Relationship between Fractal Geometry and Fractography

Fractal geometry has been used to describe irregular fracture surfaces in a quantitative way. The fractal dimensional increment has been related to the fracture toughness of the material through the elastic modulus and a characteristic structure parameter, a ₀. The study of fractography has shown the relationship between the flaw/mirror size ratio and the fracture toughness. An experimental observation has shown that the fracture toughness is related to the elastic modulus through another structure parameter, b ₀. Combining all of these relationships leads to the conclusion that the fractal dimensional increment, D ^*, is directly related to the flaw/mirror size ratio. Experimental measurements of the fractal dimension and the flaw/mirror size ratio on glasses, a glass-ceramic, polycrystalline ceramics, and a single crystal all agree with the prediction. The implication of this finding is that there is a linear scaling law in operation at fracture between the energy of crack initiation and of microbranching and is reflected in the features on the fracture surface.     

Fractal Analysis of Toughening Behavior in 3BaO.5SiO2Glass-Ceramics

The objective of this paper was to analyze the crystal aspect ratio (AR), the fracture toughness, and the fractal dimension of 3BaO^.5SiO₂ glass-ceramics and to relate the topography of fracture surfaces to fractal behavior. These analyses demonstrate that crystal morphology strongly affects the fracture path, the fracture toughness, and the fractal dimension. Fracture toughness increased from 0.7 ± 0.1 MPa^.m^1/2 for the glass to 2.2 ± 0.6 MPa^.m^1/2for the glass-ceramic with an AR of 8.1 while the fractal dimensional increment ( D *) for the glass and the glass-ceramic increased from 0.10 ± 0.01 to 0.25 ± 0.02, respectively. The materials with lower aspect ratios (AR = 1.4 and 3.6) exhibited the predicted relationship between toughness and D * while the glass-ceramic with an aspect ratio of 8.1 did not satisfy the expected relationship because of multiple toughening mechanisms.     

Relationship of energy to geometry in brittle fracture

Numerous investigators have noticed that there is a relationship between the energy of branching and the energy of initiation during a fracture event in materials that fail in a brittle manner. Usually, this is measured in terms of the stress intensities, i.e., K_B/K_c. The ratio has been reported between 3 and 4, implying a constant value. However, data suggests that it is a constant for a material, but not a universal constant. The fractal dimension of the fracture surface is related to the critical stress intensity factor. It is a measure of the tortuosity of the fracture surface. We show that the K_B/K_C ratio is directly related to the square root of the fractal dimensional increment, indicating a relationship between the energy of crack propagation and the tortuosity of the fracture surface.     

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 self-similar 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 increasing 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.     

Fractal Characteristics of Fracture Surfaces

Quantitative fractography is often used to study material failure mechanisms. During calculation of surface or profile roughness parameters, the magnification used in obtaining fractographic data is found to influence the value of the parameters. Fractal geometry has been developed into a tool capable of defining surface and profile topography without sensitivity to magnification, and several studies have related fractal dimension ( D _F) to other physical or mechanical properties. In this study, we obtained the fractal dimension of profiled fracture surfaces of one glass and three proprietary dental porcelains. The fracture toughness ( K _1c) of these materials was also measured using the indentation-strength method. Results show the surfaces to be fractal. No quantitative relationship between fractal dimension and toughness was found. Differences in K _1c were demonstrated between some materials. It is postulated that the size range within which fractal dimension can be defined as constant is dependent on the toughening mechanism, and that the relationship between K _Ic and D _F cannot be identical for all materials.     

Effect of Micmcracking on the Fracture Toughness and Fracture Surface Fractal Dimension of Lithia-Based Glass-Ceramics

The effect of thermally induced microcracks on the fracture toughness and fractal dimension of fully crystalline lithia disilicate glass-ceramics was studied. The fracture toughness, K _IC, for the nonmicrocracked lithia disilicate, 3.02 ± 0.12 MPa·m^1/2, was significantly greater than the value of 1.31 ± 0.05 MPa·m^1/2 for the microcracked specimens. The fractal dimensional increment, D *, was 0.24 ± 0.01 for nonmicrocracked lithia disilicate specimens compared with a value of 0.18 ± 0.01 for the microcracked specimens. The relationship between K _IC and D * implies that the two materials exhibit dissimilar fracture behavior because of microstructural differences. Estimates of the characteristic length involved in the fracture process, a ₀, indicate that the materials have an identical fracture process at the atomic level. This apparent contradiction may be explained by the scale on which the measurements were taken. It is suggested that fractal analysis at the atomic level would yield equivalent D * values for the two different microstructures.     

