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Chapter 3 Surfaces in Contact
48
Chapter 3
SURFACES IN CONTACT Wear is characterized as a process of surface damage due to mechanical contact of matter (see Section 1 . 3 1. Independently of any special wear mode, the type of mechanical contact is very important for all wear losses and will be discussed in the following. SURFACE TOPOGRAPHY Engineering surfaces are far from ideally smooth, and exhibit more or less roughness. The texture characteristics of surfaces are described by the arrangement, shape and size of individual elements such as asperities (hills and valleys on a microscopic scale). Figure 3-1 shows a surface topography schematically. Surface profiles can be obtained by vertical sections through the surface. Horizontal sections through the surface topography lead to information about the bearing area. Contact between two solids is generally discrete, due to surface roughnesses, i.e. it occurs at areas of individual point contacts. 3.1
Surfoce Prolile
Figure 3 - 1 .
- Schematic representation of surface irregularities.
49
Different optical and mechanical methods are available for measurement of the microscopic or macroscopic geometrical features of surfaces. In profilometry, stylus devices are widely used for obtaining surface profiles. A fine diamond stylus traverses the surface and its vertical movements are recorded. The profiles represent only one pass in a linear direction across a random three - dimensional surface. From a lot of cross - sectional areas of surface texture, contour plots can be drawn (ref. 1 5). Contour plots represent a three - dimensional image of the texture characteristics. New measuring systems (ref.6) were introduced for describing the three - dimensional characteristics of surface roughness. Figure 3-2 represents sections of profilograms of surfaces of a Ti-A1 alloy obtained by using a stylus device. The surfaces were finished by polishing, turning or milling.
0)
R,=O Ilpm , A , =0.58pm
.
Ro=0.C2~m.R t =3.21pm
Figure 3-2.
-
Profilometer traces of surfaces of a Ti-A1 alloy: (a) electrolytically polished, (b) turned and (c) m i 1led.
50
Frequently used values for characterization of surface textures are the centre - line average, or the mean arithmetic deviation of the profile (cla Ra), the root mean square (rmseR4~ 1 . 2 5 Ra) roughness value,the peak - to - valley height (Rt) or the maximum peak - to - valley height (Rmax). Surface profilometers record the irregularities on surfaces with different magnifications in the vertical and the horizontal direction (see Fig.3-2). Normally the vertical magnification is greater than the horizontal. Due to this difference in magnification, the recorded profiles d o not represent a true picture of the actual shapes of the surface irregularities. The actual picture of surfaces consists rather of broad based hills, with angles of inclinations from the base line less than 1 5 degrees (ref.41, than of sharp peaks. For evaluation of wear models it is very important to bear in mind the difference in the surface picture recorded and the actual surface.
Slol~cConloct
I F,
Sllding Contocl
I F,
Figure 3-3. - Apparent and real area of contact. Different types of experiments (ref.7,8) have shown that a large difference can exist between the apparent and real areas of contact of two flat solid surfaces pressed together (Fig.3-3).
51 The r a t i o o f r e a l t o a p p a r e n t a r e a o f c o n t a c t may b e a s l o w a s ( r e f . 9 ) a n d d e p e n d s on t h e d i s t r i b u t i o n o f s u r f a c e i r r e g u a r i t i e s , c o n t a c t f o r c e a n d y i e l d s t r e s s o f t h e s o f t e r mater a 1 i n v o l v e d . T h e r e a l a r e a of c o n t a c t i s l a r g e r i n s l i d i n g t h a n i n t h e s t a t i c state.
According t o t h e s t a t i c s t a t e i n F i g . 3 - 3
t h e real area o f con-
tact is g i v e n by: A,
n
=IA1i
(3-1
i=
is t h e real area o f c o n t a c t and Ai
w h e r e A,
the area of indivi-
dual contact spots. For t h e s t a t i c c o n t a c t of
i d e a l l y elastic
-
p l a s t i c materials,
t h e r e a l area o f c o n t a c t c a n be c a l c u l a t e d :
A,
=
FN -
(3-2)
@Y
where FN is t h e n o r m a l f o r c e o n t h e s u r f a c e s i n contact a n d p
Y t h e y i e l d p r e s s u r e o f t h e s o f t e r m a t e r i a l . A c c o r d i n g t o Bowden
and T a b o r ( r e f . l O , l l ) , s o f t e r material.
py i s a b o u t e q u a l t o t h e h a r d n e s s o f t.he
U s i n g t h e f o l l o w i n g r e l a t i o n between h a r d n e s s H
a n d y i e l d stress ay o f t h e m a t e r i a l stressed: H = C . 0
Y
(3-3 1
w i t h C = 3 f o r f e r r t i c steels, w e o b t a i n f r o m e q u a t on ( 3 - 2 ) f o r
static contact:
(3-4
T h e r a t i o o f r e a l t o a p p a r e n t a r e a of c o n t a c t i s p r o p o r t i o n a l to t h e a p p l i e d s u r f a c e p r e s s u r e d i v i d e d by t h e y i e l d stress o f t h e s o f t e r material i n c o n t a c t . L a t e r w e w i l l see t h a t r e a l a r e a o f c o n t a c t is i n c r e a s e d d u e t o s l i d i n g o f t h e c o n t a c t i n g s u r f a c e s r e l a t i v e to e a c h o t h e r (Fig.3-3).
52 3.2 CONTACT MECHANICS Friction and wear of two solid surfaces in unlubricated contact depends on the type of deformation of the surface irregularities involved. Greenwood and Williamson (ref.12) proposed a plasticity index $ which describes the transition from elastic to plastic deformation of surface asperities:
with
where H is the hardness of the softer material, E l , E2 are the Young's moduli and v1 , v2the Poisson's ratios of the two bodies in contact, R i s t h e radiusof the asperity summits (which is assumed to be the same for all asperities) and S is the standard deviation of a Gaussian distribution of the asperity heights. If $<0.6 the contact is predominantly elastic whereas if $> 1 plastic deformation dominates. Whitehouse and Archard (ref.13) introduced a more general plasticity factor$*which allows asperity peaks to have a distribution of curvatures.
E' S* rlr* = 0.69 -.H D
(3-71
where E' can be calculated from relation (3-61, S* is the rmsvalue of the surface and 13 is related to the correlation distance of the surface.According to Onions and Archard (ref.14) the equation (3-5) underestimates plasticity. This may be due to the assumption that all asperities have the same radius. Whitehouse and Archard (ref.13) showed however that the higher peaks of asperities had sharper radii than lower peaks. According to Tabor (ref.l5), the transition from elastic to elastoplastic contact at the indentation of a rigid sphere into a flat surface depends on the indentation depth. The contact becomes elastoplastic if the indentation depth h exceeds the critical value:
53
hcr = 0,89R.(H/E)*
(
3-8)
where R is the radius of the sphere, H is the hardness and E the Young's modulus of the material deformed. A more detailed review of plasticity factors is given in (ref.1 ). From equations(3-51, (3-7) and (3-8) it follows that the deformation of asperities in contact is determined mainly by the characteristics of the surface texture, hardness and elastic constants. The applied normal load or surface pressure does not directly influence the transition from elastic to plastic deformation, according to equations (3-5) and (3-7). The type of contact can change during service in a tribosystem. Starting with plastic contact, a change to elastic contact can occur after running-in. The type of contact can be expected to have considerable influence for example, in rolling contact. The rolling resistance can increase by more than two orders of magnitude when the contact alters from elastic to plastic deformation (ref. 1 6 ) .
3.2.1 Elastic Deformation The discrete nature of the contact is characteristic of all contacts between solids and is related to their surface roughnesses.Considering asperities as individual contact spots, the elastic strains and stresses at the contact area can be estimated from Hertzian formulae (ref.17). The solutions of the elastic stress fields are well known (ref.18-25). For simplicity, both bodies in contact are of the same material (El = E2 and V, = v2 = 0.31. Figure3-4 shows the pressure and stress distribution in and below the contact area for different contact configurations. Contact of a Sphere and a Plane The Hertzian pressure is hemispherically distributed on a plane contact area with the contact radius a:
54
rN\2E2 )
and the maximum contact pressure pmax:
pmax =
-
0.388
(3-10)
1’3
where FN is the normal load, R the radius of the sphere and E the Young’s modulus with El = E 2 and v , = v2 = 0.3. The maximum shear stress T~~~ occurs at a depth 2, below the contact area (Fig.34(a)):
2, = 0.47a
(3-12)
Tensile stresses occur in a region close to the surface and outside the contact circle.
b)
Figure 3-4. - Schematic stress distribution for the elastic contact of:(a) a sphere and a plane due to normal load (b) a sphere and a plane due to combined normal and tangential loads where FT=0.3FN,(c) two unlubricated rolling cylinders and (d) two lubricated rolling cylinders (elastohydrodynamic lubrication).
55 Combining a normal load with a tangential load results in a substantial increase of the tensile stress at the rear of the sphere in the surface stressed (Fig.3-4fb)). The maximum shear stress now occurs much nearer to the surface or i n the surface, depending on the coefficient of friction u between the contacting bodies. The maximum tensile stresses at the rear of the contact circle can be calculated for sliding contact (ref. 22,23):
d
tmax
1 - 2 ~ FN ( 1 + C’.ll) na2
-- - .2-
(3-13)
with
cI .
