PREDICTION OF TIlE SUBSURFACE STRESS STATE IN A
MOVING SEMIINFINITE SOLID FOR LOADING CONDITIONS
REPRESENTING TOOLWORKPIECE
comACT IN ULTRAPRECISION
MACHINING
By
ANDREW CHARLES LOEBER
Bachelor of Science
Purdue University
West Lafayette, Indiana
1988
Bachelor of Science
Montana State University
Bozeman, Montana
1992
Submitted to the Faculty ofthe
Graduate College ofthe
Oklahoma State University
in partial fulfillment of
the requirements of
the Degree of
MASTER OF SCIENCE
December, 1996
PREDICTION OF THE SUBSURFACE STRESS STATE IN A
MOVING SEMIINFINITE SOLID FOR LOADING CONDITIONS
REPRESENTING TOOLWORKPIECE
CONTACT [N ULTRAPRECISION
MACHINING
Thesis Approved:
Dean of the Graduate College
ii
ACKNOWLEDGMENTS
I wish to express my thanks to my major adviser, Dr. Don Lucca, for his guidance during the
preparation of this work. I would also like to thank my other committee members, Drs. Christopher Price,
Gary Young and Larisa Volynets.
I also thank my parents for their support and encouragement.
iii
TABLE OF CONTENTS
Chapter
I. INTRODUCTION
II. ELASTIC LOADING OF A MOVING SEMIINFINITE SOLID
Elastic Stress Field
m. ELASTIC STRESS FIELD RESULTS
Introduction
Evaluation of Numerical Solutions Near the Surface
Maximum Principal Stress
Minimum Principal Stress
Maximum Shear Stress
Conclusions
IV. ELASTOPLASTIC LOADING OF A MOVING SEMIINFINITE BODY
Elastoplastic Stress Field
Residual Stresses by the Finite Element Method
V. ELASTOPLASTIC STRESS FIELD RESULTS
Verification of Residual Stresses
Elastoplastic Stress Field Results for Surface Loading
Conditions Measured in Ultraprecision Machining
Estimated Residual Stress
VI. CONCLUSION
BIBLIOGRAPHY
iv
Page
1
4
4
14
14
16
17
17
17
18
35
35
38
40
40
42
44
57
58
UST OF TABLES
Table
5.1 Workpiece Material Properties
5.2 Measured Force Components and Contact Lengths for Orthogonal Flyeutting
v
Page
43
43
LIST OF FIGURES
11. ToolWorkpiece Interface in UltraPrecision Machining 2
12. Idealization of the ToolWorkpiece Interface as a Sliding Indentation Problem 3
21. Geometry of an Inclined Concentrated Load ~
31. A Guide to the Verification of the Elastic Stress Fields 15
32. Comparison of Closed Fonn and Numerical SOlutions for the Maximum Principal
Stress for the Elastostatic Case with Elliptical Loading, qo= 1I3Po 19
33. Comparison of Closed Fonn. and Numerical Solutions for the Minimum Principal
Stress for the Elaslostatic Case with Elliptical Loading, qo=113Po 20
34. Comparison of Closed Fonn and Numerical Solutions for the Maximum Shear
Stress for the Elastostatic Case with Elliptical Loading, qo=I/3Po 21
35. Maximum Principal Stress fOT the Elastostatic Case with Elliptical Loading, Qo= 113Po 22
3~. Maximum Principal Stress for the Elastostatic Case with Constant Loading, qo= 1I3Po 23
37. Maximum Principal Stress for the Elastodynamic Case with Constant Loading, qo= l/3Po 24
38. Comparison oft.he Maximum Principal Stress for the Elastostatic, Elliptical and Constant
Load Cases, and the Elastodynamic, Constant Load Case, qo=1/3Po 2S
39. Minimum Principal Stress for the Elastostatic Case with Elliptical Loading, qo= 1/3Po 26
310. Minimum Principal Stress for the Elastostatic Case with Constant Loading, qo=1I3Po 27
,
311. Minimum Principal Stress for the Elastodynamic Case with Constant Loading, qo=1/3Po 28
312. Comparison of the Minimum Principal Stress for the Elastostatic, Elliptical and Constant
Load Cases, and the Elastodynamic, Constant Load Case, qo= 1/3Po 29
313. Maximum Shear Stress for the Elastostatic Case with Elliptical Loading, qo=I/3Po 30
314. Maximum Shear Stress for the Elastostatic Case with Constant Loading, qo=1/3Po 31
315. Maximum Shear Stress for the Elastodynamic Case with Constant Loading, qo=1/3Po 32
316. Comparison of the Maximum Shear Stress for the Elaslostatic, Elliptical and Constant
Load Cases, and the Elastodynamic, Constant Load Case, qo=l/3Po 33
vi
317. A Contour of the von Mises Equivalent Stress, Normalized by the Yield Strength, for
Various Speeds
41. Variation of Sliding Directional Residual Stress as a Function of Depth Under a
Moving Asperity
51. Comparison of the Results from the Corrected Program Using the MerwinJohnson
Method with the FEM Solution
52. Contours of the von Mises Equivalent Stress, Normalized by the Yield Strength,
for A16061T6 for O.OllJ.ffi Uncut Chip Thickness
53. Contours of the von Mises Equivalent Stress, Nonnalized by the Yield Strength,
for AI 6061T6 for 0.llJ.m Uncut Chip Thickness
54. Contours of the von Mises Equivalent Stress, Normalized by the Yield Strength,
for TeCu for O.OllJ.m Uncut Chip Thickness
55. Contours of the von Mises Equivalent Stress, Normalized by the Yield Strength,
for TeCu for 0.llJ.m Uncut Chip Thickness
56. Contours of the von Mises Equivalent Stress for a Reduced Step Size, Nonn.alized
by the Yield Strength, for AI6061T6 for O.OIIJ.ffi Uncut Chip Thickness
57. Nonnalized Cutting Directional Residual Stresses for AI606IT6 Predicted for
O.OIIJ.ID Uncut Chip Thickness
58. Normalized Cutting Directional Residual Stresses for Al 6061T6 Predicted for
O.Ij.Ull Uncut Chip Thickness
59. Nonnalized Cutting Directional Residual Stresses for TeCu Predicted for
O.OIf.Ltn Uncut Chip Thickness
510. Nonnalized Cutting Directional Residual Stresses for TeCu Predicted for
O.llJ.m Uncut Chip Thickness
511. Nonnalized Cutting Directional ResiduaJ Stresses for AI 6061T6 Using
Uncorrected Stress Gradients
512. Nonnalized Cutting Directional Residual Stresses for AI6061T6 Using
Uncorrected Stress Gradients
vii
34
39
41
46
47
48
49
50
51
52
53
54
55
56
E
e
•• ;.p •
e ij, I;"ij, eij
f(~), ['(s)
G
k
If
P,Q
Po, qo
S;j
v
x,y,z
y
(eJ,
NOMENCLATORE
dilatation wave speed
shear wave speed
Young's modulus
dilatation
elastic, plastic and total incremental strains
cutting and thrust forces
Fourier integral transform and its inverse
shear modulus
yield strength in sbear
slider halfcontact length
contact length offlank face
vertical and horizontal force per unit depth
maximum nonnal and tangential surface stresses
distances from loading point in concentrated force model
deviatoric stress components
uncut chip thickness
speed of semiinfinite body
incremental plastic work per unit volume
Cartesian coordinates
uniaxial yield strength
residua! normal strain
stress function
residual shear strain
viii
v
p
(crij)r
(crij)'r
Lame's constants
