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dissertation entitled

DISSOLUTION WEAR: DECOMPOSITION OF TOOL
MATERIAL, AND CONCENTRATION PROFILE INTO CHIP

presented by

Tim Kong-Ping Wong

has been accepted towards fulfillment
of the requirements for the

 

 

 

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Mechanics
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6/01 c:/ClRC/DateDuo.p65—p. 1 5

DISSOLUTION WEAR: DECOMPOSITION OF TOOL MATERIAL,
AND CONCENTRATION PROFILE INTO CHIP

By

Tim Kong-Ping Wong

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Materials Science and Mechanics

2004

 

 

ABSTRACT

DISSOLUTION WEAR: DECOMPOSITION OF TOOL MATERIAL,
AND CONCENTRATION PROFILE INTO CHIP

By

Tim Kong-Ping Wong

The predictive capability of Kramer’s theory of dissolution wear has raised
questions as to whether dissolution or diffusion is the dominant mechanism underlying
tool wear. The diffusion mechanism ﬁnds support in experimentation and modeling of
mass transfer (e. g., the Molinari-Nouari model of the experimental ﬁndings of
Subramanian et al.), and ﬁts more naturally in the setting of the irreversibilities of turning
processes. Kramer’s dissolution mechanism, by contrast, is based on equilibrium
thermodynamics. Our present aim is to re-phrase the dissolution hypothesis of tool wear
as a boundary condition for species transfer within the chip’s bulk via diffusion. In this
setting, dissolution is deﬁned as the combined events of tool decomposition at the
interface and the subsequent mass transfer of decomposed elements into the chip region.
A set of equations is constructed wherein the tool is treated as a mechanically rigid, but
chemically active, thermal conductor; the chip is treated as a rigid-perfectly plastic,
thermally conducting material; and the behavior of the frictional-contact interface
between tool and chip is described by a weak coupling of thermal and mass transfer.
Chemical equilibrium is invoked for the distribution of tool species at the tool-chip
interface. The Frank-Tumbull mechanism of molecular reaction (between interstitial

impurities and vacancies to form substitutional impurities) is used as a hypothesis to

explain the concentration proﬁle of tool constituents into the chip as found by
Subramanian et al. in 1993. The present interpretation of the Frank-Tumbull mechanism
is illustrated by ﬁnite-element simulations. Physical parameters have either been
estimated or curve-ﬁtted to observed data, and the level of inexactness inherent in the
curve-ﬁt process limits the present work to a demonstration of only the sufﬁciency of the
model. The assumptions made, their implications, parameters varied, and quantities

calculated are summarized in the conclusion.

ACKNOWLEDGh/ﬂiNTS

I am deeply thankful for the factors that have collectively made this moment
possible. My thanks go to my family and friends in California, who have been the
foundation of everything I do; to the admissions committee at Michigan State, for giving
me the opportunity to pursue my goal; to the professors from whom I have taken classes,
for their honesty and energy; to the administrative staffs at the Department of Materials
Science and Mechanics and the Department of Mechanical Engineering, for their
consistent help that at times reach beyond what is required of them; to my friends in
Michigan, for all of the intangibles; and to the professors on my dissertation committee,
for their grace and generosity in giving constructive criticisms.

A semester of teaching assistantship that the Department of Materials Science and
Mechanics has provided is gratefully acknowledged. The summer internship at NIST has
been a treasured experience, not only from the point of view of learning and exploration,
but also from the friendships that I have made. I have also beneﬁted from the support of
a unique organization—the Manufacturing Research Consortium at Michigan State
University. My appreciation for what it has set out to accomplish is heart-felt, and I
thank the organizers of the Consortium, as well as the sponsors who have visited our
campus.

Most of all, I would like to thank Dr. Patrick Kwon, for so many things.

iv

TABLE OF CONTENTS

LIST OF TABLES .................................................................................. vii
LIST OF FIGURES ............................................................................... viii
KEY TO VARIABLES, PARAMETERS, AND OPERATORS .............................. x
INTRODUCTION ................................................................................... 1
CHAPTER 1
THERMOMECHANICS OF TOOL AND CHIP WITH MASS TRANSFER ............. 6
The fully coupled equations ............................................................... 6
Weak coupling of the equations of plasticity and mass transfer ....................... 10
CHAPTER 2
THE ZEROTH—ORDER PROBLEM FOR THE TEMPERATURE BACKGROUND ..12
Aim of 2D simulations .................................................................... 14
Model and governing equations ........................................................... 14
Finite element discretization ............................................................. l7
Implementing the interface conditions ................................................... 20
Uncoupled interface ................................................................... 22
Temperature continuity ................................................................ 22
Convective cooling (and ﬂux continuity) .......................................... 23
Mixed interfacial behavior .............................................................. 24
Simulation results ........................................................................... 24
Mixed interfacial behavior ............................................................ 26
Temperature ﬁelds ........................................................................ 28
Summary of the temperature problem ................................................... 29
CHAPTER 3
THE FIRST-ORDER PROBLEM OF SPECIES TRANSFER ......................... 30
The time-dependent equations of the Frank-Tumbull reaction and mass transfer .. .30
Boundary conditions for mass transfer ............................................ 32
Connection to previous work by Kramer .............................................. 33
The time-independent equations .......................................................... 34
The difference between W and Co proﬁles; non-dissolution mechanisms ......... 34
Results of the species-transfer problem .................................................. 37
Unusually low values of mass diffusivities ......................................... 41
The determination of interstitial solubility; post-machining relaxation ........ 44
Solubility as the maximum amount of dissolution wear achievable ............ 48
Amount of worn cobalt not due to dissolution wear ................................. 49
The rate of tool wear as a ﬁeld quantity ............................................ 53

Table of Contents (continued)

Measurement methods for checking the distribution of vacancies ..................... 55
N on—uniqueness of Frank-Tumbull representation .................................... 56
Notes on the numerical method .......................................................... 57
CHAPTER 4
CONCLUSIONS AND SUMMARY .......................................................... 59
A list of assumptions made, with comments ............................................ 59
A list of key parameter variations used to achieve match with experimental data . . .65
A summary of motivations, results, and quantities calculated ...................... 66
APPENDDC .................................................................................... 68
BIBLIOGRAPHY ............................................................................... 77

vi

LIST OF TABLES

Table 1: Fully coupled equations ............................................................ 8
Table 2: Weakly coupled equations .......................................................... 9
Table 3: Parameters used in the Oth-order problem (background temperature) ....... 26
Table 4: Parameters used in the lst-order problem (mass transfer) .................... 41

vii

LIST OF FIGURES

Figure 1: Concentration proﬁles of tool constituents into the chip, reproduced from
Subramanian et a]. [1993]; chips are quenched after machining to preserve proﬁle shape.

The schematic illustrates the cutting geometry. .............................................. 2
Figure 2: Example of humped distribution with arbitrarily chosen functions ......... 3
Figure 3: Sources for and location of the Frank-Tumbull reaction ....................... 4
Figure 4: Decomposition of fully coupled equations ....................................... 7
Figure 5: Schematic of cutting geometry; boundary conditions of heat transfer ........ 14
Figure 6: Ideal l-D representation of the work and tool materials ....................... 20
Figure 7: Mesh used for O-th order problem, and a magniﬁed view ..................... 25
Figures 8a, b: Comparisons of heat partitions of mixed interfacial conditions to the case

of pure temperature continuity ................................................................ 27
Figures 9a, b: Interface temperatures corresponding to Figures 8 ........................ 27
Figures 10a, b: Temperature ﬁelds (see also Figures 9a, b) ................................. 28
Figure 11: Schematic of the Frank-Tumbull reaction ....................................... 30

Figures 12a, b: Experimental concentration proﬁles, deduced from Subramanian et al
[1993]. Empty and ﬁlled circles denote data points. In the original data, the 240-m/min

curve in (b) asymptotes to the same non-zero value as the lSO-m/min curve. .......... 35
Figure 13: Variation of temperatures away from the average ............................. 38
Figure 14: Mesh used in the lst-order, species-transfer problem ......................... 38
Figures 15a, b: Sample X V and X i distributions ......................................... 39

Figures 163, b: A comparison of available values of diffusivities versus actual values
used in simlations for (a) 2.5-m/s cutting speed, (b) 4-m/s cutting speed .............. 42

Figures 17a, b: X V and X i distributions using table-lookup values of Figures 16 .....43

Figure 18: Good match in an early run that turns out wrong ............................ 45

viii

List of Figures (continued)

Figures 19a, b: Why the good match of Figure 18 does not work ........................ 45

Figures 20a, b: Attempts at matching experimental W-concentration proﬁles using only
steady-state W distributions from the model ............................................. 46

Figures 21: (a) Relaxation of X s after cutting; (b) X s after 3.6 millisecs.
of relaxation ..................................................................................... 47

Figures 223, b. A check on the exhaustion of tungsten X i during relaxation ........ 48

Figures 23a, b: A check of Kramer’s hypothesis that dissolution wear is bounded by
solubility ......................................................................................... 49

Figures 24a, b: The relaxation process. A check on the exhaustion of cobalt X i during
relaxation .......................................................................................... 50

Figures 25a, b: The W-Co ratio within the chip as residual X i ’s are exhausted during

relaxation for (a) 4-m/s cutting speed, (b) 2.5-m/s cutting speed ........................ 51
Figures 26a, b: The surface rate constant for tool decomposition and tool-interface

temperature .................................................................................. 54
Figure Al: Ideal l-D representation of the work and tool materials .................... 69

ix

 

Key to Variables, Parameters, and Operators

A, B : impurity and host material, respectively

Ai & As , BV : interstitial & substitutional
impurities, and host vacancies, respectively

b : body force per unit mass (N/kg)

bk : body force per unit mass for species k *

c : total concentration of all species (mollm3)

ck : molar concentration of species k *

6V : mass-based speciﬁc heat at ﬁxed volume
(J/(kg-K)

Ework : workpiece EV (work & chip), Oth-order
problem

étool : tool EV , 0th-order problem

C [j : ij-th component of the advection matrix in
the Oth-order problem

D : rate of deformation ( 1/s)

Dk : diffusivity, within iron, of species It (mzls) *

DOF: number of nodes in the Oth-order problem

fk : rate of mass generation of species k per unit
volume (kg/(m3-s)) *

F k : (e.g., Fs , Fv, psz ) rate of mole generation
of species k per unit volume (mol/(m3-s)) *

F i]- : rj-th component of the effective ﬂux-
coupling matrix in the 0th-order problem

Fem , chcd , Fshear , kac : cutting forces for
the Oth-order problem (N)

g1 , E1 : lst-order Gibbs free energy (see Notes)

AEactivn : change in molar Gibbs free energy of
chip—boundary activation (Jlmol)

agdccomp : change in molar Gibbs free energy
of tool decomposition

G : Gibbs free energy (I)

h : heat-transfer coefﬁcient (W/(m2-°C))

hi : bilinear shape function over node i

I : the union of the rake and ﬂank interfaces

jk : diffusive mass ﬂux for species k t with
respect to v

J k * : diffusive molar ﬂux of species k * with
respect to v"

k : thermal conductivity (W/(m'K))

kwork : workpiece thermal conductivity (work
& chip) for the Oth-order problem

ktool : tool thermal conductivity for the Oth-
order problem

kfwd : rate constant for the forward Frank-
Turnbull reaction (l/s)
kdiss : rate constant for interfacial dissolution

kV : rate constant for vacancy production
K ij : rj-th component of the thermal-conduction

matrix, Oth-order problem
(digs : length scale for interface dissolution (m)

L, Lwork , Ltool : length of the work or tool

element in the one-dimensional heat-transfer
example

Lmkc : tool-chip contact length along rake (m)

m : total mass of all species (kg)

mk : mass of species k *

M k : molecular weight of species k (kg/mol) *

M i]- : ij-th component of the heat-capacity
matrix, 0th-order problem

N : total number of species

p0: 0th-order pressure (N/mz)

q : heat ﬂux (Jl(m2's))

qo , q] : Oth- and lst-order heat ﬂuxes

qﬂank , ‘Irake : heat ﬂuxes along the ﬂank and
rake interfaces, OtIi-order problem

Qi : i-th component of effective external forcing
in the Oth-order problem (W/m)

Q: interfacial heat ﬂux in the one-dimensional
heat-transfer example

le , sz : end-node heat ﬂuxes for the work,
one-dimensional heat-transfer example

Qt] , Qtz : end-node temperatures for the tool,
one-dimensional heat-transfer example

r : rate of heat generation per unit mass (Wlkg)

’work : rate of heat generation within the
primary shear zone, 0th-order problem

R : universal gas constant (cal/(mol-K))

R,- : i-th component of effective internal forcing
in the Oth—order problem (W/m)

t : time (5)

t1, t2: end nodes of the tool in the one-
dimensional heat-transfer example

ti]- : ij-th component of tool ‘stiffness’ matrix,
one-dimensional heat-transfer example

T : Cauchy stress (N/mz)

u, r? , r7 : internal energy (J/#) (see Notes)

Key to Variables, Parameters, and Operators (continued)

130 , ii, , r71 : 0th- and lst-order internal energies
(see Notes)

v : mass-averaged velocity (m/s)

v k : velocity of particles of species k t

v*: concentration-averaged velocity

v x , v y : horizontal and vertical workpiece
velocities, 0th-order problem

cht . ered . Vshear . Vrake : cutting speeds
for the 0th-order problem

V, I? (= p.1 ), I7: volume (see Notes)

w : Galerkin weighting function
w1, w2 : end nodes of the workpiece in the one-
dimensional heat-transfer example

wij : ij—th component of workpiece ‘stiffness’

matrix, one-dimensional heat-transfer
example
W: width of system transverse to ﬂow (m)
x : horizontal coordinate (global or local)

Xk (i.e., Xi , X5 , Xv ): mole fraction of
speciesk (=ck lc) *

X k, cq : solubility, within iron, of species k t

X Co, cq : solubility of cobalt in iron

X W, eq : solubility of tungsten in iron

y : vertical coordinate (global or local)
Yk : mass fraction of species k (= mk /m )1:

a: fraction of impurities within the chip lattice
identiﬁed as interstitial impurities

,6: fraction of heat partitioned to the workpiece
interface, Oth-order problem

17, I? : entropy (J/(K°#)) (see Notes)

ﬁo , ﬁl : 0th- and lst-order entropies (see Notes)

9 6 9

a): fraction by which the tungsten solubility is
multiplied to obtain the effective solubility
of cobalt in iron during dissolution

Vpc : Debye frequency of iron lattice

p: average density (kg/m3)

pk : density of species k *

Pwork : workpiece (work & chip) density, 0th-
order problem

Ptool : tool density, 0th-order problem

9: temperature (K or °C)

00 , 01: 0th- and lst-order temperatures

6 j : j-th node temperature, Orb-order problem

Hwork : workpiece (work & chip) temperature,
0th-order problem

atool : tool temperature, Oth-order problem

6w 1 , 9w2 : end-node temperatures for the

work, one-dimensional heat-transfer
example

6“ , 60 : end-node temperatures for the tool,
one-dimensional heat-transfer example
w, r]? : Helmholtz free energy (see Notes)

#70 . V713 Oth- and lst-order Helmholtz energies

A difference operator
V spatial gradient
V- spatial divergence

V-V Laplacian

8(-)/8t partial derivative with respect to time

D(-)/Dt total (or material) derivative with
respect to time

T1 :T2 = trace(TlTT2 ), tensor product

(V :T)-c = V - (TTc), divergence of a tensor T,
where vector c is any constant vector

* Index k takes on the values of ‘i , s , or ‘V’, denoting ‘interstitials’, ‘substitutionals’, or ‘vacancies’,
respectively, within the chip lattice in the lst-order (species transfer) problem.

