

JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

44, 2, pp. 219-253, Warsaw 2006

WEAR PATTERNS AND LAWS OF WEAR – A REVIEW

Alfred Zmitrowicz

Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Gdańsk

e-mail: azmit@imp.gda.pl

Wear is a process of gradual removal of amaterial from surfaces of solids
subject to contact and sliding. Damages of contact surfaces are results
of wear. They can have various patterns (abrasion, fatigue, ploughing,
corrugation, erosion and cavitation). The results of abrasive wear are
identified as irreversible changes in body contours and as evolutions of
gaps between contacting solids. The wear depth profile of a surface is a
useful measure of the removed material. The definition of the gap be-
tween contacting bodies takes into account deformations of bodies and
evolutions ofwearprofiles.Theweardepth canbe estimatedwith the aid
of wear laws. Derived in this study, constitutive equations of anisotropic
wear are extensions of the Archard law of wear. The equations descri-
be abrasion of materials with microstructures. The illustrative example
demonstrates calculations of the abradedmass and temperatures in pin-
on-disc test rig.

Key words: contact mechanics, wear, friction, constitutive equations

1. Introduction

Everything thatmanmakeswears out, usually, as a result of sliding betwe-
en contacting and rubbing solids.Wear ofmaterials is an every-day experience
and has been observed and studied for a very long time. A large body of em-
pirical data has been collected and some phenomenological models have been
developed. The first experimental investigations of wear have been carried out
byHatchett (1803) andRennie (1829). In our days, experimental tests ofwear
intensities of materials belong to standard measurements of mechanical pro-
perties of solids. Nevertheless, it is difficult to predict and to control wear of
rubbing elements.


220 A. Zmitrowicz

Contacting elements of structures are very common in technology. Many
mechanical devices andmechanismsare constructedwith the aid of component
parts contacting one with another. Contact regions occur between tools and
workpieces in machining processes. Loads, motions and heat are transmitted
through the contacts of structures. Friction and wear accompany any sliding
contact. It has been agreed that wear cannot be totally prevented. Inmachine
technology, wear is an equally important reason of damage of materials as
fracture, fatigue and corrosion.

Themodelling of friction andwear is an important engineering problem. In
theprocess of design ofmachine elements and tools operating in contact condi-
tions, engineers need to know areas of contact, contact stresses, and they need
to predict wear of rubbing elements. Friction, wear and contact problems are
subjects of numerous experimental and theoretical studies. The very complex
nature of tribological phenomena is a reason that many problems of contact
mechanics are still not solved. Themodelling of friction and wear can be car-
ried out not only with the aid of laboratory tests but using alsomathematical
models and computer simulations. Due to computer simulation techniques,
physical andmechanical phenomena in real objects can be reconstructed with
a high degree of precision.

There is still a need for efficient and reliable computational procedures of
contact problems taking into account complexphenomenaof frictionandwear.
On the one hand, the accuracy of numerical procedures should be improved.
On the other hand, new models of friction and wear should be included in
numerical calculations. Contemporary numerical codes do not discuss how to
calculate and how to predict wear. Better understanding and control of wear
in materials can be done with the aid of newmodels of wear.

The first trials on numerical analysis of wearing out solids were given by
Grib (1982),Hugnell et al. (1996), Strömberg (1997, 1999), Szefer (1998), Chri-
stensen et al. (1998), Agelet de Saracibar and Chiumenti (1999), Franklin et
al. (2001), Ko et al. (2002), Shillor et al. (2003),McColl et al. (2004), Shillor et
al. (2004), Telliskivi (2004), Kim et al. (2005). For instance, Szefer (1998) inc-
luded in his numerical analysis the so called third body.The evolution of wear
gaps in fretting problemswas studied numerically by Strömberg (1997, 1999).
The finite elementmethodwas applied. Numerical simulations of wear shapes
due to pitting phenomena for various operating conditions have been investi-
gated byGlodež et al. (1998, 1999). Theymade use of the fracturemechanics.
In the subject literature, some authors calculate propagations of surface or
near-surface fatigue cracks induced by contact stresses, see Suh (1973), Ro-
senfield (1980), Sin and Suh (1984), Bogdański et al. (1996), Dubourg et al.


Wear patterns and laws of wear – a review 221

(2003). An original analytical approach to wear is preferred by the Russian
school, see Galin and Goryacheva (1980), Goryacheva and Dobychin (1988),
Goryacheva (1998) and Põdra and Anderson (1999). There are various com-
putational approaches to wear at this stage of investigations (finite elements,
boundary elements, fracture mechanics, molecular dynamics).

Acommonway to solvenonlinear contact problems is to adopt step-by-step
incremental procedures.Following a loadinghistory, contact constraints are sa-
tisfied at any increment step with the aid of iteration techniques. Nowadays,
formulations of contact problems (i.e. the mechanical problems with constra-
ints on solution variables) are more refined by applications of the Lagrange
multiplier method, penalty method and their generalizations (e.g. perturbed
andaugmentedLagrangianmethods,mathematical programming techniques).

This study is devoted to themodelling of wear ofmaterials.Wear patterns
and laws of wear for abrasion are the main aims of the paper. Wear laws are
classified as follows: empirical, phenomenological, based on failuremechanisms
of materials. Archard laws of wear is extended in the study and applied in
illustrative calculations.

2. Experimental observations of wear and surface damage

There are numerous advantages of contacts, e.g. transmission of loads and
motions, dissipation of energy, damping of vibrations, etc. The friction pro-
cess of solids operating in contact conditions always involves frictional heat
generation and wear of their surfaces, e.g. wear of machine component parts
(gears, bearings, brakes, clutches), wear of rubber tires, wear od shoes, wear
of clothes, etc. In general, machine component parts can fail by breakage or
by wear, the former being spectacular and sudden, the latter inconspicuous
yet insidious. Particularly high wear can occur in mechanisms which operate
in conditions of dry friction or in marginally lubricated conditions (so called
mixed friction).

Wear is a process of gradual removal of a material from surfaces of solids.
The detached material becomes loose wear debris. Nowadays, wear partic-
les are the subject of intensive studies, see Godet (1984),Williams (2005) and
Zmitrowicz (2005b). Twodimensional continuummechanics-basedmodels can
describe a thin layer of the wear particles between contacting bodies, see Zmi-
trowicz (1987, 2004, 2005b) and Szwabowicz (1998). Next results of wear can
be identifiedmacroscopically as the generation of worn contours of the bodies


222 A. Zmitrowicz

(so calledwear profiles) and as the increase of clearance spaces (gaps) between
the contacting solids.

Wear of materials is the result of many mechanical, physical and chemi-
cal phenomena. Several types of wear have been recognized, e.g. abrasive,
adhesive, fatigue, fretting, erosion, oxidation, corrosion, see Bahadur (1978),
Rabinowicz (1995), Ludema (1996) andKato (2002). Wear of solids is usually
treated as the mechanical process. However, oxidation, corrosion and other
chemical processes are exceptions of this rule. The abrasive wear and the con-
tact fatigue are the most important from the technological point of view. It
was estimated that the total wear of machine elements can be identified in
80-90% as abrasion and in 8% as fatigue wear. Contributions of other types
of wear are small.

