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A remark on diagonal streaming in plane plastic strain
I. Me&.
Phys. Solids, 1966, Vol. 14, pp. 245 to 248. Pergamon
A
Press
Ltd. Printedin GreatBritain.
REMARK ON DIAGONAL STREAMING PLANE PLASTIC STRAIN
IN
By R. HILL Department of Applied Mathematics and Theoretical Physics, University of Cambridge (Received4th April 1966) SUMMARY ATTENTION IS CALLED to some elementary, but apparently neglected, geometric properties of diagonal flow in a plastic mass under conditions of plane strain. By definition the streamlines in such flows coincide with one family of principal stress trajectories. As a practical illustration a specification is given for the design of ‘ ideal ’ dies for strip drawing, such that the final deformation is homogeneous and the work done is a minimum.
1. IT
WAS POINTED
streaming
BACKGROUND
long ago by GEIRINGER (1937, pp. 65-69) that a ‘ diagonal of instantaneous deformation is formally admitted by every
OUT
’ type
self-equilibrated field of stress in an isotropic rigid/plastic strain. By definition such a deformation has streamlines slip-line directions stress
and therefore
(as in uniform
radial
solid undergoing plane that locally bisect the
coincide with one family of trajectories
expansion
of a cylindrical
Geiringer did not pursue the matter beyond noting To my knowledge the subject has subsequently
pressure
vessel).
of principal However,
a few academic illustrations. been advanced no farther,
with one notable exception. In 1962 RICHMOND and DEVENPECK exhibited a theoretical die profile that produces purely diagonal flow in a strip drawn through it, assuming
perfect
lines scribed
lubrication
and no hardening.
They calculated
that transverse
on the strip would still be exactly
transverse after drawing, so that Further, the overall expenditure the final reduction in thickness is homogeneous. of work was shown to be no more than in monotonic uniform extension. At first sight such results seem remarkable indeed, and this impression is not lessened by the complexity of the supporting computations. Credibility demands instead a straightforward explanation from first principles. With this aim the present note calls attention to certain elementary properties common to all diagonal flows. Against this perspective Richmond and Devenpeck’s findings at once become with
‘ self-evident.’
similarly
ideal
2.
It
also becomes
characteristics,
ADMISSIBILITY
apparent
how a variety
can be designed
OF
of die profiles,
at will.
DIAGONAL STREAMING
We begin with a more explicit and suggestive re-statement of Geiringer’s observation. Every continuous plane plastic field, with uniform maximum shear stress k, admits virtual diagonal modes of flow with speeds 245
246
R. HILL w
cc exp ( & a/%)
(1)
where the positive or negative sign is taken according to whether the tension u along the streamlines is the algebraically greater or lesser principal stress. In proof, the application of Hencky’s canonical relations to the right side of (1) reveals
that
the speed varies
along any one slip-line
in the manner
z& K c*
(‘L)
where 4 is the angle of inclination of the slip-line reckoned in the sense tending to make it coincide with the local streamline. By elementary kinematics it follows from (2) that the rate of extension along any slip-line vanishes (the difference in the terminal velocities of an arc-element being purely normal when &a = ze6$). And this is a necessary and sufficient condition for the modes to be compatible with the stress field*. The constant of proportionality in (1)is arbitrarily positive; in particular, this permits replacing u by the other principal stress or by the mean hydrostatic tension. If the virtual modes are to be physically possible, the rate of working must be positive everywhere. This requirement (which Geiringer ignored) decides the sense of any actual flow. Evidently, a material line element oriented along a streamline must lengthen when 0 is the major principal stress and shorten when it is not. That is, flow must proceed in the direction of increasing or decreasing speed, respectively,
and so neighbouring
streamlines
are correspondingly
convergent
or divergent. In either case we see from (I) that the flow must be in the direction of algebraically increasing principal stresses. Whether diagonal streaming actually occurs in any particular circumstances is of course finally constraints
settled
by its compatibility
or otherwise
with externally
imposed
on the motion.