Notes: Extensive quantities such as volume V, internal energy u, entropy 7], etc., are written on either
a mass or mole basis. For uniformity in notation, an overhead caret (e.g. r3 ) is used to denote a
quantity per unit mass, whereas an overhead bar is used to denote a quantity per unit mole (e.g. 17 ).
Following standard notation, partial molar quantities is denoted by a subscript indicating the particular
species (e. g., 170, = Bu ldna ). The ‘#’ symbol denotes a wild-card unit that could either be kg or mol

depending on whether a quantity is expressed on a per-unit-mass or per-unit-moles basis. Quantities
that are treated in the numerical simulations as material constants are not italicized (e.g., thermal

conductivity, k).

 

INTRODUCTION

Dissolution as a high-temperature wear mechanism has ﬁrst been proposed by
Kramer [1979]. The key idea is to interpret the solubility of a tool material within the
work piece as a measure of the tendency for dissolution. A condition of chemical

equilibrium involving free energies [Kramer, 1979; Kramer and Suh, 1980], namely,

AGdecomposition of tool = AGdissolution of tool components into work

is introduced to estimate values of solubility. In this context, ‘dissolution’ is taken as the
placement of tool components into the work. Further, a simple argument relates the
solubility to the volumetric wear rate up to an unknown (but assumed common)
multiplicative factor, which enables the life expectancy of different tool materials to be
compared on a relative basis. The quantitative results not only conform to common
experience, but also identify certain nitrides and oxides for their dissolution-wear
resistance at high-temperatures. Hence, in contrast to the theories at the time that favor
the non-equilibrium mechanism of diffusion [Cook and Nayak, 1966; Naerheim and
Trent, 1977], the importance of an equilibrium quantity, namely the solubility, is thrust to
the fore.

Over 20 years after the seminal work of Kramer, the actual concentration proﬁles
of tool constituents dissolved into the chip have been measured and reported by

Subramanian, Ingle and Kay [1993]. Their data, which correspond to the machining of

an A181 1045 steel using a tungsten-carbide tool with cobalt binder (W C-Co), are

reproduced in Figure l:

 

 

    
 

 

 

 

 

 

 

     
  

3 E Original data: mass fraction of W in chip 0 ‘ Original data: mass fraction of Coin chip
A .J ' A .I4
g A
5 3 r f, 0.12
‘5 240 rn/min ‘5 240 m/min
a 2.5 ‘ O) 0 1
.8— 2 ._ o 08
E V .
g 1‘5 150 m/min ‘ €006 150 m/min
E 1 $0.04.
I 0.5) '33, 0.02
i a -
o 0.1 . 0.2 ‘ 0.3 0.4 0 0.1 52 0.3 0.4
depth into Chip (Hm) depth into chip (pm)
Tool-chip interface
, .. ,~ 7. , ; Fi ure 1. Concentration rofiles
éééééééﬁg’é; c 'p ow of tool constituents (tungsten (W)
// / 4,0 :
jéﬂé/éé/ ; and cobalt (00)) into the chip,
4%; 2%; i .
3%: Tool 3%? . re reduced from Subramanian et
574' 'éé p
”/3 :/x , .
igégzéﬁéééé : concentration al. [1993]; chips are quenched
éﬁé/ééy’ééﬂé . proﬁle . .
gggégggggg : after machining to preserve
¢¢¢//¢ /¢j// ; _
232%?Zggézéé 5 profile shapes. The schematic
ZW/Wﬂéw .
M/zéiéééfﬁ :

illustrates the cutting geometry.

<:Il work flow
According to the authors, the humped distribution may be due to the formation of a
dislocation channel close to the contact interface. As seen from Figure 1, such a
dislocation channel would have a width of about 1/10 of a micron and is located about
1/10 of a micron away from the tool-chip interface; this is compared with a total depth of
a chip that measures hundreds of microns. As the regions adjacent to the primary and
secondary shear zones also experience severe shear deformation, it is not easy to see why
the dislocation channel is so precisely narrow in its appointed location. Solubility alone

cannot explain the humped distribution either, for a humped solubility proﬁle would

require a humped temperature proﬁle away from the interface, which stands in contrast
with an exponentially decaying behavior that is well known (see also Chapter 2).

Using a set of coupled, reaction-diffusion equations, Giese, Stolwijk and Bracht
[2000] have shown that a humped proﬁle of copper and nickel impurities diffusing in a
germanium lattice can be explained by the Frank-Tumbull mechanism of molecular mass
transfer [Frank and Tumbull, 1956; Sturge, 1959]. Delaying the discussion of this
mechanism until later, it sufﬁces to say that the enabling idea is to derive a humped
distribution as the product of a decaying function and an increasing function, as the

following example illustrates:

20

__ 1 .5-30x+175x2

 

Figure 2. Example of humped distribution with arbitrarily chosen functions
We use the same idea to capture the data of Subramanian et al. The reaction-diffusion
equations of Giese et al. are supplemented with a simpliﬁed model of advection to
describe mass transfer in the presence of chip ﬂow. An exponentially decaying,
interstitial proﬁle is obtained by solving an advection-diffusion equation, while an
increasing vacancy distribution is chosen so that its product with the interstitial decay—
under the requirement of mass conservation of the Frank-Tumbull reaction—ﬁts the
experimental data. The result is an interpretation of the humped proﬁle via the Frank-

Tumbull mechanism. The role of chip ﬂow is to establish thermal equilibrium, and to

transport the vacancies created in the primary shear zone directly into the chip, where
they interact with the tool interstitials in the proposed Frank-Tumbull reaction; this is

summarized in Figure 3.

 

 

 

 

 

Chip U tool-chip interface:
location of source of interstitials
F rank-Tumbull /
reaction

tool
primary shear /
zone: creation of work Q

vacancies

Figure 3. Sources for and location of the Frank-Tumbull reaction

Kramer’s original idea on solubility [Kramer, 1979] is re-interpreted within the
context of the rate constant of dissolution, which in turn enters the boundary conditions
of the proposed advection-diffusion-reaction equations. It is postulated that the boundary
ﬂux of dissolved components across the tool-chip interface is proportional to the
available solubility; the mole fraction of decomposed tool constituents; and the mole
fraction of vibration-activated openings on the chip surface that are wide enough to admit
tool constituents (also called “activation complexes”). A similar rate expression has been
discussed by Suh [1986, p. 369], which is less restricted than the present one in that in
Suh’s expression, the idea of activation complexes is not needed. In the present
interpretation, dissolution is deﬁned as the combined events of the decomposition of tool
constituents at the interface, and the absorption of these constituents by the chip. That
diffusive mass transfer does not account for the entirety of tool wear can also be
motivated from the proﬁle data of Subramanian et al.; this idea is introduced in Section

3.4 and re-visited in light of the present results in Section 3.5.4.

The equations of mass transfer are incorporated into the balance laws of
mechanics in a simpliﬁed way. First, the balances of mass and linear momentum are
satisﬁed using a chosen advection ﬁeld and a simple force balance (i.e., classical
machining theory). Second, the balance of energy is decomposed additively into a
zeroth-order problem governing the background temperature, and a ﬁrst-order problem
governing the mass transfer from the tool into the chip. The additive decomposition of
energy is based on the large difference in time needed to achieve a steady state for
thermal diffusion versus mass diffusion (Section 1.2). Bridgman’s notion of generalized
entropy [Bridgman, 1950] gives an operational meaning of temperature that appears in
the energy equation (Section 1.1).

The general organization of the dissertation is as follows. Chapter 1 contains a
review of the coupled governing equations and an explanation of the decomposition into
the temperature problem and the species-transfer problem. In Chapter 2, the temperature
background problem is solved. Chapter 3 contains the main results of the dissolution-
wear problem as stated before, including a discussion of the non-uniqueness of the Frank-
Turnbull representation, and a brief survey of experimental methods that are available for
measuring the spatial distribution of vacancies. Lastly, Chapter 4 is a summary of

results.

Chapter 1

THERMOMECHANICS OF TOOL AND CHIP WITH MASS TRANSFER

A model of particle exchange from one region to another in the presence of bulk
motion and deformation involves a sizable number of coupled equations. However, for
the high—temperature phenomenon of dissolution considered here, we place focus on the
solution for the temperature ﬁeld and the tracking of species exchange, while those
equations associated with the purely mechanical aspects are only approximately satisﬁed.
The result will be a set of equations involving advection, diffusion, and reaction

(Chapters 2 and 3).

1.1. The fully coupled equations

The tool is treated as a mechanically rigid, but chemically active, thermal
conductor, and the only equation of concern is the energy equation, or Equation (p) of
Table 1 written for the tool variables. For the tool, the stress power T : D = 0 because of
rigidity. The chip is treated as a rigid-perfectly plastic, thermally conducting material,
and all equations of Table 1 are applicable. In the present approach, the thermodynamics
of the cutting process is not predicated on the accessibility of states via reversible
processes, from which entropy is constructed. Instead, by appealing to the
recoverability—via a hysteresis process—of any point in a perfectly plastic process, a

generalized entropy is postulated to be a state function [Bridgman, 1950]. Further, by

arguing that the material’s structure is unchanged by planes of perfect slip, a generalized,
system entropy is chosen to be the same one that is used for a material that is rigid at all
locations except planes of perfect slip. Irreversibility is indicated by a 100% conversion
of the plastic work into heat; in other words, T : D is treated as a heat-source term, and
strain does not enter explicitly as an argument of the energy functions. The escaped heat
from plastic work adds to the entropy of the surrounding material, and is the basis
through which an absolute temperature of the system is prescribed in the presence of
perfectly plastic deformation. Such a temperature is precisely the same one that enters

the energy equation (60f Equation (p), Table 1).

main variables: main variables:
“x, chip R90» ‘70- P0 Xi. Xv. Xs

 

 

 

 

 

 

 

 

 

 

 

’IO 0 o -:> _ I: g g Q + ’1; o c .3
work o g é I O o A
90, V0, p0 are
, order problem
Oth-order problem: let-order problem:
coupled heat-transfer only species-transfer only
(Table 1) (Table 2) (Table 2)

 

 

 

 

 

 

Figure 4. Decomposition of fully coupled equations

Table 1 summarizes the governing equations of the fully coupled problem. For
discussion and derivation of the equations, the reader is referred to the references [Mase
and Mase, 1999, Ch. 5; Bird et al., 2002, Ch. 19]. In spite of the numerous equations,
Figure 4 shows the simpliﬁcation that an additive decomposition brings and the key

variables that are of interest. This decomposition is justiﬁed in the next section.

 

Ta

 

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1.2. Weak coupling of the equations of plasticity and mass transfer
The reformulation of the fully coupled equations into two simpler problems is
given in Table 2. These two problems correspond to decomposing the variables and

functions {6,ﬁ,ﬁ,t/?,q} additively as follows:

{ﬂail/1‘1}: {Hoiﬁo’ﬁo’Woﬂoh{91,771,ﬁ1,l/71’¢11}’ (1)
where the subscript ‘0’ refers to the zeroth-order problem of determining the background
temperature, and the subscript ‘1’ refers to the ﬁrst-order problem governing molecular
mass transfer. The motivation for this additive decomposition is that ﬁrst, the rise in
system temperature is almost entirely due to plastic deformation and friction, or
equivalently, molecular mass transfer from the tool does not signiﬁcantly affect the
background temperature.l Second, for a given distance, the steady state for thermal
diffusion is established much faster than the steady state for mass diffusion, which gives
the decomposition (Equation (1)) a natural division in time scale. In fact, the thermal
diffusivity of a material is typically orders of magnitude larger than the mass diffusivities
of impurities moving through the material. For example, at 900°C and atmospheric
pressure, the thermal diffusivity of iron is on the order of 10'6 mzls, whereas the mass
diffusivities of cobalt and tungsten in iron is on the order of 10’14 mzls [Elliott et al.,
11963, p. 690 for Co, p. 695 for W].

For the zeroth-order problem, the balance of mass and the balance of linear
momentum (Equations (a) and (i) of Table l) are satisﬁed by assigning a simple

advection ﬁeld and by balancing the cutting forces at steady state, as it is done in the

 

1 The system temperature is signiﬁcantly increased by drastic changes in the tool geometry due to wear, but
this temperature increase is not due to mass-transfer coupling, but rather to changes in the cutting forces
because of a change in tool geometry.

10

classical machining theory [Shaw, 1984]. The only use of plasticity is through an
idealized, 100% conversion of plastic work and friction into heat. The rate of plastic
work is connected to the cutting forces by the work-energy theorem, Equation (e) of
Table 2. The conversion of cutting forces into friction power is discussed further in
Chapter 2.

The ﬁrst-order problem pertains to the small amount of mass transfer due to too]
wear that is postulated to have no effect on the zeroth-order temperature and density

ﬁelds. Hence, the quantities of interest are mass-transfer variations with respect to ﬁxed
background temperature 6]) and speciﬁc volume )7 = 170 , as indicated in Table 2
(Statement 0)). Equilibrium distributions are governed by the Second Law, which is also
expressed in terms of the Gibbs free energy g, = 1/71— p090 (Inequality (v) of Table 2).

The interfacial pressure p0, which can be approximated from dividing the cutting forces
by the contact area (the present work will not go farther than requiring an order-of-
magnitude approximation), are those of the zeroth-order problem.

Additional relations in Table 2 (Equations (k) thru (q)) not found in Table l are

discussed in [Bird et al., 2002, Chapter 19].

ll

Chapter 2.

THE ZEROTH-ORDER PROBLEM FOR THE TEMPERATURE BACKGROUND

Because species transfer is so closely conﬁned to a submicron region adjacent to
the tool-chip interface, emphasis is placed on the details of the interfacial temperature
distribution. However, the present calculations will also return the temperature ﬁeld in a
much wider region of the tool, work, and chip.

In turning processes, while a large portion of the contact interface between tool
and work is under intimate contact [Zorev, 1958 (Fig. 54.7 on Plate 5); Wallace and
Boothroyd, 1964 (Fig. 14, p. 84); Trent, 1967] (and hence temperature continuity holds),
usually a distinct downstream region of gross, material deposition can be found [Doyle et
al., 1979; Ackroyd et al., 2001; Kim and Kwon, 2001]. It is in this latter region of looser
contact that a temperature jump is conjectured, even at steady state conditions. The aim
is to ﬁnd plausible conditions for a cooler work interface in orthogonal turning, as it
pertains not only to chemical wear but to abrasive tool wear2 as well.

From the solution of the heat equation, it is known that an interfacial temperature
jump results if a convective-cooling condition is used with a heat transfer coefﬁcient.

Furthermore, the adiabatic condition and temperature continuity are two extremes of

 

2 It is known that the hardness of metals and their inclusions decreases exponentially with increasing
temperatures [Kramer and Judd, 1985]. In turn, abrasive wear has been shown to depend on the hardness
ratio between tool and inclusions in the work material [Rabinowicz, 1977]. Abrasive wear intensiﬁes if,
during frictional contact, the temperature drops in going from tool to work material, as the inclusions in a
cooler work material become effective abrasives.

12

behavior representing assignments of 0 and co to the heat transfer coefﬁcient,
respectively. Here, the convective medium is the moving workpiece; however, the
situation is complicated by the presence of friction. In fact, the amount of frictional heat
that is partitioned to either side of the interface becomes an independent unknown. This
is studied in the Appendix through the use of a 1D problem, where it is found that the
heat—partition function enters directly into the expression for the interface-temperature
jump (Equation (A23)). Because of the lack of exact solutions for complex ﬂux
conditions and geometry in 2D, results derived in the 1D example serve only as
guidelines in the 2D FEM simulations. It is noted that the interface conditions considered
are linear in the boundary temperature, as opposed to a non-linear condition, such as
radiation, for which the present results do not apply.