Most of wear observations are carried out indirectly (post factum). The
rubbing processmust be stopped, thewearing out elementsmust be disassem-
bled, and after that the effects of thewear process can be observed.Weighting
is the simplestway of detectingwear. It gives the total amount of the removed
mass, but thedistributionof thewear depth in the contact surface is unknown.

In order to obtain qualitative information on wear, after opening the con-
tact, visual inspection of the worn surface and wear debris is very often used.
The easiest method of the visual inspection of surface damage is to photo-
graph the surface. Furthermore, the worn surfaces can be examined with the
aid of optical microscopes and with scanning and transmission electron mi-
croscopes. The surface examination bymicroscope provides a two-dimensional
view. To determine howmuch amaterial had been removed, surface topogra-
phic measurements must be preformed with the aid of a surface profilometer.
A quantitative technique whichmonitors the worn surface is based on surface
profiles perpendicular to the wear track. The depth of the removed material
from rubbing surfaces can be obtained by surface topographic measurements,
and the amount of the material worn away can be estimated. The examina-
tion of the wear profile history is a valuable indicator of the nature of the
wear process. In every-day life, one can easy observe wear profiles and wear
debris during the abrasion, e.g.: a pencil drawing marks on paper, a piece of
chalk writing on a blackboard, a rubber eraser rubbing out pencil marks on
paper, etc.

Radioactive methods and radionuclide methods are alternative direct me-
thods of measuring wear. Sliding elements or their surfaces are made radio-
active. The following quantities are measured: (a) amount of wear debris in a
lubricant if the rubbingelements are operated in thepresence of the circulating
lubricant, (b) amount of radioactive wear debris transferred to the interface


Wear patterns and laws of wear – a review 223

or to a non-radioactive surface. It is studied bymeasuring the radioactivity of
the lubricant or the irradiated sliding element (Rabinowicz, 1995).

Wear, frictional heat, fracture and fatigue are main factors which govern
machine life time. From the point of view of machine technology, the genera-
tion and the increase of clearance spaces in fitted exactly rubbing component
parts (e.g. in sliding and rolling bearings, between teeth of gears, etc.) are im-
portant consequences of wear. Consistently, numerous disadvantageous effects
follow, i.e.: increase of friction, dynamical loads,mechanical shocks, vibrations,
irregular motions, noise, fracture or break of component parts in the case of
great reduction of theirworn cross-sections, decrease of quality of rubbing sur-
faces, loss of wear-resistance properties by near-surface materials, heating of
elements, additional contamination, machine failure, etc. Numerous machine
component partsmust be taken out of service not due to failure caused by an
exceed of the limit stress, butdue towearmanifested in the removal of thema-
terial. Wear is an important topic from the economical point of view because
it represents one of ways in whichmaterial objects lose their usefulness.

Wear cannot be eliminated completely, but it canbe reduced.The simplest
methods of reduction of friction and wear are as follows: lubrication, forma-
tion of sufficiently smooth surfaces, modification of near-surface materials of
rubbing components, correct assembling and exploitation of fitted component
parts. Friction and wear can be reduced by an optimal choice of structural,
kinematical and material parameters of mechanical systems realized by: cor-
rect choice of shapes of rubbing elements, forming of loads and motions in
adequate limits, correct choice of sliding materials.

Shapeand size changes ofmachine elements are very important consequen-
ces of wear. However, in sliding electrical contacts, small wear of the machine
elements can be compensated with the aid of a spring pressing the rubbing
element to the counterpart. Abradable turbine seals are constructed and as-
sembledwith the so called negative clearance space. During turbine operation,
the seals are submitted to the running-inprocess, andfinally theyworkas leak-
proof seals with a nearly zero clearance. Brake pads, for instance, can lose up
to 3/4 of their original thickness and still give perfect service.

Overall wear consequences are negative, however one should remember ad-
vantages following from running-in, braking-in and from many methods of
producing a surface on amanufactured object exploiting the abrasion pheno-
menon (e.g. finishing). Thanks to abrasion, a pencil, a crayon and a piece of
chalk are useful in every-day life. A rubber eraser removes carbon particles
from the pencil transferred to the paper by the wearmechanism (Rabinowicz,
1995).


224 A. Zmitrowicz

3. Patterns of wear

3.1. Profiles of worn machine component parts (abrasion patterns)

In the process of wear, a material is removed from surfaces of solids, and
dimensions of worn bodies are gradually reduced. The amount of the removed
material can be estimated with the aid of the wear profile. The wear depth
defines the removedmaterial at the given point of the contact area. The wear
profile is a function of the wear depthwith respect to positions at the contact
region. The wear depth and the wear profile describe irreversible changes in
shapes and sizes of the worn bodies, and they are useful measures of wear.

Fig. 1. The wear profile of an oscillating surface in contact with a stationary sphere
after 20 000 cycles, see Fouvry et al. (2001). This is the wear profile in the Hertzian

contact

Figure 1 shows the measured wear profile of an oscillating surface in con-
tact with a stationary sphere after 20000 cycles (Fouvry et al., 2001). This
is a wear profile in Hertzian contact, i.e. contact between an elastic sphere
and a plane. In the Hertzian contact, the normal pressure is semi-elipsoidally
distributed in the circular contact area (Johnson, 1985). Therefore, the me-
asured maximum wear depth is at the center of the contact zone, and it is
equal to several tenth micrometers. Figure 1 shows a typical fretting process,
i.e. the wear process which takes place during oscillatory sliding of small am-
plitudes. Notice, that motion producing fretting is very small (per cycle, 50
micrometers). In mechanical systems, oscillatory tangential displacements of


Wear patterns and laws of wear – a review 225

small amplitudes arise from vibrations or cyclic stressing of one of the con-
tacting bodies. Fretting contacts occur in mechanical joints, supports, screw
joints, steel wires, bearings, gears, turbine blade roots, see Hills and Nowel
(1994) and Arakere and Swanson (2001). Damaged surfaces in the fretting
process were observed by microscopy in Vingsbo et al. (1990).

Weiergräber (1983) measured the wear profile of an anvil face in a half
hot metal forming process after pressing 5000 elements. It is a good example
of the wear profile in Nonhertzian contact, i.e. contact between a rectangular
punchandahalf-plane. In theNonhertzian contact, singularities in thenormal
pressure distribution occur at edges of the contact zone (Johnson, 1985). The
maximumwear depth, near the edges of the contact zone, was equal to several
tenthmicrometers. Inpoints of themaximumwear depth, the normal pressure
was high and the sliding velocity was large. Marginally lubricated conditions
were in the contact area.

