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Geometrically self-similar deformations of a plastic wedge under combined shear and compression loading by a rough flat die
lnt, Z Mech. Sci. Vol. 22, pp. 735-742 Pergamon Press Ltd., 1980. Printed in Great Britain
GEOMETRICALLY OF A PLASTIC SHEAR A N D BY A
SELF-SIMILAR DEFORMATIONS WEDGE U N D E R COMBINED COMPRESSION LOADING ROUGH FLAT DIE
I. F. COLLINSt U.S. Steel Corporation Research Laboratories, Monroeville, PA 15146, U.S.A. (Received 5 February 1980)
Summary--A new slip-line field solution for the shearing of a plastic wedge under geometrically similar conditions is described and its relevance to the formation of wear particles from sheared asperities is discussed. The solution is an example of a recently discovered class of pseudo-steady, plane strain deformations in which a portion of the rigid/plastic material rotates rigidly. NOTATION r t v p 0 A to
position vector in physical space time velocity vector position vector in unit diagram semi-wedge angle die inclination angle angular speed too angular speed when t = I qJ,X slip-line angles /x coefficient of friction
INTRODUCTION Beginning with the investigations of G r e e n [ I , 2 ] the techniques of slip-line theory have been used by m a n y authors to study problems modelling the d e f o r m a t i o n of asperities with a view to gaining understanding of the m e c h a n i s m s of friction and wear for ductile solids in contact, G r e e n [ I , 2 ] and later Edwards and Hailing[3] considered the f o r m a t i o n and growth of junctions f o r m e d by locking asperities under plane strain conditions. These studies confirmed and amplified the well-known dry friction theories of B o w d e n and T a b o r [4]. Childs [5] has used slip-line fields to study the persistence of asperities under normal loads by investigating the interaction of the plastic d e f o r m a t i o n produced in the asperity with the ambient material and with neighboring asperities. W a n n h e i m et al. [6] have made a similar investigation and have included the effect of applying a tangential as well as normal load to the asperity. The interaction b e t w e e n tangential and normal loads on a single wedge-shaped asperity has been considered by J o h n s o n [7], who treated the problem of a rigid/plastic wedge being c o m p r e s s e d by a die to which a normal force N is first applied followed by a monotonically increasing tangential force T. Johnson discusses some slip-line solutions which a p p r o x i m a t e l y describe the progressive d e f o r m a t i o n of the wedge and which predict a growth law for the contact area A which closely a p p r o x i m a t e s that suggested by T a b o r [8], (A/Ao) 2 = 1 + or(TIN) 2
(1)
where A0 is the contact area under normal load N alone and a is a constant -~ 12. The theory was supported by e x p e r i m e n t s on hard-drawn copper and plasticene. tPermanent address; Department of Mathematics, UMIST, P.O. Box 88, Manchester, England. 735
736
I . F . COLLINS
Johnson's theory and experiments also showed that, provided the adhesion between the wedge and die faces was sufficiently strong, continuous deformation would occur beneath the die until T has increased so as to be approximately equal to N, at which point discontinuous shear occurs either at the interface or on a plane inclined at an acute angle to the die face and a sliver of material becomes detached from the main body of the wedge which would eventually break off and form a wear particle. The problem discussed in the present paper is similar to that of Johnson; a wedge of rigid/plastic material is compressed under plane strain conditions by a rigid rough-faced die which is moving with constant velocity obliquely relative to the (vertical) plane of symmetry of the wedge, so that as well as being compressed the wedge material is sheared. With this formulation the problem is one of "pseudosteady flow" in which the deforming configuration grows uniformly in time remaining geometrically similar, so that the problem can be solved using the unit diagram concept [9]. Attention is concentrated on the solution for TIN ratios of the order of unity, where as will be seen a chip forms, which takes the form of a spiral and which when eventually fractured can form a wear particle. The solution is of some fundamental interest in slip-line theory since it represents a relatively simple example of a new class of pseudo-steady flow solutions involving rotating rigid regions discussed recently by Petryk [10, 11] and Collins [12]. The solution given here amplifies Johnson's analysis and throws further light on when and how a wear particle can be expected to be formed from a virgin ductile asperity. Challen and Oxley [13] have also recently produced approximate slip-line field solutions for the deformation of asperities using models in which the hard asperity (die) has an inclined face and is moving horizontally relative to the soft (plastic) asperity. Aspects of this problem using the present spiral chip formation model will also be discussed. SLIP-LINE FIELD SOLUTION WITH SPIRAL CHIP FORMATION The proposed slip-line field solution is shown in Fig. 1. The two-dimensional plastic wedge, whose vertex was originally at 0, with semivertex angle 0, is compressed quasi-statically by a flat perfectly rough die MABN whose face is inclined at an angle )t to the "horizontal (that is, at 7r]2-)t to the vertical symmetry plane of the wedge). The die has a (constant) velocity component perpendicular to its face which causes the wedge material to compress and a (constant) tangential component which is sufficiently large to induce full friction over the interface AB. This happens when the perfectly rough face of the die is slipping (or on the point of slipping) over the wedge material. The value of this critical tangential velocity component will come out as part of the solution. O'
14" \
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G FtG. 1. Unit diagram with superposed hodograph (case illustrated is for 0 = 40°, X = 25°, I = 21.2°).
Geometrically self-similar deformations of a plastic wedge
737
The slip-line field network which grows uniformly with time consists of a centered fan ABC composed of straight radial a-lines and circular arc B-lines turning through an angle $. At the generic instant the contact length AB is denoted by p. A velocity discontinuity propagates along the a-line AC and continues along the circular arc CD which separates two rigid regions. Below ACD the material is still at rest, while above CD it is rotating rigidly about the instantaneous center of rotation G (the center of the circular arc CD) with angular speed to which is a function of time. The rigid chip BCDQB must be in equilibrium. If we make the circular arc CD identical with BC, so that it has radius p and span angle ~bthe component of force on DCB perpendicular to the direction bisecting the angle B~D and the torque about C will both be zero by symmetry. The equilibrium of the chip will hence be assured by choosing the mean pressure P at C such that the force component on DCB in the direction of the bisector of BCD is zero. As time progresses the deformation pattern must grow uniformly without change of shape due to the pseudo-steady nature of the problem. To see what restrictions this condition puts on the solution we use the unit diagram concept of Hill, Lee and Tupper [14,9]. If r is the position vector of a point in the field measured relative to O and t is the scalar "time-like" parameter measuring the stage of the deformation, which in this problem can conveniently be taken as the distance 0L in Fig. I, then the stress and velocity fields depend on r and t only through the ratio
p:rlt.
