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A strain-hardening slipline field analysis of the plastic deformation observed in a model asperity interaction experiment
J. Mech.
Phvs. SolidsVol. 38, No. 6, pp. 165-185, Printed in Great Britain.
0022-5096~90
1990.
$3.00+0.00
c 1990 Pergamon Press plc
A STRAIN-HARDENING SLIPLINE FIELD ANALYSIS OF THE PLASTIC DEFORMATION OBSERVED IN A MODEL ASPERITY INTERACTION EXPERIMENT E. M.
KOPALINSKY,~
P. L. B. OXLEY~ and H. T.
YOUNG$
tSchoo1 of Mechanical and Industrial Engineering, The University of New South Wales, P.O. Box 1, Kensington, New South Wales. 2033, Australia; IDepartment of Mechanical Engineering, National Taiwan University, No. 1 Sec. 4 Roosevelt Road, Taipei, Taiwan, 10764. Republic of China
(Received
13 July 1989)
ABSTRACT SCALED-UP model asperity experiments are described in which a hard wedge is indented into and then slid along the surface of a relatively soft material with the resulting plastic deformation measured using printed grids. A slipline field analysis of the deformation occurring once steady-state conditions have been achieved is described. This shows that tensile stresses can exist in a small region of the field when in calculating stresses account is taken in the stress equilibrium equations of variations in flow stress resulting from the strain-hardening properties of the deforming materia!:- This result could be important in considering the viability of processes such as low cycle fatigue and fracture as possible wear mechanisms.
INTRODUCTION
THE PLANE strain slipline field proposed by CHALLENand OXLEY(1979) to represent asperity interactions when a hard surface slides over a relatively soft surface is given in Fig. 1. With this model the friction force is considered to be the force needed to
Streamline
of flow
(b )
FIG 1. Wave model of asperity
deformation 765
: (a) slipline field; (b) hodograph.
E. M. K~PALINSKY CI al.
766
push waves of plastically deformed material in the soft surface ahead of asperities on the hard surface. The wave model has been shown to apply to relatively smooth well lubricated surfaces. For rougher surfaces (larger as) and less efficient lubrication (larger shear stresses along the interface DE) the wave can be torn off and for very rough surfaces material can be removed by a chip formation process. Coefficients of friction predicted from the slipline field have been shown to be in good agreement with experimental results obtained from model asperity tests using wedges (CHALLEN et al., 1984) and from tests using actual surfaces (MOALIC et al., 1987). CHALLEN er al. (1986) have used the wave model to predict the wear which occurs during sliding as a result of the repeated plastic working of the surface. They have done this by calculating from the model the plastic strain increments put into the soft surface by the passage of plastic waves across it and using these increments with a low cycle fatigue equation to determine the wear rate. The analysis yields a relationship between the coefficients of friction and wear which has been shown to be in good agreement with experimental results (CHALLEN et al., 1987). This is encouraging but in order to justify more directly the use of a low cycle fatigue mechanism to predict wear in this way it is necessary to consider the stresses involved in the deformation as well as the strains. For example, the presence of a tensile stress in part of the deforming region might well be considered essential to the process of crack initiation. Not surprisingly, considering the compressive nature of the loading under sliding contact conditions, the direct stresses in.the asperity contact regions are predominantly compressive. For the slipline field in Fig. 1 it can be shown that they are entirely compressive. In constructing this field the soft material is assumed to be rigid-perfectly plastic with the mean compressive (hydrostatic) stress calculated from the Hencky equations (stress equilibrium equations referred to sliplines) p+2kll/
= const.
p - 2kll/ = const.
along a I-line, along a II-line,
>
(1)
where p is the hydrostatic stress which acts normal to the sliplines, k is the shear flow stress (assumed constant) which acts parallel to the sliplines and $ is the anticlockwise angular rotation of the I-lines from a fixed reference axis ; the I-lines are taken as those on which the shear stress exerts a clockwise angle. To satisfy equilibrium at the free surface AE (Fig. 1) p must be equal in magnitude to k and compressive in the region ABE and Eqs (1) show that its compressive value increases in moving through the centered fan BCE into the region CDE. At no point in the field therefore is p less than k and all direct stresses are compressive. OLVER et al. (1986) have used a field similar to that in Fig. 1 with u = 0 to determine the stresses when the load is applied and also the residual stresses generated during unloading. A novel feature of their analysis is the extension of the slipline field into the non-deforming plastic region in order to calculate the stresses in this region. Their results show that although the direct stresses are compressive when the load is applied residual tensile stresses can be generated in part of the field during unloading. They confirmed experimentally the existence of residual tensile stresses and have pointed out the possible relevance of these to crack formation resulting from asperity contacts.
Analysis of an asperity interactlon experiment
161
The purpose of the present paper is to investigate the so far neglected influence of the hardening properties of the deforming material on the stress distributions in the wave formation process. Once the perfectly plastic assumption is relinquished and a rigid-plastic hardening material is assumed the stress equilibrium equations referred to sliplines become 1 $+2k$-g=O 1 I 8P -2kg dsz
along a I-line,
2 - g
I
= 0
(2) along a II-line,
where s, and s2 are distances measured along the I and II lines respectively. If k varies Eqs (2) show that the variation of p along a slipline depends not only on the angle turned through by the slipline, as is the case with (I), but also on the rate of change of k with distance along the sliplines. PALMER and OXLEY (1959), COLLINS (1978) and CONNING et al. (1984) have discussed the nature of rigid-plastic hardening solutions. They have noted that from (2) the normal derivative of k across a slipline must be continuous. Hence in hardening flows sliplines cannot propagate jumps in velocity as are frequently used in rigid-perfectly plastic_soiutions, the jump in tangential velocity across the slipline ABCD in Fig. 1 being an example. In hardening flows velocity discontinuities must, therefore, be replaced by deformation zones in which gradual changes in velocity occur. So far no analytical slipline field solutions for metal deformation processes, such as the wave formation process now considered, have been found once k is allowed to vary and would appear to be prohibitively difficult. In hardening problems it is generally necessary to determine the distribution of k as part of the solution. As a consequence it is no longer possible to determine simple geometric properties of hardening slipline fields such as the well-known ones of Hencky for the perfectly plastic case. Hardening problems are therefore intrinsically more complicated than non-hardening problems. Although no analytical solutions to hardening problems have been obtained, slipline fields have been constructed for hardening flows using visioplasticity techniques. The basis of this approach is to use an experimental flow field to determine the velocities of flow which in turn are used to determine the strainrates and hence the directions of maximum shear strain-rate. If an isotropic material is assumed then the latter can be taken to be slipline directions, i.e. directions of maximum shear stress as well as of maximum shear strain-rate, and used in conjunction with known stress boundary conditions to construct a slipline field. The distributions of strain and hence k (strain-rate and temperature effects are usually neglected) are then determined by integrating the strain-rate with respect to time along streamlines and a stress analysis is carried out using (2). Slipline fields obtained in this way clearly show the extent to which the velocity discontinuities in rigid-perfectly plastic solutions open-up in hardening flows. They also show that the distributions of p calculated from (2) can be very different from those calculated from (1). Of particular interest in the present context is the slipline field of PALMER and OXLEY (1959) for orthogonal machining which indicates a high
768
E. M. KOPALINSKY eral.
compressive value of p near the cutting edge when (1) are used to calculate stresses but a high tensile value of p when (2) are used. CONNING and OXLEY (1989) have recently given a comprehensive review of visioplasticity techniques. Their application to the wave formation process of Fig. 1 is now considered.
