10–33 (1965)] are only the special cases of the HAM. However, the HAM works well even when the perturbation methods become invalid. All of these demonstrate the validity and potential of the HAM for the Von Kármán’s plate equations, and show the superiority of the HAM over perturbation methods for highly nonlinear problems.

MSC:

74B20 Nonlinear elasticity
74K20 Plates

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References:

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