coefficient of friction
Poisson's ratio
included angles in concentrated force model
mass density
maximum and minimum principal stresses
stress on face i in direction j
residual stress
pseudoresidual stress
maximum shear stress
dimensionless coordinates for x, y, z
ix
Chapter 1
Introduction
There has been significant interest of late in developing a better understanding of the process
mechanics which govern the removal of material by cutting with su1:>micrometer depths of
cut. Recently both experimental [11] and theoretical [101 studies of the resulting force
system in the ultraprecision machining of ductile materials have been reported. It has
been observed in the experimental study of ultraprecision machining at su1:>micrometer
uncut chip thicknesses that the dominant length scale may become the contact length at
the toolworkpiece interface [111. It has been further observed that the cutting force is
much greater than the thrust force at these uncut chip thicknesses, so that sliding and
plowing, rather than chip formation, may be the dominant energy dissipative processes. As
a consequence, shearing in the shear zone and rake face friction may possibly be neglected
in the idealized model of the ultraprecision machining process. For such a case, a sliding
indentation model of the toolworkpiece interaction may be appropriate. Figure 11 is a
schematic of the toolworkpiece interface under the conditions described above. Here, to is
the uncut chip thickness, and l f is the contact length at the toolworkpiece interface.
Since the tool is much stiffer than the workpiece, and the length scale of the workpiece is
much larger than its interface with the tool, the sliding indentation model may be idealized
as a rigid slider (tool) on an elastic/elastoplastic, semiinfinite body (workpiece). Figure
12 shows this model as a stationary slider of length 2l, where l is the halfcontact length,
with a semiinfinite body moving in the negative xdirection.
This model will be used to determine the elastic stresses in a moving semiinfinite body
1
Workpiece
i Uncut Chip Thickness, to
Figure 11: ToolWorkpiece Interface in UltraPrecision Machining.
under a slider exerting a constant surface pressure. The solution of the elastic stresses will
be used as the initial conditions for a numerical model used to determine the elastoplastic
stress field and residual stress for several experimentally measured loading conditions previously
rePOrted IIIJ. These elastic stresses, as well as the elastoplastic and the resulting
residual stresses, will be calculated by a modified version of a previous, nonworking version
(the version available for use in this study) of the FORTRAN program employed, but not
included in [13]. To calculate the elastic fields correctly, it was necessary to change the
program so that all constants are calculated within subroutines, rather than in the main
program.
The present study was conducted to reconcile the residual stresses reported in 113J using
the MerwinJohnson method with the FEM results reported in 116]. The expressions used
for the elastic stress gradients in the nonworking program yielded the same residual stresses
as [13]. After modification of those expressions, which will be discussed later, the residual
stress solution agreed with the reported FEM solutionI16].
Previously reported solutions [14] for the elastic stress fields for a stationary elastic
semiinfinite body with an elliptical surface load will be used to verify the stress fields for
a stationary elastic semiinfinite body with a constant load. At a low sliding speed, the
dilatation and shear wave speeds are small relative to the sliding speed, so dynamic effects
2
 21
V .........1
SemiInfinite Body
Figure 12: Idealization of the ToolWorkpiece Interface as a Sliding Indentation Contact.
are negligible. For this case, the elastodynamic stress fields with a constant pressure can
be compared to those of a stationary semiinfinite body with a constant pressure.
According to Saint Venant's principle, given different surface pressure distributions, the
overall elastic stres~ fields should be the same at points far away from the load as long the
total applied force and the geometry of the body remain the same. Therefore, the numerical
solution for the elastostatic stress fields with constant pressure may in turn be verified by
comparing it to the elastostatic fields Wlder an elliptical surface pressure, for which there
is a closedform solution [14].
3
Chapter 2
Elastic Loading of a Moving
SemiInfinite Solid
2.1 Elastic Stress Field
This chapter presents the solutions for the elastostatic stress fields in a semiinfinite body
moving underneath a rigid slider exerting an elliptical and a constant surface pressure, and
the elastodynamic stress fields underneath a rigid slider exerting a constant surface pressure.
2.1.1 Elastostatic Stress Field
Concentrated Surface Load
Figure 21 shows a semiinfinite elastic solid loaded by a concentrated line indenter which
is infinitely long in the direction perpendicular to the x and ydirections. The force per
unit length has vertical and tangential components P and Q, respectively. An Airy's stress
function, ¢, may be expressed as [13]:
(2.1 )
where Ti and Oi , i = 1,2 are distances hom the loading point and included angles from the
direction of loading to the point of interest, respectively. Using the geometric relations of
4
p
Q
Figure 21: Geometry of an Inclined Concentrated Load.
Figure 21,
x y (h = arctan  I ()2 = arctan 
y x
¢ can be expressed in terms of x and y. The stress fields can now be determined as:
Hence,
5
(2.280)
(2.2b)
(2.3)
(2.480)
(2.4b)
(2.4c)
where PI = Q./P.
Distributed Surface LoadElliptical Surface Distribution
The elastostatic stresses due to a distributed surface load can be formulated using the stress
fields for a concentrated load. Equations 2.4 are applied to a differential length c1{ at x = ~,
integrating over the contact length. For an elliptically distributed surface load over a length
2l, the stresses acting at the boundary of the semiinfinite body are114], [16]:
a for I x I> 1
(Jyy =
( x
2
Po 1 IT) t for Ixl~l
a for I x I> 1
(Txy =
( x
2
) qo IIT ~ for I x I~ 1
(2.5a)
(2.5b)
where Po and qo are the maximum normal and tangential stresses acting at (x, y) = (0,0)
and 21 is the contact length.
