DERIVATION OF THE EQUATION EXPRESSING WEIGHT LOSS BY FRETTING CORROSION

29.7 DERIVATION OF THE EQUATION EXPRESSING WEIGHT LOSS BY FRETTING CORROSION

Assume n asperities or contact points per unit area of metal (or oxide) surface which, for mathematical convenience, are circular in shape. The average diame- ter of asperities is c and the average distance apart is s (Fig. 8.21 , Section 8.7.1 ). In the fretting process, the asperities move over a plane surface of metal at linear velocity υ , with each asperity plowing out a path of clean metal of width averag- ing c and of length depending on distance of travel. Behind each asperity on the track of clean metal, gas from the atmosphere adsorbs rapidly, followed in time by formation of a thin oxide fi lm. The next asperity, moving in the same path as the fi rst, scrapes the oxide fi lm off and leaves, in turn, a track of clean metal behind. The average time during which oxidation occurs is t . Then

s t = (29.56)

The corresponding amount of oxide W removed by one asperity plowing out a path l long and c wide depends on the amount of oxide formed in time t . For thin - fi lm oxidation, the logarithmic equation is obeyed (see Section 11.4.1 ):

472 APPENDIX

W = clk ln + 1 (29.57)

where τ and k are constants. Preliminary to oxidation, we can also consider the situation of oxygen adsorb- ing rapidly as physically adsorbed gas, followed by conversion at a slower rate to chemisorbed oxygen atoms. The chemisorbed oxygen, in turn, reacts with underlying metal to form metal oxide, a reaction that is activated mechanically by asperities moving over the metal surface. Chemisorption limits the amount of oxide that is formed in such a process, the rate of chemisorption following an equation identical in form to that of (29.57) [6] . Hence, whichever process applies, the form of the fi nal equation is essentially the same. *

Substituting (29.56) into (29.57) , we obtain s

W = clk ln

Assuming that relative motion of the two surfaces is sinusoidal, 2 l is the total length of travel in any one cycle and x , the linear displacement from the midpoint of travel at time t , is given by

x = cos θ (29.59)

If f represents constant linear frequency, this is related to constant angular veloc- ity by the expression

d θ =2 π f

dt (29.61) Therefore, the average velocity is given by

* This argument is not compelling, because the logarithmic term is eventually expanded and only the fi rst term is used. This is equivalent to a linear rate of oxidation or of gas adsorption with time. A linear rate of gas adsorption suggests that the amount of oxygen reaching the clean metal surface as physically adsorbed gas may actually be controlling, rather than its conversion to chemisorbed oxygen atoms. This possibility is given support by the observed increase of fretting weight loss as the tem- perature is lowered, corresponding to increased rate and extent of physical adsorption at lower tem- peratures. The rate of chemisorption, on the other hand, usually decreases as the temperature is lowered.

DERIVATION OF THE EQUATION EXPRESSING WEIGHT LOSS

π lf ∫ sin θθ d

V =− 0 = 2 lf (29.62)

Hence, for n contacts or asperities per unit area of interface, weight loss W per cycle caused solely by oxidation is

W = 2 nlck ln ⎛ s + corr ⎞ 1 (29.63)

⎝ 2 lf τ ⎠

To this must be added loss of metal by wear because each asperity, on the average, digs below the oxide layer and dissipates metal in an amount propor- tional to the area of contact of the asperities and the length of travel. The area of asperity, rather than the width, is important now because of the “ tearing out ” or welding action taking place during mechanical wear, in contrast to scraping off of chemical products from the surface, as discussed previously. A shearing off of asperities without welding also leads to wear that depends on total area of contact. For n circular asperities, weight loss per cycle is given by

W mechanical = 2 kn ′

π l (29.64)

But n π ( c /2) 2 , the total area of contact, is equal [7] to the load L divided by the yield pressure p m . The term p m is approximated by three times the elastic limit; hence, for mild steel, p m equals 100 kg/mm 2 or 140,000 psi. Therefore,

W mechanical = 2 k ′

lL

= k lL 2 (29.65)

where k 2 is a constant equal to 2 k ′/p m . The total wear or metal loss per cycle is the sum of the oxidation, or corrosion, term and the mechanical term

W total = W corr + W mechanical (29.66) Returning to (29.63) , the logarithmic term can be expanded according to

ln( x +=− 1 ) x +

where x is equal to s /(2 lf τ ). When the latter expression is much smaller than unity, the square and higher terms can be omitted. This condition applies particularly to high loads (small values of s ), high - frequency f , and large value of slip l .

Empirical values of the constant τ for iron range from 0.06 to 3 s. Assuming reasonable values for τ = 0.06 s, f = 10 cps, l = 0.01 cm, and s (distance between

474 APPENDIX

asperities) = 10 −4 cm, then s /(2 lf τ ) = 0.008. Therefore, whenever experimental conditions approximate those cited and higher terms of the logarithmic expan- sion can be neglected, we obtain

W corr =

ncks

This expression is equivalent to assuming, from the very start, a linear rate of oxidation or of gas adsorption on clean iron, where k / τ is the reaction - rate con- stant. The linear rate reasonably approximates the actual state of affairs for very short times of adsorption or oxidation.

Since the number of asperities along one edge of unit area is equal to n , it follows that s + c is approximated by 1/ n . Also, recalling that n π ( c /2) 2 = L/p m ,

the terms n , c , and s can be eliminated from (29.67) , or

Combining (29.65) , (29.66) , and (29.68) , we have the fi nal expression for fretting as measured by weight loss corresponding to a total of C cycles:

W total = ( kL 0 12 / − kL 1 ) + k lLC 2 (29.69)