Characteristics of impact friction and wear of polymeric materials

Characteristics of impact friction and wear of polymeric materials

Wear, 73 (1981) 213 - 234 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands C~ARAC~RISTICS OF IMPACT FRICTION POLYMERIC MATERIALS 213 ...

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Wear, 73 (1981) 213 - 234 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands




Summary The fundam~~t~ differences between friction and wear under impact and friction and wear under sliding are the pressure (i.e. the load responsible for friction) and the friction coefficient, The dynamic equivalents of both factors are determined for impact effects by specific parameters; these parameters are the particle velocity uO and the angle Q of attack. The pressure is given by iI = puo2 sin2cro and the friction f =



l.J(-j COS(YQ - Uf COSQf tte sin a o + uf sin cyF

In the range of small and medium values of eye the impact friction coefficient for elastomers decreases with an increase in the angle of attack and increases with increases in the angle of attack for plastics. For large values of (x0 the friction coefficient of all materials decreases and approaches zero at normal impact. The friction coefficient under impact is of fundamental impo~~e to wear by a flow of particles because it defines the fatigue strength of the body being worn. To increase wear resistance under impact this friction coefficient should be lowered; this is similar to the case of sliding friction and wear. For microcutting by a colliding particle, in contrast with sliding, it is advantageous to have an increased coefficient of friction under impact. When the s~mil~ity between the pressures p and B is taken into account, the expressions for sliding wear and impact wear are analogous. Because the initial process of failure is the same in both cases, the relationship between the wear resistance and the properties of the material being worn (strength, deformability, endurance and friction coefficient) is the same in both cases, In the mierocutti~g of plastics an increased hardness is import~t. In the microcutting of vuleanizates by flying particles an increased modulus of elasticity is important.


1. Introduction There are three types of external friction, namely static friction at rest, kinetic friction (in motion) and dynamic friction (under impact). Static friction and kinetic friction are associated with a permanent contact between two solid bodies. (With static friction and kinetic friction both sliding and rolling are possible. In the case of rolling, wear occurs through the sliding component [l, 21. Therefore no distinction is made between these two types of friction; they are both known as friction under a permanent contact or as its principal type, which is sliding friction.) Dissipation of energy in these cases is characterized by a friction coefficient, while in dynamic friction dissipation of energy is usually characterized by the portion of energy lost at impact. Thus under a permanent contact the value of the friction coefficient should be known for the calculation of the friction force (moment). This problem does not usually arise in impact friction because in practice impact wear is not related to the work of the friction forces [3]. The characteristics of dynamic friction and of friction under a permanent contact are similar. Logically, the quantitative effect of friction on frictional breakdown under impact should be substantially the same as that under sliding (ref. 4, p. 300) where the frictional wear results from a process of fatigue fracture as does usual wear. Thus it is essential to investigate first methods for the evaluation of the friction coefficient under impact and specific methods for the measurement of this friction coefficient. This is the subject of Section 2 of the present paper. Then, if we assume the concept of wear as a fatigue process, the relationship between impact wear and the basic parameters of loading and material properties, including the coefficient of friction under impact, should be determined. This is the subject of Section 3 of the present paper; the analysis is based on the analogy between wear under impact and wear at a permanent contact. 2. Friction

under impact

With a permanent contact it is usual to define the friction coefficient as the ratio of the friction force F to the normal load N. Similarly, the coefficient of friction at impact is defined (ref. 1, p. 239; refs. 5 and 6) as the ratio between the tangential and the normal impact impulses:

f ,F




where T* is the impact duration and 7 the elapsed time. Usually the particle speed at impact is known and therefore it is advisable to transform eqn. (1) on the basis of the equation of motion of a flying particle. 2.1. Formula for the friction coefficient under impact Using the principles of Nepomnyaschy [7], we consider the case of a glancing collision between a solid spherical particle and an elastic or a plastic half-space (Fig. 1). The particle has an initial speed uo. The vector of this speed forms an angle a0 of attack with the surface. The particle first contacts the body, at point A, and then slides over the surface while penetrating into the deformed body before it is pushed out. Along the X coordinate the particle traverses the path x, , while along the h coordinate the particle penetrates to a depth h,,, . At point B sliding stops and the particle jumps back from the surface at a final speed V. The vector of this speed is directed from the surface at a rebound angle a!.

Fig. 1. Scheme of the glancing impact of a spherical particle against a surface.

