Measure theoretic derivation of Richardson's equation

T. J. Mackin, C. T. Chen, J. J. Mecholsky, James Patrick Runt

Research output: Contribution to journalArticle

Abstract

Richardson's empirical equation relating length to measuring scale has been used extensively to assign fractal dimensions to rugged curves. The inconsistency in units, length on one side of the equation vs. fractional units of length on the other has led to considerable debate. This letter presents a derivation of Richardson's equation based upon measure theory. In this context, the constant appearing in Richardson's equation is shown to have a precise meaning; it is the Hausdorff measure of the curve.

Original languageEnglish (US)
JournalMaterials Science and Engineering A
VolumeA186
Issue number1-2
StatePublished - Jan 1 1994

Fingerprint

Fractal dimension
derivation
curves
fractals

All Science Journal Classification (ASJC) codes

  • Materials Science(all)
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

Cite this

Mackin, T. J., Chen, C. T., Mecholsky, J. J., & Runt, J. P. (1994). Measure theoretic derivation of Richardson's equation. Materials Science and Engineering A, A186(1-2).
Mackin, T. J. ; Chen, C. T. ; Mecholsky, J. J. ; Runt, James Patrick. / Measure theoretic derivation of Richardson's equation. In: Materials Science and Engineering A. 1994 ; Vol. A186, No. 1-2.
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Mackin, TJ, Chen, CT, Mecholsky, JJ & Runt, JP 1994, 'Measure theoretic derivation of Richardson's equation', Materials Science and Engineering A, vol. A186, no. 1-2.

Measure theoretic derivation of Richardson's equation. / Mackin, T. J.; Chen, C. T.; Mecholsky, J. J.; Runt, James Patrick.

In: Materials Science and Engineering A, Vol. A186, No. 1-2, 01.01.1994.

Research output: Contribution to journalArticle

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Mackin TJ, Chen CT, Mecholsky JJ, Runt JP. Measure theoretic derivation of Richardson's equation. Materials Science and Engineering A. 1994 Jan 1;A186(1-2).