Curves and Fractal Dimension

Tricot, Claude

doi:10.1007/978-1-4612-4170-6

Cited by 298 publications

(265 citation statements)

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“…This point is on the curves of the fractal radii described by Eq. (22). What is then the physical meaning of that point?…”

Section: Discussionmentioning

confidence: 99%

“…(11) follows Hölderian form: [22] vðt; εÞ ¼ sup On the other hand, as f is uniformly anti-Hölderian too, with the same exponent, there exists a positive constantc′;independent of t such that, for any ε :…”

Section: Theoretical Relation In Proposed Methodsmentioning

confidence: 99%

“…But,f being uniformly Hölderian, then cand c′do not depend on t, and therefore [22]: (12) and (14) c AðΩðl x ÞÞ l x 2 H ≥∬ ω∈ΩðlxÞ l y l x ðt; hÞ 2 ; t dω≥c′ AðΩðl x ÞÞ l x 2 H that can be rewritten by the practical forms…”

Section: δðG F þ≥2−h ð15þmentioning

confidence: 99%

“…Then from the definition of the fractal dimension related to the holder exponent [22], it can be noted that H≈logl y =logl x and therefore from Eq. (18) we can write…”

Section: δðG F þ≥2−h ð15þmentioning

confidence: 99%

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Roughness signature of tribological contact calculated by a new method of peaks curvature radius estimation on fractal surfaces

Bigerelle

Nianga²,

Najjar

et al. 2013

Tribology International

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. a b s t r a c tThis paper proposes a new method of roughness peaks curvature radii calculation and its application to tribological contact analysis as characteristic signature of tribological contact. This method is introduced via the classical approach of the calculation of radius of asperity. In fact, the proposed approach provides a generalization to fractal profiles of the Nowicki's method [Nowicki. Wear Vol. 102, p. 161-176, 1985] by introducing a fractal concept of curvature radii of surfaces, depending on the observation scale and also numerically depending on horizontal lines intercepted by the studied profile. It is then established the increasing of the dispersion of the measures of that lines with that of the corresponding radii and the dependence of calculated radii on the fractal dimension of the studied curve. Consequently, the notion of peak is mathematically reformulated. The efficiency of the proposed method was tested via simulations of fractal curves such as those described by Brownian motions. A new fractal function allowing the modelling of a large number of physical phenomena was also introduced, and one of the great applications developed in this paper consists in detecting the scale on which the measurement system introduces a smoothing artifact on the data measurement. New methodology is applied to analysis of tribological contact in metal forming process.

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“…This point is on the curves of the fractal radii described by Eq. (22). What is then the physical meaning of that point?…”

Section: Discussionmentioning

confidence: 99%

Section: Theoretical Relation In Proposed Methodsmentioning

confidence: 99%

Section: δðG F þ≥2−h ð15þmentioning

confidence: 99%

“…Then from the definition of the fractal dimension related to the holder exponent [22], it can be noted that H≈logl y =logl x and therefore from Eq. (18) we can write…”

Section: δðG F þ≥2−h ð15þmentioning

confidence: 99%

See 2 more Smart Citations

Roughness signature of tribological contact calculated by a new method of peaks curvature radius estimation on fractal surfaces

Bigerelle

Nianga²,

Najjar

et al. 2013

Tribology International

View full text Add to dashboard Cite

show abstract

“…Unfortunately, different methods estimate different values for D, especially to characterize objects with limited self-similarity, such as texture images. One of the most accurate methods to estimate the fractal dimension of a shape image is the Bouligand-Minkowski method [41]. This method is based on the study of the influence area of the object contained in the image, and it is very sensitive to structural changes of the object.…”

Section: Fractal Dimensionmentioning

confidence: 99%