Measurements of the wear profiles of rails and rail wheels with respect to
thedistance travelledwere reported in the subject literature, seeMoore (1975),
Seyboth (1987), Sato (1991), Olofsson and Telliskivi (2003). Sato (1991) pre-
sented consequent changes in transverse profiles of worn rail heads and worn
rail wheels on curved railway tracks after wheel run 1600, 27000, 79000 and
200000 km investigated onTokaido Shin kansen (Japan). Themaximumwear
depth of the worn rail was equal to several tenth millimeters. Sometimes the
rail head can be designed to have such aworn profile which coincides with the
shape of a newwheel (Sato, 1991).

Stresses in some points of the worn rail cross-section were measured in an
experimental set-up and reported by Seyboth (1987). An increase of stresses
in the head and at the foot of the worn rail cross-section was observed in
comparison with the unworn rail. The increase of stresses depended on the
wear depth. A great increase of the stresses was measured in the point near
the contact area (i.e. in the rail head, up to 149%, when the wear depth was
28 millimeters) and a less increase in the points far from the contact (i.e. in
the rail foot, up to 28%, when the wear depth was 28 millimeters). Ghonem
andKolousek (1984) presented a model describing wear of rails as a function
of the angle that a wheelset formswhen deviating from the radial direction in
relation to the track (so called angle of attack).

Measurements ofwear profiles of tools used in variousmachining processes
have been reported in the literature. König (1985) presented the formation
and development of a crater wear profile on a worn cutting tool top surface
during successive stages of increasing cutting time, from 4 to 35 minutes. An
average life time of the cutting edge usually vary from a few to several tens


226 A. Zmitrowicz

minutes depending on the cutting tool and the cutting process. Exceptionally,
it is equal to several hundred minutes. The maximum wear depth was equal
to 0.2 millimeters (König, 1985). Lueg (1950) showed themeasured profiles of
worn drawing die inserts of the wire drawing machine in dependence on the
slide length (22650m and 45000m). Two types of the die insert were taken
into account, cylindrical and non-cylindrical.
Occasionally, wear profiles of other machine component parts are presen-

ted in the literature. For instance, changes in teeth profiles of gears caused by
abrasive wear weremeasured after some number of cycles. Extensive wear oc-
curs in large-size open toothed gears operating in drivingmechanisms applied
in cement plants and in steelworks; there are emitted great amounts of dusts
which have abrasive properties. Usually toothed gears are lubricated. Wear
profiles of engine components have beenmeasured, e.g. at the contact betwe-
en piston rings, piston and cylinder liner. Sometimes, it becomes important
to study worn surfaces of journal bearings. Changes in the journal bearing
geometry due to wear strongly affect the bearing performance especially at
high operating speeds, e.g. in high speed turbomachinery.

3.2. Surface fatigue

Certain forms of surface damage do not occur slowly and continuously but
may suddenly cause large failure of rubbing surfaces. For instance, initiation
andgrowth of surface and subsurface cracks can lead to the formation of surfa-
ce defects by large particles of thematerial removed from the surface, see Suh
(1973), Rosenfield (1980), Ohmae and Tsukizoe (1980), Sin and Suh (1984),
Bogdański et al. (1996), Glodež et al. (1998, 1999), Dubourg et al. (2003). Sur-
face fatigue wear was observed during repeated loading and unloading cycles
(e.g. during repeated sliding or rolling).

3.3. Ploughing of surfaces

From the point of view of machine technology, wear is also thought as a
decrease of quality of rubbing surfaces, i.e. increase of surface roughness, fog-
ging of the surface, generation of scratches and grooves, etc. Surfaces can be
ploughed by wear particles, hard particles entrapped from the environment
and by hard asperities of the counterface. Irreversible plastic deformations in
the surfaces are themain results of ploughing.When the surfaces are ploughed
with evidence of plastic flowof amaterial, then scratches andgrooves are gene-
rated on the surface. Surfaces ploughed (scratched) by a cone-shaped indenter
and hemispherical-shaped pin were observed with the aid of microscopy by
Sakamoto et al. (1977), Kayaba et al. (1986) andMagnée (1993).


Wear patterns and laws of wear – a review 227

3.4. Surface corrugation

Changes in shapes and sizes are observed in transverse and longitudinal
profiles of worn rails. Rail corrugations are specific type of wear, since rail
corrugations are variations in the longitudinal profile of a rail. They are ty-
pically periodic, characterized by long (100-400 millimeters) and short (25-80
millimeters) pitch wavelengths, see Sato et al. (2002), Grassie (2005), Meehan
et al. (2005). The same phenomenon was observed for rail wheels. The rail
corrugations induce vibrations in trains as they pass over them, see Knothe
and Grassie (1999), Bogacz and Kowalska (2001), Meehan et al. (2005). So-
metimes re-griding of the rail and wheel profiles may be necessary to restore
the initial conditions of their co-operation.

3.5. Surface damage due to erosion and cavitation

Wear of solid surfaces as a result of bombardment by solid particles or
liquid drops is an important form of surface damage. In the case of erosion,
hard solid particles moving with some velocity impact the solid surface at a
certain angle, then slide along the surface andfinally bounce off.Moving along
the surface for a small distance, these particles cut scratches and grooves, and
producewear debris.There are high-speedand low-speedprocesses.Therefore,
surface damage and material removal may occur by the mechanism of crack
initiation and crack growth. In the case of erosive wear, it is assumed that a
volume of the removedmaterial is proportional to a high power of the particle
velocity (e.g. fifth, fourth, second powers).

Surface damage may also occur due to cavitation of liquids. Cavitation
takes place in fluid flow systems in which negative pressure regions exist.
Bubbles filled with vapor derived from themoving liquidmay be formed near
the solid surface.When the bubbles collapse, a jet forms, impacts the surface
and damages it (Rabinowicz, 1995).

4. Wear depth predictions

In the tribology literature, measures of wear have been formulated with
respect to changes of the following quantities: (a)mass of the removedmaterial
from the solid, (b) volume of the removedmaterial, (c) reduced dimensions of
the body. Themeasures of wear have non-zero values as long as an observable
amount of the material is removed. Due to this, usually, but not always, a


228 A. Zmitrowicz

significant period of the sliding time (or a great number of cycles) must be
taken into account.
Barber (see the paperbyKennedyandLing; 1974) suggested that thewear

rate (i.e. the depth of the material removed per time unit) is a prescribed
function of the normal pressure, sliding velocity and temperature. The formof
the function for a givenmaterial couldbedeterminedby laboratorywear tests.
Galin and Goryacheva (1980) recognized that besides elastic deformations of
thebodycontactingwith the rigid foundation, irreversible changes of the shape
of the body take place in the wear process. In the case of abrasive wear, the
amount of the removed material is proportional to the work of the friction
force.
Let a systemof twobodies Aand B beamodel of rubbingandwearing out

solids. It is assumedthat velocities v+c and v
−
c describemotions of thewearing

out boundaries (ΩcA,ΩcB) due to two reasons: deformations of the bodies
(vA,vB) and the wear process (v