(2)
As r traces out the slip-line field in physical space, p describes the "unit diagram" which is identical to the slip-line field at the instant t = 1. Differentiating (2) with respect to "time" t yields; t ~ -t = v - p
(3)
where v = dr/dt is the velocity of the material particle at r. As the material particle r moves in physical space its image in the unit diagram p traces out a trajectory in the unit diagram whose path is described by (3). This equation can most conveniently be interpreted geometrically by using the concept of the hodograph diagram as pointed out by Green[15]. (This concept is implicit in the arguments given by Hill, Lee and Tupper [14] but the term "hodograph" was not used at this time.) If the diagram in Fig. 1 is interpreted now as the unit diagram (t = I) and on it is superimposed the hodograph diagram of velocity image points of the slip-line field with origin at the center of distortion 0 (the usual small letter convention has been adopted for labelling hodograph image points), then equation (3) states that the rate of change of position of p in the unit diagram is equal to the vector going from p to the image point v in the superimposed hodograph diagram. Thus the path of a material element in actual space is represented by an image trajectory in the unit diagram, the tangent to which is always directed toward the current image point in the superposed hodograph diagram. In particular the unit diagram image of a free surface which consists of the same material particles must possess this property, Since the hodograph image of a rigid translating region is a single point it follows that a free surface bordering on a rigid translating region must be straight, as in the well-known solutions for wedge indentation of a half-space and normal flattening of a wedge [9]. In the present problem, however, it is postulated that the rigid region in the chip bordering on the free surface is rotating. Now as shown by Green [15] a region in physical space which is undergoing a rigid rotation is mapped in the hodograph diagram into a geometrically similar region, scaled up by a factor of to and turned through a right angle in the direction of rotation. In particular the hodograph images dq and bq of the free surfaces DQ and BQ are obtained in this way. Hence the tangent at a point on DQ passes through the corresponding point on dq, which is a curve orthogonai and geometrically similar to DQ. dq (and similarly bq) is hence the curve which is geometrically similar to its evolute DQ (BQ) in the ratio too: I (tOo being the angular velocity at the unit diagram (t = l) stage of the deformation). It is easily shown that the unique curve possessing this property is the logarithmic spiral, whose tangent makes an angle d, = arctan (too) with the radius vector from the center of the spiral. A formal proof is given here in the appendix. This development was clearly forseen by Green [15], who states this result, but without proof or example. The center of the spirals in both the unit and hodograph diagrams is Q, coincident with its hodograph image q, this being the current position of the material particle originally at the wedge vertex 0. It follows by similar reasoning that the unit diagram trajectories of all material particles in the rigid rotating chip, whether or not they lie on the free surface, are logarithmic spirals centered on Q. In passing it may be noted that Petryk [10, 1 l] has used the matrix operator concept for slip-line field construction [12, 16, 17] to develop a procedure for constructing the free surface bounding of an arbitrary plastically deforming region in a general geometrically self-similar pseudo-steady deformation. Since the scale of the velocity field is invariant in time but the scale of the slip-line field in physical space increases in proportion to t, it follows that the angular speed of rotation of the rigid chip is inversely proportional to t. It follows, therefore, that to = tOo~t,since tOo is the angular speed when t = 1. The angular speed is hence infinite at the initial instant of contact of the die with the apex of the wedge. The motion of the rigid chip in physical space can best be visualized by drawing on the well-known theorem of rigid body kimematics [18] which states that any planar rigid body motion can be generated by rolling a curve fixed in the body (the body centrode) on a curve fixed in space (the space centrode). From the continuing similarity of the slip-line field it follows that the path of any point fixed in the slip-line network (i.e. fixed p) moves along a radial straight line through 0 in physical space (since r = pt). Thus in particular the center of rotation G follows such a path in physical space so that the space centrode is a radial straight line through 0. If R is the radius of curvature of the body centrode at G, and S is the distance 0G at time t, then the speed of G in physical space is both S and RtO. Since S is proportional to t, S = So the value of S in the unit diagram, and since to = tO0/t it follows that R : S / o , : Sot~too : S/too. MS Vol. 22, No. 12--B
(4)
738
I.F. COLLINS
However, S is also the arc length of the body centrode so that (4) states that the body centrode is such that its radius of curvature is proportional to its arc length and is hence a logarithmic spiral. This spiral must pass through Q since the material particle at Q was initially at 0 on the space centrode. (Putting R = dSld~b and integrating (4) gives R = R0 exp (~,/to0) see equation 8 in Appendix.) As is illustrated in Fig. 2, the motion of the rigid chip in space can hence be generated by rolling the logarithmic spiral QGU on the radial straight line OGT. During this motion the center of the spiral Q itself moves in a radial straight line away from 0. This property of the rolling logarithmic spiral was noted by Clerk Maxwell in his classic paper on rolling curves [19]. The velocity of the wedge material adjacent to the die face A B is given by the vector 0-b in the hodograph Fig. 1. The solution is hence valid whenever the component of velocity of the die in the direction tangential to the die face is greater than or equal to the component of 0b in this direction. O
¢e~ II
1 Ix I xx U
T
FIG. 2. Motion of rigid chip is generated by rolling the body centrode Q G U on space centrode OGT. COMPUTATIONS Since the detailed geometry of the field is quite complicated a small computer program was written in order to calculate the details of the solution. The semiwedge angle 0, the slip-line span angle ~ and the die inclination angle A are taken as independent variables. The ratio tip is determined by the condition that D lies on the undeformed face of the wedge (Fig. 1). Straightforward geometry gives; t O
((cos ~ - sin ~ + sin 2qJ) - tan (0 - A)(sin ~b+ cos ~b- cos 2~)) (cos ;t tan (0 - A) + sin A + sin 0/cos (0 + A))
(5)
Since the hodograph diagram is geometrically similar to the unit diagram slip-line field in the ratio to0 : 1, it follows that the radii 0d, 0c, 0'c and 0'b, representing the magnitude of the velocity discontinuity along ACD, are all equal to tooO. From the condition that b lies on the die face it follows that ~Oo= (tip) cos M(sin ~ - cos 4J + 1)
(6)
and so COoand hence ~b = arctan too can be computed, The inclination of 0b to the horizontal is (X - A) where X = ObA and is given by tan X = (sin ~ - cos ~ + 1)/(cos tp + sin ~).