While experimental proof of temperature jumps across frictional interfaces has yet
been found, the results of the present study imply that if interfacial temperature jumps
exist, then they can be characterized by the behavior of the heat-partition function across
the interface. Assuming that temperature continuity is associated with intimate contact,
the absence of intimate contact can then be quantiﬁed as deviations in the heat-partition
function away from its value in the temperature-continuity case.

For an extensive coverage and review of analytical work on calculating
temperatures in machining, and in particular, on the relationship between temperature
continuity and heat partitioning, the reader is referred to a set of papers by Komanduri
and Hon [2000; 2001a; 2001b]. The present discussion is largely motivated by the work
of Chao and Trigger [1955; 1958]; here, we are interested in the opposite problem of how

heat partitioning can be manipulated to achieve temperature jumps.

13

The subscript ‘0’ denoting the zeroth-order problem is dropped for the remainder

of this chapter.

2.1. Aim of 2D simulations

It will be seen that 2D simulation results are consistent with exact 1D results
found in the Appendix. Namely, for the part of the contact interface where the
convective-cooling condition is Operative, if the heat partitioned to the work is less than
that for the case of temperature continuity, then the work temperature will be cooler than
the tool temperature (Inequalities (A24)). In turn, such a temperature jump will have

ramiﬁcations for dissolution wear (Chapter 3).

2.2. Model and governing equations
The two dimensional geometry of concern is shown in the following ﬁgure, which

corresponds to orthogonal turning with a 5° rake angle:

zero flux

ambient
chip j {V

rake face tool {
adiabatic
“L7 IL adiabatic

\ 13’ [

ambient 7 flank face

\7: => f zero flux

work

 

 

 

 

Figure 5. Schematic of cutting geometry, boundary conditions of heat transfer

The governing equations for the work and the tool temperatures are a re-phrasing of

Equations (c) and (h) in Table 2 as follows:

14

 

 

 

 

 

 

2 2
[kwork[a 6work + a 6work] l
a 39 3x2 2
Pworkcwork 301k :1 3y i (2)
t A 36work aHwork
" Pworkcwork Vx T + V y T + pworkr work
. aa 320 826
Ptoolctool 5:01 = ktool[ 31:30] + aytgol] (3)

Here, x is the horizontal direction in Figure 5, y is the vertical direction, r is the rate of
heating in the primary shear zone, k is the thermal conductivity, p is the density, 6 is the
speciﬁc heat (per unit mass), v is the cutting speed, and 6is the temperature. Equations
(2) and (3) are coupled through conditions at the contact interface (marked by the letter I
in Figure 5). Boundary conditions are as follows. Inﬁnitely far away from the cutting
edge and contact interfaces, the system temperature returns to the ambient. Downstream
of the contact interfaces along the rake and ﬂank faces, adiabatic boundaries are assumed;
that is, the bulk of the heat transfer is due to conduction across the contact interface I, and
radiation far away from I does not play a key role. For the moving work and chip, the
outflow boundary condition for the advection-diffusion equation is that of zero ﬂux (k -
grad6= 0), as discussed in the text of Zienkiewicz and Taylor [2000b, pp. 44-46].

The advection term in Equation (2),

.. 86 66
Pworkcwork[vx _£\;/Tork + Vy 7%] 9 (4)
describes the flow of the work piece and its chip. For convenience, Equations (2) and (3)

will be written in condensed form as

p6%q-=kV-V6—pcv-V6+pr, (5)

15

where it is understood that if the equation refers to the tool, then the velocity v is equal to
0, and the internal source r equals 0, etc. The symbol V denotes the spatial gradient.
Stress and strain ﬁelds are not direct inputs to the energy equation that is being
considered. Instead, they enter through the rate of plastic work, which is considered as a
heat-source term as stipulated in Chapter 1. In particular, the source term T : D in
Equation (h) of Table 2 is related to the external power through Equation (e) in Table 2.

The basic inputs are forces3 and velocities { Fcut , vcut } and { Ffeed, ered} in the cutting
and feed directions, which can be resolved into { Fshear» vshwr } and { Fake, Vchip} along
directions tangent to the primary shear zone and the rake face, respectively [Shaw, 1984,
pp. 21-22]. The heat generated in the primary shear zone is expressed as a body source

r work : Fshear"'shear ’ (6)
while the frictional heat generated along the secondary shear zone (tool-chip, or rake

interface) is expressed as a boundary flux,

Frakchhip

—— . (7)
Lrakew

(Irake = (q ' n)rake =
where q and n are the heat flux vectors and outward normal to the boundary,
respectively, Lrake is the length of contact, and Wis the out-of-plane thickness of the

system. The frictional heat flux qﬂank generated along the tertiary shear zone (tool-

work, or flank interface) is also expressed as a boundary flux. As implied by DeVries’

 

3 Knowing the cutting geometry and the cutting speeds, forces can be converted from speciﬁc-energy data
that is tabulated in DeVries’ text [1992, pp. 106-121] based on the classical machining theory. Here, we
simply make use of the force and average-temperature data reported by Subramanian et a1 [1993].

16

text [1992, Figure 5.3c, p. 113], the product of qﬂank and the ﬂank-wear area can be

obtained as the correction, in the presence of ﬂank wear, to the total power input

 

Fcutvcut + Ffeedvfeed’ (8)
so that
‘Iﬂank = (q 'n)nank
_ {change in chvcut + Ffeedeeed due to ﬂank wear correction} . (9)
Lﬂankw

The quantity Frake in Equation (7) is calculated from Fcut and Ffeed before the ﬂank-
wear correction is applied. The ﬂux sources qrake and qﬂank are then uniformly applied

to the contact regions regardless of whether they are prescribed to be temperature-
continuous or convective-cooling.

The interface heat-transfer coefﬁcient h is known to be highly dependent on
process conditions. An intermediate value of h = 1000 W/(m2-°C) is used in the

simulations.4

2.3. Finite-element discretization

The ﬁnite-element weak form of the governing equations is derived by pre-
multiplying Equation (5) (or Equations (2) and (3)) by a weight function w, integrating
over the respective domains of work and tool, and performing integrations by part,

resulting in

 

4 As a crude indication, a value of h = 400 W/(m2-°C) has been reported between molten aluminum and its
die (http://msewww.engin.urrrich.edu/research/groups/pehlke/publications/lppm/index_html).

17

[w pég—fdA= -[(Vw).(kva)dA-[wpcv -vadA+[wprdA+[wkv0-nds (10)
where I(-)dA denotes area integration, and is taken over the entire areas of tool and work,
and n is the outward normal vector to the boundary element ds. Because of the
simplifying assumption that the boundaries of the system are mostly adiabatic, the
boundary integral in the last term of Equation (10) include only the contribution from the
contact interface I (rake and ﬂank).

The ﬁnite-element approximation of the temperature ﬁeld is
DOF
t9= 20j(t)hj(x,y), (11)
'=1

where DOF is the number of degrees of freedom—which, in the case of the scalar

Equation (5), coincides with the number of nodes—and h j- is a bilinear interpolation

function associated with the node j. Substituting approximation (11) into the weak form

(10) using the Galerkin weighting

w(x, y) = h.- (x, y) (12)

(i = 1,. . ., DOF), then performing the required integration DOF number of different times,

results in the spatially discretized equation

2M1} 37921919; ”2019 +1? +Q.-+ZF ,- 6,, (13)

j 1' 1'
where the terms of Equation (13) are deﬁned as follows:

heat-capacity matrix M ij = Ipé hihjdA , (13a)

conduction matrix K”. = Ith" - Vh j(1A , (13b)

18

advection matrix C0. = j pcv - hthjdA, (13c)

effective internal forcing Ri = IhiP’work dA , (13d)
effective external forcing Q, = Ihiq -nds , (13c)
I
effective coupling-ﬂux matrix F”. = I 12,.(thj — q)- nds . (13f)
I

Flux conditions such as convective cooling and ﬂux continuity enter into the matrix Fr ,
the effect of which is to alter the “stiffness” matrix K ij + Cij. For the case of
temperature continuity, the convective-cooling condition is irrelevant, and Fij- vanishes.

The effective internal forcing R,- , which accounts for the power generated within the

primary shear zone, is evenly distributed to a minimal set of nodes covering the primary

shear zone at each stage of the mesh reﬁnement. We will only study the steady state, so

that the left-hand sides of Equations (5) and (10) vanish, and the nodal temperatures— 6}-

in Equations (11) and (13)—become independent of time t. The (kV h j — q) term in

Equation (13f) isolates the contribution of gap heat transfer, as will be shown in a later
example (Section 2.4.3).

The current algorithm makes use of the background material and source codes of
Kwon and Bang [2000, Examples 5.9.1 and 6.6.2]. Alterations to the source codes are

made to accommodate Ci]- , Fij , and the source term Ri. To improve accuracy in the

present case of advection-dominated ﬂow [Gresho and Lee, 1979] and for the modeling
of the primary shear zone, a subroutine is written to incorporate the mesh-creation and

mesh-reﬁnement (‘regular’lbisection) algorithms found in Matlab’s “pde toolbox.”

19

2.4. Implementing the interface conditions

A matrix equation in the form of Equation (13) for the steady state is written for
each of the tool and work regions. Enforcement of interface conditions results in the
coupling of the tool equation and the work equation. For a detailed discussion of such
coupling, it is felt that the terminology of Equation (13) would unnecessarily obscure the
essential ideas. Hence, in the following discussion, we use the notation of a 1D problem,
and illustrate how different interfacial behavior can be implemented in the ﬁnite-element
setting. For this purpose, we refer to Figure 6. The work and tool materials are
schematically represented to lie along the x-axis, and they are in contact at the location

shared by the work node w2 and the tool node t1:

frictional ﬂux Qapplied
t1 tool t2

‘ k A
v v v

M work w2

Figure 6: Ideal 1-D representation of the work and tool materials

A heat ﬂux Q(in W/mz) is applied at the interface. The energy equations governing the

work and tool domains are

 

d23 . d6
kwork jfi _ pworkcworkV if“ = 0 9 (14)
d29
km] —g’i‘ = o. (15)
dx

As before, k is the thermal conductivity, p is the density, 6 is the speciﬁc heat (per unit

mass), v is the cutting speed, Bis the temperature, and x is the distance coordinate. The

20

system is subject to two far-ﬁeld boundary conditions:
aworrlxzxwl = 6m = 0, (16)
and

6100],:th = 6a = 0. (17)

For simplicity, one element is used to cover each of the work and tool regions.

Then, the ﬁnite-element discretization of Equations (14) and (15) becomes
w w 6 t r 0
[ 11 121 w ]=[Qw1) and [11 121 11)=[Qr1] (18)&(19)
W21 W22 9w2 sz t 21 ‘22 9:2 Q12
where, for the work and the tool, respectively, Wij and ti}- (i, j = l, 2) are components of
the “stiffness” matrixes, Hwi and 6?t ,- (i = 1, 2) are the temperatures at nodes wi and ti,

and Qwi and Q,- (i = 1, 2) are corresponding components of the frictional heat ﬂuxes,

being positive for ﬂux inﬂow into a region. Standard calculations for 1st-order, linear

elements show that

kwork _ pworkéworkv _ kwork + P workaworkV

Lwork 2 Lwork 2

 

ktool §_ktool
tutu)- Ltool Ltool
and _ .............. , ............. 2 1
(’21122 _kr001§ ktool ( )
Lroolé Ltool

where Lwork = xwz —xw1 and Ltool = xa —xt1 are the element lengths.

21

2.4.1. Uncoupled interface

Accounting for the total, effective frictional ﬂux chat is applied simultaneously
to nodes w2 and t1, the uncoupled system is modeled by the following combination of
Equations (18) and (19):

(W11 W12 0 0Y0“) (le\ ( lel

W21 W22 0 0 9w2 = Qw2 = ﬂQ (22)

0 0 ’11 ‘12 911 Q11 (1‘13)Q
(0 0 ’21 122/0912) (Q12) ( Q12 )

 

 

 

 

 

 

 

Here, ,6 is the fraction of heat that is partitioned to the work. Being undetermined, ,6 is
commonly set to 0.5 to give equal heat partitioning to the work and tool. This is
equivalent to applying the two conditions

sz + Q11 = constant Q, (23)

Qw2 - Q11 = 0- (24)

2.4.2. Temperature continuity
The algorithm used here for implementing the temperature-continuity condition

follows directly that of Tay et al. [1974]. Under temperature continuity,
9,, =9,, :9, (25)
while at the same time, the total amount of applied, frictional flux is

sz + Q11 = constant Q. (26)
Equations (25) and (26) represent a redundancy in Equation (22) (ﬁrst equality) that can

be collapsed:

22

W11 W12 0 9w1 Qw1
W21 W22 +211 112 9 = Q . (27)
0 t 21 t 22 912 Q12

Hence, the application of temperature continuity to a coincident, contact-node pair
reduces the size of the uncoupled system by one [Tay et al., 1974]. In the 2D algorithm
used in this paper, the contact nodes on the work and tool boundaries coincide in location
to ease the matrix-reduction procedure.

The far-ﬁeld boundary conditions
0m = 6’12 = 0. (28)
are applied to Equation (27). Solving for 9 = 0w2 and re-expanding to Equation (22), the

heat partitioned to the work element is found to be

,6 = W22 ' 6w2 = [0‘ work [Lwork )+ (pworké5 workV/ 2)] . (29)
Q [0‘ work [Lwork )+ (P worke workV/ 2)] + (k tool/Ltool )

2.4.3. Convective cooling (and ﬂux continuity)
Convection-cooling conditions are formulated using the heat transfer coefﬁcient h

and the fraction ,6 of heat partitioned to the work as follows:

80
km 3"“ = h(611-6w2)+ﬂ<2, (30)
x=xw2
819
ktool[- 7:31] : h(6w2 _6tl) + (1"sz (31)
x=xtl

Their ﬁnite-element weak forms of which are as follows:

(M = h (611 —6w2 )+/2’Q, (32)

23

Q11 = h (6.12 - 011) +(1—ﬂ1Q. (33)

Substituting Equations (32) and (33) into the ﬁrst equality of Equation (22) gives:

 

 

 

 

 

I W11 W12 0 0 Yam) [ Qwr \
W21 (W22 + h) ‘ h 0 6'm _ ﬂQ
— . (34)
0 "h (’11 +11) ‘12 911 (l-ﬂh
\ 0 0 t21 t22 A912) \ 912 J
Referring to Equation (13), the effective coupling-ﬂux matrix in this case is
0 0 0 0
_ _ 0 h - h 0
(Pg) 0 _ 1. h 0 (35)
0 0 0 0

Flux continuity is convective cooling with Q: 0, which leads to a trivial solution
if the system boundary is held at ambient, unless a heat source is speciﬁed away from the
interface. By comparing Equations (22) and (34), it is seen that ﬂux conditions do not
reduce the size of the uncoupled system, but rather alters the “stiffness” matrix. The heat

partition ,6 becomes a parameter that can be varied.

2.4.4. Mixed interfacial behavior
In the 2D implementation of interfacial coupling, the presence of many interface
node pairs requires consistent book—keeping. The added complication, however, allows

us to consider inhomogeneous interface behavior.