3 n *4 + v 8 1-2v
Contact of two Spheres For Hertzian contact of two spheres with radii R 1 and R2, the contact radius a is calculated from:
a = i.iib(Rl
R1
.]”
. R2 +
(3-14)
and the maximum pressure due to a normal load alone:
(3-15)
The maximum shear stress at a depth Z, below the surface is obtained by : (3-16) and 2, = 0.47a
(3-17)
56
Contact of two Cylinders The contact radius a for Hertzian contact of two cylinders can be calculated from:
a = 1.52
"
F .R1'R 2 E.1 . (R1 + R 2 )
(3-18)
where 1 is the length, R1 and R 2 the radii of the cylinders and E the Young*s modulus. The maximum pressure is obtained by:
Pmax = - 0.418(2.
E.
(3-19)
The maximum shear stress at a depth 2, below the surface is obtained from: (
3-20)
and 2, = 0.78a
(3-21)
Figure 3-4(c) shows the contact of two cylinders qualitatively. Friction between the rolling cylinders results in tangential traction in the surface. Surface stresses in the contact area are compressive for a coefficient of friction = 0. The compressive stress component becomes increasingly unsymmetrical with increasing value of u. Simultaneously, the tensile stress increases at the end of the region of contact. The reference stress calculated by using the strain energy hypothesis reaches its maximum value below the surface for u = 0. The maximum reference stress occurs however in the surface when the coefficient of friction exceeds about 0.2 (ref.24). Slip in the rolling contact of two cylinders was considered by Bentall and Johnson (ref.26).In elastohydrodynamic lubrication (Fig.3-4(d)) a local peak in the pressure curve occurs just before the end of contact. Figure 3-5 (ref.20) shows the influence of the shape of ters on the distribution of pressure at the contact area.
inden-
57
I
I
FN
FN
I
, ’l
.
.’./
J’
, ,,,//’,,
Figure 3-5. - Pressure distribution curves at elastic contact for three different indenters ( sphere, flat die and cone)
.
In the contact area between a sphere and a flat surface, the maximum pressure, with a finite value, occurs in the centre. In contrast, the model of a cone or a flat die results in indeterminate stresses in the centre or at the periphery of the contact. Hence, a spherical geometry is favoured above others for calculating elastic contact problems. Dependence of the real area of contact on the normal load is an important question for friction and wear. For purely elastic deformation, the proportionality between real area of contact Ar and normal load FN is presented in Fig.3-6 for different contact geometries. It follows from the Hertzian equations (3-9) or (3-14) that the real area of contact between a sphere and a flat surface, or between two spheres,or between two general curved surfaces which result in an elliptical contact, depends on the normal load according to: (
3-22)
58
Figure 3 - 6 . - Influence of contact geometry on the dependence of the real area of contact A, on the normal load FN in purely elastic contact. For real surfaces in friction and wear, multi-asperities models are more practicable. Archard (ref.27) studied the contact between surfaces covered with spherically shaped asperities. Two contact models may be distinguished. Firstly the number of asperity contacts is independent of the normal force. An increasing normal load results in increasing deformation of each contact. Secondly the average area of each asperity contact remains constant with increasing normal loads, but the number of asperity contacts increases. According to these models, it is obtained: A r d FNm
(3-23)
59 where m = 213 for a constant number and m = 1 for an increasing number of asperity contacts with increasing normal load. Figure 3-6 shows schematically both asperity models. Other models (ref. 28) describe the elastic contact of surfaces covered with smaller asperities. It follows from these, that the smaller the asperities and the closer they are packed the more closely the factor m approaches 1. According to Adams (ref.29) and Bowden and Tabor (ref.9), the real area of contact between a hard sphere and the smooth flat surface of a polymer depends alsoon the normal load, as predicted by equation (3-23) with m between 0.1 and 0.8. The plastic contact of asperities is considered in equation (3-21, i.e. m is equal to 1. The finite element technique was recently applied for modelling of the elastic contact of three -dimensional rough surfaces (ref.30). In many friction and wear problems, the contact between two smooth elastic bodies may be influenced by adhesion (see Section 4.2.1 ).More or less attractive forces can occur between surfaces in contact, depending on the environment or condition of lubrication, surface roughness, surface layers or materials involved. Johnson et al. (ref.31,32) have shown that the radius of contact between two bodies can be substantially increased by the action of attractive surface forces. They described the surface forces of adhesion by the surface energy, i.e. the work required to separate unit area of the adhered surfaces. For two smooth spheres, they calculated the ratio of contact radius with adhesion a, and without a :
(
3-24)
where F N is the normal force, y the surface energy and R = R1R2/(Rl+RZ), R 1 and R2 being the radii of the spheres. Without adhesion, the surface energy y becomes zero and .a equal to a. Figure 3-7 shows the stress field at the contact area taking into consideration the attractive forces due to adhesion. The spheres were pressed together by the normal force FN. Then this force was reduced by AFN. The contact is maintained over the original enlarged area due to adhesion. As a result, the stresses between the surfaces are compressive at the centre but tensile at the edge of contact.
60
Tension ( 4 )
Figure 3 - 7 .
-
Elastic contact between two solids in the of surface forces.
presence
Johnson (ref.33) presented a model of the contact of rough surfaces. According to this model, the maximum contact pressure decreases and the effective contact pressure spreads over an increasing area when surface roughness is increased. The influence of adhesion depends on the value of an "adhesion index":
( 3-25
in which
and
-E'1 _
-
1
El
v1
+
1 - v
2
2
E2
where S is the standard deviation of asperity heights, y the surface energy, R = R l * R 2 / ( R l + R 2 ) , R 1 and R 2 are the radii and E l , E2 the Young's moduli of the bodies in contact. Adhesion ceases when the adhesion index exceeds a critical value (aad = 1 . 6 ) .
61
Hence, the influence of adhesion decreases w i t h increasing surface roughness or elastic modulus and with decreasing surface energy. Fuller and Tabor (ref.34) also investigated the effect of surface roughness on the adhesion of elastic solids and introduced an adhesion parameter which results in similar conclusions to those from equation (3-25). Roy Chowdhury and Pollock (ref. 35) presented a multi-asperity model for plastic contact which expresses the area of contact by:
Ar
-
)-
N
(3-26)
where FN is the applied load, H the hardness, S the standard deviation of the asperity heights and yad the work of adhesion per unit area. According to this model, a significant adhesion force can only be expected when the asperities are plastically deformed. The real area of contact is increased due to adhesion. The influence of adhesion is reduced by increasing surface hardness and surface roughness.
3.2.2 Plastic Deformation When two elastic bodies, e.g. a sphere and a flat specimen, are pressed lightly against each other, the contact is purely elastic. If the applied normal load exceeds a critical value, the elastic l i m i t , a plastic zone develops which is surrounded by elastically deformed material.The elastic limit can be calculated (ref. 36) from: p = 1.85
T
Y
(3-27)
where p is the mean contact pressure and T the yield stress in Y pure shear. In elastic contact, the peak contact pressure is 1 . 5 times the mean contact pressure. W i t h increasing load, the contact becomes elastoplastic and the pressure distribution more and more uniform. Finally the full plasticity condition exists, which is given (ref. 37) by: p
-
c+.T
Y
(3-28
62
where C * = 6 on Tresca's criterion and 5.2 on Mises' criterion.Figgure 3-8(a) shows the pressure distribution in elastic, elastoplastic and plastic contact schematically. With increasing plasticity, the hemispherically distributed pressure, with peak pressure in the centre of the elastic contact, is changed to a pressure uniformly distributed across the contact area in the fully plastic condition.
0)
Eloslic
Elostoploslic
lncreosing Plasticity
PlOSllC
ession
I
Figure 3-8. - Schematic representation of the pressure distribution at and below contact areas: (a) elastic, elastoplastic and plastic contact of a sphere and a flat surface, (b) indentation of a sharp indenter in a flat surface. After repeated loading and unloading, a steady state "shakedown limit" can be reached, i.e. the contact is quasi-elastic. This state results from changes in contact geometry due to plastic
63 flow connected with work hardening and the development of residual stresses. The shakedown limit is determined (ref.38) by: p = 3.69
T
Y
(3-29)
Elastoplastic indentation by sharp indenters (Fig.3-8(b)) w a s discussed by Perrott (ref.39).According to his analysis,the pressure (equal to hardness) for an elastic - plastic indentation by cones and Vickers pyramids is given by:
(3-30)
where a is the yield stress under simple tension, E the Young's Y modulus and u the Poisson's ratio of the test material and 0 is the semi-apical angle of conical or pyramidal indenters. From an "expanding cavity" model introduced by Hill (ref.401,Marsh (ref. 41) and Johnson (ref.42) presented similar equations. According to Johnson, the following relationship between indentation pressure and yield stress can be used:
where 0 is the semi-apical angle of conicial or pyramidal indenters. The factor cot 0 can be replaced by d / D for spherical indenters, where D is the diameter of the indenter and d the diameter of the indentation. Spherical indenters lead to an elastic deformation at low loads, in contrast to sharp indenters. Hence, equation (3-31) is only applicable for spherical indenters when the elastic limit is exceeded. The influence of work hardening during the indentation process is ignored in these models. The indentation plasticity was experimentally investigated with steel (ref.43) and glass (ref.44). The real area of contact in the plastic condition can be estimated from equation (3-2). Under the combined action of a normal and a tangential force, the real area of contact is increased. This was studied by Courtney-Pratt and Eisner by using electrical
64 resistance measurements (ref.45). McFarlane and Tabor (ref.46) calculated the real area in sliding contact from a yield criterion for junction growth: a2 + C l T 2 -
2
(3-32)
- PY
2
(3-33)
(3-34
(3-35)
from equation (3-21, where : A and A, are the real area in sliding and static contact,respectively,py the yield pressure of the softer material, o the applied normal and T the applied tangential stress and F N l F T the normal and tangential forces. C1 is a constant with a value of about 10 (ref.g).Under a normal force only, equation (3-34) is reduced to equation (3-2). The combined effect of a normal and a tangential force results in an increase of the real area of contact. In the plastic contact of metals, work hardening can become an important factor.The influence of work hardening on the real area of contact can be estimated from a yield criterion, similar to that of equation (3-32):
a2 + C p 2 =
( Py
+
LIPy) 2
and finally we obtain:
(3-36)
65
Ar * = ArJ1
+
Cl(Z)*
(3-37
APY
l+-
PY where apy is the increase in yield pressure p due to work harY dening.In genera1,equation ( 3 - 3 7 ) could also be applied when work softening ( - lapy! ) occurs. According to this model, work hardening should result in a smaller and work softening in a larger real area of contact. Rolling contact is of great interest for many practical tribosystems. Johnson (ref.16) and Collins (ref.47) analysed elasticplastic and plastic contact in rolling. This will be discussed in more detail in the Chapter 7 .
