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A slip-line field analysis of the rolling contact problem at high loads
Int. J. Mech. Sci. Vol. 25, No. 4, pp. 265-275, 1983 Printed in Great Britain.
0020-7403/83/040265-11503.00]0 Pergamon Press Ltd.
A SLIP-LINE FIELD ANALYSIS OF THE ROLLING CONTACT PROBLEM AT HIGH LOADS H . PETRYK Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland
(Received 23 August 1982) Summary--An analysis of the steady rolling of a rough rigid cylinder over the surface of a rigid-perfectly plastic solid is presented. An exact slip-line field solution is investigated by using some recently developed techniques. It is shown that the solution is not uniquely defined and infinitely many complete solutions can be constructed for given boundary conditions. The numerical results show interesting variations of the deformation pattern and of rolling resistance at high loads. The limits are predicted to the magnitudes of the applied load and braking torque for which the steady rolling is possible. NOTATION F
Q W R L to V
k f = F/kRL q = Q/kR2L w= W/kRL h r
a, li, ~,, ,L & O
~rp
horizontal translatory force torque vertical load radius of cylinder length of cylinder angular speed of cylinder advance speed of cylinder yield shear stress dimensionless horizontal force 1 dimensionless torque dimensionless load variables dimensionless vertical load depth of plastic zone radius of isolated slip-line slip-line field angles, Fig. 1 parameter of free boundary operator compressive principal stress in non-plastic stress field slip-line fan angle, Fig. 3 1. INTRODUCTION
When the vertical load on a hard cylinder rolling over the plane surface of a deformable solid is sufficiently high, large plastic strains beneath the cylinder may occur. The large strains will usually be accompanied by the appreciable deformation of the free surface of the solid in front of or behind the advancing cylinder. The shape of the deformed free surface is not specified in advance. In theoretical investigations of the steady rolling this shape has to be found such that it does not vary when the deformation proceeds. The occurrence of large deformations and unspecified boundaries makes the steady-state problem of the rolling contact at high loads difficult to analyse without introducing some simplifying a~sumptions. The aim of the present paper is to examine an exact solution to the problem which satisfies all the statical and kinematical conditions, however, at the cost of using of the highly idealized, incompressible rigid-perfectly plastic model of the material. Inertia and body forces are neglected, and the plane-strain deformation is assumed. The cylinder is taken to be perfectly rough so that no slipping is possible over the arc of contact until the shear contact stress is equal to the yield shear stress k. The analysis presented below, in spite of idealization of the material behaviour, seems still relevant to such problems of practical importance as rolling friction of metals at sufficiently high loads [1] or cohesive soil-vehicle interaction. The theoretical aspect of the problem is also of considerable interest because of untypical boundary conditions. Over the deforming free surface not only the usual zero traction condition but also the kinematical steady-state condition have to be 265
266
H. PETRYK
satisfied. On the other hand, the shape of this surface is to be found. This type of boundary conditions need not lead to a unique solution. It will be shown that there exist even infinitely many complete solutions to the rolling contact problem. Several existing examples [2-6] indicate that this is probably a general rule for plane-strain ideal plasticity problems involving unspecified boundaries, not only for steady-state ones but also for those of geometrically self-similar deformation. The analysis will be carried out by using the slip-line field theory[7]. Several attempts have been made to construct a slip-line field solution to the rolling contact problem [8-11] but only recently Collins [12] succeeded in calculating the slip-line field for this problem without making any simplifying geometrical approximations. The same slip-line field and a method of constructing has been independently proposed in [3] but without performing any computations. Collins' solution[12] is determined uniquely by the values of two load variables (e.g. the dimensionless vertical load w and horizontal force f acting on the cylinder) which fully define the loading conditions. In the present paper a more general, non-uniquely defined solution is investigated which contains Collins' solution as a particular case. The question arises how to interpret physically the arising non-uniqueness of solutions (for a brief discussion of this question see [5]). It might be postulated that only that solution may have a physical meaning which is associated with a local minimum of the pulling force (or torque) with respect to the neighbouring complete steady-state solutions obtained at the fixed values of the remaining load variables, that is torque (or pulling force) and vertical load. General criterion of this type has been derived in [13] from a postulate of stability of a deformation process against persistent disturbances. H o w e v e r , this has been done under the assumptions which are not satisfied in the case of the solution considered here. The question remains still open, and the results will be presented without reference to any criterion of selection of the physically correct solution. 2. THE NON-UNIQUE BOW-WAVE SOLUTION The form of the solution shown in Fig. 1 was previously proposed by the author[3, 14]; the present paper contains the numerical analysis of this solution. In the particular case when the angle 3' of the slip-line fan at the point B is equal to zero, the solution coincides with that considered by Collins[12]. The slip-line field and corresponding hodograph in Fig. 1 represent the steady stage of rolling of a rigid, perfectly rough cylinder on a rigid-perfectly plastic half-space. According to the assumed incompressibility condition, free surface elements before and after deformation are on the same level. The additional amount of the material forming the "bow-wave" ahead of the cylinder is thought to be pushed out of the half-space during an initial, unsteady stage of the deformation process which is not considered here.
0
b". f
E
(3 m
$1_ {a)
(b)
FIG. 1. Exact non-unique solution: (a) slip-line field, (b) hodograph.
A slip-line field analysis of the rolling contact problem at high loads
267
The deformation takes place in the region ABFDGA (Fig. la) and along the velocity discontinuity line EDFB, the isolated slip-line ED being a circular arc of radius r and S as the centre. The traction-free boundary AB is subject to compression. In the reference frame translating g,ith the cylinder the boundary AB constitutes a part of the trajectory of surface elements. The region AGDEA adhering to the cylinder is rigid and rotates together with the cylinder with the same angular speed ~o. Hence its hodograph image a*gdea* (Fig. lb) is a geometrically similar region, rotated through a right-angle in the direction of rotation and scaled up by toil5]. The instantaneous centre of rotation lies at the point O or S if the centre of the cylinder or the block of material is fixed in space, respectively. The advance speed v of the cylinder is always greater than the peripheral speed ~oR, the difference (or skid) being equal to tor, i.e. to the magnitude of the velocity discontinuity along EDFB. Note that this "skid" is entirely due to the deformation within the solid since no slipping occurs over the arc of contact. Neither the slip-line field nor the hodograph can be constructed in a straightforward manner since there are not a sufficient number of starting curves of known shape in the physical or hodograph planes. However, if one slip-line, say AC, and three field angles a, /3 and y are specified then the rest of the slip-line field and the hodograph net can be determined directly. The base slip-line AC has to be chosen such that the resulting solution describes a steady motion in which the size and shape of the free boundary AB does not vary in time. When formulated mathematically this condition leads to an integral equation for the base curve. A convenient and efficient method to find unknown base slip-lines has been proposed by Collins[16] who introduced the matrix operational formulation in place of an analytic one. This idea has been developed into a systematic computational procedure by Dewhurst and Collins[17]. The procedure is based on a double power series representation of the solution of the field equations due to Ewing[18]. In the matrix formulation the integral equation for the base curve is reduced to an algebraic matrix equation for the vector representation of this curve as unknown. It is essential for the integral equation to be linear; in that case the resulting matrix equation is of constant coefficients and can be solved by standard techniques. A general case of the steady-state deformation of a free boundary has been considered by Petryk[3, 14] who introduced the matrix boundary operator which acts on the hodograph plane and generates the velocity field automatically satisfying the steady-state condition on the free boundary (for convenience the result is quoted in Appendix 1). This free boundary operator can be used to complete the matrix formulation of the present problem. The details are given in Appendix 2 where the derivation of the matrix equation governing the shape of the base slip-line AC is given. This equation is linear provided not only the field angles a, /3 and 3, but also certain parameter ~: on which the free boundary operator depends are regarded as known. In Appendix 2 it is shown that the final solution has actually three degrees of freedom since the above-mentioned four parameters a, /3, 3' and 6 are not mutually independent. On the other hand, the loading of the cylinder is fully defined by prescribing the magnitudes of two load variables; the third one is to be predicted by the theory. Hence an infinite number of solutions may exist for the same conditions of loading of the cylinder, each associated with different geometry of the bow-wave and predicting different resistance to rolling. The numerical analysis confirms that this is really the case. The obtained family of solutions may be parametrized, e.g. by 3,, the angles a and/3 being for each 3' selected such that the given load variables have prescribed values. The permissible range of solutions has been found to be limited by: (i) Vanishing of the fan angle 3,. The corresponding limit solution has the form analysed by Collins[12]. (ii) Vanishing of the angle A at which the slip-line GA meets the cylinder surface. The shear contact stress at the point A reaches the limit value k and the slipping of the material over the cylinder surface becomes possible at this point. (iii) Overstressing of the rigid corner at the point B. From Hill's criteria[19] we obtain that the rigid corner at B is overstressed if the angle 8 of inclination of the. free boundary AB at the point B is smaller than 23,. Therefore the solution can be statically admissible only if the condition 0~<2y~<8
(1)
is satisfied. (iv) Vanishing of the fan angle//, to be interpreted as the skidding limit. For certain domain of loading, all solutions within the permissible range defined by the above limits are proved complete (see Sections 3 and 4). Vanishing of the fan angle /3 (limit iv) was found to be necessarily accompanied by simultaneous vanishing of the angles d and O. The corresponding limit solution is shown in Fig. 2. The free boundary AB is now straight and inclined to horizontal at the angle 8 = (¢r/4) + y. The solution describes the skidding of the cylinder without any rotation. The material in EABDE, pushed out of the half-space in an initial, unsteady stage of flow, adheres to the cylinder and does not undergo any further deformation. Only the shearing of the material along the straight velocity discontinuity line EB occurs, in the theory without loss
E
O
\~-
FIG. 2. Skidding limit solution. MS VoL 25, No. 4 ~ D
268
H. PETRYK
of the cohesion of the material. The relations between the load variables are easily found to be f = wl(l + 23,),
(2)
q = w ( w - 2)/(2(1 + 23,)).