EXPERIMENT The objectives of the experimental work were to obtain a flow field from which the velocities of flow could be determined for a typical plane of flow for conditions as near as possible to plane strain, steady-state conditions and to measure the associated forces. In the experiments, which were similar to those made by CHALLEN et al. (1984) and more recently by BLACK et al. (1988), a hard wedge representing a scaled-up asperity was indented vertically into the horizontal surface of a relatively soft specimen with subsequent movement of the specimen relative to the wedge in a direction parallel to its surface and normal to the edge of the wedge. Conditions were made as close to plane strain as possible by clamping the specimen between side plates with the wedge having sufficient clearance to slide freely between the plates. The wedge was free to move up or down during horizontal movement of the specimen in order to maintain a constant vertical force and to allow thewedge to return to the surface once the steadystate wave process (Fig. 1) had been achieved. The test rig used in the experiments was in the modified form described by BLACK et al. (1988) with a hydraulic “quick-stop” device for disengaging the wedge and wave at the end of a test. A Kistler dynamometer which was part of the specimen assembly was used to measure forces. Vertical displacements of the specimen relative to the wedge were measured with a linear variable differential transformer. With the measuring devices used, the signals of which were amplified and recorded on a multichannel chart recorder, it was estimated that forces could be measured with an accuracy of +25 N and displacements with an accuracy of + 1 pm. In all cases these were much smaller than the actual measured values. Experimental flow fields were obtained by using a specimen divided on what would be a typical plane of flow during wave formation with a grid printed on one of the faces. The material selected for the specimens was a cold rolled 5083-H32 1 aluminiummagnesium alloy which preliminary experiments had shown gave well-defined flow fields. The results of compression tests on this material were used to obtain the plastic stress-strain relations k=
for y d 1.0
k = 263.5
1 for y > 1.0, J
and
(3)
where k is the shear flow stress in MPa and y is the maximum shear strain. In deriving (3) plane strain and uniaxial conditions have been related in the usual way with k = a/ J3 and y = J3 E w h ere D and E are the uniaxial stress and strain. Specimens were made of two equal parts of 150 mm length, 25 mm depth and 6 mm width which gave a composite specimen of 12 mm width. These were milled from plate and then polished
Analysis of an asperity Interaction experiment
769
1200 grade paper on the test surface and on the sides to eliminate any bowing resulting from residual stresses caused by machining. A chequerboard grid pattern of 0.1 mm sides was printed on the relevant region of one of the inner faces using the photo resist technique described by FARMER and FOWLE (1979). Care was taken in printing the grid to align it with the specimen sides so that during a test one side of the grid would be parallel to the sliding direction. The wedges were machined from Bohler Amutit S tool steel, hardened and tempered to RC60, ground and then polished on 1200 grade paper in a direction normal to the edge of the wedge. In order to minimize the friction at the wedge-wave interface (DE in Fig. 1) during a test and thus reduce the risk of severe deformation and tearing in the wave in this region both the specimen and tool test faces were sprayed with a bonded lubricant (Dow Corning Molykote 321 R). A strict, repeatable procedure was used which gave a coating estimated to be 10 pm in thickness. In a test the wedge was indented into the specimen surface by applying the selected vertical force. During the subsequent horizontal movement the wedge at first was observed to dig in deeper to the specimen as the vertical force was transferred from both sides to a single side of the wedge. With further movement the wedge started to climb on plastically deformed material pushed up in front of it until once again the edge of the wedge was on the specimen surface. In the absence of fracture in the deforming material steady-state condition< were then achieved with a plastic wave pushed ahead of the wedge as depicted in Fig. 1. Once a steady-state had been achieved, as judged by the constancy of the horizontal force F (Fig. I), the “quickstop” was made. A photograph of the deformed grid obtained in this way is given in Fig. 2. The corresponding test conditions were: wedge angle c1= 20”, measured vertical and horizontal forces N = 7953 N and F = 3250 N respectively and horizontal speed U = 0.3 mm s- ’ (Fig. 1). Results were obtained for a number of wedge angles (and also for lubricants other than Molykote) but in spite of the slight imperfection in the deformed grid in Fig. 2 it was decided that this was the best defined of the flow fields obtained and therefore the most promising for the purpose of making the slipline with
field analysis.
FIG. 2. Experimental flow field ; grid 0.1 mm sides.
710
E. M. K~PALINSICY CI al SLIPLINE FIELD ANALYSIS
If carried out manually the calculation of strain-rates from an experimental flow field is extremely time consuming and for this reason a computer-aided method has been developed. The basis of the method has been given by FARMERand OXLEY(1976). Its application to the flow field in Fig. 2 is now described. In the analysis the Aow was from left to right with the wedge fixed as in Fig. 1. For experimental flow fields such as that in Fig. 2, which are for approximately plane strain, steady-state conditions, it can be assumed that the originally horizontal grid lines are streamlines and that their intersection points with the originally vertical lines (transversals) represent points of equal time interval along the streamlines. It is therefore possible to determine from the flow field the velocity gradients required in calculating strain-rates. The starting point of the present analysis was to make a line diagram tracing from an enlarged photograph of the flow field in Fig. 2 to give a clearer definition of the intersection points of the grid lines. The S.J~ coordinates of these intersection points were then digitized with the x axis taken along the surface in the direction of motion and with the JJ axis coinciding with one of the transversals in the undeformed region to the left of the wave. The digitized coordinates together with the value of the equal time interval formed the input data to the computer programme used in making the calculations. Figure 3 shows the digitized data from Fig. 2 reconstructed by replacing the curved sides of the elements with chords. It is usual to display the digitized input data in this way prior to further processing as an aid to verifying the data and ensuring that they are set up in the correct order. In finite difference form the strain-rate equations. referred to axes measured along and normal to the streamlines. are
(4)
where i and i, are the direct and shear strain-rates respectively, s and n are the distances measured along and normal to a streamline, u,~is the velocity and 0 is the angular rotation of a streamline from a fixed reference axis. In terms of these strain-rate components the magnitude and direction of the maximum shear strain-rate needed in the slipline field analysis are given by (5) and (6)
Analysis
of an asperity
interaction
FIG. 3. Reconstructed
digitized
771
experiment
data.
where I) is measured from the same fixed reference axis as 8. The net used in calculating the strain-rate components is given in Fig. 4 from which the strain-rates at the grid intersection point i, j can be expressed as
4,., =
t’s,,,+,-
vs,,,_,
Asi,,
’
K,,i-K+,,i i”,,, = 0, ‘.I b. I
L,,, =
vi,+ ,,]-“I,, An, j
-
(7)
’ I
,,,
ei,j+ + vs~.~ Aq i
FIG. 4. Net used in calculating
I -ei.
j-
strain-rates.
I
’
J
772
E.
The calculations of the velocities in the following order.
M. KOPALINSKY etc. required
er al
for substitution
in (7) were performed
(1) The streamline equations were obtained from the digitized coordinates x, y of the flow field. By starting in the undeformed region and working along each streamline second-order polynomials were fitted to sets of three consecutive grid intersection (equal time) points with overlapping so that the second and third points of the first set become the first and second points of the second set and so on. (2) The distance to the grid intersection points measured from the y axis along the streamlines were determined by progressively summing the distances between grid intersection points found by integrating streamline equations. (3) The corresponding values of time for an element to flow from they axis to the grid intersection points were found by summing equal time intervals along streamlines. (4) The distance-time equations were determined by fitting second-order polynomials to the distance s and time t values measured from they axis ; the same method was used as for the streamline equations. (5) The magnitude and direction of the velocity at the nodal points on streamlines i (Fig. 4) which coincide with the grid intersection points i, j- 1 and i, j+ 1 could now be found by differentiating the appropriate distance-time and streamline equations. (6) The required distance AS;., (Fig. 4) was found from the streamline equation. (7) The nodal points on the adjacent streamlines i- 1 and i+ 1 (Fig. 4) in general did not coincide with grid intersection points, therefore it was necessary first to determine their coordinates. The equation of the straight line normal to the streamline i at the grid intersection point i,j was derived from the equation of the streamline passing through this point. The mid-points of the parts of the line between the grid intersection point i,i and the adjacent streamlines were then determined and the equations of the lines passing through these points and normal to the adjacent streamlines were calculated. The required coordinates .u;- ,.,, & I..i and xi+ ,, i and I were found by solving simultaneously these equations and equations repreYi+ I./ senting i-j and i+j streamlines in the vicinity of the coordinates. (8) The value of AGI,~ (Fig. 4) was found by summing the chord lengths. (9) Once the coordinates of the nodal points on the adjacent streamlines were known the directions of the velocities cl_ ,,, and v,{,+,,, were found by differentiating the streamline equation fitted to the nearest grid intersection point. (10) To find the magnitudes of c:,_ ,., and u:,, i., the first step was to obtain secondorder polynomials relating the time i and .Ycoordinates at the grid intersection points along streamlines by the same method as before. The values oft for the coordinates time-x coordinates equations fitted xl- 1.j and xi+ ,, j were found from the appropriate to the nearest grid intersection point. The required magnitude of the velocity could then be found by differentiating the corresponding distance s-time equation.
FARMER and CONNING (1979) noted that large errors could result if strain-rates were determined as described above using raw data, i.e. x,y coordinates digitized directly from tracings of experimental flow fields. They attributed this to local inhomogeneity of the flow resulting from the granular structure of the metals used and to random tracing and digitizing errors. In view of this they introduced numerical procedures to remove small irregularities from the flow and so smooth the data
Analysis
of an asperity
interaction
773
experiment
sufficiently to permit reasonably accurate computations of strain-rates. In one of the methods Spoint least-squares smoothing of the streamlines and transversals is applied but this tends to give excessive flattening of sharp bends in the regions of intense deformation. They therefore developed two alternative schemes based on the required constancy of the grid element areas for plane strain, incompressible flow. In the first scheme the areas of four adjacent elements are made more nearly equai by shifting their common vertex, while in the second a grid intersection point is shifted along a streamline tangent to make the area of one of the pertinent elements nearer to the global mean. In theory a combination of adjustment of the element areas and leastsquares smoothing of streamlines and transversals can be employed in an arbitrary sequence and repeated as many times as necessary but with the changes in grid intersection point coordinates at each iteration restricted to ensure stability. However, one or two passes of area adjustment followed by least-squares smoothing first of the transversals and then of the streamlines is usually sufficient. A computer programme was used to smooth the digitized data obtained from Fig. 2 and to compute the strain-rates in magnitude and direction at grid intersection points on the streamiines in the way described above. The crosses in Fig. 5 represent the directions of maximum shear strain-rate. It can be seen that these show well defined trends in direction in those parts of the field where significant plastic deformation occurred. As would be expected nq.such trends are observed in the region adjacent to the wedge-wave interface where little if any plastic deformation took place. Although in the case of extrusion the computer-aided method for calculating strainrates described above has been extended to the construction and stress analysis of slipline fields (CONNING et al., 1984) it has not so far been extended in this way to processes such as the wave formation process now under consideration. The rest of the analysis given in this paper is therefore essentially manual. SlipiineJield construction The starting point in the construction of the slipline field was to estimate from the original flow field (Fig. 2) the boundary of the zone in which plastic deformation
FIG. 5. Slipline
field and
directions
of maximum shear strain-rate construction of field.
