Plastic deformation of metal: Theory of simulated sliding

Plastic deformation of metal: Theory of simulated sliding

Wear, 38 (1976) 43 - 72 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands PLASTIC SLIDING DEFORMATION OF METAL: THEORY 43 OF SIMULA...

2MB Sizes 0 Downloads 42 Views

Wear, 38 (1976) 43 - 72 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands







V:PIISPANEN ,Helsingint

2A, Kauniainen



(Received December 24, 1974; in final form September 25, 1975)

Summary The theory of simulated sliding has already been applied for a long time in connection with the examination of plastic deformation in continuous chip formation. The theory is also suitable for the examination of other continuous deformation processes, e.g. where new material flows uninterruptedly through the deformation area as in rolling, extrusion, drawing and in the measurement of deformation strength. The theory of simulated sliding enables mathematical equations to be derived and applied to the calculation of speed, output, torque and rolling force in flat rolling etc. An example of the application of the formulae is presented for the case of hot rolling.

1. Introduction The most important metals possess a polycrystalline microstructure, the behaviour of which in plastic deformation is illustrated by a model constructed of rigid plastic pieces (Fig. l(a)).




Fig. 1. (a) Grain of deformed steel. (b), (c) The grains in which the deformation occurred are located in narrow zones; the deformation is thus inhomogeneous.

44 rG

N/mm2 60


50 1 LO 30


20 1 10 7





r =1-h/H

Fig. 2. Calculated

values of the shear strength.

The plastic deformation of such materials takes place inhomogeneously inside the grains. Deformed steel consists of glide packages which have slipped inside a grain (Fig. l(a)) [l] . Deformation occurs almost entirely on slip surfaces whilst the material between them remains practically unaffected; deformation is thus microscopically inhomogeneous. However, the plastic deformation of rigid plastic material is also macroscopically inhomogeneous. Those grains in which deformation occurs form narrow zones and are called deformation zones. Figures l(b) [2] and l(c) [3] show two examples of plastic deformation caused by rolling. Figure l(b) shows two deformation zones and Fig. l(c) four deformation zones, this being the most common amongst the plastic deformation caused by rolling. Many researchers [4 - 61 have found that their experimental results do not coincide with known plasticity theories. Figure 2 shows values for the shear strength To calculated using the equations of Piispanen et al. presented later. Test values were taken from Wallquist [7] who used a 0.1% C steel at a temperature of.1000 ‘C; the differences between the calculated To values are small. 2. Theory

of simulated


Figures 3(a) - 3(d) are diagrams of simple plastic deformation and the drawing of steel strip where the deformation occurs in one deformation zone [ 81. In the figures, (1) is the unworked strip, (2) is the worked strip, (3) is the die and (4) the drawing force. Deformation occurs in the deformation zone J between the points A and C, and is bordered by the lines 5. Deformation takes place inside the grains where slip is presumed to have just occurred. The single grains have been drawn large for simplicity. Different grains have slipped differently. The grains which slip first are those with their weakest slip planes in the most favourable orientation with respect to the direction of the effective force. In the slip zone there can also be grains which do not slip. The plastic deformation of steel under normal circumstances takes place as translational slipping. The plastic deformation of certain metals can also happen in other



(d) Fig. 3. Schematic representation of slip during the drawing of steel strip.

..~.. --_---..-..... ...._.__,, .-_._,_.____. C

‘L_J 9'


-----.--.----lo ~/





M ...o’ MN=OP



Fig. 4. (a) Plastic deformation during the drawing of steel strip. (b) The process of (a) represented schematically.

ways, e.g. mechanical twinning. The plastic deformation of a polycrystalline substance is a microscopically complicated process. Defor’mation takes place along the line K (Figs. 3(b) and 3(c)) and the deformations occurring in the zone J have a similar general effect to deformation by slip along the line K, between the unworked material 1 and worked material 2 and in the corresponding plane.



Mcosuriny at deformation st rtngth

Continuous chlv formation


/ Ia __-.__

Fig. 5. Deform&on



.I _ ..--..._



processes and formation of deformation strips.

During the process of continuous flow of material through the strain area, the strain area retains its place in relation to the tool. In a strip being worked the strain area shifts continuously. The shifting may be simulated. It is presumed that the strain area shifts periodically (Fig. 3(d)). The material divides into lamellae and deforms by sliding between the newly formed lamellae and the unworked material. Figure 3(d) shows a lamella (6) in the process of sliding. It is attached to the worked strip 2 but is sliding on the plane K with respect to the unde,formed material 1. Figures 4(a) and 4(b) illustrate the case. Figure 4(a) is from a drawing test where a strip of 2 mm thick stainless steel was drawn to 0.8 mm thick. The roundish grains in the polished side surface of the strip gained an ellipse-like shape approximately foIlowing the drawing direction of the strip. It was observed that deformation had occurred mainly on the line AC. Figure 4(b) shows a schematic diagram of the same deformation. Before the test, circles were drawn on the side surface of the workpiece. In stage 8, a circle is still unaffected. On the deformation line AC, one-half of the circle 9’ is still unaffected but the other half has transformed to 9” due to slip of the lameflae. Point 0 has shifted towards point P, so that NM = OP. The completely deformed circle has changed into ellipse 10. This is a combination of mechanics and mathematics [ 91 similar to chip formation [ 10, 111.


