A continuum Lagrangian sensitivity analysis for metal forming processes with applications to die design problems
✍ Scribed by Nicholas Zabaras; Yangang Bao; Akkaram Srikanth; William Garth Frazier
- Publisher
- John Wiley and Sons
- Year
- 2000
- Tongue
- English
- Weight
- 479 KB
- Volume
- 48
- Category
- Article
- ISSN
- 0029-5981
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✦ Synopsis
A continuum sensitivity analysis is presented for large inelastic deformations and metal forming processes. The formulation is based on the di erentiation of the governing ÿeld equations of the direct problem and development of weak forms for the corresponding ÿeld sensitivity equations. Special attention is given to modelling of the sensitivity boundary conditions that result due to frictional contact between the die and the workpiece. The contact problem in the direct deformation analysis is modelled using an augmented Lagrangian formulation. To avoid issues of non-di erentiability of the contact conditions, appropriate regularizing assumptions are introduced for the calculation of the sensitivity of the contact tractions. The proposed analysis is used for the calculation of sensitivity ÿelds with respect to various process parameters including the die surface. The accuracy and e ectiveness of the proposed method are demonstrated with a number of representative example problems. In the die design applications, a BÃ ezier representation of the die curve is introduced. The control points of the BÃ ezier curve are used as the design parameters. Comparison of the computed sensitivity results with those obtained using the direct analysis for two nearby dies and a ÿnite di erence approximation indicate a very high accuracy of the proposed analysis. The method is applied to the design of extrusion dies that minimize the standard deviation of the material state in the ÿnal product or minimize the required extrusion force for a given reduction ratio. An open-forging die is also designed which for a speciÿed stroke and initial workpiece produces a ÿnal product of desired shape.