断口定量分析的分形几何方法

近几年来,分形(fractal)几何已广泛地应用于材料的断裂分析。本文简要介绍了分形几何的发展概况及其几个主要定义,分析了一些简单不规则图形的分形维数的计算。把断口形貌统计地看作分形表面,进行了材料的断口的分析分析。根据断口的分形维数,人们可能去定量地追溯材料断袭时的力学行为。     

内部氢对奥氏体不锈钢拉伸断口分形维数的影响

采用热充氢方法研究了1Cr18Ni9Ti、21Cr6Ni9Mn、Fe31Cr14Ni奥氏体不锈钢的氢脆性能。采用垂直截面法测量了不同热充氢时间拉伸试样的断口分形维数,分析了材料氢脆性能与分形维数的相关性。研究表明:随着充氢时间的延长,3种材料内部氢含量增加、断面收缩率降低、氢致塑性损减增加。反映在断口分形维数上,表现为断口分形维数随充氢时间的延长而降低,且断口分形维数与其断面收缩率呈线性相关。断口分形维数能够描述氢对材料性能的影响。     

多相材料断裂的分形特征

测量了多相材料复合涂层的断裂挠度和断口表面的分形维数，分析了复合涂层的断裂特征。结果表明：随着硬质相增加，复合涂层的断裂挠度和断口表面的分形维数降低。     

大气降尘轮廓投影分形与来源辨识

选取燃煤锅炉、道路扬尘、汽车燃油不完全燃烧产生的颗粒、土壤颗粒等4种颗粒,利用数字显微和图像处理技术,求取降尘颗粒轮廓投影曲线的分形维数,并根据实际采样情况,求取降尘部分源颗粒的轮廓投影曲线分形维数,根据分形维数间的关系对比分析后判断降尘颗粒的来源。结果表明:将大气降尘轮廓投影曲线分形维数值作为降尘颗粒来源的辨析参数是可行的,将轮廓投影曲线的分形维数值和投影轮廓的其他几何特征参数相互结合可以得到很好的判断结果。     

D6AC钢冲击断口形貌的分形研究

本文利用扫描电镜对含MnS夹杂物和含ZrN夹杂物的D6AC钢的冲击断口形貌进行了观察与分析;利用数字图象法测定了试样的断口分形维数,揭示了冲击断口形貌和冲击韧性与夹杂物含量之间的内在联系,探讨了材料冲击断口形貌与分形维数的关系,拓宽了分形几何学在材料领域的应用范围。     

D6AC钢冲击断口形貌的分形研究

本文利用扫描电镜对含Mn$夹杂物和含ZrN夹杂物的D6AC钢的冲击断口形貌进行了观察与分析;利用数字图象法测定了试样的断口分形维数,揭示了冲击断口形貌和冲击韧性与夹杂物含量之间的内在联系,探讨了材料冲击断口形貌与分形维数的关系,拓宽了分形几何学在材料领域的应用范围.     

分形维数与D6AC钢的韧性

本文研究了超高强度钢Ｄ６ＡＣ的断裂韧性、冲击韧性和拉伸面缩率与断口分形维数的关系。分别用二次电子线扫描和数字图像法断裂韧性试样的断口分形维数，得出试样的韧性与分形维数Ｄｓｅ，ＤＨ和ＤＬ成正比关系，即韧性随分形维数增大而增加。试样断口的粗糙度由夹杂物引起差异小于金相组织不同引起的差异时，使用数字图象法测得的分形维数与韧性的线性关系优于二次电子线扫描的结果。     

分形维数与D6AC钢的韧性

本文研究了超高强度钢Ｄ６Ａｃ的断裂韧性、冲击韧性和拉伸面缩率与断口分形维数的关系。分别用二次电子线扫描和数字图象法测定断裂韧性试样的断口分形维数，得出试样的韧性与分形维数Ｄｓｅ、Ｄ_Ｈ和Ｄ_Ｌ成正比关系，即韧性随分形维数增大而增加。试样断口的粗糙度由夹杂物引起的差异小于金相组织不同引起的差异时，使用数字图象法测得的分形维数与韧性的线性关系优于二次电子线扫描的结果。     

