A new approach to deformation zone analysis

A new approach to deformation zone analysis

Int. J. Mach.Tool Des. Res.yol. 22, No. 3, pp 215 226. 1982. Printed in Great Britain 002(~7357/82,'0302 5 12 ~3.00'0 Pergamon Press Ltd. A NEW APPR...

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Int. J. Mach.Tool Des. Res.yol. 22, No. 3, pp 215 226. 1982. Printed in Great Britain

002(~7357/82,'0302 5 12 ~3.00'0 Pergamon Press Ltd.

A NEW APPROACH TO DEFORMATION ZONE ANALYSIS M. C. SHA~' Professor of Engineering, Arizona State University, Tempe, AZ 85287, U.S.A. IReceited 29 September 1981 ; in t!nalJbrm 1 May 1982)

Abstract - T h e deformation zone approach to the solution of practical deformation processing problems in the workshop introduced by Hill and elaborated by Green et al. is a very useful concept based on two-dimensional (plane strain ) slip line field theory. A key element of this approach is an estimate of the intensification of die pressure due to inhomogeneous strain as the ratio of die contact separation (h) to die contact length (L) increases above one. Since this estimate is based on slip line field theory that assumes the work material to be rigid where it is not plastic and to act in plane strain, it cannot be used to answer questions concerning residual elastic stresses or brittle fracture (such as the central burst problem). A new approach to deformation zone geometry based on elastic rather than rigid plastic behavior is presented and compared with slip line field results. Unlike the slip line field technique, the new approach is applicable to 3-D (axisymmetric) as well as 2-D problems and in addition, provides a limit of safety beyond which a central burst or centerline residual tensile stresses may be expected to present a problem. INTRODUCTION

There is need for a reliable approximate approach to forming problems for use in the workshop for diagnostic purposes that involves relatively simple input data and calculations. What has been termed a deformation zone method [1] has gradually evolved for this purpose. A central element of this approach is the treatment of the effects of deformation zone geometry upon strain inhomogeneity and the resulting influence upon die face pressure and ultimately upon deformation energy. This paper is concerned with a new treatment of strain inhomogeneity that augments and extends the presently employed slip line field technique. However, before discussing the new approach, it may be well to review briefly several key steps in the present procedure. To focus attention, strip (2-D) and bar (axysymmetric, 3-D) drawing will be considered where it is useful to employ an example. However, the principles may be applied to rolling, extrusion, forging and other hot or cold forming operations with suitable changes in details, particularly with regard to friction and strain hardening. DEFORMATION ZONE ANALYSIS Figure 1(a) shows a strip drawing operation in which h/L is less than one. For such a case, an initially square grid will deform as shown and redundant work (work not directly associated with final change in shape) if die-friction are negligible. If the material is nonstrain-hardening, the specific energy to deform the material will be u =aylnhUhx

(1)

where af is the flow stress of the material (yield stress for non-strain-hardening material). The normal stress on the die faces (p,) will be afjust as in the case of a plane strain compression test [Fig. l(b)]. The drawing force Pl may be found by equating the internal work per unit time to external work per unit time for a unit width perpendicular to the paper in Fig. l(a).

/h \

a / l n | 7 0 | ( h l × 1)VI = P I Vt \nl /

(2)

or,

13) where I/1 is the velocity of the exiting material and Pb is the axial stress in the exiting strip (drawing stress). 215

216

M.C. SHAW

The value p~)obtained from equation (3) will be too small in general for real situations for the following reasons. 1. Strain hardening is not included. 2. Die face friction is not included. 3. Redundant energy is not included. The stress--strain curve for a strain hardening material may be expressed as follows to a good approximation : ~r=K,d'

(c
o-=A +Be

(4a)

(c>l)

(4b)

where K and n are constants (n = strain hardening index) and A =K(I

-n)

(5a)

B = Kn.