For the X and h components of equations applies: d2X m-_-z-dr2

in the case of impact the following



(2a) m $


where m is the mass of the solid particle. If eqn. (1) is transformed using eqn. (2a) we obtain f





We now integrate eqn. (2b) assuming that f is a mean value with respect to time of the friction coefficient, i.e. that f has a constant value during one impact, to obtain the following:


dh dx f -- = - + constant dr dr The boundary


@a) for eqn. (3a) are = u. sina

= ug cos(Yo


dh - = u sin ff dr

dX - = u cos Q dr

Here, as in Fig. 1, the zero subscript denotes values of the parameters of the particle motion before impact (initial conditions) and the absence of a subscript denotes these values after impact (final conditions). Putting the initial conditions into eqn. (3a) provides the integration constant: constant

= u. ( f sin ffo - cos cyo)

Substitution f=




of the final condition -

u cos

and the integration




u. sin a0 + u sin (Y

The coefficient of friction under impact may be expressed even more simply through the speed components of a solid particle on the basis of the relationship between the components of an impact impulse according to eqn, (1). The theorem on momentum of a solid particle for these eonstituents is


from which, according

to eqn. (l),

This equation, which takes account of the boundary conditions, is similar to eqn. (4a). The same relationship for f has been obtained by other workers

E81. Equation (4a) allows a mean value of the friction coefficient in the direction of sliding of a solid particle at a single impact to be determined. 2.2. Methods of determination of the friction coefficient The parameters of eqn. (4a) necessary to calculate f can be determined experimentally in many ways. High speed filming and photography with a stroboscope and a tribometer have been used for the determination off at a

glancing collision with individual beads. f has also been determined from eqn. (1) as a ratio of values proportional to impact impulses using the tensometric method. Furthermore, experiments have been performed with spherical and nonspherical bodies (beads and other particles). These solid particles are directed towards the surface of a sample either individually or in a jet (the jet is sufficiently rarefied to allow the flight of an individual particle to be traced both before and after the impact). The collision of particles in this jet with the sample surface has been recorded cinematic~ly; the collision of individual particles has been photo~aphed using a stroboscopic disc. The scheme of a tribometer for investigating glancing impact with beads is shown in Fig. 2. A bead is shot off by means of a flat spring. The initial speed u. is determined by the range of free flight of the particle. u is determined by the flight range after rebound from the sample. The rebound angle a! is determined by the point of incidence of the bead onto the scale of angles and distances which is located in the horizontal plane. Figure 3 shows diagrammatically a tensometric device used for the measurement of impact impulses. A sample 1 is secured on two steel plates 2 with electric wire sensors attached. On deformation of the plates due to the impact, a light-beam oscillograph records the amplitudes of the damping oscillations of the sample on a ~hoto~aphic film. According to known relationships, the maximum amplitudes of the oscillations are proportional to the impact impulses.



Fig. 2. Scheme of a tribometer for glancing impacts with individual beads: 1, base with scales for angles and distances; 2, flat spring; 3, cam for loading the spring; 4, cam electric motor; 5, holder with the sample; 6, bead. Fig. 3. Scheme for securing the sample (1) on flat springs (2) and the diagram of the bead impact directions: (a) glancing impact; (b) normal impact on an end face of the sample.


At glancing impact (Fig. 3(a)) the oscillations whose m~imum amplitudes are proportional to the impulse JT F d7 are recorded directly. The maximum amplitude proportional to the impulse JZN dr is determined indirectly, i.e. through the amplitude for the impulse J~,*N, dr where N, is the normal load under impact at the end face of the sample (Fig. 3(b)). Because


J N,

dr = m(uo + ufe)


where ufe is the speed after impact,









N, dr + m(v sin CY- ufe sin IQ,)


The second term of eqn. (6) has been shown experimentally to be very small compared with the first term. Thus in the determination off this term may be neglected and we may assume that T*



Nd7 * sina!a



N, d7



Using eqn. (7), the friction imate formula


is determined

by the approx-


s Q

F dr



sin 0.




A, sincro


N, dr


where A, is the initial amplitude on the oscillogram of the damping oscillations of the sample after a glancing impact and A, the corresponding value after impact at the sample end face. The value A, is propo~ion~ to the tangential component of the impact impulse J70+Fd7 and the value of A, is proportional to the impulse JFN, d7 at normal impact to the sample end face. In the experimental unit the flat springs of the tensometric device are rigid enough to prevent longitudinal strains but allow an oscillation amplitude which is large enough for convenient measurements. Measurements of the friction coefficient by the different methods have given substantially similar results. Figure 4 shows values of the friction coefficient obtained by three different methods for a polyurethane elastomer at various angles of attack.


