Deformation of an Elastic Ring under Pointwise Radial Loads

Cylindrical shell subject to a concentrated normal pressure from outside

Project

The project "Deformation of an Elastic Ring under Pointwise Radial Loads" (Deformation einer elastischen Flexlippe) is carried out at the University of Freiburg in collaboration with the machine factory "Reifenhäuser" in Troisdorf. It is directed by Prof. Dr. D. Kröner and Prof. Dr. S. Müller. The work is supported by the BMBF in connection with the project "Application Oriented Joint Projects in Mathematics".

Abstract

Thin plastic foils are produced with profile extruding machines. A special difficulty arises in the production of tube-shaped plastic foils (e.g. for seamless plastic bags). These foils are produced by pressing a special heated plastic mass through a ring-shaped extruder. Afterwards the flexible plastic tube is blown up to the desired size and winded on a take-up roller.
In order to save material and to guarantee a constant quality of the produced foils, it is very important that they have a uniform thickness. Because the melted plastic is a very inhomogeneous mass, it may happen that the thickness of the produced foil is irregular.
To overcome this problem engineers placed feed screws around the outer shell ("Flexlippe") of the extruding machine. Using these screws the thickness of the foil can be corrected. If the foil is too thick at a certain point, the associated feed screw can be pushed against the shell. The advances of the screw bolts are steered with stepping motors.
Reifenhäuser builds extruding machines with up to 120 bolts equidistantly positioned around the outer shell. The picture shows a sketch of a profile extruding machine from two different views.


profile extruding machine The aim is to construct an automatic control system which regulates the advances of the bolts. This can only be done by continuously measuring the thickness of the extruding foil and correcting the advances of the bolts simultaneously.
To build up this control system is at first necessary to study the deformation of the outer shell of the extruding machine with a given number of scew bolts. The extruding mass causes a hydrostatic pressure form inside. The deformation of the shell is so determined by the hydrostatic pressure and the advances of the bolts.
Another problem is to find out the forces at the bolts.




Mathematical Model

Mathematically the problem is to determine the deformation of an elasic cylinder subject to non-symmetric pointwise radial loads. The cylinder is clamped at the lower edge.
Since the shell is very thin compared to its radius, we can use a Kirchhoff-type shell model. The deformations are very small so we consider an entirely linear model due to Koiter [2]. The advances of the bolts are expressed by pointwise constraints to the radial deformation.
The deformed configuration can then be determined by minimizing the total elastic energy in a certain convex set of admissible deformations. The Lagrange-multipliers for the active constraints (which are always positive because of the Kuhn-Tucker condition) give the acting forces at the bolts.
The ellipticity of the problem and the convexity of the set of admissible deformations yields a unique weak solution of the problem.
A simplified model is studied in [1].

Numerical Method

Numerically we solve the problem by a finite element method with conforming elements which is preferrable in case of pointwise constraints.
A severe numerical problem in the computation of thin shells is the locking-phenomenon [4]. Because we use a Kirchhoff-type model we have no shear-locking but still membrane-locking. To avoid this, special care is needed in the choice of the right elements. In a first approach we used the SHEBA 6-element with 63 degrees of freedom [3] and the bicubic Bogner-Fox-Schmit element.
The picture on top of this page shows a cylindrical shell subject to a concentrated normal pressure from outside. The deformations are stretched to make them visible. The wave shape which can be seen is a very typical and can also be examined for the simpler model in [1].
In [1] we used a primal active set method to solve the problem with the constraints, it turned out to be a very efficient for this special problem. Our aim is to generalize this approach for the shell problem.
To make the calculations more efficient, we want to add adaptive mesh refinement and test different error-estimators. This is well known for plates and we intend to find similar algorithms for our shell-problem.

Literatur

[1] Koop, A.: Deformation einer elastischen Flexlippe, Diplomarbeit, Bonn 1993

[2] Koiter, W.T.: A consistent first approximation in the general theory of thin elastic shells, Proc. IUTAM Symposium on the Theory of Thin Elastic Shells, North-Holland, 12-33, 1960

[3] Bernadou, M.: C1-curved finite elements with numerical integration for thin plate and thin shell problems I/II, Comp. Meth. in Appl. Mech. and Eng. 102, 255-289, 289-421, 1993

[4] Chenais, D.; Paumier J.C.: On the locking phenomenon for a class of elliptic problems, Numer. Math. 67, 427-440, 1994

[5] Koop, A.; Kröner, D.; Müller, S.; Schupp, B.; Schmitz, S.; Schroeter, B.: Deformation einer elastischen Flexlippe. Erscheint im Tagungsband des Statusseminars der Anwendungsbezogenen Verbundprojekte auf dem Gebiet der Mathematik, M"unchen 1996, Springer-Verlag


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