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From a paper by Simo, et al (cited below);
“It is well known that classical solution procedures fail for problems with limit points. From a physical standpoint, such problems are characterized by load/deflection curves which exhibit a ‘softening’ effect. Thus, at some stage in the iteration process the tangent stiffness becomes indefinite, and for a load level constrained to a fixed value one cannot find an equilibrium state. To overcome this difficulty several procedures have been developed. Among them, the displacement control method of Haisler et al. and the arc-length-control procedure originally proposed by Riks and [independently] by Wempner, together with its modications develped thereafter, offer the the largest range of applications.”
The Ramm citation in the Figure 2 caption::
Ramm. E. “Strategies for tracing the nonlinear response near limit points”, in Nonlinear Finite Element Analysis in Structural Mechanics, edited by W. Wunderlich, E. Stein and K.J. Bathe, Proceedings of the Europe-U.S. Workshop, Ruhr-Universität, Bochum, Germany, July 1980
This and the next 2 slides are from:
J.C. Simo (1), Peter Wriggers (2), Karl Schweizerhof (3) and R.L. Taylor (4)
(1) Applied Mechanics, Stanford University, USA
(2) Leibniz University, Hannover, Germany
(3) Karlsruhe Institute of Technology
(4) Structural Engineering and Structural Mechanics, Department of Civil Engineering, University of California, Berkeley, California, U.S.A.
“Finite deformation post-buckling analysis involving elasticity and contact constraints”, Interntional Journal for Numerical Methods in Engineering, Vol. 23, No. 5, pp 779-800, May 1986, DOI: 10.1002/nme.1620230504
ABSTRACT: This paper is concerned with the numerical solution of large deflection structural problems involving finite strains, subject to contact constraints and unilateral boundary conditions, and exhibiting inelastic constitutive response. First, a three-dimensional finite strain beam model is summarized, and its numerical implementation in the two-dimensional case is discussed. Next, a penalty formulation for the solution of contact problems is presented and the correct expression for consistent tangent matrix is developed. Finally, basic strategies for tracing limit points are reviewed and a modificaiotn of the arc-length method is proposed. The good performance of the procedures discussed is illustrated by means of numerical examples.
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