Fig. 4. Relationship between the coefficient of friction of the polyurethane elastomer PUR-7 and the angle of attack as determined by three methods: X, tensometric; *, stroboscopic; 0, on the tribometer for glancing impact with beads.

2.3. The relationship between the friction coefficient and the angle of attack Figure 4 shows that the friction coefficient of the polyurethane elastomer decreases with increasing angle of attack. The friction coefficient for vulcanizates (Fig. 5) varies in a qualitatively similar manner. The friction coefficient for plastics at small and medium angles of attack usually increases insignificantly and then, as in the case of elastomers, decreases to zero as normal impact is approached (Fig. 5).

Fig. 5. Relationships between the coefficient of friction of rubbers, plastics and other materials and the angle of attack (curves 1 - 7, bead diameter, 2 mm; 00 = 2 m s-l ; curve 8, river sand with a particle size of 0.5 - 0.9 mm; uo = 19.5 m s-l): curve 1, vulcanizate with a hardness of 55 units; curve 2, vuleanizate with a hardness of 33 units; curve 3, the same material as for curve 2 but wetted with water; curve 4, steel Y8 with hardnesses 63 HRC (0) and 22 HRC (0); curve 5, mirror glass; curve 6, polymethylmethacrylate (PMMA); curve 7, glass plastic AG-4W; curve 8, rubber vulcanizate with a hardness of 60 units.

It is clear from Figs. 4 and 5 that the coefficient of friction at impact depends substantially on the angle of attack; this is contrary to the common practice of calculation where it has been assumed that the friction coefficient is constant (refs. 7 and 9 and ref. 10, p. 88). To explain the character of the f(ao) curves, we consider separately two ranges of attack angle: small and medium angles (up to 40” - 60”) and large angles (greater than 40” 60’). A variation in the friction coefficient within the smaller angle range is probably caused by a change in load. Indeed, with an increase in the angle of attack equivalent to a load increase, the friction coefficient for vulcanizates


is lowered as it is in normal sliding (ref. 1 and ref. 11, p. 277) while the coefficient of friction for plastics increases slightly as is usually observed for a plastic contact under increasing load [l] . From this standpoint, the friction coefficient of plastics should steadily increase with the attack angle; however, the curves of Figs. 5 and 6 show that at angles above 40” - 60” the friction coefficient begins to decrease for plastics and some other materials. It may be assumed [ 121 that the reason for this phenomenon is a shortened path X, of movement of a particle over the surface. When this path becomes smaller than the value of the preliminary displacement xpr (ref. 4, p. 181) a reduced friction force occurs (the force of adherence in ref. 4); this is manifested in a reduced coefficient of friction.

3 2 i (4


f 42

o,f 0)

IO20 30 40 50 60 70d

Fig. 6. Relationships between (a) the ratio x*/x, and (b) the friction coefficient f for plastics and the attack angle (~0 (bead diameter, 2 mm): curves 1, PMMA, II,-,= 4.3 m s-l ; curves 2, polyethylene, uo = 4 m s-l.

2.4. The preliminary displacement under impact In connection with the assumption of a shortened is the problem of the determination of the range of preliminary displacement. This range can be determined value of the ratio x,/xP,. In the case of plastic contact, over the surface is expressed by the equation x* = nh,,,

movement path there transition to a for example by the the path of a particle

(cot e. - f)


This equation contains h,,, which depends [7] on the density p and the normal component u. sin e. of the initial particle velocity: 112

h max = 2Rvo sine0


where the value of cu, is approximately equal to the Brine11 hardness. The pressure is treated in mechanics as the specific kinetic energy p u2/2; this gives p = p(uo sina0)2 f ll if the normal component of the velocity alone is taken


into account, Il denotes the dynamic pressure equivalent. Hence the depth of intrusion h,,, in eqn, (9a) is a function of the dynamic pressure equivalent :

To assess the preliminary displacement xpr, the formula suggested by Kragelsky [ 131 and Mikhin (ref. 11, p, 109) was used for permanent plastic contact in the ease of a unit indenter: 1’2

k---1 N

Xpr =



f2p2p2 - 11


It is also possible to substitute for the load N the expression containing the dynamic pressure equivalent fl : 2R n Xpr = i-4 711123c0, Then the ratio xJxP,

