Statistical topography of the set of admissible crack paths in a brittle solid (original) (raw)

On the interpretation of the fractal character of fracture surfaces

Acta Metallurgica et Materialia, 1990

To examine the usefulness of the fractal concept in quantitative fractography, a series of classical fracture surfaces, namely transgranular cleavage, intergranular fracture, microvoid coalescence, quasicleavage and intergranular microvoid coalescence, are analyzed in terms of fractal geometry. Specifically, the five brittle and ductile fracture modes are studied, from three well characterized steels (a mild steel, a low-alloy steel and a 32 wt% Mn-steel) where the salient microstructural dimensions contributing to the final fracture morphology have been measured. Resulting plots of the mean angular deviation, and Richardson (fractal) plots of the lineal roughness, as a function of the measuring step size, are interpreted with the aid of computer-simulated fracture-surface profiles with known characteristics. It is found that the ranges of resolution, over which the fractal dimension is constant, correspond to the pertinent metallurgical dimensions on the fracture surface, and thus can be related to microstructural size-scales Rrsumr--Afin d'rvaluer l'utilit6 du concept de fractal dans la fractographie quantitative, une srrie de surfaces de rupture classiques--par clivage transgranulaire, rupture intergranulaire, coalescence de microcavitrs, pseudoclivage et coalescence de microcavitrs intergranulaires--sont analysres en fonction de la grom6trie fractale. On &udie plus particuli&ement ces cinq modes de ruptures ductiles et fragiles dans trois aciers bien connus (un acier doux, un acier faiblement alli6 et un acier d 32% en poids de mangand~) o~ l'on a mesur6 les dimensions microstructurales essentielles qui contribuent ~i la morphologie finale de rupture. On interprdte los courbes qui en rrsultent pour la drviation angulaire moyenne, et les courbes de Richardson (de fractal) de la rugosit6 linraire d l'aide de profils de surfaces de rupture simulres par ordinateur d partir des caractrristiques connues. On trouve que les domaines de rrsolution pour lesquels la dimension du fractal est constante correspondent aux dimensions m&allurgiques approprires sur la surface de rupture, et peuvent done 6tre relirs fi des 6chelles de taille microstructurales.

The Fractal Nature of Fracture

Europhysics Letters (EPL), 1987

The fractal geometry of fracture patterns in materials is explored by means of a simple model which incorporates the equations of elasticity and simple rules for fracture propagation. Different isotropic media and boundary conditions are considered. Self-similar patterns with fractal dimensions nearly independent of the elastic constants are obtained.

Quantitative Analysis of Brittle Fracture Surfaces Using Fractal Geometry

Fractal geometry is a non-Euclidean geometry which has been developed to analyze irregular or fractional shapes. In this paper, fracture in ceramic materials is analyzed as a fractal process. This means that fracture is viewed as a selfsimilar process. We have examined the fracture surfaces of six different alumina materials and five glass-ceramics, with different microstructures, to test for fractal behavior. Slit island analysis and Fourier transform methods were used to determine the fractal dimension, D, of successively sectioned fracture surfaces. We found a correlation between increasing the fractional part of the fractal dimension and increasing toughness. In other words, as the toughness increases, the fracture surface increases in roughness. However, more than just a measure of roughness, the applicability of fractal geometry to fracture implies a mechanism for generation of the fracture surface. The results presented here imply that brittle fracture is a fractal process; this means that we should be able to determine processes on the atomic scale by observing the macroscopic scale by finding the generator shape and the scheme for generation inherent in the fractal process. [

Stereological analysis of fractal fracture networks

Journal of Geophysical Research, 2003

1] We assess the stereological rules for fractal fracture networks, that is, networks whose fracture-to-fracture correlation is scale-dependent with a noninteger fractal dimension D. We first develop the general expression of the probability of intersection p(l, l 0 ) between two populations of fractures of length l and l 0 . We then derive the stereological function that gives the fracture distribution seen in 2-D outcrops or 1-D scan lines for an original three-dimensional (3-D) distribution. The case of a fractal fracture network with a power law length distribution, whose exponent a is independent of D, is particularly developed, but the results can, however, be extended to any other length distributions. The analytical results were tested using a numerical model that generates 3-D discrete fractal fracture networks. The corresponding 1-D and 2-D length distributions are still described by a power law with exponents a 1-D and a 2-D that are related to the original 3-D exponent by a 3-D = a 1-D + 2 and a 3-D = a 2-D + 1, respectively, regardless of the fractal dimension. The density distributions of fractures in two or one dimensions remain fractal but with a dimension that depends on both the original 3-D distribution and the power law length exponent a. The fractal dimension of 2-D or 3-D fracture networks cannot be directly inferred from one-dimensional scan-line data sets unless a is known. We found a good adequacy between our predictions and measurements made on a few natural data sets. We propose also an original method for measuring the fractal dimension from the variations of the average number of fracture intersections per fracture.

