ALUMINUM PROFILES EXTRUSION PROCESS ANALYSIS. MECHANICAL ASPECTS

The presented paper subject is the engineering analysis of the aluminum profiles hot extrusion. Some theoretical aspects of the process as described in professional publications have unsufficient physical grounds. The imperfections of the theory supply a platform for arbitrary interpretations and distorted understanding of the process. Particularly, this relates to the billet friction problem. The paper is addressed primarily to practical engineers with the aim to give them physically consistent process imagine. Some principal solutions to the process were developed in the paper: billet friction by liner; the plastic flow model in dummy block region; the stress state and flow beginning relations in the die opening zone. General model of stable metal flow relations in the billet volume was formulated. The analysis of all mentioned above problems fulfilled by the uniform methodology with the necessity simplifications. The uniform field stress state was used as base model. Strain energy minimum principal for static relations and mass transfer energy minimum principal for plastic flow relations were used generally. Numerical estimations of given solutions were fulfilled. Principal stress component ratio in uniform stress of three dimensional compressing field was defined. Billet – liner, billet – dummy block friction forces acting model was developed. By appraisal, billet – liner friction force does not exceed about 31% of summative press force. The region of friction force acting was defined. The region restricted as rather short distance from dummy block. Surface forces accumulation effect was revealed. Probable model of plastic flow in dummy block region is presented. Investigation of stress state in die opening region was fulfilled. Additional stresses field in the region revealed. Configuration of the field was defined with numerical estimation. The field in great degree defines the necessary stress level for metal flow start and stable process. The algorithm of maximal flow stress definition was developed. The flow model with restricted shear strain was presented. Preferable flow direction is radial, relative to die opening. Essential interdependencies and general relations of billet metal flow are presented. The analyses of press load by dummy block displacement graph was fulfilled. No contradictions with the theoretically achieved results were revealed. The results experimental confirmation and further introduction to practice are proposed.


