Establishing a protocol for measurements of fractal dimensions in brittle materials

Many techniques have been used to measure the fractal dimension of brittle fracture surfaces. The purpose of this study was to create a protocol for obtaining the fractal dimension using a simplified optical technique for comparison to reported procedures. Four classes of ceramic materials were used in this study: baria silicate (a glass-ceramic), silicon nitride (a fine grain polycrystalline ceramic), zinc selenide (a large grain polycrystalline ceramic), and silicon (a single crystal ceramic). Contours were produced in perpendicular and parallel planes to the fracture surface using three techniques: slit-island (parallel), profile technique (perpendicular), and crack indentation technique (perpendicular). These contours were then analyzed using a method first introduced by Richardson. The slit-island technique produced statistically greater fractal dimensional increments, ranging from 0.08 to 0.28 for all the materials, than either the profile technique (0.01 to 0.03) or the indentation technique (0.02–0.05). This difference is due in part to the fact that many brittle fracture surfaces are self-affine objects and not self-similar objects. A list of recommendations for a protocol and sources of error for this technique are presented in the appendices.     

The fractal effect of irregularity of crack branching on the fracture toughness of brittle materials

Microstructural observations of brittle materials indicated that a variety of microdefect events can be responsible not only for inelastic behaviour, but also for macroscopic crack front irregularity. This irregularity produces an increase in the fracture toughness of the material. In this paper, this irregularity is analysed by fractal geometry in a very simple manner; a fractal model of crack branching is established. Both microscopic and macroscopic analytical results show that the toughness can be raised appreciably as a fractal geometric effect of the irregularity.

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Résumé Des observations microscopiques sur des matériaux fragiles ont montré qu'une variété d'évènements à l'échelle du microdéfaut peuvent être responsables non seulement du comportement inélastique, mais aussi de l'irrégularité du front d'une fissure macroscopique. Cette irrégularité provoque un accroissement de la ténacité à la rupture du matériau. Dans cette étude, on analyse de manière très simple cette irrégularité par fractogéométrie (Mandelbrot) et on établit un modèle fractal relatif à une fissure qui se ramifie. Les résultats de l'analyse microscopique et macroscopique montrent qu'un effet fractogéométrique de l'irrégularité du front de fissure est d'accroitre de manière appréciable la ténacité.

     

Edge chipping of brittle materials: effect of side-wall inclination and loading angle

An earlier analysis of chipping fracture in brittle solids is here extended to include the case of blocks with inclined side faces and non-normal contact loading. The simple relation P _F = β K _c h ^3/2 for the critical chipping load P _F in terms of indent location h and material toughness K _c is preserved, with angular coordinates simply incorporated into the β coefficient. Chipping fracture tests using a Vickers indenter near the edges of glass blocks with non-orthogonal faces is used to validate the analysis. Implications of the results in relation to practical engineering, biomechanical and anthropological structures are indicated. An erratum to this article can be found at     

脆性材料微裂缝无序度与尺寸效应

假定在某些脆性无序材料内含有相同的微裂缝随机分布概率密度,但具有不同无序度的情况下,建立了模拟材料力学行为的二维不连续位移法边界元数值计算模型.实现了材料微裂缝生长、扩展到最终破坏的全过程数值仿真.根据分形几何理论确定了材料断裂表面几何形貌的分形维数.得到了材料的断裂强度随微裂缝长度随机分布无序度的增加而降低的规律性.数值模拟结果符合Bazant尺寸效应定律,并进一步证实了脆性或准脆性无序材料产生尺寸效应的微观机理.     

The effect of microcracking upon the Poisson's ratio for brittle materials

Microcracking-elasticity theories typically relate a decrement in elastic moduli to the number density,N, and the mean microcrack radius 〈a〉. In this paper, four microcracking-modulus theories are rewritten in terms of the macroscopic, observable parameters of Young's modulus and Poisson's ratio, eliminating the specific dependence on the difficult to measure, microscopic quantitiesN and 〈a〉. The rewritten microcracking elasticity theories are then compared to elasticity data on a variety of microcracked, polycrystalline ceramics.     

Application of fractal geometry to damage development and brittle fracture in materials

This article represents a first attempt to apply the concepts of fractal geometry to damage development in materials. Two extremes of material behavior were considered. In the case of unstable brittle propagation of microcracks, application of fractal geometry and weakest link statistics lead to the well-known Weibull distribution for fracture stress. The Weibull slope (shape parameter) is equal to twice the fractal dimension of the flaw distribution. For the case of stable microcrack growth, a number of simplifying assumptions led to a closed-form expression for the change in effective modulus with an increment of damage. Although an oversimplification, this latter analysis illustrates the potential value of fractal geometry in modeling damage development. Future work will be directed towards developing these initial results further.     

A modified size effect model for brittle nonmetallic materials

Formulae of both existing size effect models for brittle materials, `effective volume' and `effective surface' were theoretically derived for bar specimens with rectangular and circular cross-sections, for three- and four-point bending load configuration, respectively. Additionally, a modified model called `effective shell model' is introduced. Exemplarily three- and four-point bending tests were performed on bar specimens of different sizes of a leucite reinforced glass ceramic material. The results were analyzed based on the three size effect models. It was found that the defect population of the specimens could be characterized better with the effective surface than with the effective volume model. The new effective shell model improves the statistical reliability even more.     