In this connexion the typical Cauchy boundary-value problem plays a fundamental role. Suppose that the known flow across certain segments OP, OQ of a pair of intersecting slip-lines is diagonal, and so satisfies (2). The data uniquely determines a virtual mode throughout the slip-line quadrilateral OPQR. It follows that this mode is entirely diagonal there, since a diagonal streaming consistent with the velocity at 0 certainly exists and matches the remaining data automatically.
3.
GRID DEFORMATION
IN
STEADY
DIAGONAL
STIUXAMING
Consider any curvilinear segment embedded in the material and momentarily orthogonal to the streamlines in a steady diagonal flow. In an isotropic rigid/plastic solid the principal axes of strain-rate at each point are directed along the principal stress trajectories. Consequently the orthogonality persists while the segment traverses the field. Let its successive
positions
in space be marked
at arbitrary
equal intervals
of
time. By marking also any finite set of streamlines we construct an orthogonal embedded grid which is self-mapping after each interval. And, since the material is incompressible, neighbouring grid streamlines contain cells of equal area. In *In the hodograph plane the traces of the slip-lines are always lqarithmic spirals.
A remark
particular,
on
by appropriately
diagonal streaming in plane plastic strain
2‘47
spacing
grid
the set, we can arrange that the et&e
is equi-areal* as well as self-mapping. Consider the passage of an injnitesimal subjected
to monotonic
cell through the field.
Being perpetually
or compression along the same material axes, the cell absorbs an amount 2k 1In (to/t) 1o f work per unit volume, just as in a standard tension or compression text, where Q, and t are the initial and final thicknesses of the corresponding
extension
stream tube.
But, since the motion
is steady
and the density
is constant, this amount is equal to the overall change 1~ - a~( in stress and to/t is equal to ZU/ZQ. We have thereby recovered (1) and given it an intuitive interpretation. With these preliminaries
in mind, suppose that in a certain steady-state
forming
operation the plastic flow is purely diagonal, while rigid material enters and leaves uniax:ially and smoothly (no velocity jumps). Then, of necessity, any equi-area1 grid of the type described must be uniformly rectangular both before and after Further, the change in dimensions is passing through the zone of deformation. accomplished without redundant work.
‘I.
SPECIFICATION
OF
IDEAL
As an illustration we apply the foregoing dies for strip drawing.
FORMING
principles
OPERATIONS
to the design of symmetrical
-
~___-_----FIG. 1.
In Fig. 1 AC is the axis of symmetry. Choose an identical pair of slip-lines AB concave to the exit and such as to define a continuous plastic field covering the whole quadrilateral ABCB. and the desired diagonal
Then all slip-lines there are concave in the same sense flow is automatically convergent everywhere. Next
select a symmetrical pair of principal stress trajectories such as EF. Extend the plastic field into the triangular regions ADE and CFG at entry and exit as far as the continuations DE and FG of trajectory EF, taking slip-lines AD and CG straight. Then DEFG is the profile of an ideal die. Precisely the same reduction in strip thickness is obtained with any other choice of EF; in particular the broken curves indicate the construction of the profile of the relatively ‘ shortest ’ die when AB is given. This specification generalizes Richmond and Devenpeck’s design, in which AB was a logarithmic spiral and the flow within ABCB purely radial. ‘As a by-product this furnishesan elementaryproof of SADOW~KY’S (1941) theorem that equi-arealnets can be formed from the trajectoriesof principalstressin a plane plastic state. The original proof by differential geometry is unilluminating
and masks the kinematic
significance
of the theorem.
Ft. HILL
24x
REFERENCES GEIRINGER, H.
1937
Mdm. Sci. Math.
86.
RICHMOND, 0. and DEVENPECK, M. L. 1962 SADOWSICY, M. A.
1941
Proc. 4th U.S. h’atn. Congr. Appl. Mech. 8, A74.