Introduction
The presented analysis subjects are several aspects of the aluminum profi les hot extrusion process. In order to avoid ambiguity, the process of direct extrusion fulfi lls in a hydraulic press with the following scheme and cycle elements.
The state and position of the elements are as follows: the container is pressed to die with sealing force; billet in the liner hollow after burp cycles, whole volume is fi lled with billet material; the pressure is being raised by the slow motion of the dummy block (work stroke beginning ).
The former reason for present analysis was necessity to fi nd the rationale for some extrusion technology improvements. As the work went on, theoretical contradictions (on the author's opinion) in some principal positions were revealed. Particularly, the friction infl uence on the extrusion process as presented in the theoretical model of friction in monograph [1], to which the most of other authors refer [2], [3], [4]. On the other hand, the point of view of a number of operative specialists do not match the theoretical predictions of results. Hence, because of impossibility to use even partly the existed calculation methods, the author was necessitated to start the analysis from the base positions. It was a good piece of job made before this decision. In the beginning the author's view of the problem did not differ from the traditional ones. The fi rst probe of analysis targeting polycrystalline fi bers extrusion was fulfi lled with the traditional process model. No productive results were achieved, no way to improve the process was found, no cause of worse changes of fi bers microstructure was revealed. The author also has to note, that the present analysis results are based on a huge experience achieved by specialists of the aluminum industry branch. Especially it relates to experimental elaborations. Two works [5] and [6] were more helpful to fi nd out the way of solving the friction problem with the minimum level of abstraction.
2. Friction along inside liner surface 2.1 Billet volume stress state Again we repeat the initial state. The whole container volume is fi lled with the billet material. The inner pressure is being raised by dummy block action till the value of σ z ≤ σ e -the plastic fl ow beginning ( Fig. 2.1 ).
The stress state of the disc with the width d z on any distance from the die surface is considered. Without signifi cant strains the fl at sections hypothesis considered rightful. The axis directed stresses uniformly distributed on the side disc surface. Friction stresses on the outer cylindrical surface are absent or negligible without any displacement. An element of the disc volume at any point of side surface is considered (Fig. 2.2).
The element faces are in the same as principal stresses directions: σ z, σ r, σ θ , where σ z -axis direction stress; σ r -radial stress; σ θ -considered as tangential stress. Because σ z is the active one and by axisymmetric relation the principal compressive stress σ z = σ 3 , therefore σ r and σ θ are principal stress components also. It seems illegitimately to use Generalized law of elasticity in presented three dimensional compressive stress state without any verifi cation: In the principal stress symbols: In presented components: ( 2.1) Interdependences (2.1) allow random component values. Strain formulas do not match the compatibility relations: the strain component caused by σ z = σ 3 in σ 2 and σ 1 directions required either opposite sign strain of adjacent elements or their displacement. Both are impossible in three dimentional compressive state. Adjacent disc volume elements are under the same conditions. In other words we may consider that stress fi eld of either differential part of volume is uniform. Furthermore, the liner inside surface remains fi xed. (Actually radial liner strain does exist, but it is negligible, because the liner -container fi t stiffness about 10 times exeeds the heated billet's one.) Thus, it is rightful to consider radial displacement absent -u r = 0; and tangential -u θ = 0 -as well, because of axisymmetric relation. So, the only possible stress state remaining the uniform fi eld: We have no experimental data of interdependences in three dimensional compressive state. Hence, we need to use the superposition principal to defi ne them, whenever possible without physical relations distortion. So, we have the elastic law for linear stress state, rightful up to known stress level: where E and μ -are constants or functions of stress-strain dependence in the acting stress range; -virtual feasible strain components in r and directions of action. Whereas, is negative in compressive state, than >0. Full virtual strain consists of the three components: Resulting real linear strain in three dimensional stress state with resistance: (2.6) The (2.5) is the base formula of present analysis. It defi nes compressive stress components ratio in principal directions in differential part of stress fi eld. Note, that (2.5) may be received directly from (2.1) system with formulated above relations. The (2.5) ratio still remains a hypothesis and is necessary to be explained, that all other ratios are impossible with the defi ned restrictions. was defi ned as greatest active compressive stress, ergo principal stress. Exceeding of (2.7) in one of both remaining directions means, that there is an active stress source in this direction. It causes that and its direction is not principal. The principal direction and value w i l l b e : , and .
Reactions in both remaining directions will equalize to (2.5) ratio if plastic fl ow does not occur. The (2.5) parameters: E -Strain Resistance Modulus and μ -stress components ratio fully defi ne the material behavior. Their dependence on the stress level should be defi ned experimentally.

The friction forces model
The integral load circuit applied on the billet includes ( Fig. 2.1): 1) distributed on whole surface active dummy block load in axis direction; 2) distributed on whole surface active die reaction load in inverse direction; 3) liner radial reaction along whole cylindrical outer surface; 4) acting in axis direction friction forces, caused by radial liner pressure and axial material displacement ( not shown in fi g. 2.1).
Analysis initial relations: whole liner volume fi lled with billet material; the billet loaded till plastic fl ow stress σ z =σ e , which acts to some distance z from die surface. Since there is no peripheral material displacement, the friction forces are absent too. Consider equilibrium of differential disc of width dz 1 of billet (Fig. 2.3 a) on a distance from die surface. Assuming that disc zero position by z 1 is on the border of material axial displacement start. Ergo at that point the friction forces appear.
The next step of present analysis would be the defi nition of the infl uence of friction by liner surface on the billet. Wherein the usual approach, that considers the yielding pressure acting without any change in radial direction, seems illogical.
From (2.5) the radial pressure: Accepting friction law in Coulomb form -the friction force by area unit is proportional to the normal pressure (in the same manner as in [6 ]): , (2.9) where friction factor k is considered as constant from known range [6]: 0.6…1.0. The 1.0 value matches sticking friction relations. The start friction force value by area unit from (2.8): (2.10) The elementary force in random section by z 1 from friction forces region: The billet reaction on friction force analysis was not found in known to author publications. Usually, most of them announce that the shear a) b) Figure 2.3 -a -friction forces diagram; b -reaction stresses distribution in the billet volume stresses exceed the ultimate level, thus a shear strain appears in the layers on some distance from outer surface. This does not seem right. It would be more correct to rely on the general polycrystalline materials property, including aluminum alloys, to resist to outer loads.
The friction force, acting by the surface causes reaction in the billet each section, that equal by value and inversed by direction -N z . . "Superficially" looking. lets consider for example that the reaction distributed in the layer of 10 molecules or 10 crystallites of 10 µm each. It defi nitely contradicts with continuity and isotropy relations, because requires some "jump" of properties on the border. Moreover it contradicts with minimum strain work principal: 3) tangential to two axes and r : ; with r=0, ; with r 1 = r max , ; 4) concaved ; .