Indentation Fracture Mechanics Materials of high wear resistance are frequently very hard but brittle, e.g. hardened steels, cast irons or ceramics. The contact loading of brittle solids can result not only in elastic and plastic deformation but also in microcracking at and below the stressed surfaces. 3.2.3
I
lHerlzion Crock .---.--
Figure 3 - 9 .
-
‘compression {
_ ______ /.
.A
- Formation of a Hertzian crack in an elastic stress field.
66
Cone-shaped Hertzian cracks are a well know example iref.48). A circular cone-shaped crack originates around the contact area between a sphere and the flat surface of a brittle solid when a critical load is exceeded. This crack propayates, with increasing load, from the circumference of the contact circle along the periphery of a cone into the solid (Fig.3-9). The maximum tensile stress occurs at the contact circle.
Figure 3-10. - Indentation of a Vickers diamond on surfaces of (a) WC - Co cemented carbides (light micrograph), (b) cracking at the periphery of an indentation on 0 . 9 4 % C - steel (SEM micrograph). Blunt and sharp indenters have to be distinguished when studying indentation problems. Pyramids or cones may be considered as sharp, and spheres a s blunt, indenters. Depending on the type of indenter, blunt or sharp, the contact results in predominantly elastic or plastic deformation. The theoretical solution of the elastic stress field caused by a sharp indenter shows a singularity at the centre of indentation (Fig.3-5). The stress field due to a normal point load was first described by Boussinesq (ref.49). Lawn and Wilshaw (ref.50) presented an excellent review of the principles of indentation fracture. Figure 3-10 shows the cracking that was caused by indentation by a Vickers diamond pyramid in the surface of tool steel and cemented carbides.
67
The profile of crack propagation perpendicular to the surface was measured by polishing in steps from the surface t o t h e interior of the steel (Fig.3-11 ). At the surface of the steel, cracks propagated almost perpendicularly from the edges of the indentation into the interior. The direct contact between the crack profiles and the areas of indentation was lost at a depth of about 1/3 of the total indentation depth. This is also shown i n Fig.310(b) and is predicted in Fig.3-8tb) by the tensile stress field close to the surface.
Figure 3-11. - Surface cracking in 0.94%C - steel due to indentation of a Vickers diamond by a test load of 625 N: (a) crack formation about 5 5 u m below the surface, ( b ) cracks (see arrows) about 7 5 below ~ the surface ( indentation was removed by polishing), (c) measured crack profile below the surface, (d) total crack profile.
68
According to Lawn and Swain (ref.Sl),surface loading by a point indenter results in median and lateral cracks below the stressed surface.Figure 3 - 1 2 shows schematically different types of cracks formed during loading and unloading.
lime
t
Un'ood'ng
Figure 3 - 1 2 .
-
Formation of median and lateral cracks in brittle solids due to indentation by a sharp indenter.
An increasing point load results in increasing size of a plastic zone around and below the indentation. A median crack is formed when the load exceeds a critical value F2. This crack grows in depth with increasing load. During unloading, the median crack is closed and lateral cracks are formed and propagate to surface under an applied load of less than F 5 . An immediate reloading closes the lateral cracks and reopens the median crack. The formation of lateral cracks during unloading is connected with residual stresses which are due to the plastic deformation zone.Residual surface stresses can play an important role in microcracking (ref.52-54). Residual tensile stresses increase crack lengths and reduce the critical load for microcracking. During unloading, median cracks may propagate in depth due to residual stresses. Palmqvist (ref.55) first used crack lengths at the corners of a Vickers hardness indentation as a measure for ductility and sensitivity to microcracking. This method is frequently used today for characterization of ceramics or cemented carbides.
69 Different studies (ref.56-60) have shown that the critical stress intensity factor (fracture toughness KIc) can be estimated from cracking at hardness indentations. Measurements of the lengths of the median cracks on the symmetry planes of the indentation, the normal force and the shape of the indenter allow the calculation of KIc. Figure 3 - 1 3 shows the model of cracking used for the calculation.
Figure 3 - 1 3 .
-
Palmqvist cracks compared with halfpenny-like surface cracks used as model.
The s h a p e o f t h e m e d i a n c r a c k s i s a s s u m e d t o b e a centre-loaded halfpenny configuration. For this geometry, the stress intensity factor at the crack tip of a Vickers indentation can be calculated (ref. 6 1 ) :
FN 3’2. tanR
(3-38)
where 2c is the crack diameter. FN is the indenter load and 13 the semi-apical angle of the indenter (68O for a Vickers diamond). The friction between the indenter and the specimen is not considered but can be introduced by replacing 13 by (I3 + p q , where p*is the friction angle.
70
Figure 3-14 shows stress intensity factors of surface cracks of half-penny shape for different crack lengths.
Figure 3-14. - Stress intensity factors a s a function of length of surface cracks due to indentation:(a) at variable loads FN1 or FN2 and (b) for materials of different fracture toughness KIcl or KIc2. The stress intensity factor decreases with increasing crack length, d u e to the inhomogeneous stress field at the indenter. This means that cracks propagate to a final length which is d e termined by the fracture toughness of the material studied. Figure 3-14(a) shows the final crack length caused by two different loads FN1 and FN2. A crack of initial length co propagates to the length c 1 or c2 due to the load F N 1 o r FN2, respectively. A t this crack length, the stress intensity factor is equal to the fracture toughness KIc. In Fig.3-14(b), a crack propagates under a constant load F N from an initial length co to a length c1 or c2 for materials of fracture toughness KIcl or KIc2, respectively. This means that shorter cracks occur in materials of greater fracture toughness. Initial cracks can be caused by the indentation or are already present, e.g. hardening cracks or cracks at embrittled grain boundaries.
71
The situation becomes more complicated when a tangential force acts on the indenter in addition to the normal force. Figure 3-15 shows surface cracks in a hard chromium plating which were formed by the sliding action of a diamond.The crack lengths substantially exceeded the width of the groove produced by the diamond. The extent and shape of cracking can be strongly influenced by residual stresses in the plating.
Figure 3 - 1 5 .
-
Surface cracking caused by a diamond sliding under a normal load of 3N across a hard chromium plating on an austenitic steel.
Conway and Kirchner (ref.62) described the propagation of penny -shaped cracks for different horizontal - to - vertical load ratios. According to their model, tangential forces should not influence crack initiation and propagation in the plane perpendicular to the plane of motion. Different studies have shown that indentation fracture mechanics can be successfully applied to wear problems. This will be discussed in Chapter 5 in more detail.
3.3
SURFACE TEMPERATURE In both elastic and plastic deformation during the sliding contact of t w o surfaces of solids, energy must be expended for main-
72
taining the motion. About 90% of the energy expended for deformation of the surfaces in contact is dissipated as heat, and causes an increase in surface temperature. Frequently, it is very important to estimate temperatures in the contact area since they inf luence the mechanical and microstructural properties of solids. It is well known that thermally activated processes such as recrystallization, transformation, precipitation or chemical reactions can substantially change contact conditions and hence friction and wear. In Hertzian contact, the surface pressure is reduced with increasing surface temperature due to a decreasing Young's modulus. The area of the surface in actual contact has to be considered as a heat source acting only over a very short time. The temperature distribution in the surfaces in contact is strongly dependent on surface pressure, velocity, geometry of contact, surface roughness,conductivity,surface film,lubricant etc. In the contact of individual asperities,energy is being dissipated so quickly that there is no time for substantial heat flow into regions outside the contact zone. Hence very high temperatures,the so-called flash temperatures,are induced locally which may raise the contact temperature substantially above the surface temperature for the time of asperity contact. When the asperities are out of contact, the temperature drops to an average temperature due to conduction of heat into the bulk. This average temperature may be called the surface temperature in an equilibrium state. A controversy exists as to whether the surface temperature or the contact temperature (i.e. average surface temperature plus flash temperature ATf) has to be considered the more important for friction and wear problems. It seems that the importance of these temperatures depends strongly on the tribosystem involved.The occurrence of white layers in bearing steels is due to a friction - induced martensite/austenite transformation. This process is determined by the contact temperature and is nearly independent of time. In contrast, recrystallization or precipitation depends on temperature and time. This means that the average surface temperature is the most important one, since it prevails over a sufficiently long time. Microstructural changes in surface asperities caused by contact temperatures are effective only until the uppermost surface zone is worn away. The influence of the surface temperature goes farther down into the bulk material and lastslongerin the friction and wear processes.