Residual stress and fracture toughness of sintered body of ZrO2-GO composite ceramics material

Zirconia ceramics and carbon-based materials are widely adopted in medical and dental applications due to their excellent biocompatibility and aesthetics. However, fracture toughness of ceramic materials limits their application in clinical dentistry because of the existence of residual stress. In this study, zirconia/graphene oxide (ZrO₂-GO) composite ceramics were fabricated by hot-press sintering. Residual stresses developed on the surface of ZrO₂-GO composite ceramics were evaluated by X-ray residual stress analysis and indentation techniques. The variation of surface residual stress with GO content was evaluated, and found to be consistent with that of fracture toughness. The generation of residual stress was found to be directly related to fracture toughness. Residual stress calculated by theoretical formula of indentation method was consistent with that measured by X-ray diffraction in line with the content of GO. Based on above results, it is concluded that 0.1–0.15 wt% GO composite ceramics possessed better mechanical properties.     

Quantitative Characterization of the Fracture Surface of Si Single Crystals by Confocal Microscopy

Experiments are conducted to study the dislocation nucleation conditions at the crack tip in {110}〈110〉 oriented Si single crystals. Specimens with surface cracks are first statically loaded at elevated temperatures for a prolonged period of time to initiate and move dislocations away from the crack tip, then cooled down to room temperature and loaded to fracture to measure the fracture toughness. Fractographic analysis of the fracture surfaces is performed. Distinct wavy patterns on the fracture surface at the initial cleavage crack front are observed, which is attributed to the existence of local mixed mode I/mode III stresses resulting from the inhomogeneous dislocation activity. Confocal microscopy is employed to quantify the fracture surface roughness. The results show that the increase of fracture toughness is directly associated with the increased area of the rough surface, which is characterized by the roughness number or the fractal dimension increment. Our results also demonstrate that dislocation nucleation can occur only at discrete sites. The spacing between these dislocation nucleation sources is of the order of 1 μm. A simple model is developed for the relationship between the fracture toughness and the surface roughness parameters, which is in good agreement with the experimental results.     

Rising Crack-Growth-Resistance (R-Curve) Behavior of Toughened Alumina and Silicon Nitride

R -curves for a sinter/HIPed SiC(whisker)-reinforced alumina and a sintered silicon nitride were assessed by direct measurements of lengths of cracks associated with Vickers indentation flaws. The fracture toughness measurements based on (a) initial (as-indented) crack lengths, (b) equilibrium growth of cracks during increasing far-field loading, and (c) crack lengths corresponding to unstable fracture showed definitive trends of R -curves for both materials. The fracture mechanics analyses employed an indenter-material constant that was independently estimated using a physical model for the residual driving force and a free surface correction factor that accounted for the effects of size and shape of the cracks on stress intensity. It is shown that R -curve estimations based on crack length measurements have the intrinsic advantage that crack length dependence of fracture toughness is not assumed a priori as is done in conventional analysis based on strength. The measured fracture toughness of SiC(whisker)-reinforced alumina was in agreement with the prediction of a toughening model based on crack bridging by partially debonded whiskers.     

Determination of Surface Residual Stresses in Brittle Materials by Hertzian Indentation: Theory and Experiment

Hertzian indentation has been used to determine the surface residual stress levels in brittle materials. In this method, a hard sphere is pressed into the surface of the material: at a critical load a preexisting surface-breaking crack in the neighborhood of the contact will propagate. There is a threshold load below which no such crack, of whatever size, can be propagated. The presence of a residual stress in the surface will lead to a shift in this threshold load. The effects of residual stresses on the minimum load to produce Hertzian fracture are predicted for alumina and glass, assuming that the variation of the residual stress over the length of the crack is small. Two methods of analysis (one approximate, one more general) are presented that enable the residual stress to be calculated from the shift in threshold load; the only further information required is a knowledge of the radius of the sphere, the elastic constants of the sphere and substrate, and also the fracture toughness of the substrate (or use of a stress-free specimen as a reference). No measurement of any crack length is necessary. Experimental results are presented for the residual stress levels determined in glass strengthened by ion exchange. Indenting balls of a variety of materials with a range of elastic mismatch to the glass substrate were used, so as to evaluate the effects of elastic mismatch and interfacial frictional tractions on the results obtained. The results obtained by Hertzian indentation are consistent with residual stress levels determined by differential surface refractometry. We also present results on alumina specimens with induced surface stresses.     

