Ideal forming operations for perfectly plastic solids

J.Mech.Phys.Solids,1967,VoI.15,~~. 223to227. Perganmn PressLtd. Print&in Grwt

IDEAL

FORMING

OPERATIONS PLASTIC By

Department

PERFECTLY

SOLIDS

R.

of Applied Mathematics

(Received

FOR

Britain.

HILL and Theoretical

21stFebruary

Physics,

Cambridge

1967)

SUMMARY IT IS SHOWN that steady-state forming operationsof a general kind can be designed so that in the working zone the streamlines are trajectories of principal stress, either exactly or to a close approximation. Initially transverse planes are then perpetually convected as surfaces orthogonal to the streamlines, and so finish as transverse planes. Advantages of this type of deformation in regard to a low energy consumption are discussed.

1.

CHARACTERISTICS

OF AN

IDEAL

OPERATION

THE PRESENT analysis is relevant to any steady-state drawing, ironing or extrusion process where material enters and leaves the working zone in streams of uniform cross-section. Dimensional changes being prescribed, we seek to determine die profiles that minimize the work of deformation. To make the problem precise the ma,terial is supposed isotropic, rigid/plastic, and non-hardening, while the working surfaces are assumed to be smooth. Frictional losses in practice can be kept low by choosing an ideal profile with a sufficiently short contact length. To eliminate redundant work that does not contribute to the reduction in section area, it is desirable that material elements traversing any infinitesimal streamtube should not be sheared longitudinally. This is achieved if everywhere in the working zone, and in particular along the die profile, the local streamline coincides with a principal axis of strain-rate; further, the material should enter and leave the zone smoothly, without any abrupt change of direction. Such deformation will be called ‘ ideal.’ Its distinctive feature is that the streamlines automatically admit a family of orthogonal surfaces which are convected with the material and which start and finish as transverse planes [as originally remarked by HILL (1966) for plane strain processes]. The overall deformation may of course still be inhomogeneous in other respects and may involve transverse shearing within the orthogonal surfaces. Consider the energy absorbed in ideal deformation of a material element, with unit volume say, during its passage through the working zone along a streamtube of infinitesimal area. Since the motion is steady and the material is incompressible, the total energy is equal to the incremental work of deformation (per unit volume flux) of all the material currently within the streamtube. And, by the principle of virtual velocities, this is just the difference between the longitudinal tensions at the 223

R. HILL

224

ends of the tube, since within the zone the tractions over its curved surface hypothesis purely normal and do not contribute. Symbolically, [Ol]

=

where ~1 is the tension

l1 _t (53 63 +

01

I(

along a streamline

a3

~3)

dt

=

and square

arc by

W, say,

(1)

brackets

denote

its overall

increase in the working zone; (al, us, us) and (~1, ~2, ~3) are the principal components of stress and rate of strain; t is a timelike parameter, and the integration follows the

element

different

through

the

streamlines.

fractional

change

zone.

This

However,

strainpath

since

in area, if not in shape,

applies,

strainpath

a maximum

in a streamtube

work

ei =

is convex

inequality

of total

s

E( dt

can

the

for same

is not expected

and the usual ‘ normality be written

in respect

’

of the

state.

tension

whatever

Then,

(; = 1, 2, 3)

logarithmic

(al*, us*, as*) is any yieldpoint (27, 0, 0) say.

be different

undergo

:

where are the components

necessarily

the value of the integral

to vary markedly across the workpiece. When the yield function of the material flow-rule

will generally

all tubes

strain

for a considered

For the latter

streamline,

choose, in particular,

the transverse

strain

ratio

and

a uniaxial

es/es may

be?

Thus, the actual work of deformation per unit volume exceeds the work that would be done in a tension test to the same reduction in area?. This can be asserted for every

streamtube

and hence

also for the whole workpiece.

In simple

operations

that merely reduce the cross-section without altering its shape any excess in (3) can fairly be regarded as redundant. But, when the shape is also changed, expenditure

of energy

in transverse shearing

is generally

inescapable.