73
The maximum temperature (flash temperature plus average surface temperature) due to frictional contact was analytically studied by Blok (ref.63,64).He introduced a flash temperature concept for gear design. Blok's analytical work was extended by Jaeger (ref. 65) and Archard (ref.66). An overview of this work was given b y Winer and Cheng (ref.67) and Polzer and Meissner (ref.68). The flash temperature at the frictional contact of two solids can be calculated (ref.67) from:
where u is the coefficient of friction, FN the normal load,vl,v2 the velocities of surfaces 1 and 2, k,, k2 the thermal conductivities,q ,02 the densities, c:, c; the specific heats, 2a the width of the contact area (e.g.twice the Hertzian contact radius) and 1 the length of cylindrical bodies in contact perpendicular to the motion. For practical calculation, helpful data about typical thermal properties of solids are given in (ref.67). Figure 3-16(a) shows some parameters involved in equation ( 3 - 3 9 ) .
Archord's Model
L-20-4
Figure 3-16.
-
v2
Models for calculation of temperature increase due to frictional heating: (a) for a rectangular contact area and (b) for a circular contact area.
14
The instantaneous temperature of surface asperities in contact can be calculated from the bulk temperature or average surface temperature Tb and the flash temperature ATf: Tc= Tb + ATf
(3-40)
Archard (ref.66) used a model of a circular contact area ( F i g . 3 for some special cases. From this model the following equations result for elastic or plastic contact deformation: - 1 6 ( b ) ) for calculations of mean flash temperature
(a) elastic deformation and low sliding speed (La < 0 . 1
)
(3-41 )
(b) elastic deformation and high sliding speed (La> 1 0 0 )
(3-42
(c) plastic deformation and low sliding speed ( L a < 0 . 1 )
(3-43)
(d) plastic deformation and high sliding speed (La> 1 0 0 ) u . v 21 / 2 . F N 1 / 4 ( n . p ) 3 / 4 ATf=
3 . 2 5 (k.0.c* ) 1 / 2
Y
(
3-44)
where La=
v 2 . p.c*.a
2 k
(3-45)
For elastic contact, the contact radius a is given by equation ( 3 - 9 ) and for plastic contact we obtain from equation ( 3 - 2 ) :
75
(3-461
The symbols used are: FN normal load,p coefficient of friction, v2 sliding speed, E Young's modulus, py yield pressure (about equal to hardness), pdensity, c* specific heat, k thermal conductivity and R undeformed radius of asperities. Winer and coworkers (ref.69) confirmed, in their experiments for rolling and sliding, the powerdependenceof the flash temperatureon load as predicted by equations (3-41) and (3-42). Kuhlmann-Wilsdorf (ref. 70) presented a more general evaluation of flash temperatures. Krause and Christ (ref.71) reported for rolling contact an increase in temperature with increasing load and slip. The load and slip dependence on temperature was about linear. Very high peaks of temperature occurred due to surface damage in rolling contact.
3.4 1.
2. 3. 4.
5.
6.
7. 8. 9.
REFERENCES Whitehouse,D,J.:Surface topography and quality and its relevance to wear,in Fundamentals of Tribology.Suh,N.P.and Saka , N. ,eds. ,MIT Press,Cambridge 1980, pp.17-50. Moore,D.F.:Principles and Applications of Tribology.Pergamon Press, Oxford 1975. Halling,J.,ed.:Principles of Tribology. Maxmillan Press,London 1978. Williamson,J.B.P.:Topography of solid surfaces -an interdisciplinary approach to friction and wear. NASA SP-181, 1978, pp.85-142. Mignot ,J. and Gorecki,C.: Measurement of surface roughness: comparison between a defect-of-focus optical technique and the classical stylus technique. Wear, 87 (1983) 39-49. Tsukada,T. and Sasaj ina,K.: A three-dimensional measuring technique for surface asperities. Wear, 71 (1981) 1-1 4. Bowden ,€ P and Tabor,D. :The area of contact bet ween stationary and moving surfaces.Proc.R.Soc.,l69,N 938 (1 939)391-413. Dyson,J. and Hirst,W.: The true contact area between solids, Proc.Phys.Soc.,Ser.B,67, N 412 (1954) 309-312. Bowden ,F. P. and Tabor ,D. :Fr iction,lubrication and wear:a survey of work during the last decade. Brit. J. Appl. Phys., 17
..
76 (1966) 1521-1544. (1970) 145-179. 10. Tabor,D.:Hardness of solids.Rev.Phys.Tech.,l 1 1 . Bowden,F.P.and Tabor,D.: The Friction and Lubrication of So1ids.Clarendon Press, Oxford 1954. Contact of nominal 1y 12. Greenwood ,J.A. and Will iamson,J.P.B.: flat surfaces. Proc.R.Soc., A295 (1966) 300-319. 13. Whitehouse,D.J. and Archard,J.F.: The properties of random surfaces of significance in their contact.Proc.R.Soc.London, A316 (1970) 97-121. contact of surfaces having a 14. Onions,R.A.and Archard,J.F.:The random structure. J.Phys.D,Appl.Phys.,6 (1 973) 289-304. 15. Tabor ,D. :The Hardness of Metals.Clarendon Press,Oxford 1951. 16. Johnson, K.L.: Rolling resistance of a rigid cylinder on an elastic-plastic surface. Int.J.Mech.Sci.,l4 (1 972) 145-148. 17. Hertz,H.: Uber die Beriihrung fester elastischer Korper und iiber die Harte. Sitzungsberichte des Vereins zur Forderung des Gewerbef leiRes ( 1 882 ) ,S. 449- 463. of Elasticity. McGraw 18. Timoshenko,S. P.and Goodier,J.N.:Theory Hil1,New York 1951. 19. Sarkar,A.D.: Wear of Metals. Pergamon Press,Oxford 1976. 20. Kragelsky,I.V. ,Dobychin,M.N. and Kornbalov,V.S.: Friction and Wear, Calculation Methods. Pergamon Press, Oxford 1982. 21. Habi9,K.H.: VerschleiO und Harte von Werkstoffen.Hanser Verlag, Miinchen 1980. The stress field created by 22. Harnilton,G.M. and Goodrnann,L.E.: a circular sliding contact. J.Appl.Mech.,33 (1966) 371 -376. 23. Gilroy ,D.R.and Hirst,W.:Brittle fracture of glass under nor(196911784mal and sliding loads. Brit.J.Appl.Phys.,Ser.2,2 1787. 24. Kloos,K. H .and Schrnidt,F.:Surf ace fatigue wear-causes and remedial measures,in Metallurgical Aspects of Wear. Hornbogen, E. and Zurn Gahr,K.H.,eds., DGM Verlag 1981, pp.163-182. 25. Broszeit,E.,Zwirlein,O.and Adelmann,J.: Werkstoffanstrengung i m Hertzschen Kontakt - Einflufl von Reibung und Eigenspannungen. Z.Werkstof f tech., 1 3 ( 1 982 ) 423 - 429. 26. Bental1,R.H. and Johnson,K.L.:Slip in the rolling contact of (1 967) 389two dissimilar elastic rollers.Int.J.Mech.Sci.,9 404. 27. Archard,J.F.: Contact and rubbing of flat surfaces. J.App1. Phys. ,24 (1 953) 981 -988. 28. Lodge ,A .S .and Howell ,H.G.:Friction of an elastic solid.Proc.
77
Phys.Soc. ,Ser.B,67 (1954) 89-97. 29. Adams,N.: Friction and deformation of nylon, 1-Experimental. J.Appl.Polym.Sci. I 7 (1 963) 2075-2103. 30. Francis,H.A.: The accuracy of plane strain models for the elastic contact of three-dimensional rough surfaces.Wear, 85 (1983) 239-256. 31. Johnson,K.L., Kendal1,K. and Roberts,A.D.:Surface energy and the contact of elastic solids.Proc.R.Soc.,A324(1971 )301-31 3. 32. Johnson,K.L.:A note on the adhesion of elastic solids. Brit. J.Appl.Phys.,9 (1958) 199-200. 33. Johnson,K.L.:Non-Hertzian contact of elastic spheres, in The Mechanics of the Contact Between Deformable Bodies.de Pater, A. D. and Ka 1ker ,J.J., ed s . , De 1f t Un i ver s i ty Press,Delf t 1 975, pp. 26-40. 34. Fuller,K.N.G.and Tabor,D.:The effect of surface roughness on the adhesion of elastic solids.Proc.R.Soc.,LondonIA345( 1975) 327-342. 35. Roy Chowdhury,S.K. and Pollock,H.M.: Adhesion between metal surfaces:the effect of roughness,Wear,66 (1981 307-321. 36. Hills,D.A. :Some aspects of post-yield contact problems.Wear, 85 (1983) 107-119. 37. Tabor,D.: A simple theory of static and dynamic hardness. Proc.R.Soc. ,London,A192 (1 948) 247-274. 38. Hills,D.A.and Ashelby,D.W.:A note on shakedown.WearI65(1980) 125-129. 39. Perrott,C.M.:Elastic-plastic indentation: hardness and fracture. Wear,45 (1977) 293-309. 40. Hi11,R.: The Mathematical Theory of Plasticity. Clarendon Press, Oxford 1950. 41. Marsh,D.M.: Plastic flow in glass. Proc.R.Soc., London, A279 (1964) 420-435. 42. Johnson,K.L.: The correlation of indentation experiments. J. Mech.Phys.Solids,lB (1970) 119-120. 43. Studman,C.J.and Field,J.E.:The indentation behaviour of hard metals.J.Phys.D.:Appl.Phys., 9 (1 976) 857-867. 44. Swain,M.V. and Hagan,J.T.:Indentation plasticity and the en(1 976) 2201 suing fracture of glass. J.Phys.D.:Appl.Phys.,g 2214. 45. Courtney-Pratt,J.S. and Eisner,E.:The effect of a tangential force on the contact of metallic bodies. Proc.R.Soc.,London, A238 (1957) 529-550.