(7)
The die will he travelling horizontally and the wedge material will just be on the point of slipping if ~ = A and since ,,'t"depends only on 4' from (7), this equation can be used to calculate h for this particular type of deformation. As discussed earlier the pressure at C is determined from the overall equilibrium equations of the rigid chip. The pressure through the rest of the field and hence the normal force and friction coefficient ~ on the interface A B can then readily be found from Hencky's relations. It is worth noting that this calculation is independent of wedge angle 0 and inclination A, so that the coefficient of friction #, depends only on the slip-line angle ~. The program contained three checks: (a) Area check Since the plastic deformation is isochoric the area of the chip formed must be equal to that removed by the die. Thus at any stage of the deformation the areas A B Q D C A and AODCA in Fig. 1 must be equal. This condition does not provide any new information since incompressibility and mass conservation are
Geometrically self-similar deformations of a plastic wedge
739
implicit in the basic equations defining slip-line fields and their hodograph images, but it does provide a valuable check on the correctness and accuracy of the calculations. In all cases these two areas were found to be identical to seven significant figures. (b) Overstressing check A slip-line solution is incomplete until it can be demonstrated that a statically admissible stress field exists in the proposed rigid regions. In general this is difficult to do and would appear particularly so in the present solution due to the complex shape of the rigid chip. In these circumstances the best that one can do is to check all the vertices of the proposed rigid regions using the criteria for overstressing developed by Hill [20]. In the present solution these criteria were applied to the two rigid vertices at D and to those at A and B. These conditions put a definite limit on the range of validity of the solution as discussed in the next two sections. (c) Interference check It was frequently found that for arbitrary choices of 0, ¢, and A, the position of spiral center Q was either above the die face MABN or inside the wedge face ODR. These two conditions also put limitations on the range of values of 0, ¢, and A for which a physically meaningful solution could be found, although the former condition was always preempted by overstressing at D. Accordingly, the computer program contained a check to ensure that not only Q but also no other part of the chip "penetrated" the wedge or die faces. S O L U T I O N S W I T H H O R I Z O N T A L D I E F A C E (A = 0 ) Fig. 3 shows the domain of validity of the solution in the particular case when the die face is perpendicular to the symmetry axis of the wedge (A = 0). This is the configuration considered by Johnson [7]. For a given wedge angle 0 it is seen that there is only a narrow range of values of slip-line angle ~ and hence coefficient of friction # for which a solution can be found. The lower limit on @ (or upper limit on V-) is when "interference" occurs between the chip and the undeformed wedge; the upper limit on $ corresponds to overstressing at D (in fact when the upper angle at D is less than 45° which necessarily means overstressing [20]) or at A for 0 in the range 28--32°. No solution of this type exists for semiwedge angles less than 28 or greater than 83°. This lack of uniqueness of solution for a given wedge angle is not surprising. The basic uniqueness theorem for the rigid/plastic yield point problem does not apply to pseudo-steady processes and in the simple case of normal compression of a wedge, Hill [9] discusses two distinct solutions. In the present problem this lack of uniqueness is of little practical significance due to the narrow range of permissible values of @. It is of interest to note that the critical coefficient of friction t* shown in Fig. 3, ranges from around 0.95 for sharp wedges (0 = 30°) to around 1-0 for blunt wedges (0 = 80°), (strictly/~ ~ 1.0 as ~ 0 ° ) . This hence supports Johnson's [7] conclusion that intense shearing sufficient to produce a wear particle requires a coefficient of friction of around unity. The particular solution valid for 0 = 6 0 °, @ = 11° is illustrated in Fig. 4. 25"
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+. 0.96
2o' 0.97 OVERSTRESSING AT D
g 0.98
INTERFERENCE WITH WEDGE 0.99
lo" 0.995
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20
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60
70
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FI(;. 3. Solution validity domain (A = 0°).
1.000 90
740
I . F . COLLINS O t
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- ~ ' ~
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PIG. 4. Slip-line field solution with 120° wedge angle (0 = 60°, X = 10°, I = 0°). It is instructive to compare this chip forming solution with the asymmetrical analogue of Hill's [9] solution for normal compression. This is shown in Fig. 5. Rather similar asymmetrical fields have been discussed by Devenpeck and Weinstein [21] and Shindo [22] who, however, consider the case when the wedge material is squeezing out sideways under the die face instead of being dragged along with the laterally moving die as in the present problem. The material in A B C and B D E translates as rigid blocks in directions parallel to A C and DE, respectively, while the material is deforming continuously in the fan BCD. There is a tangential velocity discontinuity along ACDE. The details of the field can easily be evaluated by modifying Hill's arguments [9] for the solution for normal compression. In this solution the deformed material remains attached to the main body of the wedge and does not form a chip: in the friction context this corresponds to a "rubbing model" as opposed to a "chip forming" or "wear model." [13]. For fixed wedge angle, the fan angle ~ ((Fig. 5) decreases with increasing friction coefficient, and the limit of validity is when 4, = 0. The values of p. at which this occurs for O = 40, 60, and 80° are found to be 0.28, 0.57, and 0.86, respectively, while the corresponding friction coefficients at which the curling chip mode of deformation first occurs are 0.96, 0.98, and 1.00. There is hence a range of friction coefficients for which a solution still has to be found, although this gap is smaller for the blunt wedges which are of main interest in the wear context. While this missing solution will almost certainly involve plastically deforming zones bordering on the free surface it is not clear whether it is of "rubbing" or "wear" type. \
0
I////.,A"
f/
,~
\
' , > t l / / ~ /' / / / / > / .~, Z ; / v ' . J " . 4 B ' / / / / / / I I / / I /
FIG. 5. "Rubbing" mode of deformation at small coefficients of friction. S O L U T I O N S F O R I N C L I N E D D I E S M O V I N G H O R I Z O N T A L L Y (h = X) An alternative configuration which is perhaps more directly relevant to the asperity deformation problem is where the die face is inclined to the horizontal (;t ¢ 0) and is translating horizontally. This models the situation of two surfaces in contact moving parallel to each other, the die face corresponding to a hard asperity on the upper surface. This is the situation considered by Challen and Oxley [13]. The results for this model when in addition the wedge material is adhering to the die face, so 0b is horizontal and ;t = X in Fig. 1 are summarized in Fig. 6. Much the same effects are observed as in the previous situation, the main
Geometrically self-similar deformations of a plastic wedge !
3o"
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741
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40" 0.90
30" 20"
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OVERSTRESSING
2o" INTERFERENCE "~. 10"
WITH
-"1,' ~ .
WEDGE
10"
' 10
' 20
I 30
, 40
I 50
610
, 70
--
ao
1.00
90
FIG. 6. Solution validity domain (A = X). difference being that the range of possible slip-line angles @(and hence of the possible friction coefficients) is widened for a specified wedge angle. This is because the chip has now more room to form and is less likely to "interfere" with the wedge face. As can be seen from Fig. 6, the friction coefficient must be in the range 0.9-1.0 for the chip forming solution to occur. The corresponding inclination of the die face A = X(@) is given by equation (7) and increases monotonically with @as shown in Fig. 6. The solution illustrated in Fig. 1 is for O= 40°, ~ = 25°, A = 21-2°.
