464 



AP « Pv-Pmin 



- ( o + CPmin) PV" ■ 



(6) 



where 



A = 



npV^-L^A^a [-(a + Cpn,in)l' 



Combining Eqs. (5) and (6), the following equation 

 is derived. 



V = X^ [-(a + Cp-.i„) 



3 

 ■2" 



(7) 



The number of cavity bubbles per unit time is then 

 given as. 



6e V 



N = 



x3l [-( a + CPn,in)]2 



(8) 



where A as the nondimensional cavity length, X = 

 X/L, &e is the nondimensional cavity thickness at 

 the end, and 6e = 6e/L. 



Here , we make the same assumption as in the 

 previous paper [Kato (1975) ] on the statistical 

 energy distribution of cavity bubbles. The distri- 

 bution is given as. 



cE exp (-aE) 



(9) 



where n is the number of bubbles per unit time 

 whose energy is between E and E + dE. Total number, 

 N, and total energy of bubbles, E^-, are given as 

 follows : 



K^ &e 



n2p2v3L3x9a2 [-(a + Cp . )]2 

 min 



In the present case the chord length is taken as a 

 suitable reference length, L. 



Equation (13) is similar to Eq. (2) , but it is 

 extended to include differences in the cavitation 

 numbers. 



The next problem is the modelling of deformation 

 of a material surface caused by the attack of 

 collapsing bubbles. For the present tests, hardness 

 seems the best property to express the erosion 

 resistance of a material. However it was found to 

 be insufficient as seen in Figures 17 and 18. 



The methods of hardness testing can be divided 

 into two types. One is the measurement of a dent 

 size caused by the static load of a sphere or a 

 pyramid on the material surface . The other method 

 is the measurement of absorbed energy from dropping 

 a certain test body on the surface. The Vickers 

 hardness test made in the present study belongs to 

 the first type. 



When a pyramidal dent whose depth is d, is 

 formed by a static load F (Figure 21) , the energy 

 used to the deformation is 



E "^ Fd 



(14) 



N = 



ndE 



c 

 72" 



(10) The hardness has the following relation by its 

 definition. 



(15) 



2c 



EndE = — T 



a^ 



The increase of surface roughness (SR) by the single 

 dent is given as 



Constants a and c can be decided by combining Eqs. 

 (4) , (8) , and (10) . 



V 

 SR = - 



(16) 



^1 



npv2L3x3[-(a + Cpi^n) ] 



K'6e 



— 9 



n2p2v3L'7xV[-(a + Cpmin)]^ 



(11) 



where Kj and K2 are constants independent of the 

 chord length, velocity and cavitation number. From 

 Eq. (9) , the distribution function of energy density, 

 f , is derived as a function of energy density, e. 

 The detailed discussion of this point is given in 

 the previous paper [Kato (1975) ] . 



Substituting Eqs. (9) and (11) into the relation 



f(e) = L^n (eL^) 



the final expression for f is 

 f = C e exp (-Ae) , 



(12) 



(13) 



where V and S are the volume of the dent and refer- 

 ence area, respectively. 



Combining equations (14) ~ (16) , 



SR 



(17) 



Diamond Pyramid 



iF 



h £ — i 



////// 



Material 



formation 



FIGURE 21. Model of Vickers hardness test method. 