Improved Fracture Behavior of Alumina Microstructural Composites with Highly Textured Compressive Layers

Alumina‐based microstructural composites combining equiaxed and textured layers were fabricated to examine how cracks propagate and the mechanical properties are affected as a function of the residual stress and volume fraction of texture in a multilayer structure. By combining equiaxed and highly textured alumina layers of varying thermal expansion, the embedded textured layers were placed under compressive residual stresses as high as ?670 MPa. Composites with a near constant maximum failure stress of up to 300 MPa were shown to be almost independent of the initial defect size as result of the compressive residual stress in the textured layers. An apparent fracture toughness of up to 10.1 MPa·m^1/2 was obtained for composites with an equiaxed to textured volume ratio of 7.4:1. The high compressive stress in the textured layers arrested cracks, whereas the weak bonding parallel to the basal surfaces of the textured alumina grains caused cracks to deflect within the textured layers. The coupling of these two mechanisms resulted in crack arrest and a maximum work of fracture of ~1200 J/m² or almost 50 times higher than equiaxed alumina. We believe that embedding textured layers having compressive stresses below the surface of multilayer composites represent an important strategy for designing flaw‐tolerant materials with pronounced crack growth resistance and a high work of fracture.     

Biomimetic micro-textures,mechanical behaviors and intermittent turning performance of textured Al2O3/TiC micro-composite and micro-nano-composite ceramics

This work was conducted to investigate biomimetic micro-textures, mechanical behaviors and intermittent turning performance of textured Al₂O₃/TiC micro-composite and micro-nano-composite ceramics. Chip characteristics and the geometry features of the structures on the cuticle of Procambarus clarkia were considered in the preparation of the laser-induced biomimetic micro-textures on the composite ceramic surfaces. Characteristics of the biomimetic micro-textures were revealed in terms of geometry, morphology and chemical composition. The connection between the thermal stress resulting from laser pulse and the fractal dimension of the micro-crack on the micro-textures was analyzed and identified. The correlation between the mechanical behaviors (damage and fracture toughness) of the textured composite ceramics and the fractal dimension of the micro-crack was fitted and revealed quantitatively. The performance of the textured composite ceramic tools was pre-evaluated by means of a proposed indicator. Damage, fracture toughness and tool stress were incorporated in the indicator. The indicator and the experimental tool wear were compared for validating the effectiveness of the proposed indicator. It was found that the micro-composite ceramic exhibited greater sensitivity to laser than the micro-nano-composite ceramic did. The damage of the textured micro-composite ceramic was larger than that of the textured micro-nano-composite ceramic when the same laser parameters were utilized in the micro-texture preparation. On the contrary, the fracture toughness of the textured micro-composite ceramic was found to be smaller. There was a negative correlation between the damage of the textured composite ceramic and the fractal dimension of the micro-crack. Conversely, there was a positive correlation between the fracture toughness and the fractal dimension. The performance of the textured micro-nano-composite ceramic tool was better than that of the textured micro-composite ceramic tool. The proposed indicator can be used to predict the combination of laser parameters that resulted in the optimum performance of the textured composite ceramic tool.     

Stress-Induced Microcracking at Second-Phase Inclusions

The critical inclusion size for microcracking due to an applied stress for an inclusion that has a residual stress field is estimated using fracture mechanics. In particular, an analysis is presented for both circumferential and radial crack formation at spherical inclusions that have a uniform misfit strain compared to the matrix. It is found that the critical size can be greatly reduced below that for spontaneous microcracking when the applied stress is of the order of the residual stress. It was predicted that the applied stress would cause extensive micro-cracking when the local fracture toughness is low and when the size of the inclusions approaches the critical size for spontaneous microcracking.     

Transformation Toughening in Large-Grain-Size CeO2-Doped ZrO2 Polycrystals

The fracture and transformation behavior of tetragonal polycrystalline ZrO₂ alloys containing 18 mol% CeO₂ (Ce-TZP) was investigated. In the absence of applied stress the tetragonal phase was found to be stable in large-grained (>30 μm) samples at room temperature. The monoclinic phase was detected in regions of high residual stress near hardness indentations although no evidence of a wake of monoclinic phase along the fracture surface was observed. The fracture toughness increased from 4 to 7 MPa · m^1/2 as density and/or grain size increased. It is proposed that the relatively high toughness of these materials is due to the occurrence of stress-driven tetragonal-to-monoclinic transformation near the crack tip, which reverses when the crack has passed.     

