Civil Engineering Reference
In-Depth Information
FIGURE 5.15
Optimal preform and subsequent forged part for the 17% upsetting problem. Optimal preform and
subsequent forged part for the 44% upsetting problem [5] (with permission).
like an equation of the principle of virtual work, the unknown being the shape sensitivity of
F
. For its
resolution, the same finite element and time discretizations are used, as for the direct analysis. However,
it must be noticed that the subsequent linear system does not exhibit the same stiffness matrix as the
direct problem, while this is the case with the discrete differentiation methods.
This method has been successfully applied to the shape optimization of extrusion dies [6] and of
simple forging preforms [5]. This last application consists of finding the shape of the preform that
eliminates the bulging of the part after upsetting (a similar problem is presented in Section 5.5).
Figure
5.15
shows the optimal preforms for two different upsetting heights. The shape is described with quintic
splines and the material is assumed hyperelastic viscoplastic. For this example, the method shows the
same level of accuracy as the discrete differentiation method, although it is slightly less efficient since the
stiffness matrix of the sensitivity problem has to be recomputed.
5.3
The Optimization Problem
Numerical Simulation of the Forging Process
The forthcoming description will be limited to the two-dimensional axisymmetrical forging problem. In
the introduction, we have justified the restriction to the forging process, and it seem that until now the
design optimization techniques have not been applied to three-dimensional non-steady-state metal
forming problems. Actually, the 3D simulation is still a difficult issue and the reliability and robustness
of the software is very recent. We shall then restrict ourselves to axisymmetrical forging. Moreover, for
the sake of simplicity, we shall describe only the hot forging problem under isothermal conditions in
order to manipulate rather simple constitutive and friction equations. The isothermal hypothesis is well
verified in standard forging processes. In fact, the forging time is too short to allow a significant heat
exchange with the dies, and the heat generated by the deformation is not enough to actually modify the
flow. The thermal calculations are necessary when thermal phenomena themselves have to be controlled
during the process. The interested reader will find more details of the 2D simulation in [34, 40, 47, 47],
and [27]. As for the 3D simulation, recent descriptions can be found in [12, 15], and [66].
Problem Equations
In most of the forming processes, elasticity effects are negligible, so the material is considered to be rigid
plastic and incompressible. The material is assumed to be homogeneous and follows the viscoplastic
Norton-Hoff law.
div
v
()
0
(5.12)
m
1
˙
˙
s
2
K
(
3
)
(5.13)