--=r x*





i 307,


[ { 2( 1 + f2)1’2}1’2 - 11



1 114

cota*--f { 2( 1 + f2)l’2)“2

- 1


Calculations have shown 1121 that the ratio x,/x,, becomes less than unity within the range (40” - 60”) of attack angle, where the reduction in the friction coefficient of plastics begins (Fig. 6). Thus the assumption of the present section is justified. The notion of a dynamic pressure equivalent introduced here will be of use when the equations of sliding wear and the equations of wear under a flow of particles are compared. 2.5. ~~rn~ur~~unbetween dynamic friction and friction under a permanent contact It appears that friction under impact and sliding friction (i.e. kinetic and static friction) are analogous. When the angle of attack is varied from 90” to 0” it is possible to pass through two regions corresponding to the states of rest and of stable sliding under a permanent contact. In the case of impact these are the regions of a preliminary displacement at large angles of attack and of stable sliding at medium and small angles of attack. Variations in f within the range of large angles of attack are related to a variation in the path of sliding of particles at the moment of impact. Within this range of angles, f increases from zero at 90” to the final value within the region of transition to preliminary displacement. Variation within the range of stable sliding, i.e. at small and medium values of (Ye, is similar to the relationship between f and the load on sliding. At an elastic contact (e.g. the frictional contact of vulcanizates) f decreases with increasing load, while at a plastic contact (e.g. the frictional contact of plastics) f increases slightly as the load increases. Figures 4 - 6 show the variation in the coefficient of fric-


tion of vulcanizates and plastics under impact. It is known (ref. 11, p. 143) that dry vulcanizate has a high friction coefficient under sliding conditions because of the considerable contribution of its adhesion component. The friction coefficient of a wetted rubber is small owing to the insignificant role of the adhesion component. A similar situation is observed in the case of impact (Fig. 5). These results make it possible to assume that the friction coefficient under impact has been measured correctly and that the role of the angle of attack is significant and understandable. 3. Wear by a glancing current

of solid particles

The observed characteristics of dynamic friction are analogous to those of friction under a permanent contact. Thus the quantitative characteristics of both wear processes should be analogous: the initiation of failure is in principle the same under impact friction and under sliding friction. Consequently, to determine specific quantitative characteristics of wear under impact it is expedient to consider the already established features of wear under sliding contact. Such an approach to the problem of impact wear has been used by Nepomnyaschy [ 71; this approach has proved to be most effective for polymers [2, 14 - 161. 3.1. The equation for wear by a glancing flow of solid spherical particles According to Nepomnyaschy [7], the mechanism of wear under impact with solid particles may be similar in nature to general fatigue, to low cycle fatigue and, as an extreme case, to the mechanism of microcutting. The following model was assumed for the derivation of the equations. The surface is worn by a flow of absolutely rigid spherical particles. These particles all have the same speed and angle of attack. Each particle acts independently. The concept of the intensity J of wear was defined for impact wear in a traditional manner as the ratio of the mass AM of the worn material to the mass M,, of solid particles causing the wear: J$




SP +rR3pq

where V is the volumetric wear of the material, pm the density of the material being worn, R the particle radius, q the number of particles causing the wear and p the density of the particles. The deduction of equations for impact wear was based on the principal assumption of the fatigue wear theory (under sliding); this assumption is associated with the proportionality of the ratio of the deformed volume Vdef to the number n of frictional acts causing breakage and elimination of this volume : dV=--_

dv,,, _ qvm n





where Vdef is the volume deformed by 4 particles during their interaction with the surface being worn, V,, is the volume deformed by a single particle at one contact spot and 1 is the diameter of the contact spot. Substitution of the expressions for 1 and n (separately for plastic and of the dimenelastic contact) and for V,, in eqn. (13) and the introduction and sliding characteristic ,$ = sionless invasion characteristic E = h/h,,, xl&ax make it possible to integrate eqn. (13). The resulting expression for the volume V of worn material when substituted in eqn. (12) gives equations defining the relationship of wear to the mechanical strength and deformation properties of the material being worn, the external parameters of the process and the coefficient of friction under impact. These equations in their general form are as follows: J = !_j!.p,(3p)t/5(V0