J. Appl.
Mech. p. 1053.
Phys. Solids, 1966, Vol. 14, pp. 245 to 248. Pergamon
A
Press
Ltd. Printedin GreatBritain.
REMARK ON DIAGONAL STREAMING PLANE PLASTIC STRAIN
IN
By R. HILL Department of Applied Mathematics and Theoretical Physics, University of Cambridge (Received4th April 1966) SUMMARY ATTENTION IS CALLED to some elementary, but apparently neglected, geometric properties of diagonal flow in a plastic mass under conditions of plane strain. By definition the streamlines in such flows coincide with one family of principal stress trajectories. As a practical illustration a specification is given for the design of ‘ ideal ’ dies for strip drawing, such that the final deformation is homogeneous and the work done is a minimum.
1. IT
WAS POINTED
streaming
BACKGROUND
long ago by GEIRINGER (1937, pp. 65-69) that a ‘ diagonal of instantaneous deformation is formally admitted by every
OUT
’ type
self-equilibrated field of stress in an isotropic rigid/plastic strain. By definition such a deformation has streamlines slip-line directions stress
and therefore
(as in uniform
radial
solid undergoing plane that locally bisect the
coincide with one family of trajectories
expansion
of a cylindrical
Geiringer did not pursue the matter beyond noting To my knowledge the subject has subsequently
pressure
vessel).
of principal However,
a few academic illustrations. been advanced no farther,
with one notable exception. In 1962 RICHMOND and DEVENPECK exhibited a theoretical die profile that produces purely diagonal flow in a strip drawn through it, assuming
perfect
lines scribed
lubrication
and no hardening.
They calculated
that transverse
on the strip would still be exactly
transverse after drawing, so that Further, the overall expenditure the final reduction in thickness is homogeneous. of work was shown to be no more than in monotonic uniform extension. At first sight such results seem remarkable indeed, and this impression is not lessened by the complexity of the supporting computations. Credibility demands instead a straightforward explanation from first principles. With this aim the present note calls attention to certain elementary properties common to all diagonal flows. Against this perspective Richmond and Devenpeck’s findings at once become with
‘ self-evident.’
similarly
ideal
2.
It
also becomes
characteristics,
ADMISSIBILITY
apparent
how a variety
can be designed
OF
of die profiles,
at will.
DIAGONAL STREAMING
We begin with a more explicit and suggestive re-statement of Geiringer’s observation. Every continuous plane plastic field, with uniform maximum shear stress k, admits virtual diagonal modes of flow with speeds 245
246
R. HILL w
cc exp ( & a/%)
(1)
where the positive or negative sign is taken according to whether the tension u along the streamlines is the algebraically greater or lesser principal stress. In proof, the application of Hencky’s canonical relations to the right side of (1) reveals
that
the speed varies
along any one slip-line
in the manner
z& K c*
(‘L)
where 4 is the angle of inclination of the slip-line reckoned in the sense tending to make it coincide with the local streamline. By elementary kinematics it follows from (2) that the rate of extension along any slip-line vanishes (the difference in the terminal velocities of an arc-element being purely normal when &a = ze6$). And this is a necessary and sufficient condition for the modes to be compatible with the stress field*. The constant of proportionality in (1)is arbitrarily positive; in particular, this permits replacing u by the other principal stress or by the mean hydrostatic tension. If the virtual modes are to be physically possible, the rate of working must be positive everywhere. This requirement (which Geiringer ignored) decides the sense of any actual flow. Evidently, a material line element oriented along a streamline must lengthen when 0 is the major principal stress and shorten when it is not. That is, flow must proceed in the direction of increasing or decreasing speed, respectively,
and so neighbouring
streamlines
are correspondingly
convergent
or divergent. In either case we see from (I) that the flow must be in the direction of algebraically increasing principal stresses. Whether diagonal streaming actually occurs in any particular circumstances is of course finally constraints
settled
by its compatibility
or otherwise
with externally
imposed
on the motion.