2.5. Simulation results
Figure 7 illustrates the reﬁned mesh used for the numerical solution. For

imposing coupling conditions, node pairs across the tool-chip interface coincide in

24

location at each step of mesh reﬁnement. A separate reﬁnement near the end of the tool-
chip contact helps to resolve the temperature jumps for loose—contact conditions. Table 3

summarizes the parameters used in the simulation.

      

 

    

  
     

 
  
     

          
  

  
 
      
   

 

 

x 10° 2X 10 , _
14 page
12 chip 1 ,5 E55?
E 10 ’E‘ 1. 5:551?
v 8 v 3:: ..
8 8
"' en
% 4 ‘6 0 VA‘
‘és'A‘
> V
2 tool > -o 5 0m
v
0 A
work '1 5‘
0 5 1 0 1 5 -1 0 1
distance (in) x 10'3 distance (m) x 10“

Figure 7. Mesh used for O-th order problem, and a magnified view

25

 

 

Table 3. Parameters used in the 0th—order problem (background temperature)

Work piecel: AISI 1045 work piece Tool’: Kennametal TPG322, K-ll Co—cemented
Thermal conductivity" kwork : 50.8 N/(s °C) Thermal conductivity‘ ktool : 84 N/(s °C)
Density pwork : 7870 kg/m3 Speciﬁc heat capacity” Ptool 43V, tool :
Speciﬁc heat" 511,111,111 = 486 J/(kg °C) 3e6 J/(m3 °C)
Uncut-chip thickness’: 173nm Rake angle’: 5 degrees

Cut-chip thickness: 519nm Ambient temperature: assumed 25°C
Out-of-plane width": 2 mm Cutting speeds“: 2.5 and 4 m/s

Calibrated cutting forces,w uncoated carbide tool, 4 m/s: cutting force = 1170N, feed force = 494N.
Calibrated cutting forcesf” uncoated carbide tool, 2.5 m/s: cutting force z 1222N, feed force z 516N.
(Cutting forces reported' for 4 m/s cutting speed and HfN-coated tool: cutting force = 800N, feed
force = 330N)

Interface contact lengths . from 0 to 1.2 mm, intimate/temperature—continuity contact; from 1.2 to 1.8
mm, loose/convective-ﬂux contact.

1.
i

ii
ii
iii
iii

11*.

taken from [Subramanian et al., 1993]

information on conductivity of carbides is scant; data is taken from CRC Materials Science and
Engineering Handbook, CRC Press, Boca Raton, FL, 2001, p. 405

taken from Properties and Selection: Irons, Steels, and High-Perfonnance Alloys, Metals
Handbook, Tenth Ed., Vol. 1, ASM International, Materials Park, OH, 1990, pp. 197-198
approximated from Tlusty, Manufacturing Processes and Equipment, Prentice Hall, Upper
Saddle River, NJ, 2000, p. 435

forces calibrated from Hﬂ‘l-coated-tool data to obtain the same average interface temperature
reported by Subramanian et al. for uncoated tools (1250°C for 4 m/s; 1100°C for 2.5 m/s)
Contact length is reported in Subramanian et al. for the uncoated carbide tool only for the case
of cutting at 4 m/s. From Figure 7 of Subramanian et al. [1993, p. 296], the contact length
increases approximately linearly from 0.8mm to 1.6mm, corresponding to 3 to 30 seconds of
cutting. Since the tool—chip interface temperature for 240 m/min stabilized after approximately
15 seconds, we assume a steady state contact length of 1.2 mm, or 1200 microns. The
development of contact length up until the ﬁnal value of approximately 1800 microns is also
taken from Subramanian et al. Because our main interest is in illustrating mass transfer, the
same contact length is used for the case of 150 m/min for simplicity.

 

For simplicity, the only coupling between the tool and the chip is through the tool-chip

contact (i.e., the ﬂank wear-land is assuming to be zero and conditions are adiabatic

there).

2.5.1. Mixed interfacial behavior

Two-thirds of the tool-chip contact interface, from 0 to 1.2 mm, is prescribed by

temperature-continuity conditions. The remaining, downstream one-third of the interface

is convectively cooled with a heat transfer coefﬁcient of h = 1e3 W/(m2 -°C).

26

 

Figures 8-9 plot comparisons of the heat-partition ﬁelds and the corresponding
interfacial temperatures. In Figures 8, the heat-partition ﬁelds used are compared against
what they would have been had temperature continuity been imposed on the entire tool-
chip interface:

Heat partitioned to work interface, ,6 Heat partitioned to work interface. ,6
for a 2.5«m/s cutting speed for a 4-rn/s cutting speed
2 r . . 2 . - .

 

 

  
     

 
 
    

temperature continuity temperature 00'1”"in

 

fraction of total friction flux
fraction of total friction flux

 

 

 

 

 

 

 

 

    
  

   
    

 

 

 

 

 

 

0.51 . I . . 0.51 . / . .
2/3 contact. temperature continuity 2/3 contact. temperature contmurty
us contact convection with 0.5 1/3 contact: convection with 0.5
O ‘ heat partition . 0 . heat partition .
0 0.5 1 1.5 0 0.5 1 1.5
Distance from cutting edge x10” Distance from cutting edge x10”
(a) along toolvchip interface (m) (b) along tool-chip interface (m)
Figures 8a, 0. Comparisons of heat partitions of mixed interfacial conditions
to the case of pure temperature continuity
Tool-chip-interface temperatures for 66% Tool-chip—interface temperatures for 66%
temperature-continuity contact on tool- temperature-continuity contact on tool-
0 C chip interface, with h = 193, [3:05. °C chip interface, with h = 1e3, 13:0.5.
Cuttin eed = 2.5 m/s Cutti eed = 4 m/s
1300 9 Sp . - 1600 . "9 Sp .
Tool-interfacw Tool-interface/
1200. temperature 1400' temperature
1100 .
22:22:23? 1 200‘ Chip-interface
1 000 temperature
1000
900
8 00 800-
700 600 1
6000 0:5 1 1:5 40‘)0 0:5 1 1:5

'3 Distance from cutting edge )110'3

along tool-chip interface (m)

Distance from cutting edge x10
(a) along tool-chip interface (m)

(b)

Figures 9a, b. Interface temperatures corresponding to Figures 8

27

Figures 9 illustrate how the chip—interface temperatures overtake the tool interface
temperatures near the end of the tool-chip interface, where the heat partition for mixed
conditions overtakes the heat partition for temperature continuity.5 The size of the
temperature jumps also correlates with the differences in the heat partition ﬁelds. The

location of the crossover point in each set of data, however, is not easy to predict.

2.5.2. Temperature ﬁelds
Finally, Figures 10a, b plot the full solution to the O-th order problem
corresponding to the two cutting speeds.

Temperature field for 2.5-m/s cutting speed Temperature field for 4-m/s cutting speed

    
     

r “.

    
     
 

a
Wé‘\’\§
%&)¢§V§
"v“ "

i

 
  

   
  

  
 
    

 
  
  
   

I‘lllll‘n. .
l r 'lgl'w":‘i ‘ I,

@333

     
  
 

'gz‘VAVAVAVAv

       

-1 -0.5 0 0.5 1 1.5
x axis (m) x 104 x axis (m)

Figures 10a, b. Temperature fields (see also Figures 9a, b)
The extremely rapid rise of the temperature ﬁeld across the primary shear zone, and a less
rapid but still substantial rise adjacent to the tool-chip interface, both contribute to the

need for mesh reﬁnement.

 

5 The crossover near 1 mm in Figure 8b is considered an imperfect attempt of the numerical scheme to
satisfy temperature continuity from 0-1.2 mm and to shift to convective conditions. The main crossover
points of interest are associated with the constant heat partition of 0.5, as this is the cause of the difference
between Figures 8a and 8b.

Considering that in the case of mixed boundary conditions, the numerical scheme receives no error
feedback in its attempt to satisfy temperature continuity, the match in Figures 8 over portions of
temperature continuity is very good.

28

2.6. Summary of the temperature problem

The nature of heat transfer across the tool-chip interface has been studied for the
uncoupled interface, the temperature-continuity interface, and the convectively cooled
interface. It has been found that the nature of interfacial temperature jumps in all cases
except temperature continuity is directly related to the interfacial heat partition. Further,
if the work’s heat partition is higher (lower) than the value that it assumes in the case of
temperature continuity, then the work’s interfacial temperature is also higher (lower) than
the tool’s interfacial temperature, although the crossover points in the two cases are
different in general. Namely, the presence of crossovers on the ,6 plots (e.g., Figures 8)
correlates with the presence of crossovers in the interface-temperature ﬁelds (e.g.,
Figures 9).

Hence, interfacial temperature jumps are directly related to the heat-partition
function in the 1D case (see Appendix), and can be numerically correlated to the behavior
of the heat-partition function in the 2D case. The missing link to the actual nature of
contact would be a relationship between the heat-partition function and the mechanical
nature of frictional contact, but this is beyond the reach of the present work. In the next
chapter, the interfacial temperature jumps found in this chapter are assumed, and their

effects on dissolution wear will be studied.

29

Chapter 3

THE FIRST-ORDER PROBLEM OF SPECIES TRANSFER

To completely deﬁne the ﬁrst-order problem, it is necessary to specify the species
number N and how mass is exchanged. It is proposed here that the equations of the
Frank-Tumbull mechanism be used. The Frank-Tumbull mechanism describes the
combination of an interstitial-A atom and a B-lattice vacancy to form a substitutional-A
impurity, or

A, + BV —> AS. (36)

This reaction is illustrated in Figure l 1:

interstitial substitutional
W0 0 o o 0/o\o/
o o 0 Ci) 0 e o o
codex oooo

vacancy

Figure 11: Schematic of the Frank-Tumbull reaction

3.1. The time-dependent equations of the Frank-Tumbull reaction and mass transfer

The governing equations are written using Equations (0) and (p) of Table 2 (with
k = ‘i’, ‘V’, and ‘s’) for the Reaction (36), where references to A impurity atoms and the
B lattice are understood, and all terms are deﬁned in the “Key to Variables, Parameters,

and Operators” on page ix:

3O

 

 

-a—-X—i-=V(Di VXi)-VXiV*—£ (37)
a: c
BXV F FV psz
-—=V'(Dv VXV)—VXV -v*-—i+—’—— (38)
at c c
8X. =_VXS -v*+£ (39)
at c
F X', XV,
‘—s=kfwd[XiXV_ ”‘1 6“ X8] (40)
C Xs,cq

where Fs lc embodies the rate of production of substitutional impurities via the Frank-

Tumbull reaction [Sturge, 1959; Giese etal., 2000]. The term dem / c is deﬁned by

F V, psz

c =Ic\,(XVdaq -XV) (41)

 

and represents the generation of vacancies within the primary shear zone only (see Figure
3). The impurity subscripts ‘i’ and ‘s’ can refer to either tungsten or cobalt. In effect,
each diffusing tool element is modeled separately as a dilute tracer component.

Additional assumptions are the constancy of concentration c, the diluteness

X k 2m Fm I = I X k (— Fs + FV, psz)| ), and the neglect of the

 

approximation (lel »
diffusive mobility of substitutionals as compared to the mobility of interstitials and
vacancies (DS « Di,DV ).

Note that no equation is written for the chip species (iron) in its role as the
substrate lattice; however, its presence implies the approximate constancy of the total

concentration c, which is an important simpliﬁcation. Solubility data from binary phase

31

diagrams are also reported relative to the chip species. Lastly, the concentration-

averaged velocity v* is approximated to be that of the chip advection velocity.

3.1.1. Boundary conditions for mass transfer

The decomposed tool components are postulated to enter the chip lattice as
interstitial impurities. Then, dissolution is controlled by the flux of interstitials entering
the chip boundary (viz., the crossing of the bold line from tool to chip in Figure 3):

3X-
Di 3;? = [disskdiss (Xi, eq — Xi ). (42)

where n is the outward normal distance from the tool interface into the depth of the chip,

ldiss is a length that is used to render a volumetric rate into an area ﬂux, and kdiss is the

rate constant for the dissolution of tool material into chip, deﬁned by

AEdecomp ( A? ' j
k ' = ex - —_ ex "' __a_Cth_n_ . 43
diss AOVFe P[ R 0 P R 0 ( )

Here, A0 is a calibration constant. According to Shewmon [1989, p. 70, pp. 74-75], given

the diluteness assumption, exp(— Agdccomp / (R0 » can be interpreted as the mole

fraction of decomposed tool components under chemical equilibrium (i.e., the equality of

Condition (v) in Table 2), and Vpe exp(— Agacuvn / (R 0)) can be interpreted as the jump

frequency of tool constituents moving into the work lattice under chemical equilibrium.6

 

6 The interpretation makes use of the idea of activated complexes (i.e., vibration-activated openings on the
chip surface through which tool constituents pass). In an earlier edition of his book, Shewmon [1963, p.
60] notes that the “precise deﬁnition of v is one of the more difﬁcult aspects of a rigorous theory.
However, it is usually taken equal to the Debye frequency.” Shewmon [ibid., p. 60] continues to say that

the entire derivation hinges on the assumption that Agacm “is a state function, that is, that the free
energy of the activated complex attained by the reversible process is the same as the free energy of the

32

In this interpretation, we can also view the interstitial boundary flux Di 3X i / an as the

product of the concentration of open sites for interstitials, times the product of the
probabilities (i.e., mole fractions) of decomposition and activation, times the frequency
that the tool constituents are allowed to jump into a lattice site within the chip. Hence,
we deﬁne dissolution wear as the combined sequence of events of decomposition at the

interface and the subsequent advection-diffusion-reaction within the chip region.

X V and X s are simply given zero mass ﬂux boundary conditions; the validity
of this choice for the advection boundaries is discussed in [Zienkiewicz and Taylor,
2000b, pp. 44-46]. The same condition applies for X i along all parts of the boundary

except the tool-chip interface.

3.2. Connection to previous work by Kramer

The role of the rate constant is to determine the rate of rise of the interstitial

distributions but not its saturation value X i, eq . In fact, simulation results using Equation

(42) shows that the solubility X i, eq is the maximum value of X i that is achievable

along the contact interface (Section 3.5.3). This is consistent with the earlier works of

Kramer. In particular, mass transfer at the contact interface is rate-controlled by the

supply of interstitials, and the solubility X i, eq governs the maximum amount of

 

activated complex that really occurs in nature. This would be true if the complex, like a vacancy, existed
for a period long enough to let the surrounding lattice completely adjust to the presence of the complex. . ..
but if the system is not in thermal equilibrium (present author’s note: i.e., chemical equilibrium), the entire
procedure may become only an approximation of doubtful value... One of the main justiﬁcations of this
analysis is that it works. It is not clear to many authors just how the assumption discussed above can be
fulﬁlled, but whenever the predictions of this theory have been experimentally checked, the theory has been
vindicated.”

33

dissolution wear. At high temperatures, the value of the impurity distribution at the

interface approaches its solubility value, and the dissolution effect approaches its peak.

3.3. The time-independent equations

To model time-independent behavior, we set both B(')/at = 0 and Fs = 0 in
Equations (37)-(40). It is implicitly assumed that the chronological order of occurrence
of (i) the achievement of a macroscopic steady state, and (ii) the local saturation of the

Frank-Tumbull reaction, is immaterial. Under these conditions, X s can be eliminated by

recasting the equations as follows:

 

0=§¥i+§§=v.(oivxi)—V(xi+Xs)-v* (44)
a: a:
F

0=§£l+3X—S=V-(DVVXV)-V(XV+XS)-v*+-l’-P-s—z- (45)

a: a: c

XS,”
WhCI‘C XS = KequXV = XiXV (46)
Xi eqXV,eq

and K eq = X 5,6,] / (X i,eqX V,eq) is the equilibrium constant. Equations of the type (44)

and (45) have been referred to by Sturge [1959, p. 298] as the equations of atom

conservation and lattice-site conservation.