]*e(t+s)/s 0



where eqn. (14) describes elastic contact and eqn. (15) describes plastic contact. In these equations t is a fatigue curve parameter, cro is a breaking stress at n = 1 for the material being worn, 13is an elastic constant equal to (1 - p2)/E, E is the modulus of elasticity, y is Poisson’s coefficient, e. is the relative elongation at rupture, u, is the yield stress, cu, is approximately equal to the hardness of the worn material, E, is the dimensionless intrusion at which sliding is stopped and [.+ is the dimensionless length of the sliding path of a particle from the moment of coming into contact to the end of sliding. In the deduction of these equations three types of behaviour of a particle on collision have been considered: (1) the particle stops its tangential movement without reaching the maximum depth h,,, of intrusion; (2) the particle stops its tangential movement after reaching h,,, while it is still intruded in the material; (3) the particle retains the tangential speed component at the moment of rebound from the surface. For each type of behaviour the solution A of the integral in eqn. (14) or eqn. (15) is given. In the first and second cases rather complicated formulae have been obtained whereas in the third case the resulting formulae are less complicated. The formulae for the third case for an elastic contact and a plastic contact are as follows: A, =





where r is the gamma function and the subscripts and the plastic contacts respectively.

e and p refer to the elastic

3.2. Simplifications The results of experiments on the glancing collision of a solid particle with various materials demonstrate that plastic contact occurs within sufficiently broad ranges of speed and particle size (initial particle speed, 1 23 m s-l ; diameter of metal beads, 1 - 5.5 mm; diameter of sand particles, 0.5 - 0.9 mm; samples of vuleanizates, plastics, metals and glass). We now determine the volume Vdrf deformed by all particles during their interaction with the surface being worn. According to eqn. (13) dVdcf = qV,,,



V D1, dx and 1 in this equation are replaced and further substitutions are performed as suggested by ~epomny~chy [ 71 in relation to V and J. After integration the following relation is obtained for elastic contact: V def = q $rRii2

hmax512(cot e. - f)


The substitution of h,,, with the expression involving the dynamic equivalent fl = p(uo sin ao)2 and the elastic constant 0 gives V def = qn2R3M(cot

Similar operations

a0 -f)


for the case of the plastic contact



(cot Qo - f)

v def = g


Corresponding equations for the volumetric wear V = V&f/n are obtained by using the following expressions for the fatigue strength of the deformed volume :

The expressions

for volumetric

wear are

and (I=) The integration of eqn. (17a) to give dVdef and the deduction of eqns. (18a) and (18b) eliminated certain difficulties encountered previously (ref. 4, p. 330, and ref. 7) in integration of the equation for dV; these difficulties had


caused complications in the final results. In our approach from the equations for wear the factor incorporated in eqns. (16a) and (16b) is excluded and related to the I’ function with its argument being represented by the variable parameter t. A further simplification of eqns. (14) and (15) is attained if the intensity J of the impact wear is defined as the ratio of the volume wear V of the material to the volume V,, of solid particles causing this wear: J= v/v,,


This simplification makes it possible to exclude the worn material density pm from the equations and to reduce the power index of the material density of the particles so that p is incorporated only in the dynamic pressure equivalent p(ua sin 01~)~. The simpli~cation obtained from the new definition of the concept of wear intensity facilitates the analysis of the role of external factors and the comparison with the characteristics of wear under a permanent contact (Table 1). Taking account of expressions (l&a) and (18b) for the deformed volume and the new definition of the wear intensity J, equations simpler than eqns. (14) - (16) are obtained; e.g. the equation for fatigue wear under elastic contact is J, =;

.,t/[email protected]/5t


For a single-act destruction, i.e. microcutting (n = l), J = VdeffVsp and J el =gnrrO(cot



Similarly, the equation for wear at a plastic contact is simplified: Jp

=;(&rt+“‘;*;t (f =$~'2jL(cOtao


f 224

Since in most cases t = 2 at a plastic contact, we obtain

(22b) For a single-act destruction n = 1 and (cot eo - f)


3.3. Comparison between theory and experiment Equations (14) - (23) were obtained for spherical particles. Although experiments are usually performed with nonspherical particles, nevertheless such particles have some parts of their projections spherical or almost spherical in shape. This has been confirmed by the direct recording of the outline and dimensions of quartz sand, crushed quartz and glass [ 17 - 191.


Forcedly elastic material; plastic contact

n=l (microcutting)

(fatigue wear)




1 (fatigue wear)


Highly elastic material; flexible




The value of J for the following

for the friction






f t




and dynamic

c pl+t/sE(4/5,t-l

J el = E







of wear under kinetic

and the scheme

of the regularities

Model of a counterbody of contact interaction

Factors --.