3. Continuous



In a continuous stabilized deformation process [8] new material flows through the strain area uninterruptedly as in the case of drawing, extrusion, rolling, measurement of deformation strength and continuous chip formation (Fig. 5). The working mechanism consists of deformation zones which can be distinguished in Fig. 5 by their dark lines. The subdivisions of deformation modes differ in the number of deformation zones; the arrow represents the deforming force. The force affects the workpiece: after the deformation area in drawing; inside the deformation area in rolling; and before the deformation area in chip formation and extrusion. 4. Rolling of sheet and strip Rolling is one of the main modes of plastic deformation

[2, 3, 8, 12 -

181. 4.1. Observations from experiments on rolling Several research rollers were constructed for the tests. The dimensions are indicated in Fig. 6(a). 4.1.1. Test (A) Figure 6(b) shows the research roller used in this test: R = 10.0 mm; H = 2.50 mm; h = 2.08 mm. The test bar was 2.50 mm wide, recrystallized annealed 0.15% C mild steel, 3.50 X 3.50 mm. During the cutting process a strain-hardened 0.5 mm layer formed on its surface was removed from each side of the bar by grinding. The bar was rolled until continuity was attained in the strain process. The emerging surface was polished but not etched so that only marks from deformation showed. Polishing powder was used between the rolls and the test bar, corresponding to the scaling which occurs in hot rolling.

t ~





Fig. 6. (a) Symbols used for rolling. (b) The miniature roller, underside upwards. When the handwheel 3 is turned, the rolls 1 rotate synchronously and the test bar 2 is rolled. R = 10 mm.





Fig. 7. (a) Start of deformation. (b) A deformation pattern resembling the letter X. (c) The process in Fig. 7(b) represented schematically.

Fig. 8. Photomicrograph

of point A (cc Fig. 7(c)), test (A).

Fig. 9. The miniature roller, R = 23.84 mm.

The roller and test bar were then placed on the specimen stage of a metallurgical microscope and the test was begun. As the rolls started turning slowly, elastic deformation occurred. However, no marks appeared on the side surface of the bar. Further turning of the rolls caused a sudden configuration of lines to appear as shown at points B and G in Fig. 7(a), indicating that the deformation process had begun. Deformation consisted of the sliding of packages inside the grains. As the rolling continued the deformation area grew but the signs of deformation appeared only in four deformation zones (Fig. 7(b)). A diagram of the process is shown in Fig. 7(c); areas 1 and 2 did not exhibit deformation marks. Figure 8 shows the center of the test bar (point A in Fig. 7(c)). Slipping extended in many directions. The roller and test bar were removed from the specimen stage and were examined from various positions, the source of light being situated in an inclined position behind the observer; a deformation figure resembling the letter X could be seen (Fig. 7(b)). The angle of the side surface plane of the test bar was about 15” to the plane of the photograph.





Fig. 10. Illustrations of test (B) (see text).

4.1.2. Test (B) A miniature roller (Fig. 9) was used: R = 23.84 mm; H = 3.374 mm; h = 3.330 mm; width B = 1.65 mm. The test bar was a recrystallization annealed 1% Cr, 1% Si steel. Figures 10(a) - 10(d) are from different stages of the test; W is the roll, V is the test bar, with C-C being the center-line of the bar. The linear reduction was slight: r = 1 -h/H = 0.013. The deformation zone S occurred suddenly, like a collapse, on the polished side of the test bar (Fig. 10(a)). The area, about 10 grains wide, was curved and located partly outside the roll gap on the curve AB (Fig. 11). Strain appeared where the subject material became overworked, i.e. on the cylindrical surfaces corresponding to the curves AB and AG. Grains with the weakest slip planes were in the most favourable orientation with respect to the effective force and slipped first, causing a chain reaction in the surrounding material. The slipped grains diminished the resistance of the other grains on curves AB and AG (Fig. 11) and a number slipped. Each grain that slipped increased the possibility of its neighbouring grain slipping and many did slip. This form of slip creates a balance by reducing the stress at the point which is under strain, thus causing slip to cease. When rolling continues, the stress grows and the area S grows longer and broader. In Fig. 10(b) the deformation zone S has reached the edge of the test bar; a deformation area T has also appeared reaching as far as the side edge of the bar. When rolling continues areas S and T become broader and join


(a) Fig. 11. Mechanism of strip rolling when the reduction is very small.


Fig. 12. Illustrations of test (C) (see text).

