Fractal dimensions for fatigue fracture surfaces performed on micro- and meso-scale levels

A methodology was developed to use digital photograph fracture for calculating the integral fractal dimension and spectrum fractal dimensions for two perpendicular directions. The fractal dimension value, which was discovered by the variation of the pixel size up until the fractal dimension became unchangeable, i.e., where d _\min,d _\max are the borders of the scale level used. The effectiveness of the methodology performed showed the basis of the fracture surface analyses for Al-based alloys BS L65 and 2024 T351 (analogous to Al alloy D16T), specimens of which were tested by pull-push with imitation fretting corrosion damage of the sur- face and with four-point bending after the laser peening, respectively.     

Correlation between crack tortuosity and fracture toughness in cementitious material

This paper proposes a new technique for the evaluation of fractal dimension (D) of fracture surface and a quantitative correlation between D and fracture toughness of cementitious materials. The experimental program has been performed on compact tension (CT) specimens (600 × 525 × 125 mm) with three different aggregate sizes (d _max=4.7 mm, 18.8 mm and 37.5 mm). The fractal geometry concept is utilized in the evaluation of fracture surface roughness. To avoid indirect or destructive experimental procedures that are prohibitively laborious and time consuming, a new non-destructive technique is presented. Results of the analysis indicate that the concept of fractal geometry provides a useful tool in the fracture surface characterization. The results also suggest that the fracture toughness can be correlated with the fractal dimension of fracture surface.     

Fractal dimensions for fatigue fracture surfaces performed on micro- and meso-scale levels

A methodology was developed to use digital photograph fracture for calculating the integral fractal dimension and spectrum fractal dimensions for two perpendicular directions. The fractal dimension value, which was discovered by the variation of the pixel size up until the fractal dimension became unchangeable, i.e., where d _\min,d _\max are the borders of the scale level used. The effectiveness of the methodology performed showed the basis of the fracture surface analyses for Al-based alloys BS L65 and 2024 T351 (analogous to Al alloy D16T), specimens of which were tested by pull-push with imitation fretting corrosion damage of the sur- face and with four-point bending after the laser peening, respectively.     

钛酸钾晶须增韧尼龙66及其断面分形研究

钛酸钾晶须用弹性体表面改性后实现了对尼龙66的有效增韧,改性剂 氧树脂用量为晶须的1．5％（质量）时,复合体系的冲击强度比纯尼66提高132％,同时弯曲,拉伸强度分别提高55％和48％,采用显微图像灰度法,对晶须增韧尼龙进行了基于断面小岛周长－面积关系的分形研究,考察了小岛选取对断面分维数测量结果的影响,研究表明,测得的分维数随选取小岛的面积阈值增加而增大,面积阈值达500η^2(η为侧量码尺）后,分维数趋于稳定,晶须增韧尼龙的力学强度与材料断面分维数之间的变化规律一致,对于从材断断裂机制与断面形貌进行了解释。     

氢致裂纹慢扩展区断裂表面的分维研究

用垂直剖面法测量了一种含氢高强度钢在静弯矩作用下由裂纹慢扩展形成的断裂表面不同部位的分维 D,发现 D 随断裂表面塑性成分的增加而增加。在平面应变条件下,裂纹慢扩展形成断裂表面的过程是一个增维过程,当分维达到某一临界值时,裂纹失稳扩展。     

The crack surface anomalous scaling and its connection with the size-scale effects

The size-scale effects is one of the most important research topics in solid mechanics. Several theories have been proposed in order to describe the scaling of mechanical properties in fracture mechanics of quasi-brittle materials such as concrete, rock, wood and a broad class of fibrous or particulate composites. In the last two decades they were investigated by means of several techniques, including renormalisation group theory, intermediate asymptotics, dimensional analysis, statistics of extremes among the others. One of the most successful approaches is the fractal one. It is based on the assumption of a fractal-like damage localization at the mesostructural level and on the linking of mechanical properties to the fractal dimensions of the damage domains. In particular, the fractal dimension of fracture surface an be linked to the scaling properties of toughness. On the other side, recent experimental researches have shown that fracture surfaces present an anisotropic propagation in the longitudinal and transverse directions. To describe such anisotropy, it does not appear sufficient to characterize the fracture surface by a single fractal dimension, but the anomalous scaling (Morel et al., Physical Review E 58, 6999–7005 [1998]) should be introduced. This approach has proved to be very effective in describing the R-curve behaviour (Morel et al., International Journal of Fracture 114, 307–325 [2002]). Dealing with the size-scaling effects, a scaling law for both fracture toughness and tensile strength has been recently proposed. In this work, we point out some inconsistencies of the proposed approach, suggesting a more consistent way to derive the scaling laws and a correction on the scaling exponent at the larger scales. The phenomenon of scaling in notched and un-notched structures is summarized in a unified framework and the anomalous scaling is applied to the case of unnotched specimens, showing how it captures correctly only the convexity of the scaling law in a bilogarithmic plane and not the real asymptotes, thus indicating that the anomalous scaling can not be considered as a satisfactory explanation to the size-scale effects.     