(represented
by crosses)
used in
774
E. ht.
KOPALINSKY
ef
cd.
occurred. This was done by noting the points on the streamlines where deviations from the horizontal first occurred at entry to the plastic zone and where the streamlines once again became horizontal at exit from the zone. Further evidence for the start and end of plastic flow was obtained from noting the points where the transversals first deviated from the vertical at entry to the zone and assumed a constant slope at exit from the zone. In general these two sets of points were in close agreement. The construction of the boundary was started at the free surface at the point where the surface streamline first deviated from the horizontal. It was assumed that the boundary would be a slipline and would therefore meet the free surface at 45. in order to satisfy equilibrium. The points representing the start and end of plastic flow were then used together with the calculated directions of maximum shear strain-rate, which were taken as slipline directions, to construct the boundary slipline AI-Al7 given in Fig. 5. This was terminated at the wedge corner to the right of which no external loads were applied. Once the boundary slipline had been established two orthogonal sets of curves representing the sliplines were constructed within the plastic region using the calculated directions of maximum shear strain-rate. The sliplines were drawn so as to meet the free surface at the required angle of 45”. In the region adjacent to the wedgewave interface the calculated directions of maximum shear strain-rate showed no clear trends and it was assumed that the field would be similar to that in Fig. 1, i.e. made up of straight sliplines. The inclination of these sliplines to the wedge-wave interface was determined from the equilibrium relation cc+ cf, = 4 cos- ’ (t/k) where r is the shear stress along the wedge-wave interface and k is the shear flow stress of the deformed material in this region and the angles are shown in Fig. 1. Because at this stage of the analysis the value of s/k was not known it was decided to use the value of 0.25 obtained by CHALLEN et al. (1984) for similar test conditions. With this value of r/k the anticlockwise angular rotation of sliplines such as E5-El6 from the wedge face was determined to be 38”. As can be seen in Fig. 5 the shpline field constructed in this way had two fan regions emanating from the leading and trailing edges of the wedge-wave interface. For convenience these fans were terminated at stress singularities as shown although it was recognised that such singularities were not acceptable in hardening flows. This point is returned to later in the paper. In Fig. 5 the I-sliplines, i.e. those on which the shear stress exerts a clockwise couple, are designated by numbers and the II-sliplines by letters. The hodograph corresponding to the slipline field of Fig. 5 was next constructed in order to check how accurately this field represented the velocities from which it had been derived. The given information was taken to be the magnitude and direction of the velocity at entry Al-A9 to and exit A9-A17 from the plastic zone and the direction of the velocity along the free surface Al-19. No conditions were imposed regarding the velocity along the wedge-wave interface which could therefore serve as a check on the accuracy of the construction. The hodograph was obtained by the following step-by-step method which makes use of the condition that for volume constancy (rigid-plastic assumption) the rate of extension along sliplines must be zero. The construction was started at the point A2 with the velocity at this point represented by the line drawn from the pole to the point A2 in the hodograph (Fig. 6). The direction of the velocity at the point B2 was then taken as the slope of the free surface at this point (Fig. 5) and a line was drawn in this direction through the
Analysis
of an asperity
interaction
experiment
115
FIG. 6. Hodograph.
pole. To find the point B2 on this line and hence the magnitude of the velocity at this point the orthogonal image of the slipline A2-B2 was drawn in the hodograph as shown. In this way the velocity at B2 relative to the velocity at A2 was made to act at right angles to the slipline A2-B2 and a zero rate of extension along this slipline was obtained as required. Once the point B2 in the hodograph had been established, the point B3 and hence the velocity at this.. point were determined using the same principle of constructing the orthogonal images of the sliplines A3-B3 and B2-B3 to find their intersection point. By working in this way throughout the field the hodograph given in Fig. 6 was obtained. As can be seen, in constructing the hodograph the slipline elements have been approximated by their equivalent chords but for the field considered this would only have introduced small errors. It should be noted that the slipline field and hodograph given in Figs 5 and 6 respectively are those finally obtained after a number of adjustments had been made to the slipline field in order to get a better agreement with experimental velocities. The results in Figs 7 and 8 are given so as to make a more direct comparison
16
13 12 11 10 9 8 7 6 5
FIG. 7. Velocity
10
directions
obtained
15
from hodograph
20
(represented
by arrows)
25
and original
streamlines.
E. M. KOPALINSKYVI al.
20 Transversal
30
10
number (a)
Jj
p
z O.lz! ._ zl r”
0
I IO
I
20 Transversal
I
I
30
40
number
FIG. 8. Velocity magnitudes obtained from hodograph (represented by symbols) and from computer programme (represented by lines): streamlines and transversals are numbered as shown in Fig. 7: (a) streamline number 16; (b) streamline number 1.5; (cf streamtine number 14; (d) streamline number 13 ; (e) streamline number 12; (f) streamline number 11.
Analysis of an asperity interaction experiment
Transversal
number fc)
Transversal
number (d)
FIG. 8 (continued).
771
778
E. M. KOPALINSKYet al.
z aI O.l-a 2 ._ El s
0
I 10
I 20 Transversal
I 30
1 CO
I 30
I LO
number (e)
2 .g 0.25 > 6 4
O.l-
2 ._ En I 10
I 20 Transversal
number IfI
FIG. 8 (continued).
Analysis
of an asperity
interaction
experiment
779
between the velocities determined from the slipline field and the original data used in constructing the field. Velocity directions obtained from the hodograph and the original streamlines re-drawn from Fig. 3 are shown in Fig. 7, while the velocity magnitudes obtained from the hodograph and those obtained from the computer programme are given in Fig. 8. It can be seen that the slipline field in Fig. 5 represents the flow quite accurately over most of the plastic region. The greatest discrepancy is at the wedge-wave interface where the direction of flow should be at 20” (wedge angle) to the horizontal but as shown on the hodograph actually varies from 23.5” at H13 to 14.5’ at E16. In spite of this it was concluded that the slipline field represented the flow from the velocity viewpoint sufficiently accurately to warrant proceeding to the stress analysis. Stress anal.vsis In order to calculate the distribution of p throughout the slipline field it was necessary first to calculate the distribution of k. This was achieved by integrating the maximum shear strain-rate along streamlines with respect to time to find the maximum shear strain at a point. The plastic stress-strain relations for the material given in (3) were then used to find the corresponding k. Contours of equal values of maximum shear strain calculated using the computer-aided method described earlier are given in Fig. 9. In obtaining these values it was assumed that the slipline field defined the region where plastic deformation occurred and that strain-rates calculated from the digitized data which fell outside the field should be treated as noise and reset to zero before carrying out the integration. At this stage of the analysis the maximum shear strain yA, which defines the yield shear stress along the entry boundary Al-A9, was not known and in making the integration yAwas taken as zero. Therefore, the results in Fig. 9 represent the increases in shear strain above yA and to find the actual shear strain at a point it is necessary to add YAto the value obtained from Fig. 9. In the analysis ?A was determined by selecting its value to give a stress distribution along the
FIG. 9. Contours
of equal maximum
shear strain values
; numbers given on contours
are y x 100/d
E. M.
780
KOPALINSKY
ef a/
boundary slipline Al-Al7 which was consistent with the experimentally measured vertical force (N = 7953 N). The method used was as follows. The distribution of p along Al-A17, which as drawn in Fig. 5 is a II-slipline, was determined by starting at Al, where to satisfy the free surface condition p is equal to k and is compressive, and applying the appropriate equation of (2). In finite difference form this can be written as Ap = okay + (Akl~~)As~~ where the strain-hardening
(8)
term can be expressed as
(Ak/As,)Asz = (Ak/Ay)(Ay/As,)Asz.