Exceptionally, the equality holds in (3) when the yield surface in stress space has a pointed vertex on the ray representing uniaxial tension (a particular example being

the

Tresca

hexagonal

cylinder).

remains within the permitted the working zone is just a constant varying

hydrostatic

It incidentally

pressure

Indeed,

provided

the

strain-rate

ratio

range for the vertex, the stress (al, 02, us) in tension 27 along streamlines, together with a

q/c3

which

does

no net

follows from (1)that the increase

work

in a solenoidal

in longitudinal

motion.

stress is now simply

[al]= 27el. That ideal profiles can actually be designed for materials of this kind is in § 2 below. Suppose that these same profiles are used to shape other materials, with convex yield surfaces-for which the deformation would generally not he ideal. The expenditure of energy should still be practically minimal. In +Ey another choice for (al*, components to deform advance).

pmportional an element

oa*, OS*), namely

the yield stress that would correspond

proved regular strictly fact it

via the flow-rule to strain-rate

to (el, ea, es), HILL (1957 5 2) deduced that the actual work also exceeds what would be needed

monotonically

under fixed strain-rate

ratios to the same final shape (which is here not known in

Ideal forming operations for perfectly plastic solids

225

would probably differ little from (4) itself, since the triaxial stress in the working zone is likely to range only over a limited portion of the yield surface. The difference is again an excess, as may be shown by a direct application of the maximum work principle for a rigid/plastic continuum. The greatest deviation is to be expected with plane strain operations (~3 = 0), but in this case ideal profiles can be constructed with comparative ease (RICHMOND and DEVENPECK 1962; HILL 1966). With ~1 $ ~2 = 0 and al - a2 = 2k, where k is the yield stress in pure shear, (1) gives W = 2k el precisely, which coincides with the value for plane uniform extension. For the common metals Ic exceeds 7 by at most about 15 per cent. 2.

PROOF OF EXISTENCE

OF

IDEAL

OPERATIONS

Consider any self-equilibrated field of stress which at every point is equivalent to a variable hydrostatic tension Q superimposed on a uniaxial tension of fixed value 27. Then the tensor components on fixed rectangular axes Q are everywhere of type aaj = a&j + 2~ 8~13 (i, j = 1, 2, 3) (5) where &j is the Kronecker delta. Two principal stresses, a2 and as say, are equal to a; 1~ is a unit tangent to the trajectory of the other, al = CT+ 27,

(6)

which is major or minor according to the sign of 7. Substitution in the equations of balance gives

where the summation convention is used. By resolving these along la and any perpendicular principal direction ma, we obtain the intrinsic equations (7)

since 162= 1, 4 &/ax3 = 0 and la rnf = 0. Note that this intrinsic formulation does not require in advance that the stress trajectories admit orthogonal surfaces (as would have to be assumed if starting instead from the standard equations in general curvilinear coordinates). By simple geometry or Gauss’s theorem the divergence in (7) can be interpreted as 22b (In a)/axt where a is the sectional area of an infinitesimal tube of trajectories. Xn agreement with (4) we therefore have the integral u -+ 2~ In a = eonst., on each trajectory. (9) Since this is valid for any three-dimensional field of the specified type, it coincides in particular with the familiar distribution in a spherically symmetric plastic state, where tube area is proportional to squared radius. Now let ‘it = vie (V > 0) be a velocity field whose streamlines are the stress trajectories. The rate of dilatation is

226

R. HILL

(10) while the typical component of shear strain-rate parallel to a streamline is

(11) The structural resemblance between (7) and (lo),and between (8) and (ll), immediately suggests considering fields in which the speed varies so that &J/V = h/27,

or

D-

27 In v = const., universally,

(12)