R-curve behavior and roughness development of fracture surfaces

We investigate the idea that the fractal geometry of fracture surfaces in quasibrittle materials such as concrete, rock, wood and various composites can be linked to the toughening mechanisms. Recently, the complete scaling analysis of fracture surfaces in quasibrittle materials has shown the anisotropy of the crack developments in longitudinal and transverse directions. The anomalous scaling law needed to describe accurately these particular crack developments emphasizes the insufficiency of the fractal dimension, usually used to characterize the morphology of fracture surfaces. It is shown that a fracture surface initiating from a straight notch, exhibits a first region where the amplitude of roughness increases as a function of the distance to the notch, and a second one where the roughness saturates at a value depending on the specimen size. Such a morphology is shown to be related to an R-curve behavior in the zone where the roughness develops. The post R-curve regime, associated with the saturation of the roughness, is characterized by a propagation at constant fracture resistance. Moreover, we show that the main consequence of this connection between anomalous roughening at the microscale and fracture characteristics at the macroscale is a material-dependent scaling law relative to the critical energy release rate. These results are confirmed by fracture experiments in Wood (Spruce and Pine).     

Characterization of grain-boundary configuration and fracture surface roughness by fractal geometry and creep-rupture properties of metallic materials

Grain-boundary configuration in heat-treated specimens and fracture surface roughness in creep-ruptured specimens of several kinds of metallic material were quantitatively evaluated on the basis of fractal geometry. Correlations between the fractal dimension of grain boundary, that of fracture surface profile, the creep-rupture properties and the fracture mechanisms of the alloys are discussed. In heat-resistant alloys, the fractal dimension of a nominally serrated grain boundary was always larger than that of a straight grain boundary in the same alloy. The relative importance of the ruggedness of grain boundaries was estimated by the fractal dimension difference between these two grain boundaries. There was a quantitative relationship between the increase of the fractal dimension of the grain boundary and the improvement of rupture ductility and rupture strength owing to grain-boundary serration in the alloy. A similar correlation was also found between the increase in the fractal dimension of the fracture surface profile and the improvement of the creep-rupture properties, since in some cases the fractal dimension of the fracture surface profile was correlated with that of the grain boundary. Both grain boundary and fracture surface profile were assumed to exhibit a fractal nature between one grain boundary length (upper bound) and an interatomic spacing (lower bound). In carbon steels with ferrite-pearlite structure, according to the increase in pearlite volume fraction, the rupture ductility decreased and the fracture mechanism changed from transgranular fracture in pure iron and low-carbon steels to intergranular fracture at ferrite-pearlite grain boundaries in medium-carbon steels, and further to intergranular fracture at pearlite grain boundaries in high-carbon steels. The correspondence between the fractal dimension of the grain boundary and that of the fracture surface was confirmed in ruptured specimens of ferrite-pearlite steels when the grain boundary was the fracture path.     

Tensile tests and analyses of notched specimens fabricated from high strength steels using a generalized yield model

This article summarises the uniaxial tension tests for 20 notched bars fabricated from high strength steel Q345 specified in Chinese National Standards. The effects of the notch radius, r, and that of the notch depth ratio, d/D , on the ductility and fracture resistance of this high strength steel are examined. The experimental data are further analyzed using a generalized yield model together with an elliptical fracture stress envelope originally proposed by the first author. The experimental results demonstrate that cracks initiate at the notched section, with the fracture surface filled with many dimples and shearing marks. Specimens with a sharper notch radius (a smaller r) and a larger notch depth (a smaller d/D ratio) show poor ductility, but high fracture strength. The stress field computed from the numerical procedure including the generalized yield model indicates that the crack initiation occurs at the centre of the notched section which experiences the highest stress triaxiality ratio (σ_m/σ_seq) . As the stresses at the notched section reach the limiting values determined from the elliptical fracture criterion, macroscopic fracture failure in the notched bar occurs.     