DISCUSSION AND CONCLUSION The p h e n o m e n a of wear are highly complex and various [23]. Thermal and oxidation effects are of equal importance to purely mechanical processes. The shearing of asperities is only one of several possible mechanisms for the formation of wear particles, ploughing, and so-called delamination processes [24-26] in which wear particles are formed from the breaking up of surface layers due to the propagation of subsurface cracks are two other well-established mechanisms. Experimental evidence for the formation of spiral chips comes from Johnson's experiments on plasticene wedges where such shaped chips are seen to form at friction coefficients 1, see Fig. 5 of Ref. [7]. (As noted earlier, the upper limit on the friction coefficient for which the chip forming solutions of Figs. 1 and 4 are valid is determined by the condition that the curling chip should not " p e n e t r a t e " the wedge asperity. It should, however, still be possible to find solutions of this general type for larger friction coefficients with the chip just touching the asperity surface but with the dimensions of the deforming zone suitably modified to ensure equilibrium of the rigid chip taking due account of the reaction at the point of contact with the asperity. This possibility has not been investigated here however). In addition, spiral or helical chips are not infrequently found among wear debris as noted by Sub [24], who, however, attributes their presence to the curving of initially thin flat sheet particles formed by delamination due to residual stresses in the sheets. One would also expect to get spiral chips from the ploughing action of hard wedge-shaped asperities analogous to the initial stages of machining. It is very difficult to study the continued deformation of a wedge asperity under general loading conditions as remarked by G r e e n [ l , 2 ] and Challen and Oxley[13]. The adoption of the particular loading program which makes the continued deformation geometrically self-similar has appreciably simplified this problem. The present analysis has largely confirmed Johnson's [7] conclusion that one needs a friction coefficient of around unity in order to generate sufficient shear for a wear particle to
742
I . F . COLLINS
form and has further demonstrated a mechanism for the formation of such a particle based on simple ideal plasticity theory. The new solution given here is of some fundamental interest in slip-line theory as it is a relatively simple example of a pseudo-steady solution involving a rigid rotating region. Similar solutions for oblique wedge indentation discussed by Petryk [10, 11] and Collins [12] belong to a class of more complex "indirect" solutions which require the solution of linear integral equations. Acknowledgement--The author is grateful to Dr. Owen Richmond for some helpful comments on this work. REFERENCES 1. A. P. GREEN, The plastic yielding of metal junctions due to combined shear and pressure, Z Mech. Phys. Solids, 2, 197 (1954). 2. A. P. GREEN, Friction between unlubricated metals: A theoretical analysis of the junction model, Proc. Roy. Soc. Ser. A228, 191 (1955). 3. C. M. EDWARDS and J. HALLING, An analysis of the plastic interaction of surface asperities and its relevance to the value of the coefficient of friction, J. Mech. Eng. Sci. 10, 101 (1968). 4. F. P. BOWDEN and D. TABOR, The Friction and Lubrication of Solids, Clarendon Press, Oxford (1950). 5. T. H. C. CmLDS, The persistence of asperities in identation experiments, Wear, 25, 3 (1973). 6. T. WANHEIM, N. BAY and A. S. PETERSEN, A theoretically determined model for friction in metal working processes, Wear, 28, 251 (1974). 7. K. L. JOHNSON, Deformation of a plastic wedge by a rigid flat die under the action of a tangential force, J. Mech. Phys. Solids, 16, 395 (1968). 8. D. TABOR, Junction growth in metallic friction: The role of combined stresses and surface contamination, Proc. Roy. Soc. Ser. A251, 378 (1959). 9. R. HILL, The Mathematical Theory of Plasticity, Clarendon Press, Oxford (1950). 10. H. PETRYK, Ustalone plaskie preplywy osrodkow idealnie plastycznych ze swobodnym brzegiem, (Doctoral Thesis), Institute of Fundamental and Technical Research, Warsaw (1977). 11. H. PETRVK, On slip-line field solutions for steady-state and self-similar problems with stress-free boundaries, Arch. Mech. 31,861 (1979). 12. I. F. COLLINS, Integral equation formulation of slipline field problems, Applications of Numerical Methods to Forming Processes, ASME, San Francisco, 129 (1978). 13. J. M. CHALLEN and P. L. B. OXLEY, An explanation of the different regimes of friction and wear using asperity deformation models, Wear, 53, 229 (1979). 14. R. HILL, E. H. LEE and S. J. TOPPER, The theory of wedge indentation of ductile metals, Proc. Roy. Soc. Ser. A188, 273 (1947). 15. A. P. GREEN, On the use of hodographs in problems of plane plastic strain, J. Mech. Phys. Solids, 2, 73 (1954). 16. I. F. COLLINS, The algebraic-geometry of slip line fields with applications to boundary value problems, Proc. Roy. Soc. Ser. A303, 317 (1968). 17. P. DEWHtJRSTand I. F. COLLINS, A matrix technique for constructing slip-line field solutions to a class of plane strain plasticity problems, Int. J. Num. Meth. Engrg 7, 357 (1973). 18. L. A. PARS, Introduction to Dynamics, Cambridge University Press (1953). 19. J. C. MAXWELL, The theory of rolling curves, In The Scientific Papers of James Clerk Maxwell, (Edited by W. D. Niven), 1, Vol. No. 4, Dover, New York (1965). 20. R. HILL, On the limits set by plastic yielding to the intensity of singularities of stress, J. Mech. Phys. Solids, 2, 278 (1954). 21. M. L. DEVENPECK and A. S. WEINSTE1N,Experimental investigation of work hardening effects in wedge flattening with relation to nonhardening theory, J. Mech. Phys. Solids, lg, 213 (1970). 22. A. SmNDO, General considerations on the compression of a wedge by a rigid flat die, Bull. JSME 5, 21 (1962). 23. D. TABOR, Wear-a critical synoptic view, Proc. Int. Conf. on Wear of Materials, St. Louis, U.S.A. 0977) ASME, l 0977). 24. N. P. SOIl, The delamination theory of wear, Wear, 25, I l I (1973). 25. N. P. Soil, An overview of the delamination theory of wear, Wear, 44, ! (1977). 26. N. P. SON, The Delamination Theory of Wear, Elsevier Sequoia S. A., Lausanne 0977). APPENDIX A For completeness we have proved the elementary result used in the text, that the curve which is geometrically similar to its evolute is a logarithmic spiral. If tr is a measure of arc length on the curve and t# is the inclination of the tangent vector to a fixed reference direction, then p = do'/d~b is the radius of curvature of the curve. An element of arc length on the evolute is dp = (d2o~/d~b2)d~b,which is also equal to dtdw0 due to the assumed similarity. We are hence led to solve the equation ( d o / d 0 ) = to0(d2tr/d02) or equivalently p = t00(dp/d0) with solution p = po exp (~/~Oo)
(8)
which defines a logarithmic spiral whose tangent vector makes an angle 4, = arctan (~o0) with the radius vector from its center.