+,v−). Notice, vA and vB are material
velocities. Two quantities mA and mB describe rates of themass flowing out
from the bodies A and B due to the wear process

mA = ρA(vA−v+c ) ·n+ =−ρAv+ ·n+
(4.1)

mB = ρB(vB −v−c ) ·n− =−ρBv− ·n−

where, ρA and ρB aremass densities, n
+ and n− are unit vectors normal to

the boundaries ΩcA and ΩcB. The following velocities normal to the bounda-
ries (ΩcA,ΩcB) define the wear process

v
+ ·n+ = v+ v− ·n− = v− (4.2)

and they are dependent variables of the constitutive equations of wear. Galin
andGoryacheva (1980) assumed that the velocity of thewearing out boundary
of the solid body is a function of the normal pressure and the sliding velocity.
According to Rabinowicz (1995), v+ defines the velocity at which the surface
ΩcA of the body A ”travels” into other surface because of its wearing away.
Kinematics of the wearing out boundary ΩcA of the solid body A is shown
in detail in Fig.2. The wear profiles are drawn for successive stages of the
increasing sliding time.
Kinematics of the two contacting bodies A and B can be described in

Euler description by displacement functions uA(xA,τ) and uB(xB,τ), where
τ is time. Velocities of displacements and the sliding velocity are given by

vA = u̇A vB = u̇B
(4.3)

VAB =(1−n+⊗n+)(u̇A− u̇B)


Wear patterns and laws of wear – a review 229

Fig. 2.Wear profiles – kinematics of a wearing out boundary of a solid body A. In
Lagrange description: XA – radius vector of the wearing out boundary of the
body A0,N

+ – unit vector normal to the boundary at the contact area Ω0
cA
,

V
+ – wear velocity. In Euler description: xA – radius vector, n

+ – unit normal
vector, v+ – wear velocity

In Fig.2, kinematical quantities describing the wear profiles in Lagrange and
Euler descriptions are shown schematically. Notations in Lagrange description
are as follows: XA is the radius vector of the wearing out boundary Ω

0
cA of

the body A0, N
+ is the unit vector normal to the boundary at the contact

area Ω0cA, V
+ is the wear velocity. In Euler description, notations are follo-

wing: xA is the radius vector, n
+ is the unit normal vector, v+ is the wear

velocity.

The total clearance gap between two deformable bodies (dn) is equal to
the sum of the initial gap (gn), the elastic deformations (u

A
n ,u
B
n ) and the

depth worn away (u+n ,u
−
n), i.e.

dn = gn−uAn +u+n +uBn −u−n on ΩcA∪ΩcB (4.4)

where

uAn(xA,τ)= (n
+⊗n+)uA ≡uAnn+ xA ∈ΩcA

uBn (xB,τ)= (n
−⊗n−)uB ≡uBnn− xB ∈ΩcB

(4.5)

Notice, dn is the distance between two points coming into contact. The depth
of the material removed in the time interval 〈0, t〉 is defined by the following
integrals


230 A. Zmitrowicz

u+n =

t∫

0

v+(xA,τ) dτ xA ∈ΩcA
(4.6)

u−n =

t∫

0

v−(xB,τ) dτ xB ∈ΩcB

Figure 3 shows principal quantities which define the wear profile and the gap
between the body A and the rigid foundation. In Table 1, definitions of the
radius vector, the wear velocity and the wear depth in Euler and Lagrange
descriptions are given.

Fig. 3. Quantities which define the contact: wear depth (u+
n
), gap between

contacting bodies (dn), initial gap (gn) and normal and tangential contact forces
(pA
n
,pA
t
)

Table 1.Definitions of the radius vector, wear velocity andwear depth in
Euler and Lagrange descriptions

Quantity Euler description Lagrange description

Radius vector X+ =X+(xA,τ) x
+ =x+(XA,τ)

of wearing xA ∈ΩcA XA ∈Ω0cA
out boundary τ ∈ 〈0, t〉 τ ∈ 〈0, t〉
Wear velocity v+(xA,τ)= V

+(XA,τ)=

=
∂x+(XA,τ)

∂τ

∣∣∣
XA=XA(xA,τ)

=
∂x+(XA,τ)

∂τ

∣∣∣
XA=const

v+ ·n+ = v+ V+ ·N+ =V+
Depth worn u+n(xA,τ)= U

+
n (XA,τ)=

away =

t∫

0

v+(xA,τ) dτ =

t∫

0

V+(XA,τ) dτ


Wear patterns and laws of wear – a review 231

In the case of wearing out solids, we have to solve the problem in the
region (A∪B) whose boundary (ΩcA∪ΩcB) is partial unknown in advance.
From the point of view of continuum mechanics, successive positions of the
wearing out boundary ΩcA of the solid A depend on the solution to the
contact problem.The shape and size of theworn body A should be calculated
together with the fields of stresses, strains and temperatures. It means that
the evolution of the wear profile can be modelled by a material continuum
with a moving boundary, see Alts and Hutter (1988, 1989). Usually, moving
boundary problems (so called Stefan problems) are solved using a mapping
of the continuum to a fixed domain (fixed-domainmapping) at any step from
the given time interval. For this reason, the pull-back technique can be used.
In the finite element analysis of contact problems, a finite elementmesh in

thematerial continuumnear the contact surface shouldbedesignedadequately
with respect to themaximumweardepth.For instance, dimensionsof thefinite
element can be equal to several micrometers. In the wear process, dimensions
of the wearing out body are gradually reduced. If the calculated wear depth
is equal to the dimension of the finite element, then this element is removed
from the mesh. Further analysis is conducted for the mesh with the reduced
number of the finite elements. This is an updating process of the contact
surface geometry.
The first trials on theoretical and numerical calculations of wear profiles

have been carried out by Galin and Goryacheva (1980), Grib (1982), Gory-
acheva and Dobychin (1988), Hugnell et al. (1996), Strömberg (1997, 1999),
Goryacheva (1998), Christensen et al. (1998), Agelet de Saracibar and Chiu-
menti (1999), Franklin et al. (2001), Ko et al. (2002), Shillor et al. (2003),
McColl et al. (2004), Shillor et al. (2004), Telliskivi (2004), Kim et al. (2005).
Paczelt andMróz (2005) discussed optimization problems with respect to the
contact surface geometry generated by wear.
Analysis of the contact problem generally requires the determination of

stresses and strainswithin contacting bodies, togetherwith information regar-
ding the distribution of displacements, velocities and stresses at the contact
region. In most cases, two following contact constraints have to be satisfied.
The contacting bodies cannot penetrate each other

dn ­ 0 onΩcA (4.7)

and they are separated or pressed on each other (no tensile normal forces)

pAn ¬ 0 onΩcA (4.8)


232 A. Zmitrowicz

Figure 3 shows components of the contact force (pA), i.e. the normal pres-
sure force (pAn) and the tangential traction (p

A
t ) given by

pA =σAn
+ onΩcA

p
A
n =(n

+⊗n+)(σAn+)≡ pAnn+

p
A
t =(1−n+⊗n+)(σAn+) (stick) (4.9)
|pAt | ¬µA|pAn | (stick)

p
A
t =−µA|pAn |

VAB

|VAB|
(slip)

where, µA is the friction coefficient, σA, is the Cauchy stress tensor in the
body A. Material constitutive relations of the elastic body A are as follows