(3)
It is worth mentioning that in this particular solution it is not essential that the arc of contact is circular. The solution shown in Fig. 2 can represent also the steady stage of shifting of a sufficiently rough, rigid prismatic body of arbitrary crossection (wedge-shaped for instance) over the surface of a rigid-perfectly plastic solid, the rotation of the body being excluded in advance by kinematical constraints. In that case the value of the fan angle 3, is no longer constrained by the equation (3), and the solution is non-uniquely defined. According to the formula (2), the value of the shifting force for a given vertical load decreases with increasing 3, and reaches its minimum value in the limit case 3, = 45°; for 3, > 45 ° the rigid corner at B of the apex angle 180° would be overstressed. 3. COMPLETENESS OF THE SOLUTION By using the method of Green[20] it is found that the rate of plastic work is certainly positive everywhere in the deforming region A B F D G A (Fig. la). This can be stated by examining only the general form of the slip-line and hodograph nets without the need of any computations. Also the work of shear along the velocity discontinuity line E D F B is positive. There remains to check whether the rigid regions are not overstressed. For the neighbourhood of the isolated segment E D of the velocity discontinuity line to be not overstressed, the pressure along this segment must vary according to Hencky relation. This follows from the theorem due to Bonneau[21,22]. In other words, the segment E D must be a slip-line and not an envelope of slip-lines. Therefore we can construct a regular slip-line field based on the slip-lines E D and DGA and extend it up to the cylinder surface (Fig. 3). The resulting normal contact stresses are evidently compressive as they should be, and the shear contact stresses are in agreement with the assumed perfect roughness of the cylinder since no slipping occurs along AE. This proves that the rigid region E D G A E is not overstressed. Below we show that the plastic stress field can also be extended in a statically admissible manner into the half-space, however, under certain restrictions on the field angles. Let us construct a singular slip-line field E D K based on the slip-line E D with singularity at E (Fig. 3), the fan angle ~b being equal to (~r/4) + 2 y - & Then by Hencky relations the pressure on the line El(, at the point E is equal to k, and we can build a regular plastic stress field between the line E K and a hypothetical stress-free surface EL. Now, similarly as in [5], we construct first a regular slip-line field based on the slip-lines D L and D B and next a singular field with singularity at B and the fan angle equal to 3', up to a line L M N B along which the pressure is constant and equal to k. Then, from every point of the line L M N B we draw a straight ray in the direction of the principal compression stress at this point. The rays constitute outside L M N B trajectories of the compressive principal stress ~rp, while the second principal stress acting across the rays is assumed to vanish identically. It can be shown[5] that the resulting stress field is statically admissible provided I~'~1 is chosen to decrease in an appropriate manner with increasing distance p from the L M N B - I i n e . Thus, if the above construction can be performed the solution is complete. There are two possible limitations to the whole procedure. The first is that before completing the stress field in the region E D K L E the radius of curvature of some slip-line reaches zero. The second is that one (or both) of the limiting rays drawn from the points B and L, outside of which the stresses are zero, goes out of the material. However, the condition (1) implies that the ray drawn from B lies inside the half-space except 23, = 6 when the ray lies exactly on the surface. The ray drawn from L will lie in the half-space if ¢J+ 20 -< (3/4) ~', i.e. if 20 ~ (w]2) + 6 - 23,.
(4)
It has been checked numerically using the matrix procedure [17] that the smallest radius of curvature of slip-line in the region E D K L E , while being always at L, is positive if (4) is satisfied. Hence we have come to the conclusion that the solution shown in Fig. I is complete if the field angles satisfy the inequalities (1) and (4). The condition (4) is not necessary for statical admissibility of the solution while (1) is. In particular, the condition 0 <~ 3, <~45° is both necessary and sufficient for completeness of the solution shown in Fig. 2.
FIG. 3. Construction of statically admissible stress field in rigid regions.
A slip-line field analysis of the rolling contact problem at high loads
269
4. NUMERICAL RESULTS The slip-line field shown in Fig. 1 was analysed numerically over its full range of validity. The computations were performed on a small ODRA 1204 computer using the matrix procedure described in [17]. The matrix equation (A8) was solved using a standard subroutine employing Gaussian elimination. The matrix dimension was taken to be e_q_yalto eight. For each set of the input values of the angles a, fl and 3, the parameter ~: was iterated until [Joel- [oa*l [ (Fig. lb) was smaller than 10-8, the scale parameters to and r being taken as unity. This was always found possible within the limits (i)-(iv) defined in Section 2, also for 3' = 0, what implies that the "skidding limits" reported by Collins[12] may be ascribed to the numerical procedure used in [12]. It is worth mentioning that the resulting value of ~: tends to 0 or 1 if the speed ratio wRlv tends to I or 0, respectively. By using a Newton-type iterational procedure the values of a, /3 or y were selected to obtain required values of the dimensionless load variables w or q with accuracy 10-5, or to reach the validity limits ;t = 0 or 2y = 8 with accuracy 10_6 [rad]. Several accuracy checks were also contained in the computer program. For example, in all cases the difference in fhe vertical coordinates of the points B and E, being theoretically zero, was smaller than R ' 10-7. Also the resultant dimensionless moment and force components acting across the line AGDFB, which should vanish for the region AGDFBA to be in equilibrium, in most cases were smaller than 10_4 and always smaller than 10 3. In Collins' paper[12] a comparison of the predictions of his analysis with previous approximate solutions and experiment is given. Since Collins' solution[12] is a particular case of the present one when 3' = 0, no such a comparison is included here. 4.1 The case o[ [orce-driven cylinder (q = 0) The range of validity of the solution in the case q = 0 can perhaps be best visualized in the (w, h)-plane (Fig. 4), where h is the depth of the plastically deformed zone. The corresponding range of field angles is shown in Fig. 5. Figure 4 shows some interesting features of the solution. Because of the non-uniqueness of the solution, not a single curve but a whole family of curves is obtained, each of them showing the relation between w and h for a different, fixed value of the fan angle 3'. The validity limits (i)-(iv) defined in Section 2 are indicated in the figure. For w > - 1-5 all solutions within these limits are complete since the inequality (4) is found to be satisfied in this range of loading. It is seen that for a fixed radius of the cylinder the maximum possible depth of plastic zone hmax= 0.182-R occurs for 3' = 0 and w ~ 2.05 and for still higher loads it decreases with increasing load. Moreover, any curve corresponding to a fixed value of 3' has a turning point which defines the maximum load w for which a solution of this value of y can be constructed. No solution of the examined type can be constructed for w > 2.355 which suggests that the cylinder cannot roll steadily under such high loads. However, this cannot be stated ultimately until the possibility of existence of other solutions is excluded. For 2.355> w > - 2.1 the permissible range of solutions is from both sides limited by vanishing of the angle I', a smooth maximum of y being attained somewhere within this range. For -2.1 > w > 2 this range splits into two families of solutions. When tends to 2 [rom above the depth of the plastic zone in one of these families tends to zero and the skidding limit is reached. This is seen more clearly in Fig. 6 where the ratio of the peripheral to advance speed of
0.~
~=o
0,16 0.14 rr
j012
to
//
/ . ~ / complete ~ " \ solutions " ~
\
.~ O.lO
/,/
overstressing
at
B
'~ 1208 "~ Qo6 Q04
X=0 skidding limit
O j02
\
ds
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dimensionless
1.~---vertical
Ioad,w
20 -
FIG. 4. Depth of plastically deformed zone (q = 0).