GEOMETRICALLY OF A PLASTIC SHEAR A N D BY A
SELF-SIMILAR DEFORMATIONS WEDGE U N D E R COMBINED COMPRESSION LOADING ROUGH FLAT DIE
I. F. COLLINSt U.S. Steel Corporation Research Laboratories, Monroeville, PA 15146, U.S.A. (Received 5 February 1980)
Summary--A new slip-line field solution for the shearing of a plastic wedge under geometrically similar conditions is described and its relevance to the formation of wear particles from sheared asperities is discussed. The solution is an example of a recently discovered class of pseudo-steady, plane strain deformations in which a portion of the rigid/plastic material rotates rigidly. NOTATION r t v p 0 A to
position vector in physical space time velocity vector position vector in unit diagram semi-wedge angle die inclination angle angular speed too angular speed when t = I qJ,X slip-line angles /x coefficient of friction
INTRODUCTION Beginning with the investigations of G r e e n [ I , 2 ] the techniques of slip-line theory have been used by m a n y authors to study problems modelling the d e f o r m a t i o n of asperities with a view to gaining understanding of the m e c h a n i s m s of friction and wear for ductile solids in contact, G r e e n [ I , 2 ] and later Edwards and Hailing[3] considered the f o r m a t i o n and growth of junctions f o r m e d by locking asperities under plane strain conditions. These studies confirmed and amplified the well-known dry friction theories of B o w d e n and T a b o r [4]. Childs [5] has used slip-line fields to study the persistence of asperities under normal loads by investigating the interaction of the plastic d e f o r m a t i o n produced in the asperity with the ambient material and with neighboring asperities. W a n n h e i m et al. [6] have made a similar investigation and have included the effect of applying a tangential as well as normal load to the asperity. The interaction b e t w e e n tangential and normal loads on a single wedge-shaped asperity has been considered by J o h n s o n [7], who treated the problem of a rigid/plastic wedge being c o m p r e s s e d by a die to which a normal force N is first applied followed by a monotonically increasing tangential force T. Johnson discusses some slip-line solutions which a p p r o x i m a t e l y describe the progressive d e f o r m a t i o n of the wedge and which predict a growth law for the contact area A which closely a p p r o x i m a t e s that suggested by T a b o r [8], (A/Ao) 2 = 1 + or(TIN) 2
(1)
where A0 is the contact area under normal load N alone and a is a constant -~ 12. The theory was supported by e x p e r i m e n t s on hard-drawn copper and plasticene. tPermanent address; Department of Mathematics, UMIST, P.O. Box 88, Manchester, England. 735
736
I . F . COLLINS
Johnson's theory and experiments also showed that, provided the adhesion between the wedge and die faces was sufficiently strong, continuous deformation would occur beneath the die until T has increased so as to be approximately equal to N, at which point discontinuous shear occurs either at the interface or on a plane inclined at an acute angle to the die face and a sliver of material becomes detached from the main body of the wedge which would eventually break off and form a wear particle. The problem discussed in the present paper is similar to that of Johnson; a wedge of rigid/plastic material is compressed under plane strain conditions by a rigid rough-faced die which is moving with constant velocity obliquely relative to the (vertical) plane of symmetry of the wedge, so that as well as being compressed the wedge material is sheared. With this formulation the problem is one of "pseudosteady flow" in which the deforming configuration grows uniformly in time remaining geometrically similar, so that the problem can be solved using the unit diagram concept [9]. Attention is concentrated on the solution for TIN ratios of the order of unity, where as will be seen a chip forms, which takes the form of a spiral and which when eventually fractured can form a wear particle. The solution is of some fundamental interest in slip-line theory since it represents a relatively simple example of a new class of pseudo-steady flow solutions involving rotating rigid regions discussed recently by Petryk [10, 11] and Collins [12]. The solution given here amplifies Johnson's analysis and throws further light on when and how a wear particle can be expected to be formed from a virgin ductile asperity. Challen and Oxley [13] have also recently produced approximate slip-line field solutions for the deformation of asperities using models in which the hard asperity (die) has an inclined face and is moving horizontally relative to the soft (plastic) asperity. Aspects of this problem using the present spiral chip formation model will also be discussed. SLIP-LINE FIELD SOLUTION WITH SPIRAL CHIP FORMATION The proposed slip-line field solution is shown in Fig. 1. The two-dimensional plastic wedge, whose vertex was originally at 0, with semivertex angle 0, is compressed quasi-statically by a flat perfectly rough die MABN whose face is inclined at an angle )t to the "horizontal (that is, at 7r]2-)t to the vertical symmetry plane of the wedge). The die has a (constant) velocity component perpendicular to its face which causes the wedge material to compress and a (constant) tangential component which is sufficiently large to induce full friction over the interface AB. This happens when the perfectly rough face of the die is slipping (or on the point of slipping) over the wedge material. The value of this critical tangential velocity component will come out as part of the solution. O'
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G FtG. 1. Unit diagram with superposed hodograph (case illustrated is for 0 = 40°, X = 25°, I = 21.2°).
Geometrically self-similar deformations of a plastic wedge
737
The slip-line field network which grows uniformly with time consists of a centered fan ABC composed of straight radial a-lines and circular arc B-lines turning through an angle $. At the generic instant the contact length AB is denoted by p. A velocity discontinuity propagates along the a-line AC and continues along the circular arc CD which separates two rigid regions. Below ACD the material is still at rest, while above CD it is rotating rigidly about the instantaneous center of rotation G (the center of the circular arc CD) with angular speed to which is a function of time. The rigid chip BCDQB must be in equilibrium. If we make the circular arc CD identical with BC, so that it has radius p and span angle ~bthe component of force on DCB perpendicular to the direction bisecting the angle B~D and the torque about C will both be zero by symmetry. The equilibrium of the chip will hence be assured by choosing the mean pressure P at C such that the force component on DCB in the direction of the bisector of BCD is zero. As time progresses the deformation pattern must grow uniformly without change of shape due to the pseudo-steady nature of the problem. To see what restrictions this condition puts on the solution we use the unit diagram concept of Hill, Lee and Tupper [14,9]. If r is the position vector of a point in the field measured relative to O and t is the scalar "time-like" parameter measuring the stage of the deformation, which in this problem can conveniently be taken as the distance 0L in Fig. I, then the stress and velocity fields depend on r and t only through the ratio
p:rlt.