(Fig. 10(c)). On further rolling, a new curved deformation area U appears (Fig. 10(d)). The rest of the deformation occurs in the same area AB (Fig. 11) when rolling is continued. The distribution of strain areas is not uniform as the reduction is small (r = 0.013). 4.1.3. Test (C) The miniature roller (Fig. 9) was used: R = 23.84 mm; H = 3.440 mm; h = 3..330 mm; r = 0.032. The test bar was a recrystallization annealed 1% Cr, 1% Si steel. The photomicrographs (Figs. 12(a) and 12(b)) are provided with the abbreviations W for the roller, V for the test bar and C-C for the centerline of the bar. The rolling of one end of the bar was continued until the continuous strain process began. The strain marks were polished off from the side surface of the bar and the test was renewed. The curved strain marks appeared on the side surface of the bar on the arcs AB and AG (Fig. 11). When rolling continued, the arcs of the deformation area broadened (Fig. 12(a)) because when the bar was being rolled in the direction of the arrow K the earlier signs of strain also shifted in the same direction and the new marks broadened the deformation area. The spot where the strain occurred also stayed the same with respect to the center-line of the roller. In Fig. 12(b), the edge of the test bar, point Y, represents the point where the roll and the test bar meet. It can be observed that strain also occurs outside the roll gap and that it happens before the roll touches the test bar. Tests (B) and (C) deal with the cases when line BAG (Fig. 11). Signs of deformation could CA and DA; here reduction was very small and DA were elastic - the packages did not slip nor marks be seen.

strain occurs only at the not be observed on the lines the strains on lines CA and could any deformation

Fig. 14. The miniature roller. These rolls had grooves.



Fig. 15. Illustrations of test (E) (see text).

4.1.4. Test (0) The roller in Fig. 6(b) was utilized. The test bars were steel St 37 in the ho&rolled condition. Two test bars, one upon the other, were rolled at the same time. The radius R of the rolls was 10.34 mm. The added thickness of the two bars was H = 1.80 mm before the test and h = 1.38 mm after. The linear reduction r was~O.233. The test bars were bent as shown in Fig. 13, i.e. following the surface of the roll. No crushing of the rolled material along the roll occurred during the rolling [19] . The rolled test bars moved at a speed near the angular velocity of the roll. 4.1.5. Test (Ii’) The test was made with the machine shown in Fig. 9, equipped with rolls with grooves as shown in Fig. 14. The two test bars 1 and 2 could be rolled side by side in the same roller. By examining the surfaces 3 between the test bars 1 and 2, events from the middle of an unsplit bar could be

Fig. 16. Illustration

of test (F) (see text).

recorded. This type of rolling in a groove has the advantage of preventing broadening of the bar from its sides and thus solely plane deformation can be examined. The test was made with the following measurements (Fig. 14): R = 24.0 mm; H = 3.43 mm and h = 3.00 mm; width B = 2.50 + 2.50 mm; reduction 0.125. The surfaces of the rollers were even and clean and were used without polishing powder - in contrast to test (A). The test bar was an annealed tin bronze Tp 107: Sn 6.7870, P 0.24%. The surfaces 3 between the bars were polished and the bars were rolled until continuous deformation occurred; Fig. 15(a) shows how the deformation took place. The line BAG (Fig. 15(b)) can be clearly differentiated, whereas the line CAD is not clear because earlier deformed material in the line BAG, which showed deformation marks, crossed over to line CAD. Figure 15(c) shows the area BAC. 4.1.6. Test(F) The test bars rolled in test (E) were removed from the roller, their surfaces 3 (Fig. 14) were repolished and the test was renewed by rolling the bars a further 0.20 mm. The results can be seen in Fig. 16; the strain areas BAG and CAD are apparent. The abbreviations are the same as for Fig. 15(b). 4.1.7. Test (G,! A roller with R = 100.0 mm was used (Fig. 17(a)). The test bar was an annealed 0.15% C mild steel. The thickness of the bar was H = 4.86 mm before the test and h = 4.22 mm after and the width B was 4.50 mm. The test bar was rolled initially until the state of continuous deformation was created. The test bar was removed from the roller and the emerging surface polished and rolled a further 0.35 mm. In the area C,AD1 (Fig. 17(b)) the deformation is not clear. In the area MCIDIN deformation is slight.





__ -I h-4.22 N






Fig. 17. (a) Research roller with

R = 100 mm. (b) Schematic representation of test (G)

(see text).

Fig. 18. Simulated deformation of rolling. Fig. 19. Drawing: mechanism of four deformation zones.

4.2. The rolling process in terms of mathematics The fundamentals of the theory have been developed for a simplified case in which no account is taken of (1) broadening of material sideways, (2) roll flattening and (3) glide between the rolls and test bar. The calculated strengths and moments are applicable only to the deformation work involved.



S’ /

Fig. 20, Sliding speeds and rolling pressure of roiling. Fig. 21. Qlculation

af peripheral speed UQof roll.

The tests in question involved deformation usually in four deformation zones: AB, CA, DA and AG (Fig. 18). In Fig. 18 the sheet which is to be rolled is marked 3 and the sheet which has been rolled 4. ~eform&tlon did not occur in areas 1 and 2, the material moving as a solid entity [16]. Such movement is only possible when 1 (and 2) move at the angular velocity of the roll. The strain occurring in the defo~ation zones can be simulated as shown in Section 2. It is assumed that the deformation lamellae are very narrow and that defecation occurs as slip at the boundary of, for example, the deformed and the undeformed material. It is also assumed that slip occurs pe~odic~ly. Figure 18 ~lus~a~s this situation. Area 1 is moving at the angular velocil;y of the roll [ 161. The thin lamella 5 is attached to area 1 and slips with respect to ares 3. S~ul~eously, the thin lamella 7 is attached to area 4 and slips with respect to area 1. Below the center-line the events are similar (thin lamellae 6 and 8). The lamellae move together with the sheet in the manner illustrated in Figs. 3(d) and 19. The abbrevia~ons used in the equations are illustrated in Fig. 20.


