Phenomenological models of cavitation erosion kinetics

4 International Cavitation Erosion Test Seminar, Sopot, 1-2 June 2000where the expression under the integral sign denotes erosion rate at the surface element uncoveredin the instant T.F.J.Heymann states that realistic erosion curves can be obtained by assuming the f(t)function to vanish in the initial period of exposure (t ≤ T 0 ), and then to be described by athree-parameter logarithmic-normal distribution[ ln( t − T ) − m]21 ⎪⎧−⎪⎫0f () t = exp⎨2 ⎬(2)σ ( t − T0 ) 2π⎪⎩ 2σ⎪⎭For the original surface layer F.J.Heymann proposes to use a distribution with parametersdifferent from those applied for the subsequent layers. However, there are no hints on themethod of their determination to be found in reference [4].Later on, J.Noskievič [5] and K.Steller [6] proposed models of erosion kinetics referringdirectly to the erosion curve pattern.J.Noskievič has noticed that the instantaneous erosion rate curve, IER =IER(t) remindsthat of damped harmonic oscillator motion. The equation of motion of such an oscillator canbe written asd2( IER) d( IER)dt22+ 2α + β IER = I . (3)dtwith α and β denoting coefficients responsible for material properties (α - work-hardeningcapability, β - cavitation resistance), and I - parameter characterising intensity of cavitationerosive activity (I = const). By assuming erosion rate and its first derivative to vanish at thebeginning of the process, the solution of equation (3) can be written down as−κtκt[( α −κ) e − ( α + κ ) e ]−αt⎧ e⎪vs+ vs⎪2κ−αt⎪vs− vs( 1+α2t)eIER()t = ⎨⎪ ⎛ α ⎞vs− vs⎜cosωt− sinωt⎟e⎪ ⎝ ω ⎠⎪⎩vs− vscosωt−αtififififα > βα = ± βα < βα = 0andα ≠ 0, (4)2 22where κ = α − β and ω = β −α2 , whence the volume loss curve is given by:∆V= vs−αt⎡ α e ⎛α+ κ kt α −κ−kt⎞⎤⎢t− 2 + ⎜ e − e ⎟⎥⎣ β 2if ⏐α⏐ > β (5a)2κ⎝α−κα + κ ⎠⎦∆ V= vs⎡ 2 ⎛ 2 ⎞⎢t− + ⎜t+ ⎟e⎣ α ⎝ α ⎠−αt⎤⎥⎦if α =±β(5b)⎧ α α t ⎡⎛⎞⎤⎫−αα ω∆V = vs⎨t− 2 2+ 2e ⎢⎜− ⎟ sin ωt+2 cosωt⎣⎝⎠⎥⎬,⎩ β β ω α⎦⎭if ⏐α ⏐< β and α ≠ 0 (5c)⎞⎜1∆V= v − s t ω t ⎟⎝ ω sin ,⎠if α = 0 (5d)