Crack-resistance behavior as a consequence of self-similar fracture topologies

The effect of the invasive fractality of fracture surfaces on the toughness characteristics of heterogeneous materials is discussed. It is shown that the interplay of physics and geometry turns out to be the non-integer (fractal) physical dimensions of the mechanical quantities involved in the phenomenon of fracture. On the other hand, fracture surfaces experimentally show multifractal scaling, in the sense that the effect of fractality progressively vanishes as the scale of measurement increases. From the physical point of view, the progressive homogenization of the random field, as the scale of the phenomenon increases, is provided. The Griffith criterion for brittle fracture propagation is deduced in the presence of a fractal crack. It is shown that, whilst in the case of smooth cracks the dissipation rate is independent of the crack length a, in the presence of fractal cracks it increases with a, following a power law with fractional exponent depending on the fractal dimension of the fracture surface. The peculiar crack-resistance behavior of heterogeneous materials is therefore interpreted in terms of the self-similar topology of the fracture domains, thus explaining also the stable crack growth occurring in the initial stages of the fracture process. Finally, extrapolation to the macroscopic size-scale effect of the nominal fracture energy is deduced, and a Multifractal Scaling Law is proposed and successfully applied to relevant experimental data.     

Influence of concrete composition on the characterization of fracture surface

An experimental method based on laser triangulation was used to measure the 3-D profile of fracture surface of concrete. The projective-covering method was established to determine the fractal characteristics of fracture surface. Based on the experiment, the influence of composition of concrete on the fractal dimension, including water–binder ratio, maximum aggregate size and aggregate type, was investigated. The variation of fractal dimension with the material composition was also analysed to make further understanding of fracture mechanism of concrete.     

Adhesive power measurements of bonds between old and new concrete

A simple procedure for measuring the adhesive bond of cement-bonded materials is introduced and tested with an old-new concrete bond. Cubic or cylindrical specimens with a notch and the interface in their middle are split under stable crack growth conditions. The load is recorded as a function of the crack opening displacement. From this curve, the maximum load (notch tensile stress) and the fracture energy (G _F) can be determined. The course of the curve characterizes the mechanical behaviour of the material bond in the crack opening mode and is an important basis for a numerical treatment of interface problems. Different pre-treatments of the old concrete surface have an important influence on the adhesion of the material compound. Good adhesive properties have been measured after sand-blast treatment and poorer properties after a dispersion emulsion treatment.     

Conversion of transgranular to intergranular fracture in NiCr steels

The paper is focused on quantification of causes and characteristics that govern the intergranular fracture initiation and propagation of this fracture micromechanism in competition with cleavage one. A NiCr steel of commercial quality and the same steel with an increased content of impurity elements, Sn and Sb, have been used for this investigation. Step cooling annealing was applied in order to induce intergranular embrittlement and brittle fracture initiation in both steels. Standard bend and the pre-cracked Charpy type specimen geometries were both tested in three-point bending to determine the fracture toughness characteristics. Charpy V notch specimens tested statically in three-point bending supported by FEM calculation have been used for local fracture stress and other local parameters determination. Relation of cleavage fracture stress and critical stress for intergranular failure has been followed showing capability of this parameter for quantification of the transgranular/intergranular fracture conversion. In order to characterise the quantitative roughness differences in fracture surfaces fractal analysis was applied. A boundary level of fractal dimension has been determined to be 1.12 for the investigated steel; the fracture surface roughness with a higher value reflects high level of intergranular embrittlement and thus fracture toughness degradation.     

Fractal in Fatigue Fractography

Fracture profile roughnesses and fractal dimensionsof fracture traces were measured on a fatigued Ti-6Al-4Valloy.It is found that although fractal dimension can wellreflect the variation of fracture traces with the measuringunits,it is difficult to apply it to quantitative analysis offractured surfaces because of the dependence of the meas-ured profile roughness on the measuring units.Based onfractal concept.an alternative equationlgR_l(η)=lgR_o-(D-1)lgη was obtained.in which we intro-duced a parameter of intrinsic profile roughness to evalu-ate fracture profile roughness without restriction of themeasuring units employed.     