陶瓷刀具材料断口形貌及裂纹扩展的分形特征

通过热压烧结工艺,制备了一种高性能的Si3N4基陶瓷刀具材料。利用X射线衍射仪和扫描电子显微镜分别对材料物相组成、微观形貌及裂纹扩展路径进行了分析。借助图像处理技术和分形理论,计算了断口形貌及裂纹扩展路径的分形维数,并揭示材料断裂机制。研究表明,Si3N4基陶瓷刀具材料表现为穿晶/沿晶的混合断裂模式,其裂纹扩展方式主要是偏转和桥联,断口形貌及裂纹扩展均具有明显的分形特征。当材料断口形貌越粗糙,裂纹扩展路线越不规则,分形维数值增大,表明断口微观结构的粗糙程度、裂纹扩展路线的不规则程度可用分形维数来刻画。     

双相钢疲劳断口的分形分析

本文研究了双相钢疲劳门槛值与门槛值附近断口形貌和分形维数的关系。初步实验结果表明:分形维数基本上能反映断口粗糙度。发现门槛值随分形维数呈线性变化,门槛值、分形维数和粗糙度都随马氏体含量呈抛物线型变化。     

Fractal fracture mechanics—A review

The revolution in geometry, which has recently created the notion of the fractional dimension of real world bodies and has formed fractal geometry, has substantially influenced fracture science. An experimentally measured crack in real materials has appeared to be substantially a fractal, that is, a geometrical object of fractional dimension. In the present paper, an attempt has been undertaken to provide an account of some achievements in this area, herein called “Fractal Fracture Mechanics”.     

形变渗氮08F钢冲击断裂表面不同方向的分形分析

本文利用垂直截面法测量了０８Ｆ钢经不同形变渗氮复合处理后冲击断裂表面的分形维数。试验结果表明，材料的冲击能ＡＫ和断裂表面的分形维数ＤＦ均随形变量的增加而增大，且变化规律也十分相似，冲击断口Ｔ、Ｓ两方向上的分形维数ＤＴ和ＤＳ均与冲击能的对数ｌｎＡＫ之间呈正变化的线性关系，且Ｄｓ〉Ｄｒ。     

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.     

Relationship between Fracture Toughness and Fractal Dimension of Fracture Surface of Steel

The relationship between fracture toughness and fractal dimension of fracture surface was analysed based on a united fractal model of microscopic fracture modes. It was indicated that fracture toughness is a multivariant function of yield strength, microstructural length, fractal dimension of fracture surface and volume fraction of inclusions, other than a monovariant function of fractal dimension of fracture surface. This new fractal model of fracture toughness was verified to fit well with a set of experimental data of a ultrahigh strength steel with equal microstructure and grain size but different volume fraction of inclusion.     

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。     

The fractal dimension of grain-boundary fracture in high-temperature creep of heat-resistant alloys

The fractal dimension of the grain-boundary fracture in high-temperature creep was estimated by the vertical section method on several creep-ruptured specimens of the cobalt-nickel- and iron-based heat-resistant alloys. Grain-boundary microcracks linked to the fracture surface were also taken into account in the present analysis by the box-counting method. In the specimens containing many grain-boundary microcracks linked to the fracture surface, the fractal dimension of the grain-boundary fracture was larger in the scale range of more than about one grain-boundary length than in the scale range less than this length. Thus, there was a cross-over in the fractal dimension of the grain-boundary fracture at about one grain-boundary length in these specimens. In the specimens containing much fewer microcracks, there was no clear cross-over in the fractal dimension of the grain-boundary fracture with regard to the scale of the analysis, irrespective of creep-ductility and grain-boundary configuration of the specimens. The fractal dimension of the grain-boundary fracture was generally larger in specimens with serrated grain boundaries than in specimens with straight grain boundaries in these heat-resistant alloys, because the fractal dimension of the grain boundary and the number of the grain-boundary microcracks were larger in the former specimen. The fractal dimension of the grain-boundary fracture did not tend to converge to unity when the scale of the analysis approached the specimen size. The inclusion of near-specimen size data with regard to the scale of the analysis did not affect the fractal dimension of the grain-boundary fracture in these alloys. Thus, the grain-boundary fracture in the creep-ruptured specimens exhibited a fractal nature, at least in the scale range below specimen size, although there was a cross-over in the fractal dimension of the grain-boundary fracture in specimens containing a large number of grain-boundary microcracks.     

水泥断口表面形貌的分形维数定量表征研究

利用分形几何学理论定量表征了几种水泥断口表面的微观形貌特征,并且考察了断口表面分形维数与其抗压强度的关系.结果表明,水泥断口的表面分形维数介于2～3之间,且线性回归分析的相关系数均大于0.98,强的相关性表明实验所选用的水泥断口具有明显的分形特征;水泥的抗压强度与其断口表面分形维数值呈正相关关系.     

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 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.     

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.