Capítulo II: Marco Teórico
2.2. Bases Teóricas
2.2.2. Silencio Administrativo
2.2.2.5. Silencio Administrativo Positivo
Lubrication and Shell (Aluminum Alloys)
Calculation of the Friction Load FR. As al-ready discussed, the stem load FSt can be as-sumed to consist of the sum of the axial load FM acting on the die, which is necessary for the de-formation of the material being worked in the primary deformation zone, and the axial load FR required to overcome the friction between the billet and the container. From Eq 3.4:
FSt艑 F Ⳮ FR M
From Fig. 3.5 the friction load FRcan be cal-culated by assuming that there is shear stress sS with which the billet shears along the boundary surface. The maximum value of the shear load
Fig. 3.12 Variation of the shear stress with the shear rate for Al99.5 at different temperatures
FR maxoccurs after the billet has been upset in the container where SSt⳱ S1at the start of the actual extrusion process given by, according to Eq 3.5:
FR max ⳱ pD (l ⳮ l )s0 0 R S
Equation 3.6 gives the variation of the load FR with stem displacement:
F (s )R St ⳱ pD • s (l ⳮ l Ⳮ s ⳮ s )0 S 0 R 1 St
The discard length is lRwhere the friction load FRis zero.
According to Eq 3.3:
D0 lR 艑
6
The stem displacement s1 needed to upset the billet in the container is obtained from the con-stant volume:
p 2 p 2
D0 • l ⳱0 DB • lB
4 4
D2B
s1⳱ l ⳮ l ⳱ l 1 ⳮB 0 B冢 D20冣 (Eq 3.8) where lB is the initial billet length, DB is the initial billet diameter, l0is the upset billet length, and D0is the container diameter.
As mentioned previously, the shear stress sS is the stress at which a specific material is sheared along a boundary surface at a specific
temperature and a specific speed. According to Eq 3.7:
sS ⳱ f(material, temperature, shear rate) If in the direct extrusion of a specific material FR max is measured along with the temperature of the billet and the ram speed, the shear stress sSis, according to Eq 3.5:
FR max
sS⳱ (Eq 3.9)
pD (l0 0ⳮ l )R
It is therefore possible to determine the shear stress sS in the direct extrusion of aluminum from the stem load stem displacement diagram.
Because the temperature is not constant over the complete length of the billet in contact with the container, there is some inaccuracy in the deter-mination of the shear stress sS.
Figure 3.12 shows the variation of the shear stress sSwith the shear rate at different tempera-tures for Al99.5, and Fig. 3.13 shows sSas a function of the temperature for shear rates be-tween 2 and 6 mm/s.
Another method of determining sSis by pierc-ing an upset billet uspierc-ing a lubricated mandrel avoiding material flow against the mandrel movement [Sie 76b].
As shown in Fig. 3.14, the billet is first upset in a container with the die closed by a sealing piece supported against the bolster. The sealing plate is then rotated through 90 so that it can be removed through the rectangular aperture of the bolster. Piercing is then carried out with a
lubri-Fig. 3.14 Determination of the shear stress sSby piercing an upset billet. A, upsetting the billet; B, opening the die; C, piercing; a, container; b, die; c, die holder; d, dummy block; e, stem; f, mandrel; g, sealing piece; h, billet; i, sealing plate
Fig. 3.13 Variation of the shear stress with temperature for shear rates between 2 and 6 mm/s for Al99.5
cated mandrel. The die diameter is only slightly larger than the mandrel diameter. Fixing the dummy block avoids the material flowing back against the piercing ram movement during pierc-ing.
The macrographs in Fig. 3.15 show that dur-ing piercdur-ing, the narrow shear zone formed is narrow enough to be designated a boundary sur-face. After piercing the core length must be equal to the upset billet length. For a piercing load FDover the mandrel displacement SDand
ignoring the friction between the billet and the mandrel:
FD⳱ s • D p(l ⳮ s )s D 0 D (Eq 3.10) where l0is the upset billet length in the container and DDthe mandrel diameter.