4.2.1. Equation for peripheral speed of the roll [3] An equal material volume flows through each flow surface section in one unit of time. If it is assumed that the width of the bar is unity, the following equation is derived (cf. Fig. 21): vzh/2 = v4u = vIH/2


v4 = vzh/2u in which v4 is the flow speed on the center-line speed v3 is given by

of area 1. The peripheral

v4 2R


The right-angled


= -2R + u

” = R + u/2

triangle gives

(R + u)~ = (R + h/2)2 + a2


2Ru + u2 = hR + h2/4 f a2 Equations

(1) and (2) give =


v2hR 2Ru + u2

eqn. (3), v2hR

v3 =


hR + h2/4 + a2

and from this U2=l+k,a2 4R v3



4.2.2. Equations for slip lines and slip speeds [3] The equation for curve CA (Fig. 22) can be derived from right-angled triangles: (R+d)2


2Rd + d2 =x2+hR+h2/4-2Ry-hy+y2 If the flow of material


is constant,

v5d + v2y = v,h/2 vg =

v&/2 -



This is due to material

= v3(R


R movement

at the angular velocity

of the roll. It gives


Fig. 22. Equation for curve CA (see text).



2~2yR -


= d(2R

Fig. 23. Equation for curve AB (see text).

+ d)


Using u3 from eqn. (4) gives

hR+>+a2 -2Ry--p--

2a2y h

= 2Rd + d2


Equations (5) and (6) give h2 i2 + hR + --2Ry-hy+y2=hR+-+a’-2Ry-----








and fram this x2+y2+



Slip line CA (Fig. 22) is part of the circumference, the center of which is in line with the centers of the rolls (line QS, Fig. 20). The radius of curve CA is R2



Slip line AB is also part of a circumference (Fig. 23) and its center is in line with the center of the rolls (line QS). The radius of curve AB is R3 = (c2/4 + a2)lj2



where c=-


h2 2a2 +-+--2R-h WH



The slip speeds in plastic rolling deformation can be obtained from Fig. -20, and the following equations can be derived: UAB

R -H/2

u; + u; - 2uius


+ h/2










4.2.3. Equations for output and torque [ 121 Deformation occurs as slip on the circumferences and on the corresponding cylinder surfaces. In each case the speed of the rolling cylinder is the same. The output required in deformation can be calculated by multiplying the circumferential speed of the cylinder by the area of strain and by the slip strength. Slip on the cylinder surface AB (Fig. 20) requires an output PAB: PAB

= vAB



The length of curve AB is nR3 (arcsin b/R3 - arcsin a/R,)

AB =



when (Rg - a2)l12 2 H/2, and AB=

nR3( 180” - arcsin b/R3 - arcsin a/R3) 180


when (Ri - a2)l12 s H/2. The radius R3 is derived from eqn. (9) and the length of b is given by b = {R(H - h) - (H - h)=/4}l’=


Slip on the cylinder surface CA requires an output PCA given by P&, =





The length of curve CA is cA = nR2(arcsin a/R,)


180 when a 1 h/2, and CA=

nR2(180° - arcsin a/R=)

when a 5 h/2.




Fig. 24. Equations for rolling pressure (see text).

Slip on the surface DA (Fig. 18) requires an output PCA, whilst slip on the surface AG requires an output PAB. The total output required for rolling deformation is Pw where pW

= 2(pAB


+ PCA)

The torque A4 required M = R P,/lOOO

from a pair of rolls in rolling deformation us

is given by (22)

The measure a (cf. Fig. 20) in the preceding equations influences the outputs PAB and PCA so that an increase in a causes an increase in PCA and a decrease in PA,. Thus the sum PA, + PCA depends on a and a certain value of a gives a minimum value to that sum. During the rolling process the slip figure tends to shape so that the rolling output is at its minimum value P w min and this is indicated: a/b = i when Pw = Pw min. The value of i can be selected by computer through application of the above equations. In the case when the determination of i is required to be more specific, consideration should be given to the fact that rocA > ToAB because strain-hardened material from the cylinder surface AB moves over to cylinder surface CA. 4.2.4. Equations for the rolling force Figure 24(a) shows half the test bar. Deformation takes place by slip on the cylinder surfaces AB and CA. In Fig. 24(a) the small vectors F1, F2 etc. indicate the slip forces on the surface AB. If the correct scale is chosen, the vectors join with the chords of the curve AB. The sum F3 of the small vectors is equivalent to the chord AB. An equal but opposite force F4 is required to overcome the force F3 (Fig. 24(b)). A force F,, is needed to bring about the force F4; this force is effective at right angles to the longitudinal axis of the bar and is defined as F AB

(b--a)’ = 2rGA~






Force FCA car1 be derived in the same way (cf. Fig. 24(c)): FCA


270cA B (h/4 + a2/h)

The total rolling force is F where



Fig. 25. Deformation strengths calculated according to eqn. (27) for hot rolling, depending on the pass sequence.