In this connexion the typical Cauchy boundary-value problem plays a fundamental role. Suppose that the known flow across certain segments OP, OQ of a pair of intersecting slip-lines is diagonal, and so satisfies (2). The data uniquely determines a virtual mode throughout the slip-line quadrilateral OPQR. It follows that this mode is entirely diagonal there, since a diagonal streaming consistent with the velocity at 0 certainly exists and matches the remaining data automatically.
3.
GRID DEFORMATION
IN
STEADY
DIAGONAL
STIUXAMING
Consider any curvilinear segment embedded in the material and momentarily orthogonal to the streamlines in a steady diagonal flow. In an isotropic rigid/plastic solid the principal axes of strain-rate at each point are directed along the principal stress trajectories. Consequently the orthogonality persists while the segment traverses the field. Let its successive
positions
in space be marked
at arbitrary
equal intervals
of
time. By marking also any finite set of streamlines we construct an orthogonal embedded grid which is self-mapping after each interval. And, since the material is incompressible, neighbouring grid streamlines contain cells of equal area. In *In the hodograph plane the traces of the slip-lines are always lqarithmic spirals.
A remark
particular,
on
by appropriately
diagonal streaming in plane plastic strain
2‘47
spacing
grid
the set, we can arrange that the et&e
is equi-areal* as well as self-mapping. Consider the passage of an injnitesimal subjected
to monotonic
cell through the field.
Being perpetually
or compression along the same material axes, the cell absorbs an amount 2k 1In (to/t) 1o f work per unit volume, just as in a standard tension or compression text, where Q, and t are the initial and final thicknesses of the corresponding
extension
stream tube.
But, since the motion
is steady
and the density
is constant, this amount is equal to the overall change 1~ - a~( in stress and to/t is equal to ZU/ZQ. We have thereby recovered (1) and given it an intuitive interpretation. With these preliminaries
in mind, suppose that in a certain steady-state
forming
operation the plastic flow is purely diagonal, while rigid material enters and leaves uniax:ially and smoothly (no velocity jumps). Then, of necessity, any equi-area1 grid of the type described must be uniformly rectangular both before and after Further, the change in dimensions is passing through the zone of deformation. accomplished without redundant work.
‘I.
SPECIFICATION
OF
IDEAL
As an illustration we apply the foregoing dies for strip drawing.
FORMING
principles
OPERATIONS
to the design of symmetrical
-
~___-_----FIG. 1.
In Fig. 1 AC is the axis of symmetry. Choose an identical pair of slip-lines AB concave to the exit and such as to define a continuous plastic field covering the whole quadrilateral ABCB. and the desired diagonal
Then all slip-lines there are concave in the same sense flow is automatically convergent everywhere. Next
select a symmetrical pair of principal stress trajectories such as EF. Extend the plastic field into the triangular regions ADE and CFG at entry and exit as far as the continuations DE and FG of trajectory EF, taking slip-lines AD and CG straight. Then DEFG is the profile of an ideal die. Precisely the same reduction in strip thickness is obtained with any other choice of EF; in particular the broken curves indicate the construction of the profile of the relatively ‘ shortest ’ die when AB is given. This specification generalizes Richmond and Devenpeck’s design, in which AB was a logarithmic spiral and the flow within ABCB purely radial. ‘As a by-product this furnishesan elementaryproof of SADOW~KY’S (1941) theorem that equi-arealnets can be formed from the trajectoriesof principalstressin a plane plastic state. The original proof by differential geometry is unilluminating
and masks the kinematic
significance
of the theorem.
Ft. HILL
24x
REFERENCES GEIRINGER, H.
1937
Mdm. Sci. Math.
86.
RICHMOND, 0. and DEVENPECK, M. L. 1962 SADOWSICY, M. A.
1941
Proc. 4th U.S. h’atn. Congr. Appl. Mech. 8, A74.
J. Appl.
Mech. p. 1053.