78 46. McFarlane,J. S.and Tabor,D.:Relation between friction and adhesion. Proc.R.Soc.,London,A202 (1 950) 244-253. 47. Collins,I.F.: A simplified analysis of the rolling of a cylinder on a rigid/perfectly plastic half -space. 1nt.J.Mech. Sci., 14 (1972) 1-14. The hertzian fracture test. J.Phys.D.: Appl. 48. Wilshaw,T.R.: Phys., 4 (1971) 1567-1581. 49 * Boussinesq,J.: Application des Potentiels a 1'Etude de 1'Equilibre et du Mouvement des Solides Elastiques,GauthierVillars ,Paris 1885. 50. Lawn,B. and Wilshaw,R.: Indentation fracture: principles and applications. J.Ma ter.Sc i., 1 0 ( 1 975 ) 1 049 - 1 08 1. beneath point indenta51. Lawn,B.R.and Swain,M.V.:Microfracture (1975) 113-122. tion in brittle solids. J.Mater.Sci.,lO The indentation of hard metals: 52. Studman,C.J. and Field,J.E.: 21 5-218. the role of residual stresses.J.Mater.Sci.,12(1977) stress effects in sharp 53. Marshal1,D.B. and Lawn,B.R.:Residual (1 979) 2001 -2012. contact cracking. J.Mater.Sci.,l4 54. Marshal 1,D. B. ,Lawn,B. R. and Chantikul ,P.:Residua 1 stress ef fects in sharp contact cracking. J.Mater.Sci.,l4(1979) 22252235. 55. Palmqvist,S.: RiRbildungsarbeit bei Vickers-Eindrucken als MaR fur die Zahigkeit von Hartmetallen.Arch.Eisenhuttenwes., 33 (1962) 629-634. 56. Evans,A.G.and Charles,E.A.:Fracture toughness determinations by indentation. J.Am.Ceram.Soc.,59 (1 976) 371 -372. simple method for evalu57. Viswanadham,R.K.and Venables,J.D.:A ating cemented carbides. Met.Trans.,8A (1977) 187-191. Zusammenhang von RiRausbreitungswiderstand 58. Zum Gahr,K.H.: beim Harteeindruck und Bruchzahigkeit von Werkzeugstahl 90MnCrV8. Z.Metallkde.,69 (1 978) 534-539. 59. Warren,R.: Measurement of the fracture properties of brittle solids by Hertzian indentation. Acta Metall., 26(1978) 17591769. 60. Exner ,E. L. ,Pickens,J .R .and Gurland,J.:A comparison of indentation crack resistance and fracture toughness of five WC-Co alloys. Met.Trans.,gA (1 978) 736-738. 61. Lawn,B.R. and Fuller,E.R.: Equilibrium penny-like cracks in indentation fracture. J.Mater.Sci.,lO (1975) 2016-2024. mechanics of crack initia62. Conway,J.C. and Kirchner,H.P.:The tion and propagation beneath a moving sharp indentor. J.
79 Mater.Sci., 1 5 (1980)2879-2883. 63. Blok,H.:Measurement of temperature flashes on gear teeth under extreme pressure conditions. Proc. General Discussion on ( 1 93711 4Lubrication and Lubricants.Inst.Mech.Eng.,LondonI2 20 and 222-235. 64. Blok,H. :The flash temperature concept. Wear,6 (1963)483-494. 65. Jaeger,J.C.: Moving sources of heat and the temperature of sliding contact. Proc.R.Soc.,N.S.W.76 (1942) 263-224. 66. Archard,J.F. :The temperature of rubbing surfaces.WearI2(1958 /59) 438-455. 67. Winer,W.O. and Chen9,H.S.: Film thickness,contact stress and surface temperatures, in Wear Control Handbook.Peterson,M.B. and Winer,W.O., eds., ASME, New York 1980, pp.81-141. 68. Polzer,G. und MeiRner,F.: Grundlaqen zu Reibung und Verschleifi. VEB Verlag, Leipzig 1979. surface tem69. Nagara j ,H. S. ,Sanborn,D.M. and W iner,W.O.:Direct perature measurement by infrared radiation in elastohydrodynamic contacts and the correlation with the Blok flash temperature theory. Wear,49 (1978) 43-59. 70. Kuhlmann-Wilsdorf,D.:Flash temperatures due to friction and Joule heat at asperity contacts. Wear,lOS (1985) 187-198. 71. Krause,H. und Christ,E.:Kontaktf lachentemperaturen bei technisch trockener Reibunq und deren Messung. VDI-Z., 118(1976) 51 7-524.
Chapter 3
SURFACES IN CONTACT Wear is characterized as a process of surface damage due to mechanical contact of matter (see Section 1 . 3 1. Independently of any special wear mode, the type of mechanical contact is very important for all wear losses and will be discussed in the following. SURFACE TOPOGRAPHY Engineering surfaces are far from ideally smooth, and exhibit more or less roughness. The texture characteristics of surfaces are described by the arrangement, shape and size of individual elements such as asperities (hills and valleys on a microscopic scale). Figure 3-1 shows a surface topography schematically. Surface profiles can be obtained by vertical sections through the surface. Horizontal sections through the surface topography lead to information about the bearing area. Contact between two solids is generally discrete, due to surface roughnesses, i.e. it occurs at areas of individual point contacts. 3.1
Surfoce Prolile
Figure 3 - 1 .
- Schematic representation of surface irregularities.
49
Different optical and mechanical methods are available for measurement of the microscopic or macroscopic geometrical features of surfaces. In profilometry, stylus devices are widely used for obtaining surface profiles. A fine diamond stylus traverses the surface and its vertical movements are recorded. The profiles represent only one pass in a linear direction across a random three - dimensional surface. From a lot of cross - sectional areas of surface texture, contour plots can be drawn (ref. 1 5). Contour plots represent a three - dimensional image of the texture characteristics. New measuring systems (ref.6) were introduced for describing the three - dimensional characteristics of surface roughness. Figure 3-2 represents sections of profilograms of surfaces of a Ti-A1 alloy obtained by using a stylus device. The surfaces were finished by polishing, turning or milling.
0)
R,=O Ilpm , A , =0.58pm
.
Ro=0.C2~m.R t =3.21pm
Figure 3-2.
-
Profilometer traces of surfaces of a Ti-A1 alloy: (a) electrolytically polished, (b) turned and (c) m i 1led.
50
Frequently used values for characterization of surface textures are the centre - line average, or the mean arithmetic deviation of the profile (cla Ra), the root mean square (rmseR4~ 1 . 2 5 Ra) roughness value,the peak - to - valley height (Rt) or the maximum peak - to - valley height (Rmax). Surface profilometers record the irregularities on surfaces with different magnifications in the vertical and the horizontal direction (see Fig.3-2). Normally the vertical magnification is greater than the horizontal. Due to this difference in magnification, the recorded profiles d o not represent a true picture of the actual shapes of the surface irregularities. The actual picture of surfaces consists rather of broad based hills, with angles of inclinations from the base line less than 1 5 degrees (ref.41, than of sharp peaks. For evaluation of wear models it is very important to bear in mind the difference in the surface picture recorded and the actual surface.
Slol~cConloct
I F,
Sllding Contocl
I F,
Figure 3-3. - Apparent and real area of contact. Different types of experiments (ref.7,8) have shown that a large difference can exist between the apparent and real areas of contact of two flat solid surfaces pressed together (Fig.3-3).
51 The r a t i o o f r e a l t o a p p a r e n t a r e a o f c o n t a c t may b e a s l o w a s ( r e f . 9 ) a n d d e p e n d s on t h e d i s t r i b u t i o n o f s u r f a c e i r r e g u a r i t i e s , c o n t a c t f o r c e a n d y i e l d s t r e s s o f t h e s o f t e r mater a 1 i n v o l v e d . T h e r e a l a r e a of c o n t a c t i s l a r g e r i n s l i d i n g t h a n i n t h e s t a t i c state.