F=FBA+FCA By assuming that a = b/2 and that 7oAB = Tot. imate equation is derived:

= To the following


rGB(2-r) 2 or


2F rG = B(2 __-r)


The validity of the approximate formula (27) has been tested by comparing the calculated values with the test results. The test values were obtained with steel St 42 [20]. Figure 25 shows the calculated values plotted as the ordinate. The sum Cr (the abscissa) is the total of all earlier deformations including the most recent: pass 2, H = 470 mm, 1100 “C; pass 10, H = 149 mm, 1050 “C. It can be seen from Fig. 25 that the value of 76 increases pass by pass. There are two reasons for this. First, at each pass the size of the crystals becomes smaller and the strength of the material increases. Second, the temperature of the material under rolling decreases from one pass to another. 4.3. Range of applicability of the equations Tests have so far proved the range of validity 0.5 < m

< 1.5

of the equations

to be (26)

The validity region A is indicated in Fig. 26; the figure is drawn on a log-log scale. The validity region is bordered by the lines m = 0.5 and m = 1.5. Area A ends at line G, where the bite angle is 20”. The equations in Section 4.2.2 have been derived without considering roll flattening. This is possible for the case of hot rolling where the strength of the material is


Fig. 26. Range of application of the equations indicated. Fig. 27. Plastic zones for the case where m

< 0.5.

small compared with the strength of the roll. In the following, two cases from outside the region of validity are presented and their mechanisms described. Figure 27 presents the case where m < 0.5. Strain occurs on the slip surfaces AiB, CAi, AsG and DAs. In the area Al As strain has occurred as compression [2,13]. In the areas G and B (Fig. 27) the case of indented slipping is presented (see Fig. 12(b), point Y). Figure 17(b) shows diagrammatically the case when m > 1.5. Strain was caused by slipping on the planes AB, C&A, AG and DIA. In the area CIDINM plastic deformation was slight. At points MC1 and ND1 roll flattening occurred. 5. Extrusion of strip The following concerns only cases where the extrusion die is solid. Cases in which the die is partly formed by the so-called dead metal of the material being extruded are not dealt with. The abbreviations are illustrated in Fig. 28. Figure 29 shows extrusion test material Al 99. The thickness H of the test bar was 21 mm before the test and 13.6 mm after. The test bar was constructed of three sheets 7 mm thick, giving a total width B of 21 mm. Equally distributed grooves were cut on the intermediate surfaces of the sheets parallel to the line AC (Fig. 27). Copper wires of 0.5 mm thickness were inserted in the grooves. The photograph in Fig. 29 was taken after the test. It can be seen that the copper wires and their intermediate distances are the same, except for the area near surface AK. Here the friction force deformed the bar considerably and bent the copper wires. The following equations apply to the extruding force:




Fig. 28. fa) Extrusion: mechanism of one deformation zone with the fixed die 3. (b) Extrusion: mechanism of four deformation zones with the fixed die 3. Fig. 29. Photograph

of the extrusion



l I I




Fig. 30. Drawing: mechanism of four deformation zones. The zones are marked with the figure 6. Simulated sliding occurs on line K and the corresponding plane.

~/~ +








sin (Y 4 - 2r - 4~0s at/i--r

J'E~ =


sin ty 4E

The friction of the internal rather indeterminate, If


walls of the extrusion

(301 die makes the deformation

F ~1 min= r(2 - r) TGHB/( 1 - r)


tan QI’A r(2 - r)/2(1 - r)





Fig. 31. Research device for investigation of chip formation, mounted on the microscope stage.

then sin (IL’= r/(2 - r)

6. Drawing


[21- 251

The abbreviations

are illustrated

in Figs. 8, 19 and 30.

6.1. Drawing of strip 6.1.1. Deformation in one zone For deformation in one zone see Figs. 3(d) and 4. The following tions can be derived: F Dl

H/h + h/H - 2~0s a =

sin LY





F D1 min = r(2 - r) 76 hB/(l - r)


tan (Y’= r(2 - r) /2( 1 .- r)


p’ = 45”-




FDlf min = 2(taIl CY’+ p) 7~ hB






(d) Fig. 32. Drawing: research equipment and test.

