Keywords: metal forming, rolling, buckling, necking, residual stresses, forming limit.

Structural instability phenomena are typically related to the load carrying behaviour of structures already in use. In contrast to this, the present contribution deals with the computational treatment of instabilities that appear when not yet in use but during the production process of the structural elements. Both structural instabilities (elastic and plastic buckling) and material instabilities (plastic localisation in the form of necking) are considered. As typical examples, (i) buckling due to rolling and heat treatment of strip metal and (ii) necking due to flaring of circular cylindrical shells, are considered in more detail.

In the metal forming community it is well known that residual stresses in thin rolled metal strips can lead to instabilities. In many cases these instabilities result from inhomogeneous plastic deformations during the rolling process. Other reasons for instabilities can, for instance, be inhomogeneous temperature distributions and phase transformations during heat treatments. In general such kinds of instabilities cause problems. The flatness of the strip is essential to the further processing stages (for example, the wave height must be lower than the openings for stamping or deep drawing machines), and to the quality of the products made from it. Furthermore, post-buckling deformations, in the form of waviness of strips, can lead to local yielding, particularly during heat treatments where temperatures are high and the yield stresses are low. Again, the quality of the strips can easily be spoiled during heat treatments.

In order to optimise the quality of strips it is necessary to have procedures and methods in place to find accurate estimates for the waviness of the strip once the residual stress distribution, resulting from the rolling process, is known and, the inverse problem, to derive estimates for the residual stresses from the wave patterns of the strip in order to control the rolling process. If the residual stress intensity in the rolled strip is below a critical value, and the strip appears to be flat, instabilities may occur if this strip undergoes heat treatment. It is shown how inhomogeneous heating and phase transformation have a strong influence on the flatness of the strip.

In other publications, such as [1], the authors presented procedures derived using analytical methods which allow them to estimate the type and the intensity of the residual stress fields appearing in the strips immediately after leaving the roller set. This approach is based on evaluating the observed buckling of the rolled strip during reduction of the strip tension. In a less accurate procedure the waviness is measured by laying the strip on a horizontal measuring table. In order to estimate error due to the influence of gravity, that is, the dead load of the strip and contact and cutting out a plate, a strongly coupled nonlinear bifurcation problem is solved by means of finite element methods. In a similar way the appearance of waviness during heat treatments (including solid state phase transformations) of the rolled strip is treated as a coupled, highly nonlinear stability problem.

Although material instabilities in the form of plastic localisation, that is, necking, also belong to the class of bifurcation problems, this requires different computational approaches. In metal forming such phenomena are related to the term "forming limit". The computation of forming limits is demonstrated by the numerical treatment of the expansion of a ring and the flaring process, in which a circular cylindrical shell is formed into a conical shell. Analytical methods were again developed by the authors [2]. These methods are based on several simplifying assumptions. The influence of these simplifications on the quality of the closed form solution is validated by means of computational approaches.

It is shown that a periodic necking pattern appears at a critical point when the necking criterion [3] is met. Depending on the material and geometrical parameters, it might happen that buckling of the cylindrical part of the shell in the form of concertina buckles appears before necking, a situation which also limits the forming process. The analytical, as well as the computational, results for calculating this forming process, and the related instability problems, are compared to experimental results, and reasonably good agreement can be achieved.

1

F.G. Rammerstorfer, F.D. Fischer, N. Friedl, "Buckling of free infinite strips under residual stresses and global tension", Journal of Applied Mechanics, 68, 399-404, 2001. doi:10.1115/1.1357519

2

F.D. Fischer, F.G. Rammerstorfer, T. Daxner, "Flaring - An analytical approach", submitted for publication. doi:10.1016/j.ijmecsci.2006.06.004

3

T.B. Stoughton, X. Zhu, "Review of a strain-based FLD model and its relevance to the stress-based FLD", In: Dislocations, Plasticity and Metal Forming (A.S. Khan, R. Kazmi, J. Zhou, Edts.) NEAT Press, Fulton, Maryland, USA, 397, 2003.

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