According t o t h e s t a t i c s t a t e i n F i g . 3 - 3
t h e real area o f con-
tact is g i v e n by: A,
n
=IA1i
(3-1
i=
is t h e real area o f c o n t a c t and Ai
w h e r e A,
the area of indivi-
dual contact spots. For t h e s t a t i c c o n t a c t of
i d e a l l y elastic
-
p l a s t i c materials,
t h e r e a l area o f c o n t a c t c a n be c a l c u l a t e d :
A,
=
FN -
(3-2)
@Y
where FN is t h e n o r m a l f o r c e o n t h e s u r f a c e s i n contact a n d p
Y t h e y i e l d p r e s s u r e o f t h e s o f t e r m a t e r i a l . A c c o r d i n g t o Bowden
and T a b o r ( r e f . l O , l l ) , s o f t e r material.
py i s a b o u t e q u a l t o t h e h a r d n e s s o f t.he
U s i n g t h e f o l l o w i n g r e l a t i o n between h a r d n e s s H
a n d y i e l d stress ay o f t h e m a t e r i a l stressed: H = C . 0
Y
(3-3 1
w i t h C = 3 f o r f e r r t i c steels, w e o b t a i n f r o m e q u a t on ( 3 - 2 ) f o r
static contact:
(3-4
T h e r a t i o o f r e a l t o a p p a r e n t a r e a of c o n t a c t i s p r o p o r t i o n a l to t h e a p p l i e d s u r f a c e p r e s s u r e d i v i d e d by t h e y i e l d stress o f t h e s o f t e r material i n c o n t a c t . L a t e r w e w i l l see t h a t r e a l a r e a o f c o n t a c t is i n c r e a s e d d u e t o s l i d i n g o f t h e c o n t a c t i n g s u r f a c e s r e l a t i v e to e a c h o t h e r (Fig.3-3).
52 3.2 CONTACT MECHANICS Friction and wear of two solid surfaces in unlubricated contact depends on the type of deformation of the surface irregularities involved. Greenwood and Williamson (ref.12) proposed a plasticity index $ which describes the transition from elastic to plastic deformation of surface asperities:
with
where H is the hardness of the softer material, E l , E2 are the Young's moduli and v1 , v2the Poisson's ratios of the two bodies in contact, R i s t h e radiusof the asperity summits (which is assumed to be the same for all asperities) and S is the standard deviation of a Gaussian distribution of the asperity heights. If $<0.6 the contact is predominantly elastic whereas if $> 1 plastic deformation dominates. Whitehouse and Archard (ref.13) introduced a more general plasticity factor$*which allows asperity peaks to have a distribution of curvatures.
E' S* rlr* = 0.69 -.H D
(3-71
where E' can be calculated from relation (3-61, S* is the rmsvalue of the surface and 13 is related to the correlation distance of the surface.According to Onions and Archard (ref.14) the equation (3-5) underestimates plasticity. This may be due to the assumption that all asperities have the same radius. Whitehouse and Archard (ref.13) showed however that the higher peaks of asperities had sharper radii than lower peaks. According to Tabor (ref.l5), the transition from elastic to elastoplastic contact at the indentation of a rigid sphere into a flat surface depends on the indentation depth. The contact becomes elastoplastic if the indentation depth h exceeds the critical value:
53
hcr = 0,89R.(H/E)*
(
3-8)
where R is the radius of the sphere, H is the hardness and E the Young's modulus of the material deformed. A more detailed review of plasticity factors is given in (ref.1 ). From equations(3-51, (3-7) and (3-8) it follows that the deformation of asperities in contact is determined mainly by the characteristics of the surface texture, hardness and elastic constants. The applied normal load or surface pressure does not directly influence the transition from elastic to plastic deformation, according to equations (3-5) and (3-7). The type of contact can change during service in a tribosystem. Starting with plastic contact, a change to elastic contact can occur after running-in. The type of contact can be expected to have considerable influence for example, in rolling contact. The rolling resistance can increase by more than two orders of magnitude when the contact alters from elastic to plastic deformation (ref. 1 6 ) .
3.2.1 Elastic Deformation The discrete nature of the contact is characteristic of all contacts between solids and is related to their surface roughnesses.Considering asperities as individual contact spots, the elastic strains and stresses at the contact area can be estimated from Hertzian formulae (ref.17). The solutions of the elastic stress fields are well known (ref.18-25). For simplicity, both bodies in contact are of the same material (El = E2 and V, = v2 = 0.31. Figure3-4 shows the pressure and stress distribution in and below the contact area for different contact configurations. Contact of a Sphere and a Plane The Hertzian pressure is hemispherically distributed on a plane contact area with the contact radius a:
54
rN\2E2 )
and the maximum contact pressure pmax:
pmax =
-
0.388
(3-10)
1’3
where FN is the normal load, R the radius of the sphere and E the Young’s modulus with El = E 2 and v , = v2 = 0.3. The maximum shear stress T~~~ occurs at a depth 2, below the contact area (Fig.34(a)):
2, = 0.47a
(3-12)
Tensile stresses occur in a region close to the surface and outside the contact circle.
b)
Figure 3-4. - Schematic stress distribution for the elastic contact of:(a) a sphere and a plane due to normal load (b) a sphere and a plane due to combined normal and tangential loads where FT=0.3FN,(c) two unlubricated rolling cylinders and (d) two lubricated rolling cylinders (elastohydrodynamic lubrication).
55 Combining a normal load with a tangential load results in a substantial increase of the tensile stress at the rear of the sphere in the surface stressed (Fig.3-4fb)). The maximum shear stress now occurs much nearer to the surface or i n the surface, depending on the coefficient of friction u between the contacting bodies. The maximum tensile stresses at the rear of the contact circle can be calculated for sliding contact (ref. 22,23):
d
tmax
1 - 2 ~ FN ( 1 + C’.ll) na2
-- - .2-
(3-13)
with
cI .
3 n *4 + v 8 1-2v
Contact of two Spheres For Hertzian contact of two spheres with radii R 1 and R2, the contact radius a is calculated from:
a = i.iib(Rl
R1
.]”
. R2 +
(3-14)
and the maximum pressure due to a normal load alone:
(3-15)
The maximum shear stress at a depth Z, below the surface is obtained by : (3-16) and 2, = 0.47a
(3-17)
56
Contact of two Cylinders The contact radius a for Hertzian contact of two cylinders can be calculated from:
a = 1.52
"
F .R1'R 2 E.1 . (R1 + R 2 )
(3-18)
where 1 is the length, R1 and R 2 the radii of the cylinders and E the Young*s modulus. The maximum pressure is obtained by:
Pmax = - 0.418(2.
E.
(3-19)
The maximum shear stress at a depth 2, below the surface is obtained from: (
3-20)
and 2, = 0.78a
(3-21)
Figure 3-4(c) shows the contact of two cylinders qualitatively. Friction between the rolling cylinders results in tangential traction in the surface. Surface stresses in the contact area are compressive for a coefficient of friction = 0. The compressive stress component becomes increasingly unsymmetrical with increasing value of u. Simultaneously, the tensile stress increases at the end of the region of contact. The reference stress calculated by using the strain energy hypothesis reaches its maximum value below the surface for u = 0. The maximum reference stress occurs however in the surface when the coefficient of friction exceeds about 0.2 (ref.24). Slip in the rolling contact of two cylinders was considered by Bentall and Johnson (ref.26).In elastohydrodynamic lubrication (Fig.3-4(d)) a local peak in the pressure curve occurs just before the end of contact. Figure 3-5 (ref.20) shows the influence of the shape of ters on the distribution of pressure at the contact area.
inden-
57
I
I
FN
FN
I
, ’l
.
.’./
J’
, ,,,//’,,
Figure 3-5. - Pressure distribution curves at elastic contact for three different indenters ( sphere, flat die and cone)
.
In the contact area between a sphere and a flat surface, the maximum pressure, with a finite value, occurs in the centre. In contrast, the model of a cone or a flat die results in indeterminate stresses in the centre or at the periphery of the contact. Hence, a spherical geometry is favoured above others for calculating elastic contact problems. Dependence of the real area of contact on the normal load is an important question for friction and wear. For purely elastic deformation, the proportionality between real area of contact Ar and normal load FN is presented in Fig.3-6 for different contact geometries. It follows from the Hertzian equations (3-9) or (3-14) that the real area of contact between a sphere and a flat surface, or between two spheres,or between two general curved surfaces which result in an elliptical contact, depends on the normal load according to: (
3-22)
58
Figure 3 - 6 . - Influence of contact geometry on the dependence of the real area of contact A, on the normal load FN in purely elastic contact. For real surfaces in friction and wear, multi-asperities models are more practicable. Archard (ref.27) studied the contact between surfaces covered with spherically shaped asperities. Two contact models may be distinguished. Firstly the number of asperity contacts is independent of the normal force. An increasing normal load results in increasing deformation of each contact. Secondly the average area of each asperity contact remains constant with increasing normal loads, but the number of asperity contacts increases. According to these models, it is obtained: A r d FNm
(3-23)
59 where m = 213 for a constant number and m = 1 for an increasing number of asperity contacts with increasing normal load. Figure 3-6 shows schematically both asperity models. Other models (ref. 28) describe the elastic contact of surfaces covered with smaller asperities. It follows from these, that the smaller the asperities and the closer they are packed the more closely the factor m approaches 1. According to Adams (ref.29) and Bowden and Tabor (ref.9), the real area of contact between a hard sphere and the smooth flat surface of a polymer depends alsoon the normal load, as predicted by equation (3-23) with m between 0.1 and 0.8. The plastic contact of asperities is considered in equation (3-21, i.e. m is equal to 1. The finite element technique was recently applied for modelling of the elastic contact of three -dimensional rough surfaces (ref.30). In many friction and wear problems, the contact between two smooth elastic bodies may be influenced by adhesion (see Section 4.2.1 ).More or less attractive forces can occur between surfaces in contact, depending on the environment or condition of lubrication, surface roughness, surface layers or materials involved. Johnson et al. (ref.31,32) have shown that the radius of contact between two bodies can be substantially increased by the action of attractive surface forces. They described the surface forces of adhesion by the surface energy, i.e. the work required to separate unit area of the adhered surfaces. For two smooth spheres, they calculated the ratio of contact radius with adhesion a, and without a :
(
3-24)
where F N is the normal force, y the surface energy and R = R1R2/(Rl+RZ), R 1 and R2 being the radii of the spheres. Without adhesion, the surface energy y becomes zero and .a equal to a. Figure 3-7 shows the stress field at the contact area taking into consideration the attractive forces due to adhesion. The spheres were pressed together by the normal force FN. Then this force was reduced by AFN. The contact is maintained over the original enlarged area due to adhesion. As a result, the stresses between the surfaces are compressive at the centre but tensile at the edge of contact.