6.1.2. Deformation in four zones Figure 31 shows the research equipment mounted on a microscope stage; Fig. 32(a) shows the underside of the equipment. Point 2 in the moving slide 1 shows the space reserved for the head of the bar being drawn. The fixed body has room 3 for the die. The die is shown in Fig. 32(b) and the test bar in Fig. 32(c). Figure 32(d) is a photomicrograph of the drawn test bar. The bar was first drawn until the steady state strain began, and then polished and drawn to 0.15 mm; Fig. 32(d) shows this situation. Four deformation zones can be differentiated. Figures 19 and 30 are diagrams of the mechanism of deformation in the four deformation zones in the drawing of strip. The slipping lamellae are indicated in black. The slip planes are marked with the letter K; the angles a and 0 are the same as in Fig. 3(d), area AC. To create one series of lamellae, the thin end of the strip has to be drawn a distance CI (Fig. 19); the thicker end then travels a distance MN. The slip distances of the material are thus BM and JI. The friction distances along the die wall are BN and CJ, these being of equal length. The drawing force calculations are based on the fact that the output done by the external force is the same as the sum of the outputs done by internal forces and on the fact that angle 13is such that the output is a minimum. The equations for deformation in the four zones are


,I 0.2



I /I /








I ’ 4 ,’ -I,‘/ /’ I/,/’ 0 ”






= r/(2 -r)



10 s"

Fig. 33. Drawing of strip: abscissa, semi-angle 01; ordinate, linear reduction r. Areas of occurrence of various deformation mechanisms.

F D4f F D4f

4 - 2r - 4~0s cx~/iT =


+ 2P

sin (YdYi77

min =





2~) TGhB


and then sin cy’ = r/(2 - r) The following



are derived (cf. Fig. 19):

h=a+fl tan p =

j -= m

(43) tana{l-r++(l-r)(l+ta.rP~)} (44)

r+ tan20

tan e( 1 - r) (45)


The areas of occurrence of the various deformation mechanisms are shown in Fig. 33. The semi-angle cr is plotted as the abscissa and the linear reduction r as the ordinate. The minimum value FDu min of eqn. (41) is obtained from the line OE. It is assumed that in each area the deformation mechanism occurs which is the easiest in that particular area. Area DOE (Fig. 33) gives eqn. (40) its smallest value, whereas area COD gives the eight-zone-mechanism formula (46) a minimum drawing force value. 6.1.3. Deformation in eight zones in the drawing of strip For area COD in Fig. 33 the equation for the required drawing force is 2 + 2x/F--

4cos (II e/ll-r

sin Q vi77 The mechanisms elsewhere [22, 231.


in areas BOC and EOI have been described

rGhB (46)


Fig. 34. Deformation mechanism for measurement of deformation strength 76. Fig. 35. The testing device for the measurement of deformation strength.

6.2. Drawing of square bar and wire [22 ] Equations are presented for the force required for the drawing of square bar and wire. 2-r 4-2r-4cosadiT FDs4f



sin cr d-




This applies for area DOE (Fig. 33). F DS8f

2 + 2Ji=7=


4cos (Y e-i=7

sin (Y 4\i7


7Gs (I+ _)” l-r


Equation (48) also applies in the area COD (Fig. 33). 7. Measurement of deformation strength 76; In an ordinary tensile test, plastic deformation occurs when the stress on the test bar is between the yield point and the ultimate tensile strength. In the deformation process of simulated slip the deformation strength is between the shear yield point and the ultimate shear strength, and the magnitude of the deformation strength depends not only on the quality of


the material (also temperature, speed of deformation etc.) but also on the strain hardening of the material. It depends on the ratio e/g where e _= g

slip distance thickness

of the lamella

of slipped lamella

(Fig. 34). Here the slip strength in the simulated slip process is called deformation strength and is designated To. It is possible to measure the deformation strength To of a material by utilizing the one zone mechanism (Figs. 3(d) and 34) characteristic of the drawing of strip [26]. Figure 35 shows an example of the construction of the measuring device. The main body 4 is fixed to the slide of the tensile testing machine. The test bar is marked 1 and is fixed to the body of the tensile testing machine. The undeformed part la of the test bar slides in the groove 2a of the part 2. Part 2 is pivoted by the pin 3 to the body 4. The test bar is guided in its movement by part 5, which moves on rolls 6. These rolls exert pressure on the main body 4 through the wedge 9. When the test bar is drawn, a force F, is produced which exerts pressure on the main body through parts 2 and 7. The drawing force F6 in the test bar is measured using the tensile testing machine. The compression force F7 in part 7 is measured with strain gauges. During the test, the force F6 starts from zero and increases. When it stops increasing the steady deformation state has been attained and the measured force F6 can be used in eqn. (49) below. It is advisable in the test to use a value of the angle e’ that fits eqn. (37) -. contraction will then not occur and the deformation cross-sectional area will not decrease during the test. The deformation strength To of a material is studied by making several tests on bars with different reductions (e.g. r = 0.1, 0.2, 0.3, 0.4 and 0.5). The angle (Y’is defined by eqn. (37) and is calculated for each test deformation strength To as follows. The forces F6 and F7 are measured for each test (each reduction) and To is given by To = (F, - F6 sin cr’)/HB












Fig. 36. Results ordinate, 76.

of a test. The material

used was tin bronze

with 7.28%

Sn. Abscissa,



Fig. 37. Schematic diagram of continuous chip formation. Fig. 38. Schematic diagram of extrusion of strip.