2.5
270
H.. PETRYK 120
=0
w=O.5
9 .0
t
_
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~
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0verstressi at B
4
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2._.?2"L.X
l
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'
i
'
6
fiet~
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angle,~,degrees
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F1G. 5. Range of field angles (q = 0).
1
>
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X=O
skidding limit
0.2 0
~
.
i
0.5
B
.
.
.
,
~
1.0 1.5 2.0 dimensionless verticQt IoQd,w
__
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FIG. 6. Ratio of peripheral to advance speed (q = 0).
the cylinder is plotted vs w. The skidding limit solution (Fig. 2), exceptionally for q = 0, is not uniquely defined by the values of w and q since for q = 0 and w = 2 the equation (3) is satisfied identically and the value of 3' is left undetermined. Thus for w = 2 two families of complete solutions can be constructed, one corresponding to rolling and the second to skidding of the cylinder. The family corresponding to rolling is shown in Fig. 7 where two limit solutions of this family and a medial one are drawn to scale. The (non-unique) relationship between the dimensionless pulling force .f and vertical load w is shown in Fig. 8. While for w < 2 the variations of the pulling force calculated for a fixed w within the permissible range of solutions are small and do not exceed 3%, they become quite large when the solutions approach the skidding limit. For a fixed w < ~ 1.5, the greater the angle 3" the smaller the pulling force so that the minimum value of f within the permissible range is obtained for the limit solution with 23, = 8. For ~1.5 < w < 2.04 the absolute minimum of [ still corresponds to the limit solution with 23, = & but also a local minimum of .f appears for 3' = 0 and for w > 2-04 it becomes the absolute one. The variations of the bow-wave profile with increasing w are illustrated in Fig. 9 where a number of the profiles corresponding to the different limit solutions are shown to scale. At higher loads rapid growth of the bow-wave occurs with decreasing load w when the skidding limit is approached, especially for the solution with 3" = 0. This corresponds to remarkable increase of the rolling resistance (see Fig. 8). The variations of the geometry of the solution within the permissible range for a fixed w may also be very large: compare four different profiles marked in Fig. 9, corresponding to the same w = 2.0. 4.2 General case (q # 0) In Fig. 10 the domain of validity of the solution in (q, /')-plane is presented. Only a half of the complete figure is shown since there is a polar symmetry with respect to the origin of the coordinate system in this plane. The figure is rather complicated due to the non-uniqueness of the solution. The entire validity domain is covered by the regions corresponding to permissible ranges of solutions for different, fixed values of the vertical load w. For several values of w such regions are indicated in the figure; for sufficiently close values of w the regions would overlap each other. The validity limit A = 0 is essential practically only for positive
A slip-line field analysis of the rolling contact problem at high loads
271
q:0 w=20
,/
,/
,~/ If: 1701" ' f=0620 (max) / ,=10" (min) If= 0"
F]o. 7. Solution range for rolling at w = 2.0 (q = 0).
2,0
,
,
I
:0 I
skidding Limit
1,5
I I
I
2o\r 25" {t u
1,0
e-
~r=o i/1
overstressing at B
I
I
I
0.5
1,0
1,5
2,0
2,5
dimensionless vertical toad, w FIG. 8. Non-unique relationship between pulling force and vertical load (q = 0).
values of torque. The maximum value of w for which a solution of the considered type can be constructed was found to be -2.43. The entire validity domain can also be thought as a topological sum of particular validity domains of the solutions with different, fixed values of the fan angle 3'. The validity domain for Y = 0 is bounded by the line OPQRSO. It generally shrinks with increasing 3' (except little expansion near PQR-line) up to a single line OM for 1, = 45°. For all permissible solutions represented by points lying to the right and down of the line ON the inequality (4) is satisfied so that all those solutions are complete. The lines OSR, OM and MTR in Fig. 10 correspond to the skidding limit reached by another limit solution with 3" = 0, 23, = 8 and it = 0, respectively. From the relations (2) and (3) we obtain that the lines OSR and OM are segments of the parabolas q = ( l / 2 ) f - f and q = (1/2)(1 + I r / 2 ) f - f, respectively; also the line MTR can be determined analytically. The skidding limit can be reached for Ioadings which are represented by the points lying in the region bounded by the line OMTRSO. If the load w is specified the relation between f and q due to skidding is linear, namely, q = (l/2)(w - 2)f from (2) and (3). Note that the region OMTRSO does not represent the entire validity domain of the solution shown in Fig. 2. This is so since the condition it = 0 is not the limiting one for this solution treated not as a limit case of the rolling solution shown in Fig. 1 but as ap independent solution.
272
H. PETRYK
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-
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ing [i w.2
F;G. 9. Variation of bow-wave profile with vertical load (q = 0).
0,4
\ ~---~ 0,2
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%, 0 -0,2 -0,4
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\
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\ \ FIG. 10. Domain of validity of solution plotted in (q, /')-plane.
It can be seen from Fig. 10 that the relation between [, q and w describing a rolling friction law is close to be unique in one part of the validity domain while in the other the variations of the load variables within the permissible range of solutions are much greater. The second part corresponds to the region in which the skidding limit can be reached. This is not surprising since the large variations in both the geometrical and load variables when the skidding limit is approached have yet been observed in the case q = 0 (see Figs. 8 and 9). No solution of the considered type can be found if the braking torque ( - q ) is sufficiently large. The critical value of the torque vs the vertical load w is plotted in Fig. 11. It looks very likely that if the braking torque exceeds this critical value the steady rolling becomes impossible. A different situation occurs in the case q > 0 (driving condition) when the solution fails with increasing q by vanishing of the angle A. This should not be interpreted as an ultimate limit to rolling but rather as a limit of validity of the assumption that no slipping occurs over the arc of contact (see the remark (ii) in Section 2). Therefore a modified slip-line field which allows the slipping on a part of the contact arc need also be examined in order to obtain a more complete view on the mechanism of rolling under driving conditions. Moreover, the bow-wave solution investigated in this paper is not the only possible form of solution, especially when q > 0 and [ < 0 (see the remark in [12], p. 444). The investigation of solutions of the other types is beyond the scope of the present paper.
A slip-line field analysis of the rolling contact problem at high loads O~
o21 ~ 0.1
.
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.
.
, 7 'Y'i '''~
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.
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FIG. 11. Range of braking torque and vertical load for which rolling is possible.
5. CONCLUSIONS (i) T h e s t e a d y - s t a t e p r o b l e m o f s u r f a c e rolling with a r o u g h cylinder has infinitely m a n y c o m p l e t e solutions in the f r a m e w o r k o f the p l a n e - s t r a i n t h e o r y o f a rigidp e r f e c t l y plastic solid. This n o n - u n i q u e n e s s is due to o c c u r r e n c e in the p r o b l e m o f a stress-free s u r f a c e o f initially u n k n o w n shape. (ii) T h e matrix t e c h n i q u e f o r c o n s t r u c t i n g slip-line field solutions o f indirect t y p e [ 1 7 ] with the use o f the free b o u n d a r y o p e r a t o r s [ 1 4 ] c a n effectively be applied to c o n s t r u c t a w h o l e range o f solutions to the s t e a d y rolling p r o b l e m r a t h e r than a single, particular solution. T h e q u e s t i o n o f selection of the solution w h i c h has a p h y s i c a l m e a n i n g r e m a i n s open. (iii) A t high loads the large variations o f b o t h the g e o m e t r y of solution and p r e d i c t e d rolling r e s i s t a n c e are o b t a i n e d within the limits o f validity o f the e x a m i n e d solution, especially w h e n the skidding limit is a p p r o a c h e d . (iv) T h e e x a m i n e d " b o w - w a v e " solution s e e m s to p r o v i d e a rather c o m p l e t e d e s c r i p t i o n o f the rolling of a r o u g h c y l i n d e r at zero or negative torque, o f c o u r s e within the limits o f the ideal plasticity t h e o r y itself. T h e case o f positive t o r q u e requires also solutions of o t h e r t y p e s to be investigated. REFERENCES I. K. L. JOHNSON,Rolling resistance of a rigid cylinder on an elastic-plastic surface. Int. Z Mech. Sci. 14, 145 (1972). 2. H. KtJDO, Some new slip-line solutions for two-dimensional steady state machining. Int. J. Mech. Sci. 7, 43 (1965). 3. H. PETRYK,Ph.D. thesis (in Polish). I.F.T.R. Rep. 46, 1977. 4. P. DEWHURST,On the non-uniqueness of the machining process. Proc. Roy. Soc. Lond. A 361), 587 (1978). 5. H. PETRYK,Non-unique slip-line field solutions for the wedge indentation problem. J. Mdc. AppL 4, 255 (1980). 6. I. F. COLLINS,Geometrically seft-similar deformations of a plastic wedge under combined shear and compression loading by a rough flat die. Int. J, Mech. Sci. 22, 735 (1980). 7. R. HILL, The Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950). 8. J. MANDEL,Rrsistance au roulement d'un cylindre indrformable sur un massif parfaitement plastique. Le Frottement et l'Usure, p. 25 (1966). 9. E. A. MARSHALL,Rolling contact with plastic deformation. J. Mech. Phys. Solids 16, 243 (1968). 10. I. F. COLLINS,A simplified analysis of the roiling of a cylinder on a rigid/perfectly plastic haft-space. Int. J. Mech. Sci. 14, 1 (1972). 11. V. M. SEGAL,Plastic contact in the motion of a rough cylinder over a perfectly plastic haft-space (in Russian). Mekh. Tv. Tela, No. 3, 184 (1971). 12. I. F. COLLINS,On the rolling of a rigid cylinder on a rigid/perfectly plastic haft-space. J. Mic. Appl. 2, 431 (1978). 13. H. PETRYK,On the stability of non-uniquely defined processes of plastic deformations. J. Mdc. Theor. Appl. Numrro sprcial, p. 187 (1982). 14. H. PETRYK, On slip-line field solutions for steady-state and self-similar problems with stress-free boundaries. Arch. Mech. 31,861 (1979).