(2)
As r traces out the slip-line field in physical space, p describes the "unit diagram" which is identical to the slip-line field at the instant t = 1. Differentiating (2) with respect to "time" t yields; t ~ -t = v - p
(3)
where v = dr/dt is the velocity of the material particle at r. As the material particle r moves in physical space its image in the unit diagram p traces out a trajectory in the unit diagram whose path is described by (3). This equation can most conveniently be interpreted geometrically by using the concept of the hodograph diagram as pointed out by Green[15]. (This concept is implicit in the arguments given by Hill, Lee and Tupper [14] but the term "hodograph" was not used at this time.) If the diagram in Fig. 1 is interpreted now as the unit diagram (t = I) and on it is superimposed the hodograph diagram of velocity image points of the slip-line field with origin at the center of distortion 0 (the usual small letter convention has been adopted for labelling hodograph image points), then equation (3) states that the rate of change of position of p in the unit diagram is equal to the vector going from p to the image point v in the superimposed hodograph diagram. Thus the path of a material element in actual space is represented by an image trajectory in the unit diagram, the tangent to which is always directed toward the current image point in the superposed hodograph diagram. In particular the unit diagram image of a free surface which consists of the same material particles must possess this property, Since the hodograph image of a rigid translating region is a single point it follows that a free surface bordering on a rigid translating region must be straight, as in the well-known solutions for wedge indentation of a half-space and normal flattening of a wedge [9]. In the present problem, however, it is postulated that the rigid region in the chip bordering on the free surface is rotating. Now as shown by Green [15] a region in physical space which is undergoing a rigid rotation is mapped in the hodograph diagram into a geometrically similar region, scaled up by a factor of to and turned through a right angle in the direction of rotation. In particular the hodograph images dq and bq of the free surfaces DQ and BQ are obtained in this way. Hence the tangent at a point on DQ passes through the corresponding point on dq, which is a curve orthogonai and geometrically similar to DQ. dq (and similarly bq) is hence the curve which is geometrically similar to its evolute DQ (BQ) in the ratio too: I (tOo being the angular velocity at the unit diagram (t = l) stage of the deformation). It is easily shown that the unique curve possessing this property is the logarithmic spiral, whose tangent makes an angle d, = arctan (too) with the radius vector from the center of the spiral. A formal proof is given here in the appendix. This development was clearly forseen by Green [15], who states this result, but without proof or example. The center of the spirals in both the unit and hodograph diagrams is Q, coincident with its hodograph image q, this being the current position of the material particle originally at the wedge vertex 0. It follows by similar reasoning that the unit diagram trajectories of all material particles in the rigid rotating chip, whether or not they lie on the free surface, are logarithmic spirals centered on Q. In passing it may be noted that Petryk [10, 1 l] has used the matrix operator concept for slip-line field construction [12, 16, 17] to develop a procedure for constructing the free surface bounding of an arbitrary plastically deforming region in a general geometrically self-similar pseudo-steady deformation. Since the scale of the velocity field is invariant in time but the scale of the slip-line field in physical space increases in proportion to t, it follows that the angular speed of rotation of the rigid chip is inversely proportional to t. It follows, therefore, that to = tOo~t,since tOo is the angular speed when t = 1. The angular speed is hence infinite at the initial instant of contact of the die with the apex of the wedge. The motion of the rigid chip in physical space can best be visualized by drawing on the well-known theorem of rigid body kimematics [18] which states that any planar rigid body motion can be generated by rolling a curve fixed in the body (the body centrode) on a curve fixed in space (the space centrode). From the continuing similarity of the slip-line field it follows that the path of any point fixed in the slip-line network (i.e. fixed p) moves along a radial straight line through 0 in physical space (since r = pt). Thus in particular the center of rotation G follows such a path in physical space so that the space centrode is a radial straight line through 0. If R is the radius of curvature of the body centrode at G, and S is the distance 0G at time t, then the speed of G in physical space is both S and RtO. Since S is proportional to t, S = So the value of S in the unit diagram, and since to = tO0/t it follows that R : S / o , : Sot~too : S/too. MS Vol. 22, No. 12--B
(4)
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I.F. COLLINS
However, S is also the arc length of the body centrode so that (4) states that the body centrode is such that its radius of curvature is proportional to its arc length and is hence a logarithmic spiral. This spiral must pass through Q since the material particle at Q was initially at 0 on the space centrode. (Putting R = dSld~b and integrating (4) gives R = R0 exp (~,/to0) see equation 8 in Appendix.) As is illustrated in Fig. 2, the motion of the rigid chip in space can hence be generated by rolling the logarithmic spiral QGU on the radial straight line OGT. During this motion the center of the spiral Q itself moves in a radial straight line away from 0. This property of the rolling logarithmic spiral was noted by Clerk Maxwell in his classic paper on rolling curves [19]. The velocity of the wedge material adjacent to the die face A B is given by the vector 0-b in the hodograph Fig. 1. The solution is hence valid whenever the component of velocity of the die in the direction tangential to the die face is greater than or equal to the component of 0b in this direction. O
¢e~ II
1 Ix I xx U
T
FIG. 2. Motion of rigid chip is generated by rolling the body centrode Q G U on space centrode OGT. COMPUTATIONS Since the detailed geometry of the field is quite complicated a small computer program was written in order to calculate the details of the solution. The semiwedge angle 0, the slip-line span angle ~ and the die inclination angle A are taken as independent variables. The ratio tip is determined by the condition that D lies on the undeformed face of the wedge (Fig. 1). Straightforward geometry gives; t O
((cos ~ - sin ~ + sin 2qJ) - tan (0 - A)(sin ~b+ cos ~b- cos 2~)) (cos ;t tan (0 - A) + sin A + sin 0/cos (0 + A))
(5)
Since the hodograph diagram is geometrically similar to the unit diagram slip-line field in the ratio to0 : 1, it follows that the radii 0d, 0c, 0'c and 0'b, representing the magnitude of the velocity discontinuity along ACD, are all equal to tooO. From the condition that b lies on the die face it follows that ~Oo= (tip) cos M(sin ~ - cos 4J + 1)
(6)
and so COoand hence ~b = arctan too can be computed, The inclination of 0b to the horizontal is (X - A) where X = ObA and is given by tan X = (sin ~ - cos ~ + 1)/(cos tp + sin ~).