60
Tension ( 4 )
Figure 3 - 7 .
-
Elastic contact between two solids in the of surface forces.
presence
Johnson (ref.33) presented a model of the contact of rough surfaces. According to this model, the maximum contact pressure decreases and the effective contact pressure spreads over an increasing area when surface roughness is increased. The influence of adhesion depends on the value of an "adhesion index":
( 3-25
in which
and
-E'1 _
-
1
El
v1
+
1 - v
2
2
E2
where S is the standard deviation of asperity heights, y the surface energy, R = R l * R 2 / ( R l + R 2 ) , R 1 and R 2 are the radii and E l , E2 the Young's moduli of the bodies in contact. Adhesion ceases when the adhesion index exceeds a critical value (aad = 1 . 6 ) .
61
Hence, the influence of adhesion decreases w i t h increasing surface roughness or elastic modulus and with decreasing surface energy. Fuller and Tabor (ref.34) also investigated the effect of surface roughness on the adhesion of elastic solids and introduced an adhesion parameter which results in similar conclusions to those from equation (3-25). Roy Chowdhury and Pollock (ref. 35) presented a multi-asperity model for plastic contact which expresses the area of contact by:
Ar
-
)-
N
(3-26)
where FN is the applied load, H the hardness, S the standard deviation of the asperity heights and yad the work of adhesion per unit area. According to this model, a significant adhesion force can only be expected when the asperities are plastically deformed. The real area of contact is increased due to adhesion. The influence of adhesion is reduced by increasing surface hardness and surface roughness.
3.2.2 Plastic Deformation When two elastic bodies, e.g. a sphere and a flat specimen, are pressed lightly against each other, the contact is purely elastic. If the applied normal load exceeds a critical value, the elastic l i m i t , a plastic zone develops which is surrounded by elastically deformed material.The elastic limit can be calculated (ref. 36) from: p = 1.85
T
Y
(3-27)
where p is the mean contact pressure and T the yield stress in Y pure shear. In elastic contact, the peak contact pressure is 1 . 5 times the mean contact pressure. W i t h increasing load, the contact becomes elastoplastic and the pressure distribution more and more uniform. Finally the full plasticity condition exists, which is given (ref. 37) by: p
-
c+.T
Y
(3-28
62
where C * = 6 on Tresca's criterion and 5.2 on Mises' criterion.Figgure 3-8(a) shows the pressure distribution in elastic, elastoplastic and plastic contact schematically. With increasing plasticity, the hemispherically distributed pressure, with peak pressure in the centre of the elastic contact, is changed to a pressure uniformly distributed across the contact area in the fully plastic condition.
0)
Eloslic
Elostoploslic
lncreosing Plasticity
PlOSllC
ession
I
Figure 3-8. - Schematic representation of the pressure distribution at and below contact areas: (a) elastic, elastoplastic and plastic contact of a sphere and a flat surface, (b) indentation of a sharp indenter in a flat surface. After repeated loading and unloading, a steady state "shakedown limit" can be reached, i.e. the contact is quasi-elastic. This state results from changes in contact geometry due to plastic
63 flow connected with work hardening and the development of residual stresses. The shakedown limit is determined (ref.38) by: p = 3.69
T
Y
(3-29)
Elastoplastic indentation by sharp indenters (Fig.3-8(b)) w a s discussed by Perrott (ref.39).According to his analysis,the pressure (equal to hardness) for an elastic - plastic indentation by cones and Vickers pyramids is given by:
(3-30)
where a is the yield stress under simple tension, E the Young's Y modulus and u the Poisson's ratio of the test material and 0 is the semi-apical angle of conical or pyramidal indenters. From an "expanding cavity" model introduced by Hill (ref.401,Marsh (ref. 41) and Johnson (ref.42) presented similar equations. According to Johnson, the following relationship between indentation pressure and yield stress can be used:
where 0 is the semi-apical angle of conicial or pyramidal indenters. The factor cot 0 can be replaced by d / D for spherical indenters, where D is the diameter of the indenter and d the diameter of the indentation. Spherical indenters lead to an elastic deformation at low loads, in contrast to sharp indenters. Hence, equation (3-31) is only applicable for spherical indenters when the elastic limit is exceeded. The influence of work hardening during the indentation process is ignored in these models. The indentation plasticity was experimentally investigated with steel (ref.43) and glass (ref.44). The real area of contact in the plastic condition can be estimated from equation (3-2). Under the combined action of a normal and a tangential force, the real area of contact is increased. This was studied by Courtney-Pratt and Eisner by using electrical
64 resistance measurements (ref.45). McFarlane and Tabor (ref.46) calculated the real area in sliding contact from a yield criterion for junction growth: a2 + C l T 2 -
2
(3-32)
- PY
2
(3-33)
(3-34
(3-35)
from equation (3-21, where : A and A, are the real area in sliding and static contact,respectively,py the yield pressure of the softer material, o the applied normal and T the applied tangential stress and F N l F T the normal and tangential forces. C1 is a constant with a value of about 10 (ref.g).Under a normal force only, equation (3-34) is reduced to equation (3-2). The combined effect of a normal and a tangential force results in an increase of the real area of contact. In the plastic contact of metals, work hardening can become an important factor.The influence of work hardening on the real area of contact can be estimated from a yield criterion, similar to that of equation (3-32):
a2 + C p 2 =
( Py
+
LIPy) 2
and finally we obtain:
(3-36)
65
Ar * = ArJ1
+
Cl(Z)*
(3-37
APY
l+-
PY where apy is the increase in yield pressure p due to work harY dening.In genera1,equation ( 3 - 3 7 ) could also be applied when work softening ( - lapy! ) occurs. According to this model, work hardening should result in a smaller and work softening in a larger real area of contact. Rolling contact is of great interest for many practical tribosystems. Johnson (ref.16) and Collins (ref.47) analysed elasticplastic and plastic contact in rolling. This will be discussed in more detail in the Chapter 7 .
Indentation Fracture Mechanics Materials of high wear resistance are frequently very hard but brittle, e.g. hardened steels, cast irons or ceramics. The contact loading of brittle solids can result not only in elastic and plastic deformation but also in microcracking at and below the stressed surfaces. 3.2.3
I
lHerlzion Crock .---.--
Figure 3 - 9 .
-
‘compression {
_ ______ /.
.A
- Formation of a Hertzian crack in an elastic stress field.
66
Cone-shaped Hertzian cracks are a well know example iref.48). A circular cone-shaped crack originates around the contact area between a sphere and the flat surface of a brittle solid when a critical load is exceeded. This crack propayates, with increasing load, from the circumference of the contact circle along the periphery of a cone into the solid (Fig.3-9). The maximum tensile stress occurs at the contact circle.
Figure 3-10. - Indentation of a Vickers diamond on surfaces of (a) WC - Co cemented carbides (light micrograph), (b) cracking at the periphery of an indentation on 0 . 9 4 % C - steel (SEM micrograph). Blunt and sharp indenters have to be distinguished when studying indentation problems. Pyramids or cones may be considered as sharp, and spheres a s blunt, indenters. Depending on the type of indenter, blunt or sharp, the contact results in predominantly elastic or plastic deformation. The theoretical solution of the elastic stress field caused by a sharp indenter shows a singularity at the centre of indentation (Fig.3-5). The stress field due to a normal point load was first described by Boussinesq (ref.49). Lawn and Wilshaw (ref.50) presented an excellent review of the principles of indentation fracture. Figure 3-10 shows the cracking that was caused by indentation by a Vickers diamond pyramid in the surface of tool steel and cemented carbides.
67
The profile of crack propagation perpendicular to the surface was measured by polishing in steps from the surface t o t h e interior of the steel (Fig.3-11 ). At the surface of the steel, cracks propagated almost perpendicularly from the edges of the indentation into the interior. The direct contact between the crack profiles and the areas of indentation was lost at a depth of about 1/3 of the total indentation depth. This is also shown i n Fig.310(b) and is predicted in Fig.3-8tb) by the tensile stress field close to the surface.