(9)
In the above Ak/Ay is the slope of the stress-strain curve at the appropriate strain and Ay/As, is the rate of change of 1:with distance normal to the slipline. In making the calculations it was necessary to represent Al-Al7 by chord segments. It was found su~ciently accurate to do this by taking the nodes at slipline intersection points as natural dividers with As? taken as the distance between them. To start the calculations a reasonable value of yA was assumed and (3) were used to obtain the value of k along Al-A9 with y = yA and also the values of k along A9-Al7 with yA added to the appropriate strain values given in Fig. 9. In applying (8) the curvature term 2kll/was evaluated by measuring All/ from Fig. 5 (taken as the difference in rl/ between nodes) and an average value of k was used where this varied along the step considered. The strain-hardening term as expressed in (9) was determined by measu~ng Ay/As, from Fig. 9 and calculating Ak/A~ for the appropriate strain from (3). For each segment average values of A~/Asi and Ak/Ay determined from the values at the adjoining nodes were used. The step-by-step calculations were performed from Al along A l-Al7 and it was found that the curvature term gave a compressive increment to p while the hardening term gave a tensile increment. Once the distributions of p and k were determined for a given yA the forces acting on each segment were calculated and resolved in the horizontal and vertical directions. These components were then summed to give the total horizontal and vertical forces. Different values of Y,.,were used and it was found that a value of 0.022 gave a vertical force of 7719 N which was within 3% of the measured value of 7953 N. This was considered to be sufficiently close and therefore yA was taken to be 0.022. The distributions of p and k corresponding to this value of yAare given in Fig. 10. Also given are the vertical and horizontal forces calculated from these stress distributions together with the line of action of their resultant which was determined by taking moments of the forces acting on each segment about A 17. After the value of ?A had been evaluated this was added to the shear strains given in Fig. 9 to obtain the actual values of shear strain throughout the plastic zone which were then used to determine the corresponding values of k from (3). The values of k at each slipline node were then found by linear interpolation and these are given in Table 1. To complete the stress analysis p was calculated throughout the slipline field by applying the appropriate stress equilibrium equation in its finite difference form as described above. The calculations were made in two ways : first by starting at the boundary slipline A 1-A 17 where p was now known and working along I-sliplines such as A2-B2 and second by working along II-sliplines from the free surface Al-19 where
Analysis of an asperity interaction experiment
781
266 MPa
FIG. 10. Calculated stresses and forces on plastic zone boundary and wedge-wave interface.
p was equaf to k and therefore are given for each slipline node The values of the normal and be determined from the known
Al
A2
A3
A4
A5
A6
known. These values, termed p, and p2 respectively, in Table 1. _ shear stresses at the wedge-wave interface could now values of p and k at the interface together with the
A?
AG
A9
Al0
All
A12
Al3
A14
A15
A16
A17
E. M.
782
KOPALINSKY
et al
inclination of the sliplines to the interface. (Two values of normal stress were found for each point on the interface. one corresponding to p, and the other to pJ but these were very close and an average value was used.) These normal and shear stress values were then used to calculate the vertical and horizontal forces acting at the interface ; these are given in Fig. 10 together with the line of action of their resultant found by taking moments about A 17. To show the importance of allowing for strain-hardening in the analysis all of the above calculations were repeated using only the curvature terms in the stress equilibrium equations with the strain-hardening terms taken to be zero. In applying the curvature terms an average value of k was used as before. The results obtained in this way are given in Fig. 11 and Table 2.
DISCUSSION It can be seen in Fig. 5 that the region of the slipline field above the slipline D& D17 is similar to the perfectly plastic slipline field given in Fig. 1. This is perhaps not surprising when it is observed from the results in Table 1 that for much of this region k had reached its saturation value (in this region y > 1 and hence from (3) k is constant), The corresponding flow would therefore have been expected to approximate closely to a perfectly plastic flow. From the values of k in Table 1 it is clear that this
Al
Al I_
Al7 k
---._\_
1264
-1.
MPa
‘\.. \ “\ q_
FIG.
II.
Calculated
stresses
-
764 two.
and forces on plastic zone boundary and wedge-wave hardening terms taken IO be zero.
interface
with strain-
Analysis TABLE
of an asperity
interaction
experiment
783
2. Stresses (in MPa) at slipline nodes with strain-hardening terms taken to he zero ; lajwut same as in Table 1
o4 . 319 . zy) . 111 .. ... .. .........
. m . M .m .10W .*082 .10'18 .1079 .wra w ... ..-..................~~.....~
has been the case for the flow in the regions of the singularities at each end of the wedge-wave interface. The problem raised earlier of these singularities existing in a hardening flow, therefore, no longer applies. The region of the slipline field below the slipline D4-D17 can be looked upon as that resulting from the opening-up of the velocity discontinuity across the slipline ABCD in Fig. 1 in order to accommodate the strain-hardening which had taken place in the process. It is in this region, particularly along the entry boundary A l-A9 (Fig. 5), that the strain-hardening terms in the stress equilibrium equations have their largest influence on the calculated values ofp. Once hardening is introduced into the analysis by using visioplasticity techniques then the results obtained are, by their very nature, approximate. Nevertheless it can be seen in Fig. 10 that the agreement between the forces at the wedge-wave interface and across the boundary slipline AI-AI 7 obtained from the analysis is good with both sets of forces in reasonable agreement with the experimental forces (N = 7953 N; F = 3250 N). Another check on the accuracy of the stress analysis can be made from the results in Table 1. For an exact solution the values of p, and pz at a slipline node should be equal and the results in Table 1 show that this condition is approximately satisfied for most of the field. When the strain-hardening terms are neglected then the results in Table 2 and Fig. 11 show that the agreement is very poor. The question of whether or not tensile stresses can be shown to exist in part of the plastic zone once hardening effects are allowed for is now considered. It can be seen
784
E. M. KOPALINSKY et al.
from the results in Table 1 that in no part of the field does the hydrostatic stress become tensile. However. in the small region adjacent to the slipline nodes A8 and A9, on the boundary slipline, p does become significantly less than k (see also Fig. 10) which indicates tensile direct stresses in this region. The results show that the maximum value of tensile stress occurs near to A9 where the principal stress acting in a direction at 45” measured clockwise from the horizontal has a tensile value of approximately 22 MPa. With the errors involved in the calculations this value might be considered to be too smaI1 to confirm the existence of a tensile stress. However, had the value of p at A9 been caiculated by starting on the free surface at 19 with p = k = 264 MPa and working along I9-A9 then the tensile stress obtained at A9 would have been 43 MPa and not 22 MPa. It can therefore be argued that the errors in the stress analysis had led to the tensile stress at A9 being underestimated and should not be used to cast doubt on its existence. It can be concluded that once strain-hardening is allowed for in the stress analysis in the way described then a mechanism exists for the generation of tensile stresses in the wave formation process considered in spite of the compressive nature of the loading. This should certainly be taken into account in considering low cycle fatigue and fracture as possible wear mechanisms in those situations where asperity interactions can be modelled by the wave formation process as described in the introduction to this paper.
The authors wish to thank their former colleague STAN CONNING for help in applying the computer programme used in calculating strain-rates etc., and the Australian Research Council for their financial support of the project. -NCES
BLACK,A. J., KOPALINSKY,E. M. and OXLEY, P. L. B. CHALLEN, J.M. and OXLEY,P. L. B. CHALLEN,J. M., MCLEAN, L. J. and OXLEY,P. L. B. CHALLEN,J. M., HOCKENHULL,B. S. and OXLEY,P.L. B. CHALLEN,J. M., KOPALINSKY,E. M. and OXLEY,P.L. B. COLLINS,1. F.
1988
Wear 123,97.
1979
Weur 53,229.
1984
Proc. R. Sot. (Land.) A394, 161
1986
Wear 111,275
1987
In Prvc. ht.
1978
Me& Engrs ht. Con$ TribologyFriction. ~ubricotion and Weor-Ftyt_v Years On, 2, 957. In IUTAM Svrtp. Metal Forming Plasticity, Tutz-
ing (edited by H. LIPPMANN),p. 227. Springer, Berlin. CONNING,S. W. and OXLEY,P. L. B.
1989
In Plosticit_v nnd Modern Metal- Forming Technolog_v (edited by T. Z. BLAZYNSKI).Elsevier Applied Science, Amsterdam.
CONNING, S. W., FARMER, L. E. and OXLEY,P. L. B.
1984
Phil. Trans. R. Sot. (Lnnd.), A311,495.
Analysis of an asperity interaction FARMER. L. E. and CONNING. S. W. FARMER. L. E. and FOWLE. R. F. FARMER, L. E. and OXLEY, P. L. B. MOALIC, H., FITZPATRICK, J. A. and TORRANCE, A. A. OLVER, A. V.. SPIKES, H. A., BOWER, A. F. and JOHNSON. K. L. PALMER, W. B. and OXLEY, P. L. B.
experiment
1979
Int. J. Med.
1979 1976
Int. J. Mech. Sci. 21, 599. J. Strain Anal. 11, 26.
1987
Proc. Inst. Mech. Engrs 201, 321.
1986
Wear 107, 151.
1959
Proc. Inst. Mech. Engrs 173, 623.