The To values can be presented as in Fig. 36. The ratio e/g is taken as the abscissa and To as the ordinate. The deformation strength ro appears as a function of the ratio e/g (magnitude of deformation in the material). The value of the ratio e/g is known for each particular case of deformation. 8. Continuous chip formation [9 - 11, 271 In the following, orthogonal cutting and the case where the width B of the workpiece is less than the width of the tool are dealt with. 8.1. Detachment of material from the workpiece Deformation in the material is presumed to occur as in the theory of simulated sliding, i.e. the material moves as lamellae. In order for the thin plate ACLM (Fig. 37) to slip it must first shear off from the workpiece along the line AM. The necessary force F1 is given by F1 =kB


8.2. Formation of detached material into a chip 8.2.1. Deformation of chip material according to the theory of simulated sliding The mechanism for the continuous formation of a chip (Fig. 37) is, in principal, the same as that in one zone deformation in extrusion (Fig. 38). In the formation of a chip the angle 0 is free to assume such a magnitude that the necessary force Fz is a minimum. Equation (29), derived for extrusion, is also applicable in continuous chip formation provided that the angle /I (and function h) are such that F2 = Ftmin; H and (Yare constant. The derivation of eqn. (29) gives Fz = 2

1 - cos (Y

sin OL



Thus the formula is applicable in both continuous chip formation and the







Fig. 39. Schematic figure of the underside of the research device: 1, workpiece; 2, chip; 3, tool; 4, body of the research device; 5, moving slide.

extrusion of strip but with the aforesaid condition. It is assumed that no friction occurs between the tool and the chip (nor between the material to be extruded and the die). Since 1-





sin (Y the equation

may be presented

F2 = 2 tan ((Y/2) To HB

in the form (53)

8.2.2. Test of continuous chip formation The test was performed with the research equipment shown in Figs. 31 and 39. Figure 39 shows the underside of the equipment diagrammatically: fixed body 4, moving slide 5, workpiece 1, chip 2 and tool 3. Figure 31 shows the device mounted on the stage of a microscope. The electric motor of the device produces a cutting speed of 0.001 mm s-l. Such a cutting speed enables a detailed examination of plastic deformation to be carried out. The test bar was St 37.11 steel in the rolling condition. The angle (Y was 65” and the cutting depth H = 0.90 mm. The bar was machined until the deformation steady state was attained. The bar was polished and the deformation marks were removed. Machining was continued by advancing the tool 0.025 mm. Figure 40(a) shows this situation, i.e. what has just happened in the test bar. The deformation area ANP has a fan-like shape and curves towards the undeformed side, i.e. into an area which has not undergone any work-hardening. Figure 40(a) shows the grains which have just slipped. As working continues the slipped grains shift in the direction NP. Figure 40(b) shows diagrammatically all the grains which have slipped in the area ANP. On the left side of the line AN there is material which is completely unaffected; beyond the line AP there is material which has all changed into chip and which is work hardened.



(b) Fig. 40. (a) Photomicrograph of continuous chip formation. (b) Diagrammatic representation of the deformation according to (a).

8.3. Equation for the force Fs required for continuous chip formation Taking into consideration the separation of chip material from the workpiece, the formation of material into a chip and the friction between chip and tool, the force F, can be derived; it has the same direction as the travelling tool. The fictional depth Hfi is used as the cutting depth (Fig. 40(b)). Through derivation of the formulae in ref. 27 the following equation is obtained: Fs =hB+2tan


Nomenclature a



42 i F AB FCA F Dl

F Dlf F D4f F D8f F DS4f F DS8f FEN





F6 F7 g

h H

Hfi i j, m k

defined in Figs. 20, 21, 23, 24(a) and 24(c) = i when Pw = Pw min in rolling length of circumference AB, Fig. 20 (mm) defined in Figs. 20, 21, 23, 24(a) (mm) width of strip or chipping width (mm) length of circumference CA, Fig. 20 (mm) defined in Fig. 22 (mm) slip distance of the lamellae, Fig. 34 (mm) F A* + FCA, force required for the whole process of rolling deformation, Fig. 20 (N) force required for slip on the surface AB, Fig. 20, 24(b) (N) force required for slip on the surface CA, Fig. 20, 24(c) (N) drawing force, one zone deformation, friction excluded, strip drawing (N) drawing force, one zone deformation, friction included, strip drawing (N) drawing force, four zone deformation, friction included, strip drawing, Figs. 19, 30 (N) drawing force, eight zone deformation, friction included, strip drawing (N) drawing force, four zone deformation, friction included, drawmg of square bar or round wire (N) drawing force, eight zone deformation, friction included, drawing of square bar or round wire (N) extrusion force, one zone deformation, friction excluded, strip extrusion, Fig. 28(a) (N) extrusion force, four zone deformation, friction excluded, strip extrusion, Fig. 28(b) (N) force necessary to shear off the chip material from the workpiece (N) force necessary for deformation of the detached material into a chip, friction excluded, Figs. 37, 38 (N) force necessary to shear off the chip material from the workpiece, to deform the detached chip material and to overcome the friction between the tool and the chip, Fig. 40(b) (N) force in Figs. 34, 35 (N) force in Fig. 35 (N) thickness of a slipped lamella, Fig. 34 (mm) thickness of strip after deformation or thickness of chip (mm) thickness of strip before deformation or chipping depth (mm) fictional chipping depth, Fig. 40(b) (mm) a/b defined in Fig. 19 (mm) detachment coefficient (N mm-‘)