Rewriting Equations 2.4 for a concentrated load acting at x = ~ and integrating with
respect to ~ from I to 1 gives the stress fields due to the elliptically distributed load:
6
Integrating Equations 2.6, the stresses in the semiinfinite solid due to the elliptically distributed
load are [14]:
flO I( 2 2 2 X 2 2 2 X Po [2 + 2x2 + 2y2 27r
(7xx =  2x 2[ 3y ),p+27r+2(1 x y )w]y[ '113X1Pl
1f [ 1 7r 1 1
(2.7a)
Po [2 + 2x2 + 2y2 27r y[ 'it    3x,p]
1f [ [
(2.7b)
(7xy = qo [([2 + 2x 2 +2y2) Y'l1  21fY  3xy,pI  Poy211'
7r 1 [ 1f
(2.7c)
The maximum normal and tangential stresses at the boundary, Po and qo , are related to
the resultant force per unit length P and Q of Equations 2.4 as:
(2.7f)
(2.7e)
(2.7g)
(2.7d)
where
2P
po=1ft
(2.8a)
7
2Q
qo=1rl
(2.8b)
The 0'xx component of the stress at the botuldary y = 0 can be derived by setting y = 0 in
Equation 2.6a and evaluating the integral [14]. This stress is given by
2qo [T  (~  1) ~ ] for x > l
O'xx = 2qo [T+ (;Y:  1) ~ ] for x < l (2.9)
for I x I:::; 1
For any finite normal elongation, the normal strain f.zz is zero for an infinite width. Therefore,
plane strain conditions apply, and:
(2.10)
8
Distributed Surface LoadConstant Surface Distribution
Although an elliptical stress distribution is often assumed to exist, even between elastoplastic
solids, the actual distribution tends to be more uniform across the interface in
elastoplastic solids when yielding occurs at the contact area [5], [7]. Therefore, a constant
surface pressure distribution is probably more appropriate to the toolworkpiece interface
in ultraprecision machining, if yielding actually occurs at this interface.
Expressions similar to Equations 2.6 can be formulated using the same approach as for
an elliptical distribution. The surface conditions for a constant distribution are:
(2.lla)
(2.l1b)
0 for 1x I> l
O'yy =
Po for Ix I:::; l
0 for 1x I> l
O'XII =
go for Ix I:::; l
The equations for the stress fields are:
(2.12a)
(2.12b)
(2.12c)
and ()zz = v(()xx +a yy) for plane strain.
The above equations were solved by Seo [13] using a tenpoint Gaussian quadrature
technique; however, results for au, a yy and axy were not reported. The same method was
used in the modified code of the present work to determine these stresses. These results are
discussed in the next chapter.
2.1.2 Elastodynamic Stress Fields: Distributed Surface LoadConstant
Surface Distribution
For the case of a semiinfinite body moving with velocity V in the negative xdirection
beneath a stationary, constant surface load (Figure 12), a Fourier integral transform method
was used to determine the elastodynamic stress fields in [13J and in the present study. This
method is often used to simplify the equations arising in these problems [1], [9J and have
been used in the a..'1alysis of anisotropic bodies [2], [8] and sliders of varying shapes [3], [4].
The body's speed is assumed to be much less than the Rayleigh wave speed in the body.
That is, elastic surface wave effects are neglected. The body is assumed continuous and
homogeneous.
The Fourier integral transform and its inverse transform are defined as:
!(E,) = 2
1 ;00 J'(s)eiS~ds
7f 00
9
(2.13a)
(2.13b)
where i = yCI.
Navier's equations of motion, neglecting body forces, are:
where ). and J.L are Lame's constants and e is the dilatation. The boundary conditions
are that the surface normal and shear stresses are the same as the applied surface normal
and shear pressures, that is, those of Equations 2.11. The integral transform is applied to
Equations 2.14 and the boundary conclitions. Ordinary differential equations then result,
which are solved in the transformed space. The inverse transform is then taken and the
complex part rejected. The stress fields are then, in nonclimensional form (~ = TI T'/ = ~):
OOij _ 02Ui ax.  Po 8t2
J
(2.14a)
(2.14b)
(2.15a)
2 2 [ 1 lCX> sin S k 100 +(M + 2k ) (j + ;)  cos s{ e ST/ds  2J.L1 sin s s.ms{ ekIl'f/]ds )
J 0 s 0 s
(2.15b)
2 [ 1 100 sin s k 100 (2  M) (j + ;)  coss{ e ll'7ds  2J.Lf sin s s.m8~ ekBTJd]s )
J 0 s 0 s
2 1 { 2 100 O{'7=C [(j+;) (2M )J.LI sicnosss{eJIlf. /ds+2k looo sinssms.~eJIl.'7}ds
7fl J 0 s 0 s
(2.15c)
{ 100 sin S k . 1 fooo sin s.  k } +2k 2J.Lf coss{eST/ds(J+;) sms{e Il'f/ds J
o S J 0 s
10
where
(2.15d)
Stress Gradients
!vI = %.' N = ~ and PJ = ~ and Cd and Cs are the dilatation and shear wave speeds in
the body. These integrals are evaluated using a tenpoint GaussLaguerre method, because
the integration limits are 0 and 00.
aO"f,f, 2 a100 sins· 2 a100  = (4k coss~eJSf/ds  2(2  M )J.LJ sinssm. s~ e_J.6f/ds aE. 1rG1 a~ 0 s aE. 0 s
(2.16a)
(M2 +2k2) [U+. 1:)a loOO sincS oss~ekSf/ds2J.La/ 1'XJ sinss.ms~ek6f/ds] )
J 8~ 0 s aE. 0 8
(2.15e)
M2
).2 __ 1 _ M2, k2 1
=  N2'
au 2 a100 sinS· 2 a100 sin8. _ . .!TI = (4k cossr. eJs"ds+2(2M )J.Lf sms~ e J6f/ds
a~ 1fG1 af. 0 s aE. 0 s
(2.16b)
2 [ 1 a100 sin S k a100 +(2 .M ) (j + : )a  cossr. e SJ)ds  2J.LIa sin Ss.m s~ eks'1ds] )
J r. 0 s r. 0 S
As they will be used to determine the elastoplastic stress field and the residual stresses, the
stress gradients (with respect to the sliding direction) of the elastodynamic stress fields were
also verified. The central difference technique, in which the quotient of two differences is
used to approximate the value of a derivative, was used to check the stress gradients. Using
this technique, the stress gradients can be calculated from the stress fields alone, providing
a independent check. This technique was used for a few points using the elastic stress field
results and compared to the gradients as obtained from Equations 2.16. They did not agree.