From Eq 10:
FD
ss⳱ (Eq 3.11)
D p(lD 0ⳮ s )D
Fig. 3.15 Macrographs of a shearing process
From the maximum piercing load FD maxat the start of piercing when SD⳱ 0:
FD max
ss⳱ (Eq 3.12)
D plD 0
It is therefore possible to experimentally deter-mine sSand show it in diagrams. In the deter-mination of the shear stress sS, it is assumed that the mean stress state in the billet (usually re-ferred to as the hydrostatic stress) does not in-fluence the shear stress. The validity of this as-sumption still has to be scientifically verified.
Calculation of the Deformation Load FM. The following discussion covers the axial load FM acting on the die necessary for the defor-mation of the material located in the primary deformation zone.
If the initial billet temperature is approxi-mately the same as the container temperature, which is the case in the hot extrusion of alumi-num, then FMis approximately constant over the stem displacement. There is only a maximum at the start of extrusion, which is associated with the temperature in the deformation zone at the start of extrusion being the initial billet tem-perature. A temperature increase occurs in the deformation zone as extrusion continues be-cause of the work of deformation. This results in decreasing deformation loads as the flow stress is reduced by the increasing temperature.
If approximately constant strain, velocity, and temperature conditions occur in the deformation
zone, a quasi-stationary deformation process is referred to. The maximum in the axial load FM occurs in the nonstationary state.
For the quasi-stationary state:
D20
FM ⳱ C • ugges • k p¯f (Eq 3.13) 4
where¯kfis the mean flow stress for the primary deformation zone,ugges is the logarithmic total strain (Eq 3.14), and C is a factor that is die specific and includes a shape efficiency factor g.
is obtained from the extrusion ratio ugges
(Eq 3.1):
A0
ugges⳱ ln V ⳱ ln (Eq 3.14)
AS
where A0 is the container cross-sectional area and AS is the section cross-sectional area. The flow stress kfis the stress at which plastic flow of a material occurs in a uniaxial stress state. The flow stress kf is a function of the material, the deformation temperature, the logarithmic prin-cipal strain ug, and the logarithmic principal strain rateu˙g.
Flow Stress kf⳱ f(material, , ug,u˙g).
The mean flow stress ¯kf in Eq 3.13 can be obtained from quasi-adiabatic flow curves found in the literature for:
● The temperature Ethat the material being deformed enters the deformation zone
● The mean logarithmic principal strain ¯ug
● The mean logarithmic principal strain rate
¯˙ug
● The specific material
and¯˙ are mean values that apply to the entire u¯g ug
material being deformed in the deformation zone.
⳱ f(material, entry temperature E, mean
¯kf
logarithmic principal strain u¯g, mean logarith-mic principal strain rate ¯˙ug).
From the literature [Sie 76a]
¯ 3
where lUis the length of the deformation zone.
In the extrusion of round bars of diameter DS, then, for a conical deformation zone with an opening angle 2␣ ⳱ 90:
D0ⳮ DS
lU⳱ (Eq 3.17)
2 tan␣
If it is assumed for the hot extrusion of a round bar the deformation zone is conical with ␣ ⳱ 45, then from Eq 3.16 and 3.17: If the extruded section does not have a round cross section, an equivalent diameterD*Sshould be used to replace DS:
4• AS
D*S ⳱
冪
p (Eq 3.19)where ASis the section cross-sectional area.
From the literature [Ake 66], [Stu 68], and [Stu 71], the approximation for direct hot extru-sion attributable to Feltham [Fel 56] is:
uSt
¯˙u ⳱ 6 •g • ugges (Eq 3.20)
D0
This approximation does, however, give signifi-cantly lower values than does Eq 3.18, which
appears to better describe the velocity behavior in the deformation zone and will therefore be used to determine the mean flow stress for the primary deformation zone.
Figure 3.16 shows for Al99.5 in a double log-arithmic format the flow stress kfas a function of the logarithmic principal strain ug.
This graph gives straight lines for kf⳱ f(ug).
These have been extrapolated from ug ⳱ 1.1.