σA =EAεA
(4.10)

εA(uA)=
1

2
(graduA+ grad

>
uA)

where, EA is the tensor of elasticity, εA is the (right) Cauchy-Green strain
tensor. The contact constraints, are in the form of inequalities (4.7) and (4.8),
and they restrict admissible displacements (uA) and stresses (σA).
The boundary conditions out of the contact region are following

uA = ûA onΩuA

σAnA = q̂A onΩqA
(4.11)

where, nA is the unit vector normal to the body boundary and directed
outward the contact region, ûA and q̂A are prescribed displacements and
loads (see Fig.3).
The contact zone (ΩcA) and the contact forces (pA) are usually unknown

in advance, and they depend on the solution to the contact problem. Contem-
porarymethodsused in contactmechanics aremainly based on computational
techniques. First conferences on computational methods in contact mechanics
were organized by Raous (1988) and Aliabadi and Brebbia (1993). Most con-
tributions in (Raous, 1988) were devoted to applications of finite elements
to contact mechanics. Studies presented in (Aliabadi andBrebbia, 1993) dealt
withboundaryelements used to solve contact problems.First bookson compu-
tational contactmechanicswere publishedbyKikuchi andOden(1988), Zhong
(1993), Laursen (2002) andWriggers (2002). In the books, finite elementswere
applied to solve nonlinear contact problems. Various techniques were used to


Wear patterns and laws of wear – a review 233

findcontact stresses and to satisfy kinematical contact constraints, i.e.method
ofLagrangemultipliers and their generalizations: JooandKwak(1986), Juand
Taylor (1988), Alart and Curnier (1991), Simo and Laursen (1992), Wriggers
(1995), Oancea and Laursen (1997); method of penalty function e.g. Laur-
sen and Simo (1993), Zhong (1993), Wriggers (1995). Incremental-iterative
methods have been commonly used, since solutions to the contact problems
depend on the loading history, see Cescotto and Charlier (1993), Buczkowski
and Kleiber (1997), Stupkiewicz and Mróz (1999). Bajer (1997) used origi-
nal space-time finite elements in contact mechanics. Zmitrowicz (2000, 2001a,
2001b) included the interfacial layer ofwear debris between contacting bodies.

5. Laws of wear

5.1. Archard’s law of abrasive wear

Friction andwear depend as much on sliding conditions (the normal pres-
sure and the sliding velocity) as on properties of materials concerned. The
normal pressure and the sliding action are necessary for wear, i.e. mechanical
wear is a result of themechanical action.Therefore, thewear process discussed
in this study depends first of all on the rubbing process.
The earliest contributions to the wear constitutive equations were made

by Holm (1946). Holm established a relationship for the volume of the ma-
terial removed by wear (W) in the sliding distance (s) and related it to the
true area of contact. Archard (1953) formulated the wear equation of the
form: the volume of thematerial removed (W) is directly proportional to the
sliding distance (s), the normal pressure (pn) and the dimensionless wear co-
efficient (k), and inversely proportional to the hardness of the surface being
worn away (H), i.e.

W = k
pns

H
(5.1)

Nowadays, it is generally recognized thatwear is related to thewear coefficient,
the pressure and the sliding distance. Pin-on-disc test experiments canbe used
to determine howwear is affected by the pressure and the sliding distance.
By analogy to Archard’s law, in themost simple case, wear velocities (4.2)

can be defined as functions of the normal pressure and the sliding velocity, i.e.

v+ =−iA|pAn | |VAB|
(5.2)

v− =−iB|pBn | |VBA|


234 A. Zmitrowicz

where

p
B
n =(n

−⊗n−)(σBn−)≡ pBnn−
(5.3)

|pAn |= |pBn | VAB =−VBA

iA and iB are wear intensities of the bodies A and B. They may depend on
fields of temperature in the bodies, i.e.

{iA, iB}= f(TA,TB, GradTA, GradTB) (5.4)

where, TA and TB are temperatures in the bodies A andB.
The dimensionless wear coefficient k in Archard’s law can be defined in

various ways depending on the physical model which is assumed in deriving
the wear equation. A simple interpretation is to consider k as a measure of
the efficiency of the material removed for the given amount of work done.
Wear equations (5.2) differ fromArchard’s law (5.1) in the omission of the

termrepresenting the inverse proportionality of the surface hardness H. In the
tribology literature, the wear equation coefficients iA and iB used in (5.2) are
called the dimensional wear constants or the specific wear rates. If iA and iB
aremultiplied by the hardness of bodies, thenwe get dimensionless intensities
of wear iAHA and iBHB. Therefore, the hardness can be easily included in
the quantitative estimates of the dimensional wear intensity coefficients iA
and iB.

5.2. Extensions of Archard’s law of wear

It can be presumed that anisotropic friction induces anisotropic wear, i.e.
the intensity of the removed mass depends on the sliding direction (Zmitro-
wicz, 1992, 1993a,b). Anisotropy of friction results from roughness anisotropy
of contacting surfaces and anisotropy of mechanical properties of materials
withmicrostructures (crystals, composites, polymers, ceramics, biomaterials).
Anisotropicwear velocities dependon the normal pressure, the sliding velocity
and the sliding direction. Notice that pAn and VAB do not define directional
properties of the anisotropic wear.
Let us assume that the wear intensity iA is a function of the sliding direc-

tion parameter αv, i.e.

iA = iA(αv) αv ∈ 〈0,2π〉 (5.5)

αv is the measure of the oriented angle between the given reference direction
at the contact surface (e.g. Ox axis) and the sliding velocity direction. We


Wear patterns and laws of wear – a review 235

postulate that the wear intensity function and the anisotropic friction force

component µ
‖
α are functions of the same type, i.e.

iA(αv)∼µ‖α(αv) (5.6)

The coefficient of the friction force component tangent to the sliding direction
is defined by

µ‖α =−
1

|pAn |
p
A
t ·v (5.7)

where, the sliding velocity unit vector is as follows

v=
VAB

|VAB|
(5.8)

We apply the constitutive equation of the anisotropic friction force (Zmi-
trowicz, 1991) in form of a trigonometrical polynomial with the second order
constant friction tensors (C1k; k=0,1,2), i.e.

p
A
t =−|pAn |[C10+C11 cos(n1αv)+C12 sin(m1αv)]v n1,m1 =0,1,2, . . .