Distribution of friction forces along billet axis (z 1 )
Let's observe the equilibrium of differential disc in friction forces region ( Fig. 2.4).
Friction force increment on the element surface, dz 1 length, equal to the billet reaction increment -dN z1 : . At z = 0, is identical to (2.27).

Numerical estimation
The extremal numerical estimation is possible to be made using the process real parameters.
Minimal appraisal at friction factor k ' = 0.6 (in real process was not observed). Radial pressure on liner surface from (2.5): ; friction force on surface unit, on distance z 1 = 1 cm: ; axis stresses increment on 1 cm of billet length: Maximal appraisal at friction factor k '' = 1: N m (1) = 0.493σ e *0.01 ; σ 1max = 0.258σ e . Since the experimental data is absent, let's assume that the summative axial stresses twice as high as the beginning fl ow stress: . Then it is possible from (2.28) to defi ne the full distance of friction forces acting: ; ; ; from last: ; with substitution . (2.30) At k ' = 0.6, z ' = 4.48cm; at k '' = 1, z '' = 2.7 cm.
For real image obtained values: -maximal linear stressesσ zmax = 2σ e = 394.2 MPa; -summative friction force per perimeter length unit -N z = 3.77 MN/m; -billet diameter -d = 7" = 0.178 m; -billet maximal length -L max = 0.9 m . The origin point by z 1 was determined in indefi nite distance from the die. Now we have to move it to the distances z ' and z '' from dummy block. Summative friction forces resistance: .
General press force: . The friction forces in this example are about F f / F g = 0.31 -31% .
As a result the friction forces act only on the small part of the billet length. Here assumed the friction force appears only on the condition of relative movement by contact surface. So, we have adjacent to dummy block short part of billet ( moving with dummy block speed ) with increased peripheral stresses. Arbitrary chosen stress level σ 1 = σ e seems suffi cient for maximal appraisal. It ensured mass transfer in stress gradient direction. Real σ 1 , σ zmax , z '' values should be confi rmed experimentally.
Until now the press force and stresses increasing to fl ow beginning level, σ e , was supposed with dummy block low speed. In real process -1…3mm/s. The stress distribution in the billet is not likely to change with such low speeds, except in two regions: 1) adjacent to dummy block region; 2) die openings region.
3. Liner and dummy block contact region 3.1 Stress state of the region Stresses acting near adjacent surfaces of dummy block and liner (shown in Fig.3.1): at μ= 0.33 , σ rmax =0,493*2σ e = 0.986σ e (3.1) Third direction does not count, because of axisymmetric relations: ɛ θ = 0; u θ = 0 -strains and displacement are absent in this direction. Stresses σ z1 distributed along radius are normal to dummy block surface. They produce friction forces on the surface in radial direction, as reaction -friction stresses in the billet, in the same manner as the friction by liner surface. Maximal friction stress from (3.1): Summative friction force without surface forces accumulation effect: Stress fi eld geometric change by front dummy block surface is not taken into account as well. In numeric estimation from (3.2): At k ' = 0.6, σ' fr = k ' (σ 1 + σ e ) = 0.6 * 2σ e = 1.2σ e ; At k '' = 1, σ'' fr = k '' (σ 1 + σ e ) = 1 * 2σ e = 2σ e .
3.2 Plastic fl ow relations at dummy block region So, there are two mutually perpendicular directions for the material to move in the region with the greatest motion resistance -along dummy block and liner contact surfaces. Whereas the randomness of numerical values used above, it is rightful to state, the fl ow direction makes a sharp angle with both surfaces. And fl ow character will not change principally in all variations of the angle (Fig. 3.3 a). The indirect verifi cation of this representation is the shape of removed from liner material in cleanout cycle. This shape resembles a pig's face and repeats constantly (Fig. 3.2) [11]: The analogue of two cycles, extrusion beginning and cleanout, may be doubt. The main difference between those cycles is that in cleanout cycle fl ow occurs in free space and extrusion start fl ow in the compressed metal space. For the present consider the analogue as hypothesis.
As an option the plastic fl ow presents as shearing "strain'' result. Therefore we choose a model of minimal length shearing surface ( Fig.3.3 b). Than a wedge shaped dead volume in the contact "point" of the two surfaces is expected. The shear will occurred by only one surface. Here we note, the invented scheme may have a simple positive continuation -it is worth to make this shape wedge on the dummy block outer ring [12]. The wedge will work as bulldozer shovel obviously to lessen friction resistance. The region of mass transfer in this model appears as a sector with speed fi eld per-