(5b)

While equation (4a) is well known, there is relatively little data in the literature for plastic strains greater than one. However, data that is available [2, 3] are in excellent agreement with equations (4) and (5). For strip drawing, equation (4a) will suffice and the mean flow stress will be [4]

ado df =

;--; .

.

dc

K ~. d c .

.

.

.

*~

K c]

1+ n

a1 --

(61 •

1+ n

)

where al and e1 are the flow stress and strain at die exit. The effect of strain hardening may be included to a good approximation by substituting o=i for o-r in equation (3) and the mean stress on the die face will be @. If the coefficient of friction (/~) along the die face is constant, the mean friction stress will be /~dy and the additional normal stress associated with friction will be/~ds cot ~ (Fig. 3), where c~ is the die semi-angle. Thus, the normal stress on the die face with friction and strain hardening will be @ (1 +/~ cot c~)instead o f a I. I f d I (1 +/~ cot c~)is substituted for a I in equation (3), friction and strain hardening effects will be well accounted for in a strip drawing case like Fig. l(a). The alteration of equation (3) to take care of inhomogeneous strain (redundant) effects requires some sort of field analysis and Hill and Tupper [5] have applied the slip line field technique to estimate the change in mean die stress due to variation of the aspect ratio of the deformation zone (A = h / L in Figs 1 and 2). The slip line field technique is limited to plane strain analysis and assumes the material to be rigid beyond the zone where it is plastic. An admissible slip line field involves fines of constant plastic shear stress associated with a compatible set of velocities (hodograph). Since the material beyond the plastic region is assumed to be rigid, the stresses involved may or may not be consistent with the actual elastic stress field that pertains. Hence, a proper slip line

h

I° 1 t. ',- 1",--i

t I I1 soI

...... t"'

Fl(i 1. (a) Strip drawing for A = h/L < I (strain approximately homogeneous). (b) Equivalent plane strain compression.

A N e w A p p r o a c h to D e f o r m a t i o n Zone Analysis

217

solution is kinematically admissible but need not be statically admissible. Such a solution represents an upper bound to the true solution just as one that is statically admissible, but not necessarily kinematically admissible, represents a lower bound to the true solution [6]. When upper and lower bound solutions coincide, they. of course, both correspond to the true solution. Hill [7] estimated the increase in mean normal stress on the die in an upsetting operation with increase in h/L by first considering a block of material resting on a frictionless surface (Fig. 4) indented by a flat frictionless punch of lateral extent L located a vertical distance h/2 from the frictionless surface. Hill's slip line field is shown in Fig. 4, which corresponds to a consistent hodograph. The mean compressive stress on the punch face (,6) for any value of h/2L may be obtained by applying Hencky's theorem in the usual way. Next, considering the plane at the centerline of a strip subjected to opposing loaded surfaces to be a principal plane (Fig. 5), Hill obtained the variations of stress intensification (C = p/2k) with h/L given in Fig. 6, where 2k is the plane strain flow stress. The rising curve was terminated at h/L = 8.74, where p/2k becomes equal to the stress intensification for a flat punch operating on a semiinfinite solid. The abrupt change in Fig. 6 at h/L = 8.74 is due to the fact that the mode of deformation predicted by slip line field analysis changes from downward flow beneath the punch to the upward flow assumed in the slip line field approach to hardness. Fig. 7 is the well known slip line field due to Hill [7] for a flat, frictionless punch loaded against a semi-infinite body. The stress intensification value (fi/2k) is equal to 1 + rt/2 = 2.57 for this case. The relation between the uniaxial flow stress (Y) and the flow stress for plane (2k) is found to be 2k=

