The study is concerned with the analysis of backward extrusion for three-dimensional shapes using the central flow models.
In general, two types of flow models are used in the analysis of backward extrusion of tubular shapes ; tubular flow model with a dead metal zone at the center and central flow model without dead metal zone. The use of central flow model has a great advantage in application in that the analysis for forward extrusion can be conviently used also for backward extrusion through adequate velocity transformations. Once the validity and efficiency of central flow model is established in axi-symmetric backward extrusion, this type of model can be effectively used in three-dimensional backward extrusions.
The study consists of the analyses and experiments for the following three extrusion models:
1) Axi-symmetric backward extrusion as a comparison between two types of flow models.
2) Three-dimensional backward extrusion of tubular shapes having geometrical similarity between internal and external contours.
3) Three-dimensional backward extrusion of arbitrary shaped tubes from the circular billets.
For the above extrusion models, kinematically admissible velocity fields are derived in order to formulate upper-bound solutions under the following basic assumptions.
(1) Entrance and exit boundaries of plastic flow in the transformed velocity fields are flat planes perpendicular to the axial direction.
(2) Distribution of axial velocity at each cross-section is uniform in the plastically deforming zone.
Flow function approach is employed in the analysis of axi-symmetric backward extrusion. In analyzing three-dimensional backward extrusion, the complicated three-dimensional boundaries are transformed into unit circles either by conformal transformation or by special transformation to facilitate the theoretical formulation. In each case, the bounding surface of plastic flow is controlled by the fourth-order profilization function. The optimal configurations of the bounding surface are determined such that the extrusion power is minimized with respect to some chosen geometrical parameters.
Some examples of tubular shapes are chosen for the calculations and experiments in order to verify the foregoing extrusion models. In the calculation, the work-hardening effect is incorporated approximately by defining the mean flow stress.
Experiments are carried out with the commercially pure aluminum billets for various shaped tubes. Comparison of the theoretical calculations with the experimental results shows that the theoretical predictions for the extrusion loads are in good agreement with the measured loads for the foregoing three extrusion models.
Through this study, it is found out that the central flow models can be conviently used in predicting the extrusion loads for the backward extrusion of three-dimensional shapes.