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H. PETRYK
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. C O L L I N S , The algebraic-geometry of slip line fields with applications to boundary value problems. Proc. Roy. Soc. A 303, 317 0%8). 17. P. DEWHURST and I. F. COLLINS,A matrix technique for constructing slip-line field solutions to a class of plane strain plasticity problems. Int. J. Num. Meth. Engng 7, 357 (1973). 18. D. J. F. EW1NG,A series-method for constructing plastic slip-line fields. J. Mech. Phys. Solids 15, 105 (1%7). 19. R. HILL, On the limits set by plastic yielding to the intensity of singularities of stress. J. Mech. Phys. Solids 2, 278 (1954). 20. A. P. GREEN, The plastic yielding of notched bars due to bending. Quart. J. Mech. Appl. Math. 6, 223 (1953). 21. M. BONNEAU, Ann. Pts et Ch., Sept-Oct. 609 (1947). 22. J. SALENqON, Th~orie de la Plasticit~ pour les Applications a la M~canique des Sols. Eyrolles, Paris (t974). APPENDIX 1 Let us consider a plastically deforming region A B C bounded by a concave segment A B of a stress-free boundary and by slip-lines A C and B C of angular range 0 (Fig. 12a). Let the configuration of the slip-line images in the corresponding hodograph diagram abe be as shown in Fig. 12(b). We assume ~hat the stress and velocity fields in A B C are described by analytic functions• The slip-line field in A B C is thought as a part of the solution to a steady-state problem. Thus in some reference frame the free boundary A B must coincide with a stream-line. It can be shown[3, 14] that this steady-state condition is satisfied if and only if the vector representations[16, 17] ~r, and w2 of the hodograph curves ac and bc are related through the matrix equation ~r2 = Hob'l,
(A1)
where the matrix "free boundary" operator H0~ has the form Hoe = RoP-*o(A6 - Q'Re).
(A2)
Ro, P-*o = (P*) I Q , in the formula (A2) are the basic matrix operators[16, 17] and
-l A~ =
1
0
0
0
• • ' |
0
• " •
0
"" "
-~
1
0
1
+~
-1
-~
-2 2
1 -2
1
The parameter ~: on which the operator H depends may take an arbitrary value from the interval (0, I]. It defines the mutual position of the hodograph net and hodograph pole o according to the formula ~: = 1 - 2X/(2)[~oo[/80,
(A3)
where So is the radius of curvature of the curve uc at the point u, and the ray ou meets the curve ac at an angle 45 °. The form of the other free boundary operator F0: A C ~ BC (Fig. 12a) which generates the stress field in A B C such that A B is traction free has been given by Dewhurst and Collins[17]. In applications it is convenient to use the equivalent form of the operator F, viz. Fo = RoP-*o(Al + Q*Ro),
(A4)
where Al = Ad~=l. The formula (A4) can be derived in an analogous way as (A2). Both the operators F and H do not depend on the actual shape of the free boundary so that they can be employed in the matrix procedure[17] to construct slip-line field solutions for steady-state problems involving deforming free boundaries of initially unknown shape.
A
bV~\~
P
0
(a) (b) FIG. 12. Plastic region with stress-free boundary: (a) slip-line field, (b) hodo~'aph.
A slip-line field analysis of the rolling contact problem at high loads
275
APPENDIX 2 The reader isreferred to [17] for a full discussion of the matrix procedure. Referring to Fig. 1, let us choose the slip-line A C as the base curve 0" (for simplicity we use the notation in which a curve is identified with its vector representation). Then the slip-line B C is F~0", and from the definition of the basic matrix operators P and Q we have G A = P~a0", C G = Q*0" and C F = Q*F~0". On using the fundamental construction of the field between two characteristics C F and CG we obtain DG --- 0"3 = (Py~Q*F~ + Q~vQ*)0"
(A5)
The hodograph images of the slip-lines DG and G A bordering the rigid r e # o n E D G A E rotating with angular speed to are geometrically similar curves scaled up by to[15], that is dg = to¢3 and ga* = toP,a0". The curve dfb* is a circular arc with radius tor, the magnitude of the velocity discontinuity along E D F B . Thus dfb* = tore, where e is the unit circle vector. The fundamental construction applied first to the curves db* and dg and next to the curves gc and ga* gives bc =- 0"2 = R~+a{P*(tore) + Q*(to~r3)}
(A6)
ac -~ 0., = P~(toP,~0.) + Qa~{P*(tore) + Q*(to¢3)}.
(A7)
Finally, according to (AI), the curve ac transformed by the operator Hoe should give the curve bc. Eliminating ~r~, ~2 and tr~ between (AI), (A5)-(A7) and rearranging we obtain the following matrix equation for 0.: K ¢ = rLc
(A8)
where K = (Ra+/3 --
* H ~ Q ~ , , ) Q *~(P:,~Q,~F,~ + Q~:,Q*) - H ~ P , ~ P ~
and L = - (R,~÷~ - H ~ Q ~ ) P * ~ .
cr can be found from (A8) once the three field angles a, /3 and 7 and the parameter ~ are prescribed (the radius r as well as the angular speed to play the role of scale parameters only). From 0. the shape of other slip-lines in the region A B F D G A and their hodograph images can be determined in a straightforward manner. The position of the hodograph pole o can be determined from (A3) what completes the construction in the hodograph plane. The construction in the physical plane can be completed using the property of geometric similarity of the regions A G D E A and a*gdea*. However, the obtained solution is usually kinematically inconsistent since the hodograph image of t._he centre of the cylinder does not in general coincide with the hodograph pole. In the correct solution Ioel must be equal to Ioa*l; this geometric condition reduces the number of independent parameters of the solution to three. If the parameter ~ is to be selected to satisfy this condition then the field angles a, fl and y represent three degrees of freedom of the final solution. Various required geometric parameters as well as torque and force components can be computed by using the series method due to Ewing[18]. The corresponding subroutines have been worked out and included to the system of standard subroutines of the matrix technique by Dewhurst and Collins [17]. In the particular case when y = 0 the equation (A8) reduces to H,,¢P.~P.~o
= - r(I + H~¢Q~,~)c,
(A9)
where I is the unit matrix. Comparison with (A7) shows that (A9) is equivalent to H~#rl = - tore. The equation (A10) could in fact be written at once since it is vector 0.2=-tore[3]. Collins' formula[25] of the paper[12] can certain equation equivalent to (A10) what can be verified directly. constructing the solution is equivalent to the present one when y apply in the general case y # 0.
(A10) simply the relation (AI) with the known be regarded as an analytic solution of This proves that Collins' method [ 12] for = 0. However, Collins' method does not
0020-7403/83/040265-11503.00]0 Pergamon Press Ltd.