The values in the range 0.4ⱕ ugⱕ 1.1 are taken from the measured values from the work [Bue 70]. These flow stress values are show for the logarithmic principal strain ratesu˙g⳱ 0.25, 4.0, and 63 sⳮ1. It can be seen that as the temperature increases, the influence of the logarithmic prin-cipal strain rate also increases.
Figure 3.17 shows for aluminum in double logarithmic format the flow stress as a function of the logarithmic principal strain rate.
These values were measured for the principal strains ug ⳱ 1, 3, and 5 [Alu]. Figure 3.18 shows that in the temperature range 350 to 500
C, the influence of the logarithmic principal stress on the flow stress is small. Flow stress values as a function of ug,u˙g, and for metallic materials can be found in the literature, e.g., [DGM 78].
The factor C in Eq 3.13 contains:
fp
C⳱ (Eq 3.21)
gF
where fpis the profile factor, and gFthe defor-mation efficiency factor.
If the axial load on the die for the extrusion of a specific section is compared with the axial load for the extrusion of a round bar of the same extrusion ratio:
FM section
fp⳱ (Eq 3.22)
FM round
The deformation efficiency factor gFis defined as the ratio of the theoretical deformation load needed FM theor to the actual measured load FM gem:
pD20
¯k • uf gges •
FM theor 4
gF ⳱ ⳱ (Eq 3.23)
FM measured FM measured
If no experimental results are available, gF⳱ 0.5–0.6 can be assumed.
If a round bar is being extruded, the factor C is given by:
Fig. 3.16 Flow stress kfof Al99.5 as a function of the logarithmic principal straingforE⳱ 20 C, 120 C, and 240 C (values in the range 0.4ⱕ ugⱕ 1.1 taken from [Bue 70])
Fig. 3.17 Flow stress kfof Al99.5 as a function of the logarithmic principal strain rate˙ug(values from [Alu])
C ⳱ 1 (Eq 3.24)
gF Example:
Material: Al99.5
Container diam D0⳱ 140 mm
Initial billet diam DB⳱ 136 mm
Initial billet length lB⳱ 500 mm
Product Round bar 19.8 mm
Initial billet temperature⳱ container temperature
400C
Ram speed 10 mm/s
Then:
Equation Parameter Result
3.1 Extrusion ratio V⳱ 50
3.14 Logarithmic principal strain ugges⳱ 3.9 3.15 Mean log. Principal strain u¯g⳱ 0.674 3.16 Mean log. Principal strain rate u˙g⳱ 8.15 sⳮ1
With ¯˙ug ⳱ 8.15 sⳮ1, u¯g ⳱ 0.674, and ⳱ 400 C, then the mean flow stress over the full deformation zone from Fig. 3.17 is:
¯k ⳱ 31 N/mmf 2
Equation Parameter Result
3.21 Factor C for fp⳱ 1 and gf⳱ 0.6 (estimated)
C⳱ 1.67
3.13 Deformation load Fm⳱ 3100 kN
3.8 Upset displacement s1⳱ 28 mm
3.8 Upset billet length l0⳱ 472 mm
3.3 Discard length lR⳱ 23 mm
With uSt ⳱ 10 mm/s, ⳱ 400 C, then for Al99.5 from Fig. 3.12, the shear stress is sS⳱ 10 N/mm2:
Eq 3.6 friction load FR ⳱ 4396 N/mm
• (477 ⳮ s ) mmSt
Eq 3.6 Stem load FSt⳱ F Ⳮ F ⳱ 3100 kNM R
Ⳮ 4396 N/mm • (477 ⳮ s ) mmSt
For the maximum of the stem load at the start of extrusion where sSt⳱ s1mm.
Then:
Max stem load FSt max ⳱ 5074 kN The stem load minimum at sSt⳱ l0Ⳮ s1ⳮ lR
gives:
Min stem load FSt min⳱ 3100 kN The result is the principal variation of the stem force over the stem displacement shown in Fig. 3.18.
3.1.1.5 Thermal Changes in Direct Hot