(5.9)
where, the components of friction force vector (5.9) and unit vector (5.8) are
given by

pit =−|pAn |[C
ij
0 +C

ij
1 cos(n1αv)+C

ij
2 sin(m1αv)]vj i,j =1,2

(5.10)

v= [cosαv,sinαv]
>

Depending on the form of the friction tensors, we get descriptions of aniso-
tropic friction and anisotropic wear with a different number of constants and
parameters.
Taking the following components of the friction tensors

C11k =µ(k)1 C
22
k =µ(k)2 k=0,1,2 (5.11)

the friction force component collinear with the sliding direction is given by

µ‖α(αv)=−{[µ(0)1+µ(1)1 cos(n1αv)+µ(2)1 sin(m1αv)]cos2αv +
(5.12)

+[µ(0)2+µ(1)2 cos(n1αv)+µ(2)2 sin(m1αv)]sin
2αv}

According to similaritypostulate (5.6),weobtain thewear intensity coefficient,
i.e.

iA(αv)= [i1+ i2 cos(nαv)+ i3 sin(mαv)]cos
2αv +

(5.13)

+[i4+ i5 cos(nαv)+ i6 sin(mαv)]sin
2αv


236 A. Zmitrowicz

where, n,m=0,1,2, . . . are two parameters, i1, i2,. . . , i6 are six constants.

Using three spherical friction tensors

C
ij
k
=µkδ

ij k=0,1,2 i,j =1,2 (5.14)

the friction force coefficient is the following trigonometrical polynomial

µ‖α =−[µ0+µ1 cos(n1αv)+µ2 sin(m1αv)] (5.15)

In this case, the wear intensity coefficient has the following form

iA(αv)= i1+ i2 cos(nαv)+ i3 sin(mαv) (5.16)

where, n,m are two parameters, i1, i2, i3 are three wear constants.

Using the first friction tensor spherical, and the second and third equal to
zero, i.e.

C
ij
0 =µ0δ

ij C
ij
1 =C

ij
2 =0 i,j =1,2 (5.17)

the friction force coefficient and the wear intensity are constant

µ‖α =µ0 iA(αv)= i1 (5.18)

where, i1 is the wear constant.

Different types of anisotropic wear can be distinguished depending on a
number of particular sliding directions: (a) neutral directions (if wear is inde-
pendent of the sense of the sliding direction)

∃αv ∈ 〈0,2〉 : iA(αv)= iA(αv +π) (5.19)

and (b) extreme value directions (if it gives extreme values of wear)

iA(αv)=min{iA(α̃v) : α̃v ∈ 〈0,2π〉} (5.20)

or

iA(αv)=max{iA(α̃v) : α̃v ∈ 〈0,2π〉} (5.21)
With the aid of symmetry groups, the following types of anisotropic wear

can be distinguished: isotropic, anisotropic, orthotropic, trigonal anisotropic,
tetragonal anisotropic, centrosymmetric anisotropic and non-centrosymmetric
anisotropic (Zmitrowicz, 1993a,b). Symmetry properties of anisotropic wear
are identified by elements of the symmetry groups i.e.: identity (+1), inver-
sion (−1), rotations (Rγn, γ = 2π/n, n = 1,2, . . .) and mirror reflections.
Themirror reflections are with respect to neutral directions (Ju) or extreme


Wear patterns and laws of wear – a review 237

value directions (Js). If anisotropic wear has a finite number of neutral direc-
tions, then the mirror reflections are with respect to the neutral directions. If
there is a finite number of extreme value directions and all sliding directions
are neutral, then the mirror reflections are with respect to the extreme value
directions.
Heterogenous wear andwear dependent on the sliding path curvature have

been investigated by Zmitrowicz (2005a).
It is postulated that for a given normal pressure and sliding velocity at

the contact of two surfaces, the resultant rate of mass flowing out is equal to
the product of a ”composition coefficient” (κ) and a sum of rates of themass
flowing out obtained for each surface taken separately, i.e.

mAB =κ(mA+mB) (5.22)

The wear intensity functions of individual surfaces can be determined expe-
rimentally by sliding a test third body with isotropic wear properties. The
resultant mass flowing out from the contact can be defined by

mAB = ρiAB(α
A
v )|pAn | |VAB| (5.23)

where ρ is the average mass density. After substituting the rates of mass
flowing out related to the individual surfaces A and B into (5.22), we obtain
the following resultant wear intensity

iAB(α
A
v )=κAiA(α

A
v )+κBiB(α

A
v −ϕ)

(5.24)

κA =
κρA
ρ

κB =
κρB
ρ

We have assumed different reference directions on both contacting surfaces.
Then, the following relation holds between the sliding direction parameters on
the surfacesA and B

αBv =α
A
v −ϕ (5.25)

where, ϕ is the angle of relative position of the contacting surfaces (i.e. the
angle between the reference directions). The symmetry group of the resultant
wear at the contact of two surfaces with different wear properties is equal to
an intersection of the symmetry groups for the surfaces A and B.

5.3. Thermodynamical restrictions

The wear constitutive equations (5.2) satisfy the axiom of material objec-
tivity, i.e. two different observers of the sliding at the contact region recognize
the same wear velocities, see Zmitrowicz (1987, 1992, 1993a,b).


238 A. Zmitrowicz

The constants in wear constitutive equations are restricted by two ther-
modynamic requirements (Zmitrowicz, 1987):

5.3.1. Restrictions following from the Second Law of Thermodynamics

The internal energies spent at the wear process are always positive, i.e.
they satisfy the following inequalities

mA�A ­ 0 mB�B ­ 0 (5.26)

where, �A and �B are internal energies (or energies consumed by formation
of a unit mass of wear debris). Substituting constitutive equation (5.2) and
definition (4.1) into thermodynamic inequality (5.26), and taking into account
that

ρA, |pAn |, |VAB|,�A ­ 0 (5.27)

we obtain the following restriction on the wear intensity

iA(αv)­ 0 αv ∈ 〈0,2π〉 (5.28)

It means that the wear intensity is positive for any sliding direction (and for
any location of the contact point).

5.3.2. Constraints of the energy dissipated at the frictional contact

Theprincipal question is: howmechanical energy of the frictionprocess can
be dissipated? In general, the whole energy dissipated in the friction process
is converted (in a priori unknown proportion) into frictional heat, the energy
spent at the wear process and other forms of energy. Therefore, the following
constraint of energy dissipated at the contact must be satisfied

p
A
t ·VAB = qAf +qBf +βA+βB (5.29)

where, qAf and q
B
f are frictional heat fluxes entering into the bodies A and

B, βA and βB are the energies spent in the wear process, i.e.

βA ≡−mA�A βB ≡−mB�B (5.30)

In equation (5.29), the frictional power is converted into frictional heat and
energies spent on thewear process (other forms of energy are neglected). Equ-
ation (5.29) does not decide which part of the friction force power appears as
frictional heat and wear process energy.


Wear patterns and laws of wear – a review 239

5.4. Other models of wear

Bahadur (1978) presented 13 analytical expressions for the prediction of
wear published in the years 1937-1974. Earlier relationships connected the
volume of the removedmaterial with hardness only.Meng andLudema (1995)
found more than 300 equations of wear published during the period 1955-
1995. The great number of independent variables that influence the wear rate
was used in those equations, namely 625 variables. Meng and Ludema (1995)
classified three general stages of deriving wear equations: empirical equations,
phenomenological equations, equations based onmaterials failure mechaisms.
Ludema (1996) classified the variables of wear equations as follows: variables
from fluidmechanics, variables from solid mechanics, variables from material
sciences, variables fromchemistry. In spite of that, at present,Archard’s law is
a quantitatively simple calculation procedure of wear, see Rabinowicz (1995),
Ravikiran (2000), Kato (2002).