In the unmodified program the expressions for the stress gradients are:
11
8a~TJ = [2(j+""71) { (2  M 2 )l£8f100 sms coss{ e]l.J'1ds + 2k8 [ si.ns sin s~ eJl.JT/}ds
8~ 1rGl J 8{ 0 s 8{ 0 s
(2.16c)
{
8 100 sins k 1 8100 +2k 21£f8  coss{ e lJTJds +(j +""7) sins sins~ eklJTJds } I
.; a S J 8{ a s
They should be:
8a 2 8 100 sins· 2 8 100 .!!!l = (4k coss{ e]81Jds  2(2  M )J.Lf sins sins'; eJS.TJds
8f. 1rGl 8f. 0 s 8{ 0 8
(2.17b)
(2  M 2 ) [u +~)~ roo sins coss{ eklJTJds _ 21£/~ roo sins sins{ eklJTJds])
J o{ Jo s 8~ Jo s
8a~ 2. 1 { 2 8 100 sin s . 8 100 sin s . .} __TJ =[(J+""7) (2M )1£/ coos{e]l1T/ds+2k sms~e]8TJds
8{ 1rGl J 8.; 0 s 8{ 0 s
(2.17c)
{
0 100 sin s k . 1 8 100 sin s . ks} +2k 2p/ coos'; e 81Jds  (J + ""7) sms{ e f/ds J
8{ 0 s J 0'; 0 s
The expressions in Equations 2.17 are the negatives of their respective expressions in
Equations 2.16.
In this chapter the equations used for the elastostatic stress fields in a semiinfinite body
beneath a rigid slider for elliptically and uniformly distributed loads have been developed.
The equations for the elastadynamic stress fields beneath a slider with a constant load have
also been developed, and the necessary corrections in the program for the stress gradients
far this case were noted. In the next chapter the stress field results for the two elastostatic
12
cases will be used to verify the stress fields for the elastodynamic case. In a later chapter,
the elastoplastic stress fields and residual stress results for loading conditions resultant from
the ultraprecision machining experiments of [13] will be presented using the corrected stress
gradients.
13
Chapter 3
Elastic Stress Field Results
3.1 Introduction
This chapter presents the elastic stress field results in the form of maximum and minimum
principal stresses and the maximum shear stress for the three cases developed in the previous
chapter: an isotropic, elastic semiinfinite solid loaded by a' stationary rigid slider exerting
an elliptical and a constant pressure distribution, and a stationary rigid slider exerting a
constant pressure distribution on a moving semiinfinite body. According to Saint Venant's
principle, given the same total loading, the stresses should be the same far from the point
of loading, though they would differ near the applied load. Also, given the same loading,
the effect of a moving semiinfinite solid should be small if the dilatation wave speed and
shear wave speed are small. An abbreviated sequence to the verification procedure for the
elastic stress fields is presented in Figure 3l.
All results are for the case l]o=~po,where Po and qo are the maximum normal and
tangential stresses at the surface tllldemeath the slider. The surface stresses are normalized
with respect to PO. Dimensions are normalized with respect to the halfcontact length l of
the slider (c; = xll I TJ = yll). They are plotted for the range of e= [3,31 and TJ = [0,4J
using a step size of 0.05 in both directions.
For all three cases the surface stresses a1/11 and axy are known. However, for the two
constant loading cases there are singularities in the surface stresses at either end of the
slider. Therefore equations similar to Equations 2.9 for the surface stress a xx can not be
Elastostatic, Elliptical Loading
Closed Form Y$. Numerical
~
Elastostatic Elastostatic
Elliptical loading Y$. Constant Loading
(Closed Form) (Numerical)
~
Constant Loading
Elastostatic Elastodynamic
Y$.
(Numerical) (Numerical)
~
I Check Dynamic Effect
Low Speed vs. High Speed I
Figure 31: Sequence Used to Verify the Elastic Stress Fields.
The stress fields for the elastostatic, constant load case are calculated using tenpoint
Gaussian quadrature, because it is suited to integrals with finite limits. Gauss's formula is
(3.1)
(3.2)
(3.380)
Tmax =
the plots for the constant loading cases were prepared without the surface (7] = 0) data.
The maximum and minimum principal stresses are given by
used. Since (Jxx (and therefore (Jzz because of plane strain) is undefined at I ( 1= 1, 7] = 0,
errors are introduced into the numerical stress field solutions at the surface. For this reason
and the maximum shear stress is
15
where Xi is the i th zero of the Legendre polynomial Pn(x) and
(3.3b)
For an arbitrary interval [a,b]'
l b b  a n
f(y)dy =  L: Wi!(Yi)
a 2 i=l
(3.3c)
and
(b a) (b + a) Yi = 2 Xi + 2 (3.3d)
The stress fields for the elastodynamic, constant load case are calculated using tenpoint
GaussLaguerre quadrature, because it is suited to integration limits of [0,00]. The
GaussLaguerre fonnwa is
16
Since near the surlace the numerical solutions are inaccurate because O"~~ (and therefore
0"(( because of plane strain) is illldefined at I { 1= 1, 1] = 0 for the constant loading cases,
Equations 26 were evaluated using tenpoint Gaussian quadrature to compare with the
results of Equations 27 through 210, the closedform solution for elliptical loading, so as
to see at what depth the numerical solutions become valid. The error in the numerical
solutions for the elliptical distribution is in general small. Underneath the slider, errors
are within 4% at a dimensionless depth of 1] = 0.35 for both principal stresses and the
maximum shear stress. Near the edges of the slider there are small zones where there are
also errors. For the maximum principal stress the error does not become less than 1%near
the trailing edge illltil { = 1.3 at a depth of 1] = 0.05, for the minimum principal stress and
(3.4a)
(3.4b)
rOO n
Jo ex f(x)dx = ~ wi!(xt}
where Xi is the i th zero of the Laguerre polynomial Ln(x) and
3.2 Evaluation of Numerical Solutions Near the Surface
the maximum shear stress on the trailing edge until ~ = 1.35 at 1] = 0.25 and ~ = 1.05
at 1] = 0.1, respectively. These stress fields are compared in Figures 32, 33 and 34.
3.3 Maximum Principal Stress
Figures 35, 36 and 37 show the contours of the maximum principal stress,which lies in
the ~  TJ plane, for the elliptical and constant distribution stationary cases and the constant
distribution dynamic case. Under the slider these stresses are compressive (negative values),
with semicircular contours shifted somewhat toward the leadingedge, but the contour of
any given stress extends somewhat deeper into the body for the constant distribution cases
than for the elliptical case. All three show tensile stresses behind the slider. The magnitude
of the stresses is greatest at the surface near the slider and becomes smaller away from the
slider and deep within the body, as is consistent with the boundary conditions. Figure 38
shows a comparison of the three solutions for two contours.
3.4 Minimum Principal Stress
Figures 39, 310 and 311 show the contours of the minimum principal stress for the three
cases, which also lie in the ~ 1] plane. As is the case with the maximum principal stress, the
stresses are compressive under the slider with semicircular stress contours shifted toward
the leading edge. As with the maximwn principal stress contours, the contours extend
somewhat deeper into the body for the constant distribution cases than for the elliptical
case. Figure 312 shows a comparison of the three solutions for two contours.