Sci. 21, 577.
785
Phvs. SolidsVol. 38, No. 6, pp. 165-185, Printed in Great Britain.
0022-5096~90
1990.
$3.00+0.00
c 1990 Pergamon Press plc
A STRAIN-HARDENING SLIPLINE FIELD ANALYSIS OF THE PLASTIC DEFORMATION OBSERVED IN A MODEL ASPERITY INTERACTION EXPERIMENT E. M.
KOPALINSKY,~
P. L. B. OXLEY~ and H. T.
YOUNG$
tSchoo1 of Mechanical and Industrial Engineering, The University of New South Wales, P.O. Box 1, Kensington, New South Wales. 2033, Australia; IDepartment of Mechanical Engineering, National Taiwan University, No. 1 Sec. 4 Roosevelt Road, Taipei, Taiwan, 10764. Republic of China
(Received
13 July 1989)
ABSTRACT SCALED-UP model asperity experiments are described in which a hard wedge is indented into and then slid along the surface of a relatively soft material with the resulting plastic deformation measured using printed grids. A slipline field analysis of the deformation occurring once steady-state conditions have been achieved is described. This shows that tensile stresses can exist in a small region of the field when in calculating stresses account is taken in the stress equilibrium equations of variations in flow stress resulting from the strain-hardening properties of the deforming materia!:- This result could be important in considering the viability of processes such as low cycle fatigue and fracture as possible wear mechanisms.
INTRODUCTION
THE PLANE strain slipline field proposed by CHALLENand OXLEY(1979) to represent asperity interactions when a hard surface slides over a relatively soft surface is given in Fig. 1. With this model the friction force is considered to be the force needed to
Streamline
of flow
(b )
FIG 1. Wave model of asperity
deformation 765
: (a) slipline field; (b) hodograph.
E. M. K~PALINSKY CI al.
766
push waves of plastically deformed material in the soft surface ahead of asperities on the hard surface. The wave model has been shown to apply to relatively smooth well lubricated surfaces. For rougher surfaces (larger as) and less efficient lubrication (larger shear stresses along the interface DE) the wave can be torn off and for very rough surfaces material can be removed by a chip formation process. Coefficients of friction predicted from the slipline field have been shown to be in good agreement with experimental results obtained from model asperity tests using wedges (CHALLEN et al., 1984) and from tests using actual surfaces (MOALIC et al., 1987). CHALLEN er al. (1986) have used the wave model to predict the wear which occurs during sliding as a result of the repeated plastic working of the surface. They have done this by calculating from the model the plastic strain increments put into the soft surface by the passage of plastic waves across it and using these increments with a low cycle fatigue equation to determine the wear rate. The analysis yields a relationship between the coefficients of friction and wear which has been shown to be in good agreement with experimental results (CHALLEN et al., 1987). This is encouraging but in order to justify more directly the use of a low cycle fatigue mechanism to predict wear in this way it is necessary to consider the stresses involved in the deformation as well as the strains. For example, the presence of a tensile stress in part of the deforming region might well be considered essential to the process of crack initiation. Not surprisingly, considering the compressive nature of the loading under sliding contact conditions, the direct stresses in.the asperity contact regions are predominantly compressive. For the slipline field in Fig. 1 it can be shown that they are entirely compressive. In constructing this field the soft material is assumed to be rigid-perfectly plastic with the mean compressive (hydrostatic) stress calculated from the Hencky equations (stress equilibrium equations referred to sliplines) p+2kll/
= const.
p - 2kll/ = const.
along a I-line, along a II-line,
>
(1)
where p is the hydrostatic stress which acts normal to the sliplines, k is the shear flow stress (assumed constant) which acts parallel to the sliplines and $ is the anticlockwise angular rotation of the I-lines from a fixed reference axis ; the I-lines are taken as those on which the shear stress exerts a clockwise angle. To satisfy equilibrium at the free surface AE (Fig. 1) p must be equal in magnitude to k and compressive in the region ABE and Eqs (1) show that its compressive value increases in moving through the centered fan BCE into the region CDE. At no point in the field therefore is p less than k and all direct stresses are compressive. OLVER et al. (1986) have used a field similar to that in Fig. 1 with u = 0 to determine the stresses when the load is applied and also the residual stresses generated during unloading. A novel feature of their analysis is the extension of the slipline field into the non-deforming plastic region in order to calculate the stresses in this region. Their results show that although the direct stresses are compressive when the load is applied residual tensile stresses can be generated in part of the field during unloading. They confirmed experimentally the existence of residual tensile stresses and have pointed out the possible relevance of these to crack formation resulting from asperity contacts.
Analysis of an asperity interactlon experiment
161
The purpose of the present paper is to investigate the so far neglected influence of the hardening properties of the deforming material on the stress distributions in the wave formation process. Once the perfectly plastic assumption is relinquished and a rigid-plastic hardening material is assumed the stress equilibrium equations referred to sliplines become 1 $+2k$-g=O 1 I 8P -2kg dsz
along a I-line,
2 - g
I
= 0
(2) along a II-line,
where s, and s2 are distances measured along the I and II lines respectively. If k varies Eqs (2) show that the variation of p along a slipline depends not only on the angle turned through by the slipline, as is the case with (I), but also on the rate of change of k with distance along the sliplines. PALMER and OXLEY (1959), COLLINS (1978) and CONNING et al. (1984) have discussed the nature of rigid-plastic hardening solutions. They have noted that from (2) the normal derivative of k across a slipline must be continuous. Hence in hardening flows sliplines cannot propagate jumps in velocity as are frequently used in rigid-perfectly plastic_soiutions, the jump in tangential velocity across the slipline ABCD in Fig. 1 being an example. In hardening flows velocity discontinuities must, therefore, be replaced by deformation zones in which gradual changes in velocity occur. So far no analytical slipline field solutions for metal deformation processes, such as the wave formation process now considered, have been found once k is allowed to vary and would appear to be prohibitively difficult. In hardening problems it is generally necessary to determine the distribution of k as part of the solution. As a consequence it is no longer possible to determine simple geometric properties of hardening slipline fields such as the well-known ones of Hencky for the perfectly plastic case. Hardening problems are therefore intrinsically more complicated than non-hardening problems. Although no analytical solutions to hardening problems have been obtained, slipline fields have been constructed for hardening flows using visioplasticity techniques. The basis of this approach is to use an experimental flow field to determine the velocities of flow which in turn are used to determine the strainrates and hence the directions of maximum shear strain-rate. If an isotropic material is assumed then the latter can be taken to be slipline directions, i.e. directions of maximum shear stress as well as of maximum shear strain-rate, and used in conjunction with known stress boundary conditions to construct a slipline field. The distributions of strain and hence k (strain-rate and temperature effects are usually neglected) are then determined by integrating the strain-rate with respect to time along streamlines and a stress analysis is carried out using (2). Slipline fields obtained in this way clearly show the extent to which the velocity discontinuities in rigid-perfectly plastic solutions open-up in hardening flows. They also show that the distributions of p calculated from (2) can be very different from those calculated from (1). Of particular interest in the present context is the slipline field of PALMER and OXLEY (1959) for orthogonal machining which indicates a high
768
E. M. KOPALINSKY eral.
compressive value of p near the cutting edge when (1) are used to calculate stresses but a high tensile value of p when (2) are used. CONNING and OXLEY (1989) have recently given a comprehensive review of visioplasticity techniques. Their application to the wave formation process of Fig. 1 is now considered.