causing both (10) and (11) to vanish. This connexion accordingly renders any such flow solenoidal [as is otherwise apparent from (9) which implies v oc l/a] and makes the streamlines also trajectories of principal strain-rate*. Conversely, given any velocity field with these properties, self-equilibrated states of stress of type (5) can be generated via (12). Since the trajectories of stress and strain-rate now coincide, the velocity fields can be regarded as particular modes of instantaneous deformation in an isotropic rigid/plastic solid whose yield surface has a vertex under uniaxial tension or compression. More particularly, the modes can be regarded as steady flows through dies, perhaps with internal mandrels, contoured like any of the streamtubes. In order that the work-rate per unit volume should be positive the direction of flow must be such that it is convergent under tension (7 > 0, decreasing a) and divergent under compression (T < 0, increasing a). The shape of the yield surface can be arbitrary, except that the ‘ strength ’ of the vertex should be enough to admit the particular range of transverse strain-rate ratios involved in a considered process. [The Tresca vertex, for example, admits any pair of transverse strain-rates with the same sign, which is necessarily opposite to that of the longitudinal strain-rate]. An alternative viewpoint is that (5) is an approximation to the presumably limited range of triaxial stress in the working zone for solids with regular yield surfaces. Under plane strain conditions the analysis is formally the same except that the subscripts i, j take values 1, 2 only. Also, as = CJbut 0s = 0 + T, while T must now be replaced by the yield stress lc in pure shear (no other ma.terial properties are involved). Alternative derivations for this case have been given by RICHMOND and DEVENPECK (1962)and HILL (1966). 3.

When the required

AXIALLY

SYMMETRIC OPERATIONS~

profile has an axis of symmetry

(e.g. forming

of round wire

*Ananalogous correspondence relation is 0 + 27 1” u = const. which, when 7 > fluids (HILL

0, is the basis of a connexion

1954)

(12’)

between the present plastic states and steady Bows of certain barntropic

: ~ij = PSij + PU'lilj

where p is pressure, p

density, and u speed. Because of the sig” difference between (12) and (12’) the fluids are compressible [in fact, the implied pressure-density law is p = T In (p/p,,) from which it can be seen that the flow is supersonic at Mach2/2] and volume elements BE sheared in the direction of flow. However, since (I - p, while the streemlines and characteristics of the solid and fluid motions are the same, there is a partial resemblance

between ideal forming

and flow through pipes. tThe original version of this section has been much shortened to avoid any overlap with a prior paper on this subject by RICHMOND and MORRISON (1967,

this issue of the Journal).

Ideal forming operations for perfectly plastic solids or tubing), equations

standard (7)

trajectories profiles

techniques (S)f.

are in principle

relations

are available

S’mce

of the maximum

operations

(e.g.

similar

some

computed

representative

curves

to those In

already

both

profiles

for

wire

relations,

strain

Hencky

methods 1963).

LOCKETT

drawing

along with their analogues

can of course also be used to confirm the theoretical flow along trajectories

of principal

situation,

the general approach

into

structure

the

for plane

respective

have

are ideal

so far

By

recently

such been

by RICHMOND and MORRXSON (1967).

The Heneky

relations.

the

are the basis of all approximate 1960;

planes

for constructing

elaborated

contexts

EASON and SHIELD short

of the hyperbolic

in meridian

stress 7, the methods

references).

SHIELD 1955;

for the integration

characteristic

shearing

very

(see preceding

the

along the characteristics

proposed means

and

227

of the

It further

other kinds

stress.

correspondence

might

components,

of axially

symmetric

even in this relatively

simple

of 0 2 above is more direct and gives greater insight

offers a convenient

of material

However,

for the velocity

possibility

between procedure

the

statical

by which

and

optimum

kinematical profiles

for

be investigated.

This work is part of a programme of research on the mechanics of materials which is supported by a grant from the Science Research Council.

REFERENCES EASON, G. and SHIELD, IL T. HILL,

R.

1980

J.

19.57 1963

Ibid. 6, 1. Itrid. 14,245. J. Mech. Phys. Solids 11, 345.

1962

Proc. Fourth U.S. Natn Cangr. Appl. Mech., p. 1058.

1967

J. Merh. Phys. Solids 15, 19.5.

1966 LOCKETT, F. .J.

2. Angew. Math. Phys. 11,33. Mech. Phys. Solids 2, 110.

1954

RICHMOND, 0.

and DEVENPECK, M. L. RICHMOND, 0.

and MORRISON, H. L.

*In this case they could be m-written 98

with similar equations for the velocities. Here 81 and ba we arc-leogthsof the stress trajectoriesin 8 meridian#me; PIand paaretheirradiiof curvature;Pis the distanceto the axis of symmetrymeasuredin the directionof the streamline tangent. The bracketedexpressionis the local meanourvstureof the surfaceof revolutionorthogonalto the streamlines.