颗粒孔结构的积木分形模型

构造了立方和四面体两种积木分形体，得到一般积木分形体模型，导出关联表面积和体积增量的3个分形表达式，并分析了表面分数维的几何意义. 实验结果表明，利用该模型的表面积与体积增量分形表达式可以从压汞和BET的实验数据计算表面分数维，相关系数较高. 对同一种颗粒，两种实验方法可以得到相同的分数维. 讨论了体积增量的计算方法.     

Fractal characterization of ceramic crack patterns after thermal shocks

This work utilized a combination of experimental evidence and fractal geometric method to assess the effect of crack extension concerning the thermal shock on residual strength of ceramics. Sintered alumina (Al₂O₃) ceramic slabs were bundled and quenched in water under different thermal shock temperatures. The fractal dimension of thermal shock crack patterns on the interior surface and the cooled surface was calculated by the Box-counting method. Fracture energy of a fractal pattern of microcracks in quasi-brittle solids was employed to explain the relationship between crack length and fractal dimensions. The results show that if the crack propagation has the same crack length but a larger fractal dimension, it will absorb more fracture energy. The thermal shock crack patterns of Al₂O₃ ceramics with different grain sizes were analyzed, and the smaller grain size ceramic had a higher fractal dimension of crack patterns than the larger one.     

Fractal Perimeters of Polishing-Induced Pull-Outs Present on Polished Cross Sections of Plasma-Sprayed Yttria-Stabilized Zirconia Coatings

Polished cross sections of plasma-sprayed yttria-stabilized zirconia coatings deposited using different process parameters were prepared with both hot- and vacuum-mounting techniques and investigated by image analysis. It was found that polishing-induced pull-outs were evidently present on the hot-mounted cross sections, and that the perimeter of these pull-outs could be described statistically by means of fractal analysis. In this work, values of the corresponding fractal dimension range from 1.45–1.54; they increase linearly while increasing fracture toughness, and decrease with the increase in porosity of the coatings. Thus, this fractal dimension may be regarded as a measure of the fracture toughness of the coatings, but only for hot-mounted samples.     

Characterizing Brittle Fracture by Modeling Crack Deflection Angles from the Microstructure

This study introduces a simple analytical model for fracture toughness to bridge the length scales from grain size to bulk thickness by assembling a virtual crack path from the angles recorded on an unfractured microstructure, which is a great challenge in fracture mechanics due to the high geometrical complexity. Good agreement is found between a crack deflection angle distribution measured from 5764 crack segments and the prediction by the model and the possible influence of residual stress is quasi quantitatively discussed. A total of 7.4% of the crack segments observed acted as crack bridges, while 7.3% was predicted by the model. A quantification of how high an angle needs to be to turn crack deflection into crack bridging is given. The ratio of fracture toughness from grain boundary to grain, G_1c(gb)/G_1c(g), was measured indirectly from all samples to be between 0.3 and 0.35.     

The effect of chromium and titanium implantation on the mechanical properties of polycrystalline alumina: The role of residual stress

Mechanical properties of ion implanted ceramics are connected with the microstructure and the residual stresses introduced by implantation. In this work, studies on the implantation of chromium and titanium in alumina are reported. The residual surface compressive stress has been determined using an indentation technique, based on the method of Lawn and Fuller which deduces stress from the size of cracks around Vickers hardness indentation. The measurement of the mechanical properties, i.e. fracture toughness, in the treated surface, was undertaken using a Vickers indentation method. Finally, these implantations have been investigated by means of SEM and SIMS to study the eventual formation of chemical compounds in the implanted zone. For the two ions implantation, the increase of K_IC has been attributed to the compressive stresses. After heating, relaxation of residual stresses occurs in the two cases and TiO₂ formation in the case of titanium implantation. The oxide formation increases the fracture toughness by a compensation of the residual stresses relaxation. For chromium implantation the relaxation decreases the toughness.     

A fractal analysis of crack branching in borosilicate glass

Many fractography techniques involve precise measurements of features on the fracture surface and can be difficult to perform in the field, or rapidly. Macroscopic crack branching observations offer a more robust and forgiving method of analysis, but often are not strongly correlated with standard fractography techniques. In this study, the crack branching patterns of annealed borosilicate glass disks previously fractured in biaxial tension were analyzed using fractal methods and compared with more typical fracture surface measurement techniques. The results confirm that the fractal dimension of macroscopic crack branching (called the Crack Branching Coefficient) increases with increasing failure stress, as has been shown with other brittle materials. In addition, the existence of a threshold stress previously reported was confirmed using new techniques. The findings herein can be used to further increase the fidelity of fractography-based failure analysis of brittle materials.