Figure 3-11. - Surface cracking in 0.94%C - steel due to indentation of a Vickers diamond by a test load of 625 N: (a) crack formation about 5 5 u m below the surface, ( b ) cracks (see arrows) about 7 5 below ~ the surface ( indentation was removed by polishing), (c) measured crack profile below the surface, (d) total crack profile.
68
According to Lawn and Swain (ref.Sl),surface loading by a point indenter results in median and lateral cracks below the stressed surface.Figure 3 - 1 2 shows schematically different types of cracks formed during loading and unloading.
lime
t
Un'ood'ng
Figure 3 - 1 2 .
-
Formation of median and lateral cracks in brittle solids due to indentation by a sharp indenter.
An increasing point load results in increasing size of a plastic zone around and below the indentation. A median crack is formed when the load exceeds a critical value F2. This crack grows in depth with increasing load. During unloading, the median crack is closed and lateral cracks are formed and propagate to surface under an applied load of less than F 5 . An immediate reloading closes the lateral cracks and reopens the median crack. The formation of lateral cracks during unloading is connected with residual stresses which are due to the plastic deformation zone.Residual surface stresses can play an important role in microcracking (ref.52-54). Residual tensile stresses increase crack lengths and reduce the critical load for microcracking. During unloading, median cracks may propagate in depth due to residual stresses. Palmqvist (ref.55) first used crack lengths at the corners of a Vickers hardness indentation as a measure for ductility and sensitivity to microcracking. This method is frequently used today for characterization of ceramics or cemented carbides.
69 Different studies (ref.56-60) have shown that the critical stress intensity factor (fracture toughness KIc) can be estimated from cracking at hardness indentations. Measurements of the lengths of the median cracks on the symmetry planes of the indentation, the normal force and the shape of the indenter allow the calculation of KIc. Figure 3 - 1 3 shows the model of cracking used for the calculation.
Figure 3 - 1 3 .
-
Palmqvist cracks compared with halfpenny-like surface cracks used as model.
The s h a p e o f t h e m e d i a n c r a c k s i s a s s u m e d t o b e a centre-loaded halfpenny configuration. For this geometry, the stress intensity factor at the crack tip of a Vickers indentation can be calculated (ref. 6 1 ) :
FN 3’2. tanR
(3-38)
where 2c is the crack diameter. FN is the indenter load and 13 the semi-apical angle of the indenter (68O for a Vickers diamond). The friction between the indenter and the specimen is not considered but can be introduced by replacing 13 by (I3 + p q , where p*is the friction angle.
70
Figure 3-14 shows stress intensity factors of surface cracks of half-penny shape for different crack lengths.
Figure 3-14. - Stress intensity factors a s a function of length of surface cracks due to indentation:(a) at variable loads FN1 or FN2 and (b) for materials of different fracture toughness KIcl or KIc2. The stress intensity factor decreases with increasing crack length, d u e to the inhomogeneous stress field at the indenter. This means that cracks propagate to a final length which is d e termined by the fracture toughness of the material studied. Figure 3-14(a) shows the final crack length caused by two different loads FN1 and FN2. A crack of initial length co propagates to the length c 1 or c2 due to the load F N 1 o r FN2, respectively. A t this crack length, the stress intensity factor is equal to the fracture toughness KIc. In Fig.3-14(b), a crack propagates under a constant load F N from an initial length co to a length c1 or c2 for materials of fracture toughness KIcl or KIc2, respectively. This means that shorter cracks occur in materials of greater fracture toughness. Initial cracks can be caused by the indentation or are already present, e.g. hardening cracks or cracks at embrittled grain boundaries.
71
The situation becomes more complicated when a tangential force acts on the indenter in addition to the normal force. Figure 3-15 shows surface cracks in a hard chromium plating which were formed by the sliding action of a diamond.The crack lengths substantially exceeded the width of the groove produced by the diamond. The extent and shape of cracking can be strongly influenced by residual stresses in the plating.
Figure 3 - 1 5 .
-
Surface cracking caused by a diamond sliding under a normal load of 3N across a hard chromium plating on an austenitic steel.
Conway and Kirchner (ref.62) described the propagation of penny -shaped cracks for different horizontal - to - vertical load ratios. According to their model, tangential forces should not influence crack initiation and propagation in the plane perpendicular to the plane of motion. Different studies have shown that indentation fracture mechanics can be successfully applied to wear problems. This will be discussed in Chapter 5 in more detail.
3.3
SURFACE TEMPERATURE In both elastic and plastic deformation during the sliding contact of t w o surfaces of solids, energy must be expended for main-
72
taining the motion. About 90% of the energy expended for deformation of the surfaces in contact is dissipated as heat, and causes an increase in surface temperature. Frequently, it is very important to estimate temperatures in the contact area since they inf luence the mechanical and microstructural properties of solids. It is well known that thermally activated processes such as recrystallization, transformation, precipitation or chemical reactions can substantially change contact conditions and hence friction and wear. In Hertzian contact, the surface pressure is reduced with increasing surface temperature due to a decreasing Young's modulus. The area of the surface in actual contact has to be considered as a heat source acting only over a very short time. The temperature distribution in the surfaces in contact is strongly dependent on surface pressure, velocity, geometry of contact, surface roughness,conductivity,surface film,lubricant etc. In the contact of individual asperities,energy is being dissipated so quickly that there is no time for substantial heat flow into regions outside the contact zone. Hence very high temperatures,the so-called flash temperatures,are induced locally which may raise the contact temperature substantially above the surface temperature for the time of asperity contact. When the asperities are out of contact, the temperature drops to an average temperature due to conduction of heat into the bulk. This average temperature may be called the surface temperature in an equilibrium state. A controversy exists as to whether the surface temperature or the contact temperature (i.e. average surface temperature plus flash temperature ATf) has to be considered the more important for friction and wear problems. It seems that the importance of these temperatures depends strongly on the tribosystem involved.The occurrence of white layers in bearing steels is due to a friction - induced martensite/austenite transformation. This process is determined by the contact temperature and is nearly independent of time. In contrast, recrystallization or precipitation depends on temperature and time. This means that the average surface temperature is the most important one, since it prevails over a sufficiently long time. Microstructural changes in surface asperities caused by contact temperatures are effective only until the uppermost surface zone is worn away. The influence of the surface temperature goes farther down into the bulk material and lastslongerin the friction and wear processes.
73
The maximum temperature (flash temperature plus average surface temperature) due to frictional contact was analytically studied by Blok (ref.63,64).He introduced a flash temperature concept for gear design. Blok's analytical work was extended by Jaeger (ref. 65) and Archard (ref.66). An overview of this work was given b y Winer and Cheng (ref.67) and Polzer and Meissner (ref.68). The flash temperature at the frictional contact of two solids can be calculated (ref.67) from:
where u is the coefficient of friction, FN the normal load,vl,v2 the velocities of surfaces 1 and 2, k,, k2 the thermal conductivities,q ,02 the densities, c:, c; the specific heats, 2a the width of the contact area (e.g.twice the Hertzian contact radius) and 1 the length of cylindrical bodies in contact perpendicular to the motion. For practical calculation, helpful data about typical thermal properties of solids are given in (ref.67). Figure 3-16(a) shows some parameters involved in equation ( 3 - 3 9 ) .
Archord's Model
L-20-4
Figure 3-16.
-
v2
Models for calculation of temperature increase due to frictional heating: (a) for a rectangular contact area and (b) for a circular contact area.
14
The instantaneous temperature of surface asperities in contact can be calculated from the bulk temperature or average surface temperature Tb and the flash temperature ATf: Tc= Tb + ATf
(3-40)
Archard (ref.66) used a model of a circular contact area ( F i g . 3 for some special cases. From this model the following equations result for elastic or plastic contact deformation: - 1 6 ( b ) ) for calculations of mean flash temperature
(a) elastic deformation and low sliding speed (La < 0 . 1
)
(3-41 )
(b) elastic deformation and high sliding speed (La> 1 0 0 )
(3-42
(c) plastic deformation and low sliding speed ( L a < 0 . 1 )
(3-43)
(d) plastic deformation and high sliding speed (La> 1 0 0 ) u . v 21 / 2 . F N 1 / 4 ( n . p ) 3 / 4 ATf=
3 . 2 5 (k.0.c* ) 1 / 2
Y
(
3-44)
where La=
v 2 . p.c*.a
2 k
(3-45)
For elastic contact, the contact radius a is given by equation ( 3 - 9 ) and for plastic contact we obtain from equation ( 3 - 2 ) :
75
(3-461
The symbols used are: FN normal load,p coefficient of friction, v2 sliding speed, E Young's modulus, py yield pressure (about equal to hardness), pdensity, c* specific heat, k thermal conductivity and R undeformed radius of asperities. Winer and coworkers (ref.69) confirmed, in their experiments for rolling and sliding, the powerdependenceof the flash temperatureon load as predicted by equations (3-41) and (3-42). Kuhlmann-Wilsdorf (ref. 70) presented a more general evaluation of flash temperatures. Krause and Christ (ref.71) reported for rolling contact an increase in temperature with increasing load and slip. The load and slip dependence on temperature was about linear. Very high peaks of temperature occurred due to surface damage in rolling contact.
3.4 1.
2. 3. 4.
5.
6.
7. 8. 9.
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