2Y _ = 1.15Y

,,/3

(7)

by application of the von Mises flow criterion. For an axisymmetric indentor f~/Y = 1.15 [1 + (re/2)] = 2.97 which is in excellent agreement with experiment. Earlier, Hill and Tupper [5] presented the slip line field solution shown in Fig. 8 for frictionless plane strain drawing of a non-strain-hardening strip under conditions where strain inhomogeneity was significant. As A = h/L increased, it was found that the mean stress on the die face increased (in accordance with Fig. 6 for static upsetting). It was subsequently shown by Green [8] that the intensification of die pressure due to strain inhomogeneity given in Fig. 6 for plane strain upsetting would also pertain for steady flow as in strip drawing. Green also considered the role of friction on die face pressure and found that there was an important interaction between friction, strain hardening and strain inhomogeneity. When h/L > 1 this could be taken care of by use of a strain hardening correction factor to obtain the mean normal stress on the die face (f,) approximately as follows: j,, = ~ (

(8)

where d R is related to the area under the uniaxial stress strain curve out to a strain of [C In (ho/h ,)] instead of In (ho/hl) as for the case where C = 1 (i.e. h/L > 1). The drawing stress (PD) may be estimated by replacing the flow stress (c~a-)in equation (3) by d~ and C In (ho/hi)for the strain, that is PD = or* C In

(ho/h I )

{v,< /'/ ,/ ~

ho

±

FI(i. 2. Strip d r a w i n g for ~ - t7 g > 1.

(9)

218

M. C. SHAW

/zc~ FIG. 3. Approximate increase in normal stress on die face due to friction.

t.

/I//

////////

FIG. 4. Slip line field for punch deforming ideal plastic resting on frictionless surface (after Hill, [7]).

L

FIG. 5. Slip line field for ideal plastic subjected to opposing loads.

c=~II~TT i

I

Z

I

i i

I I I o

I

I I

I

l 2

A

I 4

6

B

8,74

IO

A : h/L Fl¢~. 6. Variation of stress intensity factor C =

p,,'2k

with A =

h,,'L

(after Hill. [7]).

A New Approach to Deformation Zone Analysis

219

P

Fl(;. 7. Slip line field for plane strain hardness indentation (after Hill, [7]).

FL~;. 8. Slip line field for strip drawing through frictionless die (after Hill and Tupper, [511.

-~

I0

]

IO

2

3 IO

IO

IO

4

I0

5

6

I0

R FIc;. 9. Variation of drag coefficient ('~; with Reynolds number R for two dimensional flow past a

circular cylinder with axis normal to flow direction tafter Eisner, Third Int. Cong. Appl. Mech. Stockholm, 1930 [9]).

where C is the factor by which the mean die pressure is increased due to strain inhomogeneity. The value o-~ is the mean flow stress, including strain hardening out to the equivalent strain (C In ho/hl). SIMILARITY

TO

FLUID

MECHANICS

The approximate m e t h o d of estimating mean die and drawing stresses is in principle analogous to that employed for estimating drag in fluid mechanics. F r o m dimensional analysis, the drag D on a b o d y of projected area A in a fluid of density p flowing with a mean velocity 1' is found to be equal to [9] [)

D = CD2 12,4

(10)

where C1~ is a non-dimensional quantity called the drag coefficient which is a function of the

220

M.C. SHA~A'

shape of the body and the nature of the flow as characterized by a non-dimensional Reynold's number (R). It is found experimentally that CDvaries with R in a manner similar to that of Fig. 9 for bodies of all shapes, a change in shape resulting only in a change in the coordinate scales. It is further found that at low values of R, inertia forces are negligible relative to viscous forces and that the opposite is true for high values of R. It is thus possible to formulate greatly simplified theories for the low and high Reynold's number regimes since either inertia forces or viscous forces may be considered to be insignificant and both need not be considered simultaneously. For example, when R is below about one in Fig. 9, the theoretical Stokes law (inertia forces ignored)is in excellent agreement with experiment. Fortunately, many of the important engineering applications involve either a high or low Reynold's number so that the more complex problems involving both inertia and viscous forces are not frequently encountered and when they are, a bounded analytical solution proves to be better than no solution at all. Fig. 6 is a non-dimensional curve that may be used in an analogous way to estimate the mean stress on the face of a forming die. The quantity (A = h/L) which characterizes the shape of the plastic zone is the counterpart of the Reynold's number. When A is less than one the effect of strain inhomogeneity is negligible, C may be taken to be one and only the effects of friction and strain hardening are significant. The mean stress on the die face will then be

k

/"=

In \, h

i +;i

<1 +,cot~).