M PAB PCA PW Ik R2 R3 s u/2 VAB vCA Vl VP

v3 v4 v5 xv


cl , p” P’ x I-1 P TG rGAB ‘GCA

torque required in rolling deformation (N m) output required for slip on surface AB (N m s-l) output required for slip on surface CA (N m s-l) required for deformation by rolling (N m s- ‘) PAB + PCA Y output 1 - h/H, linear reduction radius of roll, Figs. 6(a), 17(b), 18, 20 - 24(a) (mm) radius of circumference CA, Figs. 20, 22, 23 (mm) radius of circumference AB, Figs. 20, 23 (mm) cross-sectional area across the drawn square bar or round wire (mm2) defined in Fig. 21 (mm) speed at which the material slips on the surface AB, Fig. 20 ( m s-l) speed at which the material slips on the surface CA, Fig. 20 (m s-l) speed of the strip when going between the rolls, Fig. 21 (m s- ’ ) speed of the strip when coming from between the rolls, Fig. 21 (m s-l) peripheral speed of the rolls, Fig. 21 (m s- ‘) speed of material in Fig. 21 (m s-l) speed of material in Fig. 22 (m s- ‘) coordinates in Figs. 22, 23 (mm) angle in Figs. 3(d), 28(a), 28(b), 30, 33, 34, 35, 37, 38, 40(a) and 40(b) (“) size of angle when force is a minimum (’ ) angle in Figs. 3(d), 19, 28(a), 30, 37 and 38 (“) size of angle when the force is a minimum (“) angle defined in Fig. 19 (“) mean coefficient of friction between the die and the material to be drawn frictional angle between chip and tool, Fig. 40(b) (“) mean deformation strength (N mmW2) mean deformation strength on the surface AB, Fig. 20 (N mm-‘) mean deformation strength on the surface CA, Fig. 20 (N mmm2)

References 1 V. Piispanen, 0. Piispanen, R. Piispanen and R. Eriksson, Schweiz. Techn., 29 (1963) 402 - 405. 2 R. Piispanen, R. Eriksson and 0. Piispanen, Blinder Bleche Rohre, 3 R. Piispanen and 0. Piispanen, Tek. Aikak., 55 (6) (1965) 67 - 69. 4 P. W. Whitton, Wire Ind., (1958) 735 - 748. 5 R. B. Sims and H. Wright, J. Iron Steel Inst., London, 201 (1963) 274 - 280. 6 T. Lehmann, VDI Z., 103 (1961) 1249 - 1253. 7 G. Wallquist, Jernkontorets Ann., 138 (1954) 539 - 572. 8 V. Piispanen, Arch. Eisenhiittenwes., 44 (4) (1973) 261 - 265. 9 V. Piispanen, Tek. Aikak., 27 (1937) 315 - 322. 10 E. M. Merchant, J. Appl. Phys., 16 (1945) 267 - 275. 11 V. Piispanen, J. Appl. Phys., 19 (1948) 876 - 881. 12 0. Piispanen, R. Piispanen and R. Eriksson, Bander Bleche Rohre,

Arch. Angew. Wiss. 8 (1967)

819 - 821.

261 - 269,

7 (1966)

189 - 193.

12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27

R. Piispanen and 0. Piispanen, Konepajamies, Finland, (1966) 347 - 348. R. Piispanen, Ann. C.I.R.P., 17 (1969) 349 - 351. R. Eriksson and T. Huhtelin, Stand. J. Metail., 1 (1972) 69 - 73. J. M. Alexander, Proc. Inst. Mech. Eng., London, 169 (1955) 1021 - 1030. F. A. A. Crane and J. M. Alexander, J. Inst. Met., 91 (1962 - 63) 188 - 189, 320. H. Ford and J. M. Alexander, J. Inst. Met., 92 (1963 - 64) 397 - 404. Taschenbuch fiir Eisenhiittenleute, 5th edn., Wilhelm Ernst, Berlin, 1961, p. 745; Hubert Hoff and Theodor Dahl, Grundlagen des Walzverfahrens, 2nd edn., Stahleisen, Dusseldorf. 1955. P. Diestel, Stahl Eisen-Ztg., 78 (1958) 1536 - 1546. V. Piispanen and R. Piispanen, Wire Ind., (1966) 55 - 58. R. Piispanen and 0. Piispanen, Draht-Welt, 52 (1966) 792 - 795. 0. Piispanen and R. Piispanen, Wire Ind., May (1967). 0. Piispanen, B&rder Bleche Rohre, 9 (1968) 480 - 481. 0. Piispanen, R. Piispanen and R. Eriksson, Bander Bleche Rohre, 6 (1965) 580 - 582. 0. Piispanen, Jernkontorets Ann., 154 (1970) 211 - 213. R. Piispanen, Ann. C.I.R.P., (1969) 561 - 563.