3.5 Maximum Shear Stress
Figures 313, 314 and 315 show the maximum shear stress contours for the three cases.
As with the principal stresses, under the slider these contours are generally semicircular
and slanted toward the leading edge and extend somewhat deeper into the body for the
constant distribution cases. However, a few of the contours fold onto themselves somewhat
under the sljder and at the trailing edge. This appears not to be an artifact of the numerical
17
mod@ling, since it is ohsen'ed in the plot fur the clt..sed ~ dliptieal dLwibution e~ &S
"""elias the I;\>,'O COIbtant distribution ca.o:es. Figure 316 stK..'lWSQ Ct..~iwn of h~ t~
solut,ions for tWQ Ct..utours.
3.6 Conclusions
The stress contours for the elliptical and constant distribution stationary C~ ~ similar
and become more alike away from the slider 3Ild deeper into the body} as they should fur
the s.'UUe total load. So it is concluded that the et..>nstant distribution solution is ~t.
For the dilatation and shear wa",oe speeds invuhed in. the co:nstant distributiQn dynamic
ca..."€~ there should be very litt.le dynamic ~t. The contOUl'S fur the coust4mt distribution
stationary and dynamic cases are nearly identical. To demonstRite that there troly is a
dynamic effect, a contour of the von Mise> equi"\ralent stress has been plottoo. in Figu:ru
317. For the contour plotted the dynamic effect is only present at "'''eI}" high spe«ls.. It is
therefore concluded that the dynamic sol'ution is also consct.
18
O....,.........,...~~_..,.,....,....30...,.."_+~...,,,..~O"'__,
0.2
0.2
0.4
o
0.6
0.8;
0.2
Closed Form: Numerical:
1 o
Figure ~2: Comparison of Closed Form and Numerical Solutions for the Ma.ximum Principal
Stress for the Elastostatic Case with Elliptical Loading, qo = 1/3Po.
19
"
o
0.2
0.4
0.6
0.8
Closed Fam:
NunericaI:
1.0
1
, I
o
Figure 33: Comparison of Closed Fonn and Numerical Solutions for the Minimum Principal
Stress for the Elastostatic Case with Elliptical Loading, qo = 1/3po.
20
0.4
0.6
0.8
1
0.2
Closed Form:
Numerical:   
Figure 34: Comparison of Closed Form and Numerical Solutions for the Maximum Shear
Stress for the Elastostatic Case with Elliptical Loading, qo = 1/3Po·
21

11 2
3
4 L....:....~l....l.....:.....:.:......L.'..:....:...''..l.......l'...l.~:..... '','.:.':..........'~'"""":
3 2 1 a 1 2 3
E
Figure 35: Maximum Principal Stress for the Elastostatic Case with Elliptical Loading,
qo = 1/3po.
22
Or==!..~....3.::I.~
0.
TJ 2
3
2 1 a 2 3
4 L...:....:....:....:...':...1__l..__l..__l............L..:..':..'1..'__l..''~..........__l..__l......L__l..__l..........................L......:..__l..__l..__l........J
3
Figure 36: Maximum Principal Stress for the Elastostatic Case with Constant Loading,
qo = 1/3po.
23

Or~>'''..3..~:L.__,
0.1
o 0.1 a
11 2
3
2 1 a 2 3
4L..'__.l.._''..l.'_''~.l.._.l___'___.l__J._.l.._.l.._.l.._.l.._J._.l.._.l.._"___1._1._.L._'__'___'
3
Figure 37: Maximum Principal Stress for the Elastodynarnic Case with Constant Loading,
qo = 1/3po.
24
o 2 3
25
.... r' trf
Elastostatic, Elliptical Load:
Elastostalic. Constant Load:
Etastodynarnic, Constant Load:
2 1
O.!..r;~~:~__=;_:::..___;:77_Jr__;_,
3
4
3
Figure 38: Comparison of the Maximum Principal Stress for the Elastostatic, Elliptical
and Constant Load Cases, and the Elastodynamic, Constant Load Case, qo = 1/3po.

3
2 1 0 2 3
~
4 ll..l..l..~~':"""""~'~':'..!.':,.':,.'..:.'l....J:"""':""....L.........l...l"":"''''':''''''':''''''':'''''''"
3
Figure 39: 1vlinimum Principal Stress for the Elastostatic Case with Elliptical Loading,
(}o = 1/3po.
26
4
3
I ,
2 1
I ,
a 3
Figure 310: Minlmmn Principal Stress for the Elastostatic Case with Constant Loading,
qo = 1/3po.
27
2 1 o 2 3
4 L.....~~~~~'.:....:......l',I"':"L.:'':::''':'......I.:..:..:..:..!..'l.',,L....L:.:..:..:..J
3
Figure 311: Minimum Principal Stress for the Elastodynamic Case with Constant Loading,
qo = 1/3Poo
28
3
..........
2
• I ••
"""'  
.(l.6
a
Elastostatic, Elliptical Load:
Elastostatic, Constant Load:
Elastodynamic, Constant Load:
2 1
29
r
./.
/0'
/'
/."
/:
/ ..
/ :
I
/
/
I
I
I
I
\
\ .
\ '
\ '
\ .
O.....,......r."~":~"""""...""""::::':__.._____,
3
4
3
" 2
Figure 312: Comparison of the Minimum Principal Stress for the Elastostatic, Elliptical
and Constant Load Cases, and the Elastodynamic, Constant Load Case, qo = 1/3po.

0.2
3
2 1 0 2 3
4 L..l..l.l...l...'l...''~I.___J~':...._':...._:....:...',,,,,,,,,,,,',','.....'.......
3
Figure ~13: Maximwn Shear Stress for the Elast05tatic Case with Elliptical Loading,
qo = 1/3po.
30
,>
01
03
Q2
O...........LJ..~~"_:__,
01
3
'1 2
o 2 3
~
2 1
4L...L...L...L..U.L.."'..L.."'..L.....L...L..l...l...l...l....L"'"'"'"'""""""".l"'"""
3
Figure 314: Maximum Shear Stress for the Elastostatic Case with Constant Loading,
qo = 1/3Po·
31
2 3
0.2
2 1 o
3
41.''.l...".1.''''.1.''.1..1..1.'''''''''""""''
3
Figure 315: Maximum Shear Stress for the Elastodynamic Case with Constant Loading,
qo = 1/3po.
32

Figure 316: Comparison of the Maximum Shear Stress for the Elastostatic, Elliptical and
Constant Load Cases, and the Elastodynamic, Constant Load Case, qo = 1/3po.