3.4. The difference between W and Co proﬁles; non-dissolution mechanisms
The difference in W and Co concentrations in Figure l is not likely due to
electrical mechanisms. WC-Co is metallically bonded, and when it is dissolved into Fe,

the components W and Co are most likely charge-neutral interstitials.

34

Comparisons of atomic radii and decomposition energies do not account for the
difference in the W and Co proﬁles, either. Since Co and W have similar atomic radii as
iron, the activation energy required for Co and W to enter the chip lattice (Agacﬁvn )
should be similar. In addition, the free energies of decomposition entering the

calculations for W and Co are one and the same (viz., Agdecomp of WC —) W + C).

Referring to the impurity-ﬂux condition, Equation (42), the only remaining factor

for explaining the difference between W and Co proﬁles is the solubility X i, eq . While

the solubility of Coin Fe is much higher than the solubility of W in Fe [Hansen et al.,
1958, p. 472], the actual amount of Co dissolved in the chip is capped by the supply of
decomposed tool material, and hence should be in the same molar proportion to W as
found in the tool. To check the W-Co ratios before and after cutting, the data of
Subramanian et a] [1993] is converted into molar bases in Figures 12a, b:

Original data: mole fraction of w in chip Original data: mole fraction of Co in chip

 

_‘
J;

 

0-8 240 rn/min

240 m/min

0.6

    
 

 

 

 

 

 

mole fraction ( mol/(100 mol of chip) )

 

mole fraction ( mol/(100 mol of chip) )

 

0.4 150 m/min
0.2
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
depth into chip (pm) depth into chip (pm)
(a) (b)

Figures 12a, b. Experimental concentration profiles, deduced from Subramanian
et al. [1993]. Empty and filled circles denote data points. in (b),
the 240-m/min curve asymptotes to the same non-zero value as
the 150-m/min curve.

35

Incidentally, the maxima and minima of the concentration proﬁles for the W and Co for a
given speed occur at the same depths, which shows that Co has similar mobility as W as
both constituents move through the chip lattice. We now check the molar ratio of W and
Co within the chip and compare it to the corresponding molar ratio within the fresh tool
material. The areas under both the W and the Co proﬁles are assumed to be
representative of the total moles per unit length in the chip ﬂow direction. For the case of
240-m/min cutting speed, we ﬁnd that the W-to—Co ratio of absorbed moles per unit
length (i.e., the ratio of the W and Co areas for the case of 240-m/min cutting speed) is
5.34-to-l. On the other hand, it turns out that the molar/atomic composition of the tool is
45.6% W, 45.6% C, and 8.1% Co (with a trace of TaC-NbC), or a W-to-Co ratio of 5.63-
to-l. Considering that the 5.34-to-l ratio is calculated from a single sectioned proﬁle, it
is considered a good match to the 5.63-to-1 ratio. Hence, we ﬁnd consistency with the
hypothesis that Co maintains its molar ratio with W before and after cutting. For the case
of lSO-m/min cutting speed, the same process of calculating the W-to-Co molar ratio per
unit length from Figures 12a, b returns a low value of 2. The cause of this low value is
the non-vanishing tail in the Co proﬁle after a depth of 0.1um in Figure 12b. We choose
to ignore this tail on the hypothesis that the detected Co traces are related to the W traces
in roughly the same molar ratio inherited from the fresh tool, and the fact that the W
proﬁles decay to zero.7 Using maximum values from the proﬁles instead, we obtain a W-
to-Co molar ratio per unit length of 6.25. This ratio is consistent with an increased

amount of interface wear (e. g., abrasion) not accountable by dissolution, and it is

 

7 The other interpretation would be that the W proﬁle decays asymptotically to a small, but nonzero value,
but this is judged less likely because such a non-vanishing tail with zero flux would require us to introduce
a condition of mass buildup at some pre-determined depth. This issue will be revisited later.

36

explained as follows. Suppose that, for every 100 moles of tool constituents decomposed
(i.e., prior to diffusion and advection), 0.96 moles each of W and Co are removed due to
various interactions at the interface, and are made inaccessible to diffusion into the chip.
Then the W-to-Co molar ratio would change from 45.6-to-8.1 prior to cutting, to (45.6 —
0.96)-to-(8.1 — 0.96) as detected within the chip afterwards, or equivalently, 6.25-to—1. In
Section 3.5.4, this calculation is revised to make use of the concentration ﬁelds that have
been extrapolated from experimental data using the Frank-Tumbull interpretation.

To summarize, our interpretation of the proﬁle data of Subramanian et al.
indicates that the W-Co molar ratio found in the chip reflects, in the main, the W-Co
molar ratio found the tool. Differences in the W-Co molar ratios before and after cutting
can be used to gauge the amount of worn cobalt that is not attributed to dissolution wear.
Among non-dissolution-based wear mechanisms are the deposition of tool constituents
onto and subsequent advection by the chip (e. g., asperity exchange favored by surface
energy changes [Rabinowicz, 1965], 2-body abrasion and deposition through applied
pressure, plowing [Suh, 1986], etc), the sweeping action by loose debris in rolling friction
(i.e., 3-body abrasion [Rabinowicz et al., 1961; Rabinowicz, 1977]), the deposition of
tool constituents onto rolling debris at high pressure—with the number of mechanisms

likely limited only by the imagination.

3.5. Results of the species-transfer problem

The interfacial temperature ﬁeld is gotten from the analysis of Chapter 2, and the

result for the 4-m/s cutting speed is reproduced in Figure 13. It is seen that the interfacial

37

temperatures can vary signiﬁcantly from its average value. The temperature jump in the

region near the end of contact is due to a hypothesized region of loose contact.

'C Extension of Average Interfaclal Temperature Data
1600 ' '

 

tool average: 1265’C \_

  
 

1 400
chip average: 1235‘C

\

Subramanian et al.: 1250'C

   

12007'
1 000 >
800*
600
0 as 1 15
Distance along interface from cutting edge (m) x 10{3

 

 

 

Figure 13. Variation of temperatures away from the average

Figure 14 shows the region of interest and the numerical mesh used for the

species-transfer problem:

depth into chip (m)

 

0 1 2 3
distance along flow direction (m) X 106

Figure 14. Mesh used in the 1st-order, species-transfer probelm

The x axis from 0 to 1.8 mm corresponds to the tool-chip interface parallel to the ﬂow
direction where interstitial impurities are introduced. The y axis corresponds to a very

small portion of the primary shear zone where vacancies are generated. The y direction is

38

the chip-depth direction along which concentration proﬁles vary, and is plotted on a
highly magniﬁed scale. The depth distances are so small that the temperature, and hence
the material properties, only vary in the x direction.

A product of interstitials and vacancies leads to the humped concentration proﬁle
that is the basis of the present application of the Frank-Tumbull mechanism; for this
purpose, a special vacancy distribution is chosen. An example of one such distribution,

plotted topographically on the mesh of Figure 14, is shown in Figure 15a:

Steady-state X i (W-lnterstitial)
concentration in iron, 4-m/s

cutting speed

Steady-state Xv concentration
in iron. 4-m/s cutting speed

.0
0
00
l
I
I
I

           
     
    

2 E
‘5 . 2
TE .- .‘\\\“ :‘ E a
:0-02- .101 ~ ‘ *1 C
,s‘.\\l\11\\\“mg. 1
V _ haw , ‘1 v
.5 0.01 1 1 “ : L3
.2 0 . ‘ -
g 4 3 g
E 7 "’" 1 E -7 4
x 10 depth 0 0 x 10 depth

0
ﬂow direction (m)

x 10
. ' flow direction (m) . .
dlrectron (m) direction (rn)

(a) 02)
Figures 15a, b. Sample XV and X; distributions

The rise of X V in the depth-direction (y—direction) is obtained empirically by targeting
the iterated solutions of the Frank—Tumbull nonlinear equations (Equations (44)-(46))
towards a match with the rise of the proﬁles found by Subramanian et al., and by
retroactively adjusting the input supply of vacancies within the primary shear zone
(Equation 7) until the match is within tolerance. The iterated steady-state solution, based

on the method of quasi-linearization [Kwon and Bang, 2000, p. 539], is set up so that the

39

value of X S after the ﬁrst iteration is a simple product of X i and X V ; subsequent
iterations account for species transfer.

In Equation (41), kv is set to be a hyperbolic tangent function of y that is used to
introduce a quick rise of vacancies within the primary shear zone, and X V, eq controls

the height of the vacancy distribution. The form of vacancy generation in Equation (41)
is numerically less demanding than a Dirichlet condition, and it also has the property that
as the advection speed increases, the height of the vacancy distribution decreases. The
underlying assumption is that the amount of vacancies generated is bounded, and the
height of the plateau generated by advection is determined by a competition between the
vacancy-generation rate and the rate at which vacancies are carried away. A smooth rise
in the x direction is also introduced to avoid the need for a highly dense mesh.

Figure 15b shows the interstitial distribution (solution to Equation (44)) with
which the vacancy distribution interacts. For convenience, the parameters governing the
boundary condition of the interstitial distribution are collected in Table 4.

Before ﬁnishing the discussion on the interstitial distribution which leads to the
main results, a digression is taken to explain the two most important parameters, namely,

diffusivity and solubility.

40

 

Table 4. Parameters used in the lst-order problem (mass transfer)

3X - Agdecom A- '
M355 flux Dr a_nl= Idiss [ AOVFe CXP['—'7z'E—E‘ exp Tiféﬂ '(Xi.cq — Xi)

A0: I (assumed; as long as this number is of order 1, the sensitivity of the steady-state results to
its variations is low)

ldiss : 2.5 X 10—8 m (2: lattice distance; D. Gaskell, Introduction to Metallurgical
Thermodynamics, 2nd ed., Hemisphere Publishing Corporation, 1981, pp. 117-118)

VFe : 1X 10+13s-l (z Debye frequency; D. Gaskell, Introduction to Metallurgical
Thermodynamics, 2nd ed., Hemisphere Publishing Corporation, 1981, p. 118)
Agdccomp : 9000 cal/mol (from the reaction WC—)W+C; O. Kubaschewski, E. L. Evans, and C.

B. Alcock, Metallurgical thermochemistry, 4th edition, Pergamon Press, Oxford, 1967, p.
429. Actually, it is 9000—0.4t9cal/mol, but exp(— (9000 - 0.46)/(Rt9)) becomes exp(+0.4)-

exp(- 9000 / (R 6)) , and the exp(+0.4) term is grouped into the constant A0 .)

Macaw, : 18000 cal/mol (4-m/s cutting speed); 21000 cal/mol (2.5-m/s cutting speed) (curve-ﬁt
value that controls the maximum of X s (adjusted after abelow is determined»

X i, gq = aX W, eq and X 5, eq = (1— a)X W, eq , where a = 0.0375 (4-rn/s cutting speed), and
a = 0.0150 (2.5-m/s cutting speed) (ais the ﬁrst curve-ﬁt value determined)

Xw’ eq = 9x10-11TC3-1.23x10_7T62. + 7.6X10—5 Tc -9.8)<10”3 , where TC is temperature
in Celsius. (Value of X W, eq taken from solubility curve of W-Fe binary phase diagram, p.

734 of Constitution of Binary Alloys, 2nd edition. Metallurgy and Metallurgical Engineering
Series, McGraw-Hill, New York, 1958. Cubic best ﬁt from Microsoft Excel)

X Co, eq = a} X W, eq : at: 0.125 (4-m/s cutting speed); at: 0.185 (2.5-m/s cutting speed) (§3.5.4)
Remaining parameters:

X V, cq = 0.035 (curve-ﬁt value that affects the rise of X 5 through its inﬂuence on X V ’5 rise)

Di : 1x10"10'925 m2/s (4 m/s-cutting speed); 1x 10’11'725 m2/s (2.5-mls cutting speed)
(curve-ﬁt value that controls the spread of X s (adjusted after ahas been determined»

Dv : 1x10"14 m2/s (curve-ﬁt value that prevents X V from losing its shape due to advection)

Ds : neglected in this analysis

kv : quick rise to a value of 0.25 (curve-ﬁt function that controls the rise of X V , and hence X s ,
within the primary shear zone)

0+5 s'

kfwd : 1x1 1 (curve-ﬁt value that controls the Frank-Turnbull reaction rate during

relaxation)

p0 : 0.3 x 10+9 N/m2 (force/contact area; needed for studying effect on diffusivities, both in
decreases in magnitude and variations across the ferrite-austenite transition temperature)

 

 

 

3.5.1. Unusually low values of mass diffusivities

41

Figures 16a, b show a comparison of the tungsten-interstitial diffusivity

DW-in-Fe and iron-vacancy diffusivity DV used in the simulations versus those found

from tablessz

 

 

    
    

       

 

 

 

 

 

 

 

 

 

 

 

Comparison of diffusivities in Comparison of diffusivities in
iron, 2.5-m/s cutting speed iron, 4-m/s cutting speed
c
-u . . . .4 - . .
A DW-in—F93 table-lockup : own“: table-lockup
A $ ‘6 > \
& -8> .
NE W tablelookup "E by: table-lockup
.. s -8»
‘< -10 2:
.E‘ 22 j
.2 g -10*
s .12 -—- — -*——- a ~ —-—-—— -~—-
% Dw-in-;:e: used in simulation 3 -12 DW-in-Fe3 used in simulation .
V 9
0244* " —\ 3 -14
—°— DVI used in SimUIBIion \ 0V: used in simulation
'1"0 075 1 115 4‘0 015 1 115
distance from cutting edge x 10'3 distance from cutting edge x 10'3
along tool-chip interface (m) along tool-chip interface (m)

(a) (bl
Figures 16a, b. A comparison of available values of diffusivities versus actual
values used in simlations for (a) 2.5-m/s cutting speed, (b) 4-m/s
cutting speed

A small DV prevents the vacancy distribution from losing its shape downstream of the

primary shear zone, which is needed to model the quick rise (within a 0.1-um distance) of
the concentration proﬁle of Subramanian et al. [1993] (Figures 1 and 12 in this work).

Also, Di must be small enough to force a quick decay (also within a distance of 0.1 pm).

Values of diffusivities available in the literature are simply too large to achieve these

 

8 Spatial dependence of mass diffusivity is due to the background temperature and pressure ﬁelds from the
0th-order problem. The sharp drops in mass diffusivities near the origin correspond to the ferrite-to-
austenite transition point, which, at a pressure of 0.33e9 Pa, is approximately 718°C. Table-lookup values
are taken from Elliot et al. [1963] as Arrhenius functions of the absolute temperature. Constant values of
mass diffusivities are adopted in the simulation for a practical reason: a sharp drop in diffusivity requires
mesh reﬁnement for resolution and numerical stability, especially in the presence of strong advection.
Physically, however, most tool wear occurs downstream of the cutting edge (e.g., the boundary values of
interstitials in Figure 15b) where the mass diffusivity is almost constant, hence the use of constants.

As mentioned in Section 3.4 (discussion of Figures 12), the diffusivity of Co is rate-limited to that of W.

42

effects. In fact, the use of the table-lockup diffusivities leads to vacancy and interstitial

distributions that are shown Figures 17a, b:

Xv concentration In iron using table- Tungsten-Xi concentration in iron using
IOOkUp diffusivity, 2.5-m/s cutting speed table-lockup drffusivrty, 2.5-W8 cutting

mole fraction ( moV(1 mol of chip) )
mole fraction ( moV(1 mol of chip) )

   

Figures 17a, b. X V and Xi distributions using table-lockup values of Figures 16
In contrast to the distributions of Figures 15, neither a rapid—rise function nor a decaying
function is present, and the product of the distributions in Figures 17a, b will not result in
a humped distribution in the y-direction as required.

