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Research Papers

Diffusion Theory Applied to Tool-Life Stochastic Modeling Under a Progressive Wear Process

[+] Author and Article Information
Marcello Braglia

Dipartimento di Ingegneria Civile e Industriale,
Università di Pisa,
Via Bonanno Pisano 25/B,
Pisa 56126, Italy
e-mail: marcello.braglia@dimnp.unipi.it

Davide Castellano

Dipartimento di Ingegneria Civile e Industriale,
Università di Pisa,
Via Bonanno Pisano 25/B,
Pisa 56126, Italy
e-mail: davide.castellano@for.unipi.it

1Corresponding author.

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received June 11, 2013; final manuscript received February 7, 2014; published online March 26, 2014. Assoc. Editor: Allen Y. Yi.

J. Manuf. Sci. Eng 136(3), 031010 (Mar 26, 2014) (12 pages) Paper No: MANU-13-1256; doi: 10.1115/1.4026841 History: Received June 11, 2013; Revised February 07, 2014

In this paper, a novel approach to the derivation of the tool-life distribution, when the tool useful life ends after a progressive wear process, is presented. It is based on the diffusion theory and exploits the Fokker–Planck equation. The Fokker–Planck coefficients are derived on the basis of the injury theory assumptions. That is, tool-wear occurs by detachment of small particles from the tool working surfaces, which are assumed to be identical and time-independent. In addition, they are supposed to be small enough to consider the detachment process as continuous. The tool useful life ends when a specified total volume of material is thus removed. Tool-life distributions are derived in two situations: (i) both Fokker–Planck coefficients are time-dependent only and (ii) the diffusion coefficient is neglected and the drift is wear-dependent. Theoretical results are finally compared to experimental data concerning flank wear land in continuous turning of a C40 carbon steel bar adopting a P10 type sintered carbide insert. The adherence to the experimental data of the tool-life distributions derived exploiting the Fokker–Planck equation is satisfactory. Moreover, the tool-life distribution obtained, when the diffusion coefficient is neglected and the drift is wear-dependent, is able to well-represent the wear behavior at intermediate and later times.

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References

Figures

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Fig. 1

Ratio Rn(t) = pn(t)/pn(n/ν) between the (Poisson) probability of n events in [0,t] and the probability of n events in [0,n/ν], assuming ν constant. We put z = νt/n, so that Rn(z) = znexp{n(z-1)}.

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Fig. 2

Definition of the parameter VB used to evaluate the flank wear level (from Ref. [2])

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Fig. 3

Experimental results (from Refs. [30] and [31])

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Fig. 4

The dotted lines correspond to third-degree curves fitting the experimental points shown in Fig. 3. Curve A: mu(t) fitted with a cubic. Curve B: σu2(t) fitted with a sixth-degree polynomial.

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Fig. 5

Curve A: mu(t). Curve B: σu2(t). Curve C: m·u(t). Curve D: σ·u2(t).

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Fig. 6

Quality analysis of the experimental data fitting

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Fig. 7

Tool-wear distributions providing the best fitting of the results in Fig. 4. Each density is a two-parameter Weibull distribution.

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Fig. 8

Tool-life distributions fitting the histograms deduced from Fig. 4 in the cases (a) uf = 20×10-2mm, and (b) uf = 30×10-2mm. Density A: Weibull. Density B: log-normal. Density C: Gaussian.

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Fig. 9

Tool-life distributions. Comparison between Eq. (33) (full curve), log-normal density (dashed curve), and Weibull density (dotted curve), in the cases (a) uf = 20×10-2mm, and (b) uf = 30×10-2mm.

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Fig. 10

Tool-life distributions. Comparison between Eq. (33) (full curve), Eq. (34) with ν and δ adapted to have the best agreement with the experimental data (dotted curve), and Eq. (34) with uf≅νδt (dashed curve), in the cases (a) uf=20×10-2mm, and (b) uf=30×10-2mm.

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Fig. 11

Tool-wear distributions given by Eq. (24) in correspondence to different initial conditions, i.e., u0 from 0.9 to 1.1 with step 0.025

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Fig. 12

Comparison among different densities of the form Eq. (40) relevant to various values of n, for (a) t = 20 min, and (b) t = 25 min. Other parameters values: α = 10-3, m0 = 6.7, and σ0 = 0.72.

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Fig. 13

Comparison among different densities of the form Eq. (40) with n = 1.5 and relevant to various time instants. Other parameter values: α = 10-3, m0 = 6.7, and σ0 = 0.72.

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Fig. 14

Tool-wear distributions fitting the experimental data in Fig. 3. Each density is a three-parameter Weibull distribution.

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Fig. 15

Tool-life distributions given by Eq. (42), with n = 1.5, α = 10-3, m0 = 6.7, and σ0 = 0.72, relevant to different values of the limit wear land uf

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