A SLIP-LINE FIELD ANALYSIS OF THE ROLLING CONTACT PROBLEM AT HIGH LOADS H . PETRYK Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland
(Received 23 August 1982) Summary--An analysis of the steady rolling of a rough rigid cylinder over the surface of a rigid-perfectly plastic solid is presented. An exact slip-line field solution is investigated by using some recently developed techniques. It is shown that the solution is not uniquely defined and infinitely many complete solutions can be constructed for given boundary conditions. The numerical results show interesting variations of the deformation pattern and of rolling resistance at high loads. The limits are predicted to the magnitudes of the applied load and braking torque for which the steady rolling is possible. NOTATION F
Q W R L to V
k f = F/kRL q = Q/kR2L w= W/kRL h r
a, li, ~,, ,L & O
~rp
horizontal translatory force torque vertical load radius of cylinder length of cylinder angular speed of cylinder advance speed of cylinder yield shear stress dimensionless horizontal force 1 dimensionless torque dimensionless load variables dimensionless vertical load depth of plastic zone radius of isolated slip-line slip-line field angles, Fig. 1 parameter of free boundary operator compressive principal stress in non-plastic stress field slip-line fan angle, Fig. 3 1. INTRODUCTION
When the vertical load on a hard cylinder rolling over the plane surface of a deformable solid is sufficiently high, large plastic strains beneath the cylinder may occur. The large strains will usually be accompanied by the appreciable deformation of the free surface of the solid in front of or behind the advancing cylinder. The shape of the deformed free surface is not specified in advance. In theoretical investigations of the steady rolling this shape has to be found such that it does not vary when the deformation proceeds. The occurrence of large deformations and unspecified boundaries makes the steady-state problem of the rolling contact at high loads difficult to analyse without introducing some simplifying a~sumptions. The aim of the present paper is to examine an exact solution to the problem which satisfies all the statical and kinematical conditions, however, at the cost of using of the highly idealized, incompressible rigid-perfectly plastic model of the material. Inertia and body forces are neglected, and the plane-strain deformation is assumed. The cylinder is taken to be perfectly rough so that no slipping is possible over the arc of contact until the shear contact stress is equal to the yield shear stress k. The analysis presented below, in spite of idealization of the material behaviour, seems still relevant to such problems of practical importance as rolling friction of metals at sufficiently high loads [1] or cohesive soil-vehicle interaction. The theoretical aspect of the problem is also of considerable interest because of untypical boundary conditions. Over the deforming free surface not only the usual zero traction condition but also the kinematical steady-state condition have to be 265
266
H. PETRYK
satisfied. On the other hand, the shape of this surface is to be found. This type of boundary conditions need not lead to a unique solution. It will be shown that there exist even infinitely many complete solutions to the rolling contact problem. Several existing examples [2-6] indicate that this is probably a general rule for plane-strain ideal plasticity problems involving unspecified boundaries, not only for steady-state ones but also for those of geometrically self-similar deformation. The analysis will be carried out by using the slip-line field theory[7]. Several attempts have been made to construct a slip-line field solution to the rolling contact problem [8-11] but only recently Collins [12] succeeded in calculating the slip-line field for this problem without making any simplifying geometrical approximations. The same slip-line field and a method of constructing has been independently proposed in [3] but without performing any computations. Collins' solution[12] is determined uniquely by the values of two load variables (e.g. the dimensionless vertical load w and horizontal force f acting on the cylinder) which fully define the loading conditions. In the present paper a more general, non-uniquely defined solution is investigated which contains Collins' solution as a particular case. The question arises how to interpret physically the arising non-uniqueness of solutions (for a brief discussion of this question see [5]). It might be postulated that only that solution may have a physical meaning which is associated with a local minimum of the pulling force (or torque) with respect to the neighbouring complete steady-state solutions obtained at the fixed values of the remaining load variables, that is torque (or pulling force) and vertical load. General criterion of this type has been derived in [13] from a postulate of stability of a deformation process against persistent disturbances. H o w e v e r , this has been done under the assumptions which are not satisfied in the case of the solution considered here. The question remains still open, and the results will be presented without reference to any criterion of selection of the physically correct solution. 2. THE NON-UNIQUE BOW-WAVE SOLUTION The form of the solution shown in Fig. 1 was previously proposed by the author[3, 14]; the present paper contains the numerical analysis of this solution. In the particular case when the angle 3' of the slip-line fan at the point B is equal to zero, the solution coincides with that considered by Collins[12]. The slip-line field and corresponding hodograph in Fig. 1 represent the steady stage of rolling of a rigid, perfectly rough cylinder on a rigid-perfectly plastic half-space. According to the assumed incompressibility condition, free surface elements before and after deformation are on the same level. The additional amount of the material forming the "bow-wave" ahead of the cylinder is thought to be pushed out of the half-space during an initial, unsteady stage of the deformation process which is not considered here.
0
b". f
E
(3 m
$1_ {a)
(b)
FIG. 1. Exact non-unique solution: (a) slip-line field, (b) hodograph.
A slip-line field analysis of the rolling contact problem at high loads
267
The deformation takes place in the region ABFDGA (Fig. la) and along the velocity discontinuity line EDFB, the isolated slip-line ED being a circular arc of radius r and S as the centre. The traction-free boundary AB is subject to compression. In the reference frame translating g,ith the cylinder the boundary AB constitutes a part of the trajectory of surface elements. The region AGDEA adhering to the cylinder is rigid and rotates together with the cylinder with the same angular speed ~o. Hence its hodograph image a*gdea* (Fig. lb) is a geometrically similar region, rotated through a right-angle in the direction of rotation and scaled up by toil5]. The instantaneous centre of rotation lies at the point O or S if the centre of the cylinder or the block of material is fixed in space, respectively. The advance speed v of the cylinder is always greater than the peripheral speed ~oR, the difference (or skid) being equal to tor, i.e. to the magnitude of the velocity discontinuity along EDFB. Note that this "skid" is entirely due to the deformation within the solid since no slipping occurs over the arc of contact. Neither the slip-line field nor the hodograph can be constructed in a straightforward manner since there are not a sufficient number of starting curves of known shape in the physical or hodograph planes. However, if one slip-line, say AC, and three field angles a, /3 and y are specified then the rest of the slip-line field and the hodograph net can be determined directly. The base slip-line AC has to be chosen such that the resulting solution describes a steady motion in which the size and shape of the free boundary AB does not vary in time. When formulated mathematically this condition leads to an integral equation for the base curve. A convenient and efficient method to find unknown base slip-lines has been proposed by Collins[16] who introduced the matrix operational formulation in place of an analytic one. This idea has been developed into a systematic computational procedure by Dewhurst and Collins[17]. The procedure is based on a double power series representation of the solution of the field equations due to Ewing[18]. In the matrix formulation the integral equation for the base curve is reduced to an algebraic matrix equation for the vector representation of this curve as unknown. It is essential for the integral equation to be linear; in that case the resulting matrix equation is of constant coefficients and can be solved by standard techniques. A general case of the steady-state deformation of a free boundary has been considered by Petryk[3, 14] who introduced the matrix boundary operator which acts on the hodograph plane and generates the velocity field automatically satisfying the steady-state condition on the free boundary (for convenience the result is quoted in Appendix 1). This free boundary operator can be used to complete the matrix formulation of the present problem. The details are given in Appendix 2 where the derivation of the matrix equation governing the shape of the base slip-line AC is given. This equation is linear provided not only the field angles a, /3 and 3, but also certain parameter ~: on which the free boundary operator depends are regarded as known. In Appendix 2 it is shown that the final solution has actually three degrees of freedom since the above-mentioned four parameters a, /3, 3' and 6 are not mutually independent. On the other hand, the loading of the cylinder is fully defined by prescribing the magnitudes of two load variables; the third one is to be predicted by the theory. Hence an infinite number of solutions may exist for the same conditions of loading of the cylinder, each associated with different geometry of the bow-wave and predicting different resistance to rolling. The numerical analysis confirms that this is really the case. The obtained family of solutions may be parametrized, e.g. by 3,, the angles a and/3 being for each 3' selected such that the given load variables have prescribed values. The permissible range of solutions has been found to be limited by: (i) Vanishing of the fan angle 3,. The corresponding limit solution has the form analysed by Collins[12]. (ii) Vanishing of the angle A at which the slip-line GA meets the cylinder surface. The shear contact stress at the point A reaches the limit value k and the slipping of the material over the cylinder surface becomes possible at this point. (iii) Overstressing of the rigid corner at the point B. From Hill's criteria[19] we obtain that the rigid corner at B is overstressed if the angle 8 of inclination of the. free boundary AB at the point B is smaller than 23,. Therefore the solution can be statically admissible only if the condition 0~<2y~<8
(1)
is satisfied. (iv) Vanishing of the fan angle//, to be interpreted as the skidding limit. For certain domain of loading, all solutions within the permissible range defined by the above limits are proved complete (see Sections 3 and 4). Vanishing of the fan angle /3 (limit iv) was found to be necessarily accompanied by simultaneous vanishing of the angles d and O. The corresponding limit solution is shown in Fig. 2. The free boundary AB is now straight and inclined to horizontal at the angle 8 = (¢r/4) + y. The solution describes the skidding of the cylinder without any rotation. The material in EABDE, pushed out of the half-space in an initial, unsteady stage of flow, adheres to the cylinder and does not undergo any further deformation. Only the shearing of the material along the straight velocity discontinuity line EB occurs, in the theory without loss
E
O
\~-
FIG. 2. Skidding limit solution. MS VoL 25, No. 4 ~ D
268
H. PETRYK
of the cohesion of the material. The relations between the load variables are easily found to be f = wl(l + 23,),
(2)
q = w ( w - 2)/(2(1 + 23,)).