Pressure and damage are two possible reasons for the seeming prevalence of low
values of diffusivities. Although pressure does not enter explicitly in the equilibrium
condition (equality of Condition (v) in Table 2), it tends to decrease the values of

diffusivities. An expression by Shewmon [1989, pp. 84-85] illustrates this relationship:

36 1n 2 D ____‘7actiRy;tion , (47)
p0 ldiss VFe 6ﬁxed

where activation is the activation volume (in m3/mol), and it is known to be smaller than

the molar volume of the lattice metal (viz., iron). Assuming constancy of all terms except

D for simplicity, we have that

43

D (P0) = exp[— Foil—activation J (48)

D(1 atmosphere) R 6
For a pressure of p0 of 0.33 X109 N / m2 at 1000K and a maximum fir-activation value of

M Fe + PFe z 7 x1076m3 / mol , we arrive at a maximum drop in diffusivity by about a
factor of 3.

Damage mechanisms may be needed to achieve the levels of reduction in
diffusivities as suggested by Figure 16. That is, the concentration proﬁles from
experiments may reflect shear damage so severe that the mobility of impurities is
signiﬁcantly impaired. However, no simple, independent test of this hypothesis is

available at this time, and the mass diffusivities are considered curve-ﬁt parameters.

3.5.2. The determination of interstitial solubility; post-machining relaxation

In an early development of the model for tungsten impurities, X i, eq and X 5, eq
are both set equal to X W, eq because the fractions of interstitials and substitutionals
among a given amount of impurities are unknown. Figure 18 shows the result of a
simulation for the steady-state, substitutional distribution under the assumption X i, cq =
X W, eq = X S, eq for the 2.5-m/s cutting speed. The section taken—where comparison is

made with the proﬁle from the data of Subramanian et al.—corresponds to the chosen
distance of x = 1.8 mm; this is the end of the tool-chip contact where impurity

concentrations are near or at their maximum for a variety of loading conditions.

Comparison of w profiles in
x 104 iron, 2.5-m/s cutting speed

 

solid: experiment
dash: computed )6»,

 

 

 

 

mole fraction ( moV(1 mol of chip) )

1 2 a 4
depth into chip (m) x 10'7
Figure 18. Good match in an early run that turns out wrong

Despite a good match, it is remembered that the measured concentration also
includes the residual interstitial distribution at the end of cutting (i.e. steady state). The
residual X i ﬁeld is given as a topographical plot in Figure 19a, and its sectioned proﬁle
is superposed onto Figure 18 and presented in Figure 19b:

Steady-state X i (W-lnterstitial) Comparison of W profiles in iron at

concentration in iron, 2.5-m/s 10‘3 x = 1.8 mm, 2.5-m/s cutting speed
x

a

U

 

 
  
  
     

steady-state X i (or residual
interstitials after machining)

solid: experimental
profile of W in Fe

dash: steady-state X5

mole fraction
( moV(1 mol of chip) )

‘ \\\\\\\\\\\\\\\\

.s,
it“
I

\\\\
\\\\\\\\\\\\\\\ 1

\

\\\\‘& x u. R \\ \\\\\\

\\\\\\\\\ \\\‘x\‘ _ \ \\\\\\\\\ “\-

\\:§:§§‘\‘\e¢s\e it‘sxxsx‘ssxsgt

\\\\ _ \‘ .,

\\‘\:\\\\\‘\\§‘C\ .. w,
a

/
Ill/l,

mole fraction ( moV(1 mol of chip) )

 

 

-1 1 . L

 

2 3 4
depth into chip (m) x 10'7

 

Figures 19a, b. Why the good match of Figure 18 does not work

The sum of the steady—state X s and X i distributions no longer agrees with data.
Evidently, X i, eq , which controls the maximum of X i along the tool-chip

interface, must be decreased, as reflected by the multiplicative factor ain Table 4. In

45

fact, the value of afrom the curve ﬁt represents the fraction of impurities that can be

identiﬁed as interstitial impurities, as seen from the following equations:
qu = axwﬂq and ngq =(1—a)Xw,eq (49)
Figure 20a show that decreasing X i, eq forces the steady-state X 8 distribution to decay

too rapidly. A completely analogous situation occurs for the case of 4 rnls cutting speed:

Comparison of W profiles in iron at Comparison of W profiles in iron at
x 104 x = 1.8 mm, 2.5-m/s cutting speed x 106 x = 1.8 mm, 4-m/s cutting speed

N
O

 

 

 
 
 
  
    
 

steady-state Xi steady-state x;

d
9'

    
   

.5
9

experimental profile of i

/WinFe

steady-state X3

experimental profile of
/ W in Fe .

steady-state X3

 

 

 

 

mole fraction ( moV(1 mol of chip) )
mole fraction ( mol/(1 mol of chip) )
A

db

 

 

 

 

I
(.11

1 a {3 4 1 a a 4
depth into chip (m) x 10‘7 depth into chip (m) x 10'7
(a) (b)

Figures 20a, b. Attempts at matching experimental W-concentration profiles using
only steady-state W distributions from the model

Increasing Di delays the decay of X s in Figures 20, but the spread of the distribution is

unacceptably increased (not shown in Figures 20) beyond the experimental-data curve.
At this point it is thought that a relaxation process may take place after cutting—

but before quenching—such that the tail of the steady proﬁle would ‘ﬁll up’ to match the

experimental curve. To test this hypothesis, we model Equations (37)-(39) with time

dependence, zero advection speed, homogeneous ﬂux boundary conditions, and with the

X i , X V , X S distributions inherited from the steady-state calculation. Also, the

temperature ﬁeld is preserved as quenching has not yet occurred. The result of such a

46

relaxation run for the case of 4—m/s cutting speed is shown in Figure 21a. The solid X 5
curves correspond to the relaxation times of 0, 0.9, and 3.6 milliseconds; it is found that
the peak of X s rises to a maximum value at approximately 0.6 milliseconds before it
decreases thereafter. For a choice of a uniform and steady reaction-rate constant kfwd ,
the match to experimental data is considered reasonably good.9 Figure 21b shows the
relaxed X 5 ﬁeld at the ﬁnal time; a practical choice of the coarsest mesh possible leads

to Wiggles on order of 3% of the maximum value of X s :

 

   
   

 
  
    
        

 

 

 

Xs-relaxation profiles of tungsten in Relaxed Xs concentration in iron

A 4 AX 10-3 iron after cutting at 4~m/s cutting after cutting at 4-m/s cutting speed
A Iu r » — _ _ \ . .
o. , . ,
E 2 2 [I 4I_ _ _ ‘/
O 8 1 . . Q I /
3 dash: experimental data E 3 k
E: 6 1: no relaxation is X 10' I
,_ 2: 0.9 millisec. relaxation —° 8 I L“ Ar
‘5 . . - - E , ,. ,
g 4 3.3.6 millisec. relaxation . r, 6 I, :1 /
v g 4 , .,;/,/,///
‘9 s 2 ’
7: o 1 "5 o
_ r0
2 4 . . . ‘5

‘0 1 2 3 a

.7 E

depth into chip (m) x 10
(a) (b)
Figures 21. (a) Relaxation of X5 after cutting; (b) Xs after 3.6 millisecs. of relaxation

Figure 22a shows the ﬁnal outcome of the relaxation simulation (starting from the state of

Figure 20b) at 3.6 milliseconds. Figure 22b plots the volume under the X s plot as a

 

9 The use of a non-uniform kfwd can improve match but extra physical mechanisms have to be
introduced. The size of kfwd determines the speed of relaxation; for example, the elapse time for the
relaxation process to nearly exhaust the tungsten X i in Figures 19b is roughly 3.6 milliseconds,
corresponding to kfwd = 1e5 s". The uniformity of kfwd implies uniformity in reaction rates given a
uniform driving force; however, because a low value of DV maintains a vacancy—depletion zone near the
interface, there are no renewed, Frank-Turnbull reactions there. Consequently, the rise of the X s hump is

not signiﬁcantly altered during post-machining relaxation.
The elapsed time after machining but before quenching of the chips is not given in the original data.

47

function of relaxation time after cutting. It is seen that the residual X i ﬁeld is essentially

exhausted through its interaction with X V .

 

 

 

 

 

 

 

 

Comparison 0‘ W profiles in iron at Evolution of volume under tungsten
x = 1.8 mm, for a relaxation time of x8 plot during relaxation and after

A x 1 04; 3.6 millisem‘atter 4—mlsvcuttlng 1 4x 10.12 imam at a speed of 4 ntls

E o

‘5 8- experimental profile

:5 of W in Fe "E 1.3.

E e» .5

T: relaxed X3 ‘8: 1.2.

E t

v 4» 2

5.5 , E

o relaxed X . 1.1

s 2'

E

(E) G A 1 . . .

0 1 2 3 4 0 1 2 3
depth into chip (m) x 10'7 relaxation time (S) x 10'3

(a) (b)

Figures 22a, b. A check on the exhaustion of tungsten X i during relaxation

3.5.3. Solubility as the maximum amount of dissolution wear achievable

Figures 23a, b show how the equilibrium solubility X i, eq (as deﬁned by Table 4)

places bounds on the amount of a tool constituent that is made available for dissolution in

the chip.10 Plots are shown for tungsten; since X i, eq for cobalt is some fraction of
X i, eq for tungsten (viz., the product of aand (p as discussed in Sections 3.5.2 and 3.5.4),

the plots for cobalt are exactly analogous. If Agactivn is very large within the expression

for the rate constant of dissolution kdiss (the second Arrhenius exponential in Equation

 

to The wiggles in X i near the origin reﬂects the difﬁculty with which the coarse mesh tries to resolve a

sudden increase. Fortunately, these wiggles do not pose a signiﬁcant problem in the convergence of
numerical solutions, but they do affect accuracy. The tradeoff in mesh coarseness is necessary, as the
relaxation runs cost well over 8 hours to run on a computer with a 1.2 GHz processor.

48

(43)), then the spatial increase of X i towards its maximum amount is delayed; as

agacﬁvn increases indeﬁnitely, X i tends uniformly to zero because the boundary ﬂux

 

 

does.
Comparisonotboundarngand Comparisonoiboundarngand
X i. sq (tungsten) at a cutting X i. .q (tungsten) at a cutting
12x104r speedot2.5 ml: T 5x10: speed ot4mls

.5
9

  
 

  
 

     
       

 

 

 

 

 

 

.3 .3

'5 ‘5

”5 "6

3 > 4 _

E 8 3 3» l

Q 6~ , E

8 E 2

V 4 . V

C C 1 _

3g 2. steady-state X; at % steady-state X; at

g interface boundary g 0 interface boundary

2 O 2

'2 a i 23 e». '1 a a é 5

distance in the ﬂow direction (m) x 10° distance in the flow direction (m) x 10°

(8) (b)
Figures 23a, b. A check of Kramer's hypothesis that dissolution wear is
bounded by solubility

It is seen that in going from 2.5 m/s to 4 m/s, the maximum of the boundary
supply of interstitials increases by 40 times. While the Arrhenius dependence on
temperature of the boundary interstitial ﬂux is inﬂuential (Equations (42) and (43)),
Figures 23a, b show that solubility is the controlling factor. Further, the sudden change
in slope at 1.2 mm reﬂects the drop in the interfacial temperature of the chip in relation to

the tool in the last thirds of contact (see Figure 13).

3.5.4. Amount of worn cobalt not due to dissolution wear

By comparing the molar ratios of W to Co within the chip and within the tool, the

amount of worn cobalt not due to dissolution wear can be estimated. Relaxation of the

49

tungsten proﬁle after cutting at 4 m/s has already been shown in Figure 22b. Figure 24a

shows snapshots of the relaxation process for cobalt impurity atoms after cutting at 4 m/s,

and Figure 24b is a check on the exhaustion of X i -cobalt during relaxation:

Evolution of volume under cobalt X3

- i m '
Xs re axation pro es of cobalt in plot during relaxation and after

4 iron after cutting at 4-rn/s cutting

 

 

  
  
  

 

 

 

 

 

 

 

 

A x10 1M1043 . Cutting‘ataspeedot-tm/s'
.9 15 '
6 i — 2
‘5 1 dash: experimental data 1,3-
TE) 10» 1: no relaxation ‘ NE
5 2: 0.7 millisec. relaxation .5 1.7.
E 3: 4 millisec. relaxation :3
(U

E 5» E 1.6»
E \- 22 l 15
2 O» 1
O

0 1 2 3 4 1 40 1 2 3 4

distance into chip (m) x 10" relamtion time (s) x 10‘3

(a) (b)
Figures 243, b. The relaxation process. A check on the exhaustion of cobalt Xi

during relaxation

Concentration data suggests that the mobility of Co is similar to the mobility of W as tool
constituents diffuse into the chip lattice (Section 3.4). For simplicity, then, the mass
diffusivities of Co and W are set equal. The solubility of Co is source-limited to the

amount that is originally present in the tool; in particular, the original W-Co molar ratio

within the tool is approximately preserved. In the simulations, X Co, cq is set equal to
¢X W, eq , with (oas given in Table 4.1' Hence, with the exception of w , all parameters

in the W and Co simulations are the same. Given these assumptions, the agreement is

deemed reasonable. The thick tail of the experimental curve has been discussed in

 

H That 47 has to be empirically determined may be a reﬂection that not all Co particles enter the chip.

50

Section 3.4 (Figure 12b and discussion).‘2 The initial dip in the experimental data of
Figure 24a (dashed curve) may represent residual X i -cobalt due to quick quenching;
however, no independent conﬁrmation is available at this point.

The quotient of the integrated X s ~tungsten evolution in Figure 22b and the

integrated X s ~cobalt evolution in Figure 24b is taken to be the evolution of the total W-

Co mole fraction within the chip. This W-Co mole-fraction evolution, corresponding to

the case of 4-m/s cutting speed, is shown in Figure 25a:

 

 

 

 

 

 

 

 

Evolution of W-Co mole fraction Evolution of W-Co mole fraction
during relaxation (after cutting at during relaxation (after cutting at
a speed of 4 m/s) a speed of 2.5 m/s)
2 7.35 . . a 2 5.2 - ﬂ
8 8
.5 1 .2.
E .
z 725* J z 5'15
E 3
3 mi -
g E, 5.1 . l
v 7.151 v
s a
.. i 'o-a
’3 7" g 5.05- 4
1.3 7.05i 3
o o
E 7 . . . E 5 - . . .
0 1 2 3 O 1 2 3 4 5
relaxation time (s) x 10“ relaxation time (s) x10"i

(a) 0))
Figures 25a, b. The W-Co ratio within the chip as residual X i ’s are exhausted

during relaxation tor (a) 4-m/s cutting speed, (b) 2.5-m/s cutting
speed.

The X i of both species approaches exhaustion when the amount of X 8 stops increasing;

hence, we take the ﬁnal W-Co mole fraction within the chip to be approximately 7.35-to-

1 as indicated by Figure 25a. We note that the molar composition of the tool is 45.6% W,

 

12 That is, we disregard the thick tail as an artifact because non-zero asymptotes at large depths do not
reconcile with the usual picture of exponential decay. If need be, the thick tail can be modeled either by a
mass buildup at some depth y, or by changing the way the vacancy distribution increases with depth. The
present vacancy distribution saturates to a plateau as seen in Figure 153.