粉末NBR-PVC共混型热塑性弹性体断裂面的分形维数

本文利用扫描电镜对粉末NBR-PVC共混型热塑性弹性体拉伸断面形貌进行了观察与分析;基于断面小岛周长-面积关系,测定了试样的断口分形维数,考察了NBR含量、拉伸断面分形维数、力学性能(拉伸强度、断裂伸长率、邵氏硬度)之间的关系。结果表明,NBR-PVC弹性体拉伸断面的分形维数随着NBR含量的增加而增加,当NBR含量超过29份后其分形维数呈下降趋势;分形维数与材料的拉伸强度、断裂伸长率和邵氏硬度变化关系一致。     

金属断裂表面分形维数与组织、性能之间的关系

借助分形研究了拉伸条件下,分形维数 D_F 与拉伸性能之间的关系,讨论了显微组织对分形维数的影响,并把冲击韧性与分形维数之间的关系和拉伸性能与分形维数之间的关系进行了比较。实验结果表明,拉伸性能与分形维数只存在定性关系。延伸率与分形维数成正比,强度σ_b、σ_s与分形维数成反比,显微组织对分形维数有重要影响。断裂方式对分形维数、断裂性质以及两者的对应关系也有重要影响。     

Fractal structure of precursors and phase transformations in the sol-gel synthesis of nanoparticulate <Emphasis Type="Italic">M</Emphasis>-type barium hexaferrite

We have studied the influence of sol-gel process conditions on the fractal structure of precursors, the phase transformations during the synthesis process, and the morphology of the resulting barium hexaferrite particles and determined the optimal fractal structure of sol-gel derived precursors for the solid-state synthesis of barium hexaferrite. The activity of the precursors is shown to correlate with the fractal dimension of their first structural aggregation level. Nanoparticulate barium hexaferrite can be prepared in the form of anisometric particles: platelets (d _av ∼ 65 nm) and nanorods (d _av ∼ 30 nm, l _av ∼ 70 nm).     

A Fractal Study on Brittle-Ductile Transition forTi-24Al-11Nb Alloy

The mechanical behaviours of Ti-24Al-11Nb alloy in the brittle-ductile transition (BOT) have been investigated by using three-point bending tests. The temperature dependence of the fractal dimensions and fractal characterization of fracture surfaces are presented. The probable mechanism controlling BDT of intermetallic alloys are proposed according to fractal geometry.Additionally. it is found that there is a positive relationship between the fractal dimension and fracture toughness in BDT for Ti-24Al-11Nb alloy     

Evaluation of Tablet Formation of Different Lactoses by 3D Modeling and Fractal Analysis

ABSTRACT

The aim of this study was to use 3D modeling to differentiate not only among the four different types of lactose α-lactose monohydrate, spray-dried lactose, agglomerated lactose and lactose anhydrous but also between products from different manufacturers. Further “box-counting” fractal analysis of SEM images was done to gain additional information on tableting characteristics and tablet properties which can be found in the fractal structure. Twelve different materials from different manufacturers were analyzed for their powder-technological and physicochemical properties. They were tableted on an eccentric tableting machine at graded maximum relative densities and the recorded data, namely force, time, and displacement were analyzed by the 3D modeling technique. Tablet properties such as, elastic recovery, crushing force and morphology were analyzed. The results show that 3D modeling can precisely distinguish deformation behavior for different types of lactose and also for the same type of material produced with a slightly different technique. Furthermore, the results showed that the amorphous content of the lactose determined the compactibility of the material, which is due to a reversible exceeding of the glass transition temperature of the material. The three fractal dimensions D_BW (box surface dimension), D_WBW (pore/void box mass dimension), and D_BBW (box solid mass dimension) are capable of describing morphological differences in lactose materials. Multivariate regression analysis showed that the fractal surface structure of the lactose-based materials is strongly correlated to tableting characteristics and tablet properties. Especially with regards to 3D modeling, it was found that the fractal indices can describe the parameters time plasticity d, pressure plasticity e, and fast elastic decompression, which is the inverse of ω. In addition, the 3D parameters are able to describe the powder and tablet fractal indices. In conclusion, the 3D modeling is not only able to characterize the compression process but it can also provide information on the final tablet morphology.     

Fraunhofer diffraction by cantor fractals with variable lacunarity

Abstract

Optical diffraction by fractal openings is increasingly being studied because it allows the properties and parameters that characterize these objects to be determined. Allain and Cloitre published the first results showing that the resulting analysis of the distribution of intensity obtained from diffraction experiments through fractal deterministic pupils permits the self-similar dimension and other geometrical characteristic of these structures to be obtained directly. In this work the lacunarity effect ?, dimension d and order k of growth of the Cantor fractal about the distribution of intensities of the diffraction pattern are studied, solved analytically and characterized. In particular we note the influence of lacunarity because this is one of the first studies in which this geometric fractal parameter is taken into consideration. The selfsimilarity of the diffraction figure at different orders is also proved. The results of this study allow us to say that an intimate relation exists between the distribution of the diffracted waves and the parameters that describe this kind of fractal geometry.