(3)
It is worth mentioning that in this particular solution it is not essential that the arc of contact is circular. The solution shown in Fig. 2 can represent also the steady stage of shifting of a sufficiently rough, rigid prismatic body of arbitrary crossection (wedge-shaped for instance) over the surface of a rigid-perfectly plastic solid, the rotation of the body being excluded in advance by kinematical constraints. In that case the value of the fan angle 3, is no longer constrained by the equation (3), and the solution is non-uniquely defined. According to the formula (2), the value of the shifting force for a given vertical load decreases with increasing 3, and reaches its minimum value in the limit case 3, = 45°; for 3, > 45 ° the rigid corner at B of the apex angle 180° would be overstressed. 3. COMPLETENESS OF THE SOLUTION By using the method of Green[20] it is found that the rate of plastic work is certainly positive everywhere in the deforming region A B F D G A (Fig. la). This can be stated by examining only the general form of the slip-line and hodograph nets without the need of any computations. Also the work of shear along the velocity discontinuity line E D F B is positive. There remains to check whether the rigid regions are not overstressed. For the neighbourhood of the isolated segment E D of the velocity discontinuity line to be not overstressed, the pressure along this segment must vary according to Hencky relation. This follows from the theorem due to Bonneau[21,22]. In other words, the segment E D must be a slip-line and not an envelope of slip-lines. Therefore we can construct a regular slip-line field based on the slip-lines E D and DGA and extend it up to the cylinder surface (Fig. 3). The resulting normal contact stresses are evidently compressive as they should be, and the shear contact stresses are in agreement with the assumed perfect roughness of the cylinder since no slipping occurs along AE. This proves that the rigid region E D G A E is not overstressed. Below we show that the plastic stress field can also be extended in a statically admissible manner into the half-space, however, under certain restrictions on the field angles. Let us construct a singular slip-line field E D K based on the slip-line E D with singularity at E (Fig. 3), the fan angle ~b being equal to (~r/4) + 2 y - & Then by Hencky relations the pressure on the line El(, at the point E is equal to k, and we can build a regular plastic stress field between the line E K and a hypothetical stress-free surface EL. Now, similarly as in [5], we construct first a regular slip-line field based on the slip-lines D L and D B and next a singular field with singularity at B and the fan angle equal to 3', up to a line L M N B along which the pressure is constant and equal to k. Then, from every point of the line L M N B we draw a straight ray in the direction of the principal compression stress at this point. The rays constitute outside L M N B trajectories of the compressive principal stress ~rp, while the second principal stress acting across the rays is assumed to vanish identically. It can be shown[5] that the resulting stress field is statically admissible provided I~'~1 is chosen to decrease in an appropriate manner with increasing distance p from the L M N B - I i n e . Thus, if the above construction can be performed the solution is complete. There are two possible limitations to the whole procedure. The first is that before completing the stress field in the region E D K L E the radius of curvature of some slip-line reaches zero. The second is that one (or both) of the limiting rays drawn from the points B and L, outside of which the stresses are zero, goes out of the material. However, the condition (1) implies that the ray drawn from B lies inside the half-space except 23, = 6 when the ray lies exactly on the surface. The ray drawn from L will lie in the half-space if ¢J+ 20 -< (3/4) ~', i.e. if 20 ~ (w]2) + 6 - 23,.
(4)
It has been checked numerically using the matrix procedure [17] that the smallest radius of curvature of slip-line in the region E D K L E , while being always at L, is positive if (4) is satisfied. Hence we have come to the conclusion that the solution shown in Fig. I is complete if the field angles satisfy the inequalities (1) and (4). The condition (4) is not necessary for statical admissibility of the solution while (1) is. In particular, the condition 0 <~ 3, <~45° is both necessary and sufficient for completeness of the solution shown in Fig. 2.
FIG. 3. Construction of statically admissible stress field in rigid regions.
A slip-line field analysis of the rolling contact problem at high loads
269
4. NUMERICAL RESULTS The slip-line field shown in Fig. 1 was analysed numerically over its full range of validity. The computations were performed on a small ODRA 1204 computer using the matrix procedure described in [17]. The matrix equation (A8) was solved using a standard subroutine employing Gaussian elimination. The matrix dimension was taken to be e_q_yalto eight. For each set of the input values of the angles a, fl and 3, the parameter ~: was iterated until [Joel- [oa*l [ (Fig. lb) was smaller than 10-8, the scale parameters to and r being taken as unity. This was always found possible within the limits (i)-(iv) defined in Section 2, also for 3' = 0, what implies that the "skidding limits" reported by Collins[12] may be ascribed to the numerical procedure used in [12]. It is worth mentioning that the resulting value of ~: tends to 0 or 1 if the speed ratio wRlv tends to I or 0, respectively. By using a Newton-type iterational procedure the values of a, /3 or y were selected to obtain required values of the dimensionless load variables w or q with accuracy 10-5, or to reach the validity limits ;t = 0 or 2y = 8 with accuracy 10_6 [rad]. Several accuracy checks were also contained in the computer program. For example, in all cases the difference in fhe vertical coordinates of the points B and E, being theoretically zero, was smaller than R ' 10-7. Also the resultant dimensionless moment and force components acting across the line AGDFB, which should vanish for the region AGDFBA to be in equilibrium, in most cases were smaller than 10_4 and always smaller than 10 3. In Collins' paper[12] a comparison of the predictions of his analysis with previous approximate solutions and experiment is given. Since Collins' solution[12] is a particular case of the present one when 3' = 0, no such a comparison is included here. 4.1 The case o[ [orce-driven cylinder (q = 0) The range of validity of the solution in the case q = 0 can perhaps be best visualized in the (w, h)-plane (Fig. 4), where h is the depth of the plastically deformed zone. The corresponding range of field angles is shown in Fig. 5. Figure 4 shows some interesting features of the solution. Because of the non-uniqueness of the solution, not a single curve but a whole family of curves is obtained, each of them showing the relation between w and h for a different, fixed value of the fan angle 3'. The validity limits (i)-(iv) defined in Section 2 are indicated in the figure. For w > - 1-5 all solutions within these limits are complete since the inequality (4) is found to be satisfied in this range of loading. It is seen that for a fixed radius of the cylinder the maximum possible depth of plastic zone hmax= 0.182-R occurs for 3' = 0 and w ~ 2.05 and for still higher loads it decreases with increasing load. Moreover, any curve corresponding to a fixed value of 3' has a turning point which defines the maximum load w for which a solution of this value of y can be constructed. No solution of the examined type can be constructed for w > 2.355 which suggests that the cylinder cannot roll steadily under such high loads. However, this cannot be stated ultimately until the possibility of existence of other solutions is excluded. For 2.355> w > - 2.1 the permissible range of solutions is from both sides limited by vanishing of the angle I', a smooth maximum of y being attained somewhere within this range. For -2.1 > w > 2 this range splits into two families of solutions. When tends to 2 [rom above the depth of the plastic zone in one of these families tends to zero and the skidding limit is reached. This is seen more clearly in Fig. 6 where the ratio of the peripheral to advance speed of
0.~
~=o
0,16 0.14 rr
j012
to
//
/ . ~ / complete ~ " \ solutions " ~
\
.~ O.lO
/,/
overstressing
at
B
'~ 1208 "~ Qo6 Q04
X=0 skidding limit
O j02
\
ds
~
dimensionless
1.~---vertical
Ioad,w
20 -
FIG. 4. Depth of plastically deformed zone (q = 0).