(ll)

When A is greater than one, strain inhomogeneity is important and the effect of friction is relatively unimportant. The mean stress on the die face may then be expressed

Ck ft. =

l I' 7 In -

1 +n

~12)

The quantity in braces is the mean stress (based on the area under the stress strain curve) to a strain of In ho/h 1 in the first case and C In ho/h ~ in the second case. The sudden drop in CDat a Reynold's number of about 5 x 105 in Fig. 9 is due to a sudden decrease in pressure drag when the boundary layer becomes turbulent. The counterpart of this is the central burst discussed below that often occurs when drawing at values of A that are large. Developments in fluid mechanics have led those in plasticity timewise by several decades. For example, the Reynold's number was identified as a useful unifying concept in the 1880s while Hill did not illustrate the importance ofA = h/L until the late 1940s. In view of this, it is to be expected that fluid mechanics will provide still other hints of how best to simplify complex plasticity problems. Since the quantity A = h/L plays such an important role in the solution of problems in engineering plasticity, it deserves a name and it would appear appropriate to call it the Hill number (H = A). Also, since the relation between the stress intensity factor C and tt A is so important, it deserves further study and the remainder of this paper is devoted to a lower bound approach to augment the upperbound one that already exists. (Fig. 6) LOWER BOUND APPROACH An alternative theory for indentation hardness to the upper bound slip line field approach has been presented by Shaw and DeSalvo [10]. This assumes that the stress on the indenting punch may be estimated from considerations of elasticity. During indentation, the material flows plastically within a volume extending to the elastic plastic boundary determined by the flow criterion (von Mises or Tresca). After flow ceases, the material will again be elastic and the stress on the indenter may be computed from elasticity theory, assuming that the yield

A New Approach to Deformation Zone Analysis

221

shear stress pertains on the elastic-plastic boundary. Since velocity considerations during plastic flow are ignored and static equilibrium is satisfied by the theory of elasticity, the resulting solution may be considered to be a lower bound one. It should be noted that the elastic stress field after initial plastic flow will be somewhat different from that before plastic deformation. This effect is ignored here as it is in fracture mechanics when estimating the size and shape of the elastic plastic boundary at the tip of a crack. The effect of ignoring the change in stress distribution in the plastic zone both here and in fracture mechanics is to predict an elastic-plastic boundary that is too small. In this lower bound approach, two well known elastic solutions for a loaded semi-infinite elastic body have been employed that due to Hertz [11] for a distributed load and that due to Boussinesq [ 12] for a concentrated line load of intensity P per unit length. Figure 10 shows lines of constant shear stress for a Hertzian distribution of stress beneath a plane strain indenter (left side) and an equal concentrated load solution according to Boussinesq [right side). The Boussinesq concentrated line load is displaced a distance b = 0.167 (where L is the width of contact of the plane strain indenter) above the plane surface so that the lines of constant shear stress coincides at a reasonable distance from the applied load. The values of M in Fig. 10 correspond to M =

(13)

21~

where/~is the mean stress on the punch face and % is the maximum plane strain shear stress. Reasons for introducing the Boussinesq solution are (1) that it provides a convenient means for determining the line of maximum shear stress corresponding to the elastic-plastic boundary and (2) it provides a convenient expression for the vertical principal stress at a point on the load line distance d from the applied load. Since the material beneath the punch must be plastic all the way to the edge of the punch E, the elastic plastic boundary will be the Boussinesq circle passing through E [heavy dotted circle for which M = 0.3 in Fig. 10). The only non-zero principal stress at D will be 2P ~r3 = ~ t

i/

3~

IU

2S 0

,, !