EXPERIMENT The objectives of the experimental work were to obtain a flow field from which the velocities of flow could be determined for a typical plane of flow for conditions as near as possible to plane strain, steady-state conditions and to measure the associated forces. In the experiments, which were similar to those made by CHALLEN et al. (1984) and more recently by BLACK et al. (1988), a hard wedge representing a scaled-up asperity was indented vertically into the horizontal surface of a relatively soft specimen with subsequent movement of the specimen relative to the wedge in a direction parallel to its surface and normal to the edge of the wedge. Conditions were made as close to plane strain as possible by clamping the specimen between side plates with the wedge having sufficient clearance to slide freely between the plates. The wedge was free to move up or down during horizontal movement of the specimen in order to maintain a constant vertical force and to allow thewedge to return to the surface once the steadystate wave process (Fig. 1) had been achieved. The test rig used in the experiments was in the modified form described by BLACK et al. (1988) with a hydraulic “quick-stop” device for disengaging the wedge and wave at the end of a test. A Kistler dynamometer which was part of the specimen assembly was used to measure forces. Vertical displacements of the specimen relative to the wedge were measured with a linear variable differential transformer. With the measuring devices used, the signals of which were amplified and recorded on a multichannel chart recorder, it was estimated that forces could be measured with an accuracy of +25 N and displacements with an accuracy of + 1 pm. In all cases these were much smaller than the actual measured values. Experimental flow fields were obtained by using a specimen divided on what would be a typical plane of flow during wave formation with a grid printed on one of the faces. The material selected for the specimens was a cold rolled 5083-H32 1 aluminiummagnesium alloy which preliminary experiments had shown gave well-defined flow fields. The results of compression tests on this material were used to obtain the plastic stress-strain relations k=
for y d 1.0
k = 263.5
1 for y > 1.0, J
and
(3)
where k is the shear flow stress in MPa and y is the maximum shear strain. In deriving (3) plane strain and uniaxial conditions have been related in the usual way with k = a/ J3 and y = J3 E w h ere D and E are the uniaxial stress and strain. Specimens were made of two equal parts of 150 mm length, 25 mm depth and 6 mm width which gave a composite specimen of 12 mm width. These were milled from plate and then polished
Analysis of an asperity Interaction experiment
769
1200 grade paper on the test surface and on the sides to eliminate any bowing resulting from residual stresses caused by machining. A chequerboard grid pattern of 0.1 mm sides was printed on the relevant region of one of the inner faces using the photo resist technique described by FARMER and FOWLE (1979). Care was taken in printing the grid to align it with the specimen sides so that during a test one side of the grid would be parallel to the sliding direction. The wedges were machined from Bohler Amutit S tool steel, hardened and tempered to RC60, ground and then polished on 1200 grade paper in a direction normal to the edge of the wedge. In order to minimize the friction at the wedge-wave interface (DE in Fig. 1) during a test and thus reduce the risk of severe deformation and tearing in the wave in this region both the specimen and tool test faces were sprayed with a bonded lubricant (Dow Corning Molykote 321 R). A strict, repeatable procedure was used which gave a coating estimated to be 10 pm in thickness. In a test the wedge was indented into the specimen surface by applying the selected vertical force. During the subsequent horizontal movement the wedge at first was observed to dig in deeper to the specimen as the vertical force was transferred from both sides to a single side of the wedge. With further movement the wedge started to climb on plastically deformed material pushed up in front of it until once again the edge of the wedge was on the specimen surface. In the absence of fracture in the deforming material steady-state condition< were then achieved with a plastic wave pushed ahead of the wedge as depicted in Fig. 1. Once a steady-state had been achieved, as judged by the constancy of the horizontal force F (Fig. I), the “quickstop” was made. A photograph of the deformed grid obtained in this way is given in Fig. 2. The corresponding test conditions were: wedge angle c1= 20”, measured vertical and horizontal forces N = 7953 N and F = 3250 N respectively and horizontal speed U = 0.3 mm s- ’ (Fig. 1). Results were obtained for a number of wedge angles (and also for lubricants other than Molykote) but in spite of the slight imperfection in the deformed grid in Fig. 2 it was decided that this was the best defined of the flow fields obtained and therefore the most promising for the purpose of making the slipline with
field analysis.
FIG. 2. Experimental flow field ; grid 0.1 mm sides.
710
E. M. K~PALINSICY CI al SLIPLINE FIELD ANALYSIS
If carried out manually the calculation of strain-rates from an experimental flow field is extremely time consuming and for this reason a computer-aided method has been developed. The basis of the method has been given by FARMERand OXLEY(1976). Its application to the flow field in Fig. 2 is now described. In the analysis the Aow was from left to right with the wedge fixed as in Fig. 1. For experimental flow fields such as that in Fig. 2, which are for approximately plane strain, steady-state conditions, it can be assumed that the originally horizontal grid lines are streamlines and that their intersection points with the originally vertical lines (transversals) represent points of equal time interval along the streamlines. It is therefore possible to determine from the flow field the velocity gradients required in calculating strain-rates. The starting point of the present analysis was to make a line diagram tracing from an enlarged photograph of the flow field in Fig. 2 to give a clearer definition of the intersection points of the grid lines. The S.J~ coordinates of these intersection points were then digitized with the x axis taken along the surface in the direction of motion and with the JJ axis coinciding with one of the transversals in the undeformed region to the left of the wave. The digitized coordinates together with the value of the equal time interval formed the input data to the computer programme used in making the calculations. Figure 3 shows the digitized data from Fig. 2 reconstructed by replacing the curved sides of the elements with chords. It is usual to display the digitized input data in this way prior to further processing as an aid to verifying the data and ensuring that they are set up in the correct order. In finite difference form the strain-rate equations. referred to axes measured along and normal to the streamlines. are
(4)
where i and i, are the direct and shear strain-rates respectively, s and n are the distances measured along and normal to a streamline, u,~is the velocity and 0 is the angular rotation of a streamline from a fixed reference axis. In terms of these strain-rate components the magnitude and direction of the maximum shear strain-rate needed in the slipline field analysis are given by (5) and (6)
Analysis
of an asperity
interaction
FIG. 3. Reconstructed
digitized
771
experiment
data.
where I) is measured from the same fixed reference axis as 8. The net used in calculating the strain-rate components is given in Fig. 4 from which the strain-rates at the grid intersection point i, j can be expressed as
4,., =
t’s,,,+,-
vs,,,_,
Asi,,
’
K,,i-K+,,i i”,,, = 0, ‘.I b. I
L,,, =
vi,+ ,,]-“I,, An, j
-
(7)
’ I
,,,
ei,j+ + vs~.~ Aq i
FIG. 4. Net used in calculating
I -ei.
j-
strain-rates.
I
’
J
772
E.
The calculations of the velocities in the following order.
M. KOPALINSKY etc. required
er al
for substitution
in (7) were performed
(1) The streamline equations were obtained from the digitized coordinates x, y of the flow field. By starting in the undeformed region and working along each streamline second-order polynomials were fitted to sets of three consecutive grid intersection (equal time) points with overlapping so that the second and third points of the first set become the first and second points of the second set and so on. (2) The distance to the grid intersection points measured from the y axis along the streamlines were determined by progressively summing the distances between grid intersection points found by integrating streamline equations. (3) The corresponding values of time for an element to flow from they axis to the grid intersection points were found by summing equal time intervals along streamlines. (4) The distance-time equations were determined by fitting second-order polynomials to the distance s and time t values measured from they axis ; the same method was used as for the streamline equations. (5) The magnitude and direction of the velocity at the nodal points on streamlines i (Fig. 4) which coincide with the grid intersection points i, j- 1 and i, j+ 1 could now be found by differentiating the appropriate distance-time and streamline equations. (6) The required distance AS;., (Fig. 4) was found from the streamline equation. (7) The nodal points on the adjacent streamlines i- 1 and i+ 1 (Fig. 4) in general did not coincide with grid intersection points, therefore it was necessary first to determine their coordinates. The equation of the straight line normal to the streamline i at the grid intersection point i,j was derived from the equation of the streamline passing through this point. The mid-points of the parts of the line between the grid intersection point i,i and the adjacent streamlines were then determined and the equations of the lines passing through these points and normal to the adjacent streamlines were calculated. The required coordinates .u;- ,.,, & I..i and xi+ ,, i and I were found by solving simultaneously these equations and equations repreYi+ I./ senting i-j and i+j streamlines in the vicinity of the coordinates. (8) The value of AGI,~ (Fig. 4) was found by summing the chord lengths. (9) Once the coordinates of the nodal points on the adjacent streamlines were known the directions of the velocities cl_ ,,, and v,{,+,,, were found by differentiating the streamline equation fitted to the nearest grid intersection point. (10) To find the magnitudes of c:,_ ,., and u:,, i., the first step was to obtain secondorder polynomials relating the time i and .Ycoordinates at the grid intersection points along streamlines by the same method as before. The values oft for the coordinates time-x coordinates equations fitted xl- 1.j and xi+ ,, j were found from the appropriate to the nearest grid intersection point. The required magnitude of the velocity could then be found by differentiating the corresponding distance s-time equation.
FARMER and CONNING (1979) noted that large errors could result if strain-rates were determined as described above using raw data, i.e. x,y coordinates digitized directly from tracings of experimental flow fields. They attributed this to local inhomogeneity of the flow resulting from the granular structure of the metals used and to random tracing and digitizing errors. In view of this they introduced numerical procedures to remove small irregularities from the flow and so smooth the data
Analysis
of an asperity
interaction
773
experiment
sufficiently to permit reasonably accurate computations of strain-rates. In one of the methods Spoint least-squares smoothing of the streamlines and transversals is applied but this tends to give excessive flattening of sharp bends in the regions of intense deformation. They therefore developed two alternative schemes based on the required constancy of the grid element areas for plane strain, incompressible flow. In the first scheme the areas of four adjacent elements are made more nearly equai by shifting their common vertex, while in the second a grid intersection point is shifted along a streamline tangent to make the area of one of the pertinent elements nearer to the global mean. In theory a combination of adjustment of the element areas and leastsquares smoothing of streamlines and transversals can be employed in an arbitrary sequence and repeated as many times as necessary but with the changes in grid intersection point coordinates at each iteration restricted to ensure stability. However, one or two passes of area adjustment followed by least-squares smoothing first of the transversals and then of the streamlines is usually sufficient. A computer programme was used to smooth the digitized data obtained from Fig. 2 and to compute the strain-rates in magnitude and direction at grid intersection points on the streamiines in the way described above. The crosses in Fig. 5 represent the directions of maximum shear strain-rate. It can be seen that these show well defined trends in direction in those parts of the field where significant plastic deformation occurred. As would be expected nq.such trends are observed in the region adjacent to the wedge-wave interface where little if any plastic deformation took place. Although in the case of extrusion the computer-aided method for calculating strainrates described above has been extended to the construction and stress analysis of slipline fields (CONNING et al., 1984) it has not so far been extended in this way to processes such as the wave formation process now under consideration. The rest of the analysis given in this paper is therefore essentially manual. SlipiineJield construction The starting point in the construction of the slipline field was to estimate from the original flow field (Fig. 2) the boundary of the zone in which plastic deformation
FIG. 5. Slipline
field and
directions
of maximum shear strain-rate construction of field.