2.5
270
H.. PETRYK 120
=0
w=O.5
9 .0
t
_
1.5
~
80[ .u ~ 60 "u
0verstressi at B
4
2.0
"
limit
2._.?2"L.X
l
2 0 ~ O~
2
'
i
'
6
fiet~
8
angle,~,degrees
10
12
F1G. 5. Range of field angles (q = 0).
1
>
8-:0
Q~
3
o" o£ ~. Q4
X=O
skidding limit
0.2 0
~
.
i
0.5
B
.
.
.
,
~
1.0 1.5 2.0 dimensionless verticQt IoQd,w
__
25
FIG. 6. Ratio of peripheral to advance speed (q = 0).
the cylinder is plotted vs w. The skidding limit solution (Fig. 2), exceptionally for q = 0, is not uniquely defined by the values of w and q since for q = 0 and w = 2 the equation (3) is satisfied identically and the value of 3' is left undetermined. Thus for w = 2 two families of complete solutions can be constructed, one corresponding to rolling and the second to skidding of the cylinder. The family corresponding to rolling is shown in Fig. 7 where two limit solutions of this family and a medial one are drawn to scale. The (non-unique) relationship between the dimensionless pulling force .f and vertical load w is shown in Fig. 8. While for w < 2 the variations of the pulling force calculated for a fixed w within the permissible range of solutions are small and do not exceed 3%, they become quite large when the solutions approach the skidding limit. For a fixed w < ~ 1.5, the greater the angle 3" the smaller the pulling force so that the minimum value of f within the permissible range is obtained for the limit solution with 23, = 8. For ~1.5 < w < 2.04 the absolute minimum of [ still corresponds to the limit solution with 23, = & but also a local minimum of .f appears for 3' = 0 and for w > 2-04 it becomes the absolute one. The variations of the bow-wave profile with increasing w are illustrated in Fig. 9 where a number of the profiles corresponding to the different limit solutions are shown to scale. At higher loads rapid growth of the bow-wave occurs with decreasing load w when the skidding limit is approached, especially for the solution with 3" = 0. This corresponds to remarkable increase of the rolling resistance (see Fig. 8). The variations of the geometry of the solution within the permissible range for a fixed w may also be very large: compare four different profiles marked in Fig. 9, corresponding to the same w = 2.0. 4.2 General case (q # 0) In Fig. 10 the domain of validity of the solution in (q, /')-plane is presented. Only a half of the complete figure is shown since there is a polar symmetry with respect to the origin of the coordinate system in this plane. The figure is rather complicated due to the non-uniqueness of the solution. The entire validity domain is covered by the regions corresponding to permissible ranges of solutions for different, fixed values of the vertical load w. For several values of w such regions are indicated in the figure; for sufficiently close values of w the regions would overlap each other. The validity limit A = 0 is essential practically only for positive
A slip-line field analysis of the rolling contact problem at high loads
271
q:0 w=20
,/
,/
,~/ If: 1701" ' f=0620 (max) / ,=10" (min) If= 0"
F]o. 7. Solution range for rolling at w = 2.0 (q = 0).
2,0
,
,
I
:0 I
skidding Limit
1,5
I I
I
2o\r 25" {t u
1,0
e-
~r=o i/1
overstressing at B
I
I
I
0.5
1,0
1,5
2,0
2,5
dimensionless vertical toad, w FIG. 8. Non-unique relationship between pulling force and vertical load (q = 0).
values of torque. The maximum value of w for which a solution of the considered type can be constructed was found to be -2.43. The entire validity domain can also be thought as a topological sum of particular validity domains of the solutions with different, fixed values of the fan angle 3'. The validity domain for Y = 0 is bounded by the line OPQRSO. It generally shrinks with increasing 3' (except little expansion near PQR-line) up to a single line OM for 1, = 45°. For all permissible solutions represented by points lying to the right and down of the line ON the inequality (4) is satisfied so that all those solutions are complete. The lines OSR, OM and MTR in Fig. 10 correspond to the skidding limit reached by another limit solution with 3" = 0, 23, = 8 and it = 0, respectively. From the relations (2) and (3) we obtain that the lines OSR and OM are segments of the parabolas q = ( l / 2 ) f - f and q = (1/2)(1 + I r / 2 ) f - f, respectively; also the line MTR can be determined analytically. The skidding limit can be reached for Ioadings which are represented by the points lying in the region bounded by the line OMTRSO. If the load w is specified the relation between f and q due to skidding is linear, namely, q = (l/2)(w - 2)f from (2) and (3). Note that the region OMTRSO does not represent the entire validity domain of the solution shown in Fig. 2. This is so since the condition it = 0 is not the limiting one for this solution treated not as a limit case of the rolling solution shown in Fig. 1 but as ap independent solution.
272
H. PETRYK
Q:
F
I
,
.............
.
I
& _ _
W
.~ ~
..... -
\
,
-
-
-
-
-
It=0 2I'= 6" ~.=0
\
s
ing [i w.2
F;G. 9. Variation of bow-wave profile with vertical load (q = 0).
0,4
\ ~---~ 0,2
7t:0 ~-=0 p
N/~._ comptete sotutions .
skidding [imit, ~.:0
%, 0 -0,2 -0,4
sk,dd,ng timit, 1"=45" ~ ~ " c 4 : 2 " ~ - "
\
s
_/~.skidding tirnit. r=0
\ \ FIG. 10. Domain of validity of solution plotted in (q, /')-plane.
It can be seen from Fig. 10 that the relation between [, q and w describing a rolling friction law is close to be unique in one part of the validity domain while in the other the variations of the load variables within the permissible range of solutions are much greater. The second part corresponds to the region in which the skidding limit can be reached. This is not surprising since the large variations in both the geometrical and load variables when the skidding limit is approached have yet been observed in the case q = 0 (see Figs. 8 and 9). No solution of the considered type can be found if the braking torque ( - q ) is sufficiently large. The critical value of the torque vs the vertical load w is plotted in Fig. 11. It looks very likely that if the braking torque exceeds this critical value the steady rolling becomes impossible. A different situation occurs in the case q > 0 (driving condition) when the solution fails with increasing q by vanishing of the angle A. This should not be interpreted as an ultimate limit to rolling but rather as a limit of validity of the assumption that no slipping occurs over the arc of contact (see the remark (ii) in Section 2). Therefore a modified slip-line field which allows the slipping on a part of the contact arc need also be examined in order to obtain a more complete view on the mechanism of rolling under driving conditions. Moreover, the bow-wave solution investigated in this paper is not the only possible form of solution, especially when q > 0 and [ < 0 (see the remark in [12], p. 444). The investigation of solutions of the other types is beyond the scope of the present paper.
A slip-line field analysis of the rolling contact problem at high loads O~
o21 ~ 0.1
.
I//,
.
.
, 7 'Y'i '''~
273
.
///A
YVVVVVUVVV// dimensionless verticol t o a d . w
25
FIG. 11. Range of braking torque and vertical load for which rolling is possible.