ВЕСТНИК ОРЕНБУРГСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА 2018 № 1 (213)
Некоторые аспекты механики процесса экструзии алюминиевых ... Борис Нудельман pendicular to radial (of the sector) direction. The sector revolve center on the distance R c = z c , less than friction force region -z max . The sector angle defi nes the end of shearing surface -welded metal region beginning. Regarding the fl ow as stable and regular it is possible to consider mass transfer as sector simple revolving relative to the center till the border of shearing.
4. Flow start. Die entrance region 4.1 Stress state in the die entrance region It seems reasonable to remind initial conditions to avoid uncertainty. Pressure in the billet volume and in die region rises by dummy block action till plastic fl ow through die opening starts. The billet equilibrium (Fig. 4.1) includes: distributed dummy block force in moving direction -P z ; distributed by die surface reaction in inverse direction -R z ; friction forces in limited region near dummy block -F f . In accordance with scheme (fi g. 4.1): . Still we have no signifi cant strains. Therefore the fl at sections hypothesis considered rightful. That allows analysis with a static scheme. Consider round opening in the die with radius r 2 . Stresses σ e act by whole die surface except opening area. To explore the opening region consider a virtual cylinder with forming line as the opening edge and by now indefi nite height -h (Fig. 4.2).
Distributed axial pressure σ e (stresses) acts on cylinder left face. Stresses on right face are absent. Radial stresses by cylindrical surface from (2.5): (4.1) Naturally, the only way to obtain the cylinder volume equilibrium is reaction of adjacent die surface part. Additional stresses fi eld appearance between two fl at surfaces is natural as well. With geometrical relations additional stresses should act in some angle relative to the axis. This is the signed above basic property of crystalline bodies to resist to applied loads. And always the inevitable outcome is the additional strain in stress direction. In some moment a dome shaped convex appears in the opening and limited material displacement in axis direction. The given situation relates to the fi rst billet to be extruded. For second and the following billets with material fi lled die the conditions differ, mostly by increased resistance to the fl ow.
Let's observe an element in drawing area rz of one unit thickness for more detailed exploring (Fig. 4.3).  Here purposely geometrical relations were distort for simplifi cation -sector with radius r 2 replaced with strip of one unit thickness. That is done in the same manner as above when friction forces by liner -billet, billet -dummy block was analyzed. Additional fi eld confi guration may be defi ned with the only criterion -strain energy minimum principal. As fi rst approach let's consider a design scheme with uniformly distributed on the length L die reaction, (Fig.4.4 a). From (4.2) load per length unit: ; strain: .
Strain energy formula: . (4.3) As for die reaction, the energy dependence we have got has no minimum and inversely pro-portional to L. Minimal value at maximal L -. We note, that (4.3) formula not changes for the radial direction. As the second approach let's consider ununiformed distribution of die reaction (Fig. 4.4 b). Whereas stresses on the additional fi eld borders are at zero level, so their maximum does exist, but do not exceed the maximum in outer region -σ e .
Consider p max in the distance not less than from opening edge and triangle shaped load distribution.