1 + h’f eo2 1 - k’f

714 1 --


0 _f_



(em. t 21))

(eqn. (20))



(cot a0 - f) (eqn. (22b))



vo sin 1110+ v sin (Y

v. cos cro - u cos CY

(cot erg - f)

nl+t/[email protected]/5)t-1

a ;






= Ti



pve” sin2ao




The impact of such a projection is thus equivalent to an impact with a particle having a radius equal to the radius of curvature of the projection top and a mass equal to the total mass m of the particle [20]. This means that there is now an apparent variation in the density with respect to the wear. The apparent density papp is given by PaPP




where rme is a mean radius of curvature of the projections of a non-spherical particle. Consequently, equations for wear by non-spherical particles should incorporate the apparent density papp of the particles instead of the density p which is present in the equation for wear by spherical particles. The effect of the properties of the material being worn and of the external factors should remain unchanged. The above considerations make it possible to compare the results of experiments for wear by non-spherical particles with the characteristics which can be derived from eqns. (20) - (23). The few experimental data available on the effect on the wear of the strength and deformation properties of the material being worn qualitatively correspond in principle to our equations. Owing to the variations in the strength and deformation properties of polymers, correlation of wear is difficult because the modification of polymeric materials changes the whole spectrum of their properties. Various workers [17, 211 have reported a qualitative relationship of wear with the properties of plastics which is not in contradiction with eqns. (22) and (23). Many attempts have been made to find a relationship between wear and the hardness of metals or alloys. For commercially pure metals there is a tendency towards reduced wear with increased hardness [22] . A linear relationship between the wear resistance of commercially pure metals and their hardness has been obtained for wear by suspensions of quartz and corundum at a jet speed of 36 m s-l [ 231. This relationship is in good agreement with eqn. (23). Many researchers have studied the relationship between wear and the speed of solid particles. The speeds in such studies were sufficiently high (dozens of metres per second) for wear to occur because of microcutting or to be of a combined character. The power index h in the equation J = auOk for plastics should be in the range 2.5 - 3.5 [22,23]. These values (Table 2) have been obtained in the majority of experiments performed by Arumyae [ 171. For the wear of vulcanizates, according to eqns. (20) and (21) the power index h should equal 2 for microcutting and should be greater than 2 for combined cutting and fatigue wear. Many workers have been interested in the relationship between wear and the angle of attack. Generally this relationship has the form of a curve with a maximum. The maximum wear for vulcanizates occurs within the range of attack angles 10” - 30”. This has been found experimentally (Figs.



The power index h for the wear of plastics attack angles 01~ [ 181


and steel at different




20 45

2.8 2.8


20 45

2.1 2.8

20 45

2.5 2.6


20 45

3.0 2.9


20 45

2.5 2.6



resin ED-5

7(a) - 7(c)) [ 10, 24,251. This is also seen from eqn. (20) if it is transformed into the following equation and interpreted graphically using the variable f: a0 - f)

J, = k, sin2(f’5+1),o, f’(cot

For microcutting Jel

= k,,

the equation


(24) to eqn. (21) is

sin2ao (cot 01~-f)

where k, and heI are functions angle of attack: k, = &app, k el = a(Pappr

of parameters

(25a) which are independent

~0, t, GO,@)

of the

( 25b)

uo, 0)

Graphical interpretations of eqns. (24) and (25a) are shown in Fig. 7(d). From Fig. 7 it can be seen that the curves calculated on the basis of eqn. (24) are substantially the same as the experimental curves. The ranges of maximum wear are also the same. The maximum wear of plastics and of metals which are not very hard is within the range of attack angles 30” - 60” [ 17,19, 21, 221. To compare these experimental results with the theory, eqns. (22a) and (23) are expressed in the following form: JP = k, sin(*+ 5)‘2~o


J Pl = kpl sin512010 (cot a0 - f) where k, and kpl angle of attack:

k, = etkappr k Pl

are functions JJO,4

of parameters

(27a) which are independent

of the


= Pa (Papp, uo, cud



(b) 6


4 2 i?