(represented
by crosses)
used in
774
E. ht.
KOPALINSKY
ef
cd.
occurred. This was done by noting the points on the streamlines where deviations from the horizontal first occurred at entry to the plastic zone and where the streamlines once again became horizontal at exit from the zone. Further evidence for the start and end of plastic flow was obtained from noting the points where the transversals first deviated from the vertical at entry to the zone and assumed a constant slope at exit from the zone. In general these two sets of points were in close agreement. The construction of the boundary was started at the free surface at the point where the surface streamline first deviated from the horizontal. It was assumed that the boundary would be a slipline and would therefore meet the free surface at 45. in order to satisfy equilibrium. The points representing the start and end of plastic flow were then used together with the calculated directions of maximum shear strain-rate, which were taken as slipline directions, to construct the boundary slipline AI-Al7 given in Fig. 5. This was terminated at the wedge corner to the right of which no external loads were applied. Once the boundary slipline had been established two orthogonal sets of curves representing the sliplines were constructed within the plastic region using the calculated directions of maximum shear strain-rate. The sliplines were drawn so as to meet the free surface at the required angle of 45”. In the region adjacent to the wedgewave interface the calculated directions of maximum shear strain-rate showed no clear trends and it was assumed that the field would be similar to that in Fig. 1, i.e. made up of straight sliplines. The inclination of these sliplines to the wedge-wave interface was determined from the equilibrium relation cc+ cf, = 4 cos- ’ (t/k) where r is the shear stress along the wedge-wave interface and k is the shear flow stress of the deformed material in this region and the angles are shown in Fig. 1. Because at this stage of the analysis the value of s/k was not known it was decided to use the value of 0.25 obtained by CHALLEN et al. (1984) for similar test conditions. With this value of r/k the anticlockwise angular rotation of sliplines such as E5-El6 from the wedge face was determined to be 38”. As can be seen in Fig. 5 the shpline field constructed in this way had two fan regions emanating from the leading and trailing edges of the wedge-wave interface. For convenience these fans were terminated at stress singularities as shown although it was recognised that such singularities were not acceptable in hardening flows. This point is returned to later in the paper. In Fig. 5 the I-sliplines, i.e. those on which the shear stress exerts a clockwise couple, are designated by numbers and the II-sliplines by letters. The hodograph corresponding to the slipline field of Fig. 5 was next constructed in order to check how accurately this field represented the velocities from which it had been derived. The given information was taken to be the magnitude and direction of the velocity at entry Al-A9 to and exit A9-A17 from the plastic zone and the direction of the velocity along the free surface Al-19. No conditions were imposed regarding the velocity along the wedge-wave interface which could therefore serve as a check on the accuracy of the construction. The hodograph was obtained by the following step-by-step method which makes use of the condition that for volume constancy (rigid-plastic assumption) the rate of extension along sliplines must be zero. The construction was started at the point A2 with the velocity at this point represented by the line drawn from the pole to the point A2 in the hodograph (Fig. 6). The direction of the velocity at the point B2 was then taken as the slope of the free surface at this point (Fig. 5) and a line was drawn in this direction through the
Analysis
of an asperity
interaction
experiment
115
FIG. 6. Hodograph.
pole. To find the point B2 on this line and hence the magnitude of the velocity at this point the orthogonal image of the slipline A2-B2 was drawn in the hodograph as shown. In this way the velocity at B2 relative to the velocity at A2 was made to act at right angles to the slipline A2-B2 and a zero rate of extension along this slipline was obtained as required. Once the point B2 in the hodograph had been established, the point B3 and hence the velocity at this.. point were determined using the same principle of constructing the orthogonal images of the sliplines A3-B3 and B2-B3 to find their intersection point. By working in this way throughout the field the hodograph given in Fig. 6 was obtained. As can be seen, in constructing the hodograph the slipline elements have been approximated by their equivalent chords but for the field considered this would only have introduced small errors. It should be noted that the slipline field and hodograph given in Figs 5 and 6 respectively are those finally obtained after a number of adjustments had been made to the slipline field in order to get a better agreement with experimental velocities. The results in Figs 7 and 8 are given so as to make a more direct comparison
16
13 12 11 10 9 8 7 6 5
FIG. 7. Velocity
10
directions
obtained
15
from hodograph
20
(represented
by arrows)
25
and original
streamlines.
E. M. KOPALINSKYVI al.
20 Transversal
30
10
number (a)
Jj
p
z O.lz! ._ zl r”
0
I IO
I
20 Transversal
I
I
30
40
number
FIG. 8. Velocity magnitudes obtained from hodograph (represented by symbols) and from computer programme (represented by lines): streamlines and transversals are numbered as shown in Fig. 7: (a) streamline number 16; (b) streamline number 1.5; (cf streamtine number 14; (d) streamline number 13 ; (e) streamline number 12; (f) streamline number 11.
Analysis of an asperity interaction experiment
Transversal
number fc)
Transversal
number (d)
FIG. 8 (continued).
771
778
E. M. KOPALINSKYet al.
z aI O.l-a 2 ._ El s
0
I 10
I 20 Transversal
I 30
1 CO
I 30
I LO
number (e)
2 .g 0.25 > 6 4
O.l-
2 ._ En I 10
I 20 Transversal
number IfI
FIG. 8 (continued).
Analysis
of an asperity
interaction
experiment
779
between the velocities determined from the slipline field and the original data used in constructing the field. Velocity directions obtained from the hodograph and the original streamlines re-drawn from Fig. 3 are shown in Fig. 7, while the velocity magnitudes obtained from the hodograph and those obtained from the computer programme are given in Fig. 8. It can be seen that the slipline field in Fig. 5 represents the flow quite accurately over most of the plastic region. The greatest discrepancy is at the wedge-wave interface where the direction of flow should be at 20” (wedge angle) to the horizontal but as shown on the hodograph actually varies from 23.5” at H13 to 14.5’ at E16. In spite of this it was concluded that the slipline field represented the flow from the velocity viewpoint sufficiently accurately to warrant proceeding to the stress analysis. Stress anal.vsis In order to calculate the distribution of p throughout the slipline field it was necessary first to calculate the distribution of k. This was achieved by integrating the maximum shear strain-rate along streamlines with respect to time to find the maximum shear strain at a point. The plastic stress-strain relations for the material given in (3) were then used to find the corresponding k. Contours of equal values of maximum shear strain calculated using the computer-aided method described earlier are given in Fig. 9. In obtaining these values it was assumed that the slipline field defined the region where plastic deformation occurred and that strain-rates calculated from the digitized data which fell outside the field should be treated as noise and reset to zero before carrying out the integration. At this stage of the analysis the maximum shear strain yA, which defines the yield shear stress along the entry boundary Al-A9, was not known and in making the integration yAwas taken as zero. Therefore, the results in Fig. 9 represent the increases in shear strain above yA and to find the actual shear strain at a point it is necessary to add YAto the value obtained from Fig. 9. In the analysis ?A was determined by selecting its value to give a stress distribution along the
FIG. 9. Contours
of equal maximum
shear strain values
; numbers given on contours
are y x 100/d
E. M.
780
KOPALINSKY
ef a/
boundary slipline Al-Al7 which was consistent with the experimentally measured vertical force (N = 7953 N). The method used was as follows. The distribution of p along Al-A17, which as drawn in Fig. 5 is a II-slipline, was determined by starting at Al, where to satisfy the free surface condition p is equal to k and is compressive, and applying the appropriate equation of (2). In finite difference form this can be written as Ap = okay + (Akl~~)As~~ where the strain-hardening
(8)
term can be expressed as
(Ak/As,)Asz = (Ak/Ay)(Ay/As,)Asz.
