# Models for numeric simulation of extrusion stamping processes for metallic powders sintered pre-shapes

## Description

The numeric simulation of extrusion stamping processes for pre-shapes sintered from metallic powders is used for industrial purposes in technological parameters optimization and tools computer aided-designing. The achieved model consists of: general equation of particle flowing trajectories in a material in strain area, the constitutive elastic-plastic equations for big strains based on multiplier decomposition of gradient strain, non-linear models for contact and friction (static and dynamic)., travel speeds variations, periods of time necessary for furnace foad by the volume elements on each flowing trajectory.

Solution

The mathematics models taken into consideration for the modeling of material flowing deemes that the material particles move into the strain focus upon curvilinear trajectories, closed to a cosine curve branch.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         The cosine curve-amplitude decreases according as the material elements that come into the focus are closer of blank axis, and proportionally, with the amplitude decrease, the period increases. For mathematics transposition of flowing trajectories, all stamping assembly was reported to a system of cylindrical coordinates Z, R, Q.

SOLUTION

WORKING OF THE PROBLEM TO BE MAXIMALIZED
In order to analyse and maximize the technological parametres, knowing the equation of the material flouring trajectories into the strain focus it is possible further to simulate on computer the stamping process by extrusion. One of directions which may be flowed is to maximize the technological parametres so that the strain lack of uniformity on stamping by extrusion be minimum. As we already know, the quality of the obtained products is conditioned mainly, by the value of strain lack of uniformity, in longitudinal or cross wise, it’s the strain lack of uniformity is lower, as higher is the part mechanical characteristics. In order to establish the strain lack of uniformity it is necessary, that besides, the flowing trajectories into the focus, the variation of moving speed V, which the particles have during their moving on these trajectories, be know. As we already know, during the stamping by extrusion, the flowing speed increases from value Vi (equal to the stamp speed) up to the exit speed of the material from the focus Vz. The relation between these speeds is given by the relation:
(2)
What is interesting is the variation way of the speed V between the two limit values Vi Vf, variation that depends on the shape of flowing trajectories. In order to simplify the calculations and to obtain some dimensional data, following notations were used:
(3)
(4)

where Ri, Rf, Ro, L(Ro), R and Z have the some meaning to those used in flowing trajetories determination.Based on relation (2) and taking into accourent the flowing trajectory equation (1) after calculation, resulted:
(5)
where: η = f(ηo, ξ), G = f(ηo, ξ), F = f(ηo, ξ). The components of V speeds on the three coordinate axis Z, R, Q, depending on known elements ηo and ξ, were determined on the (3) relation based, obtaining:
(6)
It was to be mentioned that V speed as well as it components on Z and R directions, being out of dimension, represents the increase intensity of this one from Vi to Vf. The determination of the effective speed is made according to the relations:
(7)
Knowing the evolution of Vz speed into the focus, the determination of time needed that the volume elements pass through the strain focus on each flowing trajectory was possible, according to:
(8)

with ti = L(Ro)/(nvZi), where n represents the number of intervals where the lenght L(Ro) may be divided in order to analyse the Vz speed evolution. Further, taking as reference, the time covered by the volum elements on the marginal trajectory, tr, the difference between the referrence time and the time whit which the volume elements on the other traiectory reaches the exit from the stamp, was calculated:
Δt(Ro) = tr – t(Ro (9)
Taking into account that after the exit of the stamp, all the volume elements have the same Vf speed, irrespective of the traiectory they flow into the strain focus, the difference which is to be covered into the focus or the strain lack of uniformity, is proportional whith the time gap Δt(Ro):
Δl = Δt(Ro) vf

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