4 5 0




(cl 0




Fig. 7. Relationship between the wear of rubbers and the angle of attack: (a) wear of vulcanizates sand blasted with rounded and finely divided quartz calculated according to ref. 26: curves 1, vulcanizate with a hardness of 80 - 90 units; curves 2, vulcanizate with a hardness of 58 - 63 units; (b) wear of vuleanizates sand blasted on a laboratory stand (ref. 27, p. 453; particle speed, 100 m s-l): curve 1, vuleanizate of natural rubber with a hardness of 40 units; curve 2, vulcanizate of butadiene-styrene rubber BSR with a hardness of 60 units; (c) wear of vulcanizates using river sand with a particle size of 0.9 1,2 mm on a laboratory centrifuge stand (ref. 10, p, 85; collision speed, 63 m s-l): curve 1, vulcanizate of natural rubber with a hardness of 46 units; curve 2, vulcanizate of a mixture of rubbers SRD with a hardness of 65 units; (d) calculated curves of wear of vulcanizates with values of f taken from Fig. 5, curve 8; curve 1, calculated from eqn. (24) with t = 3; curve 2, calculated from eqn. (24) with t = 9; curve 3, calculated from eqn. (25). Fig. 8. Relationship between the wear of plastics and the angle of attack: (a) wear of plastics by a flow of river sand using the data of the present workers: curve 1, polyethylene, ve = 17.3 m s-l ; curve 2, polyethylene, ue = 23 m s-l; curve 3, impaetresistant polystyrene, ve = 17.3 m s-l ; curve 4, impact-resistant polystyrene, v. = 23 m ; (b) curves for wear at a plastic contact calculated using eqn. (26) with t = 2: curve 1, S-’ f = 0.15 - 0.20; curve 2, f = 0.17 - 0.22; curve 3, f = 0.13 - 0.18; (e) curves for wear under impact microcutting calculated with eqn. (27): the numerals on the curves denote the values off used for the calculation.

Equations (26) and (27a) are interpreted graphically in Figs. 8(b) and 8(c) respectively. In the calculations the f values corresponding to those obtained experimentally have been taken into account. Comparison of the calculated and experimental curves shows that these are similar, i.e. they belong to the same type. The ranges of maximum wear obtained by calculation and experiment are similar. The results show that eqns. (20) - (23) correspond to the experimental results. (The theory does not cover the cases of brittle destruction of hard metals, alloys and similar materials which have m~imum wear at e. = 90” and within the range of angles close to 90” .) In Table 1 these equations are


represented in a form convenient for comparison with the wear sliding conditions. This comparison indicates which factors (external conditions, properties of the friction couple) that affect wear are similar in these cases, which factors are the same and which factors are different. It is possible to use the results obtained for sliding conditions to aid in the selection and manufacture of materials which are wear resistant in a jet of particles. 3.4. Comparison under conditions

of the effects of the properties of sliding and of impact

of the body

being worn

From Table 1, the relationship of the wear intensity to the properties (expressed by E, u, and of the worn body such as strength uo, deformability e,) and fatigue endurance t is the same for both sliding and impact wear. In both cases wear depends on the combination of these properties in the following ways. For fatigue wear at an elastic contact J=E




t 00

t (28) !

For fatigue wear at a plastic contact J-


1 +I?‘f

cos7/4e02 1 -- k’f


For the microcutting of an elastic material, J = l/E; for the microcutting of a plastic material, J * l/co,. The above equations include the friction coefficient f; the role off in sliding and impact is discussed below. 3.5. Comparison of the effects of external factors Owing to the differences in the counterbodies and the schemes of contact interaction, each of the types of wear compared here has its own external factors. In sliding, the main external factor is pressure at the contact spot, while under impact the external factors are the speed of the flying particle and the angle of attack. The angle of attack causes the principal difference between wear by a glancing blow and wear by the sliding contact of a body. It is not possible to ignore sliding by assuming that a0 = 0 in the equations, since the normal components of speed, load, friction force and wear are then equal to zero. Neither is it possible to assume that a0 = 90”) since there is no sliding in this case. From the laws of mechanics it is known that the pressure p is equivalent to the kinetic energy, i.e. pvo2/2. Taking account of the normal component of speed, p is equal to about p (v. sin a0)2 = ll. Consequently, the combination of the most characteristic factors for impact has a clear equivalent in sliding. This is obvious from Table 1: in all cases the pressure p at the contact is incorporated in the corresponding equations and has the same exponent as its dynamic equivalent II in the equation for impact.


The effects of external criterial quantities of impact endurance II of this volume, tions. The scheme by which outlined in Fig. 9.

factors and the friction coefficient on the basic wear, i.e. on the deformed volume Vdef and the are considered here and in the following secthe role of each factor is analysed ~div~dually is


Fig. 9. The effect of the angle of attack on basic wear factors under a glancing flow of solid particles calculated for elastic contact (values of f for curves b, c, d, e and g are taken from Fig. 5, curve 2, with t = 4): curve b, atik, = f sin215&g; curve c, n&k, = (l/f *) sin2*f5~o; curve d, x,/k, = sin 4~5cto (cot ~lg - f); curve d’, h/kh = sin4i5ao; curve e, according to eqn. (18); curve g, according to eqn. (24).