45*

oF MAX

(14)

J

\

/

,M Fk~. 10. Comparison of lines of maximum shear stress for two elastic solutions Hertz left of centerline and Boussinesq right of center line. Line through D is the elastic-plastic boundary (after Shaw and DeSalvo [10]).

222

M, C. SHMA'

where P is the concentrated Boussinesq line load per unit length and

P = IYL.

(15)

At point D on the elastic-plastic boundary, a3 = 2k where 2k = plane strain flow stress according to the yon Mises flow criterion (2k = (2/x/3) Y = 1.15 Y, where Y is the flow stress in a uniaxial compression test). By substituting M = 0.3 and r m = k into equation (13) for point D on the elastic-plastic boundary and 2k for a 3 in equation (14), equations (13), (14) and (15) may be rearranged to yield (Shaw and DeSalvo [10]): b = 0.167 L d = 1.667 L (16) L

= 1.500

P = 2.62 2k

and

This value of/~/2k is very close to that predicted by the slip line field analysis of Hill (2.57), considering the degree of approximation of both theories. While the slip line field analysis for indentation hardness of a semi-infinite body is quite accurate once it has been assumed that the material will be rigid where it is not plastic, the fact is that real materials are not rigid but actually elastic. As shown experimentally by Shaw and DeSalvo [10], the assumed slip line flow zone is in excellent agreement with that observed for a composite consisting of a soft layer on a hard substrate (a plastic-rigid combination) but in poor agreement with the observed flow zone for a real homogeneous material (a plastic~lastic situation). The value of C = f/2k = 2.57 for a semi-infinite body (A = o0) thus appears to be one that should not be applied to a homogeneous material on the basis of a complete lack of agreement between the assumed and observed plastic flow zones. Since the constraint factor for indentation hardness is primarily determined by the extent of the elastic plastic boundary which is assumed nonexistent when the material is considered to act in a rigid plastic manner, the relatively good agreement between the observed and calculated (2.57) constraint factors in the case of a slip line field analysis appears to be fortuitous. This explains why the limiting lower bound result (ff/2k = 2.62) is slightly higher than the limiting upper bound result (f/2k = 2.57). However, the lower bound approach presented here is also approximate in that it assumes the same elastic field to hold before and after plastic flow. The lower bound (elastic) approach may also be used to estimate the increase in mean die pressure (/Y)with change in deformation zone geometry, i.e. change (A = h/L). Fig. 11 shows two opposing plane strain punches of axial extent L separated a distance (h) such that the normal stress (2k) is just reached at point C according to Boussinesq or Hertz. The horizontal centerline of the strip constitutes a surface that reflects identical stress patterns on each side. The stress at all points on the horizontal centerline will thus be twice as great as the values corresponding to a singly loaded semi infinite body the same distance from the centerline. Thus, to ensure plastic flow at the centerline, stress o-3 at point C given by equation (14) must equal k. Therefore, from equations (14) and (15)

2pg rrd fi rc(1.667) or . . . . . . . . . 2k 4 O-3 = k

=

2p ~z(1.667) 1.31.

(17)

A New Approach to Deformation Zone Analysis

223

FI(;. I 1. Strip subjected to two opposing loads under plane strain conditions such that the material at C is just plastic IA = h/L = 3.001.

The corresponding value of

h/L may h L

=

be found as follows : 2e L

= 2(1.50) = 3.00.

This point which represents the limiting geometry for which the material at the center of the strip will go plastic in the absence of an induced axial tensile stress is shown at A on Fig. 6. For values ofA = h/L greater than three, an axial tensile stress must be present at the centerline of the strip in order to have plastic flow across the entire cross section. When this axial tensile stress reaches a critical value, a central burst will occur. The value h/L = 3 may be taken as a very conservative limit below which a central burst will not occur. Values of fi/2k may be similarly determined for cases such as Fig. 12 where the elastic plastic boundary does not reach the horizontal centerline. The compressive stress at D will just reach the flow stress value (2k) due to the combined effects of loads Pt and P2. Thus, from equations (14) and (15)

¢7I?,=G 1 + 0 2 =

2P1L1 ~ -

PLASTIC

h

rrd I

2fi2L 2 -~- _ _ . _

red2

Fl(;. 12. Strip subjected to two opposing loads under plane strain conditions such that the material at the centerline remains elastic (k = h/L = 6,00). Point D is on the elastic-plastic boundary in the absence of the axial tensile stress required for the material to go plastic across the entire section.