From (4.2)
, ; with relation the variation possibility exists. Now current value may be defi ned as: .
Axial strain of line segment dl: .
Strain energy, calculated separately of two strips CG and GE: ; ; . Both energy parts are decreasing functions of L, hence the function minimum is achieved on the limit defi ned as natural length restriction.
Except the presented two made above approaches, it is rightful to solve the problem in general terms: to defi ne function of integral force distribution by die surface (or by random parallel to die surface, that belongs to additional stress field). Perhaps the solution already exists (or it is possible to make it) -minimum of functional: , with relation: ; where -continuous smooth function with at least one boundary point, for example -. Let's consider the second approach scheme with triangle reaction stress distribution for additional stress fi eld exploration ( Fig. 4.5 ), with reaction length restriction -L = 2r 2 ; with maximal feasible stress value σ max : . For simplifi cation let's consider that resultant force applied points of distributed force P = σ e r 2 in each radial section belong to a straight line that connects the two resultant forces points on both left and die borders: F and H. Point With all considerations only one parameter remains undefi ned -h, the distance from die surface to DA area. . .

Борис Нудельман
The value (4.7) we've got approximately the feasible minimal. It may be used in case of stretched die opening with parallel edges also. With additional stress fi eld confi guration defi ned it is possible to continue the analysis. Out of virtual cylinder the uniform axial stress fi eld acts σ z = σ e . Two fi elds superposition makes the resulted fi eld. From (4. 9) . Stress level at point J defi nes, that it should coincide to plastic fl ow start line.

General notes
As shown above, in the die opening region some volume is formed with several times lower stresses relative to whole billet volume. The volume practically unloaded on the die side. Whole stresses distribution in the volume is presented in fi gures 4.6 a and b. Before next step of analysis it is necessary to verify all assumptions we've made till now, because the result we've got is too unexpected (to the author also ).
1. The superposition principal was used at the analysis start. In further calculations it was used repeatedly. 2. We supposed existence of singlevalued smooth function E stress and relative strain interdependence on the extrusion stresses level ≤ 650MPa . 3. Existence of µ -function, that defi nes straight to inverse strains ratio on the same stresses level was considered also. The function in present case defi nes principal stresses ratio in single axis stress state. 4. The strain energy minimum principal as general relation was used. 5. Basic relations: billet material continuity and isotropy were used as mandatory restrictions.
With signed not in all proved with experiments consumptions there is always a risk to get a wrong result. Two trivial conclusions from this may be noted. In fi rst, it is worth to make such experiments, particular for defi ning functions E and μ.
In second, present analysis may in the meanwhile be considered as subject for discussion. Last note relates to following conclusions also. 4.3. Plastic fl ow beginning Stresses fi eld near the die opening is defi ned above (Figures 4.6 a and b). Maximal axial stresses near point J growing to some level will lead to plastic fl ow beginning in radial direction to opening axis. Growing axial stresses , produced by radial stresses , will act as wedge and lead to widening of plastic fl ow region. At some phase of the process unloaded part of volume should be pushed out of opening. The process described here we name as the "tunnel effect" by analogue to potential barrier overcoming.
Let's consider that the radial stresses and axial ones proportionally decrease to zero at opening axis: ; .
Than from (4.9) : . At this point of calculation geometrical simplifi cations have made above are not right if we want to avoid results distortion -real values should be defi ned. Section area element on feasible radius -r*dθ *dr; the stress acts in the element -.
Summative force in sector dθ element -.
Summative force that acts to pull out the volume (plug) of diameter 32 mm and height 11.2 mm: Despite the coarse appraisal, we should note, that the result is close to real.