1 2
0.2
o
~
Elastostatic, Elliptical Load:
Elastostatic, Constant Load:
Elastodynamic, Cons.tant Load:
2 1
~ .. , , :'"}',
'" t
,', ./.,: ~'
,/ ; i
/: :\
/ ,: :\
/ .. '.\
I .' '~\
/ ..' ',\
/.. .,\.
I
0.1 : '~
I ' ' :' , ., /7
\ ' '.. _.....' .. . ....
\ "
\ "
\ "
\ "
4
3
3
T1 2
33
Or....,.....L~~...,;:L~__.....:::___.....,
"
2 3
\
I
)
j
I
I
/
/
/
/
0.3
v = 102. 104 mmls:
V = 106 mmls:

1 o
1
(
I
(
\
\
\
\.
'
"' ........
2
3
4L..'''J:::'J'''''............:.L.'"'''''....:.....:.....:.....:...................'''''"'
3
TJ 2
Figure 317: A Contour of the von Mises Equivalent Stress, Normalized by the Yield
Strength, for Various Speeds.
J)
34
Chapter 4
Elastoplastic Loading of a Moving
SemiInfinite Body
4.1 Elastoplastic Stress Field
4.1.1 Method of Merwin and Johnson
The elastoplastic stress field is detennined by solving the PrandtIReuss equation using a
RungeKutta method. The solution method is that of [12], as modified by [6j. A description
of this approach, as presented in [161, is summarized below.
When the subsurface stress state reaches the yield condition, a plastic stress and strain
relation may be applied. It is assumed the material behavior is elasticperfectly plastic,
that plane strain conditions exist, and the elastic deviatoric stress and strain are the initial
conditions for the plastic stress and strain fields.
The total incremental strain is the sum of the incremental elastic strain and plastic
strain:
(4.1)
The incremental plastic strain is obtained from the LevyMises equation:
(4.2)
35

.p
where w is the incremental plastic work per unit volume, Sij are the deviatoric stress
components, and k is the yield strength in shear.
Using Hooke's law for the elastic strains, and the LevyMises equation for the plastic
strains, the PrandtlReuss equation in terrns of the incremental deviatoric stress and strain
IS;
(4.3)
where G is the shear modulus. The energy rate per unit volume can be represented by the
. . p
plastic energy, i.e., w=w . The deviatoric incremental stress is then:
(4.4)
It is convenient to transform the time rates of change to gradients with respect to ~ as
follows:
d (. . .. ,J:)) a(. . ..,.J:))
 Sij, eij, 7J.T = V a Sij, eij I 7J.T
dt ~
(4.5)
At steady state, the time derivatives in Equations 4.5 vanish, so the speed V is eliminated
from the equation.
4.1.2 Residual Stresses and Strains
The incremental deviatoric stress can be fOWld with the RWlgeKutta method using the
elastic stress fields starting from the first yield point at a given depth in the body. This
stress field is used to calculate the stress field for the next point. At the end of a step
at a given depth, the calculated stress may not satisfy the equilibrhun condition, which is
then satisfied by introducing residual stresses. For the calculation of residual stresses, Sub's
procedure was used [6], [16):
1. Initialize residual stresses and strains to zero,
2. Calculate elastic stresses along ~ax1s at fixed TJ.
3. When the stress state reaches the von Mises yield criterion, the PrandtlReuss equations
are used to calculate the stresses for the subsequent point, assuming the total strains
are the same a3 that gi'~ by the elastic solution.
36
)..
4. The PrandtlReuss equations are integrated using a fourthorder RungaKutta scheme.
Starting from the first yield point, the stress rates are found from Equations 4.5, using the
already calculated stress gradients. These stress rates are used to predict the stress components
of the next point.
5. If the yield criterion is not satisfied or the rate of plastic work becomes negative,
plastic deformation ends. The stress at the next point is calculated from the elastic equation.
6. The final calculated state of stress violates the equilibriwn condition. At the end of
each iteration, the stresses are relaxed elastically to satisfy equilibrium, and residual stresses
are calculated.
7. Steps 26 are repeated for the same point using the residual stresses from the previous
iteration until a steady state is reached, where the residual stresses and strains are not
significantly different from those of the previous iteration.
8. Step 7 is continued in the 7]direction.
The possible residual stress components are independent of ~ due to the nature of the
sliding problem and can be written as functions of 7] alone:
The equilibrium equations for residual stress are:
(4.6)
(4.7)
(4.8)
Substituting Equations 4.6 into Equations 4.7,
where O2 and 0 3 are constants.
The boundary conditions for the residual stresses are:
(4.9)
37
From these boundary conditions, O2 and 0 3 are zero, and the possible residual stresses
for plane strain are:
(4.10)
where ( is the dimensionless width (i.e., ( = z/l). Since the equilibrium and residual stress
boundary conditions are not satisfied at the end of each iteration, the state of stress at the
end of each iteration gives nonzero "pseudoresidual stresses" for (0"'1'1)~ and ((J~'1)~ . These
pseudoresidual stress components are the difference between the elastoplastic stress and
the elastic stress at the point where the elasticplastic boundary ends at the trailing edge
at each iteration. The corresponding strains are:
() 1  2v ( )' () (a~'1)~
c1'/1'/ r =  2(1 _ v)G (J'1'1 r' /~~ r = C
Using the stress from each iteration, the residual stresses are:
(4.11)
(4.12)
•....... ,r.
These residual stresses are used as initial conditions for the iteration, lUltil there is no
change in (O"~dr and ((J(()r' As was the case with the elastic stress fields, the singularities
at the ends of the slider cause error in the residual stresses at depths near the surface.
4.2 Residual Stresses by the Finite Element Method
Suh [16] reports the use of the finite element method by H.C. Sin (Ph.D. thesis, Massachusetts
Institute of Technology, 1981) to model the plastic deformation of a semiinfinite
elasticperfectly plastic solid Wlder cyclic loading by asperities (sliders). The residual stress
((J{()r, shown in Figure 41 (from [16)) after one, two, three and four passes of an asperity,
exerting an elliptical load on the surface, is tensile near the surface, then compressive
and larger at a greater depth, with smaller tensile stress at a still greater depth. After
four passes, there is negligible difference in the residual stress. The results obtained from
38
this method have generally been in good agTeement with those obtained from the MerwinJohnson
method. Repeated cyclic loading would be similar to the conditions found in
ultraprecision machining.
'.'
Figure 41: Variation of Sliding Directional Residual Stress as a F\mction of Depth Under
a Moving Asperity.
1.0 .u ... 'I .....\.. ...•
\ First
Second
1.0
2.0
o
/7""...../"'/
~ /
{ I
l~
I ~'\\,.+ Third and
\ \\ fOLJ~h
\ II
,~
'..::......
