Kounadis, A. N. Material-dependent stability conditions in the buckling of nonlinear elastic bars. (English) Zbl 0611.73063 Acta Mech. 67, 209-228 (1987). A thorough study of the critical and post-critical large displacement response of simple discrete and continuous systems made from a nonlinear elastic material, is presented. Simple material-dependent stability conditions are established, whose application does not require the solution of the intractable nonlinear differential equations of equilibrium. The predominent role of the nonlinear component of the curvature on the buckling mechanism of the foregoing systems, is revealed. Moreover, it is found that elastic systems which were considered as exhibiting post-buckling strength might be imperfection sensitive, if the effect of material nonlinearity is taken into account. An approximate but very efficient analytical approach leading to very reliable results is derived for establishing the large displacement behavior of a nonlinear elastic cantilever bar. The degree of accuracy of this approach is checked by comparing it with the ”exact” numerical solution of Runge Kutta. The numerical results presented herein contribute to our understanding of this problem and at the same time show the efficiency, reliability and range of applicability of the proposed approach. Cited in 1 ReviewCited in 1 Document MSC: 74G60 Bifurcation and buckling 34A45 Theoretical approximation of solutions to ordinary differential equations 74B20 Nonlinear elasticity 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:critical; post-critical large displacement response; simple discrete; continuous systems; material-dependent stability conditions; post- buckling strength; imperfection sensitive; large displacement behavior; cantilever bar Citations:Zbl 0611.73062; Zbl 0572.73052 PDFBibTeX XMLCite \textit{A. N. Kounadis}, Acta Mech. 67, 209--228 (1987; Zbl 0611.73063) Full Text: DOI References: [1] Simitses, G. J., Kounadis, A. N.: Buckling of imperfect rigid-jointed frames. J. of Eng. Mech. Div., ASCE, EM3, 569-586 (1978). [2] Oden, J. T., Childs, S. B.: Finite deflections of a nonlinearly elastic bar. J. of Appl. Mech. Trans. ASME37, 48-52 (1970). · doi:10.1115/1.3408488 [3] Kyriakides, S., Babcock, C. D.: Large deflection collapse analysis of an inelastic inextenional ring under external pressure, Int. J. of Solids and Structures17, 981-993 (1984). · Zbl 0476.73032 · doi:10.1016/0020-7683(81)90036-6 [4] Antman, S. S., Rosenfeld, G.: Global behavior of buckled states of nonlinearly elastic rods, Siam Review20 (3), 513-565 (1978). · Zbl 0395.73039 · doi:10.1137/1020069 [5] Kounadis, A. N.: On a simple and very efficient approximate technique for solving, nonlinear boundary-value problems. Scientific papers of Civil Engineering Dep., Nat. Techn. University of Athens Vol. 9, No. 3-4, 1985. [6] Budiansky, B.: Theory of buckling and postbuckling behavior of elastic structures. Advances in Applied Mechanics, Academic Press,14, 1-64 (1974). · doi:10.1016/S0065-2156(08)70030-9 [7] Thomson, J. M. T., Hunt, G. W.: A general theory of elastic stability, pp. 5-9. London: John Wiley and Sons, 1973. · Zbl 0351.73066 [8] Poston, T., Stewart, I.: Catastrophe theory and its applications, pp. 301-305. Pitman 1978. · Zbl 0382.58006 [9] Bucknall, C. B., Partridge, I. K., Ward, M. V.: Rubber-toughening of plastics, J. Mat. Sciences19, 2064-2072 (1984). [10] Pflüger, A.: Stabilitätsprobleme der Elastostatik, pp. 9-11. Berlin: Springer 1950. [11] Timoshenko, St., Gere, J.: Theory of elastic stability. New York: McGraw-Hill Book Co. 1961. [12] Britvec, S. J.: The stability of elastic systems, p. 190. Oxford: Pergamon Press 1973. · Zbl 0426.73047 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.