51

45.6% C, and 8.1% Co (with a trace of TaC-NbC), or a W-to-Co ratio of 5.63-to-1. To

produce a 7.35-to—l ratio within the chip, we hypothesize that for every 100 moles of tool
decomposed, Aw moles of W, Ac moles of C, and AC0 moles of Co are lost due to

non-dissolution mechanisms; that is, the apparent W-Co molar ratio as seen by the chip is

4i6—Aw

—8-1-Aco =7.35. (50)

Aw , Ac , and AC0 are ﬁxed by one extra piece of information from measurements:
When the mole fraction of intact WC lost to non-dissolution wear is equated to the
residual WC found (the chips are dissolved in hydrochloric acid), it is found to be
@34un) + (3517ttg/Mo) = 0.066 [Subramanian et al., 1993, Table 1], where M0 is the
molecular weight of WC. Alternatively, the total number of moles of WC lost to non-
dissolution wear out of 100 moles is

Aw + AC = ZAW = 6.6 moles. (51)

The solution to Equations (50) and (51) is that Aw = 3.3 moles and AC0 = 2.345 moles.

Hence, out of every 100 moles of tool material decomposed, about 2Aw + AC0 z 9

moles are unavailable to the dissolution mechanism for the 4-m/s experiment. A similar
calculation is carried out for the case of 2.5-m/s cutting speed, with ‘7.35’ in Equation
(50) replaced by ‘5.18’ (Figure 25b), and with ‘6.6’ in Equation (51) replaced by ’33.6’
[Subramanian et al., 1993, Table 1]. The corresponding solution is that, out of every 100
moles of tool material decomposed, 16.8 moles of W, 16.8 moles of C, and 2.54 moles of

Co are lost to non-dissolution wear at a cutting speed of 2.5 m/s. More sophisticated

52

models of non-dissolution wear can be used if information is known about the amount of

tungsten lost beyond the amount that is lost as intact WC grains.

3.5.5. The rate of tool wear as a ﬁeld quantity
An analogous relation to Equation (42) can be written to describe the ﬂux of

material leaving the tool interface

 

3X-
" [Di 3 l J = [decompkdecomp (X i, eq — X i ), (52)
"‘001 tool

where ntoo] is the outward normal distance to the tool interface, ldecomp is used to

render a volumetric rate into an area ﬂux, and kdecomp is the rate constant for the

decomposition of tool material deﬁned by

(53)

Agdecomp ]
Rgtool

kdecomp = BOVtool exp[-

Figure 26a shows a plot of [decompkdecomp for ldecomp z 2.5X10_8m , Bo = 1,

Vtool z 1x1013s'1 for the case of 4-m/s cutting speed, and Figure 26b shows the

corresponding boundary temperature:

53

Value of surface rate constant Tool-interface temperature,

of decomposition for 4-rn/s

 

 

 

 

 

 

 

 

 

4 mls cutting speed
10‘ cutting speed
2.5x . recs
2 A1400»
81200
17; 1.5 g
E ‘97 1000
V 1 8
g 800
“o 0.5 1 1.5 “”0 0:5 1 1.5
distance from cutting edge at 10‘3 distance from cutting edge x 106
along tool-chip interface (m) along tool-chip interface (at)

Figures 26a, b. The surface rate constant for tool decomposition and tool-
interface temperature

The similarity in functional forms suggests that the contribution to the wear contour is a
direct consequence of the interfacial temperature proﬁle, which is recognized early on by
Chao and Trigger [1955, p. 1117].

Recently, Molinari and Nouari [2002] have also reported calculations showing
that experimental wear proﬁles on the tool are consistent with the diffusion mechanism.
Instead of using a boundary flux condition as reﬂected by Equation (42) or Equation (52),
Molinari and Nouari use Dirichlet—type conditions. While the model of Molinari and
Nouari is sufﬁcient to describe the tool-interface contour, the use of a single diffusion
equation for the chip region with a Dirichlet boundary condition leads to exponentially
decaying proﬁles of tool components into the chip; this stands in contrast with the
findings of Subramanian et al. The question of whether Dirichlet or ﬂux conditions are
the proper ones to be applied at the interface may well be decided by details at the

interface. However, it is noted that the boundary condition of Equation (42) behaves as a

54

Dirichlet condition when the dissolution rate constant is high enough to guarantee that the

interstitial solubility is reached along the interface (e.g., Section 3.5.3).

3.6. Measurement methods for checking the distribution of vacancies

Macroscopic measurement methods of vacancies such as dilatometry [Paris et al.,
1975] measure bulk changes in volume and are not suitable for charting spatial
distributions. Evidently, highly sensitive methods operating at the level of
microstructures are required. Two such methods for measuring the distribution of
vacancies are ﬁeld ion microscopy and the tracking of positron annihilation (which
includes positron lifetime spectroscopy and gamma-ray angular correlation). Although
experimental work involving these two methods has yet been applied to the machined
chip, a brief description is included here for completeness.

In ﬁeld ion microscopy, the specimen is sharpened to a submicron-size point and
a high voltage is applied to create an electric ﬁeld at the tip that alternately polarizes and
attracts, then ionizes and repels, the atoms of an imaging gas (e.g., neon). The gas
molecules are energized according to the protrusions and indentations of the
microstructure at the specimen tip, and are propelled in a radial direction towards a
phosphor screen to produce an image. The ﬂuorescent image thus created is then a
magniﬁed picture (approximately x106) of the intersection of the specimen
microstructure with the surface of the needle point. Useful information is extracted
through careful comparisons with crystallographic models. Lattice defects such as
vacancies (and in general, point, line, and planar defects) are discernible through changes

in image brightness and contrast. A mass spectrometer can also be used to identity

55

impurity atoms. Although powerful, the use of ﬁeld ion microscopy requires many
control experiments to eliminate visual artifacts; a general discussion and a list of
references are given in [Miller and Smith, 1989; Miller et al., 1996].

In the positron method, a stream of positrons created from a radioactive source is
directed into a solid. Within the bulk of the solid, these positrons are trapped by lattice
imperfections. As positrons are annihilated upon contact with electrons, gamma radiation
is released, and a map of the traps, and hence the lattice imperfections, is created.
Because positrons are point particles, the resolution of this non-intrusive method is
extremely high. However, as positrons are not trapped by interstitials, this method is
limited to detection of “open-volume defects” such as vacancies [Hautojarvi, 1979;

Krause-Rehberg and Leipner, 1999].

3.7. Non-uniqueness of Frank-Tumbull representation

The most important of the parameters for curve ﬁt are a, aged,“ , and the
diffusivities (Sections 3.5.1, 3.5.2, 3.5.3). Of these, Agacﬁvn is the parameter with the

most amount of variability. Referring again to Figures 23, when Afg‘mm is less than
about 35000 cal/mole, the entire contact interface from 0 to 1.8 mm achieves saturation

(i.e., X i = X i.eq uniformly). As Maw,“ increases, X i becomes less than X i.eq

beginning from the cutting edge and moves downstream, but curve ﬁt at the x = 1.8-mm

section can still be performed. As mentioned earlier, Mam," can be increased to levels
where X i becomes uniformly less than X i, eq along the contact interface, until it

becomes difficult to match the peaks of the concentration proﬁles of Subramanian et al.

56

To better estimate the value of Agacﬁvn , experimental concentration proﬁles are needed

at different distances away from the cutting edge.

The previous discussion is based on the idea that one single proﬁle can be
approached spatially in an inﬁnite number of ways. The calibration constant A0 in Table
4 is also non-unique by at least an order of magnitude, mainly because the rate constant is
dominated by the exponential terms. Further non—uniqueness of representation may arise
from the non-linear nature of the governing equations. However, restrictions regarding
the size of a(Section 3.5.2) and the values of the diffusivities (Section 3.5.1) are

expected to hold in all circumstances.

3.8. Notes on the numerical method

The ﬁnite element source code is modiﬁed from 2D programs for the heat
equation taken from the text of Kwon and Bang [2000, Examples 5.9.1 and 6.6.2]. In the
Oth-order problem, the main additions are the advection ﬁeld and coupling between
interfacial tool and chip temperatures. In the lst-order problem, the program is enlarged

to model 3 scalar ﬁelds that mutually interact through advection, diffusion, and reaction.

In the transient problem of relaxation, X 5 cannot be explicitly expressed as an

algebraic function of X i and X V ; hence, it cannot be eliminated as in the steady-state

problem. The Newton-Raphson method (Crank-Nicolson time integration with
nonlinearity iteration on the solution at the new time step [Zienkiewicz and Taylor,
2000a, pp. 6-8]) proves stable not only for the differing magnitudes of coefﬁcient

matrices, but also for handling sharply varying gradients at small regions of the mesh

57

where wiggles in the solution may be present. Small wiggles are tolerated as a matter of
practicality due to the use of the coarsest meshes possible for the quickest solution

speeds, but with a compromise in accuracy. In separate tests, wiggles are minimized by

local smoothing of any sudden changes in the way kdiss varies near the cutting edge, but

the impurity proﬁles and distributions do not show signiﬁcant changes.

Because the region of interest in the mass-transfer problem measures roughly 0.2
microns by 1800 microns, elements severely elongated in the ﬂow direction (Jr-direction)
have been used. Computationally, the effect is equivalent to neglecting the x-diffusion
term, which implies that the relaxation results are only valid for time scales much shorter
than that which is necessary for mass-transfer in the ﬂow-direction x. This is deemed
acceptable for the geometry considered. Also, in the present case where most boundary
conditions are of the flux type, the desired level of mesh reﬁnement to minimize or avoid
wiggles can be obtained by trial and error (as opposed to the case of Dirichlet boundary
conditions, where limits are placed on the element Peclet number [Gresho and Lee,

1979]).

58

Chapter 4

CONCLUSIONS AND SUMMARY

In ascribing to the experimental concentration proﬁles a physical interpretation
using the Frank-Tumbull mechanism, we have made use of many assumptions, varied a
number of parameters to achieve curve-ﬁt, and calculated quantities that reﬂect the

implications of the model. Here we provide a summary.

4.1. A list of assumptions made, with comments

(1) The work/chip material is assumed incompressible.

Implication: The energy equation does not involve a change in volume term,
mainly for simplicity.

(2) Balances of linear momentum and mass are satisﬁed using the assumptions of
classical machining theory. In particular, uniform work and chip ﬂow are assumed
sufﬁcient for the present emphasis on high-temperature phenomena.

(3) Bridgman’s generalized entropy is assumed for a perfectly plastic work piece.
Implication 1: Accessibility of states via reversible processes is not required.
Implication 2: The stress power T : D is completely converted into expelled heat

during plastic deformation, which is used to make sense of the meaning of

absolute temperature appearing in the energy equation.

59

Implication 3: The thermodynamic state functions do not depend on kinematic
parameters of plasticity, as T : D is treated as a heat source.

(4) Weak coupling is assumed between thermal ﬁelds and mass-transfer ﬁelds, which
allows separate consideration of the equations of energy and mass-transfer.

Note: This assumption is justiﬁed by the difference in the sizes of thermal and
mass diffusivities, but its applicability is limited to regimes where the tool has
not worn so much that the background temperature is altered.

Implication: Insular irreversibility [Bridgman, 1950] is limited to the mechanical
aspect of the problem governing the background temperature ﬁeld. A steady
temperature ﬁeld governs the values of constituent solubilities within the chip
in the mass-transfer problem.

(5) Chemical equilibrium is invoked in the description of mass transfer—namely, in the
phrasing of the boundary ﬂux in terms of the probabilities of tool decomposition
and chip-boundary activation.

Note: Our results are consistent with the work of Kramer [1979] in the following
sense. The Arrhenius relation enters the ﬂux boundary conditions as
probabilities altering the value of the rate constant to the interfacial reaction.
In turn, the rate constant governs the speed of rise (spatially and temporally)
of the interstitial distribution towards its saturation value. However, the
saturation value itself, namely the solubility, is unaffected. As a measure of
the maximum amount of dissolution wear possible under all machining

conditions, the solubility remains the most important factor.

60

(6) The order in which the following events occur is immaterial: (i) the system
achieving a steady state in global mass transfer, and (ii) the Frank-Tumbull reaction

reaching local saturation.

Implication: 30/6: terms and F3 are set to 0 for the time-independent equations.

(7) The concentration proﬁles of Subramanian et al. are calibrated in our simulations to
correspond to a cut section at the end of the contact interface, which is about 1800
microns from the cutting edge. This is the location where the concentration proﬁles
are close to achieving their maximum levels. However, simulation results can be
recalibrated to match data at other sections along the tool-chip interface.

(8) The concentrations of defects and impurities are considered small compared to the
chip-material concentration, and the total concentration c is assumed constant.
Implication: c is eliminated in the mass transfer equations in favor of mole

fraction X, and data from binary phase diagrams can be used directly.

(9) Tracer diffusion is assumed for W and Co within the Fe lattice.

Note: This assumption is made in light of a scarcity of data with ternary and
multi-component mixtures [Shewmon, 1989, Section 4.6], but it is fortunately
justiﬁed by the low concentration levels actually observed.

Implications: diffusivities are independent of impurity concentrations;
equilibrium mole fractions of impurities are directly proportional to Arrhenius
exponential terms [Shewmon, 1989, Section 2.4]; forcing terms in the species-

transfer equations are simpliﬁed (Section 3.1).

61

(10) A vacancy distribution is assumed to rise from a very low value near the interface
and jump quickly to a saturation value, mirroring the ﬁrst rise of the concentration
proﬁle of Subramanian et al.

Implication: The assumed vacancy distribution is essential in bringing about the
low entry value of X s before the ﬁrst rise of the humped distribution, and it

reﬂects the statement that dissolution is “vacancy-controlled for small depth,
but interstitial-controlled for large depth.”

(1 l) The C0 binder atoms enter the chip region such that the original W-Co molar ratio
within the tool is approximately preserved: variations are assumed to be incurred
by material loss at the interface.

Justiﬁcation: That the W-Co molar ratio is approximately preserved is checked
directly using experimental proﬁles (Section 3.4).

Effect on simulations: The solubility of Co is related to the solubility of W by
XCo,eq = ¢XW,eq (¢<1)-

Implication: Knowing that the cobalt content is restricted by the availability of
tungsten within the chip allows the amount of cobalt lost to mechanisms other
than dissolution to be estimated.

(12) The W concentration proﬁle is assumed to decay asymptotically to zero and forces
the Co concentration proﬁle to vanish along with it (by Assumption 11).

Note: This is simply a modeling decision that frees us from having to impose

special assumptions to generate a buildup of X s at large depths into the chip.

62

(13) Pressure and damage are two independent ideas that have potential to explain the

drastic reduction in the mass diffusivities Di and DV from table look-up values.

Note: The drastic reduction in diffusivity values is necessary to maintain a
rapidly-rising vacancy distribution and rapidly-decayin g interstitial
distribution; in turn, the product of these distributions gives rise to the humped

X 5 distribution that is matched with the experimental data of Subramanian.

It is noted that the mass diffusivities used for the two cutting speeds do not
differ substantially (Table 4), and are not changed for the post-machining
relaxation simulations. This is viewed to be consistent with damage.

(14) Because W and Co have similar atomic radii to Fe, they are assumed to exist more

favorably as substitutional impurities rather than interstitial impurities. Hence,

X 8, eq for W (or C0) is approximately equal to but is less than X W, eq (or
X Co, eq ), whereas X i, eq for W (or C0) is a small remainder, say aX W, eq (or
aX Co, eq ). ahappens to be a parameter of curve-ﬁt that directly controls the

initial condition of the post-machining relaxation (Figure 20 and its discussion).
(15) The effect of pressure and vacancy concentration on the solubility of interstitials

and substitutionals is not expected to change the present results drastically.