3.6. The role of the angle of attack The angle of attack defines the ratio of the normal and to the tangential speed components. Variation in the attack angle affects all factors on which the deformed volume Vdef , the endurance n of the deformed volume and hence the wear J depend. The scheme and the graphs (Fig. 9) show the sequence and character of the effect of the angle of attack on the friction coefficient (factor a), the specific tangential stress (factor b) (ref. 2, p. 22), the endurance of the deformed volume (factor c), the sliding path under impact (factor d), the depth of intrusion (factor d’), the deformed volume (factor e) and the wear (factor g). (The calculation of curves b - e and curve g in Fig. 9 is based on the relationship f(aca) as represented by Fig. 5, curve 2.) The normal component u. sin or0 of the speed directly affects the friction coefficient (factor a), the specific stress (factor b) and the depth of intrusion of a particle (factor d’). The tangential component u. cos cyo of the speed mainly affects the path of sliding under impact (factor d). The scheme shows the complex nature of the effect provided by the angle of attack on two principal factors of the fatigue wear: V def and n. The deformed volume, which depends on the angle of attack, passes through a maximum (Fig. 9,


curve e). For this reason, the relationship of wear intensity to the angle of attack has a point of inflexion corresponding to this maximum. A shift in the maximum is possible as a result of a variation in the friction coefficient under impact or in the parameter t; this is discussed further in Section 3.8. 3.7. The role of the speed of solid particles In contrast with the angle of attack, increasing or decreasing the speed clearly changes both the tangential and the normal components. The effects of this variation on the deformed volume and its endurance are represented in Fig. 9. For a permanent contact, the sliding speed does not directly define the deformed volume or its endurance. The problem is similar to that of the calculation of wear per unit path length in the case of sliding. In contrast, for glancing impact the speed of sliding of a particle, i.e. ue cos cue, affects the deformed volume Vdef . In eqns. (20) - (23) this is accounted for by the last factor, which contains cot eo. According to eqns. (20) - (23), the relationship between wear and speed has a power character, The power indices of the speed u,, in eqns. (20) - (23) are close to those obtained experimentally. 3.8. The role of the friction coefficient The coefficient of friction in impact wear plays the same fundamental role as in surface wear by a counterbody: the fatigue durability (endurance) of the deformed volume depends on the coefficient of friction (Fig. 9). The role of the friction coefficient is significant in the wear of highly elastic polymers, for which n - l/f’ It is known that the power index t in this case may be as high as 3 I 4 and it can even exceed 10 (ref. 4, p. 292, and ref. 26). The friction coefficient defining the endurance n under wear is incorporated in the equations for wear at glancing impact as it is in the equations for permanent contact (see Table 1). However, under impact the role of the friction coefficient is different, i.e. it affects the sliding path X* of a particle over the surface of the body (Fig. 9) and hence the deformed volume. For this reason, for a single-act destruction where n = 1, the friction coefficient is absent in the equations for wear. In contrast, the friction coefficient is retained in the equations for microcutting under glancing impact; this can be seen from Table 1 and eqns. (21) and (23). However, the role of friction in this case is opposite to its general role, since the higher the friction coefficient is at impact, the shorter is the path of a particle (eqn. (9)), the smaller is the deformed volume (eqns. (Ma) and (18b)) and the lower is the wear (eqns. (21) and (23)). Owing to the variation in the friction coefficient (which is similar to the variation in t, as shown in Fig. 7(d)), the position of maximum impact wear changes. From the graphical presentation of eqns. (23) - (27), with increasing


coefficient of friction the maximum wear should be shifted towards smaller angles of attack (Figs. 8(b) and 8(c)) and this is observed experimentally. For example, the coefficient of friction is increased when plastification occurs and when the temperature of plastics increases. In both cases, the maximum wear is shifted towards the region of smaller angles of attack [171. With vulcanizates, maximum wear is shifted in the same direction with decreasing hardness or increasing elasticity (ref. 10, p. 85). This phenomenon is observed for metals, where it is apparently due to an increased area of effective contact and hence an increased coefficient of friction. With decreasing f, according to the theory [ 271, the maximum wear should be shifted towards higher values of (Ye. This is observed experimentally in the comparison of the position of maximum wear for non-extended and extended polymers [ 17, 211 and soft and rigid vulcanizates [lo] .

4. Conclusions The study of impact interaction and dynamic friction has made it possible to simplify the known [ 71 method of calculation of equations for wear by a stream of solid spherical particles and to interpret graphically the quahtative relationship between wear and the angle of attack and the relationship with the experimental results. The observed displacement of maximum wear as a result of the variation in the frictional properties of the body being deformed has also been explained.

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