Plastic fl ow start. Continue
The push out process at fi rst stage is characterized by stresses growing and then after "pushing out", axial stresses σ z ' area increasing -smooth decreasing till full pushing plug out, and stable plastic fl ow forms. More detailed process description seems not right with our coarse model. Particularly, the key questions to be answered are: does the additional stresses fi eld reproduce? Or it's confi guration changes after stable fl ow appearance? -Answers may be achieved (on the author's opinion) in special experiments.Though, as it was made in part 2, it is possible to try to invent feasible fl ow model of our axisymmetric scheme. For example, scheme shown in fi g. 4.7. With continuous thick curve line signed the surface of maximal in each line segment speed. Adjacent curves -surfaces of approximate proportionally decreased plastic fl ow speeds. (Surely, this model was built with infl uence of known experimental results, particularly, experiments with coordinate grids. But author can not to make straight references to other's researches, because their authors give signifi cantly different interpretation of their results. For example as illustration in [13]).
Ensure from stability condition, speed gradient in each section perpendicular to fl ow, not overcome some critical value. It has the meaning -gradient value for each real condition, which not lead to continuity interruption. In traditional representa- tion this means shear strain limitation. Obviously, on the author's opinion, stable fl ow condition may occur without shear strain at all. (Here it worth to add, one more cycle element exists in real extrusion process for fi rst billet, which we did not consider -dummy block stops after profi le exit from die. For the fi rst quality check and profi le clamp in puller jaw.) With considerations signed above there are possible stable plastic fl ow criteria to formulate. Of course with other factors infl uence -temperature, alloy structure and process speed, which we did not consider.
5. Metal plastic fl ow 5.1. General note As it was in part 2, devoted to liner -billet "friction", for plastic fl ow we have no reliably defi ned relations. As to the theory of plasticity criteria, they were formulated with some restricted assumptions and with such a level of abstraction that makes impossible their application to extrusion process analysis [14], [15], [16]. (The author is aware that he intrudes to theory of plasticity territory with such coarse tools as ax and sledgehammer. However, to leave this subject out of examination is unreasonable. And if not a full proof, but at least explanation of general method used for analysis has to be presented.)

Flow relations. Basic dependencies
Let's observe a differential cubic element of billet volume. The cube has edge -dl and its fl ats are oriented by the principal directions ( Fig. 5.1).
Supposed that billet material is in the plastic state (independently to either criterion of plasticity beginning). General range of stress components are: and . (5.1) We should note, that due to above made consideration, with accuracy to volume constant relation, in case: metal fl ow is absent (despite the deviator has signifi cant value ). This means that plastic fl ow possible with the only relation: and in the increment dσ 1 direction. Where dσ 1 is positive and differential: dl.
(5.2) (5.2) has simple physical meaning -stresses gradient. One more characteristic, which cannot be forgotten: direction from σ 1 to (σ 1 + dσ 1 ) is collinear to mass transfer minimal energy. Theoretically possible case: -we do not consider, because it is not realized in the axisymmetric scheme. One more conclusion may be rightful -plastic fl ow speed is the increased function of difference and stresses gradient: ; ;
Adjacent by side surfaces elements are under the same conditions. Stress increment dσ 1 distribution in dl direction accepted as proportional to length. Each element fully counterbalanced by side surfaces with adjacent ones and unbalanced by rear and front surfaces. Hence, the fl ow is possible only from rear to front surface with constant speed. The speed value may be represented as function: , (5.4) where K v -fl ow function by analogue to compliance and inversely: , is only a wish. According to physical sense it may be a higher range function. This mean we have added one more assumption to all made before.