224

M. C. SHAW

but

/)1 =/)2, LI d~ = A + 2bl

L

dl and L -- 1.667

= L2

dl - A + 2(0.167) - 1.667 L

L

= A - 1.333 and hence

aD = 2k = ~ ~1.667 + & - i.333 fi

or

2k

7[

= 2[0.60 + (1/A - 1.333)]'

(18)

Figure 13 shows the lower bound curve corresponding to equation (18) superimposed on the upper bound curve of Fig. 6. This lower bound curve is asymptotic to the horizontal line corresponding to the hardness solution (#/2k = 2.62) which it should reach only when A = zc. It should be noted that the lower bound approach does not predict upward bulging of the material in either a hardness test or for strip drawing at a large value of A(A ~ 10). If a hardness test is performed on a semi-infinite body, little or no upward bulging is observed--the material displaced by the indenter is accommodated by an increase in density of the huge elastic region surrounding the plastic zone. Figure 14 shows curves corresponding to Fig. 13 for axisymmetric (3-D) situations. AXISYMMETRIC

CASES

While the upper bound slip line field approach to inhomogeneous strain holds only for plane strain situations, such as strip drawing, the lower bound approach holds equally well for plane strain and axisymmetric cases. Equation (18) and Fig. 12 will hold for axisymmetric cases simply by replacing (p/2K) with p/Y. CENTRAL

BURST PROBLEM

Beyond a value of A of three an axial tensile stress must develop at the center of a bar in order that the plastic flow criterion be satisfied. This is necessary since plastic flow must

3I

2.62

2K

J 2

~..s

""

J.. --, - -

"

I

Y O-T y

i1+~/"

J

l--m-

,

~

I

% ~z/

Y 0

I

2

_

o-T I I 4

6 A

8

8-74

I0

= hlL

FIG. 13. Variation of die face stress intensification factor C (p/'2k) for plane strain with aspect ratio of deformation zone (A = h/L). Solid line is upper bound curve and dashed line is lower b o u n d curve.

225

A New A p p r o a c h to D e f o r m a t i o n Zone Anatvsis

extend across the entire cross-section. The magnitude of this tensile stress will depend upon the balanced biaxial compressive stress (a,,) acting at the center of the bar where 2fL

4ff

From equation (18) and noting that 2k --+ Y for a bar

[0.60

(201

+1/(6-1.333)]I(~)+0.167 ]

The required axial tensile stress (a r) will be ( Y - ]vcl) according to either the Tresca or Von Mises flow criteria and therefore from equation {20) ~r = 1 Y

ac Y

1

2 [0.60 + 1/(A -- 1.333)]

+ 0.167

]

(21)

~rc/Y and c, f l Y are also shown plotted against A in Fig. 13. The fracture that occurs in a central burst is chevron shaped, the chevrons pointing in the direction of drawing, and has the appearance of a tensile type fracture. While a tensile stress is present in the axial direction for values of A > 3, this stress is far too low to account for fracture directly according to the maximum tensile stress criterion that should obtain. The source of the central burst tensile fracture will be considered in a later paper. DISCUSSION

The slip line field approach to problems involving plastic flow is very useful for estimating energy required, but is not as reliable where the distribution of stresses is involved as in questions concerning fracture. The slip line field technique assumes the material to be rigid where it is not plastic. While this assumption has little to do with the kinematics of flow and integrated values of stress, since elastic strains are generally negligible relative to plastic strains, it has a good deal to do with the equilibrium aspects that govern the details of stress distribution. In evaluating a slip line field, the stress must be known or assumed at some starting point. In the case of Fig. 8, Hill and Tupper assumed a stress at D such that the horizontal stress integrated from A to F was zero as it should be for strip drawing without a back stress. However, this represents a necessary but insufficient requirement for obtaining a 297 3