Some researchers try to define the so called wear criterion. They define
a place in the body boundary where the wear process is initiated. Usually it
depends on the existence and development of the sub-surface plastic zones.
Kennedy and Ling (1974) assumed that the wear criterion is a modified crite-
rion for plastic flow. If the second invariant J2 of the stress deviator equals or
exceeds the critical value W (i.e. the yield stress of the material) then wear
will occur

J2 ­W (5.31)

where, the second invariant of the deviatoric part of the stress tensor and the
critical value are given by

J2 =
1

2
trS2 S =σ−

1

3
(trσ)1 W = k

2
(5.32)

where, k
2
is the yield stress of the softer material. If J2 < W no wear will

occur. It is assumed that when the accumulated strain energy reaches the
critical value at which the yielding occurs, most of this energy is transformed
into production of wear particles. In such a place, the amount of wear is
calculated, i.e. the volume of the removed material or the wear depth.

Yang et al. (1993) assumed that wear is first expected to take place at
the point with the maximum critical stress. The wear equation can be de-
rived from the maximum stress principle criterion (e.g. von Mises criterion)
and yield strength. According to Ting and Winer (1988), wear is assumed to
result from the yielding ofmaterials. The yielding condition is governed by the
temperature dependent yield strength and the stress field of a material. The


240 A. Zmitrowicz

stress field is combined from the isothermal stress field generated by surface
traction and the thermal stress field induced by frictional heating.

Ohmae andTsukizoe (1980) simulated numerically large wear particle for-
mation (so called delamination wear) using the finite element method and
theory of elastic-plastic deformations. Ohmae and Tsukizoe (1980) divided
the contact surface into very small finite elements. It was assumed that each
element represented a dislocation cell. The typical cell size was found to be
0.5µm.Whenthe equivalent stress of the element exceeds the failure stress (i.e.
the fracture stress), a void of failure is generated. From these voids, cracks are
propagated along directions ofmaximumplastic stresses. Penetrating into the
bulk of the body, the cracks cause generation of wear particles. Ohmae (1987)
recognized that large plastic deformations that lead to the void nucleation can
only be dealt with properly used theory of large strains and displacements.
Furthermore, Ohmae (1987) discussed different wear criteria used by various
researchers: stress intensity factor, J-integral (proposed by Rice), etc.
Different approaches to modelling of wear are present in the literature.

In the opinion of some researchers, wear laws should not be accepted as po-
stulates, but as a consequence of both a geometry of surface micro-regions
(asperities) and elastic-plastic deformations of interlocking surface asperities,
see Kopalinsky andOxely (1995), Torrance (1996).

6. Calculations of the abraded mass and fields of temperatures in
the pin-on-disc test rig

The pin-on-disc testing machine is a typical device used to study friction
andwearofmaterials.Astationarypin ispressedagainst a rotatingdisc,Fig.4.
After some timeof sliding, surfacedamages canbeobserved, andwearparticles
can be seen inside and outside thewear track. Pałżewicz and Jabłoński (1993)
investigated a short duration contact between the pin and the steel disc under
high pressure conditions. The removed mass and fields of temperature in the
pin are calculated in this study.

We assume that the cylindrical pin has small dimensions in the cross-
section but it has a finite dimension along its axis. Therefore, the pin can
be described by a one-dimensional continuum y ∈ 〈0,+∞), see Fig.4. In
the model, the contact region, the normal pressure and the sliding velocity
are known in advance. The normal pressure is given by pn. The disc mo-
ves with the sliding velocity V with respect to the pin. The wear velocity
of the pin is defined by v+. The depth of the material removed u+n in the


Wear patterns and laws of wear – a review 241

time of sliding t is defined by equation (4.6)1. Abraded mass from the pin is
given by

mpin = ρp

∫

Ap

u+n dS (6.1)

where, ρp is the mass density of the pin, Ap is the area of contact (the cross-
section area of the pin). The law of wear is assumed in the following form

v+ = ippnV (6.2)

where, ip is the wear intensity of the pin.

Fig. 4. The wear profile of the pin contacting with the moving disc: u+
n
– wear

profile, pn – normal pressure, V – disc velocity

When two solids slide one against another, most of the energy dissipated
in the friction process appears as heat and energy of the wear process. It is
assumed that 10% of the friction power is transformed into the wear process,
the rest part of the friction power is converted into heat.Theheat generated at
or very close to the contact surface is transferred away into the bulk of rubbing
solids, and it is conducted into surrounding. Equation of energy dissipated in
contact (5.29) reduces to the following relation given in the scalar notation

−ptV = qf︸︷︷︸
90%

+ρpv
+�p︸ ︷︷ ︸
10%

(6.3)

where, pt is the isotropic friction force (the inclination angle between the
friction force and the sliding direction is equal to zero), �p is the internal
energy of the pin, 90% is the fraction of the friction power dissipated into
heat, 10% is the fraction of the friction power dissipated into wear of the pin.


242 A. Zmitrowicz

Most of the wear occurs only on the pin. Assuming that 90% of the friction
power converts into frictional heat and the rest of the power transforms into
wear of the pin, we get the following relations

−0.9ptV = qf −0.1ptV = ρpv+�p (6.4)

After substituting the isotropic friction force, i.e.

pt =µppn (6.5)

andwear velocity (6.2) into (6.4)2, we obtain a quantitative restriction on the
wear intensity coefficient

ip =0.1
µp
ρp�p

(6.6)

where, µp is the friction coefficient of the pin. Rabinowicz (1995, p.162) sug-
gested that in estimations of �p one should consider the sliding body ”loaded
to the limit”.
The material constants used in the calculations are collected in Table 2.

Three types of bearing materials have been investigated: steel, aluminium
(Al), tin (Sn). The friction coefficients were obtained by experimental testing.
The wear intensities were identified with the aid of wear coefficient tables, see
Rabinowicz (1995), i.e.

isteel =0.03 ·10−6 MPa−1 iAl =0.5 ·10−6 MPa−1
(6.7)

iSn =2.0 ·10−6 MPa−1

Table 2. Material constants used in calculations of the pin-on-disc test
set-up

Type of Coefficient Wear Mass Thermal Coefficient of
material of friction intensity density diffusivity heat transfer

µp ip [MPa
−1] ρp [kgm

−3] K [m2s−1] b [s−1]

Steel 0.1 0.03 ·10−6 7.9 ·103 13.8 ·10−6 2.26 ·10−3
Al 0.07 0.5 ·10−6 2.7 ·103 85.9 ·10−6 3.46 ·10−3
Sn 0.026 2.0 ·10−6 7.3 ·103 39.8 ·10−6 4.96 ·10−3

Equation of the abraded mass from the pin (6.1) reduces to the following
form

mpin = ρpippnVApt (6.8)


Wear patterns and laws of wear – a review 243

Thecross-sectionof thepin is equal to Ap =200·10−6m2.Thenormalpressure
and the sliding velocity are shown inFig.5-Fig.7. The abradedmass from the
pin after 4 seconds, is as follows: 2.12 · 10−6kg for steel, 12.07 · 10−6kg for
aluminium, 130.5 · 10−6kg for tin. Figure 4 shows the removed mass from
the pin with respect to the sliding time. For a short duration contact, the
influence of temperature on the wear intensity coefficient ip can be neglected.
Coefficients (6.7) deal with the fixed room temperature 20◦C.