Fracture toughness and absorbed energy measurements in impact tests on brittle materials

Previous work on impact testing has shown that the energy/unit area (w) normally measured in notched impact tests is dependent on specimen geometry. A fracture mechanical analysis has now been developed to account for the observed dependence ofw on notch size. A correction factor () has been derived to accommodate notch effects and this allows for the calculation of the strain energy release-rateG directly from the measured fracture energies.Tests on PMMA have shown that corrected results are independent of specimen geometry and theG _c for PMMA has been evaluated as 1.04 × 10³ J m^–2. The experimental results show that there is an additional energy term which must be accounted for and this has been interpreted here as being due to kinetic energy losses in the specimens. A conservation of momentum analysis has allowed a realistic correction term to be calculated to include kinetic energy effects and the normalized experimental results show complete consistency between all the geometries used in the test series.It is concluded that the analysis resolves many of the difficulties associated with notched impact testing and provides for the calculation of realistic fracture toughness parameters.     

The effect of high hydrostatic pressures on the cohesive strength of brittle materials

Experiments have been conducted to determine the fracture threshold for cast iron and steel when subjected to hydrostatic pressures up to 11300 kg/cm². General agreement with a maximum elastic strain theory of failure is found.

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Zusammenfassung Es wurden Versuche durchgefuehrt zur Bestimmung der Bruchschwelle von Gusseisen and Stahl unter hydrostatischen Drucken his zu 11300 kg/cm²: Allgemeine Uebereinstimmung mit einer Bruchtheorie maximaler elastischer Dehnung wurde gefunden.

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Résumé Des expériences ont été effectuées pour déterminer le seuil de rupture des fontes et des aciers soumis à des pressions hydrostatiques jusqu' à 11300 kg/cm². Une concordance générale avec la théorie de la tension elastique maximum de rupture a été trouvée.

     

The effect of crack propagation mechanism on the fractal dimension of fracture surfaces in steels

The paper reports on the results of a fractal analysis of fracture surfaces of Ni–Cr steel in two different states of heat treatment simulating embrittlement. The change in the fractal dimension of the fracture surface demonstrates a wavy character and dispersion depending on the microstructural state of the tested steel. The results of the fractal analysis in the crack growth direction and across the entire crack front were used as the basis for a reconstruction of the geometry of the fracture surface, providing a new geometric tool for fractographic analysis. The competing effects of transgranular and intergranular brittle fracture may lead to increased roughness of the fracture surface and its fractal dimension. The threshold value of the fractal dimension of the sections perpendicular to the fracture surface, indicating the transition from transgranular to intergranular fracture, is 1.12.     

Simple bond energy approach for non-destructive measurements of the fracture toughness of brittle materials

A simple bond breakage model for computing the fracture surface energy and toughness of a wide variety of brittle materials is presented and correlated against values reported in the literature for the single crystalline forms of these same materials. The correlation shows that this simple model can provide an accurate estimate for both the fracture surface energy and toughness of these materials. It is further shown that this simple model can be extended to amorphous materials with reasonable accuracy by normalizing the fracture surface energy of the crystalline material by the ratio of the density of the amorphous material vs. the density of the single crystalline material. Applications to thin film low-k materials and capabilities for non-destructive measurements are also discussed.     

无序材料微裂缝分形几何与尺寸效应的微观机理

针对一些含有相同的微裂缝随机分布概率密度但无序度不同的材料,建立了模拟材料断裂力学行为的二维不连续位移法边界元数值计算模型,实现了材料微裂缝的生长、扩展到最终破坏的全过程数值模拟.从分形几何的新视角深入地揭示了脆性或准脆性无序材料产生尺寸效应的微观机理.材料断裂力学行为的数值模拟结果与Bazant尺寸效应定律相符,不仅与微缺陷的密度有关,更与微缺陷大小随机分布的无序度相关,无序度越大的材料其尺寸效应越明显.得到了用初始分形维数D0表示的关于材料断裂强度的分形维数Dσ经验公式,可以更深入地解释材料的微观尺寸效应机理和断裂过程.     

The development of rational design criteria for brittle materials

Very brittle materials are finding increasing usage in critical engineering components. Howver, their fundamental property of brittle behaviour requires very accurate design methods if reliable applications are to result. In this paper the development of rational design criteria for these materials are described. Three major topics are discussed. The statistical techniques which enable the correlation of experimental data, and the effects of size and stress gradient to be taken into account are described, and the development of these methods to cases of non-uniform stresses through a unit volume concept are assessed. The phenomenon of static fatigue, in which failure occurs, often after a considerable time, under an applied load which is less than that required to produce instantaneous failure is considered. Recent applications of fracture mechanics techniques to the determination of the crack growth rate are described which leads to the development of methods for the estimation of component lifetimes under conditions of subcritical crack growth. Finally, the experimental results obtained from investigations into the fracture strength of various brittle materials when subjected to multiaxial stress states are compared. It is concluded that a great deal of further experimental evidence is required before fully reliable design criteria are available on which engineering decisions may be based.     

The significance of impact data for brittle non-metallic materials

Experimental data for the impact energy of a number of brittle materials are reviewed and their significance discussed in terms of material properties and test conditions. For each material and test method several interpretations need to be considered but it is not always possible to extract meaningful information from the data. At one extreme, for strong, very brittle materials like ceramics, the impact strength may be controlled by the elastic energy in the specimen at the instant of fracture initiation. At the other extreme, for weaker or tougher materials like graphite and fibre reinforced plastics, the impact strength may be controlled by the work of fracture of the specimen. However, in many cases, the situation is less clear and it is emphasized that great care should be taken in the interpretation of test data.