(9)
In the above Ak/Ay is the slope of the stress-strain curve at the appropriate strain and Ay/As, is the rate of change of 1:with distance normal to the slipline. In making the calculations it was necessary to represent Al-Al7 by chord segments. It was found su~ciently accurate to do this by taking the nodes at slipline intersection points as natural dividers with As? taken as the distance between them. To start the calculations a reasonable value of yA was assumed and (3) were used to obtain the value of k along Al-A9 with y = yA and also the values of k along A9-Al7 with yA added to the appropriate strain values given in Fig. 9. In applying (8) the curvature term 2kll/was evaluated by measuring All/ from Fig. 5 (taken as the difference in rl/ between nodes) and an average value of k was used where this varied along the step considered. The strain-hardening term as expressed in (9) was determined by measu~ng Ay/As, from Fig. 9 and calculating Ak/A~ for the appropriate strain from (3). For each segment average values of A~/Asi and Ak/Ay determined from the values at the adjoining nodes were used. The step-by-step calculations were performed from Al along A l-Al7 and it was found that the curvature term gave a compressive increment to p while the hardening term gave a tensile increment. Once the distributions of p and k were determined for a given yA the forces acting on each segment were calculated and resolved in the horizontal and vertical directions. These components were then summed to give the total horizontal and vertical forces. Different values of Y,.,were used and it was found that a value of 0.022 gave a vertical force of 7719 N which was within 3% of the measured value of 7953 N. This was considered to be sufficiently close and therefore yA was taken to be 0.022. The distributions of p and k corresponding to this value of yAare given in Fig. 10. Also given are the vertical and horizontal forces calculated from these stress distributions together with the line of action of their resultant which was determined by taking moments of the forces acting on each segment about A 17. After the value of ?A had been evaluated this was added to the shear strains given in Fig. 9 to obtain the actual values of shear strain throughout the plastic zone which were then used to determine the corresponding values of k from (3). The values of k at each slipline node were then found by linear interpolation and these are given in Table 1. To complete the stress analysis p was calculated throughout the slipline field by applying the appropriate stress equilibrium equation in its finite difference form as described above. The calculations were made in two ways : first by starting at the boundary slipline A 1-A 17 where p was now known and working along I-sliplines such as A2-B2 and second by working along II-sliplines from the free surface Al-19 where
Analysis of an asperity interaction experiment
781
266 MPa
FIG. 10. Calculated stresses and forces on plastic zone boundary and wedge-wave interface.
p was equaf to k and therefore are given for each slipline node The values of the normal and be determined from the known
Al
A2
A3
A4
A5
A6
known. These values, termed p, and p2 respectively, in Table 1. _ shear stresses at the wedge-wave interface could now values of p and k at the interface together with the
A?
AG
A9
Al0
All
A12
Al3
A14
A15
A16
A17
E. M.
782
KOPALINSKY
et al
inclination of the sliplines to the interface. (Two values of normal stress were found for each point on the interface. one corresponding to p, and the other to pJ but these were very close and an average value was used.) These normal and shear stress values were then used to calculate the vertical and horizontal forces acting at the interface ; these are given in Fig. 10 together with the line of action of their resultant found by taking moments about A 17. To show the importance of allowing for strain-hardening in the analysis all of the above calculations were repeated using only the curvature terms in the stress equilibrium equations with the strain-hardening terms taken to be zero. In applying the curvature terms an average value of k was used as before. The results obtained in this way are given in Fig. 11 and Table 2.
DISCUSSION It can be seen in Fig. 5 that the region of the slipline field above the slipline D& D17 is similar to the perfectly plastic slipline field given in Fig. 1. This is perhaps not surprising when it is observed from the results in Table 1 that for much of this region k had reached its saturation value (in this region y > 1 and hence from (3) k is constant), The corresponding flow would therefore have been expected to approximate closely to a perfectly plastic flow. From the values of k in Table 1 it is clear that this
Al
Al I_
Al7 k
---._\_
1264
-1.
MPa
‘\.. \ “\ q_
FIG.
II.
Calculated
stresses
-
764 two.
and forces on plastic zone boundary and wedge-wave hardening terms taken IO be zero.
interface
with strain-
Analysis TABLE
of an asperity
interaction
experiment
783
2. Stresses (in MPa) at slipline nodes with strain-hardening terms taken to he zero ; lajwut same as in Table 1
o4 . 319 . zy) . 111 .. ... .. .........
. m . M .m .10W .*082 .10'18 .1079 .wra w ... ..-..................~~.....~
has been the case for the flow in the regions of the singularities at each end of the wedge-wave interface. The problem raised earlier of these singularities existing in a hardening flow, therefore, no longer applies. The region of the slipline field below the slipline D4-D17 can be looked upon as that resulting from the opening-up of the velocity discontinuity across the slipline ABCD in Fig. 1 in order to accommodate the strain-hardening which had taken place in the process. It is in this region, particularly along the entry boundary A l-A9 (Fig. 5), that the strain-hardening terms in the stress equilibrium equations have their largest influence on the calculated values ofp. Once hardening is introduced into the analysis by using visioplasticity techniques then the results obtained are, by their very nature, approximate. Nevertheless it can be seen in Fig. 10 that the agreement between the forces at the wedge-wave interface and across the boundary slipline AI-AI 7 obtained from the analysis is good with both sets of forces in reasonable agreement with the experimental forces (N = 7953 N; F = 3250 N). Another check on the accuracy of the stress analysis can be made from the results in Table 1. For an exact solution the values of p, and pz at a slipline node should be equal and the results in Table 1 show that this condition is approximately satisfied for most of the field. When the strain-hardening terms are neglected then the results in Table 2 and Fig. 11 show that the agreement is very poor. The question of whether or not tensile stresses can be shown to exist in part of the plastic zone once hardening effects are allowed for is now considered. It can be seen
784
E. M. KOPALINSKY et al.
from the results in Table 1 that in no part of the field does the hydrostatic stress become tensile. However. in the small region adjacent to the slipline nodes A8 and A9, on the boundary slipline, p does become significantly less than k (see also Fig. 10) which indicates tensile direct stresses in this region. The results show that the maximum value of tensile stress occurs near to A9 where the principal stress acting in a direction at 45” measured clockwise from the horizontal has a tensile value of approximately 22 MPa. With the errors involved in the calculations this value might be considered to be too smaI1 to confirm the existence of a tensile stress. However, had the value of p at A9 been caiculated by starting on the free surface at 19 with p = k = 264 MPa and working along I9-A9 then the tensile stress obtained at A9 would have been 43 MPa and not 22 MPa. It can therefore be argued that the errors in the stress analysis had led to the tensile stress at A9 being underestimated and should not be used to cast doubt on its existence. It can be concluded that once strain-hardening is allowed for in the stress analysis in the way described then a mechanism exists for the generation of tensile stresses in the wave formation process considered in spite of the compressive nature of the loading. This should certainly be taken into account in considering low cycle fatigue and fracture as possible wear mechanisms in those situations where asperity interactions can be modelled by the wave formation process as described in the introduction to this paper.
The authors wish to thank their former colleague STAN CONNING for help in applying the computer programme used in calculating strain-rates etc., and the Australian Research Council for their financial support of the project. -NCES
BLACK,A. J., KOPALINSKY,E. M. and OXLEY, P. L. B. CHALLEN, J.M. and OXLEY,P. L. B. CHALLEN,J. M., MCLEAN, L. J. and OXLEY,P. L. B. CHALLEN,J. M., HOCKENHULL,B. S. and OXLEY,P.L. B. CHALLEN,J. M., KOPALINSKY,E. M. and OXLEY,P.L. B. COLLINS,1. F.
1988
Wear 123,97.
1979
Weur 53,229.
1984
Proc. R. Sot. (Land.) A394, 161
1986
Wear 111,275
1987
In Prvc. ht.
1978
Me& Engrs ht. Con$ TribologyFriction. ~ubricotion and Weor-Ftyt_v Years On, 2, 957. In IUTAM Svrtp. Metal Forming Plasticity, Tutz-
ing (edited by H. LIPPMANN),p. 227. Springer, Berlin. CONNING,S. W. and OXLEY,P. L. B.
1989
In Plosticit_v nnd Modern Metal- Forming Technolog_v (edited by T. Z. BLAZYNSKI).Elsevier Applied Science, Amsterdam.
CONNING, S. W., FARMER, L. E. and OXLEY,P. L. B.
1984
Phil. Trans. R. Sot. (Lnnd.), A311,495.
Analysis of an asperity interaction FARMER. L. E. and CONNING. S. W. FARMER. L. E. and FOWLE. R. F. FARMER, L. E. and OXLEY, P. L. B. MOALIC, H., FITZPATRICK, J. A. and TORRANCE, A. A. OLVER, A. V.. SPIKES, H. A., BOWER, A. F. and JOHNSON. K. L. PALMER, W. B. and OXLEY, P. L. B.
experiment
1979
Int. J. Med.
1979 1976
Int. J. Mech. Sci. 21, 599. J. Strain Anal. 11, 26.
1987
Proc. Inst. Mech. Engrs 201, 321.
1986
Wear 107, 151.
1959
Proc. Inst. Mech. Engrs 173, 623.
Sci. 21, 577.
785