### 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 language | English (US) |
---|---|

Journal | Materials Science and Engineering A |

Volume | A186 |

Issue number | 1-2 |

State | Published - Jan 1 1994 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Materials Science and Engineering A*,

*A186*(1-2).

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*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.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Measure theoretic derivation of Richardson's equation

AU - Mackin, T. J.

AU - Chen, C. T.

AU - Mecholsky, J. J.

AU - Runt, James Patrick

PY - 1994/1/1

Y1 - 1994/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0028516704&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028516704&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0028516704

VL - A186

JO - Materials Science & Engineering A: Structural Materials: Properties, Microstructure and Processing

JF - Materials Science & Engineering A: Structural Materials: Properties, Microstructure and Processing

SN - 0921-5093

IS - 1-2

ER -