5. CONCLUSIONS (i) T h e s t e a d y - s t a t e p r o b l e m o f s u r f a c e rolling with a r o u g h cylinder has infinitely m a n y c o m p l e t e solutions in the f r a m e w o r k o f the p l a n e - s t r a i n t h e o r y o f a rigidp e r f e c t l y plastic solid. This n o n - u n i q u e n e s s is due to o c c u r r e n c e in the p r o b l e m o f a stress-free s u r f a c e o f initially u n k n o w n shape. (ii) T h e matrix t e c h n i q u e f o r c o n s t r u c t i n g slip-line field solutions o f indirect t y p e [ 1 7 ] with the use o f the free b o u n d a r y o p e r a t o r s [ 1 4 ] c a n effectively be applied to c o n s t r u c t a w h o l e range o f solutions to the s t e a d y rolling p r o b l e m r a t h e r than a single, particular solution. T h e q u e s t i o n o f selection of the solution w h i c h has a p h y s i c a l m e a n i n g r e m a i n s open. (iii) A t high loads the large variations o f b o t h the g e o m e t r y of solution and p r e d i c t e d rolling r e s i s t a n c e are o b t a i n e d within the limits o f validity o f the e x a m i n e d solution, especially w h e n the skidding limit is a p p r o a c h e d . (iv) T h e e x a m i n e d " b o w - w a v e " solution s e e m s to p r o v i d e a rather c o m p l e t e d e s c r i p t i o n o f the rolling of a r o u g h c y l i n d e r at zero or negative torque, o f c o u r s e within the limits o f the ideal plasticity t h e o r y itself. T h e case o f positive t o r q u e requires also solutions of o t h e r t y p e s to be investigated. REFERENCES I. K. L. JOHNSON,Rolling resistance of a rigid cylinder on an elastic-plastic surface. Int. Z Mech. Sci. 14, 145 (1972). 2. H. KtJDO, Some new slip-line solutions for two-dimensional steady state machining. Int. J. Mech. Sci. 7, 43 (1965). 3. H. PETRYK,Ph.D. thesis (in Polish). I.F.T.R. Rep. 46, 1977. 4. P. DEWHURST,On the non-uniqueness of the machining process. Proc. Roy. Soc. Lond. A 361), 587 (1978). 5. H. PETRYK,Non-unique slip-line field solutions for the wedge indentation problem. J. Mdc. AppL 4, 255 (1980). 6. I. F. COLLINS,Geometrically seft-similar deformations of a plastic wedge under combined shear and compression loading by a rough flat die. Int. J, Mech. Sci. 22, 735 (1980). 7. R. HILL, The Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950). 8. J. MANDEL,Rrsistance au roulement d'un cylindre indrformable sur un massif parfaitement plastique. Le Frottement et l'Usure, p. 25 (1966). 9. E. A. MARSHALL,Rolling contact with plastic deformation. J. Mech. Phys. Solids 16, 243 (1968). 10. I. F. COLLINS,A simplified analysis of the roiling of a cylinder on a rigid/perfectly plastic haft-space. Int. J. Mech. Sci. 14, 1 (1972). 11. V. M. SEGAL,Plastic contact in the motion of a rough cylinder over a perfectly plastic haft-space (in Russian). Mekh. Tv. Tela, No. 3, 184 (1971). 12. I. F. COLLINS,On the rolling of a rigid cylinder on a rigid/perfectly plastic haft-space. J. Mic. Appl. 2, 431 (1978). 13. H. PETRYK,On the stability of non-uniquely defined processes of plastic deformations. J. Mdc. Theor. Appl. Numrro sprcial, p. 187 (1982). 14. H. PETRYK, On slip-line field solutions for steady-state and self-similar problems with stress-free boundaries. Arch. Mech. 31,861 (1979).
274
H. PETRYK
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. C O L L I N S , The algebraic-geometry of slip line fields with applications to boundary value problems. Proc. Roy. Soc. A 303, 317 0%8). 17. P. DEWHURST and I. F. COLLINS,A matrix technique for constructing slip-line field solutions to a class of plane strain plasticity problems. Int. J. Num. Meth. Engng 7, 357 (1973). 18. D. J. F. EW1NG,A series-method for constructing plastic slip-line fields. J. Mech. Phys. Solids 15, 105 (1%7). 19. R. HILL, On the limits set by plastic yielding to the intensity of singularities of stress. J. Mech. Phys. Solids 2, 278 (1954). 20. A. P. GREEN, The plastic yielding of notched bars due to bending. Quart. J. Mech. Appl. Math. 6, 223 (1953). 21. M. BONNEAU, Ann. Pts et Ch., Sept-Oct. 609 (1947). 22. J. SALENqON, Th~orie de la Plasticit~ pour les Applications a la M~canique des Sols. Eyrolles, Paris (t974). APPENDIX 1 Let us consider a plastically deforming region A B C bounded by a concave segment A B of a stress-free boundary and by slip-lines A C and B C of angular range 0 (Fig. 12a). Let the configuration of the slip-line images in the corresponding hodograph diagram abe be as shown in Fig. 12(b). We assume ~hat the stress and velocity fields in A B C are described by analytic functions• The slip-line field in A B C is thought as a part of the solution to a steady-state problem. Thus in some reference frame the free boundary A B must coincide with a stream-line. It can be shown[3, 14] that this steady-state condition is satisfied if and only if the vector representations[16, 17] ~r, and w2 of the hodograph curves ac and bc are related through the matrix equation ~r2 = Hob'l,
(A1)
where the matrix "free boundary" operator H0~ has the form Hoe = RoP-*o(A6 - Q'Re).
(A2)
Ro, P-*o = (P*) I Q , in the formula (A2) are the basic matrix operators[16, 17] and
-l A~ =
1
0
0
0
• • ' |
0
• " •
0
"" "
-~
1
0
1
+~
-1
-~
-2 2
1 -2
1
The parameter ~: on which the operator H depends may take an arbitrary value from the interval (0, I]. It defines the mutual position of the hodograph net and hodograph pole o according to the formula ~: = 1 - 2X/(2)[~oo[/80,
(A3)
where So is the radius of curvature of the curve uc at the point u, and the ray ou meets the curve ac at an angle 45 °. The form of the other free boundary operator F0: A C ~ BC (Fig. 12a) which generates the stress field in A B C such that A B is traction free has been given by Dewhurst and Collins[17]. In applications it is convenient to use the equivalent form of the operator F, viz. Fo = RoP-*o(Al + Q*Ro),
(A4)
where Al = Ad~=l. The formula (A4) can be derived in an analogous way as (A2). Both the operators F and H do not depend on the actual shape of the free boundary so that they can be employed in the matrix procedure[17] to construct slip-line field solutions for steady-state problems involving deforming free boundaries of initially unknown shape.
A
bV~\~
P
0
(a) (b) FIG. 12. Plastic region with stress-free boundary: (a) slip-line field, (b) hodo~'aph.
A slip-line field analysis of the rolling contact problem at high loads
275
APPENDIX 2 The reader isreferred to [17] for a full discussion of the matrix procedure. Referring to Fig. 1, let us choose the slip-line A C as the base curve 0" (for simplicity we use the notation in which a curve is identified with its vector representation). Then the slip-line B C is F~0", and from the definition of the basic matrix operators P and Q we have G A = P~a0", C G = Q*0" and C F = Q*F~0". On using the fundamental construction of the field between two characteristics C F and CG we obtain DG --- 0"3 = (Py~Q*F~ + Q~vQ*)0"
(A5)
The hodograph images of the slip-lines DG and G A bordering the rigid r e # o n E D G A E rotating with angular speed to are geometrically similar curves scaled up by to[15], that is dg = to¢3 and ga* = toP,a0". The curve dfb* is a circular arc with radius tor, the magnitude of the velocity discontinuity along E D F B . Thus dfb* = tore, where e is the unit circle vector. The fundamental construction applied first to the curves db* and dg and next to the curves gc and ga* gives bc =- 0"2 = R~+a{P*(tore) + Q*(to~r3)}
(A6)
ac -~ 0., = P~(toP,~0.) + Qa~{P*(tore) + Q*(to¢3)}.
(A7)
Finally, according to (AI), the curve ac transformed by the operator Hoe should give the curve bc. Eliminating ~r~, ~2 and tr~ between (AI), (A5)-(A7) and rearranging we obtain the following matrix equation for 0.: K ¢ = rLc
(A8)
where K = (Ra+/3 --
* H ~ Q ~ , , ) Q *~(P:,~Q,~F,~ + Q~:,Q*) - H ~ P , ~ P ~
and L = - (R,~÷~ - H ~ Q ~ ) P * ~ .
cr can be found from (A8) once the three field angles a, /3 and 7 and the parameter ~ are prescribed (the radius r as well as the angular speed to play the role of scale parameters only). From 0. the shape of other slip-lines in the region A B F D G A and their hodograph images can be determined in a straightforward manner. The position of the hodograph pole o can be determined from (A3) what completes the construction in the hodograph plane. The construction in the physical plane can be completed using the property of geometric similarity of the regions A G D E A and a*gdea*. However, the obtained solution is usually kinematically inconsistent since the hodograph image of t._he centre of the cylinder does not in general coincide with the hodograph pole. In the correct solution Ioel must be equal to Ioa*l; this geometric condition reduces the number of independent parameters of the solution to three. If the parameter ~ is to be selected to satisfy this condition then the field angles a, fl and y represent three degrees of freedom of the final solution. Various required geometric parameters as well as torque and force components can be computed by using the series method due to Ewing[18]. The corresponding subroutines have been worked out and included to the system of standard subroutines of the matrix technique by Dewhurst and Collins [17]. In the particular case when y = 0 the equation (A8) reduces to H,,¢P.~P.~o
= - r(I + H~¢Q~,~)c,
(A9)
where I is the unit matrix. Comparison with (A7) shows that (A9) is equivalent to H~#rl = - tore. The equation (A10) could in fact be written at once since it is vector 0.2=-tore[3]. Collins' formula[25] of the paper[12] can certain equation equivalent to (A10) what can be verified directly. constructing the solution is equivalent to the present one when y apply in the general case y # 0.
(A10) simply the relation (AI) with the known be regarded as an analytic solution of This proves that Collins' method [ 12] for = 0. However, Collins' method does not