Fractal Properties of Fracture Surfaces: Roughness Indices and Relevant Lengthscales

1994

Abstract Experiments aiming at the measurement of the roughness index ζ of “rapid” fracture surfaces are briefly reviewed. For rapid crack propagation, measured values of ζ are close to 0.8, which seems to be a universal exponent. However, it is argued, by re-writing the Griffith criterion for a self-affine crack, that the self-affine correlation length ξ might depend upon the microstructure, and hence on the fracture toughness.

Relationship between fractography, fractal analysis and crack branching

Journal of the European Ceramic Society, 2020

A critical part of failure analysis is to understand the fracture process from initiation through crack propagation. Crack propagation in brittle materials can produce crack branching patterns that are fractal in nature, i.e., the crack branching coefficient (CBC). There is a direct correlation between the CBC and strength, σ f : ∝ σ CBC f. This appears to be in conflict with the fractal dimensional increment of the fracture surface, D * , which is independent of strength and related to the fracture toughness of the material, K c : = K E a D c 0 1/2 * 1/2 , where E is the elastic modulus and a 0 , a characteristic dimension. How can D * be constant in one case and CBC be a variable in another case? This paper demonstrates the relationship between D * and CBC in terms of fractographic parameters. Examples of fractal analysis in analyzing field failures, e.g., that involve comminution, incomplete fractures of components, and potential processing problems will be demonstrated.

Fractal patterns of fractures in granites

Earth and Planetary Science Letters, 1991

Fractal measurements using the Cantor's dust method in a linear one-dimensional analysis mode were made on the fracture patterns revealed on two-dimensional, planar surfaces in four granites. This method allows one to conclude that: (1) The fracture systems seen on two-dimensional surfaces in granites are consistent with the part of fractal theory that predicts a repetition of patterns on different scales of observation, self similarity. Fractal analysis gives essentially the same values of D on the scale of kilometres, metres and centimetres (five orders of magnitude) using mapped, surface fracture patterns in a Sierra Nevada granite batholith (Mt. Abbot quadrangle, Calif.). (2) Fractures show the same fractal values at different depths in a given batholith. Mapped fractures (main stage ore veins) at three mining levels (over a 700 m depth interval) of the Boulder batholith, Butte, Mont. show the same fractal values although the fracture disposition appears to be different at different levels. (3) Different sets of fracture planes in a granite batholith, Central France, and in experimental deformation can have different fractal values. In these examples shear and tension modes have the same fractal values while compressional fractures follow a different fractal mode of failure. The composite fracture patterns are also fractal but with a different, median, fractal value compared to the individual values for the fracture plane sets. These observations indicate that the fractal method can possibly be used to distinguish fractures of different origins in a complex system. It is concluded that granites fracture in a fractal manner which can be followed at many scales. It appears that fracture planes of different origins can be characterized using linear fractal analysis.

Fractal characterization of fractured surfaces

Acta Metallurgica, 1987

It has recently been claimed [B. B. Mandelbrot, D. E. Passoja and A. J. Pullay, Nature 308, 721 (1984)] that fractured surfaces are fractal in nature, i.e. "self similar" over a wide range of scale and that the fractal dimensions of the surfaces correlate well with the toughness of the material. We have investigated these concepts be measuring the fractal dimensions of fractured surfaces of two titanium alloys and attempting to correlate them with dynamic tear energy. We find that, although fractured surfaces can be classified as approximately fractal, the previous conclusions were too optimistic. An approximate correlation between fractal dimension and dynamic tear energy is obtained. R&sum&--II a Bt& r&cemment avan& [B. B. Mandelbrot, D. E. Passoja et A. J. Pullay, Nature 308, 721 (1984)] que les surfaces de rupture sont de nature fractale, c'est-&dire "autosimilaires" sur une vaste Bchelle et que les dimensions fractales de la surface correspondent bien au durcissement du mat&au. Nous avons ttudit ces concepts en mesurant les dimensions fractales des surfaces de rupture de deux alliages de titane et en essayant de les corrkler avec l'energie dynamique d'arrachement. Nous trouvons que, bien que l'on puisse donner aux surfaces de rupture un caracttre fractal approcht, les conclusions pr&dentes itaient trop optimistes. Nous obtenons une corrtlation approchee entre la dimension fractale et l'energie dynamique d'arrachement. Zusammenfassung-Vor kurzem wurde behauptet [B. B. Mandelbrot, D. E. Passoja und A. J. Pullay, Nature 308, 721 (1984)], daO Bruchfllchen fraktales Verhalten aufweisen, d.h. da13 sie iiber einen weiten MaDstabsbereich 'selbst-iihnlich' sind und dal3 die fraktalen Dimensionen der OberflHche gut mit der ZIhigkeit des Materiales korrelieren. Wir haben diese Konzepte iiberpriift, indem wir die fraktalen Dimensionen der BruchflIchen zweier Titanlegierungen gemessen haben und die Ergebnisse versuchsweise mit der Energie des dynamischen ZerreiDens korreliert haben. Wir finden, daB die friiheren Schliisse zu optimistisch waren, wenngleich Bruchfl&hen als niherungsweise fraktal klassifiziert werden kiinnen. Es ergibt sich eine nzherungsweise Korrelation zwischen fraktaler Dimension und der Energie des dynamischen ZerreiDens.