Ununiformed speed fi eld
Consider plastic fl ow state with speed difference in one of the directions (Fig.5.3).
For example, there is a speed difference in σ 3 direction. And in third direction speed difference is absent as in our case of axisymmetric problem.
I.e. in tangential direction σ 2 = σ Q speed difference is absent. Absolute speed difference in dl 3  By the assumed terms, active stress does not change. Hence, the virtual difference (5.7) compensates with fl ow in l 3 direction through element boarder inside it's volume. Here it is relevant to remind, we assumed continuity, isotropy and constant volume relations. So the fl ow difference through front element surface consists of the sum: , (5.8) where Q 1F -summative fl ow difference through front surfaces; Q 1R -fl ow difference through rear surfaces; Q 3 -inversed fl ow inside element in l 3 direction ( Fig.5.4 ).
Mass forces will not be taken in account. Each component of fl ow has energy characteristic as follows: The work spends to transfer Q 1r from rear to front surface W 1 = Q 1r * dl*σ 1 .
The work spends to transfer Q 3 consists of two addendums: fi rst -fl ow transfer inside the element with σ 3 resistance and different displacement value, second -transfer to front surface with σ 1 resistance. The transfer works equality may be used to defi ne two fl ow components ratio. We choose on the side The work for transferring unit of fl ow q 1 to point M: The work for transferring unit of fl ow q 3 to point M: w 3 = q 3 σ 3 y.
Equality w 1 = w 3 defi nes line and volume section to work equality ratio. Remaining part of displacement to front surface is the same for both fl ow components with the same resistance -σ 1 . The line equation: In considered model with accuracy to differential deviation rightful the ratio: .
From above examined example -μ = 0.33, . At x = dl, ; than approximately: According to appraisal about one fourth of the fl ow difference comes from σ 3 direction. 5.5 Several notes regarding results 5.5.1 (The note) As signed above we not consider the state with third direction -σ 2 speed difference. It would not add any sense to process physical essential understanding. Regarding to numerical estimations it would not make any sense also without values experimental verifi cation.

(Essential relation)
The principal stress component ratio is basic for presented plastic fl ow model (2.5). In ideal case, performed transforma-tion accurateness to ensure -the μ(σ) variability should be small-scale. Otherwise we get anisotropy in principal directions. That leads to large deviation of components fl ow ratio. Although, general fl ow estimation will not change signifi cantly. The μ function should not be close to value μ=0.5. This corresponds to ideal plasticity relation, and transform quality alloy to some jelly, which meet the Pascal law for liquids.

(Inapplicability)
The model is inapplicable in unstable plastic fl ow condition. The most probable cause of fl ow instability may be critical speed gradient value exceeding. With considered here model, this is the result of critical stress gradient exceeding dσ 1 /dl >Grad, which leads to material rupture between crystallites with opposite unconnected surfaces appearance, pressed with σ 3 stress, and slides relative to each other. The described above "shear strain" process often exists in practice. Undoubtedly it leads to profi le structure damages and macro defects. The phenomena often cause is an unjustifi ed speed increasing by press operator to enlarge process productivity.
Two regions of enlarged plastic fl ow speeds appear in extrusion process: fi rst -contact of dummy block periphery and liner surfaces where maximal stresses act, and as signed above shear type fl ow is less dangerous; second -die opening region, where shear with rupture is especially harmful.