" "''''\
\\
\ \
\ I
3.0 'I
" II
II
I(
/I
4.0 JI
"II
II
1.0
39
Chapter 5
Elastoplastic Stress Field Results
This chapter presents a verification of the residual stresses from the MerwinJohnson method
as implemented in the corrected program upon changing the stress graclients from Equations
2.16 to Equations 2.17. Data obtained from the orthogonal flycutting of Al 6061T6 and
TeCu [13] are then used to predict the elastoplastic stress contours and residual stresses in
the cutting direction using the corrected program.
5.1 Verification of Residual Stresses
Figure 51 shows the residual stresses in the cutting direction as predicted by the corrected
program for an elastodynamic, constant surface load using the MerwinJohnson method
with a step size of 0.05 in the ( and 7]directions for comparison with those of Figure
41. Shown are the results after five iterations, after which there is a negligible difference.
The material properties used are (16): isotropic, slightly work hardening (slope of the
workhardening region = 104 E, where E is Young's modulus), E=1.96xl05 MPa =2xl04
kg/mm2 , v = 0.28, and the yield strength in shear, k, is 25.0 kg/mm2 . The normal and
tangential loads are 4k and k, respectively. The stresses are normalized with respect to k.
Since the material density is used to calculate elastodynamic stresses and was not specified
in [16], the density of TeCu, 9.14x10 10 was used. A speed of 125 mm/s was used.
Compared to the curve marked "third and fourth" in Figure 41, it can be seen that the
trend is the same: tensile stresses near the surface, becoming compressive at approximately
40
Figure 51: Comparison of the Results from the Corrected Program Using the MerwinJohnson
Method with the FEM Solution.
1] = 0.5 and zero at 1] = 2.25. The maximum compressive stress occurs at 1] = 1.2, and this
maximum stress is within rougWy 33% of that predicted by the finite element model. This
difference is smaller at greater depths. Overall the results of the MerwinJohnson method
using an elastodynamic, constant surface load are in general agreement with those of the
FEM solution using an elliptical surface load.
MerwinJohnson
with corrected
stress gradients
1.0 1.0
41
L

5.2 Elastoplastic Stress Field Results for Surface Loading
Conditions Measured in Ultraprecision Machining
Tables 5.1 and 5.2 show the material properties and data for the ultraprecision machining
experiments of [13], in which a cutting speed of 125 rnm/s was used. Since the sliding
indentation model is considered applicable only in those su~rnicrometer cases where the
cutting force is less than the thrust force, only the O.Olj.Lm and O.lpm Wlcut chip thickness
cases will be considered. In the modeling of elastoplastic stress fields and residual stresses,
the thrust (Fd and cutting (Fc) forces replace the vertical and horizontal forces P and Q
of Figure 21. That is, J.Lf = ~ in Equations 2.15 and 2.17.
42
...
:~
Al 6061T6 Workpiece
YOWlg'S Modulus (E)
Poisson's ratio (11)
Mass density (p)
Yield Strength (Y)
TeCu 'Workpiece
Young's Modulus (E)
Poisson's Ratio (1I)
Mass density (p)
Yield Strength (Y)
Properties
72.4 (GPa)
0.33
2.821x 10 10 (kg/mm2 )
363 (MFa)
Properties
120 (CPa)
0.3
9.14xlO 1O (kg/mm2)
225 (MFa)
Table 5.1. Workpiece Material Properties
.~
Uncut Chip Fc Ft l :J
Thickness (pm) (N/rom) (N/nun) (f.lm)
:3
'.'"4. A16061T6 '4 :)
0.01 0.173 0.531 0.515 :1
I
'I
0.1 0.563 0.586 0.537
TeCu
0.01 0.214 0.547 1.0
0.1 0.319 0.491 1.0
Table 5.2. Measured Force Components and Contact Lengths for Orthogonal Flycutting
43
Figures 52 through 55 show the von Mises elastoplastic stress fields for Al 6061T6
and TeCu for the two uncut chip thicknesses, normalized with respect to the uniaxial yield
strength. The plots were prepared using a step size of 0.05 in both the { and 17directions
and cover the range ~ = [3,31 and 17 = [0,41.
A plot of the von Mises elastoplastic stress field using a step size of 0.01 in the ~ and
17directions for Al 6061T6 for the O.OlpIn uncut chip thickness case in shown in Figure
56. This plot shows little difference from that using the 0.05 step size, so the other plots
for a step size of 0.01 are not included here.
The plots show the elasticplastic boundary (aeqlY = 1). In each of the four cases the
bOlll1dary is shifted toward the leading edge. TeCu being of lower yield strength, exhibits a
plastic deformation zone which is much larger for a given uncut chip thickness. The actual
depth of the predicted plastic zone is somewhat different than that shown by these contours
as a result of the way the plotting software smooths the data. For uncut chip thicknesses of
0.01 and O.lpm the depth of the plastic zone is 0.6517 and 1.617 for Al 6061T6 and 0.417 and
0.5517 for TeCu. For diamond turning of an aluminum alloy, it has been found [15] that the
"workaffected" layer is on the order of submicrometers using Xray diffraction analysis.
This is consistent with the results found here.
5.3 Estimated Residual Stresses
'" '4 '.4 ;)
·1
I
'I
Using the MerwinJohnson method, residual stresses for the cutting direction were predicted
in the plastic layers of the Al 6061T6 and TeCu workpieces at O,Olpm and O.lpm uncut
chip thicknesses, as shown in Figures 57 through 510. These stresses are normalized
with respect to the yield strength. For these cases ten iterations were used; only five were
required in the verification of the method above, in which case the forces involved were
roughly four times as large. \Vhile it was unnecessary to use a step size smaller than 0.05
for the elastoplastic stress fields, the use of that step size yielded plots of very few points
for shallow plastic zone cases. Therefore, the residual stress plots were prepared using a
step size of 0.01. Figures 511 and 512 show the residual stress results from [13], with
44
I""'""
the uncorrected stress gradients. It can be seen that the trend is much unlike that shown
in Figure 51. At O.OIf.Lm and a.lf.Lm uncut chip thiclmesses the ma"<.irnum compre sive
residual stresses for Al 6061T6 are 29 MPa and 102 MPa,and 16 ),IPA and 27 wIPA for
TeCu.
As they were above, the cuttingdirectional residual stresses are tensile near the surface
and then become compressive. Only the shallow tensile and deeper compressive plastic
residual stresses are calculated by the program. To agree with the trend of the FEN!
solution, the residual stresses at greater depths should be tensile and elastic. The general
trend in the residual stresses for the Al 6061 T6 more closely matches that of the verification
above because the plastic zones are deeper. It should be noted that the residual stresses
near the surface may actually be elastic for the proper combination of applied loads and
material properties [6], [16J, resulting in a subsurface plastic zone. Therefore it is not only
unclear that the magnitudes of the residual stresses predicted near the surface are correct,
but also whether they are actually plastic at these depths.