(7)
The die will he travelling horizontally and the wedge material will just be on the point of slipping if ~ = A and since ,,'t"depends only on 4' from (7), this equation can be used to calculate h for this particular type of deformation. As discussed earlier the pressure at C is determined from the overall equilibrium equations of the rigid chip. The pressure through the rest of the field and hence the normal force and friction coefficient ~ on the interface A B can then readily be found from Hencky's relations. It is worth noting that this calculation is independent of wedge angle 0 and inclination A, so that the coefficient of friction #, depends only on the slip-line angle ~. The program contained three checks: (a) Area check Since the plastic deformation is isochoric the area of the chip formed must be equal to that removed by the die. Thus at any stage of the deformation the areas A B Q D C A and AODCA in Fig. 1 must be equal. This condition does not provide any new information since incompressibility and mass conservation are
Geometrically self-similar deformations of a plastic wedge
739
implicit in the basic equations defining slip-line fields and their hodograph images, but it does provide a valuable check on the correctness and accuracy of the calculations. In all cases these two areas were found to be identical to seven significant figures. (b) Overstressing check A slip-line solution is incomplete until it can be demonstrated that a statically admissible stress field exists in the proposed rigid regions. In general this is difficult to do and would appear particularly so in the present solution due to the complex shape of the rigid chip. In these circumstances the best that one can do is to check all the vertices of the proposed rigid regions using the criteria for overstressing developed by Hill [20]. In the present solution these criteria were applied to the two rigid vertices at D and to those at A and B. These conditions put a definite limit on the range of validity of the solution as discussed in the next two sections. (c) Interference check It was frequently found that for arbitrary choices of 0, ¢, and A, the position of spiral center Q was either above the die face MABN or inside the wedge face ODR. These two conditions also put limitations on the range of values of 0, ¢, and A for which a physically meaningful solution could be found, although the former condition was always preempted by overstressing at D. Accordingly, the computer program contained a check to ensure that not only Q but also no other part of the chip "penetrated" the wedge or die faces. S O L U T I O N S W I T H H O R I Z O N T A L D I E F A C E (A = 0 ) Fig. 3 shows the domain of validity of the solution in the particular case when the die face is perpendicular to the symmetry axis of the wedge (A = 0). This is the configuration considered by Johnson [7]. For a given wedge angle 0 it is seen that there is only a narrow range of values of slip-line angle ~ and hence coefficient of friction # for which a solution can be found. The lower limit on @ (or upper limit on V-) is when "interference" occurs between the chip and the undeformed wedge; the upper limit on $ corresponds to overstressing at D (in fact when the upper angle at D is less than 45° which necessarily means overstressing [20]) or at A for 0 in the range 28--32°. No solution of this type exists for semiwedge angles less than 28 or greater than 83°. This lack of uniqueness of solution for a given wedge angle is not surprising. The basic uniqueness theorem for the rigid/plastic yield point problem does not apply to pseudo-steady processes and in the simple case of normal compression of a wedge, Hill [9] discusses two distinct solutions. In the present problem this lack of uniqueness is of little practical significance due to the narrow range of permissible values of @. It is of interest to note that the critical coefficient of friction t* shown in Fig. 3, ranges from around 0.95 for sharp wedges (0 = 30°) to around 1-0 for blunt wedges (0 = 80°), (strictly/~ ~ 1.0 as ~ 0 ° ) . This hence supports Johnson's [7] conclusion that intense shearing sufficient to produce a wear particle requires a coefficient of friction of around unity. The particular solution valid for 0 = 6 0 °, @ = 11° is illustrated in Fig. 4. 25"
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1.000 90
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PIG. 4. Slip-line field solution with 120° wedge angle (0 = 60°, X = 10°, I = 0°). It is instructive to compare this chip forming solution with the asymmetrical analogue of Hill's [9] solution for normal compression. This is shown in Fig. 5. Rather similar asymmetrical fields have been discussed by Devenpeck and Weinstein [21] and Shindo [22] who, however, consider the case when the wedge material is squeezing out sideways under the die face instead of being dragged along with the laterally moving die as in the present problem. The material in A B C and B D E translates as rigid blocks in directions parallel to A C and DE, respectively, while the material is deforming continuously in the fan BCD. There is a tangential velocity discontinuity along ACDE. The details of the field can easily be evaluated by modifying Hill's arguments [9] for the solution for normal compression. In this solution the deformed material remains attached to the main body of the wedge and does not form a chip: in the friction context this corresponds to a "rubbing model" as opposed to a "chip forming" or "wear model." [13]. For fixed wedge angle, the fan angle ~ ((Fig. 5) decreases with increasing friction coefficient, and the limit of validity is when 4, = 0. The values of p. at which this occurs for O = 40, 60, and 80° are found to be 0.28, 0.57, and 0.86, respectively, while the corresponding friction coefficients at which the curling chip mode of deformation first occurs are 0.96, 0.98, and 1.00. There is hence a range of friction coefficients for which a solution still has to be found, although this gap is smaller for the blunt wedges which are of main interest in the wear context. While this missing solution will almost certainly involve plastically deforming zones bordering on the free surface it is not clear whether it is of "rubbing" or "wear" type. \
0
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' , > t l / / ~ /' / / / / > / .~, Z ; / v ' . J " . 4 B ' / / / / / / I I / / I /
FIG. 5. "Rubbing" mode of deformation at small coefficients of friction. S O L U T I O N S F O R I N C L I N E D D I E S M O V I N G H O R I Z O N T A L L Y (h = X) An alternative configuration which is perhaps more directly relevant to the asperity deformation problem is where the die face is inclined to the horizontal (;t ¢ 0) and is translating horizontally. This models the situation of two surfaces in contact moving parallel to each other, the die face corresponding to a hard asperity on the upper surface. This is the situation considered by Challen and Oxley [13]. The results for this model when in addition the wedge material is adhering to the die face, so 0b is horizontal and ;t = X in Fig. 1 are summarized in Fig. 6. Much the same effects are observed as in the previous situation, the main
Geometrically self-similar deformations of a plastic wedge !
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WITH
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, 70
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FIG. 6. Solution validity domain (A = X). difference being that the range of possible slip-line angles @(and hence of the possible friction coefficients) is widened for a specified wedge angle. This is because the chip has now more room to form and is less likely to "interfere" with the wedge face. As can be seen from Fig. 6, the friction coefficient must be in the range 0.9-1.0 for the chip forming solution to occur. The corresponding inclination of the die face A = X(@) is given by equation (7) and increases monotonically with @as shown in Fig. 6. The solution illustrated in Fig. 1 is for O= 40°, ~ = 25°, A = 21-2°.