3.00

,

2

0

i

I I I I I

I I I I 2

4

6

8

t

I0

8.74

Fl(,. 14. Variation of die face stress intensification factor (_.t = p. y for a x i s y m m e t r i c deformation with aspect ratio dx = h/'k Solid line is u p p e r b o u n d curve and d a s h e d line is lower b o u n d curve.

226

M.C. SHrew

correct distribution of stress along A D F . Similarly in the case of Fig. 4, the horizontal c o m p o n e n t of force at S was taken to be zero, which is the case when the flat cross-hatched surface is frictionless. However, the slip line field of Fig. 4 will correspond to but one configuration (value of A) in strip drawing that where the material just reaches the plastic state at the centerline (A = 3 according to the lower bound approach). For values of A greater than about three the material at the center line of a specimen will remain elastic. The lower b o u n d solution presented here is not without some limitations. It has already been pointed out that the elastic stress field will be altered somewhat as soon as a plastic zone develops beneath the load and this has been ignored. A second source of approximation is that the Boussinesq solution is for a semi-infinite b o d y and a strip of finite height (h) subjected to opposing loads is not a semi-infinite b o d y relative to either load considered separately. However, the presence of the two opposing stress fields largerly cancels this objection since an opposing stress field will play essentially the same role as the missing material in opposing elastic deformation. Nevertheless, the lower bound approach presented here is believed to offer a reasonable approximation to the distribution of the stresses that pertain. It would appear prudent to use the a p p r o a c h (upper b o u n d or lower bound) that is most conservative in engineering applications. This corresponds to the upper b o u n d for estimating mean die face stress and mean drawing stress but the lower b o u n d approach for estimating the stresses that arise within a product as it is worked. An important application of this is the central burst problem that sometimes arises at large values of A. In conclusion, it should be again pointed out that while strip and rod drawing have been taken to illustrate the concepts involved, the basic approach may be applied to many other forming operations. However, in doing this, the treatment of friction (involving the so-called friction hill in forging and rolling) and strain hardening will be quite different.

The author wishes to acknowledge a grant from the National Science Foundation Division (Dr. S. R. Mosier, project director) that has made it possible to have valuable discussions with Professor Eiji Usui of the Tokyo Institute of Technology concerning some of the basic concepts. He also wishes to acknowledge valuable discussion from one of the reviewers of this paper, whoever she or he may be. Acknowledqement

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

REFERENCES W. A. Backofen, Deformation Processiny, Chapter 7. Addison-Wesley, New York (19721. T. Z. Blazynski and I. M. Cole, Proc. Inst. nwch. Enqrs U.K. 174, 797 (1960). G. Lankford and M. Cohan, Trans. Am. Soc. Met. 62, 623 (1969). J. A. Schey, Introduction to .~anufacturinq Processes, McGraw-Hill (1977j. R. Hill and S. J. Tupper, J. Iron Steel Inst. 154, 353 (1948). J. M. Alexander and R. C. Brewer, Manu]iwturin.q Propertie~ ¢?]Materials. Van Nostrand Reinhold, London (1963). R. Hill, The Mathematical Theory of Plasticity. Oxford University Press (1950). A. P. Green, Proc. Inst. mech. Engrs U.K. 174, 847 (1960). J. C. Hunsaker and B. G. Rightmire, Engineering Applications ( f Fluid Mechanics, McGraw-Hill (1947). M. C. Shaw and G. J. DeSalvo, Trans. Am. Soc. mech. En~lrs 92, 469 (1970). H. Hertz, Gesammelte Werke, Leipzig (1896). J. Boussinesq, C.r. hebd. S#anc. Acad. Sci. Paris 114, 1510, (1892).