45
·.o.f
...
••.4
,)
I
II
o 1 2 3
0.4
2 1
4LJlIl....:........L..._.~__.I.__l..._:._...L_..l.._l__l. .........._._..._.J....JL.....i..___:........:__.:._....!.._.J.__L..__l.__l...J
3
3
o
" 2
Figure 52: Contours of the von Mises Equivalent Stress, Normalized by the Yield Strength,
for AI 6061T6 for 0.01}.Lm Uncut Chip Thickness.
46
2 1 o 2 3
0.1
3
4''''L....:____:.__"_~__:...........:...._l.._~_:.__:.__'__~oI__l._....l____:.__"__..L__'_~.......:...._'__....!......:._....:..._'L..J
3
o
T\ 2
Figure 53: Contours of the von Mises Equivalent Stress, Normalized by the Yield Strength,
for Al 6061T6 for O.lj.Lm Uncut Chip Thickness.
47
....
1 o 2 3
0.3
2
3
o
4"'~'~""""'~"":"J.l....L..L...:::>I,..':"''...L.l...J''L.L'....I...J.;.......l~~.:.......:....J
3
T1 2
Figure 54: Contours of the von Mises Equivalent Stress, Normalized by the Yield Strength,
for TeCu for O.Ol?£m Uncut Chip Thickness.
48
o
1
" 2
3
4
3 2 1 o 2 3
4•l
r
Figure 55: Contours of the von Mises Equivalent Stress, Normalized by the Yield Strength,
for TeCu for a.1l1m Uncut Chip Thickness.
49
" 2 ••tlI
3 03\ 0.4
4
3 2 1 0 2 3
~
Figure 56: Contours of the von Mises Equivalent Stress for a Reduced Step Size of 0.01,
Normalized by the Yield Strength, for Al 6061..T6 for O.Olp.ffi Uncut Chip Thickness.
50
~.1 ~.os 0 0.05 0.1 015 Or.:.,.:.:.,........:....;
0.1
0.2
/
.,0.3 \~\
004
\
0.5
0.8
0.7'
Figure 57: Normalized Cutting Directional Residual Stress for Al 6061T6 Predicted for
O.Olj.£ffi Uncut Chip Thickness.
51
(,,~}Y
.Q.4 .Q.3 .Q.2 .Q.l 0 0.1 0.2 0.3 0.4 0.5 0.6 07
0, "'I" 1""1 ,,",' I 'j
0.2 1
0.4
0.6
0.8
1.0
1.2
1.4
1.6
,
J
1.61 '
Figure 58: Normalized Cutting Directional Residual Stress Predicted for Al 6061T6 for
O.lILID Uncut Chip Thickness.
52
(~l,IY
~.15 ~., ~05 0 0.05 0.1 0.15 0.2 0.25
0 , I I • I • I
I
~ ~ ~L
I »
r I 0.05~
0.1
0.15
:'f /
0.25 r
0.3
0.35
0.4
0.45'.1
Figure 59: Normalized Cutting Directional Residual Stresses for TeCu for 0.01 Uncut Chip
Thickness.
53
l
L 0.1 I
0.2
~ 0.3
0.4
0.5
(,,~),IY
0.1 0.2 0.3 0.4 O.S 0.6
r i I i I I
0.6 L _
Figure 510: Normalized Cutting Directional Residual Stress for TeCu for O.lJ.Lm Uncut
Chip Thickness.
54
0.0
0.5
1.0
1.5
 
Depth oCeut
0.01 J.1lD.
 0.1 ~

,,
,,
2.0
0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3
Normalized Cutting Directional Residual Stress. (o~~)r
Figure 511: Normalized Cutting Directional Residual Stresses for Al 6061T6 Using Uncorrected
Stress Gradients.
55
0.0
0.5
s:
od' ~Q. 1.0
Q)'
~
"tj Q) 1.5
..to..!
"i
E 2.0
0 Depth of Cut
Z
2.5 0.01 J.lIIl
 0.1 J.lIIl
3.0 1.....ILJl....L..1...lIl.LJ...I...l.l...LLJl....L.J......JIl.LJl....L.'lJ
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
Normalized Cutting Directional Residual Stress, (J~~\
Figure 512: Normalized Cutting Directional Residual Stresses for TeCu Using Uncorrected
Stress Gradients.
56
Chapter 6
Conclusion
The calculation of elastoplastic stress fields and residual stresses in an elasticplastic halfspace
by the MerwinJolmson required accurate elastic stress fields and slidingdirection
stress gradients. A constant surface pressure distribution being the most likely to model
the toolworkpiece interaction in ultraprecision machining, it was used in the elastodynamic
model and in the elastoplastic model for residual stresses. These stresses were verified as
follows:
1. The stress fields for the constant surface pressure elastostatic case were compared
against those for elliptical surface pressure for the case given in 114J and were found to be
similar except near the slider, in accord with Saint Venant's principle.
2. For the small shear and dilatation wave speeds encountered in the experiments of
1131, the dynamic effects should be small and a comparison with the static case is possible.
At these speeds the elastostatic and elastodynamic constant surface pressure stress fields
were nearly identical away from the surface, and so the elastodynamic stress fields were
judged to be correct.
Having corrected the expressions for slidingdirection stress gradients, the MerwinJohnson
method was used to predict the depth of the plastic layer and the cutting (sliding)
direction residual stresses for Al 6061T6 and TeCu at 0.01 and 0.1 uncut chip thicknesses.
57
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58
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59
VITA
Andrew Charles Loeber
Candidate for the Degree of
Master of Science
Thesis: PREDICTION OF TIlE SUBSURFACE STRESS STATE IN A MOVING SEMIINFINITE
SOLID FOR LOADING CONDITIONS REPRESENTING TOOLWORKPIECE
CONTACT IN ULTRAPRECISION MACHINING
Major Field: Mechanical Engineering
Biographical:
Education: Graduated from Richmond Senior High School, Richmond, Indiana in May, 1981;
received Bachelor of Science degree in Industrial Engineering from Purdue University, West
Lafayette, Indiana, in May, 1988; received Bachelor of Science degree in Mechanical
Engineering from Montana State University, Bozeman. Montana, in May, 1992. Completed the
requirements for the Master of Science with a major in Mechanical Engineering at Oklahoma
State University in December, 1996.
Experience: Employed by Perfect Circle, 1988·1989 as an industrial engineer; employed by Montana
State University 19911992 and Oklahoma State University, 1994 as a teaching assistant.
Professional Memberships: American Society of Mechanical Engineers, Alpha Pi Mu and Tau Beta Pi
engineering honoraries.