DISCUSSION AND CONCLUSION The p h e n o m e n a of wear are highly complex and various [23]. Thermal and oxidation effects are of equal importance to purely mechanical processes. The shearing of asperities is only one of several possible mechanisms for the formation of wear particles, ploughing, and so-called delamination processes [24-26] in which wear particles are formed from the breaking up of surface layers due to the propagation of subsurface cracks are two other well-established mechanisms. Experimental evidence for the formation of spiral chips comes from Johnson's experiments on plasticene wedges where such shaped chips are seen to form at friction coefficients 1, see Fig. 5 of Ref. [7]. (As noted earlier, the upper limit on the friction coefficient for which the chip forming solutions of Figs. 1 and 4 are valid is determined by the condition that the curling chip should not " p e n e t r a t e " the wedge asperity. It should, however, still be possible to find solutions of this general type for larger friction coefficients with the chip just touching the asperity surface but with the dimensions of the deforming zone suitably modified to ensure equilibrium of the rigid chip taking due account of the reaction at the point of contact with the asperity. This possibility has not been investigated here however). In addition, spiral or helical chips are not infrequently found among wear debris as noted by Sub [24], who, however, attributes their presence to the curving of initially thin flat sheet particles formed by delamination due to residual stresses in the sheets. One would also expect to get spiral chips from the ploughing action of hard wedge-shaped asperities analogous to the initial stages of machining. It is very difficult to study the continued deformation of a wedge asperity under general loading conditions as remarked by G r e e n [ l , 2 ] and Challen and Oxley[13]. The adoption of the particular loading program which makes the continued deformation geometrically self-similar has appreciably simplified this problem. The present analysis has largely confirmed Johnson's [7] conclusion that one needs a friction coefficient of around unity in order to generate sufficient shear for a wear particle to
742
I . F . COLLINS
form and has further demonstrated a mechanism for the formation of such a particle based on simple ideal plasticity theory. The new solution given here is of some fundamental interest in slip-line theory as it is a relatively simple example of a pseudo-steady solution involving a rigid rotating region. Similar solutions for oblique wedge indentation discussed by Petryk [10, 11] and Collins [12] belong to a class of more complex "indirect" solutions which require the solution of linear integral equations. Acknowledgement--The author is grateful to Dr. Owen Richmond for some helpful comments on this work. REFERENCES 1. A. P. GREEN, The plastic yielding of metal junctions due to combined shear and pressure, Z Mech. Phys. Solids, 2, 197 (1954). 2. A. P. GREEN, Friction between unlubricated metals: A theoretical analysis of the junction model, Proc. Roy. Soc. Ser. A228, 191 (1955). 3. C. M. EDWARDS and J. HALLING, An analysis of the plastic interaction of surface asperities and its relevance to the value of the coefficient of friction, J. Mech. Eng. Sci. 10, 101 (1968). 4. F. P. BOWDEN and D. TABOR, The Friction and Lubrication of Solids, Clarendon Press, Oxford (1950). 5. T. H. C. CmLDS, The persistence of asperities in identation experiments, Wear, 25, 3 (1973). 6. T. WANHEIM, N. BAY and A. S. PETERSEN, A theoretically determined model for friction in metal working processes, Wear, 28, 251 (1974). 7. K. L. JOHNSON, Deformation of a plastic wedge by a rigid flat die under the action of a tangential force, J. Mech. Phys. Solids, 16, 395 (1968). 8. D. TABOR, Junction growth in metallic friction: The role of combined stresses and surface contamination, Proc. Roy. Soc. Ser. A251, 378 (1959). 9. R. HILL, The Mathematical Theory of Plasticity, Clarendon Press, Oxford (1950). 10. H. PETRYK, Ustalone plaskie preplywy osrodkow idealnie plastycznych ze swobodnym brzegiem, (Doctoral Thesis), Institute of Fundamental and Technical Research, Warsaw (1977). 11. H. PETRVK, On slip-line field solutions for steady-state and self-similar problems with stress-free boundaries, Arch. Mech. 31,861 (1979). 12. I. F. COLLINS, Integral equation formulation of slipline field problems, Applications of Numerical Methods to Forming Processes, ASME, San Francisco, 129 (1978). 13. J. M. CHALLEN and P. L. B. OXLEY, An explanation of the different regimes of friction and wear using asperity deformation models, Wear, 53, 229 (1979). 14. R. HILL, E. H. LEE and S. J. TOPPER, The theory of wedge indentation of ductile metals, Proc. Roy. Soc. Ser. A188, 273 (1947). 15. A. P. GREEN, On the use of hodographs in problems of plane plastic strain, J. Mech. Phys. Solids, 2, 73 (1954). 16. I. F. COLLINS, The algebraic-geometry of slip line fields with applications to boundary value problems, Proc. Roy. Soc. Ser. A303, 317 (1968). 17. P. DEWHtJRSTand I. F. COLLINS, A matrix technique for constructing slip-line field solutions to a class of plane strain plasticity problems, Int. J. Num. Meth. Engrg 7, 357 (1973). 18. L. A. PARS, Introduction to Dynamics, Cambridge University Press (1953). 19. J. C. MAXWELL, The theory of rolling curves, In The Scientific Papers of James Clerk Maxwell, (Edited by W. D. Niven), 1, Vol. No. 4, Dover, New York (1965). 20. R. HILL, On the limits set by plastic yielding to the intensity of singularities of stress, J. Mech. Phys. Solids, 2, 278 (1954). 21. M. L. DEVENPECK and A. S. WEINSTE1N,Experimental investigation of work hardening effects in wedge flattening with relation to nonhardening theory, J. Mech. Phys. Solids, lg, 213 (1970). 22. A. SmNDO, General considerations on the compression of a wedge by a rigid flat die, Bull. JSME 5, 21 (1962). 23. D. TABOR, Wear-a critical synoptic view, Proc. Int. Conf. on Wear of Materials, St. Louis, U.S.A. 0977) ASME, l 0977). 24. N. P. SOIl, The delamination theory of wear, Wear, 25, I l I (1973). 25. N. P. Soil, An overview of the delamination theory of wear, Wear, 44, ! (1977). 26. N. P. SON, The Delamination Theory of Wear, Elsevier Sequoia S. A., Lausanne 0977). APPENDIX A For completeness we have proved the elementary result used in the text, that the curve which is geometrically similar to its evolute is a logarithmic spiral. If tr is a measure of arc length on the curve and t# is the inclination of the tangent vector to a fixed reference direction, then p = do'/d~b is the radius of curvature of the curve. An element of arc length on the evolute is dp = (d2o~/d~b2)d~b,which is also equal to dtdw0 due to the assumed similarity. We are hence led to solve the equation ( d o / d 0 ) = to0(d2tr/d02) or equivalently p = t00(dp/d0) with solution p = po exp (~/~Oo)
(8)
which defines a logarithmic spiral whose tangent vector makes an angle 4, = arctan (~o0) with the radius vector from its center.