Fig. 5. The progress of the abradedmass from the pin after 0 to 4 seconds for three
types of materials: steel, aluminium, tin

Fig. 6. Temperature distributions in the pin with respect to the distance from
contact for three types of materials


244 A. Zmitrowicz

Fig. 7. The progress of the contact temperatures for the pin; the plots correspond to
the real time of sliding from 0 to 4 seconds and for three types of materials

The field of temperature induced by friction is inseparable behavior of the
contact phenomenon.Therefore, it is important to consider the loss of thema-
terial and frictional heat together. Frictional heat is generated at the contact
between the pin and the disc. The fields of temperature and contact tempe-
ratures were investigated theoretically in numerous papers, e.g. Carslaw and
Jaeger (1959), Kennedy (1981), Kikuchi (1986), Barber andComninou (1989),
Vick et al. (2000). The equation of heat conduction in a one-dimensional con-
tinuum with included heat transfer into the surrounding has the following
form

∂T

∂τ
=K
∂2T

∂y2
− b(T −Ta)

(6.9)

T =T(y,τ) y∈ 〈0,∞) τ ∈ 〈0, t〉

where, T is temperature of the pin, y is the coordinate along the pin axis,
Ta is room temperature (20

◦C). K is the thermal diffusivity coefficient, i.e.

K =
k

cρp
(6.10)

where, k is the thermal conductivity, c is the specific heat. In equation (6.9)1,
the Newton rule of heat transfer into the surrounding is assumed. The coeffi-
cient of Newton’s heat transfer rule b is as follows

b=
hg

cρpAp
(6.11)


Wear patterns and laws of wear – a review 245

where, h is the coefficient of heat transfer, g is the circumference of the pin
surface (g=50 ·10−3m).
The initial and boundary conditions of the heat conduction problem are

given by

T(τ =0)=Ta k
∂T

∂y
(y=0)= qf lim

y→∞
T(y,τ)=Ta (6.12)

The frictional heat flux vector is defined by

qf =wpnV (6.13)

where, w is the frictional heat intensity coefficient. The frictional heat qf is
divided into two parts, a part of heat which is transferred into the pin (γ),
and a part of heat transferred into the disc (γ−1), i.e.

qf = γqf +(1−γ)qf (6.14)

where, γ is the coefficient of heat partition between the pin and the disc. It is
assumedthat the90% isdivided into twoparts, i.e. 37%(it is the frictionpower
converted into heat and transferred into the pin) and 53% (it is the friction
power converted into heat and transferred into the disc). The temperature
distribution along the pin axis and the contact temperature are given by

Tpin =T(y,t)=Ta+
wpnV

k

√
K

π

t∫

0

1√
τ
e
−

(
bτ+

y2

4Kτ

)

dτ

(6.15)

Tcontact =T(y=0, t)

Calculated temperatures in the pin andat the contact surface are shown in
Fig.6 and Fig.7. Figure 6 presents temperature plots in the pin with respect
to the length coordinate of the pin, i.e. with respect to the distance from
contact.The temperatures stronglydecrease along thepinaxis since theheat is
conducted into the surrounding.Thehighest temperatures occurat the contact
surface.Figure 7 illustrates contact temperatures for thepin sliding in thedisc.
The contact temperatures increase with respect to time of sliding. Notice, the
highest contact temperature is for the steel pin and the least abradedmass is
for the steel pin. All contact temperature plots correspond to the real time of
sliding. We do not consider cooling effects.


246 A. Zmitrowicz

7. Conclusions

• Contact surface damages can have various patterns: abrasion, fatigue,
ploughing, corrugation, erosion and cavitation. Irreversible changes in
body shapes and the increase of gaps between contacting solids are prin-
cipal results of abrasive wear. The sizes of the wearing out bodies are
gradually reduced in the process of abrasion.

• The amount of the removed material can be estimated with the aid of
the wear depth and the wear profile. They are useful measures of the
wear process, and they can be described with the aid of the wear laws.
The definition of the gap includes deformations of bodies and evolutions
of wear profiles.

• The measures of wear have non-zero values as long as an observable
amount of thematerial is removed. Usually, but not always, a significant
period of the sliding timemust be taken into account. Due to this, rather
long simulation times are needed.

• The constitutive equations of anisotropic wear describe abrasion of ma-
terials with microstructures. The equations are derived in the frame of
Archard’s law of wear. In the present time, Archard’s law is commonly
used in engineering calculations.

The proposed theoretical descriptions of wear can be used in numerical
codes for strength calculations of structures and materials. The calculations
can be useful for improvement of wear-resistance of materials and durability
of rubbing and wearing out machine parts and tools.

References

1. Agelet de SaracibarC., ChiumentiM., 1999,On the numericalmodeling
of frictional wear phenomena, Computer Methods in Applied Mechanics and
Engineering, 177, 401-426

2. Alart P., Curnier A., 1991,Amixed formulation for frictional contact pro-
blems prone to Newton like solution methods, Computer Methods in Applied
Mechanics and Engineering, 92, 3, 353-375

3. Aliabadi M.H., Brebbia C.A. (edit.), 1993, Computational Methods in
Contact Mechanics, ComputationalMechanics Publications, Southampton


Wear patterns and laws of wear – a review 247

4. AltsT.,HutterK., 1988, 1989,Continuumdescription of the dynamics and
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Formy zużycia i prawa zużycia: przegląd

Streszczenie

Zużycie jest to stopniowe usuwanie materiału z powierzchni stykających i ślizga-
jących się ciał stałych.Następstwem zużycia są uszkodzenia powierzchni styku.Mogą
oneprzyjmowaćróżną formę(ścierania, zmęczenia,bruzdowania,korugacji, erozji i ka-
witacji). Konsekwencjami procesu zużycia ściernego są nieodwracalne zmiany kształ-
tówciał orazpowiększanie luzów(szczelin)między stykającymi się ciałami.Użyteczną
miarą usuniętegomateriału jest profil głębokości zużycia powierzchni. Definicja luzu
między stykającymi się ciałami uwzględnia odkształcenia ciał oraz ewolucję profili
zużycia. Głębokość zużyciamoże być oszacowana z pomocą praw zużycia.Wyprowa-
dzone w niniejszej pracy równania konstytutywne anizotropowego zużycia są rozsze-
rzoniemprawa zużyciaArcharda.Opisują one ścieraniemateriałów zmikrostrukturą.
W przykładzie ilustracyjnym obliczono ubytekmasy i temperaturę w stanowisku do-
świadczalnym typu pin-on-disc.

Manuscript received December 1, 2005; accepted for print December 28, 2005