Press force integral graph at extrusion
Typical graph of aluminum profi les extrusion presented practically in each monograph regarding the topic. Furthermore, it may be observed in each cycle, if the press is equipped with modern control system. Hence, it presented here without references to any source, because it does not hurt copyrights of anybody ( Fig. 6.1). As illustration typical graph shape in [12], [17].
In the same manner as it done in most other sources we subdivide it to specifi c pieces with possible explanation of each one [5]. First graph piece -quick force increasing; linear part of graph -elastic billet compressing and part press metal structure (ram) elastic deformation; includes part of friction force; performed with dummy block slow-motion.
Second graph piece -nonlinear, with slowing force increasing to maximal value -plastic fl ow start. In accordance to present analysis results two regions of billet exist, where the plastic fl ow begins earlier than in the rest of the billet volume: fi rst -dummy block and liner contact region, friction force acting region; second -die opening entrance region, additional stress fi eld acting region, where with stresses increasing the region of "tunnel effect" plastic fl ow. Perhaps one more, third component of force growing to maximum existswhole billet volume conversion to plastic state, which requires additional energy to perform. According to the dislocation theory the energy spends for -redistribution, duplication and relocation of the dislocation towards crystallites surfaces [14], [18]. We still consider this component unknown. Third graph piece -force smooth decreasing till graph infl ection point -dummy block slow-motion continuation -conversion to stable plastic fl ow.
Fourth graph piece -dummy block speed increasing by press operator to the prescribed value. Fifth graph piece -dummy block motion with constant work speed -stable extrusion process. Press force smoothly decrease practically proportionally to dummy block displacement till the work stroke end [12], stop at the butt thickness distance from die. Press force decreasing in most of publications is explained as a result of friction force de-creasing. Where friction forces are distributed by whole liner surface [15], [19]. The only word may be added, to avoid unwanted critique is that, it is not proved. The friction force in presented model very restricted, proportionally decreasing during work stroke, acts on a short distance from dummy block. Almost the whole volume of the billet, during the work stroke is not only in a plastic state but even in plastic fl ow state. Flow state as was shown above is performed with energy and force spending. The billet volume decreases proportionally to dummy block displacement. Hence, the force has to decrease in the same manner. The indirect verifi cation of it may be obtained by comparison with indirect extrusion graph, where plastic fl ow concentrated near moveable die. Except for restricted fl ow volume, the process characterized by the similar components as direct extrusion: billet friction on die and liner contact surfaces, additional stresses fi eld and "tunnel" fl ow near die openings. Unfortunately, author have no own experience in indirect extrusion, and does not consider himself rightful to use other authors results by the same cause as was signed above. Metal fl ow on the fi fth graph piece is ununiformed with changing fl ow directions dependent on dummy block position, with uneven speed distribution by section. That is an evidence of stresses uneven distribution. There are both force components in integral scheme axial and radial directed as well. Liner to billet axial reaction always exists. It may be considered as "static frictional force".
In general, the force -stroke (pressure -ram displacement) graph should be examined as a huge 7.2 Billet -liner, billet dummy block friction forces acting model was developed. The presented model does not contradicts with physical relations. The level of friction forces infl uence on extrusion process appraised. By appraisal, billet -liner friction force does not exceed about 31% of summative press force. The region of friction force acting was defi ned. The region restricted as rather short distance from dummy block, not more than the distance equal to liner radius.
7.3 Probable model of plastic fl ow in dummy block region is presented. It is characterized of single and minimal area shear surface.
7.4 Investigation of stress state in die opening region was fulfi lled. Additional stresses fi eld in the region revealed. Confi guration of the fi eld was defi ned with numerical estimation. The fi eld in great degree defi nes the necessary stress level for metal fl ow start and stable process. The algorithm of maximal fl ow stress defi nition was developed. In real example fl ow stress about six times exceeds the ultimate stress level. 7.5 The fl ow model with restricted shear strain was presented. The model was built on example of round die opening. General "tunnel" type fl ow scheme was defi ned, bypassing axial direction. (The name "tunnel" considered in analogue of potential barrier overcoming in tunnel effect). Preferable fl ow direction is radial, relative to die opening. 7.6 Essential interdependencies and general relations of billet metal fl ow are presented. 7.7 The analyses of press load by dummy block displacement graph was fulfi lled. No contradictions with the theoretically achieved results were revealed. The results experimental confi rmation and further introduction to practice are proposed. 18.12.2017