Reasoning: X S is source-limited by the boundary supply of interstitials, or

X 1, eq . Therefore, X s is of the same order as the maximum of X i, eq ,
implying that X S « X 5’ eq . What is being assumed here is that in spite of

changes to X 3, eq due to pressure and vacancy concentration, X S is still

63

expected to be small in comparison to its ceiling value of X 5, eq . The
parameter of importance is judged to be the curve-ﬁt value of X i, eq at the
boundary (viz., the value of a). Error is incurred if changes in X S, eq due to

pressure and vacancy concentration are so drastic that the X 5 distribution is
signiﬁcantly changed by the changes in the solubility limit.

Implication: X 8, eq is based on the solubility of tungsten in iron, which is
obtained directly from a W-Fe binary phase diagram without regard to the
prevalent pressure and vacancy concentration.

(16) X V, eq is treated as a variable parameter of ﬁtting with experimental data in the

absence of an explicit molecular model explaining the formation of vacancies.

(17) The slow decay of X S is hypothesized to be an effect of relaxation after machining.

Note: Given the constraint of a small value of X i, cq , the matching the steady-

state X 8 distribution to the experimental data has proved difﬁcult. The
hypothesis is then developed from the realization that the measured impurity

proﬁle is the sum of the steady-state X 5 distribution and the residual X i

distribution. Computationally, relaxation gives us one more chance to
improve the curve-ﬁt. Physically, a post-machining relaxation actually occurs
in the short time between the end of cutting and the quenching of the chips,
although the exact time elapsed until quenching has not been reported in the

experiments of Subramanian et a] [1993].

(18) Interfacial temperature jumps are controlled by the heat-transfer coefﬁcient ﬁeld, as

discussed and justiﬁed in Chapter 2 and the Appendix.

4.2. A list of key parameter variations used to achieve match with experimental data

(1) X i, cq (the maximum boundary value of impurity concentration) is chosen so that

the residual X i distribution provides a desired alteration of the X s decay during

the relaxation after the steady-state has stopped. This is controlled by the fraction 0'
in Table 4.

(2) X W, eq and X C0, eq are related by a fraction ¢ (Section 3.5.4).

(3) Agactivn mainly controls the variation of X i along the tool-chip interface, in other
words, how much too] components are admitted into the chip lattice. In turn, this
affects how X S varies along the ﬂow direction.

(4) Di controls the spread of interstitials, and hence the location of the maximum of
the X 8 distribution; DV is set so that given a cutting speed, the vacancy

distribution applied within the primary shear zone is advected relatively unchanged

across the length of the contact interface. These values of diffusivities—much

lower than table lookup values—help to achieve the rise and fall of the X 5

distribution, the location Of its maximum, and its spatial spread.

65

(5) The saturation value of X V, eq affects the X 5 distribution only insofar as it affects
the manner in which the X V, eq function rises, but it does not affect the tail of the
X 5 distribution.

(6) For post-machining relaxation, the value of kfwd determines the rate at which the

Frank-Tumbull reaction occurs in relation to the rate of the diffusing interstitial

species. A uniform reaction constant kfwd produces fairly good results;

improvement is possible if the uniformity of kfwd is not required.

4.3. A summary of motivations, results, and quantities calculated

(1) The functional form of the rate constant kdiss for the dissolution of tool

constituents inside the chip is obtained by a model of dissolution that is consistent
with Kramer’s earlier work. Undetermined parameters in the expression for kdiss

are estimated via a curve-ﬁtting process using the concentration proﬁles of

Subramanian et al. [1993].

(1a) Dissolution is deﬁned as the combined events of tool decomposition at the
contact interface and subsequent mass transfer and exchange (advection,
diffusion, reaction) into the chip.

(lb) The boundary supply of tool constituents into the chip region is capped by the
solubility of tool material within the chip. It is also inﬂuenced spatially by
Arrhenius terms governing tool decomposition and the jump frequency of tool

atoms across the contact interface.

66

(2)

(3)

(4)

(5)

(6)

(1c) Low temperatures hinder dissolution. In particular, dissolution wear is
decreased wherever the chip interface is cooler due to loose contact.

The dissolved, concentration ﬁeld within the chip at steady state is calculated using

the proposed Frank-Tumbull mechanism, and curve ﬁt is improved using a

proposed, mass-transfer relaxation that occurs after cutting.

(2a) The proportions of interstitial impurities and substitutional impurities among
the total amount of impurities are given by a curve-ﬁt parameter of the
relaxation simulation.

Table-lockup values of diffusivities for W and Co within Fe are too high to achieve

a humped concentration proﬁle at the observed space scales. Curve-ﬁt values are

used instead, and it is proposed that damage may be the cause for the drastic

reductions in diffusivities.

Table-lockup values of the solubility of Co in Fe are too high to achieve the

boundary concentration of Co observed. To cap the value of Co solubility, we

hypothesize that cobalt remains in roughly equal molar proportions to tungsten
before and after cutting.

With information on the amount of WC grains that remain intact after machining

[Subramanian et al., 1993, Tables 1 & 2], the proportions of worn cobalt attributed

to dissolution mechanisms and non-dissolution mechanisms are estimated.

The functional form of the rate of wear for the tool is given, and it is shown to obey

the same variations as the tool-interface temperature.

67

Appendix

RELATIONSHIP OF INTERFACIAL TEMPERATURE JUMPS
TO THE HEAT PARTITION FIELD USING A 1D PROBLEM

Al. Introduction

When one material slides against another, it is commonly assumed that the
steady—state temperatures of the two bodies match at their contact interface. In this
Appendix, we derive conditions under which interfacial temperature continuity does not
hold using a 1D example. The simplicity of the 1D problem allows us to study the cases
of an uncoupled interface, an interface obeying temperature continuity, and a
convectively cooled interface. It will be shown that when an interfacial-temperature
jump does occur, its magnitude can be expressed in terms of the heat partitioned to either
side of the interface. Further, for the convectively cooled interface where frictional ﬂux
is generated, if the heat partitioned to the work is less than that for the case of
temperature continuity, then the work temperature will be cooler than the tool

temperature (Inequalities (A24)).

A2. The 1D problem
In Figure Al, the work and tool materials are schematically represented to lie
along the x-axis, and they are in contact at the location shared by the work node w2 and

the tool node tl:

68

frictional flux Qapplied
t1 tool t2

A ‘
v v

M work w2

Figure A1: Ideal 1-D representation of the work and tool materials

A heat ﬂux Q(in W/mz) is applied at the interface. The energy equations governing the

work and tool domains are

 

 

2
d 0 .. d0
kwork 61:20“ “pworkcworkv xrk =0: (A1)
2
d 9
km] “2°01 =0. (A2)
dx

Here, k is the thermal conductivity, p is the density, 6 is the speciﬁc heat (per unit mass),

v is the cutting speed, Bis the temperature, and x is the distance coordinate. The system

is subject to two far-ﬁeld boundary conditions:
aworklxzxW1 = wl = 0, (A3)

and

0tool|x=m = 9,, = 0, (A4)

the application of which to the solution of Equations (A1) and (A2) yields

 

 

 

 

 

 

‘ ~ 1
exp[Pwol:kcwlprkV (x-xw1)]"l x _x
mek : Aw p WC; v and 6tool = At[ Z ] . (A5), (A6)
exp[ wol:k work Lwork J _ l 001
L work _

69

Here, Aw and At are constants to be determined, Lwork = xwz —xw1 , and Ltoo] =

xa — xtl. The complete solution (for AW and At) depends on the interfacial conditions

applied.

A2. 1. The uncoupled interface

At the interface, the conservation of energy requires that

36 work

d6
kwork + ktool [" AOL] = Q - (A7)
ax x=xw2 dx x=xn

In the absence of additional information on interfacial behavior, the assignment of heat
partition is arbitrary. For convenience, an equal partitioning of heat to both work and

tool is usually chosen, which is equivalent to the condition

a 6 work

kwork " ktoolETiqm‘) = 0 ° (A8)
8x x=xw2 dx Jr=xt1

The system of Equations (A1) and (A2), subjected to conditions (A3), (A4), (A7) and

(A8), yields an exact solution of

- -

 

 

 

 

 

 

A \
exp[pworkcworkv (x "xwl) -l
9 k ___ Q kwork 1 (A9)
wor .
2PworkC workv p[1)workéworkv )
ex Lw k
L kwork or 1 _

gtool = —Q (th — x) (A10)

2ktool

As mentioned, the heat partition function ,6, which for concreteness is deﬁned to be the

fraction of heat ﬂowing into the work, is arbitrarily set to 0.5:

70

ﬂwork = ﬂ= 05. (Al 1)

191001 = 1 — ﬂ: 0.5. (A12)

A22. Temperature continuity

The interfacial conditions in this case are

kwork aawork + ktoolL— M) = Q 1 (A13)
ax x=xw2 x=xt1
aworklx=xw2 _ 6tool|x=xﬂ = w2 _ atl : 0 . (A14)

which, together with conditions (A3) and (A4), result in the following exact solution of

the system of Equations (Al) and (A2):

r Q .1
f " pworke workV 1 W
ex Lwork

kwork + ktool

Pworke workv P 6 V Lt 1
exp[ wo]:k work Lwork J _ l 00
_ work _ j t (A15)

- e 1
ex;{ pworkcworkv (x _ xw1)] _1

kwork

6 v
exp( pwork work Lwork ] _ 1
i.

k work

 

 

 

 

 

 

6work = 1 K

 

 

 

 

 

 

 

- J

Q th‘x

gtool = f — [
exp£9workcworkv Lwork] W W [“001

k work + k tool

Pworkéworkv 6 V Lt
exp[ 9 wolzk work Lwork ] _ 1 001
( _ work _ j

] (A16)

 

 

 

 

 

 

Heat partition functions can be calculated from the ﬂuxes evaluated at the interface, and

they are

71

1 86
.6 = E[kwork gxmk

P work 6 work V
Lwork

kwork

P work C work V

 

 

6 v
Lexp[ Pwork work Lwork J _1

. (A17)

 

— -

kwork
exp( P work6 work V Lthk

.. kwork
P work c work V
exp[
L

l

Lwork J “'1

P workc work V

 

 

 

d

k tool

ﬂtoolzl-ﬂ‘: LL00]

 

 

p-

k work
P workéS work V

kwork

e V
exp( Pwork work
L

Lwork ]

P work C work V

k work

 

A2.3. Convective cooling

Lwork ] “l

q

 

and

(A18)

+ ktool

Ltool

The convective-cooling interface is modeled with a heat-transfer coefﬁcient h as

follows. From the work side:

39mm = “(Tu ‘Tw2)+.5Q
8x

x=xw2

k WOT

From the tool side:

36001

ktool(__atT') = h(Tw2 _ t1) + (I‘mQ _
x=xu

72

(A19)

(A20)

The four conditions (A3), (A4), (A19) and (A20),13 when applied to the system of
Equations (Al) and (A2), yield the following solution, which for simplicity, is given in
terms of the constants Aw and At of Equations (A5) and (A6):

ﬂ-—::°°‘ +h
Aw = w2 = 9“ -Q.(A21)

/ x
( ex({ P workC work V Lwork ] W

W k
[h + ktoo} h + P workéworkv ' p[ work
ex

 

J

 

 

Ltool J

 

 

 

 

 

P k5 k"
M: wor Lwork J T 1
work )

 

( f . ‘1 )
C V
exr{ P work work I ork J

kwork + h

 

 

 

 

(1" :6) Pworkeworkv '
CX[{

P k5 kV
wol: wor Lwork] _ 1
work )

ex{Pworkéworkv Lwork) W W

kt IN A kwork
[h + _go_ h + PworkcworkV °

LI 1 6 v
00 ) ex Pwoli'k work Lwork _1
K L ( work }

The equality of the constants AW and At to the interface temperatures of the two regions

 

A. = 611= ) ; 'Q .(A22)

 

 

 

 

 

 

 

 

 

U

can be seen by inspection of Equations (A5) and (A6).

From Equations (A21) and (A22), it is seen that setting h = 0 corresponds to the
uncoupled-interface solution (Equations (A9) and (A10) with ,6 = 0.5), while setting h =
00 corresponds to the temperature-continuity solution (Equations (A15) and (A16)).
Hence, the heat transfer coefﬁcient h can be used to describe intermediate behavior; it is

presently used to quantify the temperature jump between tool and work:

 

13 The case of Q: 0 corresponds to ﬂux continuity across the interface and can also be considered;
however, the present boundary and interface conditions must be altered for the solution to be non-trivial.

73

 

 

 

 

 

At ’Aw = gtl _6w2
.. N
C V
( exr{ Pwolzk wifrk Lwork] k
(l-ﬂ)' PworkeworkV' p ewor v ‘ﬂ'ﬁ
exp[ work work Lwork] -l 00
_ \ kwork J . Q (A23)

 

 

 

 

 

/ f .. \
ex PworkcworkV Lw W
ktool \ .. kwork ork 2
h +£— h‘i'PworkaorkV' e V “h
001 ) exp( Pwol:k work Lwork ] _ 1
K ( work ) )

 

 

 

Since the denominator of the last expression in Equation (A23) is always positive, the

inequality

holds as long as

(

PworkcworkV '

 

K

P workcwork V '

 

K

8X

9w2 -<- 6t]

.1
.1
i

k work

k work

P work 6 work V
k Lwork
work

P orke okV
wk w r Lwork
work

P work 5 work V
Lwork

P work 6 work V Lwork

J-l Ltool

(A24),

) )
lb

] + ktool

 

(A24)2

Y

 

)

Note that the right-hand side of Inequality (A24); is just the heat partition for the case of

temperature continuity (Equation (Al7)). Hence, under the assumption of a convective-

cooling interface, if the work absorbs less heat than it would have in the case of

temperature continuity (Inequality (A24)2), then the work will be at a lower temperature

than the tool at the interface (Inequality (A24);). The precise amount of the temperature

74

jump is given by Equation (A23). A similar result can be obtained if the direction of the

previous inequalities is reversed; namely, am 2 (9” if ,62 {right-hand side of (A24); }.

The heat partition ,8 in the case of convective cooling is viewed as a process
parameter that depends on material properties and the details of contact. While ,6 cannot
be solved explicitly in the present setup of convective cooling, its limiting behavior can
be deduced. Suppose in the steady state, the tool and work temperatures are postulated to

match at the interface if the work is stationary, even under convective cooling. In other
words, in the limit where v = 0, we set Aw = “,2 = 6,1 = A , and from Equation (A23),

arrive at

kwork
lim ,8 = LW" (A25)

v—)O (1(on + ktoo/ ) .
Lwork Ltool

On the other hand, as v —> co, the interfacial temperature jump reduces to

 

[1— lim )6)
lim At —Aw = lim Tu —Tw2 = v_’°° -Q 20 whenever 13$ 1. (A26)

001

 

Note that for temperature continuity, limv_,°° ,6 =1 (from Equation (Al7)). For the
interfacial temperature to drop from tool to work, the condition limv_,°° ,6 <1 is needed

(for consistency with Inequalities (A24)); the plausibility of such a condition can be
shown in the case of a heated object sliding across a cold surface, but will not be pursued

here.

75

A.3. Summary

Using the concept of the heat transfer coefﬁcient, we have concluded that, as long
as the heat partitioned to the work is less than that for the case of temperature continuity,
then the work temperature will always be cooler than the tool temperature in the 1D

problem.

76

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